![C H A P T E R
1
Introduction
Interfaces are the boundaries separating two phases and define all objects in the three-dimensional world. Depend-
ing on the strength of cohesion forces and binding energies between atoms and molecules, the phases can be gases,
liquids, and solids, defining the physical states of matter. When the cohesion energies between the constituting atoms
and molecules are stronger than randomizing effects of the thermal energy, the physical state changes from gas to a
condensed phase of matter, liquid, or solid. The Boltzmann distribution gives the probability P that a system will be in
a certain state as a function of the state’s energy and temperature:
P eE=kT
kT factor is often used as a scale energy factor in the molecular interactions. The cohesive energies per atom or molecule
at 298K can vary from several kT between gas atoms, between 9 and 23kT in liquid Hg (the liquid with the strongest
cohesive energy, 57.9 kJ/mol [1]), and 50kT in solids up to 342kT in W (1kT4.051021
J), the metal with the high-
est melting point. The kT energy scale factor is introduced and discussed in detail in Chapter 2. Because the most
important interactions between material interfaces take place in the liquid, or between material interfaces and liquids,
the solid-liquid, liquid-liquid, and liquid-air interfaces deserve special attention. The overall balance between the
repulsive and attractive forces between solutes and colloidal objects in liquids must be comparatively equal or larger
than 9–23kT to have aggregation, adsorption, self-assembly, etc., and below 9kT to obtain stable dispersions and col-
loids. As mentioned, liquids form at T¼298K, when the cohesive energy between the constituting atoms and mole-
cules is larger than 9kT. While in the bulk of a liquid the interaction forces of a molecule or atom are fully symmetric at
interfaces, in contrast, in the topmost layer of molecules or atoms the interaction forces are asymmetric. Due to this
asymmetry, a certain tension/force arises in the plane of the interface. The stronger the interfacial tension, the stronger
the asymmetry. At contact between two phases, the topmost layer of molecules at the phase boundary also interacts
with the molecules from the other phase, this is called adhesion. The adhesion forces and energies counterbalance the
asymmetry of the forces acting on the topmost molecular layer, i.e., the stronger the adhesion force, the smaller the
interfacial tension. If the adhesion force is stronger than the cohesion force, then the interfacial tension disappears,
the interface disappears, and the phases become fully miscible, as discussed in Chapter 3. This interfacial tension
has also the character of an energy density, and for pristine interfaces this is causally related to the cohesion energy
in the bulk material; interfacial energy density is about half the cohesion energy in bulk. Surface and interfacial tension
of liquid-gas and liquid interfaces, as well as interfacial and surface energy of solids-liquid and solid-gas interfaces, are
thoroughly discussed in Chapter 3. The effects of the interface tension can be seen in small liquid droplets or molten
metals, as the shape of the droplet itself is modeled by this interfacial tension. The small world of insects and bugs are
particularly affected by the interfacial tension. Because their size is comparable to the capillary length, when the shape
of the liquids is fully determined by interfacial tension, not by gravitation, they have a different perception of the sur-
rounding world than humans do. Interfacial tension can have a devastating effect on insects; some drown as they can-
not escape the surface tension, but some have adapted to take full advantage of it. For example, small water droplets
can be manipulated and transported by ants without any need for bottles or glasses, and some mosquitos have adapted
on water to straddle along the smooth water surface, etc. (Fig. 1).
Intuitively, the interfacial tension is the 2D equivalent of the cohesion energy in 3D. Interfacial tension is discussed in
detail in Chapter 3.
However, when the surface and the interface are chemically modified, e.g., with surfactant adsorbates, the inter-
facial tension and energy density of interfaces do not reflect anymore the cohesive energy between the molecules
in the bulk phase. Thus, the interface itself can be treated as a thermodynamic system on its own, as discussed in
Chapter 7. The interfacial tension and interfacial energy density between phases are now an exclusive reflection of
1
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![wetting, wettability, and contact angle as the main measurement methods for macroscopic and nanoscale surfaces.
Chapter 5 also introduces the several functional surfaces.
In Chapter 6, a series of equations permitting the calculation of unknown surface tension, energy, work of adhesion,
etc. from known measurable macroscopic parameters have been grouped under the name “fundamental equation of
interfaces.” Their versatility in predicting the values of many interfacial parameters, for example, interfacial tension,
wettability, polarity of the surface, etc. from contact angle makes them extremely useful in practice. In Chapter 7, the
surface and interfacial tension are introduced via thermodynamic treatment of the interfacial layer. Although this
treatment has no direct practical implications, it gives the theoretical background necessary for the interpretation
of interfacial adsorption isotherms and interfacial tension vs concentration curves for surfactants and amphiphiles.
Chapter 8 treats surface functionalization that can be achieved in different ways, by physical methods such as
roughening of the surface, or photolithographic nanopatterning, and by chemical methods, by adsorption of surfactant
molecules. The adsorption of surfactant molecules on solid surfaces involves either chemical or physical bonding,
resulting in the formation of a self-assembled monolayer. Several types of chemical bonding and substrates are
reviewed. In addition, a surfactant monolayer can be prepared first at the water-air interface and then transferred onto
the surface of the solid via the Langmuir-Blodgett and dip-coating methods.
Solid-solid interfaces also have practical relevance, especially in layered electronic devices. Solid-solid interface, in
particular the metal-organic interface, is the locus of another type of phenomena of practical importance, namely the
electron transfer. In the previous chapters, the interfaces were the place where different forces met. In Chapter 9, the
metal-organic interfaces are treated as the contact point between electron energy levels of a metal, material with delo-
calized electron energy levels called bands, and the organic molecules and polymers whose energy levels are discrete
and localized. Understanding electron transfer between metal electrodes and organic conductors is of practical impor-
tance, especially for the manufacturing of organic photovoltaics, organic light emitting diodes, and other organic elec-
tronic devices. Any of these devices requires at least several layers of electroactive organic materials, and knowledge of
adhesion, wettability, and interfaces is required for their development and manufacturing.
Chapters 10 and 11 deal with the interaction forces and energies between interfaces in different media. These inter-
action forces can be repulsive or attractive and they are the same forces governing the molecular interactions. The bal-
ance between the attractive and repulsive interaction forces is of practical importance, controlling the phenomena of
particle aggregation, colloid stability, particle adsorption on surfaces, self-assembly of nanoparticles, etc. Chapter 12
introduces colloids, which are the oldest type of nanomaterials known and are today encountered in the food industry,
pharma, and many other consumer products. Colloids are constituted from finely divided particles, nanoparticles, or
liquid droplets dispersed into a continuous medium. Because their surface-to-volume ratio is very high, their behavior
is governed almost exclusively by their surface and interfacial properties. Synthesis of colloids as well as stability cri-
teria is discussed.
As a continuation on the topic of colloids, but deserving special attention, Chapter 13 introduces the synthesis of
polymeric nanoparticles and polymeric nanostructured interfaces via emulsion polymerizations. As expected, the
interfacial aspects determine the types of emulsions and nature of the nanomaterials that can be synthesized. The types
of emulsions and conditions of formation are briefly reviewed. A case study covers some examples of synthesis of
nanostructured interfaces, polymerization of the emulsions stabilized by amphiphilic particles.
Some nanoparticles, depending on their surface properties, can also spontaneously adsorb at interfaces; they can
form monolayers and stabilize emulsions. The factors responsible for why some particles can adsorb at liquid-liquid,
liquid-gas, and solid-liquid interfaces are discussed in Chapter 14. Once adsorbed at the interfaces the particle-particle
interactions leads to the decrease in the interfacial tension. Responsible for this is their lateral interaction, which is
governed by the same types of forces as in case surfactants, and in addition by particle specific interactions, capillary
floatation, or immersion forces. In fact, in recent times, nanoparticles have been used in the synthesis of photonic crys-
tals via the Langmuir-Blodgett method and other self-assembly structures.
The last chapter of this book discusses the role of interfaces in integrated circuit manufacturing via photolithogra-
phy. Photolithography is the only top-down preparation method of nanomaterials and nanostructured surfaces. In the
past few years, it evolved into the most precise technique to prepare with large machines, structures as small as 7nm
(the gate of the field-effect transistor). In practice, the photolithographic manufacturing process of chips and processors
requires in-depth knowledge and control of interfacial phenomena such as adhesion, wetting, capillary forces, and
interfaces.
Reference
[1] G. Kaptay, G. Csicsovszki, M.S. Yaghmaee, An absolute scale for the cohesion energy of pure metals, Mater. Sci. Forum. 414–415 (2003) 235–240.
https://doi.org/10.4028/www.scientific.net/MSF.414-415.235.
3
Reference](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-11-320.jpg)
![ρz
i ¼ ρ0
i exp
μz
i μ0
i
kT
¼ ρ0
i exp
mgz
ð Þ
kT
(2.5)
where m is the molecular mass and g is the gravitational acceleration. Note that the potential energy of the air mol-
ecules mgz is “compared” to the kT at any height above the Earth’s surface. With the increase in the potential energy of
the molecules compared to kT, less molecules are found at higher altitudes (Fig. 2.1). In other words, if mgz is small
compared to kT, then the thermal energy would uniformize the distribution of molecules with the altitude such that
little variation in the number density of air molecules would be registered.
The same distribution applies to ions that, for example, carry a charge e between two different regions that have
different potentials ψ1 and ψ2:
ρ2
i ¼ ρ1
i exp
e ψ2 ψ1
ð Þ
kT
(2.6)
and this is known as the Nernst equation. It is nonetheless important to note that interactions are additive; for example,
if the difference in energy between two regions is given by potential, potential energy, and chemical potential, then the
exponent will be the sum of all these contributions.
The above equations also give us the possibility to gauge the strength of interaction between molecules. For exam-
ple, if a liquid is in equilibrium with its vapors at standard conditions of pressure 1atm and temperature 298K, then
1 mol of gas will occupy approximately 22.4m3
and a mole of liquid approximately 0.02m3
. Then the difference in
energy between the liquid and gas states will be [1]:
μ
0 gas
i μ
0 liquid
i kT ln
X
gas
i
X
liq
i
kT ln
22:4
0:02
7kT (2.7)
where 1kT is approximately the energy of the thermal fluctuations. Therefore, it can be said that if the interaction
strength between molecules in a gas phase at temperature T is larger than 7 kT, then it condenses into liquid. Con-
versely, if the cohesion strength between the molecules of a liquid become smaller than 7kT, then it transforms into
gas as the cohesion energy is simply too low to hold the molecules together. This alludes to what is known as the
Trouton rule, which states that the entropy of vaporization is roughly the same for different kinds of liquids, about
85JK1
mol1
, which is roughly 9.5kT.
FIG. 2.1 The bottle was capped (left) in the mountain and brought to the ground level (right).
6 2. Thermal energy scale kT](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-13-320.jpg)
![The denominator in the above equation is the partition function 1/Z:
1
Z
¼
1
X
n
j
exp
Ej
kT
(2.11)
Reference
[1] J. Israelachvili, Intermolecular and Surface Forces, third ed., Academic Press, San Diego, CA, 2011.
8 2. Thermal energy scale kT](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-15-320.jpg)
![C H A P T E R
3
Surfaces and interfaces
An interface is the boundary between two immiscible phases in contact, such as liquid-liquid, liquid-solid, liquid-
air, etc. Immiscibility arises when the constituent molecules interact stronger with the molecules from the same phase
than with the molecules from the other phase, i.e., the “cohesion forces” are stronger than the “adhesion forces.” The
force of cohesion is defined as the sum of the forces that act between the molecules of the superficial layer and the bulk,
while the forces of adhesion are defined as the forces that act between the superficial layer and the molecules of the next
phase.
The interface is characterized by a certain thickness, which is intuitively taken as the thickness of the last layer of
molecules at the surface of the phase that enter in “contact” or “feel” the influence of the molecules in the other phase. It
has been the subject of intense research where exactly lies the borderlines defining the interface between two phases in
contact. This can be simplistically defined as two monolayers thick, one monolayer at the interface belonging to one
phase and the other monolayer to the next phase (Fig. 3.1). This is probably the most satisfactory way to intuitively
understand the interface thickness. However, this not very rigorous, because the molecules from subsequent layers
also feel the presence of molecules from the other phase via “longer ranged” forces that operate and whose intensity
decays with the distance from the interface. Michael C. Petty stated that “if the molecules are electrically neutral, then
the forces between them will be short-range and the surfaces layer will be no more than one or two molecular diam-
eters; in contrast, the Coulombic forces associated with the charged species can extend the transition region over con-
siderable distances” [1].
The experimental studies of the neat liquid-liquid interfacial thickness revealed that the hexadecane-water thickness
is about 6Å by X-ray reflectivity and 15Å by neutron reflectivity [2]. The apparent discrepancy comes from the
limitation of the both methods which include two contributions, namely the intrinsic width of the interface “that char-
acterizes the crossover from one bulk composition to the other and a statistical width due to thermally induced cap-
illary wave fluctuations (ripples) of the interface” [2]. This also reflects the difficulty of the experimental methods to
probe the interfaces at molecular length scales. The X-ray reflectivity studies of the thickness of the mercury-water
interface was 5Å, which is comparable to that of mercury-vapor interface of 5Å and that of pure water-vapor inter-
face 3Å [3]. These and other studies have revealed that the liquid-liquid and liquid-vapor interfaces are at least two
monolayers of molecules or atoms.
Recently, combined surface vibrational spectroscopy and molecular dynamics revealed an even more complex
aspect of interfaces; in addition to interfacial thickness variation, molecular structuring by ordering and layering of
molecules near interface were observed [4]. The characteristic molecular vibrations were probed at the water-
chloroform and water-dichloromethane interfaces as a function of interfacial depth. From the concentration profiles
of both water and organic solvent molecules it was observed that both the dichloromethane and water extended dee-
per into the opposite phase forming a thicker interface, while the water-chloroform interface was sharper. Near the
water-chloroform interface water monomers were detected, not associated via H-bonds and the concentration profile
of chloroform deeper into the bulk organic phase is oscillatory suggesting the CCl4 molecules are layered near the
interface, not observed for dichloromethane. Numerous examples of ordering and structuring of molecules near
the interface were reported, as well as the consequences, for example, surface freezing of the top molecular layer
of alkane at alkane-vapor interface is 2–3°C higher than the freezing temperature, while this was not observed at
an alkane-water interface, which suggests an increased ordering of alkane molecules in the former case [2].
9
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![Molecular modeling also helped in gaining insight into the interfacial boundaries. It was found that the water-
hexane interface is very sharp and is about two monolayers thick [5]; the water molecules near the hydrophobic inter-
face are oriented such that the molecular plane and the dipole moment are parallel to the plane of the interface and the
long axis of the hexane molecules is also parallel to the interface [5, 6]. In addition, some of the water molecules at the
hydrophobic interfaces are incapable of hydrogen bonding, about one in four molecules exhibit dangling hydrogen
bonds, which gives rise to a large interfacial energy [7]. On the other hand, water molecules near strongly polar inter-
faces such as quartz are capable of hydrogen bonding and are oriented in an ice-like structure and no dangling bonds
were observed in surface vibrational studies with vibrational sum-frequency spectroscopy [7].
The consequence of the molecular orientational ordering, layering, reduced capability of molecular bonding, and
interfacial mixing lead to a change in the solvent properties near the interface. For example, second-harmonic gener-
ation in combination with solvatochromic surfactants of different lengths known as “molecular rulers” were able to
probe solvent polarity with depth near the weakly and strongly associating water-organic solvent interfaces. For exam-
ple, the solvent polarity near the weakly associating water-cyclohexane interface quickly converges from the aqueous
to the organic limit in less than 9Å, while the strongly associating water-1-octanol interface revealed a transition region
of ordered octanol molecules at the interface giving rise to a hydrophobic barrier [8]. The chemical structure and the
molecular dimensions greatly affect the thickness of the interface. Further systematic studies performed with
“molecular rulers” revealed that at the water-organic solvent interfaces the interface thickness and polarity strongly
depend on the molecular structure [9, 10].
The conclusion that can be drawn from experimental evidence is that the interface can be visualized as a sheet or as a
thin “membrane” with certain thickness. The thickness of the interface depends on the ability of the phases to interact
given by the balance between the adhesion and cohesion forces. For weakly interacting phases the interface thickness is
nearly two monolayers thick (Fig. 3.1A), while for strongly interacting phases the interface will be thicker than two
molecular monolayers (Fig. 3.1B). In addition, the polarity gradient across the interface can change due to molecular
ordering at the interface, which propagates to a certain depth in bulk, depicted as a color gradient in Fig. 3.1B.
The membrane separating two immiscible phases has therefore boundaries and is an open thermodynamic system
because it can exchange matter and energy with the neighboring phases. Molecules move continuously to and from
interface to bulk. A thermodynamic system is everything that has boundaries, an object constitutes a thermodynamic
system, be that a car, a grain of salt, or an interface, and all possess a certain internal energy, U. The energy of the
interface between the phases, also most commonly referred to as “interfacial energy,” is highest when the cohesion
energy is much larger than the adhesion energy. Interfacial energy decreases to negligible values when the adhesion
forces become comparable to cohesion forces and the phases begin to mix. Water-hexane interface is an example of a
high interfacial energy interface while water-ethanol interface has a zero interfacial energy, i.e., completely miscible.
FIG. 3.1 (A) The ideal sharp inter-
face between two weakly interacting
phases α and β can be imagined as a
thin membrane, two monolayers
thick with a sharp molecular density
profile that separates two phases.
(B) The thicker interface between
two phases that are strongly interact-
ing with diffuse density profile,
thicker than two monolayers. The
oscillation in the β phase indicate
ordering. The solvent polarity near
this interface changes due to order-
ing and loss or gain in bonding
capability.
10 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-17-320.jpg)
![The surface tension of various liquids changes with temperature. The magnitude of the surface tension reflects the
strength of interaction forces between the composing molecules and atoms, i.e., the cohesion forces (Table 3.1). For
neon the strength of the interaction forces between atoms are very weak and the surface tension reaches 5.2mN/m
at 247°C. The cohesion forces are comparatively stronger for oxygen molecules than for neon atoms. Metallic bonds
between mercury atoms are considerably stronger than the van der Waals forces between neon atoms, therefore even
at room temperature the surface tension value is comparatively large.
3.2 Predictive models for calculating the surface tension of liquids
There were attempts to apply theoretical models to calculate the surface tension from the energy of cohesion or
enthalpy of vaporization. For example, Stefan’s equation has been often used to do this [11]:
γL ¼
ZS
Z
ΔHvapρ
2
=3
M
2
=3N
1
=3
A
2
4
3
5 (3.2)
where ΔHvap is the enthalpy of vaporization of the liquid (kJ/mol) in standard conditions of pressure and temperature
of 105Pa and 273.15K, respectively, M is the molecular weight (g/mol), NA is the Avogadro number, and ρ is the
density of the liquid (g/cm3
). ZS
Z is the ratio between the coordination number of molecules at the surface with respect
to bulk; this cannot be determined directly by experiment but can be calculated [12]. The ratio of the coordination num-
bers for the compounds ranges from 0.0559 to 0.1784, whereas a value 0.25 for the ratio ZS
Z has been obtained for many
organic substances [12].
Numerical example 3.3
Calculate the surface tension of water using Eq. (3.2) knowing that ΔHvap ¼41 kJ/mol, M ¼18g/mol, ρ¼1g/cm3
, and ZS
Z ¼
0:13 is the average value of all the determined values by Strechan et al. [12].
Solution
γwater ¼ 0:13
41104
6:868:44107
KJ
mol
mol
g
2=3
g
m3
2=3
mol
molecules
1=3
!
¼ 76
mJ
m2
(3.3)
The obtained value slightly overestimates the experimentally determined value of water surface tension 72.4 mJ/m2
at room temperature.
Therefore, the correct calculation of ZS
Z is important for obtaining accurate values of the surface tension.
Significant effort has been dedicated to modeling the coordination number ratio ZS
Z .
Another extendedly used empirical model, especially for determining the surface tension of molten metals, is called
the bond-broken model or the bond-cutting model where the surface energy is calculated based on the energy of cohe-
sion Ecohesion [13]:
γL ¼
ZZS
Z
Ecohesion (3.4)
3.3 Interfacial tension between liquids
Surface tension is only a particularization of interfacial tension and it is used only when referring to liquid-gas (vapor)
interface. Interfacial tension refers to liquid-liquid or solid-liquid interfaces, but because it is a more general concept
than the surface tension it can be used throughout, also for liquid-gas interfaces. For liquid-liquid interfaces, the top-
most layers of molecules from each phase are in contact (Fig. 3.5). In contrast to the liquid-gas case, the topmost layer at
the liquid-liquid interface will now be under the action of two forces, forces of cohesion with the molecules from the
bulk of the same phase and forces of adhesion with the molecules from the other phase. Consequently, the topmost
layer of liquid is not as strongly compressed as at the liquid-gas interface. The stronger the forces of adhesion, the
stronger the attraction of the topmost layer to the second liquid phase.
14 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-21-320.jpg)
![Two extreme situations can be distinguished, if Fa is very high and comparable to the cohesion forces, then γ12 !0 and
the two liquids can easily mix and will not form an interface. If, however, the molecules between the two phases are
incapable of “bonding” or Fa is extremely low, then γ12 !γ1 +γ2, consequently it is more difficult to expand or deform
this interface.
3.4 Relating surface tension to surface energy
The surface tension is a force per unit length. However, the surface tension can also be related to energy per surface
area, or energy density. Consider a wire frame with one mobile side, which can slide on the U-shaped frame without
friction (Fig. 3.7). The mobile side has a length l. On this frame we have a membrane of water, a thin film. If we try to
pull the mobile side to increase the area of the membrane by Δx, then the work done will be
W ¼ FΔx (3.6)
but F is 2γl, the factor 2 comes from the fact that there are two sides of the surface. Therefore, surface tension is also the
energy per unit area:
γ ¼
W
2lΔx
≡
Energy
Area
J=m2
(3.7)
The surface tension can thus be redefined as the energy required to increase the surface area with one unit.
3.5 Surface and interfacial energy of solids
Surface and interfacial tensions have units of force per unit length N/m or energy per unit area J/m2
and the two
forms are perfectly equivalent. For the solid-gas interfaces instead of interfacial tension one uses the concept of “surface
energy.” Similarly, for the solid-liquid interfaces “interfacial energy” is used instead of “interfacial tension.” The sur-
face energy of solids arises because of all unsaturated or dangling bonds per unit area of surface of a solid. Surfaces of
metals, for example, have high energy associated with them because the atoms in the first surface layer have fewer of
neighbors than in bulk and therefore unsatisfied capacity to metallic bonding. The cohesive energy of metals is given
by the enthalpy of atomization ΔHa
(equivalent to its bond strength), which is 418kJ/mol for Fe, 844kJ/mol for W,
368kJ/mol for Au, and 327kJ/mol for Al [14]. A great amount of work is needed to form and shape metals of high
cohesive energy. The surface energy is taken as a fraction of the cohesion energy, γ ¼fcohesion energy, 0f1. In
calculations of the surface energy from energy of cohesion, f could arbitrarily be chosen as 0.5 (see Fig. 3.3). There are
roughly 1.61019
atoms/m2
on the surface of an Fe and the surface energy can be estimated from the cohesion energy:
SE 0:5 418
kJ
61023
atoms
1:61019 atoms
m2
5535
mJ
m2
(3.8)
The surface density of metal atoms Nd was calculated from the density of the metal: Fe (ρ¼7850kg/m3
, Nd ¼1.61019
),
W (ρ¼19,600kg/m3
, Nd ¼1.321019
), Au (ρ¼19,320kg/m3
, Nd ¼1.251019
), and Al (ρ¼2712kg/m3
,
FIG. 3.7 (A) U-shaped wire
frame holding a thin liquid film
membrane which has a mobile side
of length l that can slide under the
action of an external force; (B) the
cross section of the liquid mem-
brane film having two surface ten-
sion forces, corresponding to each
interface of the liquid membrane,
opposing the expansion of the sur-
face area under the action of the
pulling force F.
16 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-23-320.jpg)
![Nd ¼1.271019
). The calculated surface energy using the above equation is 5535mJ/m2
for Fe, 9294mJ/m2
for W,
3833mJ/m2
for Au, and 3462mJ/m2
for Al. These calculated values give the expected trend but the magnitude differs
significantly from the experimental and calculated values of the surface energies of the corresponding metals, listed in
Tables 3.3 and 3.4. The source of this difference is found in the value of the factor f used in the calculations of the above
formula, under the initial assumption that the surface energy is about half of the cohesion energy. This factor plays a
similar role with the coordination number seen in Stefan’s Eq. (3.2). If value of the factor is taken as 0.16 instead of 0.5,
which is close to that used for calculating the surface tension of water, see Eq. (3.3), the recalculated value for the sur-
face energy of metals is 1772 for Fe, 2974mJ/m2
for W, 1226mJ/m2
for Au, and 1108mJ/m2
for Al, which are very close
in magnitude to the experimental data and those calculated by more advance models, Tables 3.3 and 3.4 [22]. This
shows that the magnitude of the surface energy originates in the cohesion energy of the condensed phase. It is, how-
ever, difficult to predict a priori the value of the factor f used in such calculations, it can be done empirically just as in
the case of the coordination number of Stefan’s equation or the use of more complex theories could provide a deeper
explanation and a method for calculating such factors from fundamentals.
Unlike liquids where the action of surface tension force is visible, especially in small droplets that acquire spherical
shape under its action, in the case of solids the action of surface energy has no visible mechanical action. For example, if
a solid is cut into smaller pieces then they will not suddenly change their shape due to the action of surface tensions.
However, the action of surface energy/tension becomes visible if the solid can be melted at high temperatures. For
example, a molten metal, when in liquid state at high temperatures above the melting point, behaves just as any liquid,
will occupy the smallest volume so it will squeeze in spherical droplets, upon cooling the metal is shaped, formed,
extruded, or pulled, so its surface remains “frozen” in a metastable state, therefore its surface possesses a high energy.
A metastable state is a state in which a system can spend an extended time in a configuration other than the system’s
state of least energy.
Surface energy of materials is of great technological importance, as, for example, in material fatigue and stress anal-
ysis. The cracks in materials can propagate and produce failures, whose detection is critical in aircraft construction and
safe operation. The crack produces in a material to relax the elastic stress in the vicinity of the crack. The crack equi-
librium length depends on the balance of forces between the elastic stress and increase in the surface energy of the
material. Yet another area in which knowledge of the surface energy plays an important role is the wettability of
“reservoir rocks” for the oil extraction and recovery. The natural oil reservoirs were usually classified as oil-wet,
water-wet, or intermediate based on the affinity of the rock’s surface to oil or toward water. Wettability of the rock
to water affects the reservoir production and the performance of enhanced oil recovery processes [23].
Another relevant field is that of sealants and adhesives. For optimum adhesion, an adhesive must completely wet
and cover out the surface to be bonded. Wetting is necessary for an adhesive to cover a surface to maximize the contact
area and the attractive forces between the adhesive and bonding surface. For a good performance, the surface energy of
the adhesive must be substantially lower than the surface energy of the substrate to be bonded, as, for example, regular
adhesives bond very poorly on the low surface energy Teflon, or polyethylene but bond very well on higher surface
energy glass or metal surfaces. This is on the other hand also the principles behind nonstick coatings of the pans, such
as Teflon. Surface energy of several polymer surfaces is presented in Table 3.2.
To improve the adhesion of paints and coatings the surface must be thoroughly washed and degreased. Grease and
wax have a low surface energy material and prevent a good adhesion when present on surfaces. To improve adhesion
on low energy surface, different methods can be applied, which include plasma treatment, UV-light exposure, chem-
ical oxidation with piranha solution, etc. Exposure of a surface to UV light will generate ozone and singlet oxygen that
oxidizes the surface. It has been proposed that UV-generated ozone or singlet oxygen will insert in the CdH bonds of a
hydrocarbon surface and create polar functional groups such as dCOOH, dCO, dCOH, etc. For example, polyeth-
ylene (PE) surface energy is roughly 31mN/m, and increases after the treatment with various methods: after expo-
sure to UV 33mN/m, after flame treatment 39mN/m, and after etching with chromic acid 40mN/m [24].
On the other hand, surface treatment for lowering the surface energy is also possible by surface modification with
hydrophobic compounds. For example, a very popular hydrophobizing agent used often to lower the surface energy of
glass, or silicon wafers in semiconductor manufacturing technology is the 1,1,1,3,3,3-hexamethyldisilazane (HMDS);
upon reaction with the fused silica or silicone surface and HMDS the surface is fully covered by a monolayer of hydro-
phobic trimethylsilyl groups (TMS-FS). The TMS-FS monolayer can be degraded by exposure to UV light to fine-tune
the surface hydrophobicity. The photodegradation kinetics of the TMS-FS was characterized by measuring the water
contact angle as function of UV irradiation time (Fig. 3.8) [25].
Most of the experimental data of surface energies of metals come from surface tension measurements in molten state
extrapolated to zero temperature [26]. The surface energy of metals can also be computed from the first principles.
Computational and experimental work have also shown that the surface energy of metals depends on the orientation
17
3.5 Surface and interfacial energy of solids](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-24-320.jpg)
![of the surface facets, see Table 3.3. For a polycrystalline metal surface, it can be expected that its surface energy is some-
what an average of these.
