![I. INTRODUCTION
In many areas of contemporary science and technology one is confronted with
the problem of miniaturizing parts of the system of interest in order to control
processes on very short length and time scales [1]. For example, to study the ki-
netics of certain chemical reactions, reactants have to be mixed at a sufficiently
high speed. By miniaturizing a continuous-flow mixer, Knight et al. have re-
cently shown that nanoliters can be mixed within microseconds, thus permitting
one to study fast reaction kinetics on time scales unattainable with conventional
mixing technology [2]. The importance of designing and constructing micro-
scopic machines gave rise to a new field in applied science and engineering
known as “microfabrication technology” or “microengineering” [3]. A central
problem in the operation of such small mechanical machines is posed by fric-
tion between movable machine parts and wear. Lubricants consisting of, say, or-
ganic fluids can be employed to reduce these ultimately destructive phenomena.
Their functioning depends to a large extent on the nature of the interaction be-
tween the fluid and the solid substrate it lubricates [4,5]. In the case of micro-
machines the lubricant may become a thin confined film of a thickness of only
one or two molecular layers. The impact of such severe confinement is perhaps
best illustrated by the dramatic increase of the shear viscosity in a hexadecane
film of a thickness of two molecular layers, which may exceed the bulk shear
viscosity by four orders of magnitude [6].
Understanding the effect of confinement on the phase behavior and materi-
als properties of fluids is therefore timely and important from both a fundamen-
tal scientific and an applied technological perspective. This is particularly so
because the fabrication and characterization of confining substrates with pre-
scribed chemical or geometrical structures on a nano-to micrometer length scale
can nowadays be accomplished in the laboratory with high precision and by a
variety of techniques. For example, by means of various lithographic methods
[3,7] or wet chemical etching [8] the surfaces of solid substrates can be en-
dowed with well-defined nanoscopic lateral structures. In yet another method
the substrate is chemically patterned by elastomer stamps and, in certain cases,
subsequent chemical etching [9-12].
The development of a host of scanning probe devices such as the atomic
force microscope (AFM) [13-17] and the surface forces apparatus (SFA) [18-
22], on the other hand, enables experimentalists to study almost routinely the
behavior of soft condensed matter confined by such substrates to spaces of mo-
lecular dimensions. However, under conditions of severe confinement a direct
study of the relation between material properties and the microscopic structure
of confined phases still remains an experimental challenge.
2 Schoen
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-20-320.jpg)
![Computer simulations, on the other hand, are ideally suited to address this
particular question from a theoretical perspective. Generally speaking, computer
simulations permit one to pursue the motion of atoms or molecules in space and
time. Since the only significant assumption concerns the choice of interaction
potentials, the behavior of condensed matter can be investigated essentially in a
first-principles fashion. At each step of the simulation one has instantaneous ac-
cess to coordinates and momenta of all molecules. Thus, by applying the laws
of statistical physics, one can determine the thermomechanical properties of
condensed matter as well as its underlying microscopic structure. In many cases
the insight gained by computer simulations was and is unattainable by any other
theoretical means. Perhaps the most prominent and earliest example in this re-
gard concerns the prediction of solid-fluid phase transitions in hard-sphere flu-
ids at high packing fraction [23].
However, because of limitations of computer time and memory required to
treat dense many-particle systems, computer simulations are usually restricted
to microscopic length and time scales (with hard-sphere fluids, which may be
viewed as a model for colloidal suspensions [24] (this volume, chapter by M
Schmidt), and Brownian dynamics [25] as two prominent exceptions). This lim-
itation can be particularly troublesome in investigations of, say, critical phe-
nomena where the correlation length may easily exceed the microscopic size of
the simulation cell. In confinement, on the other hand, a phase may be physi-
cally bound to microscopically small volumes in one or more dimensions by the
presence of solid substrates so that computer simulations almost become a natu-
ral theoretical tool of investigation by which experimental methods can be com-
plemented. It is then not surprising that the study of confined phases by
simulational techniques is still flourishing [26], illustrated here for one particu-
lar aspect, namely the relation between microscopic structure and phase transi-
tions in confined fluids. In Sec. II an introduction to equilibrium theory of
confined phases will be given. Sec. III s devoted to formal and technical aspects
of computer simulations. In Sec. IV the microscopic structure of confined
phases will be analyzed for a number of different systems. The chapter con-
cludes in Sec. V with a description of phase transitions that are unique to phases
in confined geometry.