Depending on the material type, the type of interactions between the constituting atoms or molecules in the material
and the surface energy varies wildly, as highlighted in Table 3.4. As already mentioned, the materials with strong
cohesion have a high surface energy, which is a fraction of the energy of cohesion. This is also the basis for the cal-
culation of surface energies directly from the number of uncompensated or broken bonds, the “broken bonds” concept
[19, 20]. Rough tendencies for surface energies of solids can be sketched: polymers 0.1J/m2
, ionic solids 1J/m2
,
metals 1J/m2
, covalent solids 1J/m2
. Surface energy is an important parameter used for evaluating the resistance
of materials to mechanical stress also known as resistance to mechanical failure. Under mechanical stress solid mate-
rials can suffer brittle or ductile failure. The ductile failure occurs slowly and usually can be predicted because the
material suffers plastic deformation ahead of appearance of a crack and the crack is relatively stable and doesn’t prop-
agate quickly. Most of the metals at normal temperatures suffer ductile failure. In contrast, the brittle failure of mate-
rials is less desired due to its unpredictability. The material suffering a brittle failure will shatter with no deformation,
with the crack propagating rapidly. Ceramics, ice, and cold metals shatter rapidly without increase in mechanical
stress. A crack appears in a material when the stress exceeds the energy of creating two new surfaces. Brittle metals
are typically associated with high surface energy values. For example, Au is very ductile (defined as percent elongation
before cracking) and can be drawn in thin wires very easily (1g Au can be drawn in thin wire up to 2.4km in length),
whereas W is hard but brittle. Also, diamond is hard yet very brittle. This is however not to be generalized to all types
of materials, for example, ionic solids have low surface energies yet are very brittle. In making such statement we con-
sider the surface energy in vacuum as a reflection of the cohesion strength between the atoms in the bulk material. Once
the surface of the material has been exposed to air or chemically modified (compare the surface energies for diamond in
vacuum and after treatment with gases in Table 3.4) it does not reflect the bulk properties of the material anymore. In
the latter case the “broken bond” concept cannot be applied to calculate the surface energy of a material whose surface
was not freshly cut under vacuum.
TABLE 3.3 Surface energies (SE) of some 3d transitional metals calculated by the full charge density (FCD)
method in generalized gradient approximation (GGA).
Metal Surface plane SE-FCD (J/m2
) SE-experiment (J/m2
)
Li (110)
(100)
(111)
0.556
0.522
0.590
0.522
Fe (110)
(100)
(111)
2.430
2.222
2.733
2.417
Al (111)
(100)
(110)
1.199
1.347
1.271
1.143
Ni (111)
(100)
(110)
2.011
2.426
2.368
2.380
Cu (111)
(100)
(110)
1.952
2.166
2.237
1.790
Au (111)
(100)
(110)
1.283
1.627
1.700
1.500
W (110)
(100)
(111)
4.0
4.635
4.452
3.265
For comparison the experimental data are included.
Adapted from L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollar, The surface energy of metals, Surf. Sci. 411 (1998) 186–202.
19
3.5 Surface and interfacial energy of solids](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-26-320.jpg)
![Numerical example 3.4
Calculate the contact angle of water with diamond (in vacuum), graphite, and soda glass. Comment on the values obtained:
what can be said about these surfaces? Can this knowledge in the value of the contact angle can be used to differentiate between
or identify different materials?
TABLE 3.4 Surface energies of various high-energy solids at room temperature, experimental or theoretically
calculated values.
Material Surface energy (mJ/m2
) Reference
Hg (liquid) 438 Exp. [15]
Sn 680 Exp. [15, 16]
Zn 896 Exp. [15]
Ag 1140–1250 Exp. [15, 16]
Al 1140 Exp. [15, 16]
Au 1370–1500 Exp. [15, 16]
Ti 1700 Exp. [15, 16]
Cu 1710–1830 Exp. [15, 16]
Pd 2050 Exp. [15]
Pt 2300–2480 Exp. [15, 16]
Ir 3000 Exp. [15]
Mo 2200–3000 Exp. [15, 16]
W 2900–3680 Exp. [15, 16]
CaF2 450 Exp. [16]
KCl 110 Exp. [16]
MgO 120 Exp. [16]
NaCl 370 Exp. [16]
Mica (in vacuum) 4500 Exp. [16, 17]
Mica (in air) 310 Exp. [16, 17]
Ice 110 Exp. [16]
Soda Glass 460 Exp. [16]
Al2O3 (sapphire) 638, 9 Calc. [18], exp. [17]
Graphite 1174, 1250 Exp. [17], calc. [18]
Diamond (111) 5650 Calc. [19]
Diamond (100) 9820 Calc. [19]
Diamond (hydrogenated surface) 47 Exp. [20]
Diamond (oxidized) 65 Exp. [20]
Quartz 50–69 Exp. [21]
Si (110) 58–61 Exp. [21]
Si (111) 34–38 Exp. [21]
PTFE 15–24 Exp. [21]
PE 22–35 Exp. [21]
Nylon 6 37–41 Exp. [21]
20 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-27-320.jpg)
![Solution
Calculate the contact angle of water w/diamond.
γwater ¼ 72:4 mN=m;γdiamond ¼ 5650mJ=m2
1 + cos θ
ð Þγwater ¼ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γwaterγdiamond
p
; 1 + cos θ
ð Þ γwater ¼ work of adesion
ð Þ
cos θ ¼
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
565072:4
p
72:4
1
θ¼arccos (16.7), where arccos is not valid in this range which means the equation cannot be used to predict the contact angle for
such high energy surfaces, i.e., not applicable. As a curiosity, it is known that diamond has high affinity to fat, so it can be
extracted with oil.
There are several methods that can be used to determine directly the surface energy of solids.
Fracture method: The fracture surface energy is defined as the energy absorbed during the propagation of a crack
over a unit of surface area [27]. For this experiment the samples are prepared as long rectangular bars, which are pre-
pared from single crystals or polycrystalline materials. The prepared specimens are then cracked with a special device
[28] that applies tensile forces as depicted in Fig. 3.9, the load is increased until the crack appears and a “double
cantilever” forms [29]. The experimental methods available for measuring the fracture surface energy include the dou-
ble cantilever method, notched-beam technique, work of fracture method, and the compliance method [27]. The work
dW done by the applied force F to increase the fracture by dx is equal to the strain energy dU (elastic energy stored in the
specimen) of the “leaf springs” and the surface energy, dS [28, 30]:
dW ¼ dU + dS
where dS¼γwdx is the surface energy per unit area and external work dW¼Fdy, where y is the displacement of one side
of the cantilever, solving for the surface energy gives [28]
γ ¼
6F2
x2
Ew2t3
(3.9)
Measuring Young’s modulus, E, the load, and the lengths x, y, see Fig. 3.9A, gives γ.
The measurements of the surface energy by the fracture methods can be done in air or vacuum. Because adsorbates
present in atmosphere will modify the surface energy of the material the length of the crack is affected. For example, air
molecules can adhere to the newly formed surface during the fracturing process reducing the surface energy; the sur-
face energy of mica in vacuum and in air is compared in Table 3.4.
3.6 Solid-liquid interfaces
When a water droplet touches the surface of freshly cleaned glass it spreads until total coverage is achieved. Respon-
sible for this is the force of adhesion between glass and water molecule, which is larger than the forces of cohesion
between the molecules of water. Therefore, the increase in the surface area of the liquid is compensated by the strong
FIG. 3.9 (A) Geometry of a double cantilever crack and the parameters x, y, and d describing the crack. (B) Geometry of a crack produced by
indentation.
21
3.6 Solid-liquid interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-28-320.jpg)
![adhesion forces (Fig. 3.10). If the adhesion force is comparable in magnitude to the cohesion force, then the spreading
will not be complete.
The angle formed between the tangent at the surface of the liquid and the plane of the surface is called “contact
angle,” θ. At equilibrium, for θ 0, the contour of the droplet where all three phases, liquid, solid, and gas meet is
called “three-phase” line. The forces acting at this three-phase line are illustrated in Fig. 3.11 and correspond to all
the interfacial tension vectors: γSG, γSL, and γLG. The orientation of these vectors is such that they minimize the inter-
facial area they represent.
By convention the interface is indicated by capital letters in the subscript and the denser phase is mentioned first. In
addition, the letter “G” in the subscript indicating the solid-gas interface or liquid-gas interface can be dropped, e.g., γS
and γL.
Based on the representation in the cartoon one can predict if the droplet will spread on the surface function of the
relative magnitude of each interfacial tension vectors. It is intuitively clear that if the γSG vector is very large in com-
parison to γSL then the surface will be wetted by the liquids because, γSG is pulling stronger on the three-phase line. The
forces acting against γSG are γSL and the projection of the γLG vector on the horizontal axis. On the other hand, if the force
of the adhesion Fa between liquid and solid is very large then γSL will be extremely small, according to Eq. (3.5), and the
expansion of the liquid droplet takes place and surface is fully wetted. Oppositely, if Fa is rather very small then
γSL γSG, compression or de-wetting of the surface will take place. The three interfacial tension vectors acting at
the three-phase line are related through Young’s equation:
γSG ¼ γSL + γLG cosθ (3.10)
The above equation was first described in words by the American scientist Young and later put in this form by Bang-
ham and Razouk [31]. The vertical components due to vertical projection of the γSL sinθ vector cancels out with the
surface strain vector created in the solid under the three-phase line, for an interesting discussion of this aspect consult
the critical review by Good [32]. It is important to note that Eq. (3.10) is only valid for the “dry” wetting, which is the
ideal case of a wetting liquid with zero vapor pressure. However, this situation is not realistic, and in most cases, the
wetting liquid will evaporate ahead of the three-phase line creating a situation of “wet” wetting, due to adsorption of a
FIG. 3.10 (A) The action of the adhesion force between the molecules in the
surface layer and glass will lead to the expansion of the solid-liquid interface
depicted in (B).
FIG. 3.11 Balance of interfacial forces depicted at equilibrium at the three-phase line
of a liquid droplet resting on a solid surface; the surface and interfacial tension vectors
are always tangent to the interface and oriented to minimizing it.
22 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-29-320.jpg)
![monolayer or submonolayer of molecules on the solid substrate ahead of the three-phase line. But the “wet” wetting
can be also caused by other factors, for example, by the humidity in the atmosphere or adsorption of other volatile
components in the environment on the surface of the solid, effectively lowering the surface energy of the solid and
even that of the liquid. In normal atmospheric conditions, we deal with the “wet” wetting situation.
In Young’s equation, we have used the interfacial tension of the pure solid and the liquid with the gas. Thus in
order to account for the case of the liquid evaporation ahead of the three-phase line, or adsorption from atmosphere,
the “effective” surface energy of the solid and surface tension of the liquid can be related to the energy and tension
of the pure solid and liquid in “vacuum,” γS and γL by the following relations first proposed by Bangham and
Razouk [31]:
γS γSG ≡ πeSG (3.11)
γL γLG ≡ πeLG (3.12)
where the pressures πeSG and πeLG are the equilibrium film pressures of the adsorbate, which is a monolayer or less,
whose contact angle on the surface is zero. πeLG is analog to the surface pressure in the Langmuir-Blodgett monolayer,
discussed in the later chapters.
Including the above corrections, Young’s equation becomes
γS πeSG ¼ γSL + γL πeLG
ð Þcosθ (3.13)
3.7 Scaling effects: When surface tension dominates gravity
At the surface of the Earth, the gravity deforms the liquid droplets when their diameters exceed a certain threshold
value lc. This value is called the capillary length, which depends on the nature of the liquid and the gravitational
constant.
P ¼
G
A
¼
mg
A
¼
ρV g
A
ρl3
c g
l2
c
¼ ρlc g (3.14)
Very often we can visualize water droplets smaller than 1mm in diameter that have perfectly round shape, when sit-
ting, for example, on a leaf of a plant. The bigger droplets are deformed and flattened by the gravitation. The reason the
spherical droplets can be seen when the water droplets are small is that below a certain size the surface tension dom-
inates gravity (Fig. 3.12). The capillary length below which surface tension dominates gravity can be calculated, from
the balance of surface tension and gravitational forces acting on a liquid droplet. The Laplace pressure acts to keep the
liquid droplet spherical:
P ¼
γ
lc
(3.15)
When the two pressures are balanced:
γ
lc
¼ ρlc g
lc ¼
ffiffiffiffiffiffiffiffiffiffi
γ
ρg
r
(3.16)
where lc is known as the capillary length or capillary constant.
3.7.1 Case study: How surface tension of liquids affects life at small scale and in outer space?
The capillary length scale is the crossover point between the hydrostatic pressure and the Laplace pressure, below
which the surface tension dominates the gravity, depicted in Fig. 3.13. In addition, the change in the capillary length on
Earth (g¼9.81m/s2
) and on Mars (g¼3.7207m/s2
) is significant so if there were water and vegetation on Mars the
landscape would look different, pearls of water on the plant leaves will be twice as large as on the Earth surface
(Fig. 3.12), ensuring an more unusual landscape. On the other hand, on Jupiter (g¼25.85m/s2
), where the gravitation
is significantly stronger than on Earth, the droplet of liquid would probably be less visible with the eye.
Interestingly, the size of the water droplets seems to be significantly larger than a few centimeters and appear to be
floating in imponderability on the international space station (ISS) which orbits only around 450km away from the
23
3.7 Scaling effects: When surface tension dominates gravity](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-30-320.jpg)
![Small creatures, such as insects, comparable in size to the capillary length have learned to adapt and smartly use the
surface tension of water for locomotion or to acquire special abilities such as ability to collect water in desert or dive in
and breathe under water. Due to the surface tension dominating the gravity below the capillary length, the small crea-
tures can manipulate liquid or air droplets similarly to the way we manipulate solid objects. For example, ants and
similar size insects can carry water droplets as we carry cups or balloons filled with water. Some aquatic insects such
as diving beetles can carry bubbles of air with them when diving into water, this air bubble is held in place by special
hairs that are attached to the body; the role of the attached air bubble is to provide the insect with oxygen for breathing
in this way without the need for gills [33, 34]. Some insects have more advanced adaptations called “plastron” that aids
breathing, which is a special array of grids that create an air cushion around the body [34]; when the insect breathes it
consumes the oxygen from the air cushion which lowers the partial pressure of oxygen which is then replenished by
the dissolved oxygen from the surrounding water [33]. Their life on other habitable planets of the size of Jupiter or Mars
would require new adaptation due to this critical capillary length. Water striders, insects living on the surface of the
water, use surface tension for locomotion. They can move very quickly on the surface of water and do not get wet due
to the hydrophobic hairs covering their entire legs and body surface; the insects’ weight is supported by the surface
tension force and they propel themselves by moving their legs in a sculling motion [35]. Recent studies of the biome-
chanics of water surface locomotion [36] revealed that the propulsion mechanism involves momentum transfer
through surface-generated hemispherical vortices (drag) generated by their leg stroke on the water surface and not
by capillary waves as initially believed [37]. The striking force of the water surface by the insects’ leg ranges between
0.1 and 2mN/cm when walking and jumping and depends on the size and type of the insect [38, 39]. Other insects such
as Microvelia use a different propulsion mechanism on the surface of the water, it uses the surface tension gradients for
propulsion, the Marangoni effect, by releasing the surfactant-like body fluids [36]. For larger creatures, compared to
capillary length with large Baudin number Ba¼Mg/γP≫1, where M is mass, P is the wetted perimeter, like iguanas
that are able to walk short distances on the surface of water to escape predators [36], the surface tension cannot keep
them afloat, so to walk on the water surface they commonly use high driving power and speed to generate inertial
forces [39]. Other creatures such as the “pistol shrimp” are able to hunt and communicate by releasing jet streams
of bubbles that travel as fast as 6.5m/s [40].
Plants have developed natural strategies for water repellency, and this will be mentioned throughout the current
work, this constitutes a source of inspiration for creating new material surfaces through biomimetics [41].
3.8 Capillary rise
Capillary rise phenomenon can be observed on immersing a glass capillary of radius r in a water container, water
will rise in the capillary. Capillary action is the process used by plants to take water and minerals from the ground.
Between water and the clean glass walls of the capillary there is strong adhesion. If the force of the adhesion Fa between
liquid and solid is very large, then according to Eq. (3.5), γSL will be extremely small, γSL ≪γSG, and the liquid is pulled
FIG. 3.14 NASA astronaut watches a water bubble float on
the International Space Station (ISS). From NASA astronaut
Chris Cassidy, Expedition 36 flight engineer, watches a water
bubble float freely between him and the camera, showing his
image refracted, in the Unity node of the International Space
Station., NASA Image Video Libr. (2013). https://images.
nasa.gov/details-iss036e018302.html (Accessed 18 September
2019).
25
3.8 Capillary rise](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-32-320.jpg)
![Note that if one of the radii is very large or infinity this reduces to the previous form of the equation. Also, menisci can
have one curvature positive 1/R1 0 and one negative 1/R2 0, in which case the above equation must be correctly
modified.
The Young-Laplace equation could also be easily deduced from the capillary rise, that is, the difference of pressure
pushing from the concave side of the meniscus must be equilibrated by the surface tension:
ΔPπR2
meniscus ¼ γ2πRmeniscus (3.28)
Oppositely, it can also be used to calculate the height of the liquid column in a capillary and obtain Jurin’s law,
Eq. (3.19).
Numerical example 3.5
What are the consequences of Laplace pressure? If one connects balloons with different radii through a tube, what will be the
radii of the balloons at the end, after the balloons have freely exchanged the gas? (see Fig. 3.19). Explain why?
3.9.2 Case study: Emulsions and foams
Emulsions are typically produced from two immiscible liquids, such as oil and water, by shaking or shear. One of
the phases will become the dispersed phase spreading in the form of droplets in the bulk of the other liquid. There are
two main types of emulsions: oil-in-water (o/w) and water-in-oil (w/o) [42]. The emulsion type in pure immiscible
phases is mainly determined by the volumetric fraction, where typically the phase with the lower volumetric fraction
tends to be the dispersed phase. For example, upon shaking of the biphasic system of water and heptane, where hep-
tane has a low volumetric fraction, an instantaneous emulsion is produced where large heptane droplets are observed
for a short time, but the phases quickly separate. When high shear stress is applied, by ultrasonication or high-speed
homogenization, to the same biphasic system one can observe that the heptane droplets become slightly smaller and it
takes a longer time for the phase separation to occur. In other words, in emulsion formation it takes external energy
input to break the droplets and opposing this is the Laplace pressure which is directly proportional to the interfacial
energy. The interfacial energy between heptane and water is around 40mN/m. To break the droplets with less energy
and obtain a better stability for the emulsions, surfactants are used. Surfactants play a dual role, firstly, they lower the
interfacial tension which helps with obtaining smaller droplets and secondly, they stabilize the emulsion and prevent
coalescence. When surfactants are present, the emulsion type is determined mainly by the nature of surfactant, accord-
ing to Bancroft’s rule [43], which states that o/w emulsions are obtained if the surfactants have a higher affinity for
the water phase and w/o emulsions are obtained are obtained if surfactants have a higher affinity for the oil phase.
The Bancroft rule applies also to Pickering emulsions and other type of amphiphiles such as Janus nanoparticles or
FIG. 3.19 Effect of the Laplace pressure on the balloon of different radii connected through a tube which allows for free flow of gas.
29
3.9 Capillary number](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-36-320.jpg)
![pseudo-amphiphilic nanoparticles which stabilize emulsions [44–47]. Emulsions are thermodynamically unstable sys-
tems as they carry a nonzero interfacial energy. Emulsions have only kinetic stability, meaning that they will phase
separate. Emulsions are found in many food products [48], cosmetics, paint and coating formulations, drugs, syrups,
and also find uses in agriculture and in high-tech fields [49]. The droplet characteristic size is typically above 100nm.
A special type of emulsion is microemulsion, when the dispersed phase is well below 100nm, typically between
10 and 50nm. Unlike emulsions the microemulsions are thermodynamically stable, the interfacial energy (Gibbs free
energy) between the immiscible phases with the addition of surfactant (often with a co-surfactant) is almost zero [50].
Although regular emulsions have surfactants, only in special cases with special types of surfactants microemulsions
are obtained. Arguably, it is believed that due to surfactant adsorption and structuring at the interface, the interfacial
energy is virtually zero [50–52]. Unlike regular emulsions, microemulsions form spontaneously or by very light shak-
ing. Due to the small size of the dispersed phase they appear clear and transparent, the dispersed phase is so small that
it does not scatter light, while regular emulsions appear milky and white. Surfactant micellar solutions are examples of
pseudo-microemulsions. Microemulsions are of interest for their use in enhanced oil recovery, due to their good oil
solubilization efficiency [53].
Similarly, with emulsion formation, the liquid foam formation is the dispersion of air bubbles in the water phase. In
this case the magnitude of the interfacial tension plays a role in bubble breaking and foam formation. Surfactants are
added to lower the surface tension of the liquid and they play a crucial role in foam stabilization [54].
3.10 Kelvin equation
From the Young-Laplace equation we have learned that the pressure inside a droplet of liquid is higher than that
outside, due to the action of the surface tension. In 1870 Lord Kelvin showed how the vapor pressure of a liquid is
affected by curvature of the interface. Due to Laplace pressure ΔP in a liquid droplet the evaporation of molecules
is faster than those from a liquid in a large container with a flat surface. In other words, the vapor pressure is greater
in the former case. Oppositely, the vapor pressure inside a gas bubble formed in a liquid is lower than that above flat
surface liquid, due to the negative Laplace pressure ΔP on the liquid side, convex side. Therefore, liquid may con-
dense inside the gas bubble leading to bubble collapse. To summarize, if the curvature is concave on the liquid side,
then Pcurved Pflat. If the curvature is convex on the liquid side, then Pcurved Pflat (where Pflat is the vapor pressure
when the surface has zero curvature).
The derivation of the Kelvin equation can be done by applying the hydrostatic principles or the more abstract ther-
modynamic ones [55].
The thermodynamic approach follows the change in the free energy ΔG upon curving the surface. Considering the
vapor pressure from a flat surface Pflat as the initial state and the vapor pressure from a spherical liquid bubble Pcurved
we obtain
ΔG ¼ RT ln
Pcurved
Pflat
(3.29)
The change in the free energy can also be determined from the equation:
dG ¼ VdPSdT (3.30)
at the constant temperature, the change of the free energy per mole of liquid becomes
ΔG ¼
ðΔP
0
VmdP ¼ VmΔP ¼
2γVm
Rcurved
(3.31)
where 1/Rcurved is the curvature of the surface and is positive on the convex side and negative on the concave side.
Vm is the molecular volume that is constant regardless of the curvature.
Bringing the two equations together we obtain Kelvin’s equation:
RT ln
Pcurved
Pflat
¼
2γVm
rcurved
Pcurved ¼ Pflat exp
2γVm
RTrcurved
(3.32)
30 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-37-320.jpg)
![The above equation has important consequences. It shows that the vapor pressure from a bubble liquid is larger than
that from a flat surface: this phenomenon is responsible for cloud stabilization—condensation and reevaporation of the
water from tiny liquid droplets. A liquid aerosol consists of tiny droplets of different sizes and different Laplace pres-
sures. In smaller aerosol droplets of a cloud, the evaporation of liquid is faster than that in the larger ones. Therefore, in
proper conditions of pressure and temperature the larger droplets in clouds continue to grow due to condensation at
the expense of smaller ones (Ostwald ripening) eventually falling as rain. Eq. (3.1) shows that the size of the droplet has
a significant effect on the vapor pressure of the liquid below 10nm, Table 3.5. For these calculations, it was assumed
that the surface tension of liquid remains constant with the change in the radius of the droplet, but it has been shown
that the surface tension of water in pores as tiny as 10nm, when there are so few liquid molecules, decreases from
72.8mN/m, the known value in bulk, to 55mN/m at 20°C.
3.11 Case study: Surface tension of liquids at the nanoscale and in nanopores
At a given temperature, surface and interfacial tensions of a planar interface is a constant characteristic of the liq-
uids. However, as the characteristic sizes of the liquids, in the form of droplet, bubbles, etc., decrease below 10nm,
significant differences in the value of the surface/interfacial tension can be observed. Experimentally, it was found
that the surface tension of liquids in nanopores deviates from that of a flat surface [56]. Surface tension changes only
when the liquid meniscus of a liquid achieves very large curvatures. Theoretically, the relationship between surface
tension and the curvature of the liquid was derived by Richard C. Tolman using arguments from Gibb’s thermody-
namic theory of the interfaces [57]. Others calculated, using models different from that of Tolman, the change in the
surface tension of water and other liquids (cyclohexane, benzene, etc.) with the curvature and noted that the surface
tension increases with concavity (bubbles) and decreases with convexity (droplets) [58]. For example, in the case of a
spherical water droplet with a radius of (radius of the curvature in this case) 5nm, the surface tension dropped to
67mN/m and for a radius of the curvature of 2nm the surface tension was 58mN/m. It therefore follows that when
using Kelvin’s equations to calculate the vapor pressure around a liquid droplet smaller than 10nm corrections to sur-
face tension must be made. The relationship between the surface tension and the droplet curvature can be given by the
following expression:
γ
γ∞
¼ 1 +
2δ
r
1
where γ∞ is the surface tension of a planar surface (with an infinite curvature), γ is the surface tension of the liquid, δ is
Tolman’s length on the order of the molecular diameter [59], and r is the radius of the curvature. It is generally assumed
that δ0 for spherical droplets and δ0 for bubbles in a liquid. It is worth noting that the expressions obtained by
Tolman and later by Ahn et al. [57, 58] are essentially the same using different arguments. Such experiments on drop-
lets and bubbles that are below 10nm are, however, very difficult to carry out. Therefore, measurements were done on
liquids contained in nanopores, for example. Due to the very few liquid molecules contained in the nanopore, the nano-
pore walls strongly influence the surface tension. The density and the surface tension of the water in pores of a meso-
porous silica, with a pore radius between 1.55 and 3.90nm, were determined to be lower than those of bulk liquid
water. This anomalous change in the density and surface tension of the water was attributed to the hydrogen bond
interaction between liquid water molecules and the surface hydroxyl groups on silica surface, which led to some level
of molecular ordering and structuring in the fluid [56]. The surface energy of other materials, such as metals, crystals,
TABLE 3.5 Calculated equilibrium pressure ratios for droplets and bubbles as a function of their radius.
Radius (nm) Pcurved
Pflat
for droplets Pcurved
Pflat
for bubbles
1000 1.001 0.999
100 1.011 0.989
10 1.114 0.898
1 2.950 0.338
31
3.11 Case study: Surface tension of liquids at the nanoscale and in nanopores](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-38-320.jpg)
![alloys, was also shown to be size-dependent. Establishing rigorous models to calculate and predict interfacial energy
values for materials in the nanoscale is of vital importance [59]. Jiang and Lu have recently attempted to model the
evolution of surface energy of different materials and found that solid-vapor interface energy, liquid-vapor interface
energy, solid-liquid interface energy, and solid-solid interface energy of nanoparticles and thin films decrease with the
decrease in their dimensions to several nanometers, while the solid-vapor interface is size-independent and equals the
corresponding bulk value [59].
3.12 Methods for measuring the surface and interfacial tensions of liquids
Capillary rise is arguably the oldest method for the measuring the surface tension of liquids. A thin capillary of
known radius is immersed in a liquid and due to the interaction of forces of the liquid with the capillary walls, the
liquid rises in the capillary. By measuring the height of the capillary and using Jurin’s equation, Eq. (3.13), one can
determine the surface tension of the liquid.
Stalagmometer method or the drop volume method is based on the weighing of one or several drops of liquid formed
at the end of a capillary and allowed to drop in a weighing pan. The pendant drops formed at the tip of the capillary
start to detach when its weight reaches the magnitude balancing the surface tension. Therefore
mg ¼ 2πrγ (3.33)
where r is the radius of the capillary and m is the mass of the single droplet. The measurement of several droplets can
make the method more accurate. The limitations of the method come from the fact that not the entire droplet at the end
of the capillary falls, and this depends on the liquid properties; large errors can be produced this way unless correction
factors are introduced.
Wilhelmy plate method was initially proposed by Ludwig Wilhelmy in the 19th century and it is based on immersion
of a Pt plate of known dimension with a roughened surface into a liquid to determine its surface tension. The Pt plate is
suspended from a balance so that the total weight can be measured. The total weight is the contribution of the surface
tension and the weight of the plate given by
Ftotal ¼ Weight ¼ mg + Pγcosθ (3.34)
where P is the wetted perimeter of the Pt plate (widthlength) and θ is the contact angle of the liquid with the Pt
surface. The depth of the immersion must be adjusted so that the effect of buoyancy is eliminated. If the contact angle
is zero the determination of the surface tension can be very accurate by this method.
Du No€
uy ring method is similar to the plate method but instead a Pt ring is submersed in the liquid. Upon submersion
the ring is pulled out slowly until completely detached from the liquid surface. The maximum force needed to detach
the ring is measured and it is equal to
Ftotal ¼ Wring + 22πrγ (3.35)
where Wring is the weight of the ring and r is its inner and outer radius which are considered equal because the ring is
very thin, made from a very thin wire and thus multiplied by factor 2 as the surface tension acts on both sides. With this
method liquid-liquid interfacial tension can also be measured.
Drop shape analysis of a pendant drop is a new method where the surface tension of a liquid and the interfacial tension
between two liquids can be determined with an optical system that captures the shape of a pendant drop and analyzes
the contour. The setup of such a system is depicted in Fig. 3.20 and can be done with a modern contact angle goni-
ometer system and the contour of the droplet can be automatically extracted and analyzed with the software.
The shape of a pendant drop or hanging droplet of fluid in air or in another liquid are determined by the action of
two forces, the gravitation and the surface tension. While the surface tension acts to minimize the surface area of the
droplet the gravitation tends to pull it down and thus stretch/elongate the droplet. The Young-Laplace Eq. (3.27) tell us
that there is a pressure difference between the exterior and the interior of a curved interface, the higher pressure is
always higher at the interior of the droplet. We note this pressure difference as ΔP. If the drop is perfectly spherical
then the pressure difference will be constant everywhere in the droplet. For a pendant drop, however, the gravitation
causes an elongation along the z coordinate, which can be arbitrarily chosen in a plane cutting the long axis of the drop
in the middle in the vertical direction (Fig. 3.20B). The elongation of the droplet causes variation in the Laplace pressure
along the z axis, causing ΔP(z) to change, from that at the apex ΔP0, and this can be written as
32 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-39-320.jpg)
![ΔP z
ð Þ ¼ ΔP0 Δρgz (3.36)
which at the apex takes the following expression:
ΔP 0
ð Þ ¼
2γ
R0
¼ 2κ0
where R0 is the radius of the droplet at the apex and κ0 ¼1/R0 is the curvature at the apex. The meridional κs and the
circumferential κΦ curvatures (where κs
1
and κΦ
1
are the corresponding radii sweeping in the plane of the paper and
perpendicular to the plane of the paper, respectively, see Fig. 3.20B) of the droplet will change at any point away from
the apex [60].
The Laplace-Young equation further away from the apex becomes
ΔP z
ð Þ ¼ γ κs + κΦ
ð Þ ¼ ΔP 0
ð ÞΔρgz (3.37)
where Δρ is the density difference between fluids inside and outside of the droplet; when the fluid outside of the drop-
let is air, the density can be taken as 1.18kg/m3
, which is the density of air at sea level and 23°C.
The above equation can be reparametrized by introducing the arc length ds. Based on geometric arguments it can be
found that
dx
ds
¼ cosΦ and
dz
ds
¼ sinΦ
FIG. 3.20 (A) Typical setup for measuring the surface and interfacial tensions of a liquid with optical methods. (B) The drop shape contour anal-
ysis in cylindrical coordinates and the parameters with the reference point at the apex of the pendent droplet.
33
3.12 Methods for measuring the surface and interfacial tensions of liquids](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-40-320.jpg)
![with the boundary conditions at the apex s¼0, x(s¼0)¼z(s¼0)¼Φ(s¼0)¼0, and x(s¼L)¼D/2, where L is the full arc
length and D is the diameter of the needle from which the droplet is hanging.
From this it results that the circumferential curvature kΦ ¼sinΦ/x and the meridional curvature ks ¼dΦ/ds. Inserting
these curvatures in the Young-Laplace Eq. (3.37) we obtain
γ
dΦ
ds
+
sinΦ
x
¼
2γ
R0
Δρgz
which yields the final form of the shape equation of a pendant droplet at any point along the z axis:
dΦ
ds
¼
2
R0
sinΦ
x
Δρgz
γ
(3.38)
The above differential equation can be solved by numerical procedures. The above equation can be rewritten in dimen-
sionless form by replacing the dimensions x with dimensionless reduced variables X. One way to do that is to multiply
the above equation by a length-scale factor equal to the capillary length lc ¼(γ/Δρg)1/2
, or the capillary constant.
Thus, all the variables of length will become dimensionless X¼x/lc, ¼z/lc, S ¼ s
lc
, and B¼R0/lc Thus, the above set
of equations can be rewritten such that it contains the new dimensionless variables:
dΦ
dS
¼
2
B
sinΦ
X
Z (3.39)
The shape of the pendant drop is therefore dependent on a dimensionless quantity, namely the Bond number ¼B2
,
after the English physicist Wilfrid Noel Bond. The shape of an axisymmetric droplet depends only on parameter β.
The bond number can be best interpreted as a measure of the relative magnitude of gravitational force to surface ten-
sion force for determining the shape of the drop. Gravitational force will elongate the pendent droplet (maximizing the
surface area) while the surface tension forces will make the pendent droplet more spherical (minimize the surface area).
For example, when the bond number β≪1, the surface tension force dominates, and the drop is nearly spherical, and
for β1, the gravitational force dominates and the pendant drop becomes elongated [61]. When β for a particular pen-
dant drop geometry can be determined as well as the drop radius R0 at the apex, the interfacial tension γ can be cal-
culated with the relationship [61]:
β ¼ B2
¼ R2
0
Δρg
γ
(3.40)
Historically, Bashforth and Adams [62] were the first to solve Eq. (3.42) numerically and that the same authors also
tabulated the solutions calculated by hand for the bond numbers β that show the deviation of the drop from the ideal
profile of a sphere and researchers used this tables to identify the profile of the drop and obtain the surface tension
values this way. However, in modern computerized systems the integration is performed automatically for any value
of β and after extraction of the contour of the droplet in the digital image it finds the best solution for the Young-Laplace
equation.
An automated contour shape analysis software can search the best match of the experimental drop profiles, with the
theoretically calculated profiles using the surface tension as one of the adjustable parameters. The numerical solution
of the Young-Laplace equation yield additional data, such as drop volume V and droplet surface area A:
V ¼ π
ð
X2
sinΦdS
A ¼ 2π
ð
XdS
This procedure is called the axisymmetric drop shape analysis method (ADSA) and is one of the possible algorithms
that can be used.
The advantages of this method over all other techniques is the rapidity and accuracy. Also it is a great to study time
dependence evolution of the surface and interfacial tensions as well as “aging effects.”
Maximum bubble pressure method for measuring dynamic surface tension
In this method gas bubbles are produced at a constant rate in a fluid through a capillary of precisely known radius.
The pressure inside the bubble continues to increase and the maximum value is achieved when the bubble has a hemi-
spherical shape and thus its radius coincides with the radius of the capillary. As the bubble continues to grow the
pressure inside the drop decreases again.
34 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-41-320.jpg)
![The vibrating jet method for measuring the dynamic surface tension is a very creative yet very cost effective to implement,
although more challenging for the user. Its theory was already worked out by Lord Rayleigh. The method is based on
the measurement of the wavelength of an oscillatory jet of liquid emerging through an elliptical orifice as it progresses
in time and space (Fig. 3.22) [63]. The wavelength of the oscillating jet as it departs from the orifice becomes longer and
longer until it disappears (Fig. 3.22).
The setup consists of a liquid reservoir, a set of flow regulators, and most importantly an oval orifice, which can be
made simply by deforming a Pasteur pipette (Fig. 3.22). The exact dimensions of the orifice, the long and short axis
must be precisely known for accurate calculations of the surface tension. The surface tension values are calculated from
the wavelength using the following expression:
γ ¼ Caρ
V
λ
2
where Ca is the capillary number and can be determined with pure water, ρ is the density of the solution, V is the flow
rate in mL/s, and λ is the wavelength of the wave. The time of corresponding to a certain wave can be calculated from
the distance d of the midpoint of the wave from the orifice (Fig. 3.22), and the velocity of the jet v, with the equation:
t s
ð Þ ¼
d
v
where the velocity of the jet can be calculated with the equation:
v ¼
V
a
where a is the area of the cross section of the elliptical orifice. This method is suitable for measuring the instantaneous
dynamic surface tension for times in the range of 10–400ms. Variants of the oscillating jet methods where waves in the
jet are produced by excitation can probe the instantaneous interfacial tension to time intervals as short as 0.1ms [64].
Note that the stalagmometer method can also be used to determine the dynamic surface tension where the droplet
formation rate can be increased by a peristaltic pump. In this case the droplet size at lower surface tension will be
smaller.
3.12.1 Case study: Aerosol spray coatings and the importance of the dynamic surface tension
Measuring the dynamic surface tension is particularly useful in dynamic processes. For example, formulations of
aerosol paint contain surfactants or surface-active agents which play the role of wetting agents and ensure an even
FIG. 3.22 (A) Principle of the oscillating method and general measurement setup. (B) Photograph of the oscillating jet of an aqueous solution
containing surfactants, which emerges from an oval orifice, the actual jet and its shadow with clearly separated waves and a reference ruler can
be observed. Waves of different wavelengths can be clearly seen. At shorter times, closer to the orifice (A) the wavelength is short, because the sur-
factants did not have time to fully adsorb on the surface. At later times, from the sixth wave the difference in wavelength become less obvious.
36 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-43-320.jpg)
![spread of the coated paint on a surface. In industrial coatings the quality of the paint must meet particularly tight
requirements, the surface finish must be smooth and free of cratering or pits. Pits are created on the surface when,
for example, the distance of the spray nozzle is too close to the surface to be painted. In such situation, the time of
flight for the droplet from the nozzle to surface is much shorter than the time it takes for the surfactant to saturate
and reach the surface of the aerosol droplet and when it lands on the surface it will not properly coat the surface
(Fig. 3.23). Zhang and Basaran [65] have studied the role of surfactant in spray coating and concluded that the sur-
factant dynamics and the dynamic surface tension play a major role, firstly with respect to the ability of the aerosol
droplet landing on the surface to wet and spread on the surface and secondly due to gradients in surface tension of the
paint that induce a Marangoni flow causing local stress.
In inkjet printing surfactants are added to improve the wetting properties of the ink. The time the ink droplets take
from the moment they exit the printing nozzle and to the moment they reach the printed surface is about 1ms [66].
Therefore, very fast dynamic surface tensions are needed that can be achieved at high surfactant concentrations and
special design for the surfactant structures. Dynamic surface tension is an important parameter in wastewater treat-
ment, flotation of minerals, and other industrial processes [67].
3.13 Measuring ultralow interfacial tension—The spinning drop tensiometer
The spinning drop tensiometer is an instrument used to accurately measure ultralow interfacial tension values, typ-
ically 1mN/m, between a light and a dense phase. The pendant drop tensiometer, or the force tensiometer, can mea-
sure well values of the interfacial tension that are larger than approximately 5mN/m, but cannot be used for ultralow
interfacial tension. The spinning drop tensiometer can measure low values of the interfacial tension such as in micro-
emulsions with good accuracy, depending on the instrument this can reach a 5103
mN/m. The spinning drop
tensiometer consists of a capillary tube filled with a dense phase and a droplet of a light phase, with a difference
in density of Δρ. The capillary is then rotated on its long axis with a high angular velocity, ω; during the spin the spher-
ical droplet contained into the phase will elongate due to the action of centrifugal forces, which push the denser liquid
to the exterior and the lighter fluid (droplet) toward the central axis of the cylinder, therefore the droplet will acquire a
cylindrical shape, of radius R. The relationship between the centrifugal forces and the interfacial tension are given by
Vonnegut’s equation:
γ ¼
Δρ ω2
4
R
FIG. 3.23 (Left) Long time of flight for aerosol droplets and the finish of the painted coat surface. (Right) Short time of flight and pitted surface of
paint coat.
37
3.13 Measuring ultralow interfacial tension—The spinning drop tensiometer](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-44-320.jpg)
![The main assumption is that the drop shape acquires a fully developed cylindrical shape, which is true for L/R4. All
droplets acquire eventually a fully cylindrical shape if the spinning speed is sufficiently large, however, this puts a limit
to the instrument’s ability to measure very high values of the interfacial tension. Recent software algorithms developed
based on the drop shape analysis using the Young-Laplace equation lift these restrictions, so larger values for the inter-
facial tension can be measured.
3.13.1 Case study: Role of interfacial tension in enhanced oil recovery
Primary, secondary, and tertiary oil recovery phases refer to the method applied for the oil extraction from the res-
ervoir. For primary extraction recovery, the oil is extracted from the natural pressure built up over time in the reservoir,
where the oil is naturally pushed out of the reservoir, therefore a set of valves and pipes are enough. For the secondary
oil recovery, pressure is built up in the reservoir by adding water, waterflooding, or gases. After first and second recov-
ery phases have taken most of the oil, there are still significant oil reserves left in the reservoir and if economically
justified the tertiary (or enhanced) oil recovery can proceed. In the tertiary oil recovery, chemicals or gases are used
to displace oil that is trapped in the pores of the rock, due to viscous, gravity, and capillary forces. The tertiary oil
recovery is especially used for heavy oil and tar sand extraction. Heavy oils and tar sands can have a significant per-
centage in oil reservoirs but are poorly displaced in primary and secondary recovery. Therefore, the bulk production of
these comes from the tertiary oil recovery [68]. Chemical flooding is among the methods used to enhance oil recovery,
in which different chemicals such as surfactants, polymers, alkalis, biopolymers, and combinations thereof are used to
improve the oil displacement from the rock (microscopic efficiency) and to improve the volumetric sweep efficiency
(macroscopic efficiency) [69]. The macroscopic efficiency refers to the increase in the volume of oil brought to the sur-
face. Because of the large viscosity difference between the oil and water the mobility of the water phase is much larger
than that of the oil phase, therefore the pumped water may flow around the oil, leaving the oil phase behind. To
increase the viscosity of the water phase, polymers and biopolymers are used and a polymer flood is also performed
to increase the mobility of the oil phase. Mathematically the mobility is expressed as
M ¼
λdisplacing
λdisplaced
¼
krw=μw
kro=μo
where kro and krw are the effective permeability of oil and water, respectively, and μo and μw are the viscosities of oil and
water phases, respectively. A mobility value of 1 is considered ideal because the displaced phase moves at the same
speed as the displacing phase, but for a mobility of 10 the water moves 10 times faster than the oil.
Chemical flooding into the oil well is performed to improve the oil displacement from the rock (microscopic effi-
ciency). The principle is to reduce the interfacial tension between the water phase and the oil phase to as low as possible
to increase the capillary number Ca¼ηU/γ where U is the linear velocity of the injected phase (m/s) and η is the vis-
cosity of the injected phase. Ca correlates closely to oil recovery and residual oil saturation Sr0 (S0 oil saturation is the
volume fraction of oil within the pore volume) [69]. Ca is in the range of 107
to 106
for typical water flooding and by
increasing the capillary number to 104
and 103
the oil saturation can be reduced to 90% [70] and residual oil satu-
ration approaches zero if capillary number reaches 102
[71]. This can be achieved by decreasing the oil/water inter-
facial tension (IFT) from 10–40mN/m to 102
to 103
. Such low interfacial tensions can be achieved in formulations
using surfactants, alkali surfactants, and polymer/alkali/surfactants [69, 72, 73]. The alkali is used for the saponifica-
tion of potential product in oil and to achieve in this way ultralow interfacial tension values. Both ionic and nonionic
surfactants have been used since 1970s [73]. Petroleum-derived sulfonate surfactants are the most economical surfac-
tants used to lower the interfacial tension between the water and the oil and to alter the wettability of the porous rock
from oil-wet to water-wet.
Numerical example 3.7
What is the capillary number for a brine oil interfacial tension γ ¼10mN/m, injected phase velocity U ¼3.5106
m/s, and
viscosity η ¼1mPas? Calculate the value of the oil/water interfacial tension needed to achieve a capillary number of 102
, for a
full oil recovery from the well, for the same conditions?
As already mentioned, surfactants also influence the amount of residual oil recovered via other mechanisms, such as
emulsification of oil and changing the wettability of rock [69]. However, the adsorption of surfactant on the rock cre-
ates losses that reduces the concentration of surfactants, reducing the chemical flooding to water flooding, losing
38 3. Surfaces and interfaces](https://image.slidesharecdn.com/31582-250401100042-e0fc4799/85/Chemistry-of-Functional-Materials-Surfaces-and-Interfaces-Fundamentals-and-Applications-Andrei-Honciuc-45-320.jpg)
![therefore oil recovery efficiency. Surfactant formulations in chemical flooding are done with the spinning drop tensi-
ometer to measure the minimum concentration of surfactants needed to achieve IFT values of 103
mN/m.
3.14 Surface and interfacial tensions with temperature
The surface or interfacial tension with increase in temperature always decreases. This has been observed experimen-
tally and the meaning of it can be understood from the Gibbs-Duhem equation treated in the next chapters. Essentially
it can be easily demonstrated that the surface entropy for a 1m2
surface area and constant pressure is
Ssurface ¼
dγ
dT
(3.42)
This means that the surface excess entropy increases with the increase in temperature since dγ/dT is always negative.
Because the surface excess is positive it indicates that the molecules at the interface have more entropy, are more dis-
ordered than in bulk. As we will see the surface excess thermodynamic functions is generally the amount of energy,
concentration, or in this case entropy possessed by the surface as compared to bulk.
There have been attempts over the years to predict the surface tension of the liquids at different temperatures.
Eőtvős’ empirical equation relates molecular volume Vm, temperature T, surface tension γ, and critical temperature, Tc
(the temperature at which the phase boundaries vanish and the liquid coexists with its vapors and the γ ¼0):
γ Vm
ð Þ2=3
¼ k Tc T
ð Þ (3.43)
where k is a constant, Vm ¼Mw/ρL, ρL is the density of the liquid, and Mw is the molecular weight. The Eőtvős constant k is
a measure of the entropy of formation of surface, in other words the entropy change induced by bringing the liquid
molecules from the bulk to the surface [74]. The constant k takes a value of 2.12 for nonpolar liquids. For H-bonding,
liquids have a lower value for k, for example, it ranges 0.7–1.5 for alcohols; 0.9–1.7 for organic acids; and for water, k
varies between 0.9 and 1.2, according to the measurement temperature range. There are other empirical relationships
proposed but they are not discussed in this chapter.
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![C H A P T E R
4
Surfactants and amphiphiles
4.1 Introduction
Amphiphilicity shows a property that can be shared by surfactant molecules, macromolecules, molecular assem-
blies, and nanoscopic objects, such as Janus nanoparticles [1, 2] that, inter alia, partition spontaneously at the boundary
between two phases, such as liquid-liquid, liquid-gas, or solid-liquid interfaces. In doing so, the amphiphiles lower the
interfacial tension and interfacial energy between phases. Amphiphiles can self-assemble into monolayers and supras-
tructures of different shapes and forms dictated by the geometry and orientation of the building blocks. Amphiphi-
licity implies the existence of chemical functional groups that in water are hydrophilic and hydrophobic (lipophilic)
connected by chemical or physical bonds and form distinct and spatially segregated regions in space with contrasting
surface polarity (Fig. 4.1), such as in surfactants, block copolymers, and Janus particles. Pseudo-amphiphiles are sur-
faces, homogeneous particles, or copolymers that are constituted by both hydrophilic and hydrophobic groups mixed
at the molecular scale (Fig. 4.1), without spatial segregation and which are wetted by both water and oil. Due to the lack
of well-defined spatial segregation between polar (hydrophilic) and nonpolar (hydrophobic) functional groups,
pseudo-amphiphiles do not self-assemble into well-defined structures and are not as effective as the amphiphiles
at lowering the interfacial energy between phases. Amphiphilicity is a scalable property, being active at the molecular
level, at the nanoscale and could presumably be extended into the microscale. The scalability of the amphiphilic prop-
erty is however fundamentally an open question and it is a subject of interest in fundamental research [1–3].
Surfactants are the best-known class of amphiphiles and consist of a hydrophilic (polar) and hydrophobic (nonpo-
lar) chemical functional group connected by chemical bonds, usually represented as in Fig. 4.2A. The hydrophobic part
is usually a hydrocarbon chain and the hydrophilic part can be an anionic, cationic, or nonionic functional group. Sur-
factants can self-assemble into a variety of structures, such as micelles, bilayers, monolayers, vesicles, etc. (Fig. 4.2B).
4.2 Brief historical account of surfactants
Soaps were made and used from immemorial times, with the first records traced back to antiquity, in Mesopotamia
[4]. Soaps are fatty acids surfactants, and in the past were obtained with rudimentary synthetic methods from basic
materials available in nature such as tallow fat, olive, argan, or palm oils, and alkali-rich ashes, remnant residue from
wood burning. The soap-making activities spread later on throughout the European continent (Fig. 4.3). The soap may
have contributed to an improved quality of life, eliminated, or reduced the diseases in the densely populated regions,
and enabled urban life. The reaching of new standards of living was marked by the appearance and mass production of
personal care products, i.e., toiletries and cosmetics. Many Mediterranean regions prospered from the soap and cos-
metics manufacturing activity because of their rich natural olive oil resources, such as the case of Provence in the south
of France, or Castile in Spain, Florence in Italy [5]. Olive oil is rich in saturated palmitic, stearic and unsaturated oleic,
linoleic, and linolenic acids (Fig. 4.3), which leads to good quality soaps. Sulfated oils were the first synthetic surfac-
tants prepared after soaps, in 1834 by Runge, by mixing olive oil and sulfuric acid and in 1875 sulfated castor oil also
known as “Turkey red” was prepared and used as dyeing additives, mordants, in textile industry [6]. A sulfated oil is
not a pure surfactant but a mixture of sulfate esters, water, and fatty acid surfactants. In 1935, Colgate-Palmolive intro-
duced the first soap-free shampoo, using sulfated mono- and di-glyceride surfactants (Fig. 4.3) [6].
43
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- 6. CHEMISTRY OF FUNCTIONAL MATERIALS SURFACES AND INTERFACES Fundamentals and Applications ANDREI HONCIUC
- 7. Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-821059-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Kayla Dos Santos Editorial Project Manager: Rafael G. Trombaco Production Project Manager: Vignesh Tamil Cover Designer: Victoria Pearson Typeset by SPi Global, India
- 8. Preface In these times it is undeniable that most industries deal increasingly more often than ever with surface and inter- facial phenomena. Chemists, physicists, and material sci- entists, with background and training in materials, surfaces, and interfaces are in great demand. From my experience in both industry and academia, I have observed that students attending courses of a general chemistry degree program encounter rather late in their curriculum courses dealing with interfacial phenomena and chemistry of interfaces. Some curricula have included the chemistry of interfaces under different for- mats, at the undergraduate level, some only at master’s level, in specialized modules. This can in part be explained by the fact that Chemistry has become an enor- mously vast array of scientific domains, branching into biochemistry, organic chemistry, physical chemistry, catalysis, industrial chemistry, inorganic chemistry, ana- lytical chemistry, materials chemistry, nanotechnology, polymer chemistry, petroleum chemistry, etc. Due to fecund research in the past two decades Chemistry of functional materials and interfaces covers a multitude of intertwined interdisciplinary subjects from the nano- scale, such as synthesis of polymeric and inorganic nano- particles, to macroscopic phenomena such as manufacturing of functional surfaces, food, and con- sumer products such as cosmetics, detergents, heteroge- neous catalysts, etc. The amplitude and the amount of information in each of these fields put pressure on stu- dents, more so than several decades ago; it is now harder for the students to keep track of the newest advances, even less so acquire a relevant practical experience. Nav- igating through the maze of scientific literature and infor- mation hard to decipher can be extremely intimidating for the future chemists. The same is true in industry; working in industry one realizes that time is of the essence. Chemists and laboratory technicians are expected by the company to be innovative and thrive in interdisciplinary fields, learn on the go, and become experts in the shortest amount of time, on the job. There- fore, I feel that this book would be useful as a textbook for students, chemists working in industry, and laboratory technicians first encountering the chemistry of interfaces, interfacial phenomena, colloids, nanotechnology, poly- mer nanoparticle synthesis, etc. I have used myself part of this material in my teachings both in academia and training of technicians from industry. While this material used as a coursework material at master’s level has ini- tially included much more theory and formula, I could feel the students had difficulties grasping these, due to the pressure, lack of time, and an extremely burdening curriculum. I thus preferred to make the hard choice of reducing the material only to essential theories and adopt a more descriptive and intuitive presentation. One of the leitmotifs of the book is the emphasis on practical appli- cations of such theories. After several years in refining this material I believe it came to a format well received by the students. In addition, to make it more useful for chemists performing interfacial experiments, in industry or academia, I have tried to add experimental details or hints on data interpretation from my own experience. vii
- 9. C H A P T E R 1 Introduction Interfaces are the boundaries separating two phases and define all objects in the three-dimensional world. Depend- ing on the strength of cohesion forces and binding energies between atoms and molecules, the phases can be gases, liquids, and solids, defining the physical states of matter. When the cohesion energies between the constituting atoms and molecules are stronger than randomizing effects of the thermal energy, the physical state changes from gas to a condensed phase of matter, liquid, or solid. The Boltzmann distribution gives the probability P that a system will be in a certain state as a function of the state’s energy and temperature: P eE=kT kT factor is often used as a scale energy factor in the molecular interactions. The cohesive energies per atom or molecule at 298K can vary from several kT between gas atoms, between 9 and 23kT in liquid Hg (the liquid with the strongest cohesive energy, 57.9 kJ/mol [1]), and 50kT in solids up to 342kT in W (1kT4.051021 J), the metal with the high- est melting point. The kT energy scale factor is introduced and discussed in detail in Chapter 2. Because the most important interactions between material interfaces take place in the liquid, or between material interfaces and liquids, the solid-liquid, liquid-liquid, and liquid-air interfaces deserve special attention. The overall balance between the repulsive and attractive forces between solutes and colloidal objects in liquids must be comparatively equal or larger than 9–23kT to have aggregation, adsorption, self-assembly, etc., and below 9kT to obtain stable dispersions and col- loids. As mentioned, liquids form at T¼298K, when the cohesive energy between the constituting atoms and mole- cules is larger than 9kT. While in the bulk of a liquid the interaction forces of a molecule or atom are fully symmetric at interfaces, in contrast, in the topmost layer of molecules or atoms the interaction forces are asymmetric. Due to this asymmetry, a certain tension/force arises in the plane of the interface. The stronger the interfacial tension, the stronger the asymmetry. At contact between two phases, the topmost layer of molecules at the phase boundary also interacts with the molecules from the other phase, this is called adhesion. The adhesion forces and energies counterbalance the asymmetry of the forces acting on the topmost molecular layer, i.e., the stronger the adhesion force, the smaller the interfacial tension. If the adhesion force is stronger than the cohesion force, then the interfacial tension disappears, the interface disappears, and the phases become fully miscible, as discussed in Chapter 3. This interfacial tension has also the character of an energy density, and for pristine interfaces this is causally related to the cohesion energy in the bulk material; interfacial energy density is about half the cohesion energy in bulk. Surface and interfacial tension of liquid-gas and liquid interfaces, as well as interfacial and surface energy of solids-liquid and solid-gas interfaces, are thoroughly discussed in Chapter 3. The effects of the interface tension can be seen in small liquid droplets or molten metals, as the shape of the droplet itself is modeled by this interfacial tension. The small world of insects and bugs are particularly affected by the interfacial tension. Because their size is comparable to the capillary length, when the shape of the liquids is fully determined by interfacial tension, not by gravitation, they have a different perception of the sur- rounding world than humans do. Interfacial tension can have a devastating effect on insects; some drown as they can- not escape the surface tension, but some have adapted to take full advantage of it. For example, small water droplets can be manipulated and transported by ants without any need for bottles or glasses, and some mosquitos have adapted on water to straddle along the smooth water surface, etc. (Fig. 1). Intuitively, the interfacial tension is the 2D equivalent of the cohesion energy in 3D. Interfacial tension is discussed in detail in Chapter 3. However, when the surface and the interface are chemically modified, e.g., with surfactant adsorbates, the inter- facial tension and energy density of interfaces do not reflect anymore the cohesive energy between the molecules in the bulk phase. Thus, the interface itself can be treated as a thermodynamic system on its own, as discussed in Chapter 7. The interfacial tension and interfacial energy density between phases are now an exclusive reflection of 1 Chemistry of Functional Materials Surfaces and Interfaces https://doi.org/10.1016/B978-0-12-821059-8.00004-1 Copyright © 2021 Elsevier Inc. All rights reserved.
- 10. the lateral interactions between surfactant molecules, polymers, or particles adsorbed at the interface. In fact, under- standing how to change the interfacial tension and energy density between phases was one of the key enabling ele- ments in the development of most technological advances in the 20th and 21st centuries, ranging from detergency, oil, and ore extraction to the advanced manufacturing of processors and advanced electronic devices (see Chapter 15). Surfactants and amphiphiles are molecules, polymers, and other building blocks of matter that adsorb spontane- ously at interfaces. Surfactants lower the interfacial tension and energy density between phases (water-oil, water-gas, solid-water) independently of their cohesion energy. This enables the formation of emulsions and foams and increase in surface wettability. Earlier, it was mentioned that when the adhesion forces are stronger than the cohesion forces between two liquids, the interfacial tension vanishes, and the liquids become miscible. The fact that, in the presence of a surfactant at interfaces, the interfacial tension is not anymore a true reflection of the bulk cohesion energy of the phases can be understood from the following example. If the interfacial tension between two water and oil phases becomes vanishingly small due to the addition of a surfactant, then the two phases do not mix, but this time they form emulsions consisting of very fine oil droplets dispersed into water. Chapter 4 gives an introduction into the vast field of surfactant chemistry. Emphasis is given on surfactant classification, surfactant design, and structure activity relationship. In sim- ple words, what makes a surfactant effective and how is this reflected in different physicochemical parameters? Chapter 4 also introduces other amphiphiles, such as Janus nanoparticles and supra-amphiphiles, noting that amphi- philicity is a scalable property, being active well beyond the molecular scale, well into the nano- and microscales. Amphiphiles and surfactants have an important property, which is to self-assemble into suprastructures. This enables the creation of smart, reconfigurable, or “environmentally aware” materials, bottom up, via self-assembly processes. Most of the surfaces we interact with on a daily basis are solid, such as the screen of the smartphone, the cup of coffee, the wheel of the car, etc. The tactile feel, the adhesion, is determined by the interfacial energy between our skin and these surfaces. In the modern world, the concept of functional surfaces is gaining more popularity and it becomes a requirement in the consumer products. Functional surfaces can be defined as surfaces that perform a function, such as self-cleaning windows, or have a superior property, such as antiadherent, omniphobic antifingerprint in smartphone screens, for example, while others are icephobic, or antifogging, etc. The key concepts in understanding the phenom- ena behind functional surfaces and interfaces are adhesion and wetting. Surface wetting refers mainly to the interaction of a liquid with a solid surface. Earlier, it was mentioned that when the adhesion forces are stronger than the cohesion forces between two liquids, the interfacial tension vanishes, and the liquids become miscible. The interfacial tension or energy between a solid and a liquid can also be altered, for example, with surfactants; however, when the interfacial energy between a solid and a liquid becomes vanishingly small, the solid surface becomes fully wetted by the liquid. The converse is true: when the interfacial energy is large, the surface becomes nonwetted, and the liquid pearls up on the surface of the solid. Scientists have learned that, in addition to interfacial energy between the solid and liquid, the geometry of the interface is key to designing functional surfaces. Finding inspiration in nature, scientists found out that hierarchical structuring of the surface of the solid can lead to a variety of functional surfaces, such as superhydropho- bic, superhydrophilic, icephobic, omniphobic, self-cleaning, etc. Chapter 5 gives an overview of the phenomena of FIG. 1 (A) Ant drinking water (https://www.shutterstock.com/image-photo/ant-drinking-water-505718482); (B) mosquito striding on the sur- face of water (https://www.shutterstock.com/image-photo/water-bug-standing-on-surface-calm-1732352752). 2 1. Introduction
- 11. wetting, wettability, and contact angle as the main measurement methods for macroscopic and nanoscale surfaces. Chapter 5 also introduces the several functional surfaces. In Chapter 6, a series of equations permitting the calculation of unknown surface tension, energy, work of adhesion, etc. from known measurable macroscopic parameters have been grouped under the name “fundamental equation of interfaces.” Their versatility in predicting the values of many interfacial parameters, for example, interfacial tension, wettability, polarity of the surface, etc. from contact angle makes them extremely useful in practice. In Chapter 7, the surface and interfacial tension are introduced via thermodynamic treatment of the interfacial layer. Although this treatment has no direct practical implications, it gives the theoretical background necessary for the interpretation of interfacial adsorption isotherms and interfacial tension vs concentration curves for surfactants and amphiphiles. Chapter 8 treats surface functionalization that can be achieved in different ways, by physical methods such as roughening of the surface, or photolithographic nanopatterning, and by chemical methods, by adsorption of surfactant molecules. The adsorption of surfactant molecules on solid surfaces involves either chemical or physical bonding, resulting in the formation of a self-assembled monolayer. Several types of chemical bonding and substrates are reviewed. In addition, a surfactant monolayer can be prepared first at the water-air interface and then transferred onto the surface of the solid via the Langmuir-Blodgett and dip-coating methods. Solid-solid interfaces also have practical relevance, especially in layered electronic devices. Solid-solid interface, in particular the metal-organic interface, is the locus of another type of phenomena of practical importance, namely the electron transfer. In the previous chapters, the interfaces were the place where different forces met. In Chapter 9, the metal-organic interfaces are treated as the contact point between electron energy levels of a metal, material with delo- calized electron energy levels called bands, and the organic molecules and polymers whose energy levels are discrete and localized. Understanding electron transfer between metal electrodes and organic conductors is of practical impor- tance, especially for the manufacturing of organic photovoltaics, organic light emitting diodes, and other organic elec- tronic devices. Any of these devices requires at least several layers of electroactive organic materials, and knowledge of adhesion, wettability, and interfaces is required for their development and manufacturing. Chapters 10 and 11 deal with the interaction forces and energies between interfaces in different media. These inter- action forces can be repulsive or attractive and they are the same forces governing the molecular interactions. The bal- ance between the attractive and repulsive interaction forces is of practical importance, controlling the phenomena of particle aggregation, colloid stability, particle adsorption on surfaces, self-assembly of nanoparticles, etc. Chapter 12 introduces colloids, which are the oldest type of nanomaterials known and are today encountered in the food industry, pharma, and many other consumer products. Colloids are constituted from finely divided particles, nanoparticles, or liquid droplets dispersed into a continuous medium. Because their surface-to-volume ratio is very high, their behavior is governed almost exclusively by their surface and interfacial properties. Synthesis of colloids as well as stability cri- teria is discussed. As a continuation on the topic of colloids, but deserving special attention, Chapter 13 introduces the synthesis of polymeric nanoparticles and polymeric nanostructured interfaces via emulsion polymerizations. As expected, the interfacial aspects determine the types of emulsions and nature of the nanomaterials that can be synthesized. The types of emulsions and conditions of formation are briefly reviewed. A case study covers some examples of synthesis of nanostructured interfaces, polymerization of the emulsions stabilized by amphiphilic particles. Some nanoparticles, depending on their surface properties, can also spontaneously adsorb at interfaces; they can form monolayers and stabilize emulsions. The factors responsible for why some particles can adsorb at liquid-liquid, liquid-gas, and solid-liquid interfaces are discussed in Chapter 14. Once adsorbed at the interfaces the particle-particle interactions leads to the decrease in the interfacial tension. Responsible for this is their lateral interaction, which is governed by the same types of forces as in case surfactants, and in addition by particle specific interactions, capillary floatation, or immersion forces. In fact, in recent times, nanoparticles have been used in the synthesis of photonic crys- tals via the Langmuir-Blodgett method and other self-assembly structures. The last chapter of this book discusses the role of interfaces in integrated circuit manufacturing via photolithogra- phy. Photolithography is the only top-down preparation method of nanomaterials and nanostructured surfaces. In the past few years, it evolved into the most precise technique to prepare with large machines, structures as small as 7nm (the gate of the field-effect transistor). In practice, the photolithographic manufacturing process of chips and processors requires in-depth knowledge and control of interfacial phenomena such as adhesion, wetting, capillary forces, and interfaces. Reference [1] G. Kaptay, G. Csicsovszki, M.S. Yaghmaee, An absolute scale for the cohesion energy of pure metals, Mater. Sci. Forum. 414–415 (2003) 235–240. https://doi.org/10.4028/www.scientific.net/MSF.414-415.235. 3 Reference
- 12. C H A P T E R 2 Thermal energy scale kT At the nanoscale, the interaction energies are generally expressed in multiples of kT, also referred to as the thermal energy scale. The average kinetic energy of a gas atom with three degrees of freedom is 3/2kT, is roughly the energy of thermal fluctuations at a given temperature 1 kT. The thermal energy has a randomizing effect contributing to an increase in the entropy of the thermodynamic system. By expressing the energy of intermolecular interactions or nano- particle interactions as multiples of kT, the interaction strength can be compared with the randomizing effect of temperature. Next, it is instructive to follow the kT in several different contexts as well as its origin. A thermodynamic system will tend to move toward a lower energy state when available. When applied to chemical systems, for example, a solute in a solution or a gas has a chemical potential defined as the rate of change of the Gibbs free energy with the number of species in the system, at constant temperature and pressure: μi ¼ δG δNi T,P (2.1) Therefore, the change in chemical potential of a gas or a solute in a solution changes with the change in concentration. Chemical potentials are important in describing the equilibrium in physicochemical processes such as evaporation, melting, boiling, solubility, interfacial adsorption, liquid-liquid extraction, etc. The reason why the chemical potentials are so important in the equilibrium chemistry is that when the two chemical systems are open and can exchange mol- ecules or atoms, the rate of change of their free energy would be equal when equilibrium is established. Take, for exam- ple, the molecules in the vapor and the liquid phase at equilibrium; by equating the chemical potentials of the molecule of type i in two phases at equilibrium, or two regions 1 and 2, we obtain μ1 i + kT ln X1 i ¼ μ2 i + kT ln X2 i (2.2) At equilibrium between n different phases, the above equality must be satisfied for all phases: μn i + kT ln Xn i ¼ constant (2.3) where Xi n is the molecular fraction, volume fraction, or concentration of solute in phase n. For pure solution, this is usually taken as unity. The factor k lnX is known under different names, such as the entropy of mixing, configuration entropy, entropy of confining the molecules, etc. Eq. (2.2) gives us the possibility to calculate the distribution of molecules between two phases, or two regions of space at equilibrium, for example, a liquid in equilibrium with its vapors, or the distribution of the molecules of gas in the atmosphere due to changes in the gravitational potential with altitude. For example, the number density ρz of the molecules of gas in the Earth’s atmosphere changes with the altitude z and the mathematical function that gives us the possibility to predict this change is μz i + kT ln ρz i ¼ μ0 i + kT ln ρ0 i (2.4) where ρi z is the number density of molecule i at altitude z and ρi 0 is the number density of molecules of gas i at the surface of the Earth z¼ 0. Rearranging the above formula gives us the barometric formula or barometric law that gives the density at the altitude z as a function of the number density of air molecules at the sea level ρi 0 : 5 Chemistry of Functional Materials Surfaces and Interfaces https://doi.org/10.1016/B978-0-12-821059-8.00015-6 Copyright © 2021 Elsevier Inc. All rights reserved.
- 13. ρz i ¼ ρ0 i exp μz i μ0 i kT ¼ ρ0 i exp mgz ð Þ kT (2.5) where m is the molecular mass and g is the gravitational acceleration. Note that the potential energy of the air mol- ecules mgz is “compared” to the kT at any height above the Earth’s surface. With the increase in the potential energy of the molecules compared to kT, less molecules are found at higher altitudes (Fig. 2.1). In other words, if mgz is small compared to kT, then the thermal energy would uniformize the distribution of molecules with the altitude such that little variation in the number density of air molecules would be registered. The same distribution applies to ions that, for example, carry a charge e between two different regions that have different potentials ψ1 and ψ2: ρ2 i ¼ ρ1 i exp e ψ2 ψ1 ð Þ kT (2.6) and this is known as the Nernst equation. It is nonetheless important to note that interactions are additive; for example, if the difference in energy between two regions is given by potential, potential energy, and chemical potential, then the exponent will be the sum of all these contributions. The above equations also give us the possibility to gauge the strength of interaction between molecules. For exam- ple, if a liquid is in equilibrium with its vapors at standard conditions of pressure 1atm and temperature 298K, then 1 mol of gas will occupy approximately 22.4m3 and a mole of liquid approximately 0.02m3 . Then the difference in energy between the liquid and gas states will be [1]: μ 0 gas i μ 0 liquid i kT ln X gas i X liq i kT ln 22:4 0:02 7kT (2.7) where 1kT is approximately the energy of the thermal fluctuations. Therefore, it can be said that if the interaction strength between molecules in a gas phase at temperature T is larger than 7 kT, then it condenses into liquid. Con- versely, if the cohesion strength between the molecules of a liquid become smaller than 7kT, then it transforms into gas as the cohesion energy is simply too low to hold the molecules together. This alludes to what is known as the Trouton rule, which states that the entropy of vaporization is roughly the same for different kinds of liquids, about 85JK1 mol1 , which is roughly 9.5kT. FIG. 2.1 The bottle was capped (left) in the mountain and brought to the ground level (right). 6 2. Thermal energy scale kT
- 14. The kT criterion can be generalized to gauge the interaction strength between molecules; as stated above, if the interaction between molecules in a medium is larger than 9.5 kT at a given temperature, then this interaction will dominate over the thermal fluctuations and form a condensed phase, due to aggregation, adsorption, or self-assembly. For interaction energies, the use of the kT energy scale is convenient, as 1kT equals the thermally induced 3D Brownian motion energy of a molecule (surfactant, or solute, or particle), which provides a reference value of interaction energies for molecules sticking together vs fly apart, binding vs unbinding, etc. (Fig. 2.2). Fig. 2.2 provides a variety of inter- action energies represented on the kT energy scale. Similarly, the kT factor is also met in kinetics. For example, the Arrhenius equation per molecule is k ¼ Ae Ea kT (2.8) where Ea is the activation energy barrier and k is the rate constant of the reaction. If, for example, Ea is much larger than kT, then the reaction rate is also very small. On the other hand, if the energy barrier is comparable to kT, then the reaction rate is high, and the reaction can be activated by the thermal energy. The kT factor is also encountered in the Boltzmann distribution, which is a probability distribution that gives the probability of a state to exist function of the state’s energy and temperature and it is given by P ¼ exp Ei kT X n j exp Ej kT (2.9) where P is the probability of state i, of the energy Ei, and n is the total number of accessible states of corresponding energies Ej (j¼1n). The Boltzmann distribution describes the distribution of particles, such as atoms or molecules, over all accessible energy states. In a system consisting of many particles, the probability of picking a random particle with the energy Ei is equal to the number of particles in state i divided by the total number of particles in the system, that is, the fraction of particles occupying the state i: Pi ¼ Ni Ntotal ¼ exp Ei kT X n j exp Ej kT (2.10) FIG. 2.2 Various interaction energies on the kT scale. 7 2. Thermal energy scale kT
- 15. The denominator in the above equation is the partition function 1/Z: 1 Z ¼ 1 X n j exp Ej kT (2.11) Reference [1] J. Israelachvili, Intermolecular and Surface Forces, third ed., Academic Press, San Diego, CA, 2011. 8 2. Thermal energy scale kT
- 16. C H A P T E R 3 Surfaces and interfaces An interface is the boundary between two immiscible phases in contact, such as liquid-liquid, liquid-solid, liquid- air, etc. Immiscibility arises when the constituent molecules interact stronger with the molecules from the same phase than with the molecules from the other phase, i.e., the “cohesion forces” are stronger than the “adhesion forces.” The force of cohesion is defined as the sum of the forces that act between the molecules of the superficial layer and the bulk, while the forces of adhesion are defined as the forces that act between the superficial layer and the molecules of the next phase. The interface is characterized by a certain thickness, which is intuitively taken as the thickness of the last layer of molecules at the surface of the phase that enter in “contact” or “feel” the influence of the molecules in the other phase. It has been the subject of intense research where exactly lies the borderlines defining the interface between two phases in contact. This can be simplistically defined as two monolayers thick, one monolayer at the interface belonging to one phase and the other monolayer to the next phase (Fig. 3.1). This is probably the most satisfactory way to intuitively understand the interface thickness. However, this not very rigorous, because the molecules from subsequent layers also feel the presence of molecules from the other phase via “longer ranged” forces that operate and whose intensity decays with the distance from the interface. Michael C. Petty stated that “if the molecules are electrically neutral, then the forces between them will be short-range and the surfaces layer will be no more than one or two molecular diam- eters; in contrast, the Coulombic forces associated with the charged species can extend the transition region over con- siderable distances” [1]. The experimental studies of the neat liquid-liquid interfacial thickness revealed that the hexadecane-water thickness is about 6Å by X-ray reflectivity and 15Å by neutron reflectivity [2]. The apparent discrepancy comes from the limitation of the both methods which include two contributions, namely the intrinsic width of the interface “that char- acterizes the crossover from one bulk composition to the other and a statistical width due to thermally induced cap- illary wave fluctuations (ripples) of the interface” [2]. This also reflects the difficulty of the experimental methods to probe the interfaces at molecular length scales. The X-ray reflectivity studies of the thickness of the mercury-water interface was 5Å, which is comparable to that of mercury-vapor interface of 5Å and that of pure water-vapor inter- face 3Å [3]. These and other studies have revealed that the liquid-liquid and liquid-vapor interfaces are at least two monolayers of molecules or atoms. Recently, combined surface vibrational spectroscopy and molecular dynamics revealed an even more complex aspect of interfaces; in addition to interfacial thickness variation, molecular structuring by ordering and layering of molecules near interface were observed [4]. The characteristic molecular vibrations were probed at the water- chloroform and water-dichloromethane interfaces as a function of interfacial depth. From the concentration profiles of both water and organic solvent molecules it was observed that both the dichloromethane and water extended dee- per into the opposite phase forming a thicker interface, while the water-chloroform interface was sharper. Near the water-chloroform interface water monomers were detected, not associated via H-bonds and the concentration profile of chloroform deeper into the bulk organic phase is oscillatory suggesting the CCl4 molecules are layered near the interface, not observed for dichloromethane. Numerous examples of ordering and structuring of molecules near the interface were reported, as well as the consequences, for example, surface freezing of the top molecular layer of alkane at alkane-vapor interface is 2–3°C higher than the freezing temperature, while this was not observed at an alkane-water interface, which suggests an increased ordering of alkane molecules in the former case [2]. 9 Chemistry of Functional Materials Surfaces and Interfaces https://doi.org/10.1016/B978-0-12-821059-8.00009-0 Copyright © 2021 Elsevier Inc. All rights reserved.
- 17. Molecular modeling also helped in gaining insight into the interfacial boundaries. It was found that the water- hexane interface is very sharp and is about two monolayers thick [5]; the water molecules near the hydrophobic inter- face are oriented such that the molecular plane and the dipole moment are parallel to the plane of the interface and the long axis of the hexane molecules is also parallel to the interface [5, 6]. In addition, some of the water molecules at the hydrophobic interfaces are incapable of hydrogen bonding, about one in four molecules exhibit dangling hydrogen bonds, which gives rise to a large interfacial energy [7]. On the other hand, water molecules near strongly polar inter- faces such as quartz are capable of hydrogen bonding and are oriented in an ice-like structure and no dangling bonds were observed in surface vibrational studies with vibrational sum-frequency spectroscopy [7]. The consequence of the molecular orientational ordering, layering, reduced capability of molecular bonding, and interfacial mixing lead to a change in the solvent properties near the interface. For example, second-harmonic gener- ation in combination with solvatochromic surfactants of different lengths known as “molecular rulers” were able to probe solvent polarity with depth near the weakly and strongly associating water-organic solvent interfaces. For exam- ple, the solvent polarity near the weakly associating water-cyclohexane interface quickly converges from the aqueous to the organic limit in less than 9Å, while the strongly associating water-1-octanol interface revealed a transition region of ordered octanol molecules at the interface giving rise to a hydrophobic barrier [8]. The chemical structure and the molecular dimensions greatly affect the thickness of the interface. Further systematic studies performed with “molecular rulers” revealed that at the water-organic solvent interfaces the interface thickness and polarity strongly depend on the molecular structure [9, 10]. The conclusion that can be drawn from experimental evidence is that the interface can be visualized as a sheet or as a thin “membrane” with certain thickness. The thickness of the interface depends on the ability of the phases to interact given by the balance between the adhesion and cohesion forces. For weakly interacting phases the interface thickness is nearly two monolayers thick (Fig. 3.1A), while for strongly interacting phases the interface will be thicker than two molecular monolayers (Fig. 3.1B). In addition, the polarity gradient across the interface can change due to molecular ordering at the interface, which propagates to a certain depth in bulk, depicted as a color gradient in Fig. 3.1B. The membrane separating two immiscible phases has therefore boundaries and is an open thermodynamic system because it can exchange matter and energy with the neighboring phases. Molecules move continuously to and from interface to bulk. A thermodynamic system is everything that has boundaries, an object constitutes a thermodynamic system, be that a car, a grain of salt, or an interface, and all possess a certain internal energy, U. The energy of the interface between the phases, also most commonly referred to as “interfacial energy,” is highest when the cohesion energy is much larger than the adhesion energy. Interfacial energy decreases to negligible values when the adhesion forces become comparable to cohesion forces and the phases begin to mix. Water-hexane interface is an example of a high interfacial energy interface while water-ethanol interface has a zero interfacial energy, i.e., completely miscible. FIG. 3.1 (A) The ideal sharp inter- face between two weakly interacting phases α and β can be imagined as a thin membrane, two monolayers thick with a sharp molecular density profile that separates two phases. (B) The thicker interface between two phases that are strongly interact- ing with diffuse density profile, thicker than two monolayers. The oscillation in the β phase indicate ordering. The solvent polarity near this interface changes due to order- ing and loss or gain in bonding capability. 10 3. Surfaces and interfaces
- 18. The molecules at the interface have energy higher than those in the bulk because they are not symmetrically sur- rounded by other “alike” molecules in a perfectly balanced sphere. This imbalance of attraction forces and suppression in the ability of the molecules to bond lead to more energetic molecules at the interface. 3.1 Surface tension of liquids In the bulk, a molecule or an atom can be surrounded by a maximum 12 neighboring molecules (6 in the same plane and 3 on each side of the neighboring planes) and experience a symmetric attraction from all sides in the 3D space. On the other hand, at the surface of a liquid the molecules are about only half-way surrounded by molecules, thus expe- rience an asymmetric attraction toward the bulk of the liquid (see Fig. 3.2A). The forces of cohesion act asymmetrically on the interfacial layer and the topmost layers of molecules of a phase are compressed (Fig. 3.2B). In addition to the cohesion forces that act perpendicularly on the surface plane, the surface tension forces act in the plane of the surface and oppose any action to increase the surface area. Surface tension can be intuitively understood as a unit vector force. To visualize this, an imaginary line can be drawn, of length l, on the surface of the liquid; this imag- inary line splits the row of two molecules (Fig. 3.3) If it were possible, by pulling apart the two rows of molecules on the surface of the liquid, a resisting tension force would arise because the molecules from each row attract each other gen- erating a tension opposing the split (Fig. 3.3A). If the length of this dividing line is l, then the force with which the pair of molecules in the two rows attracts each other is. F ¼ γ l (3.1) where γ is the unit tension force (N/m), i.e., the surface tension. Therefore, the surface tension γ is the unit force acting on the surface plane to minimize the surface area. To better understand the origin of the surface tension, we imagine a cross section through the surface of a liquid (Fig. 3.3). At equilibrium no net force is acting on this horizontal line (plane) of molecules. If it were possible, the pull of only one molecule out of this line (plane) would be countered by an opposing tension force trying to minimize the area of the surface layer. The unit surface tension forces act left and right on the molecule being pulled out of the line and the sum of these is the cohesion unit force (Fig. 3.3C). It costs energy to bring new molecules from the bulk to surface. When surface is expanded, more “bonds” from the bulk are “broken” as new molecules are brought to occupy the “holes” in the newly created surface, represented by the dotted circles in Fig. 3.3B. The energy of the molecules in bulk of the liquid with maximum of 12 neighbors is lower than the energy of the molecules at the surface with the maximum 6 neighbors. The energy required to bring a molecule from the bulk to increase the area of the surface is the energy of uncompensated bonds; at the surface, a molecule has about half of the neighbors of a molecule in the bulk, half of its “physical bonds” remain uncompensated (Fig. 3.3B). Therefore, the surface energy is only a fraction of the cohesion energy as explained later in more detail. FIG. 3.2 (A) Molecular interactions in a liquid; (B) compression of the topmost surface layer of molecules due to the force of cohesion. 11 3.1 Surface tension of liquids
- 19. Numerical example 3.1 Draw the surface tension force vectors acting on a steel needle with a hydrophobic surface floating on the surface of water depicted in Fig. 3.4. The needle does not penetrate the surface. Calculate the maximum radius of a steel needle r, of length l ¼1cm that can be held on the surface of the water without sinking. The density of steel is 8000kg/m3 , ρwater ¼1kg/dm3 , γwater ¼73mN/m. FIG. 3.3 (A) Surface of a liquid on which an imaginary line of length l divides two parallel rows of mole- cules; when trying to pull sideways the two rows of molecules by apply- ing a force F on each side, the surface tension forces oppose the distancing of molecules; the surface tension unit vectors are oriented perpendicular to the imaginary line. (B) Deformation of the liquid-gas interface by pulling only one molecule out of the surface; in this case two new empty “holes” (dotted circles) are created by the expansion of the liquid-gas interface. Two new molecules must be brought in from the bulk liquid to occupy the empty holes, depicted by the green curved arrows. The total energy of the interface will increase by an amount equal to the energy of uncompensated bonds of the new molecules occupying the holes. (C) Same situation as in (B) with the depiction of the opposing surface ten- sion forces resisting the deformation. The vectorial sum of the surface ten- sion vectors acting on the molecule being pulled out of the interface is the cohesion force acting on the molecule. FIG. 3.4 Forces acting on a steel needle, with a hydrophobic surface, floating on the surface of water without penetration. 12 3. Surfaces and interfaces
- 20. Solution When the steel needle does not penetrate the surface, it means that the water does not wet the surface of the needle, in which case the water contact angle with the surface of the needle is θ¼180 degrees. In addition, the water surface will suffer a certain deformation such that the resultant of the forces acting at the three-phase line will be oriented vertically, against the pulling gravitational force acting on the needle. At equilibrium the surface tension force balances the gravitational force: 2LγL ¼ mg Therefore, R can be directly calculated: R ¼ 2γL πρg 1=2 ; R ¼ 20:0728 π 80009:81 1=2 ¼ 0:77mm: Numerical example 3.2 A wooden stick of the length l floats on the surface of pure water. If we lower the surface tension of water by adding a droplet of soap on the right side of the stick, in which direction will the wooden stick move? Under the action of which force? Write down the equation for this force. TABLE 3.1 Surface tension of several liquids and molten metals to compare the surface tension to the strength of intermolecular interactions. Liquid Surface tension (mN/m) Temperature (°C) Neon 5.2 247 Oxygen 15.7 193 Ethyl alcohol 22.3 20 Olive oil 32.0 20 Water 58.9 100 66.2 60 72.8 20 75.6 0 Mercury 465 20 Silver 800 970 Gold 1000 1070 Copper 1100 1130 13 3.1 Surface tension of liquids
- 21. The surface tension of various liquids changes with temperature. The magnitude of the surface tension reflects the strength of interaction forces between the composing molecules and atoms, i.e., the cohesion forces (Table 3.1). For neon the strength of the interaction forces between atoms are very weak and the surface tension reaches 5.2mN/m at 247°C. The cohesion forces are comparatively stronger for oxygen molecules than for neon atoms. Metallic bonds between mercury atoms are considerably stronger than the van der Waals forces between neon atoms, therefore even at room temperature the surface tension value is comparatively large. 3.2 Predictive models for calculating the surface tension of liquids There were attempts to apply theoretical models to calculate the surface tension from the energy of cohesion or enthalpy of vaporization. For example, Stefan’s equation has been often used to do this [11]: γL ¼ ZS Z ΔHvapρ 2 =3 M 2 =3N 1 =3 A 2 4 3 5 (3.2) where ΔHvap is the enthalpy of vaporization of the liquid (kJ/mol) in standard conditions of pressure and temperature of 105Pa and 273.15K, respectively, M is the molecular weight (g/mol), NA is the Avogadro number, and ρ is the density of the liquid (g/cm3 ). ZS Z is the ratio between the coordination number of molecules at the surface with respect to bulk; this cannot be determined directly by experiment but can be calculated [12]. The ratio of the coordination num- bers for the compounds ranges from 0.0559 to 0.1784, whereas a value 0.25 for the ratio ZS Z has been obtained for many organic substances [12]. Numerical example 3.3 Calculate the surface tension of water using Eq. (3.2) knowing that ΔHvap ¼41 kJ/mol, M ¼18g/mol, ρ¼1g/cm3 , and ZS Z ¼ 0:13 is the average value of all the determined values by Strechan et al. [12]. Solution γwater ¼ 0:13 41104 6:868:44107 KJ mol mol g 2=3 g m3 2=3 mol molecules 1=3 ! ¼ 76 mJ m2 (3.3) The obtained value slightly overestimates the experimentally determined value of water surface tension 72.4 mJ/m2 at room temperature. Therefore, the correct calculation of ZS Z is important for obtaining accurate values of the surface tension. Significant effort has been dedicated to modeling the coordination number ratio ZS Z . Another extendedly used empirical model, especially for determining the surface tension of molten metals, is called the bond-broken model or the bond-cutting model where the surface energy is calculated based on the energy of cohe- sion Ecohesion [13]: γL ¼ ZZS Z Ecohesion (3.4) 3.3 Interfacial tension between liquids Surface tension is only a particularization of interfacial tension and it is used only when referring to liquid-gas (vapor) interface. Interfacial tension refers to liquid-liquid or solid-liquid interfaces, but because it is a more general concept than the surface tension it can be used throughout, also for liquid-gas interfaces. For liquid-liquid interfaces, the top- most layers of molecules from each phase are in contact (Fig. 3.5). In contrast to the liquid-gas case, the topmost layer at the liquid-liquid interface will now be under the action of two forces, forces of cohesion with the molecules from the bulk of the same phase and forces of adhesion with the molecules from the other phase. Consequently, the topmost layer of liquid is not as strongly compressed as at the liquid-gas interface. The stronger the forces of adhesion, the stronger the attraction of the topmost layer to the second liquid phase. 14 3. Surfaces and interfaces
- 22. Opposing deformation of the interface between two liquids, as depicted in Fig. 3.6A, is the interfacial tension unit vector γ12. It can be imagined that γ12 is exactly the sum of the surface tensions of the two liquids, γ1 and γ2. In fact, the force of adhesion between the two liquids makes γ12 smaller than the sum of the surface tension corresponding to each liquid. It costs less energy to bring a molecule from the bulk to the liquid-liquid interface than to the liquid-gas inter- face. This is because breaking of the “bonds” of cohesion of the bulk molecules to come at the interface will be partially compensated by the adhesion “bonds.” Therefore, γ12 is the sum of two surface tension minus twice the force of adhe- sion (Fig. 3.6B and C): γ12 ¼ γ1 + γ2 2Fa (3.5) FIG. 3.5 Balance of the forces of cohesion with the force of adhesion at the liquid-liquid interfaces. FIG. 3.6 (A) The surface tension vectors corre- sponding to liquid 1 and liquid 2 oppose the defor- mation of the liquid 1-liquid 2 interface. Interfacial tension vector is depicted here as the sum of the sur- face tension vectors of the pure liquids, but does not sufficiently describe the real situation. (B) The force needed to deform or expand the interface is lower because the opposing force due to the surface tension is now minimized by the adhesion forces with the molecules in the second phase. In other words, to promote a molecule from bulk to interface it costs much less energy at the liquid-liquid interface because the intermolecular bonds in bulk are now partially compensated at the liquid-liquid interface by the adhesion bonds with the molecules from the surface of the second phase. (C) The balance of forces at equilibrium, the surface tension force is compen- sated by the adhesion force. 15 3.3 Interfacial tension between liquids
- 23. Two extreme situations can be distinguished, if Fa is very high and comparable to the cohesion forces, then γ12 !0 and the two liquids can easily mix and will not form an interface. If, however, the molecules between the two phases are incapable of “bonding” or Fa is extremely low, then γ12 !γ1 +γ2, consequently it is more difficult to expand or deform this interface. 3.4 Relating surface tension to surface energy The surface tension is a force per unit length. However, the surface tension can also be related to energy per surface area, or energy density. Consider a wire frame with one mobile side, which can slide on the U-shaped frame without friction (Fig. 3.7). The mobile side has a length l. On this frame we have a membrane of water, a thin film. If we try to pull the mobile side to increase the area of the membrane by Δx, then the work done will be W ¼ FΔx (3.6) but F is 2γl, the factor 2 comes from the fact that there are two sides of the surface. Therefore, surface tension is also the energy per unit area: γ ¼ W 2lΔx ≡ Energy Area J=m2 (3.7) The surface tension can thus be redefined as the energy required to increase the surface area with one unit. 3.5 Surface and interfacial energy of solids Surface and interfacial tensions have units of force per unit length N/m or energy per unit area J/m2 and the two forms are perfectly equivalent. For the solid-gas interfaces instead of interfacial tension one uses the concept of “surface energy.” Similarly, for the solid-liquid interfaces “interfacial energy” is used instead of “interfacial tension.” The sur- face energy of solids arises because of all unsaturated or dangling bonds per unit area of surface of a solid. Surfaces of metals, for example, have high energy associated with them because the atoms in the first surface layer have fewer of neighbors than in bulk and therefore unsatisfied capacity to metallic bonding. The cohesive energy of metals is given by the enthalpy of atomization ΔHa (equivalent to its bond strength), which is 418kJ/mol for Fe, 844kJ/mol for W, 368kJ/mol for Au, and 327kJ/mol for Al [14]. A great amount of work is needed to form and shape metals of high cohesive energy. The surface energy is taken as a fraction of the cohesion energy, γ ¼fcohesion energy, 0f1. In calculations of the surface energy from energy of cohesion, f could arbitrarily be chosen as 0.5 (see Fig. 3.3). There are roughly 1.61019 atoms/m2 on the surface of an Fe and the surface energy can be estimated from the cohesion energy: SE 0:5 418 kJ 61023 atoms 1:61019 atoms m2 5535 mJ m2 (3.8) The surface density of metal atoms Nd was calculated from the density of the metal: Fe (ρ¼7850kg/m3 , Nd ¼1.61019 ), W (ρ¼19,600kg/m3 , Nd ¼1.321019 ), Au (ρ¼19,320kg/m3 , Nd ¼1.251019 ), and Al (ρ¼2712kg/m3 , FIG. 3.7 (A) U-shaped wire frame holding a thin liquid film membrane which has a mobile side of length l that can slide under the action of an external force; (B) the cross section of the liquid mem- brane film having two surface ten- sion forces, corresponding to each interface of the liquid membrane, opposing the expansion of the sur- face area under the action of the pulling force F. 16 3. Surfaces and interfaces
- 24. Nd ¼1.271019 ). The calculated surface energy using the above equation is 5535mJ/m2 for Fe, 9294mJ/m2 for W, 3833mJ/m2 for Au, and 3462mJ/m2 for Al. These calculated values give the expected trend but the magnitude differs significantly from the experimental and calculated values of the surface energies of the corresponding metals, listed in Tables 3.3 and 3.4. The source of this difference is found in the value of the factor f used in the calculations of the above formula, under the initial assumption that the surface energy is about half of the cohesion energy. This factor plays a similar role with the coordination number seen in Stefan’s Eq. (3.2). If value of the factor is taken as 0.16 instead of 0.5, which is close to that used for calculating the surface tension of water, see Eq. (3.3), the recalculated value for the sur- face energy of metals is 1772 for Fe, 2974mJ/m2 for W, 1226mJ/m2 for Au, and 1108mJ/m2 for Al, which are very close in magnitude to the experimental data and those calculated by more advance models, Tables 3.3 and 3.4 [22]. This shows that the magnitude of the surface energy originates in the cohesion energy of the condensed phase. It is, how- ever, difficult to predict a priori the value of the factor f used in such calculations, it can be done empirically just as in the case of the coordination number of Stefan’s equation or the use of more complex theories could provide a deeper explanation and a method for calculating such factors from fundamentals. Unlike liquids where the action of surface tension force is visible, especially in small droplets that acquire spherical shape under its action, in the case of solids the action of surface energy has no visible mechanical action. For example, if a solid is cut into smaller pieces then they will not suddenly change their shape due to the action of surface tensions. However, the action of surface energy/tension becomes visible if the solid can be melted at high temperatures. For example, a molten metal, when in liquid state at high temperatures above the melting point, behaves just as any liquid, will occupy the smallest volume so it will squeeze in spherical droplets, upon cooling the metal is shaped, formed, extruded, or pulled, so its surface remains “frozen” in a metastable state, therefore its surface possesses a high energy. A metastable state is a state in which a system can spend an extended time in a configuration other than the system’s state of least energy. Surface energy of materials is of great technological importance, as, for example, in material fatigue and stress anal- ysis. The cracks in materials can propagate and produce failures, whose detection is critical in aircraft construction and safe operation. The crack produces in a material to relax the elastic stress in the vicinity of the crack. The crack equi- librium length depends on the balance of forces between the elastic stress and increase in the surface energy of the material. Yet another area in which knowledge of the surface energy plays an important role is the wettability of “reservoir rocks” for the oil extraction and recovery. The natural oil reservoirs were usually classified as oil-wet, water-wet, or intermediate based on the affinity of the rock’s surface to oil or toward water. Wettability of the rock to water affects the reservoir production and the performance of enhanced oil recovery processes [23]. Another relevant field is that of sealants and adhesives. For optimum adhesion, an adhesive must completely wet and cover out the surface to be bonded. Wetting is necessary for an adhesive to cover a surface to maximize the contact area and the attractive forces between the adhesive and bonding surface. For a good performance, the surface energy of the adhesive must be substantially lower than the surface energy of the substrate to be bonded, as, for example, regular adhesives bond very poorly on the low surface energy Teflon, or polyethylene but bond very well on higher surface energy glass or metal surfaces. This is on the other hand also the principles behind nonstick coatings of the pans, such as Teflon. Surface energy of several polymer surfaces is presented in Table 3.2. To improve the adhesion of paints and coatings the surface must be thoroughly washed and degreased. Grease and wax have a low surface energy material and prevent a good adhesion when present on surfaces. To improve adhesion on low energy surface, different methods can be applied, which include plasma treatment, UV-light exposure, chem- ical oxidation with piranha solution, etc. Exposure of a surface to UV light will generate ozone and singlet oxygen that oxidizes the surface. It has been proposed that UV-generated ozone or singlet oxygen will insert in the CdH bonds of a hydrocarbon surface and create polar functional groups such as dCOOH, dCO, dCOH, etc. For example, polyeth- ylene (PE) surface energy is roughly 31mN/m, and increases after the treatment with various methods: after expo- sure to UV 33mN/m, after flame treatment 39mN/m, and after etching with chromic acid 40mN/m [24]. On the other hand, surface treatment for lowering the surface energy is also possible by surface modification with hydrophobic compounds. For example, a very popular hydrophobizing agent used often to lower the surface energy of glass, or silicon wafers in semiconductor manufacturing technology is the 1,1,1,3,3,3-hexamethyldisilazane (HMDS); upon reaction with the fused silica or silicone surface and HMDS the surface is fully covered by a monolayer of hydro- phobic trimethylsilyl groups (TMS-FS). The TMS-FS monolayer can be degraded by exposure to UV light to fine-tune the surface hydrophobicity. The photodegradation kinetics of the TMS-FS was characterized by measuring the water contact angle as function of UV irradiation time (Fig. 3.8) [25]. Most of the experimental data of surface energies of metals come from surface tension measurements in molten state extrapolated to zero temperature [26]. The surface energy of metals can also be computed from the first principles. Computational and experimental work have also shown that the surface energy of metals depends on the orientation 17 3.5 Surface and interfacial energy of solids
- 25. FIG. 3.8 Water contact angle of hydrophobized glass surface with hexam- ethyldisilazane (HMDS) after exposure to the UV light for different periods of time. From A. Honciuc, D.J. Baptiste, D.K. Schwartz, Hydrophobic interaction microscopy: mapping the solid/liquid interface using amphiphilic probe mol- ecules, Langmuir. 25 (2009) 4339–4342. Copyright 2009 American Chemical Society. TABLE 3.2 Solid surface energy data (SFE) for common polymers. Name CAS Ref.- No. Surface energy at 20°C (mJ/m2 ) Dispersive component (mJ/m2 ) Polar component (mJ/m22 ) Polystyrene PS 9003-53-6 40.7 34.5 6.1 Polytrifluoroethylene P3FEt/PTrFE 24980-67-4 23.9 19.8 4.1 Polytetrafluoroethylene PTFE 9002-84-0 20 18.4 1.6 (Teflon) 24980-67-4 23.9 19.8 4.1 Polyvinylchloride PVC 9002-86-2 9002-85-1 41.5 45 39.5 40.5 2 4.5 Polyvinylacetate PVA 9003-20-7 25087-26-7 36.5 41 25.1 29.7 11.4 10.3 Polymethylacrylate (polymethacrylic acid) PMAA 9002-86-2 9002-85-1 41.5 45 39.5 40.5 2 4.5 Polyethylacrylate PEA 9003-32-1 87210-32-0 37 41.1 30.7 29.6 6.3 11.5 Polymethylmethacrylate PMMA Polyethylmethacrylate PEMA 9003-42-3 35.9 26.9 9 Polybutylmethacrylate PBMA 25608-33-7 31.2 26.2 5 Polyisobutylmethacrylate PIBMA 9011-15-8 30.9 26.6 4.3 Poly(t-butylmethacrylate) PtBMA – 30.4 26.7 3.7 Polyhexylmethacrylate PHMA 25087-17-6 30 27 3 Polyethyleneoxide PEO 25322-68-3 42.9 30.9 12 Polyethyleneterephthalate PET 25038-59-9 44.6 35.6 9 Polyamide-6,6 PA-66 32131-17-2 46.5 32.5 14 Polyamide-12 PA-12 24937-16-4 40.7 35.9 4.9 Polydimethylsiloxane PDMS 9016-00-6 19.8 19 0.8 Polycarbonate PC 24936-68-3 34.2 27.7 6.5 Polyetheretherketone PEEK 31694-16-13 42.1 36.2 5.9 Data from Solid surface energy data (SFE) for common polymers, (n.d.). http://www.surface-tension.de/solid-surface-energy.htm (Accessed 18 September 2019). 18 3. Surfaces and interfaces
- 26. of the surface facets, see Table 3.3. For a polycrystalline metal surface, it can be expected that its surface energy is some- what an average of these. Depending on the material type, the type of interactions between the constituting atoms or molecules in the material and the surface energy varies wildly, as highlighted in Table 3.4. As already mentioned, the materials with strong cohesion have a high surface energy, which is a fraction of the energy of cohesion. This is also the basis for the cal- culation of surface energies directly from the number of uncompensated or broken bonds, the “broken bonds” concept [19, 20]. Rough tendencies for surface energies of solids can be sketched: polymers 0.1J/m2 , ionic solids 1J/m2 , metals 1J/m2 , covalent solids 1J/m2 . Surface energy is an important parameter used for evaluating the resistance of materials to mechanical stress also known as resistance to mechanical failure. Under mechanical stress solid mate- rials can suffer brittle or ductile failure. The ductile failure occurs slowly and usually can be predicted because the material suffers plastic deformation ahead of appearance of a crack and the crack is relatively stable and doesn’t prop- agate quickly. Most of the metals at normal temperatures suffer ductile failure. In contrast, the brittle failure of mate- rials is less desired due to its unpredictability. The material suffering a brittle failure will shatter with no deformation, with the crack propagating rapidly. Ceramics, ice, and cold metals shatter rapidly without increase in mechanical stress. A crack appears in a material when the stress exceeds the energy of creating two new surfaces. Brittle metals are typically associated with high surface energy values. For example, Au is very ductile (defined as percent elongation before cracking) and can be drawn in thin wires very easily (1g Au can be drawn in thin wire up to 2.4km in length), whereas W is hard but brittle. Also, diamond is hard yet very brittle. This is however not to be generalized to all types of materials, for example, ionic solids have low surface energies yet are very brittle. In making such statement we con- sider the surface energy in vacuum as a reflection of the cohesion strength between the atoms in the bulk material. Once the surface of the material has been exposed to air or chemically modified (compare the surface energies for diamond in vacuum and after treatment with gases in Table 3.4) it does not reflect the bulk properties of the material anymore. In the latter case the “broken bond” concept cannot be applied to calculate the surface energy of a material whose surface was not freshly cut under vacuum. TABLE 3.3 Surface energies (SE) of some 3d transitional metals calculated by the full charge density (FCD) method in generalized gradient approximation (GGA). Metal Surface plane SE-FCD (J/m2 ) SE-experiment (J/m2 ) Li (110) (100) (111) 0.556 0.522 0.590 0.522 Fe (110) (100) (111) 2.430 2.222 2.733 2.417 Al (111) (100) (110) 1.199 1.347 1.271 1.143 Ni (111) (100) (110) 2.011 2.426 2.368 2.380 Cu (111) (100) (110) 1.952 2.166 2.237 1.790 Au (111) (100) (110) 1.283 1.627 1.700 1.500 W (110) (100) (111) 4.0 4.635 4.452 3.265 For comparison the experimental data are included. Adapted from L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollar, The surface energy of metals, Surf. Sci. 411 (1998) 186–202. 19 3.5 Surface and interfacial energy of solids
- 27. Numerical example 3.4 Calculate the contact angle of water with diamond (in vacuum), graphite, and soda glass. Comment on the values obtained: what can be said about these surfaces? Can this knowledge in the value of the contact angle can be used to differentiate between or identify different materials? TABLE 3.4 Surface energies of various high-energy solids at room temperature, experimental or theoretically calculated values. Material Surface energy (mJ/m2 ) Reference Hg (liquid) 438 Exp. [15] Sn 680 Exp. [15, 16] Zn 896 Exp. [15] Ag 1140–1250 Exp. [15, 16] Al 1140 Exp. [15, 16] Au 1370–1500 Exp. [15, 16] Ti 1700 Exp. [15, 16] Cu 1710–1830 Exp. [15, 16] Pd 2050 Exp. [15] Pt 2300–2480 Exp. [15, 16] Ir 3000 Exp. [15] Mo 2200–3000 Exp. [15, 16] W 2900–3680 Exp. [15, 16] CaF2 450 Exp. [16] KCl 110 Exp. [16] MgO 120 Exp. [16] NaCl 370 Exp. [16] Mica (in vacuum) 4500 Exp. [16, 17] Mica (in air) 310 Exp. [16, 17] Ice 110 Exp. [16] Soda Glass 460 Exp. [16] Al2O3 (sapphire) 638, 9 Calc. [18], exp. [17] Graphite 1174, 1250 Exp. [17], calc. [18] Diamond (111) 5650 Calc. [19] Diamond (100) 9820 Calc. [19] Diamond (hydrogenated surface) 47 Exp. [20] Diamond (oxidized) 65 Exp. [20] Quartz 50–69 Exp. [21] Si (110) 58–61 Exp. [21] Si (111) 34–38 Exp. [21] PTFE 15–24 Exp. [21] PE 22–35 Exp. [21] Nylon 6 37–41 Exp. [21] 20 3. Surfaces and interfaces
- 28. Solution Calculate the contact angle of water w/diamond. γwater ¼ 72:4 mN=m;γdiamond ¼ 5650mJ=m2 1 + cos θ ð Þγwater ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γwaterγdiamond p ; 1 + cos θ ð Þ γwater ¼ work of adesion ð Þ cos θ ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 565072:4 p 72:4 1 θ¼arccos (16.7), where arccos is not valid in this range which means the equation cannot be used to predict the contact angle for such high energy surfaces, i.e., not applicable. As a curiosity, it is known that diamond has high affinity to fat, so it can be extracted with oil. There are several methods that can be used to determine directly the surface energy of solids. Fracture method: The fracture surface energy is defined as the energy absorbed during the propagation of a crack over a unit of surface area [27]. For this experiment the samples are prepared as long rectangular bars, which are pre- pared from single crystals or polycrystalline materials. The prepared specimens are then cracked with a special device [28] that applies tensile forces as depicted in Fig. 3.9, the load is increased until the crack appears and a “double cantilever” forms [29]. The experimental methods available for measuring the fracture surface energy include the dou- ble cantilever method, notched-beam technique, work of fracture method, and the compliance method [27]. The work dW done by the applied force F to increase the fracture by dx is equal to the strain energy dU (elastic energy stored in the specimen) of the “leaf springs” and the surface energy, dS [28, 30]: dW ¼ dU + dS where dS¼γwdx is the surface energy per unit area and external work dW¼Fdy, where y is the displacement of one side of the cantilever, solving for the surface energy gives [28] γ ¼ 6F2 x2 Ew2t3 (3.9) Measuring Young’s modulus, E, the load, and the lengths x, y, see Fig. 3.9A, gives γ. The measurements of the surface energy by the fracture methods can be done in air or vacuum. Because adsorbates present in atmosphere will modify the surface energy of the material the length of the crack is affected. For example, air molecules can adhere to the newly formed surface during the fracturing process reducing the surface energy; the sur- face energy of mica in vacuum and in air is compared in Table 3.4. 3.6 Solid-liquid interfaces When a water droplet touches the surface of freshly cleaned glass it spreads until total coverage is achieved. Respon- sible for this is the force of adhesion between glass and water molecule, which is larger than the forces of cohesion between the molecules of water. Therefore, the increase in the surface area of the liquid is compensated by the strong FIG. 3.9 (A) Geometry of a double cantilever crack and the parameters x, y, and d describing the crack. (B) Geometry of a crack produced by indentation. 21 3.6 Solid-liquid interfaces
- 29. adhesion forces (Fig. 3.10). If the adhesion force is comparable in magnitude to the cohesion force, then the spreading will not be complete. The angle formed between the tangent at the surface of the liquid and the plane of the surface is called “contact angle,” θ. At equilibrium, for θ 0, the contour of the droplet where all three phases, liquid, solid, and gas meet is called “three-phase” line. The forces acting at this three-phase line are illustrated in Fig. 3.11 and correspond to all the interfacial tension vectors: γSG, γSL, and γLG. The orientation of these vectors is such that they minimize the inter- facial area they represent. By convention the interface is indicated by capital letters in the subscript and the denser phase is mentioned first. In addition, the letter “G” in the subscript indicating the solid-gas interface or liquid-gas interface can be dropped, e.g., γS and γL. Based on the representation in the cartoon one can predict if the droplet will spread on the surface function of the relative magnitude of each interfacial tension vectors. It is intuitively clear that if the γSG vector is very large in com- parison to γSL then the surface will be wetted by the liquids because, γSG is pulling stronger on the three-phase line. The forces acting against γSG are γSL and the projection of the γLG vector on the horizontal axis. On the other hand, if the force of the adhesion Fa between liquid and solid is very large then γSL will be extremely small, according to Eq. (3.5), and the expansion of the liquid droplet takes place and surface is fully wetted. Oppositely, if Fa is rather very small then γSL γSG, compression or de-wetting of the surface will take place. The three interfacial tension vectors acting at the three-phase line are related through Young’s equation: γSG ¼ γSL + γLG cosθ (3.10) The above equation was first described in words by the American scientist Young and later put in this form by Bang- ham and Razouk [31]. The vertical components due to vertical projection of the γSL sinθ vector cancels out with the surface strain vector created in the solid under the three-phase line, for an interesting discussion of this aspect consult the critical review by Good [32]. It is important to note that Eq. (3.10) is only valid for the “dry” wetting, which is the ideal case of a wetting liquid with zero vapor pressure. However, this situation is not realistic, and in most cases, the wetting liquid will evaporate ahead of the three-phase line creating a situation of “wet” wetting, due to adsorption of a FIG. 3.10 (A) The action of the adhesion force between the molecules in the surface layer and glass will lead to the expansion of the solid-liquid interface depicted in (B). FIG. 3.11 Balance of interfacial forces depicted at equilibrium at the three-phase line of a liquid droplet resting on a solid surface; the surface and interfacial tension vectors are always tangent to the interface and oriented to minimizing it. 22 3. Surfaces and interfaces
- 30. monolayer or submonolayer of molecules on the solid substrate ahead of the three-phase line. But the “wet” wetting can be also caused by other factors, for example, by the humidity in the atmosphere or adsorption of other volatile components in the environment on the surface of the solid, effectively lowering the surface energy of the solid and even that of the liquid. In normal atmospheric conditions, we deal with the “wet” wetting situation. In Young’s equation, we have used the interfacial tension of the pure solid and the liquid with the gas. Thus in order to account for the case of the liquid evaporation ahead of the three-phase line, or adsorption from atmosphere, the “effective” surface energy of the solid and surface tension of the liquid can be related to the energy and tension of the pure solid and liquid in “vacuum,” γS and γL by the following relations first proposed by Bangham and Razouk [31]: γS γSG ≡ πeSG (3.11) γL γLG ≡ πeLG (3.12) where the pressures πeSG and πeLG are the equilibrium film pressures of the adsorbate, which is a monolayer or less, whose contact angle on the surface is zero. πeLG is analog to the surface pressure in the Langmuir-Blodgett monolayer, discussed in the later chapters. Including the above corrections, Young’s equation becomes γS πeSG ¼ γSL + γL πeLG ð Þcosθ (3.13) 3.7 Scaling effects: When surface tension dominates gravity At the surface of the Earth, the gravity deforms the liquid droplets when their diameters exceed a certain threshold value lc. This value is called the capillary length, which depends on the nature of the liquid and the gravitational constant. P ¼ G A ¼ mg A ¼ ρV g A ρl3 c g l2 c ¼ ρlc g (3.14) Very often we can visualize water droplets smaller than 1mm in diameter that have perfectly round shape, when sit- ting, for example, on a leaf of a plant. The bigger droplets are deformed and flattened by the gravitation. The reason the spherical droplets can be seen when the water droplets are small is that below a certain size the surface tension dom- inates gravity (Fig. 3.12). The capillary length below which surface tension dominates gravity can be calculated, from the balance of surface tension and gravitational forces acting on a liquid droplet. The Laplace pressure acts to keep the liquid droplet spherical: P ¼ γ lc (3.15) When the two pressures are balanced: γ lc ¼ ρlc g lc ¼ ffiffiffiffiffiffiffiffiffiffi γ ρg r (3.16) where lc is known as the capillary length or capillary constant. 3.7.1 Case study: How surface tension of liquids affects life at small scale and in outer space? The capillary length scale is the crossover point between the hydrostatic pressure and the Laplace pressure, below which the surface tension dominates the gravity, depicted in Fig. 3.13. In addition, the change in the capillary length on Earth (g¼9.81m/s2 ) and on Mars (g¼3.7207m/s2 ) is significant so if there were water and vegetation on Mars the landscape would look different, pearls of water on the plant leaves will be twice as large as on the Earth surface (Fig. 3.12), ensuring an more unusual landscape. On the other hand, on Jupiter (g¼25.85m/s2 ), where the gravitation is significantly stronger than on Earth, the droplet of liquid would probably be less visible with the eye. Interestingly, the size of the water droplets seems to be significantly larger than a few centimeters and appear to be floating in imponderability on the international space station (ISS) which orbits only around 450km away from the 23 3.7 Scaling effects: When surface tension dominates gravity
- 31. Earth’s surface and the gravitational acceleration is still about 90% of the value from the Earth (g¼8.6m/s2 ). For this value of the gravitational acceleration the capillary length is only slightly larger than that on the Earth. By the size of the water droplet generated one would have expected that the capillary length is much larger. However, the reason for observing such centimeter large water droplet is that the ISS and all the objects in it orbit the Earth and experience a free fall which creates the imponderability. In fact, the large water droplets generated are due to this imponderability and free fall (Fig. 3.14). FIG. 3.12 (Top) Water droplets on a leaf of a plant. (Bottom) Water droplets on a plant leaf with different sizes. The larger droplets are deformed by the gravitation, while the smaller droplets remain spherical. FIG. 3.13 The crossover point between the Laplace pressure of a liquid and the hydrostatic pressure as a function of the characteristic scale on the surface of Jupiter, Earth, International space station (ISS), and Mars. 24 3. Surfaces and interfaces
- 32. Small creatures, such as insects, comparable in size to the capillary length have learned to adapt and smartly use the surface tension of water for locomotion or to acquire special abilities such as ability to collect water in desert or dive in and breathe under water. Due to the surface tension dominating the gravity below the capillary length, the small crea- tures can manipulate liquid or air droplets similarly to the way we manipulate solid objects. For example, ants and similar size insects can carry water droplets as we carry cups or balloons filled with water. Some aquatic insects such as diving beetles can carry bubbles of air with them when diving into water, this air bubble is held in place by special hairs that are attached to the body; the role of the attached air bubble is to provide the insect with oxygen for breathing in this way without the need for gills [33, 34]. Some insects have more advanced adaptations called “plastron” that aids breathing, which is a special array of grids that create an air cushion around the body [34]; when the insect breathes it consumes the oxygen from the air cushion which lowers the partial pressure of oxygen which is then replenished by the dissolved oxygen from the surrounding water [33]. Their life on other habitable planets of the size of Jupiter or Mars would require new adaptation due to this critical capillary length. Water striders, insects living on the surface of the water, use surface tension for locomotion. They can move very quickly on the surface of water and do not get wet due to the hydrophobic hairs covering their entire legs and body surface; the insects’ weight is supported by the surface tension force and they propel themselves by moving their legs in a sculling motion [35]. Recent studies of the biome- chanics of water surface locomotion [36] revealed that the propulsion mechanism involves momentum transfer through surface-generated hemispherical vortices (drag) generated by their leg stroke on the water surface and not by capillary waves as initially believed [37]. The striking force of the water surface by the insects’ leg ranges between 0.1 and 2mN/cm when walking and jumping and depends on the size and type of the insect [38, 39]. Other insects such as Microvelia use a different propulsion mechanism on the surface of the water, it uses the surface tension gradients for propulsion, the Marangoni effect, by releasing the surfactant-like body fluids [36]. For larger creatures, compared to capillary length with large Baudin number Ba¼Mg/γP≫1, where M is mass, P is the wetted perimeter, like iguanas that are able to walk short distances on the surface of water to escape predators [36], the surface tension cannot keep them afloat, so to walk on the water surface they commonly use high driving power and speed to generate inertial forces [39]. Other creatures such as the “pistol shrimp” are able to hunt and communicate by releasing jet streams of bubbles that travel as fast as 6.5m/s [40]. Plants have developed natural strategies for water repellency, and this will be mentioned throughout the current work, this constitutes a source of inspiration for creating new material surfaces through biomimetics [41]. 3.8 Capillary rise Capillary rise phenomenon can be observed on immersing a glass capillary of radius r in a water container, water will rise in the capillary. Capillary action is the process used by plants to take water and minerals from the ground. Between water and the clean glass walls of the capillary there is strong adhesion. If the force of the adhesion Fa between liquid and solid is very large, then according to Eq. (3.5), γSL will be extremely small, γSL ≪γSG, and the liquid is pulled FIG. 3.14 NASA astronaut watches a water bubble float on the International Space Station (ISS). From NASA astronaut Chris Cassidy, Expedition 36 flight engineer, watches a water bubble float freely between him and the camera, showing his image refracted, in the Unity node of the International Space Station., NASA Image Video Libr. (2013). https://images. nasa.gov/details-iss036e018302.html (Accessed 18 September 2019). 25 3.8 Capillary rise
- 33. up into the capillary under the action of γSG (Fig. 3.15A). Therefore, the total force acting to raise the liquid in the cap- illary is the resultant of the surface tensions multiplied by the circular perimeter of the capillary: F ¼ 2πr γSG γSL ð Þ ¼ 2πrγLG cosθ (3.17) The second relationship is Young’s equation and is valid at equilibrium. At the molecular level the expansion of the solid-liquid interface is driven by the adhesion of more water molecules to the walls of the capillary (Fig. 3.15A). The maximum height (h) at which the liquid rises in the capillary is achieved when the resultant of the interfacial forces are balanced by the weight of the liquid in the capillary (Fig. 3.15B): ρghπ r2 ¼ γ 2π r cos θ (3.18) h ¼ 2γ cos θ rρg (3.19) Eq. (3.19) is also known as Jurin’s law. Eq. (3.19) can be deduced from the difference in pressure above (on the concave side) and below the meniscus (on the convex side). For the case when Fa is very small, according to Eq. (3.5), γSL will be large, γSL ≫γSG, and the liquid is pulled down from the capillary under the action of γSL (Fig. 3.16A). Opposing the resultant of the surface tension forces 2πr(γSG γSL) is the volume of water displaced (Fig. 3.16B). Liquid rise in fine capillaries is among others responsible for the water uptake by plants, wicking action of textiles, functioning of fountain pens, chromatography, and many other liquid and water transport phenomena. 3.9 Capillary number The capillary number is a dimensionless quantity that results from the balance between drag forces in a fluid that tend to deform a moving bubble or a droplet and the interfacial tension forces that oppose this deformation. Ca ¼ μV γ ¼ viscous drag forces surface tension forces where μ is the dynamic viscosity of the fluid, V is the velocity of the fluid, and γ is the interfacial tension of the fluid. The capillary numbers should not be confused with the capillary length or constant, lc. FIG. 3.15 (A) Before equilibrium, water rises in the capillary due to the interfacial tension forces acting at the three-phase line. When Fa is large, γSL will be small and γSL ≪γSG. (B) At equilibrium, the surface tension of water opposes the gravitation. Note that the horizontal component of the surface tension force cancels out. Here R is the radius of the capillary. 26 3. Surfaces and interfaces
- 34. 3.9.1 Curved liquid surfaces, Laplace pressure, Young-Laplace equation If the interface dividing two phases is planar then the pressure experienced in each phase will be equal on both sides of the plane. Laplace pressure ΔP is the pressure difference that arises between two phases separated by a curved interface under the action of interfacial tension. The relationship between ΔP and γ is given by Young-Laplace equation that will be derived next in its simplest form. The curvature is defined as the deviation from planarity, 1/R, where R is the radius of the circle describing the bend. In a perfectly spherical soap bubble the pressure is greater inside the bubble because the surface tension acts at the surface of the soap bubble to decrease its surface area, thus compressing the gas inside. Similarly, the pressure is greater inside a gas bubble flowing in the sparkling water, or in an oil droplet in emul- sion due to the action of surface tension. Therefore, the pressure is larger on the concave side of the interface (Fig. 3.17). To derive the Young-Laplace equation in its simplest form we consider a gas bubble at equilibrium in a liquid. The force acting from the interior of the bubble is Finterior ¼ AbubblePinterior ¼ 4πR2 spherePinterior (3.20) The force acting from the exterior of the bubble is Fexterior ¼ AbubblePexterior + FIFT ¼ 4πR2 spherePexterior + FIFT (3.21) FIG. 3.16 The case when Fa ≪Fc leading to an expansion of the surface area of the liquid due to the stronger cohesion forces, leading to the low- ering the level of the liquid in the capillary due to the action of the surface tension. FIG. 3.17 Depiction of concave and convex sides of a surface. 27 3.9 Capillary number
- 35. The last term on the right-hand side of Eq. (3.21) is the force due to the surface tension and can be calculated, as described in Section 3.4, from the work needed to increase the area and the radius of the bubble by an infinitesimal amount, dR and dA: FIFT ¼ γ dA dR ¼ 4πγ dR2 sphere dR ! ¼ 8πγRsphere (3.22) The balance of forces is 4πR2 spherePinterior ¼ 4πR2 spherePexterior + 8πγRsphere (3.23) 4πR2 sphereΔP ¼ 8πγRsphere (3.24) The final form of the Young-Laplace equation in a cylindrical capillary is ΔP ¼ 2γ Rsphere (3.25) In the above equation the curvature was taken positive 1/Rsphere 0, but if the meniscus has a negative curvature then 1/Rsphere 0 and ΔP0. A simpler derivation is to equate the force on the meniscus’s surface due to the atmospheric pressure and the surface tension force acting on the three-phase line perimeter: πR2 sphere ΔP ¼ 2πRsphere γ (3.26) The Young-Laplace equation shows that the difference in pressure between the bubble decreases with the increase in the radius of the sphere, that is, if R is infinitely large, the surface has no curvature and the difference in pressure vanishes. A more general form of the Young-Laplace equation includes curved interfaces that are not spheres because the example considered above, a sphere, is only a particular case of more complex shapes. For example, the curved surface may have different curvature along direction x direction, which is described by the radius R1; in y direction, this cur- vature could be described by the radius R2 (see Fig. 3.18). For such case the Young-Laplace becomes ΔP ¼ 2γ R1 + R2 (3.27) FIG. 3.18 (Left) A curved surface whose curvature is described by R1 in the x direction and R2 in the y direction. In this case both curvatures are positive. Such situations can be met in liquids contained in rectangu- lar cuvettes or trays (Middle). (Right) An example of a liquid bridging between the tips of two pipettes showing one positive and one nega- tive curvature. 28 3. Surfaces and interfaces
- 36. Note that if one of the radii is very large or infinity this reduces to the previous form of the equation. Also, menisci can have one curvature positive 1/R1 0 and one negative 1/R2 0, in which case the above equation must be correctly modified. The Young-Laplace equation could also be easily deduced from the capillary rise, that is, the difference of pressure pushing from the concave side of the meniscus must be equilibrated by the surface tension: ΔPπR2 meniscus ¼ γ2πRmeniscus (3.28) Oppositely, it can also be used to calculate the height of the liquid column in a capillary and obtain Jurin’s law, Eq. (3.19). Numerical example 3.5 What are the consequences of Laplace pressure? If one connects balloons with different radii through a tube, what will be the radii of the balloons at the end, after the balloons have freely exchanged the gas? (see Fig. 3.19). Explain why? 3.9.2 Case study: Emulsions and foams Emulsions are typically produced from two immiscible liquids, such as oil and water, by shaking or shear. One of the phases will become the dispersed phase spreading in the form of droplets in the bulk of the other liquid. There are two main types of emulsions: oil-in-water (o/w) and water-in-oil (w/o) [42]. The emulsion type in pure immiscible phases is mainly determined by the volumetric fraction, where typically the phase with the lower volumetric fraction tends to be the dispersed phase. For example, upon shaking of the biphasic system of water and heptane, where hep- tane has a low volumetric fraction, an instantaneous emulsion is produced where large heptane droplets are observed for a short time, but the phases quickly separate. When high shear stress is applied, by ultrasonication or high-speed homogenization, to the same biphasic system one can observe that the heptane droplets become slightly smaller and it takes a longer time for the phase separation to occur. In other words, in emulsion formation it takes external energy input to break the droplets and opposing this is the Laplace pressure which is directly proportional to the interfacial energy. The interfacial energy between heptane and water is around 40mN/m. To break the droplets with less energy and obtain a better stability for the emulsions, surfactants are used. Surfactants play a dual role, firstly, they lower the interfacial tension which helps with obtaining smaller droplets and secondly, they stabilize the emulsion and prevent coalescence. When surfactants are present, the emulsion type is determined mainly by the nature of surfactant, accord- ing to Bancroft’s rule [43], which states that o/w emulsions are obtained if the surfactants have a higher affinity for the water phase and w/o emulsions are obtained are obtained if surfactants have a higher affinity for the oil phase. The Bancroft rule applies also to Pickering emulsions and other type of amphiphiles such as Janus nanoparticles or FIG. 3.19 Effect of the Laplace pressure on the balloon of different radii connected through a tube which allows for free flow of gas. 29 3.9 Capillary number
- 37. pseudo-amphiphilic nanoparticles which stabilize emulsions [44–47]. Emulsions are thermodynamically unstable sys- tems as they carry a nonzero interfacial energy. Emulsions have only kinetic stability, meaning that they will phase separate. Emulsions are found in many food products [48], cosmetics, paint and coating formulations, drugs, syrups, and also find uses in agriculture and in high-tech fields [49]. The droplet characteristic size is typically above 100nm. A special type of emulsion is microemulsion, when the dispersed phase is well below 100nm, typically between 10 and 50nm. Unlike emulsions the microemulsions are thermodynamically stable, the interfacial energy (Gibbs free energy) between the immiscible phases with the addition of surfactant (often with a co-surfactant) is almost zero [50]. Although regular emulsions have surfactants, only in special cases with special types of surfactants microemulsions are obtained. Arguably, it is believed that due to surfactant adsorption and structuring at the interface, the interfacial energy is virtually zero [50–52]. Unlike regular emulsions, microemulsions form spontaneously or by very light shak- ing. Due to the small size of the dispersed phase they appear clear and transparent, the dispersed phase is so small that it does not scatter light, while regular emulsions appear milky and white. Surfactant micellar solutions are examples of pseudo-microemulsions. Microemulsions are of interest for their use in enhanced oil recovery, due to their good oil solubilization efficiency [53]. Similarly, with emulsion formation, the liquid foam formation is the dispersion of air bubbles in the water phase. In this case the magnitude of the interfacial tension plays a role in bubble breaking and foam formation. Surfactants are added to lower the surface tension of the liquid and they play a crucial role in foam stabilization [54]. 3.10 Kelvin equation From the Young-Laplace equation we have learned that the pressure inside a droplet of liquid is higher than that outside, due to the action of the surface tension. In 1870 Lord Kelvin showed how the vapor pressure of a liquid is affected by curvature of the interface. Due to Laplace pressure ΔP in a liquid droplet the evaporation of molecules is faster than those from a liquid in a large container with a flat surface. In other words, the vapor pressure is greater in the former case. Oppositely, the vapor pressure inside a gas bubble formed in a liquid is lower than that above flat surface liquid, due to the negative Laplace pressure ΔP on the liquid side, convex side. Therefore, liquid may con- dense inside the gas bubble leading to bubble collapse. To summarize, if the curvature is concave on the liquid side, then Pcurved Pflat. If the curvature is convex on the liquid side, then Pcurved Pflat (where Pflat is the vapor pressure when the surface has zero curvature). The derivation of the Kelvin equation can be done by applying the hydrostatic principles or the more abstract ther- modynamic ones [55]. The thermodynamic approach follows the change in the free energy ΔG upon curving the surface. Considering the vapor pressure from a flat surface Pflat as the initial state and the vapor pressure from a spherical liquid bubble Pcurved we obtain ΔG ¼ RT ln Pcurved Pflat (3.29) The change in the free energy can also be determined from the equation: dG ¼ VdPSdT (3.30) at the constant temperature, the change of the free energy per mole of liquid becomes ΔG ¼ ðΔP 0 VmdP ¼ VmΔP ¼ 2γVm Rcurved (3.31) where 1/Rcurved is the curvature of the surface and is positive on the convex side and negative on the concave side. Vm is the molecular volume that is constant regardless of the curvature. Bringing the two equations together we obtain Kelvin’s equation: RT ln Pcurved Pflat ¼ 2γVm rcurved Pcurved ¼ Pflat exp 2γVm RTrcurved (3.32) 30 3. Surfaces and interfaces
- 38. The above equation has important consequences. It shows that the vapor pressure from a bubble liquid is larger than that from a flat surface: this phenomenon is responsible for cloud stabilization—condensation and reevaporation of the water from tiny liquid droplets. A liquid aerosol consists of tiny droplets of different sizes and different Laplace pres- sures. In smaller aerosol droplets of a cloud, the evaporation of liquid is faster than that in the larger ones. Therefore, in proper conditions of pressure and temperature the larger droplets in clouds continue to grow due to condensation at the expense of smaller ones (Ostwald ripening) eventually falling as rain. Eq. (3.1) shows that the size of the droplet has a significant effect on the vapor pressure of the liquid below 10nm, Table 3.5. For these calculations, it was assumed that the surface tension of liquid remains constant with the change in the radius of the droplet, but it has been shown that the surface tension of water in pores as tiny as 10nm, when there are so few liquid molecules, decreases from 72.8mN/m, the known value in bulk, to 55mN/m at 20°C. 3.11 Case study: Surface tension of liquids at the nanoscale and in nanopores At a given temperature, surface and interfacial tensions of a planar interface is a constant characteristic of the liq- uids. However, as the characteristic sizes of the liquids, in the form of droplet, bubbles, etc., decrease below 10nm, significant differences in the value of the surface/interfacial tension can be observed. Experimentally, it was found that the surface tension of liquids in nanopores deviates from that of a flat surface [56]. Surface tension changes only when the liquid meniscus of a liquid achieves very large curvatures. Theoretically, the relationship between surface tension and the curvature of the liquid was derived by Richard C. Tolman using arguments from Gibb’s thermody- namic theory of the interfaces [57]. Others calculated, using models different from that of Tolman, the change in the surface tension of water and other liquids (cyclohexane, benzene, etc.) with the curvature and noted that the surface tension increases with concavity (bubbles) and decreases with convexity (droplets) [58]. For example, in the case of a spherical water droplet with a radius of (radius of the curvature in this case) 5nm, the surface tension dropped to 67mN/m and for a radius of the curvature of 2nm the surface tension was 58mN/m. It therefore follows that when using Kelvin’s equations to calculate the vapor pressure around a liquid droplet smaller than 10nm corrections to sur- face tension must be made. The relationship between the surface tension and the droplet curvature can be given by the following expression: γ γ∞ ¼ 1 + 2δ r 1 where γ∞ is the surface tension of a planar surface (with an infinite curvature), γ is the surface tension of the liquid, δ is Tolman’s length on the order of the molecular diameter [59], and r is the radius of the curvature. It is generally assumed that δ0 for spherical droplets and δ0 for bubbles in a liquid. It is worth noting that the expressions obtained by Tolman and later by Ahn et al. [57, 58] are essentially the same using different arguments. Such experiments on drop- lets and bubbles that are below 10nm are, however, very difficult to carry out. Therefore, measurements were done on liquids contained in nanopores, for example. Due to the very few liquid molecules contained in the nanopore, the nano- pore walls strongly influence the surface tension. The density and the surface tension of the water in pores of a meso- porous silica, with a pore radius between 1.55 and 3.90nm, were determined to be lower than those of bulk liquid water. This anomalous change in the density and surface tension of the water was attributed to the hydrogen bond interaction between liquid water molecules and the surface hydroxyl groups on silica surface, which led to some level of molecular ordering and structuring in the fluid [56]. The surface energy of other materials, such as metals, crystals, TABLE 3.5 Calculated equilibrium pressure ratios for droplets and bubbles as a function of their radius. Radius (nm) Pcurved Pflat for droplets Pcurved Pflat for bubbles 1000 1.001 0.999 100 1.011 0.989 10 1.114 0.898 1 2.950 0.338 31 3.11 Case study: Surface tension of liquids at the nanoscale and in nanopores
- 39. alloys, was also shown to be size-dependent. Establishing rigorous models to calculate and predict interfacial energy values for materials in the nanoscale is of vital importance [59]. Jiang and Lu have recently attempted to model the evolution of surface energy of different materials and found that solid-vapor interface energy, liquid-vapor interface energy, solid-liquid interface energy, and solid-solid interface energy of nanoparticles and thin films decrease with the decrease in their dimensions to several nanometers, while the solid-vapor interface is size-independent and equals the corresponding bulk value [59]. 3.12 Methods for measuring the surface and interfacial tensions of liquids Capillary rise is arguably the oldest method for the measuring the surface tension of liquids. A thin capillary of known radius is immersed in a liquid and due to the interaction of forces of the liquid with the capillary walls, the liquid rises in the capillary. By measuring the height of the capillary and using Jurin’s equation, Eq. (3.13), one can determine the surface tension of the liquid. Stalagmometer method or the drop volume method is based on the weighing of one or several drops of liquid formed at the end of a capillary and allowed to drop in a weighing pan. The pendant drops formed at the tip of the capillary start to detach when its weight reaches the magnitude balancing the surface tension. Therefore mg ¼ 2πrγ (3.33) where r is the radius of the capillary and m is the mass of the single droplet. The measurement of several droplets can make the method more accurate. The limitations of the method come from the fact that not the entire droplet at the end of the capillary falls, and this depends on the liquid properties; large errors can be produced this way unless correction factors are introduced. Wilhelmy plate method was initially proposed by Ludwig Wilhelmy in the 19th century and it is based on immersion of a Pt plate of known dimension with a roughened surface into a liquid to determine its surface tension. The Pt plate is suspended from a balance so that the total weight can be measured. The total weight is the contribution of the surface tension and the weight of the plate given by Ftotal ¼ Weight ¼ mg + Pγcosθ (3.34) where P is the wetted perimeter of the Pt plate (widthlength) and θ is the contact angle of the liquid with the Pt surface. The depth of the immersion must be adjusted so that the effect of buoyancy is eliminated. If the contact angle is zero the determination of the surface tension can be very accurate by this method. Du No€ uy ring method is similar to the plate method but instead a Pt ring is submersed in the liquid. Upon submersion the ring is pulled out slowly until completely detached from the liquid surface. The maximum force needed to detach the ring is measured and it is equal to Ftotal ¼ Wring + 22πrγ (3.35) where Wring is the weight of the ring and r is its inner and outer radius which are considered equal because the ring is very thin, made from a very thin wire and thus multiplied by factor 2 as the surface tension acts on both sides. With this method liquid-liquid interfacial tension can also be measured. Drop shape analysis of a pendant drop is a new method where the surface tension of a liquid and the interfacial tension between two liquids can be determined with an optical system that captures the shape of a pendant drop and analyzes the contour. The setup of such a system is depicted in Fig. 3.20 and can be done with a modern contact angle goni- ometer system and the contour of the droplet can be automatically extracted and analyzed with the software. The shape of a pendant drop or hanging droplet of fluid in air or in another liquid are determined by the action of two forces, the gravitation and the surface tension. While the surface tension acts to minimize the surface area of the droplet the gravitation tends to pull it down and thus stretch/elongate the droplet. The Young-Laplace Eq. (3.27) tell us that there is a pressure difference between the exterior and the interior of a curved interface, the higher pressure is always higher at the interior of the droplet. We note this pressure difference as ΔP. If the drop is perfectly spherical then the pressure difference will be constant everywhere in the droplet. For a pendant drop, however, the gravitation causes an elongation along the z coordinate, which can be arbitrarily chosen in a plane cutting the long axis of the drop in the middle in the vertical direction (Fig. 3.20B). The elongation of the droplet causes variation in the Laplace pressure along the z axis, causing ΔP(z) to change, from that at the apex ΔP0, and this can be written as 32 3. Surfaces and interfaces
- 40. ΔP z ð Þ ¼ ΔP0 Δρgz (3.36) which at the apex takes the following expression: ΔP 0 ð Þ ¼ 2γ R0 ¼ 2κ0 where R0 is the radius of the droplet at the apex and κ0 ¼1/R0 is the curvature at the apex. The meridional κs and the circumferential κΦ curvatures (where κs 1 and κΦ 1 are the corresponding radii sweeping in the plane of the paper and perpendicular to the plane of the paper, respectively, see Fig. 3.20B) of the droplet will change at any point away from the apex [60]. The Laplace-Young equation further away from the apex becomes ΔP z ð Þ ¼ γ κs + κΦ ð Þ ¼ ΔP 0 ð ÞΔρgz (3.37) where Δρ is the density difference between fluids inside and outside of the droplet; when the fluid outside of the drop- let is air, the density can be taken as 1.18kg/m3 , which is the density of air at sea level and 23°C. The above equation can be reparametrized by introducing the arc length ds. Based on geometric arguments it can be found that dx ds ¼ cosΦ and dz ds ¼ sinΦ FIG. 3.20 (A) Typical setup for measuring the surface and interfacial tensions of a liquid with optical methods. (B) The drop shape contour anal- ysis in cylindrical coordinates and the parameters with the reference point at the apex of the pendent droplet. 33 3.12 Methods for measuring the surface and interfacial tensions of liquids
- 41. with the boundary conditions at the apex s¼0, x(s¼0)¼z(s¼0)¼Φ(s¼0)¼0, and x(s¼L)¼D/2, where L is the full arc length and D is the diameter of the needle from which the droplet is hanging. From this it results that the circumferential curvature kΦ ¼sinΦ/x and the meridional curvature ks ¼dΦ/ds. Inserting these curvatures in the Young-Laplace Eq. (3.37) we obtain γ dΦ ds + sinΦ x ¼ 2γ R0 Δρgz which yields the final form of the shape equation of a pendant droplet at any point along the z axis: dΦ ds ¼ 2 R0 sinΦ x Δρgz γ (3.38) The above differential equation can be solved by numerical procedures. The above equation can be rewritten in dimen- sionless form by replacing the dimensions x with dimensionless reduced variables X. One way to do that is to multiply the above equation by a length-scale factor equal to the capillary length lc ¼(γ/Δρg)1/2 , or the capillary constant. Thus, all the variables of length will become dimensionless X¼x/lc, ¼z/lc, S ¼ s lc , and B¼R0/lc Thus, the above set of equations can be rewritten such that it contains the new dimensionless variables: dΦ dS ¼ 2 B sinΦ X Z (3.39) The shape of the pendant drop is therefore dependent on a dimensionless quantity, namely the Bond number ¼B2 , after the English physicist Wilfrid Noel Bond. The shape of an axisymmetric droplet depends only on parameter β. The bond number can be best interpreted as a measure of the relative magnitude of gravitational force to surface ten- sion force for determining the shape of the drop. Gravitational force will elongate the pendent droplet (maximizing the surface area) while the surface tension forces will make the pendent droplet more spherical (minimize the surface area). For example, when the bond number β≪1, the surface tension force dominates, and the drop is nearly spherical, and for β1, the gravitational force dominates and the pendant drop becomes elongated [61]. When β for a particular pen- dant drop geometry can be determined as well as the drop radius R0 at the apex, the interfacial tension γ can be cal- culated with the relationship [61]: β ¼ B2 ¼ R2 0 Δρg γ (3.40) Historically, Bashforth and Adams [62] were the first to solve Eq. (3.42) numerically and that the same authors also tabulated the solutions calculated by hand for the bond numbers β that show the deviation of the drop from the ideal profile of a sphere and researchers used this tables to identify the profile of the drop and obtain the surface tension values this way. However, in modern computerized systems the integration is performed automatically for any value of β and after extraction of the contour of the droplet in the digital image it finds the best solution for the Young-Laplace equation. An automated contour shape analysis software can search the best match of the experimental drop profiles, with the theoretically calculated profiles using the surface tension as one of the adjustable parameters. The numerical solution of the Young-Laplace equation yield additional data, such as drop volume V and droplet surface area A: V ¼ π ð X2 sinΦdS A ¼ 2π ð XdS This procedure is called the axisymmetric drop shape analysis method (ADSA) and is one of the possible algorithms that can be used. The advantages of this method over all other techniques is the rapidity and accuracy. Also it is a great to study time dependence evolution of the surface and interfacial tensions as well as “aging effects.” Maximum bubble pressure method for measuring dynamic surface tension In this method gas bubbles are produced at a constant rate in a fluid through a capillary of precisely known radius. The pressure inside the bubble continues to increase and the maximum value is achieved when the bubble has a hemi- spherical shape and thus its radius coincides with the radius of the capillary. As the bubble continues to grow the pressure inside the drop decreases again. 34 3. Surfaces and interfaces
- 42. Numerical example 3.6 Can you explain why the pressure inside an air bubble decreases after the bubble increases in size beyond the hemispherical shape? The pressure changes in the bubble are monitored and plotted over time. The evolution of the bubble at the end of the cap- illary is depicted in the cartoon of Fig. 3.21. At the point of the maximum bubble pressure, the bubble has a hemispherical shape of radius equal to the radius of the capillary, R, the surface tension can be determined using the Laplace equation: γ ¼ RΔPmax 2 (3.41) As we will see later, this method is most commonly used in determining the dynamic surface tension of a formulation containing surfactants and thus their adsorption dynamics at interfaces. A pure liquid has a negligible change in sur- face tension over time. Other methods used to determine the dynamic surface tension is the oscillating liquid jet method, which will be discussed later. As with any other methods for the surface tension measurements the capillaries must be kept absolutely clean. The advantage of this method is the accuracy, speed, and it can be applied to a variety of fluids, even biological fluids as it requires rather small amounts of the liquid sample. Spinning drop method is yet another way to measure the interfacial tension between two immiscible liquids. A small droplet of the lighter phase liquid is suspended in a heavier phase liquid and then placed in a horizontal rotat- ing capillary. The shape of the drop is deformed by rotating the capillary at a certain rotational velocity. The shape of the drop will be deformed, elongate with the long axis perpendicular to the axis of rotation, and its long axis radius r will depend on the interfacial tension, the angular frequency of the rotation ω, and the density difference between phases Δρ. Thus, the interfacial tension can be calculated from Vonnegut’s equation: γ ¼ r3 ω2 Δρ 4 The advantage of this method is that it can determine accurately very low interfacial tensions as it is the case in micro- emulsions, or design of surfactants formulations for the enhanced oil recovery. Enhanced or tertiary oil recovery is applied after almost 40% of the oil has been extracted from the well in the primary and secondary recovery processes. The enhanced oil recovery can be achieved by adding surfactants and detergent-like polymers in the aqueous liquid pumped into the well so that the wettability of rock improves and consequently the oil can be displaced from the pores of the rock and instantly emulsified due to the very low interfacial tension. FIG. 3.21 Evolution of the pressure in an air bubble produced in a liquid with surface tension to be determined. The maximum pressure inside the air bubble is achieved when it achieves a perfectly hemispherical shape of radius equal to that of the capillary. 35 3.12 Methods for measuring the surface and interfacial tensions of liquids
- 43. The vibrating jet method for measuring the dynamic surface tension is a very creative yet very cost effective to implement, although more challenging for the user. Its theory was already worked out by Lord Rayleigh. The method is based on the measurement of the wavelength of an oscillatory jet of liquid emerging through an elliptical orifice as it progresses in time and space (Fig. 3.22) [63]. The wavelength of the oscillating jet as it departs from the orifice becomes longer and longer until it disappears (Fig. 3.22). The setup consists of a liquid reservoir, a set of flow regulators, and most importantly an oval orifice, which can be made simply by deforming a Pasteur pipette (Fig. 3.22). The exact dimensions of the orifice, the long and short axis must be precisely known for accurate calculations of the surface tension. The surface tension values are calculated from the wavelength using the following expression: γ ¼ Caρ V λ 2 where Ca is the capillary number and can be determined with pure water, ρ is the density of the solution, V is the flow rate in mL/s, and λ is the wavelength of the wave. The time of corresponding to a certain wave can be calculated from the distance d of the midpoint of the wave from the orifice (Fig. 3.22), and the velocity of the jet v, with the equation: t s ð Þ ¼ d v where the velocity of the jet can be calculated with the equation: v ¼ V a where a is the area of the cross section of the elliptical orifice. This method is suitable for measuring the instantaneous dynamic surface tension for times in the range of 10–400ms. Variants of the oscillating jet methods where waves in the jet are produced by excitation can probe the instantaneous interfacial tension to time intervals as short as 0.1ms [64]. Note that the stalagmometer method can also be used to determine the dynamic surface tension where the droplet formation rate can be increased by a peristaltic pump. In this case the droplet size at lower surface tension will be smaller. 3.12.1 Case study: Aerosol spray coatings and the importance of the dynamic surface tension Measuring the dynamic surface tension is particularly useful in dynamic processes. For example, formulations of aerosol paint contain surfactants or surface-active agents which play the role of wetting agents and ensure an even FIG. 3.22 (A) Principle of the oscillating method and general measurement setup. (B) Photograph of the oscillating jet of an aqueous solution containing surfactants, which emerges from an oval orifice, the actual jet and its shadow with clearly separated waves and a reference ruler can be observed. Waves of different wavelengths can be clearly seen. At shorter times, closer to the orifice (A) the wavelength is short, because the sur- factants did not have time to fully adsorb on the surface. At later times, from the sixth wave the difference in wavelength become less obvious. 36 3. Surfaces and interfaces
- 44. spread of the coated paint on a surface. In industrial coatings the quality of the paint must meet particularly tight requirements, the surface finish must be smooth and free of cratering or pits. Pits are created on the surface when, for example, the distance of the spray nozzle is too close to the surface to be painted. In such situation, the time of flight for the droplet from the nozzle to surface is much shorter than the time it takes for the surfactant to saturate and reach the surface of the aerosol droplet and when it lands on the surface it will not properly coat the surface (Fig. 3.23). Zhang and Basaran [65] have studied the role of surfactant in spray coating and concluded that the sur- factant dynamics and the dynamic surface tension play a major role, firstly with respect to the ability of the aerosol droplet landing on the surface to wet and spread on the surface and secondly due to gradients in surface tension of the paint that induce a Marangoni flow causing local stress. In inkjet printing surfactants are added to improve the wetting properties of the ink. The time the ink droplets take from the moment they exit the printing nozzle and to the moment they reach the printed surface is about 1ms [66]. Therefore, very fast dynamic surface tensions are needed that can be achieved at high surfactant concentrations and special design for the surfactant structures. Dynamic surface tension is an important parameter in wastewater treat- ment, flotation of minerals, and other industrial processes [67]. 3.13 Measuring ultralow interfacial tension—The spinning drop tensiometer The spinning drop tensiometer is an instrument used to accurately measure ultralow interfacial tension values, typ- ically 1mN/m, between a light and a dense phase. The pendant drop tensiometer, or the force tensiometer, can mea- sure well values of the interfacial tension that are larger than approximately 5mN/m, but cannot be used for ultralow interfacial tension. The spinning drop tensiometer can measure low values of the interfacial tension such as in micro- emulsions with good accuracy, depending on the instrument this can reach a 5103 mN/m. The spinning drop tensiometer consists of a capillary tube filled with a dense phase and a droplet of a light phase, with a difference in density of Δρ. The capillary is then rotated on its long axis with a high angular velocity, ω; during the spin the spher- ical droplet contained into the phase will elongate due to the action of centrifugal forces, which push the denser liquid to the exterior and the lighter fluid (droplet) toward the central axis of the cylinder, therefore the droplet will acquire a cylindrical shape, of radius R. The relationship between the centrifugal forces and the interfacial tension are given by Vonnegut’s equation: γ ¼ Δρ ω2 4 R FIG. 3.23 (Left) Long time of flight for aerosol droplets and the finish of the painted coat surface. (Right) Short time of flight and pitted surface of paint coat. 37 3.13 Measuring ultralow interfacial tension—The spinning drop tensiometer
- 45. The main assumption is that the drop shape acquires a fully developed cylindrical shape, which is true for L/R4. All droplets acquire eventually a fully cylindrical shape if the spinning speed is sufficiently large, however, this puts a limit to the instrument’s ability to measure very high values of the interfacial tension. Recent software algorithms developed based on the drop shape analysis using the Young-Laplace equation lift these restrictions, so larger values for the inter- facial tension can be measured. 3.13.1 Case study: Role of interfacial tension in enhanced oil recovery Primary, secondary, and tertiary oil recovery phases refer to the method applied for the oil extraction from the res- ervoir. For primary extraction recovery, the oil is extracted from the natural pressure built up over time in the reservoir, where the oil is naturally pushed out of the reservoir, therefore a set of valves and pipes are enough. For the secondary oil recovery, pressure is built up in the reservoir by adding water, waterflooding, or gases. After first and second recov- ery phases have taken most of the oil, there are still significant oil reserves left in the reservoir and if economically justified the tertiary (or enhanced) oil recovery can proceed. In the tertiary oil recovery, chemicals or gases are used to displace oil that is trapped in the pores of the rock, due to viscous, gravity, and capillary forces. The tertiary oil recovery is especially used for heavy oil and tar sand extraction. Heavy oils and tar sands can have a significant per- centage in oil reservoirs but are poorly displaced in primary and secondary recovery. Therefore, the bulk production of these comes from the tertiary oil recovery [68]. Chemical flooding is among the methods used to enhance oil recovery, in which different chemicals such as surfactants, polymers, alkalis, biopolymers, and combinations thereof are used to improve the oil displacement from the rock (microscopic efficiency) and to improve the volumetric sweep efficiency (macroscopic efficiency) [69]. The macroscopic efficiency refers to the increase in the volume of oil brought to the sur- face. Because of the large viscosity difference between the oil and water the mobility of the water phase is much larger than that of the oil phase, therefore the pumped water may flow around the oil, leaving the oil phase behind. To increase the viscosity of the water phase, polymers and biopolymers are used and a polymer flood is also performed to increase the mobility of the oil phase. Mathematically the mobility is expressed as M ¼ λdisplacing λdisplaced ¼ krw=μw kro=μo where kro and krw are the effective permeability of oil and water, respectively, and μo and μw are the viscosities of oil and water phases, respectively. A mobility value of 1 is considered ideal because the displaced phase moves at the same speed as the displacing phase, but for a mobility of 10 the water moves 10 times faster than the oil. Chemical flooding into the oil well is performed to improve the oil displacement from the rock (microscopic effi- ciency). The principle is to reduce the interfacial tension between the water phase and the oil phase to as low as possible to increase the capillary number Ca¼ηU/γ where U is the linear velocity of the injected phase (m/s) and η is the vis- cosity of the injected phase. Ca correlates closely to oil recovery and residual oil saturation Sr0 (S0 oil saturation is the volume fraction of oil within the pore volume) [69]. Ca is in the range of 107 to 106 for typical water flooding and by increasing the capillary number to 104 and 103 the oil saturation can be reduced to 90% [70] and residual oil satu- ration approaches zero if capillary number reaches 102 [71]. This can be achieved by decreasing the oil/water inter- facial tension (IFT) from 10–40mN/m to 102 to 103 . Such low interfacial tensions can be achieved in formulations using surfactants, alkali surfactants, and polymer/alkali/surfactants [69, 72, 73]. The alkali is used for the saponifica- tion of potential product in oil and to achieve in this way ultralow interfacial tension values. Both ionic and nonionic surfactants have been used since 1970s [73]. Petroleum-derived sulfonate surfactants are the most economical surfac- tants used to lower the interfacial tension between the water and the oil and to alter the wettability of the porous rock from oil-wet to water-wet. Numerical example 3.7 What is the capillary number for a brine oil interfacial tension γ ¼10mN/m, injected phase velocity U ¼3.5106 m/s, and viscosity η ¼1mPas? Calculate the value of the oil/water interfacial tension needed to achieve a capillary number of 102 , for a full oil recovery from the well, for the same conditions? As already mentioned, surfactants also influence the amount of residual oil recovered via other mechanisms, such as emulsification of oil and changing the wettability of rock [69]. However, the adsorption of surfactant on the rock cre- ates losses that reduces the concentration of surfactants, reducing the chemical flooding to water flooding, losing 38 3. Surfaces and interfaces
- 46. therefore oil recovery efficiency. Surfactant formulations in chemical flooding are done with the spinning drop tensi- ometer to measure the minimum concentration of surfactants needed to achieve IFT values of 103 mN/m. 3.14 Surface and interfacial tensions with temperature The surface or interfacial tension with increase in temperature always decreases. This has been observed experimen- tally and the meaning of it can be understood from the Gibbs-Duhem equation treated in the next chapters. Essentially it can be easily demonstrated that the surface entropy for a 1m2 surface area and constant pressure is Ssurface ¼ dγ dT (3.42) This means that the surface excess entropy increases with the increase in temperature since dγ/dT is always negative. Because the surface excess is positive it indicates that the molecules at the interface have more entropy, are more dis- ordered than in bulk. As we will see the surface excess thermodynamic functions is generally the amount of energy, concentration, or in this case entropy possessed by the surface as compared to bulk. There have been attempts over the years to predict the surface tension of the liquids at different temperatures. Eőtvős’ empirical equation relates molecular volume Vm, temperature T, surface tension γ, and critical temperature, Tc (the temperature at which the phase boundaries vanish and the liquid coexists with its vapors and the γ ¼0): γ Vm ð Þ2=3 ¼ k Tc T ð Þ (3.43) where k is a constant, Vm ¼Mw/ρL, ρL is the density of the liquid, and Mw is the molecular weight. The Eőtvős constant k is a measure of the entropy of formation of surface, in other words the entropy change induced by bringing the liquid molecules from the bulk to the surface [74]. The constant k takes a value of 2.12 for nonpolar liquids. 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- 49. C H A P T E R 4 Surfactants and amphiphiles 4.1 Introduction Amphiphilicity shows a property that can be shared by surfactant molecules, macromolecules, molecular assem- blies, and nanoscopic objects, such as Janus nanoparticles [1, 2] that, inter alia, partition spontaneously at the boundary between two phases, such as liquid-liquid, liquid-gas, or solid-liquid interfaces. In doing so, the amphiphiles lower the interfacial tension and interfacial energy between phases. Amphiphiles can self-assemble into monolayers and supras- tructures of different shapes and forms dictated by the geometry and orientation of the building blocks. Amphiphi- licity implies the existence of chemical functional groups that in water are hydrophilic and hydrophobic (lipophilic) connected by chemical or physical bonds and form distinct and spatially segregated regions in space with contrasting surface polarity (Fig. 4.1), such as in surfactants, block copolymers, and Janus particles. Pseudo-amphiphiles are sur- faces, homogeneous particles, or copolymers that are constituted by both hydrophilic and hydrophobic groups mixed at the molecular scale (Fig. 4.1), without spatial segregation and which are wetted by both water and oil. Due to the lack of well-defined spatial segregation between polar (hydrophilic) and nonpolar (hydrophobic) functional groups, pseudo-amphiphiles do not self-assemble into well-defined structures and are not as effective as the amphiphiles at lowering the interfacial energy between phases. Amphiphilicity is a scalable property, being active at the molecular level, at the nanoscale and could presumably be extended into the microscale. The scalability of the amphiphilic prop- erty is however fundamentally an open question and it is a subject of interest in fundamental research [1–3]. Surfactants are the best-known class of amphiphiles and consist of a hydrophilic (polar) and hydrophobic (nonpo- lar) chemical functional group connected by chemical bonds, usually represented as in Fig. 4.2A. The hydrophobic part is usually a hydrocarbon chain and the hydrophilic part can be an anionic, cationic, or nonionic functional group. Sur- factants can self-assemble into a variety of structures, such as micelles, bilayers, monolayers, vesicles, etc. (Fig. 4.2B). 4.2 Brief historical account of surfactants Soaps were made and used from immemorial times, with the first records traced back to antiquity, in Mesopotamia [4]. Soaps are fatty acids surfactants, and in the past were obtained with rudimentary synthetic methods from basic materials available in nature such as tallow fat, olive, argan, or palm oils, and alkali-rich ashes, remnant residue from wood burning. The soap-making activities spread later on throughout the European continent (Fig. 4.3). The soap may have contributed to an improved quality of life, eliminated, or reduced the diseases in the densely populated regions, and enabled urban life. The reaching of new standards of living was marked by the appearance and mass production of personal care products, i.e., toiletries and cosmetics. Many Mediterranean regions prospered from the soap and cos- metics manufacturing activity because of their rich natural olive oil resources, such as the case of Provence in the south of France, or Castile in Spain, Florence in Italy [5]. Olive oil is rich in saturated palmitic, stearic and unsaturated oleic, linoleic, and linolenic acids (Fig. 4.3), which leads to good quality soaps. Sulfated oils were the first synthetic surfac- tants prepared after soaps, in 1834 by Runge, by mixing olive oil and sulfuric acid and in 1875 sulfated castor oil also known as “Turkey red” was prepared and used as dyeing additives, mordants, in textile industry [6]. A sulfated oil is not a pure surfactant but a mixture of sulfate esters, water, and fatty acid surfactants. In 1935, Colgate-Palmolive intro- duced the first soap-free shampoo, using sulfated mono- and di-glyceride surfactants (Fig. 4.3) [6]. 43 Chemistry of Functional Materials Surfaces and Interfaces https://doi.org/10.1016/B978-0-12-821059-8.00011-9 Copyright © 2021 Elsevier Inc. All rights reserved.
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- 52. Le papier rouge. Le silence est profond; aucun bruit ne monte de la rue: rien que le froissement du parchemin qui glisse sous le pouce et la plume qui crie. Lorsque je tournai la première feuille du registre pour 1437, je crus que j'étais devenu, moi aussi, clerc criminel de monseigneur le prévôt. Les procès étaient signés: AL. CACHEMARÉE. L'écriture de ce clerc était belle, droite, ferme; je me figurai un homme énergique, d'aspect imposant afin de recevoir les dernières confessions avant le supplice. Mais je cherchai vainement l'affaire des Bohémiens et de leur chef. Il n'y avait qu'un procès de sorcellerie et de vol dressé contre «une qui a nom princesse du Caire». Le corps de l'instruction montrait qu'il s'agissait d'une fille de la même bande. Elle était accompagnée, dit l'interrogatoire, d'un certain «baron, capitaine de ribleurs». (Ce baron doit être le Haro Pani de la chronique manuscrite.) Il était «homme bien subtil et affiné», maigre, à moustaches noires, avec deux couteaux dans la ceinture, dont les poignées étaient ouvragées d'argent; «et il porte ordinairement avec lui une poche de toile où il met la drone, qui est un poison pour le bétail, dont les bœufs, vaches et chevaux soudain meurent, qu'ils ont mangé du grain mélangé avec la droue, par étranges convulsions». La princesse du Caire fut prise et menée prisonnière au Châtelet de Paris. On voit par les questions du lieutenant criminel qu'elle était «âgée de vingt- quatre ans ou environ»; vêtue d'une cotte de drap quelque peu semée de fleurs, à ceinture tressée de fil en manière d'or; elle avait des yeux noirs d'une fixité singulière, et ses paroles étaient accompagnées de gestes emphatiques de sa main droite, qu'elle ouvrait et refermait sans cesse, en agitant les doigts devant sa figure. Elle avait une voix rauque et une prononciation sifflante, et elle injuriait violemment les juges et le clerc en répondant à l'interrogatoire. On voulut la faire dévêtir pour la mettre à la question, «afin de connaître ses crimes par sa bouche». Le petit tréteau étant préparé, le lieutenant criminel lui ordonna de se mettre toute nue. Mais elle refusa, et il fallut lui tirer de force son surcot, sa cotte et sa chemise, «qui paraissait de soie, aussi marquée du sceau de Salomon». Alors elle se roula sur les carreaux du Châtelet; puis, se relevant brusquement, elle présenta son entière nudité aux juges stupéfaits. Elle se dressait comme une statue de chair dorée. «Et lorsqu'elle fut liée sur le petit tréteau, et qu'on eut jeté un peu d'eau sur elle, la dite
- 53. princesse du Caire requit d'être mise hors de la dite question et qu'elle dirait ce quelle savait.» On la mena chauffer au feu des cuisines de la prison, «où elle semblait trop diabolique ainsi éclairée de rouge». Lorsqu'elle fut «bien en point», les examinateurs s'étant transportés dans les cuisines, elle ne voulut plus rien dire et passa au travers de sa bouche ses longs cheveux noirs. On la fit alors ramener sur les carreaux et attacher sur le grand tréteau. Et «avant qu'on eut jeté peu ou point d'eau sur elle ou qu'on l'eût fait boire, elle qui parle requit instamment et supplia d'être déliée, et qu'elle confesserait la vérité de ses crimes». Elle ne voulut se revêtir sinon de sa chemise magique. Quelques-uns de ses compagnons avaient dû être jugés avant elle, car maître Jehan Mautainct, examinateur au Châtelet, lui dit qu'il ne lui servirait de rien si elle mentait, «car son ami le baron était pendu, aussi plusieurs autres». (Le Registre ne contient pas ce procès.) Alors, elle entra dans une éclatante fureur, disant que «ce baron était son mari ou autrement, et duc d'Égypte, et qu'il portait le nom de la grande mer bleue d'où ils venaient (Baro pani, signifie en roumi «grande eau» ou «mer»). Puis elle se lamenta et promit vengeance. Elle regarda le clerc qui écrivait, et supposant, d'après les superstitions de son peuple, que l'écriture de ce clerc était le formulaire qui les faisait périr, elle lui voua autant de crimes qu'il aurait «peint ou autrement figuré par artifice» de ses compagnons sur le papier. Puis, s'avançant soudain vers les examinateurs, elle en toucha deux à l'endroit du cœur et à la gorge, avant qu'on put lui saisir les poignets et les attacher. Elle leur annonça qu'ils souffriraient de terribles angoisses dans la nuit, et qu'on les égorgerait par traîtrise. Enfin, elle fondit en larmes, appelant ce «baron» à diverses reprises «et pitoyables»; et, comme le lieutenant-criminel continuait l'interrogatoire, elle avoua de nombreux vols. Elle et ses gens avaient pillé «et robé» tous les bourgs du pays parisien, notamment le Montmartre et Gentilly. Ils parcouraient la campagne, s'établissant la nuit, en été, dans les foins, et en hiver dans les fours à chaux. Passant le long des haies, ils les «défleurissaient», c'est-à-dire qu'ils en ôtaient subtilement le linge qu'on y mettait à sécher. Le midi, campant à l'ombre, les hommes raccommodaient les chaudrons ou tuaient leurs poux; certains, plus religieux, les jetant au loin, et, en effet, bien qu'ils n'aient aucune croyance, il existe parmi eux une ancienne tradition que les hommes habitent, après leur mort, dans le corps des bêtes. La princesse du
- 54. Caire faisait mettre à sac les poulaillers, emporter la vaisselle d'étain des hôtelleries, creuser les silos pour prendre le grain. Dans les villages d'où on les chassait, les hommes revenaient, par son ordre, la nuit, jeter la «droue» dans les mangeoires, et dans les puits des paquets noués avec du «drap linge», gros comme le poing, pour empoisonner l'eau. Après cette confession, les examinateurs, tenant conseil, furent d'avis que la princesse du Caire était «très forte claironnasse et meurtrière et qu'elle avait bien desservi d'être à mort mise; et à ce la condamna le lieutenant de monseigneur le prévôt; et que ce fût en la coutume du royaume, à savoir qu'elle fût enfouie vive dans une fosse». Le cas de sorcellerie était réservé pour l'interrogatoire du lendemain, devant être suivi, s'il y avait lieu, d'un nouveau jugement. Mais une lettre de Jehan Mautainct au lieutenant-criminel, copiée dans le registre, apprend qu'il se passa dans la nuit d'horribles choses. Les deux examinateurs que la princesse du Caire avait touchés se réveillèrent au milieu de l'obscurité, le cœur percé de douleurs lancinantes; jusqu'à l'aube ils se tordirent dans leurs lits, et, au petit jour gris, les serviteurs de la maison les trouvèrent pâles, blottis dans l'encoignure des murailles, avec la figure contractée par des grandes rides.
- 56. Le papier rouge. On fit venir aussitôt la princesse du Caire. Nue devant les tréteaux, éblouissant des dorures de sa peau les juges et le clerc, tordant sa chemise marquée au sceau de Salomon, elle déclara que ces tourments avaient été envoyés par elle. Deux «bourreaux» ou crapauds étaient dans un endroit secret, chacun au fond d'un grand pot de terre; on les nourrissait avec de la mie de pain trempée dans du lait de femme. Et la sœur de la princesse du Caire, les appelant par les noms des tourmentés, leur enfonçait dans le corps de longues épingles: tandis que la gueule des crapauds bavait, chaque blessure retentissait au cœur des hommes voués. Alors le lieutenant criminel remit la princesse du Caire aux mains du clerc Alexandre Cachemarée avec ordre de la mener au supplice sans plus loin procéder. Le clerc signa le procès de son paraphe accoutumé. Le registre du Châtelet ne contenait rien de plus. Seul, le Papier-Rouge pouvait me dire ce qu'était devenue la princesse du Caire. Je demandai le Papier-Rouge, et on m'apporta un registre couvert d'une peau qui semblait teinte avec du sang caillé. C'est le livre de compte des bourreaux. Des bandes de toile scellées pendent tout le long. Ce registre était tenu par le clerc Alexandre Cachemarée. Il comptait les gratifications de maître Henry, tourmenteur. Et, en regard des quelques lignes ordonnant l'exécution, maître Cachemarée, pour chaque pendu, dessinait une potence portant un corps au visage grimaçant. Mais au-dessous de l'exécution d'un certain «baron d'Égypte et d'un larron étranger», où maître Cachemarée a griffonné une double fourche avec deux pendus, il y a une interruption et l'écriture change. On ne trouve plus de dessins, ensuite, dans le Papier-Rouge, et maître Étienne Guerrois a inscrit la note suivante: «Aujourd'hui 13 janvier 1438 fut rendu de l'official maître Alexandre Cachemarée, clerc, et par ordre de monseigneur le prévôt, mené au dernier supplice. Lequel étant clerc criminel et tenant ce Papier-Rouge, figurant en manière de passe-temps les fourches des pendus, fut pris soudain de fureur. Dont il se leva et alla au lieu des exécutions défouir une femme qui avait été là enterrée le matin et n'était pas morte; et ne sais si ce fut à son instigation ou autrement, mais la nuit alla dans leurs chambres couper la gorge à deux examinateurs au Châtelet. La femme a nom princesse du Caire; elle est de présent sur les champs, et on n'a pu la saisir. Et a ledit Al. Cachemarée confessé ses crimes
- 57. sans toutefois son dessein, dont il n'a rien voulu dire. Et ce matin fut traîné aux fourches de notre sire pour y être pendu et mis à mort, et illec fina ses jours.»
- 58. LE LOUP L'homme et la femme, qui traînaient leurs pieds sur la route des Sables, s'arrêtèrent en écoutant des coups espacés et sourds. Ils avaient été poursuivis par les deux mâtins de Tournebride, et le cœur leur sautait dans le ventre. À gauche, une ligne sanglante coupait la bruyère, avec des bosses noires de place en place. Ils s'assirent dans le fossé; l'homme rapetassa ses brodequins troués avec du fil poissé; la femme gratta les plaques blanches de terre poussiéreuse qui écaillaient ses mollets. Le gars était «moëlleux», poignes solides, des nœuds aux bras; l'autre tirait sur la quarantaine, une «gerce de rempart». Mais des yeux luisants et mouillés, la peau encore assez fraîche, malgré le hâle. Il grommela en se rechaussant: —On croûte encore des briques, à ce soir. C'est pas saignant que tous les cagnes du patelin, des cabots de malheur, viennent vous agricher les fumerons, quand on a le ventre vide? J'y foutrais rien un ferme-gueule, au patron, si je l'dégotais. La femme lui dit doucement: —Ne crie pas, mon petit homme. C'est que tu ne sais pas leur causer aux cabzirs. On les laisse venir comme ça... petit... petit... et puis quand ils sont là, tout près, t'as plus qu'à les gonfler.
- 59. —C'est bon, dit le gars. On va pas plumer ici. Ils longèrent la route en boitant. Le soleil était couché, mais les coups sonnaient toujours. Des lumières jaunes sautaient parmi les bosses noires, éclairant çà et là des masses rougeâtres. —En voilà, des briques à croûter, dit la femme. Chez les casseux d'cailloux. On voyait maintenant des ombres se mouvoir sur les terre-pleins. Il y en avait qui piochaient la terre, courbés comme des houes, tirant des cailloux rouges. D'autres les éclataient en tas, avec des masses. Des enfants en bourgeron portaient des lanternes. Les travailleurs avaient un calot enfoncé sur la tête, et des lunettes mistraliennes, à verres bleus; leurs sabots étaient empâtés de glaise sanguine. Un grand maigre travaillait d'attaque, le crâne plongeant dans son bonnet jusqu'aux oreilles; il avait la figure couverte d'un loup en fil de fer noirci; il devait être vieux:—deux pointes de moustaches grises débordaient sous le grillage. Dans le pays on craignait les carriers. C'étaient des hommes mystérieux qui creusaient, masqués, dans la terre rouge pendant le jour et une partie de la nuit. Les entrepreneurs gageaient ce qui leur arrivait—généralement des repris de justice, des terrassiers ou des puisatiers qui variaient leur travail en luttant dans les foires, des hercules falots en carnaval forcé. Les mioches édentés qui venaient piétiner dans les retroussis de terres volaient les poules et saignaient les cochons. Les rôdeuses de grand'route fuyaient le long de la carrière; sans quoi les masques leur roulaient la tête dans les brousses et leur barbouillaient le ventre de terre mouillée. Mais les deux cheminots s'approchèrent du trou illuminé, cherchant la soupe et le gîte. Devant eux un môme balançait sa lanterne en chantant. L'homme au loup s'appuya sur sa pioche et releva la tête. On ne voyait de sa figure que le menton luisant à la lumière; une tache noire bouchait le reste. Il claqua de la langue et dit: —Ben quoi, le trimard, ça boulotte? Quand on est deux, comme ça, on n'a pas froid au ventre. N'en faudrait, pour la tierce, des poules comme la tienne. On a de la misère, nous autres—ça serait assez rupin. Les hommes se mirent à crier: —Ohé, Nini, lâch' ton mari.—Ohé, ohé, viens te coucher.—T'es bien leste, Ernest, à enl'ver l' reste.—T'es bien pressé d'aller t' plumer.—Dis donc, Étienne, c'est-il la tienne?—Sacré mâtin, v'là des rondins.
- 60. Et puis les gosses piaillèrent: —Oh! c'te cafetière! Elle l'a épousé pour ses croquenots. Ils sont bat. Ça coûte cher, des paffes comme ça, parce que ça paye des portes et fenêtres. Le gars «moëlleux» arriva sur l'homme au loup en balançant ses poings. Il lui dit tranquillement: —Toi, j'te vas asseoir du coup. J'te vas foutre un transfèrement que le mur de ton trou t'en rendra un autre. Et il lui envoya sous le menton deux brusques poussées. L'homme au loup chancela, prit sa pioche et la balança. L'autre regarda en dessous et crocha un pic à moitié enfoncé dans un tas de cailloux. —T'en veux? dit le carrier maigre. J'te fais claquer la tirelire. Mon nom, c'est La Limande; je suis Parigo, de Belleville; je me suis lavé les pieds à la Nouvelle pour une gonzesse que je n'avais pas assez à la bonne; ça fait qu'un soir j'ai crevé une boutique et j'ai été paumé sur un fric-frac. Je reviens de loin; j'ai tiré quinze longes. Je m'en fous, je vais te tomber. Alors la femme sauta sur le gars et cria: —Tu entends, je te défends la batterie. Il va te crever; je le connais, je ne veux pas que tu te battes.... Je ne veux pas... je ne veux pas.... Le gars «moëlleux» la poussa de côté. —Moi, dit-il, j'ai pas de nom. Je me suis pas connu de dabe; paraît qu'il a été sapé. C'était un maigre, mais il m'a fait solide. On y va? La femme criant toujours, les camarades l'enfermèrent dans un cercle. Elle déchirait les bourgerons, pinçait et mordait. Deux terrassiers lui tinrent les poings. Les combattants se carrèrent, l'outil levé. L'homme au loup abattit sa pioche. Le gars sauta de côté. Le pic retombant rencontra le fer de la pioche, qui rendit un son clair. Puis ils tournèrent autour d'un monticule, sautant de ci, de là, frappant à côté, écumants. Ils enfonçaient à mi-jambes dans la terre rouge; l'homme au loup y laissa ses sabots. Le pic et la pioche se croisaient. Quelquefois des étincelles jaillissaient dans la nuit, quand les ferrures battaient le briquet. Mais le gars avait de la moelle. Quoique l'autre eut de longs bras au bout desquels la pioche tournoyait, terrible, du pic il parait les coups de tête et
- 61. envoyait de furieux revers dans les jambes. L'homme au loup abattit sa pioche en terre et leva les bras. —J'vas prendre mes galoches, dit-il. On a la chemise trempée.
- 62. Le loup.
- 63. T'es un gars solide. J'te fais pardon et excuse, moi. La Limande. En se retournant, il passa dans le cercle des carriers et regarda la femme sous le nez. Alors il cria un coup et sauta de nouveau sur sa pioche en hurlant: —Ah! le paillasson! Ah! tu m'as gamellé! Je te reconnais bien: je vas te crever ton homme! La femme tomba en arrière, les yeux blancs. Ses bras raidis se collèrent aux hanches, son cou gonfla; et elle battait alternativement le sol de ses deux tempes. Le gars «moëlleux» avait repris sa parade. Mais l'homme au loup attaquait avec fureur. Les fers heurtés tintaient. Et le carrier maigre criait: —C'est le trou sanguin ici. Tu y passeras. À toi ou à moi, il faut qu'on y cloue le chêne. T'es venu pour acheter ma tête, avec ta poule. Tu entends, cette femme-là, elle est à moi, à moi seul. Je veux l'emplâtrer après que je t'aurai tombé. Je l'habillerai de noir. Et le gars à la femme disait, parmi les ahans du pic: —Grand cadavre, viens donc que je te défonce. Viens la prendre, ma femme, vilain masque. T'es trop vioque pour me ceinturer! Comme il l'appelait «vieux», son pic se ficha dans le crâne de l'homme maigre. Le fer grinça sur la toile du loup, qui glissa et tomba. Le carrier s'abattit en arrière, son grand nez au vent, ses moustaches grises frissonnantes. Sur le calot noir, une tache rouge s'agrandissait, suintant par le trou du front. Tous les travailleurs crièrent: —Holà! La femme se roule vers le bruit, et, rampante, vint regarder l'homme démasqué. Quand elle eut vu le profil maigre, elle pleura: —T'as tué ton daron, mon homme, t'as tué ton daron! Dans la minute, ils furent sur leurs pieds et s'enfuirent vers la nuit, laissant derrière eux la ligne sanglante de la carrière.
- 64. CONTE DES ŒUFS Il était une fois un bon petit roi (n'en cherchez plus—l'espèce est perdue) qui laissait son peuple vivre à sa guise: il croyait que c'était un excellent moyen de le rendre heureux. Et lui-même vivait à la sienne, pieux,
- 65. débonnaire, n'écoutant jamais ses ministres, puisqu'il n'en avait pas, et tenant conseil seulement avec son cuisinier, homme d'un grand mérite, et avec un vieux magicien qui lui tirait les cartes pour le désennuyer. Il mangeait peu, mais bien; ses sujets faisaient de même; rien ne troublait leur sérénité; chacun était libre de couper son blé en herbe, de le laisser mûrir, ou de garder le grain pour les prochaines semailles. C'était vraiment là un roi philosophe, qui faisait de la philosophie sans le savoir; et ce qui montre bien qu'il était sage sans avoir appris la sagesse, c'est le cas très merveilleux où il pensa se perdre, et son peuple avec lui, pour avoir voulu s'instruire dans les saines maximes. Il advint qu'une année, vers la lin du carême, ce bon roi fit venir son maître d'hôtel, qui avait nom Fripesaulcetus ou quelque chose d'approchant, afin de le consulter sur une grave question. Il s'agissait de savoir ce que Sa Majesté mangerait le dimanche de Pâques. —Sire, dit le ministre de l'intérieur du monarque, vous ne pouvez faire autrement que de manger des œufs. Or les évêques de ce temps-là avaient meilleur estomac que ceux d'aujourd'hui, en sorte que le carême était fort sévère dans tous les diocèses du royaume. Le bon roi n'avait donc guère mangé que des œufs pendant quarante jours. Il fit la moue et dit: —J'aimerais mieux autre chose. —Mais, sire, dit le cuisinier, qui était bachelier ès lettres, les œufs sont un manger divin. Savez-vous bien qu'un œuf contient la substance d'une vie tout entière? Les Latins croyaient même que c'était le résumé du monde. Ils ne remontaient jamais au déluge—mais ils parlaient de reprendre les choses à l'œuf, ab ovo. Les Grecs disaient que l'univers naquit d'un œuf pondu parla Nuit aux ailes noires; et Minerve sortit tout armée du crâne de Jupiter, à la façon d'un poulet qui crèverait à coups de bec la coquille d'un œuf trop avancé. Je me suis souvent demandé, pour ma part, si notre terre n'était pas simplement un gros œuf, dont nous habitons la coque; voyez combien cette théorie s'accommoderait avec les données de la science moderne: le jaune de cet œuf gigantesque ne serait autre que le feu central, la vie du globe. —Je me moque de la science moderne, dit le roi: mais je voudrais varier mes repas.
- 66. Sire, dit le ministre Fripesaulcetus, rien n'est plus facile. Il est nécessaire que vous mangiez des œufs à Pâques; c'est une manière de symboliser la résurrection de Notre-Seigneur. Mais nous savons dorer la pilule. Les voulez-vous durs, brouillés, en salade, en omelette au rhum, au truffes, aux croûtons, aux lines herbes, aux pointes d'asperges, aux haricots verts, aux confitures, à la coque, à l'étouffée, cuits sous la cendre, pochés, mollets, battus, à la neige, à la sauce blanche, sur le plat, en mayonnaise, chaperonnés, farcis? voulez-vous des œufs de poule, de canard, de faisan, d'ortolan, de pintade, de dindon, de tortue? désirez-vous des œufs de poisson, du caviar à l'huile, avec une vinaigrette? faut-il commander un œuf d'autruche (c'est un repas de sultan) ou de roc (c'est un festin de génie des Mille et une Nuits), ou bien tout simplement de bons petits œufs frits à la poêle, ou en gâteau avec une croûte dorée, hachés menu avec du persil et de la ciboule, ou liés avec de succulents épinards? aimez-vous mieux les humer crus, tout tièdes?—ou enfin daignerez-vous goûter un sublimé nouveau de ma composition où les œufs ont si bon goût, qu'on ne les reconnaît plus,—c'est d'un délicat, d'un éthéré,—une vraie dentelle.... —Rien, rien, dit le roi. Il me semble que vous m'avez dit là, si je ne me trompe, quarante manières d'accommoder les œufs. Mais je les connais, mon cher Fripesaulcetus—vous me les avez fait goûter pendant tout le carême. Trouvez-moi autre chose. Le ministre, désolé, voyant que les affaires de l'intérieur allaient si mal, se frappa le front pour chercher une idée—mais ne trouva rien. Alors le roi, maussade, fit appeler son magicien. Le nom de ce savant était Nébuloniste, si j'ai bonne mémoire; mais le nom ne fait rien à l'affaire. C'était un élève des mages de la Perse; il avait digéré tous les préceptes de Zoroastre et de Chakyâmouni, il était remonté au berceau de toutes les religions et s'était pénétré de la morale suprême des gymnosophistes. Mais il ne servait ordinairement au roi qu'à lui tirer les cartes. —Sire, dit Nébuloniste, il ne faut faire apprêter vos œufs d'aucune des manières qu'on vous a dites; mais vous pouvez les faire couver. —Parbleu, répondit le roi, voilà une bonne idée: au moins je n'en mangerai pas. Mais je ne vois pas bien pourquoi. —Grand roi, dit Nébuloniste, permettez-moi de vous conter un apologue. —À merveille, répondit le monarque, j'adore les histoires, mais je les aime claires. Si je ne comprends pas, puisque tu es magicien, tu me l'expliqueras. Commence donc.
- 67. —Un roi du Népal, dit Nébuloniste, avait trois filles. La première était belle comme un ange; la seconde avait de l'esprit comme un démon; mais la troisième possédait la vraie sagesse. Un jour qu'elles allaient au marché pour s'acheter des cachemires, elles quittèrent la grande route et prirent un chemin de traverse par les rizières qui tapissent les rives du fleuve. Le soleil passait obliquement entre les épis penchés et les moustiques dansaient une ronde parmi ses rayons. À d'autres endroits les hautes herbes entrelacées formaient des bosquets où flottait une ombre délicieuse. Les trois princesses ne purent résister au plaisir de se nicher dans l'un d'eux: elles s'y blottirent, causèrent quelque temps en riant, et finirent par s'endormir toutes trois, lassées par la chaleur. Comme elles étaient de sang royal, les crocodiles qui prenaient le frais au ras de l'eau, sous les glaives ondulés des épis trempés dans la rivière, n'eurent garde de les déranger. Ils venaient seulement les regarder de temps en temps et avançaient leur mufle de corne brune pour les voir dormir. Tout à coup ils replongèrent sous l'eau bleue, avec un grand clapotement, ce qui réveilla les trois sœurs en sursaut.
- 68. Le contes des œufs.
- 69. Elles aperçurent alors devant elles une petite vieille ratatinée, toute ridée, toute cassée, qui trottinait en sautillant, appuyée sur une canne à béquille. Elle portait un panier couvert d'une toile blanche. —Princesses, dit-elle d'une voix chevrotante, je suis venue pour vous faire un cadeau. Voici trois œufs entièrement semblables; ils contiennent le bonheur qui vous est réservé dans votre vie; chacun d'eux en renferme une égale quantité; le difficile, c'est de le tirer de là. Disant ces mots, elle découvrit son panier, et les trois princesses virent en se penchant trois grands œufs d'une blancheur immaculée, reposant sur un lit de foin parfumé. Quand elles relevèrent la tête, la vieille avait disparu. Elles n'étaient pas fort surprises; car l'Inde est un pays de sortilèges. Chacune prit donc son œuf et s'en revint au palais en le portant soigneusement dans le pan relevé de son voile, rêvant à ce qu'il en fallait faire. La première s'en alla droit à la cuisine, où elle prit une casserole d'argent. «Car, se disait-elle, je ne puis rien faire de mieux que de manger mon œuf. Il doit être excellent.» Elle le prépara donc suivant une recette hindoue et le savoura au fond de son appartement. Ce moment fut exquis; elle n'avait rien goûté d'aussi divinement bon; jamais elle ne l'oublia. La seconde prit dans ses cheveux une longue épingle d'or dont elle perça deux petits trous aux deux bouts de l'œuf. Puis elle y souffla si bien quelle le vida et le suspendit à une cordelette de soie. Le soleil passait à travers la coque transparente, qu'il irisait de ses sept couleurs; c'était un scintillement, un chatoiement continuels; à chaque seconde la coloration changeait et on avait devant les yeux un nouveau spectacle. La princesse se perdit dans cette contemplation et y trouva une joie profonde. Mais la troisième se souvint qu'elle avait une poule de faisant qui couvait justement. Elle fut à la basse-cour glisser doucement son œuf parmi les autres; et, le nombre de jours voulu s'étant écoulé, il en sortit un oiseau extraordinaire, coiffé d'une huppe gigantesque, aux ailes bariolées, à la queue parsemée de taches étincelantes. Il ne tarda pas à pondre des œufs semblables à celui d'où il était né. La sage princesse avait ainsi multiplié ses plaisirs, parce qu'elle avait su attendre. La vieille n'avait d'ailleurs pas menti. L'aînée des trois sœurs s'éprit d'un prince beau comme le jour, et l'épousa. Il mourut bientôt; mais elle se contenta d'avoir trouvé dans cette vie un moment de bonheur.
- 70. La puînée chercha ses plaisirs dans les beaux-arts et les travaux de la pensée. Elle composa des poèmes et sculpta des statues; son bonheur était ainsi continuellement devant elle, et elle put en jouir jusqu'au jour de sa mort. La cadette fut une sainte qui sacrifia toutes les distractions de cette vie aux joies du Paradis. Elle ne réalisa aucune de ses espérances dans ce monde passager afin de les laisser éclore dans l'existence future, qui est, comme vous le savez, éternelle. Là-dessus, Nébuloniste se tut. Le roi, pensif, réfléchit longtemps. Puis sa figure s'éclaira, et il s'écria d'un ton joyeux: —Voilà qui est merveilleux; mais ce qu'il y a de plus étonnant, c'est que j'ai compris du premier coup. Cela veut dire qu'il faut mettre couver mes œufs. Le grand magicien s'inclina devant la sagacité du roi, et tous les courtisans battirent des mains. Les gazettes ne manquèrent pas de vanter l'esprit de Sa Majesté qui avait ainsi démêlé la morale d'un profond apologue. La conséquence fut que le bon roi ne voulut pas être le seul heureux. Il s'enferma pendant trois heures et élucubra le premier décret de son règne. De par tout le royaume il était désormais interdit de manger des œufs. On les ferait couver. Le bonheur des sujets serait assuré inévitablement de cette manière. Des peines sévères sanctionnaient l'exécution de la loi. Le premier inconvénient du nouveau régime fut que le roi, occupé contre son habitude des affaires du royaume, en perdit la tête et oublia de commander son déjeuner pour le dimanche de Pâques. Il le regretta bien ce jour-là. Puis il y eut aussitôt des hommes politiques pour commenter le décret. L'apologue de Nébuloniste s'était répandu par les journaux et l'on vit dans la loi du prince un mythe ingénieux qui commandait aux hommes de vivre en cénobites. Le pauvre roi se trouva ainsi avoir établi, sans le savoir, une religion d'État. Ce furent alors de grandes querelles dans le royaume. Beaucoup d'hommes préfèrent trouver leur bonheur dans ce monde que dans l'autre; ceux-là firent la guerre à ceux qui voulaient faire couver leurs œufs. Le pays fut ensanglanté, et le bon roi s'arrachait les cheveux. Son cuisinier le tira de peine bien ingénieusement et prit du coup sa revanche sur le magicien. Il lui conseilla de faire couver tous ses œufs,
- 71. puisqu'il ne voulait pas les manger,—mais de laisser ses sujets, comme auparavant, libres de ne pas être heureux. Tout heureux de cette solution, le roi décora son ministre et révoqua son unique décret. Mais les couveurs d'œufs ne furent point contents. Comme ils ne pouvaient plus faire des prosélytes de par la loi, ils émigrèrent du royaume, où on ne les laissa jamais rentrer. Ils parcoururent alors l'univers entier, où, depuis, ils ont forcé bien des gens à être heureux dans l'autre monde. Quant au roi, il finit par s'ennuyer de sa nouvelle vie; il prit exemple sur ses sujets, et le malin Fripesaulcetus acheva de le déconvertir en lui servant, l'année suivante, des œufs accommodés à la quarante et unième manière pour terminer le carême—des œufs rouges.
- 72. LE ROI AU MASQUE D'OR Le roi masqué d'or se dressa du trône noir où il était assis depuis des heures, et demanda la cause du tumulte. Car les gardes des portes avaient croisé leurs piques et on entendait sonner le fer. Autour du brasier de bronze s'étaient dressés aussi les cinquante prêtres à droite et les cinquante bouffons à gauche, et les femmes en demi-cercle devant le roi agitaient leurs mains. La flamme rose et pourpre qui rayonnait par le crible d'airain du brasier faisait briller les masques des visages. À l'imitation du roi décharné, les femmes, les bouffons et les prêtres avaient d'immuables figures d'argent, de fer, de cuivre, de bois et d'étoffe. Et les masques des bouffons étaient ouverts par le rire, tandis que les masques des prêtres étaient noirs de souci. Cinquante visages hilares s'épanouissaient sur la gauche, et sur la droite cinquante visages tristes se renfrognaient. Cependant les étoiles claires tendues sur les têtes des femmes mimaient des figures éternellement gracieuses, animées d'un sourire artificiel. Mais le masque d'or du roi était majestueux, noble, et véritablement royal. Or, le roi se tenait silencieux et semblable par ce silence à la race des rois dont il était le dernier. La cité avait été gouvernée jadis par des princes qui
- 73. portaient le visage découvert; mais dès longtemps s'était levée une longue horde de rois masqués. Nul homme n'avait vu la face de ces rois, et même les prêtres ignoraient la raison du secret. Cependant l'ordre avait été donné, depuis les âges anciens, de couvrir les visages de ceux qui s'approchaient de la résidence royale; et cette famille de rois ne connaissait que les masques des hommes. Et tandis que les ferrures des gardes de la porte frémissaient et que leurs armes sonores retentissaient, le roi les interrogea d'une voix grave: —Qui ose me troubler, aux heures où je siège parmi mes prêtres, mes bouffons et mes femmes! Et les gardes répondirent, tremblants: —Roi très impérieux, masqué d'or, c'est un homme misérable, vêtu d'une longue robe; il paraît être de ces mendiants pieux qui errent par la contrée, et il a le visage découvert. —Laissez entrer ce mendiant, dit le roi. Alors celui des prêtres qui avait le masque le plus grave se tourna vers le trône et s'inclina: —O roi, dit-il, les oracles ont prédit qu'il n'est pas bon pour ta race de voir les visages des hommes. Et celui des bouffons dont le masque était crevé par le rire le plus large tourna le dos au trône et s'inclina: —O mendiant, dit-il, que je n'ai pas encore vu, sans doute tu es plus roi que le roi au masque d'or, puisqu'il est interdit de te regarder. Et celle des femmes dont la fausse figure avait le duvet le plus soyeux joignit ses mains, les écarta et les courba comme pour saisir les vases des sacrifices. Or, le roi, penchant ses yeux vers elle, craignait la révélation d'un visage inconnu. Puis un désir mauvais rampa dans son cœur. —Laissez entrer ce mendiant, dit le roi au masque d'or. Et parmi la forêt frissonnante des piques, entre lesquelles jaillissaient les lames des glaives comme des feuilles éclatantes d'acier, éclaboussées d'or vert et d'or rouge, un vieil homme à la barbe blanche hérissée s'avança jusqu'au pied du trône, et leva vers le roi une figure nue où tremblaient des yeux incertains.
- 74. —Parle, dit le roi. Le mendiant répliqua d'une voix forte: —Si celui qui m'adresse la parole est l'homme masqué d'or, je répondrai, certes; et je pense que c'est lui. Qui oserait, avant lui, élever la voix? Mais je ne puis m'en assurer par la vue—car je suis aveugle. Cependant je sais qu'il y a dans cette salle des femmes, par le frottement poli de leurs mains sur leurs épaules; et il y a des bouffons, j'entends des rires; et il y a des prêtres, puisque ceux-ci chuchotent d'une façon grave. Or, les hommes de ce pays m'ont dit que vous étiez masqués; et toi, roi au masque d'or, dernier de ta race, tu n'as jamais contemplé des visages de chair. Écoute: tu es roi et tu ne connais pas les peuples. Ceux-ci sur ma gauche sont les bouffons—je les entends rire; ceux-ci sur ma droite sont les prêtres,—je les entends pleurer; et je perçois que les muscles des visages de ces femmes sont grimaçants. Or, le roi se tourna vers ceux que le mendiant nommait bouffons, et son regard trouva les masques noirs de souci des prêtres; et il se tourna vers ceux que le mendiant nommait prêtres, et son regard trouva les masques ouverts de rire des bouffons; et il baissa les yeux vers le croissant de ses femmes assises, et leurs visages lui semblèrent beaux. —Tu mens, homme étranger, dit le roi; et tu es toi-même le rieur, le pleureur, et le grimaçant; car ton horrible visage, incapable de fixité, a été fait mobile afin de dissimuler. Ceux que tu as désignés comme les bouffons sont mes prêtres, et ceux que tu as désignés comme les prêtres sont mes bouffons. Et comment pourrais-tu juger, toi dont la figure se plisse à chaque parole, de la beauté immuable de mes femmes? —Ni de celle-là, ni de la tienne, dit le mendiant à voix basse, car je n'en puis rien savoir, étant aveugle, et toi-même tu ne sais rien ni des autres ni de ta personne. Mais je suis supérieur à toi en ceci: je sais que je ne sais rien. Et je puis conjecturer. Or, peut-être que ceux qui te paraissent des bouffons pleurent sous leur masque; et il est possible que ceux qui te semblent des prêtres aient leur véritable visage tordu par la joie de te tromper; et tu ignores si les joues de tes femmes ne sont pas couleur de cendre sous la soie. Et toi-même, roi masqué d'or, qui sait si tu n'es pas horrible malgré ta parure? Alors celui des bouffons qui avait la plus large bouche fendue de gaieté poussa un ricanement semblable à un sanglot; et celui des prêtres qui avait
- 75. le front le plus sombre dit une supplication pareille à un rire nerveux, et tous les masques des femmes tressaillirent. Et le roi à la figure d'or fit un signe. Et les gardes saisirent par les épaules le vieil homme à la figure nue et le jetèrent par la grande porte de la salle. La nuit se passa et le roi fut inquiet pendant son sommeil. Et le matin il erra par son palais, parce qu'un désir mauvais avait rampé dans son cœur. Mais ni dans les salles à coucher, ni dans la haute salle dallée des festins, ni dans les salles peintes et dorées des fêtes, il ne trouva ce qu'il cherchait. Dans toute l'étendue de la résidence royale il n'y avait pas un miroir. Ainsi l'avait fixé l'ordre des oracles et l'ordonnance des prêtres depuis de longues années. Le roi sur son trône noir ne s'amusa pas des bouffons et n'écouta pas les prêtres et ne regarda pas ses femmes: car il songeait à son visage. Quand le soleil couchant jeta vers les fenêtres du palais la lumière de ses métaux sanglants, le roi quitta la salle du brasier, écarta les gardes, traversa rapidement les sept cours concentriques fermées de sept murailles étincelantes, et sortit obscurément dans la campagne par une basse poterne. Il était tremblant et curieux. Il savait qu'il allait rencontrer d'autres visages, et peut-être le sien. Dans le fond de son âme, il voulait être sur de sa propre beauté. Pourquoi ce misérable mendiant lui avait-il glissé le doute dans la poitrine? Le roi au masque d'or arriva parmi les bois qui cerclaient la berge d'un fleuve. Les arbres étaient vêtus d'écorces polies et rutilantes. Il y avait des fûts éclatants de blancheur. Le roi brisa quelques rameaux. Les uns saignaient à la cassure un peu de sève mousseuse, et l'intérieur restait marbré de taches brunes; d'autres révélaient des moisissures secrètes et des fissures noires. La terre était sombre et humide sous le tapis varicolore des herbes et des petites fleurs. Le roi retourna du pied un gros bloc veiné de bleu, dont les paillettes miroitaient sous les derniers rayons; et un crapaud en poche molle s'échappa de la cachette vaseuse avec un tressaut effaré. À la lisière du bois, sur la couronne de la berge, le roi, émergeant des arbres, s'arrêta, charmé. Une jeune fille était assise sur l'herbe; le roi voyait ses cheveux tordus en hauteur, sa nuque gracieusement courbée, ses reins souples qui faisaient onduler son corps jusqu'aux épaules; car elle tournait
- 76. entre deux doigts de sa main gauche un fuseau très gonflé, et la pointe d'une quenouille épaisse s'effilait près de sa joue. Elle se leva, interdite, montra son visage, et, dans sa confusion, saisit entre ses lèvres les brins du fil qu'elle pétrissait. Ainsi ses joues semblaient traversées par une coupure de nuance pâle. Quand le roi vit ces yeux noirs agités, et ces délicates narines palpitantes, et ce tremblement des lèvres, et cette rondeur du menton descendant vers la gorge caressée de lumière rose, il s'élança, transporté, vers la jeune fille et prit violemment ses mains. —Je voudrais, dit-il, pour la première fois, adorer une figure nue; je voudrais ôter ce masque d'or, puisqu'il me sépare de l'air qui baise ta peau; et nous irions tous deux émerveillés nous mirer dans le fleuve. La jeune fille toucha avec surprise du bout des doigts les lames métalliques du masque royal. Cependant le roi défit impatiemment les crochets d'or; le masque roula dans l'herbe, et la jeune fille, tendant les mains sur ses yeux, jeta un cri d'horreur. L'instant d'après elle s'enfuyait parmi l'ombre du bois en serrant contre son sein sa quenouille emmaillotée de chanvre. Le cri de la jeune fille retentit douloureusement au cœur du roi. Il courut sur la berge, se pencha vers l'eau du fictive, et de ses propres lèvres jaillit un gémissement rauque. Au moment où le soleil disparaissait derrière les collines brunes et bleues de l'horizon, il venait d'apercevoir une face blanchâtre, tuméfiée, couverte d'écailles, avec la peau soulevée par de hideux gonflements, et il connut aussitôt, au moyen du souvenir des livres, qu'il était lépreux. La lune, comme un masque jaune aérien, montait au-dessus des arbres. On entendait parfois un battement d'ailes mouillées au milieu des roseaux. Une traînée de brume flottait au fil du fleuve. Le miroitement de l'eau se prolongeait à une grande distance et se perdait dans la profondeur bleuâtre. Des oiseaux à tête écarlate froissaient le courant par des cercles qui se dissipaient lentement. Et le roi, debout, gardait les bras écartés de son corps, comme s'il avait le dégoût de se toucher.
- 77. Il releva le masque et le plaça sur son visage. Semblant marcher en rêve, il se dirigea vers son palais. Il frappa sur le gong, à la porte de la première muraille, et les gardes sortirent en tumulte avec leurs torches, lis éclairèrent sa face d'or; et le roi avait le cœur étreint d'angoisse, pensant que les gardes voyaient sur le métal des écailles blanches. Et il traversa la cour baignée de lune; et sept fois il eut le cœur étreint de la même angoisse aux sept portes où les gardes portèrent les torches rouges à son masque d'or. Cependant la peine croissait en lui avec la rage, comme une plante noire enroulée d'une plante fauve, lit les fruits sombres et troubles de la peine et de la rage vinrent sur ses lèvres, et il en goûta le suc amer. Il entra dans le palais, et le garde à sa gauche tourna sur la pointe d'un pied, ayant l'autre jambe étendue, en se couronnant avec un cercle lumineux de son sabre; et le garde à sa droite tourna sur la pointe de l'autre pied, avant étendu sa jambe opposée, en se coiffant d'une pyramide éblouissante par de rapides tourbillons de sa masse diamantée. Et le roi ne se souvint même pas que c'étaient les cérémonies nocturnes; mais il passa en frissonnant, ayant imaginé que les hommes d'armes voulaient abattre ou fendre sa hideuse tête gonflée. Les halles du palais étaient désertes. Quelques torches solitaires brûlaient bas dans leurs anneaux. D'autres s'étaient éteintes et pleuraient des larmes froides de résine. Le roi traversa les salles des fêtes où les coussins brodés de tulipes rouges et de chrysanthèmes jaunes étaient encore épars, avec des balanceuses d'ivoire et des sièges mornes d'ébène rehaussés d'étoiles d'or. Des voiles gommés et peints d'oiseaux à pattes diaprées, à bec d'argent, pendaient du plafond où s'enchâssaient des gueules de bêtes en bois de couleur. Il y avait des flambeaux de bronze verdâtre, faits d'une pièce, et percés de trous prodigieux laqués en rouge, où une mèche de soie écrue passait au centre de rondelles tassées d'un noir huileux. Il y avait des fauteuils longs, bas et cambrés, où on ne pouvait s'étendre sans que les reins fussent soulevés, comme portés par des mains. Il y avait des vases fondus de métaux presque transparents, et qui sonnaient sous le doigt d'une manière aiguë, comme s'ils étaient blessés. À l'extrémité de la salle, le roi saisit une torchère d'airain qui dardait ses langues rouges dans les ténèbres. Les gouttelettes flamboyantes de résine
- 78. s'abattirent en frémissant sur ses manches de soie. Mais le roi ne les remarqua pas. Il se dirigea vers une galerie haute, obscure, où la résine laissa un sillon parfumé. Là, aux parois coupées de diagonales croisées, on voyait des portraits éclatants et mystérieux: car les peintures étaient masquées et surmontées de tiares. Seulement le portrait le plus ancien, écarté des autres, représentait un jeune homme pâle, aux yeux dilatés d'épouvante, le bas du visage dissimulé par les ornements royaux. Le roi s'arrêta devant ce portrait et l'éclaira en soulevant la torchère. Puis il gémit et dit: —Ô premier de ma race, mon frère, que nous sommes pitoyables! Et il baisa le portrait sur les yeux. Et devant la seconde figure peinte, qui était masquée, le roi s'arrêta et déchira la toile du masque en disant: —Voilà ce qu'il fallait faire, mon père, second de ma race. Et ainsi il déchira les masques de tous les autres rois de sa race, jusqu'à lui- même. Sous les masques arrachés, on vit la nudité sombre de la muraille. Puis il arriva dans les salles des festins où les tables luisantes étaient encore dressées. Il porta la torchère au-dessus de sa tête, et des lignes pourpres se précipitèrent vers les coins. Au centre des tables était un trône à pieds de lion, sur lesquels s'affaissait une fourrure tachetée; des verreries semblaient amoncelées aux angles, avec des pièces d'argent poli et des couvercles percés d'or fumeux. Certains flacons miroitaient de lueurs violettes; d'autres étaient plaqués à l'intérieur avec de minces lames translucides de métaux précieux. Comme une terrible indication de sang, un éclat de la torchère fit scintiller une coupe oblongue, taillée dans un grenat, et où les échansons avaient coutume de verser le vin des rois. Et la lumière caressa aussi de vermeil un panier d'argent tressé où étaient rangés des pains ronds à croûte saine. Et le roi traversa les salles des festins en détournant la tête. —Ils n'ont pas eu honte, dit-il, de mordre sous leur masque dans le pain vigoureux, et de toucher le vin saignant avec leurs lèvres blanches! Où est celui qui, sachant son mal, interdit les miroirs de sa maison? Il est parmi ceux dont j'ai arraché les faux visages: et j'ai mangé du pain de son panier, et j'ai bu du vin de sa coupe....
- 79. On arrivait par une étroite galerie pavée de mosaïque aux salles à coucher, et le roi y glissa, portant devant lui sa torche sanglante. Un garde s'avança, saisi d'inquiétude, et sa ceinture d'anneaux larges flamboya sur sa tunique blanche; puis il reconnut le roi à sa face d'or et se prosterna. D'une lampe d'airain suspendue au centre, une lumière pâle éclairait une double file de lits de parade; les couvertures de soie étaient tissées avec des filaments de nuances vieilles. Un tuyau d'onyx laissait couler des gouttes monotones dans un bassin de pierre polie.
- 80. Le roi au masque d'or.
- 81. D'abord le roi considéra l'appartement des prêtres; et les masques graves des hommes couchés étaient semblables pendant le sommeil et l'immobilité. Et, dans l'appartement des bouffons, le rire de leurs bouches endormies avait juste la même largeur. Et l'immuable beauté de la figure des femmes ne s'était pas altérée dans le repos; elles avaient les bras croisés sur la gorge, ou une main sous la tête, et elles ne paraissaient pas se soucier de leur sourire qui était aussi gracieux quand elles l'ignoraient. Au fond de la dernière salle s'étendait un lit de bronze, avec des hauts reliefs de femmes courbées et de fleurs géantes. Les coussins jaunes y gardaient l'empreinte d'un corps agité. Là aurait dû reposer, dans cette heure de la nuit, le roi au masque d'or; là ses ancêtres avaient dormi pendant des années. Et le roi détourna la tête de son lit: —Ils ont pu dormir, dit-il, avec ce secret sur leur face, et le sommeil est venu les baiser au front, comme moi. Et ils n'ont pas secoué leur masque au visage noir du sommeil, pour l'effrayer à jamais. Et j'ai frôlé cet airain, j'ai touché ces coussins où s'abattaient jadis les membres de ces honteux.... Et le roi passa dans la chambre du brasier, où la flamme rose et pourpre dansait encore, et jetait ses bras rapides sur les murs. Et il frappa sur le grand gong de cuivre un coup si sonore qu'il y eut une vibration de toutes les choses métalliques d'alentour. Les gardes effrayés s'élancèrent mi-vêtus, avec leurs haches et leurs boules d'acier hérissées de pointes, et les prêtres parurent, endormis, laissant traîner leurs robes, et les bouffons oublièrent tous les bonds d'entrée sacramentels, et les femmes montrèrent au coin des portes leurs visages souriants. Or le roi monta sur son tronc noir et commanda: —J'ai frappé sur le gong afin de vous réunir pour une chose importante. Le mendiant a dit vrai. Vous me trompez tous ici. Ôtez vos masques. On entendit frissonner les membres et les vêtements et les armes. Puis, lentement, ceux qui étaient là se décidèrent et découvrirent leurs visages. Alors le roi au masque d'or se tourna vers les prêtres et considéra cinquante grosses faces rieuses avec de petits yeux collés par la somnolence: et, se tournant vers les bouffons, il examina cinquante figures hâves creusées par la tristesse avec des yeux sanguinolents d'insomnie; et,
- 82. se baissant vers le croissant de ses femmes assises, il ricana,—car leurs visages étaient pleins d'ennui et de laideur et enduits de stupidité. —Ainsi, dit le roi, vous m'avez trompé depuis tant d'années sur vous- mêmes et sur tout le monde. Ceux que je croyais sérieux et qui me donnaient des conseils sur les choses divines et humaines sont pareils à des outres ballonnées de vent ou de vin; et ceux dont je m'amusais pour leur continuelle gaieté étaient tristes jusqu'au fond du cœur; et votre sourire de sphinx, ô femmes, ne signifiait rien du tout! Misérables vous êtes! mais je suis encore le plus misérable d'entre vous. Je suis roi et mon visage parait royal. Or, en réalité, voyez: le plus malheureux de mon royaume n'a rien à m'envier. Et le roi ôta son masque d'or. Et un cri s'éleva des gorges de ceux qui le voyaient; car la flamme rose du brasier illuminait ses écailles blanches de lépreux. —Ce sont eux qui m'ont trompé—mes pères, je veux dire, cria le roi, qui étaient lépreux comme moi, et m'ont transmis leur maladie avec l'héritage royal. Ils m'ont abusé, et ils vous ont contraints au mensonge. Par la grande baie de la salle, ouverte vers le ciel, la lune tombante montra son masque jaune. —Ainsi, dit le roi, cette lune qui tourne toujours vers nous le même visage d'or a peut-être une autre face obscure et cruelle, ainsi ma royauté a été tendue sur ma lèpre. Mais je ne verrai plus l'apparence de ce monde, et je dirigerai mon regard vers les choses obscures. Ici, devant vous, je me punis de ma lèpre, et de mon mensonge, et ma race avec moi. Le roi leva son masque d'or; et, debout sur le trône noir, parmi l'agitation et les supplications, il enfonça dans ses yeux les crochets latéraux du masque, avec un cri d'angoisse; pour la dernière fois, une lumière rouge s'épanouit devant lui, et un flot de sang coula sur son visage, sur ses mains, sur les degrés sombres du trône. Il déchira ses vêtements, descendit les marches en chancelant, et, écartant avec des tâtonnements les gardes muets d'horreur, il partit seul dans la nuit. Or le roi lépreux et aveugle marchait dans la nuit. Il se heurta aux sept murailles concentriques de ses sept cours, et contre les arbres anciens de la résidence royale, et il se fit des plaies aux mains en touchant les épines des haies. Lorsqu'il entendit sonner ses pas, il connut qu'il était sur la grande
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