II. EQUILIBRIUM THEORY OF CONFINED PHASES
A. Thermodynamics
1. Experiments with the Surface Forces Apparatus
The force exerted by a thin fluid film on a solid substrate can be measured with
nearly molecular precision in the SFA [27]. In the SFA a thin film is confined
3
Structure and Phase Behavior of Soft Condensed Matter
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-21-320.jpg)
![between the surfaces of two cylinders arranged such that their axes are at right
angles [27]. In an alternative setup the fluid is confined between the surface of a
macroscopic sphere and a planar substrate [28]. However, crossed-cylinder and
sphere-plane configurations can be mapped onto each other by differential-geo-
metrical arguments [29]. The surface of each macroscopic object is covered by
a thin mica sheet with a silver backing, which permits one to measure the sepa-
ration h between the surfaces by optical interferometry [27]. The radii are
macroscopic so that the surfaces may be taken as parallel on a molecular length
scale around the point of minimum distance. In addition, they are locally planar,
since mica can be prepared with atomic smoothness over molecularly large
areas. This setup is immersed in a bulk reservoir of the same fluid of which the
film consists. Thus, at thermodynamic equilibrium temperature T and chemical
potential µ are equal in both subsystems (i.e., film and bulk reservoir). By ap-
plying an external force in the direction normal to both substrate surfaces, the
thickness of the film can be altered either by expelling molecules from it or by
imbibing them from the reservoir until thermodynamic equilibrium is reestab-
lished, that is, until the force exerted by the film on the surfaces equals the ap-
plied normal force. Plotting this force per radius R, F/R, as a function of h
yields a damped oscillatory curve in many cases (see, for instance, Fig. 1 in Ref.
[30]).
In another mode of operation of the SFA a confined fluid can be exposed to
a shear strain by attaching a movable stage to the upper substrate (i.e., wall) via
a spring characterized by its spring constant k [6,31,32] and moving this stage at
some constant velocity in, say, the x direction parallel to the film-wall interface.
Experimentally it is observed that the upper wall first “sticks” to the film, as it
were, because the upper wall remains stationary. From the known spring con-
stant and the measured elongation of the spring, the shear stress sustained by the
film can be determined. Beyond a critical shear strain (i.e., at the so-called
“yield point” corresponding to the maximum shear stress sustained by the film)
the shear stress declines abruptly and the upper wall “slips” across the surface
of the film. If the stage moves at a sufficiently low speed the walls eventually
come to rest again until the critical shear stress is once again attained so that the
stick-slip cycle repeats itself periodically.
This stick-slip cycle, observed for all types of film compounds ranging from
long-chain (e.g., hexadecane) to spheroidal [e.g., octamethylcyclotetra-siloxane
(OMCTS)] hydrocarbons [21], has been attributed by Gee et al. [30] to the for-
mation of solid-like films that pin the walls together (region of sticking) and
must be made to flow plastically in order for the walls to slip. This suggests that
the structure of the walls induces the formation of a solid film when the walls
are properly registered and that this film “melts” when the walls are moved out
of the correct registry. As was first demonstrated in Ref. 33, such solid films
4 Schoen
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-22-320.jpg)
![may, in fact, form in “simple” fluids between commensurate walls on account
of a template effect imposed on the film by the discrete (i.e., atomically struc-
tured) walls. However, noting that the stick-slip phenomenon is general, in that
it is observed in every liquid investigated, and that the yield stress may exhibit
hysteresis, Granick [21] has argued that mere confinement may so slow me-
chanical relaxation of the film that flow must be activated on a time scale com-
parable with that of the experiment. This more general mechanism does not
necessarily involve solid films which can be formed only if the (solid-like)
structure of the film and that of the walls possess a minimum geometrical com-
patibility.
2. The Fluid Lamella
For a theoretical analysis of SFA experiments it is prudent to start from a some-
what oversimplified model in which a fluid is confined by two parallel sub-
strates in the z direction (see Fig. 1). To eliminate edge effects, the substrates
are assumed to extend to infinity in the x and y directions. The system in the
thermodynamic sense is taken to be a lamella of the fluid bounded by the sub-
strate surfaces and by segments of the (imaginary) planes x = 0, x = sx, y = 0,
and y = sy. Since the lamella is only a virtual construct it is convenient to asso-
ciate with it the computational cell in later practical applications (see Secs. IV,
5
Structure and Phase Behavior of Soft Condensed Matter
FIG. 1 Schematic of two atomically structured, parallel surface planes (from Ref. 134).
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-23-320.jpg)
![energy must be minimum in a state of thermodynamic equilibrium. In mathe-
matical terms, an infinitesimal virtual transformation that would take the system
from this state must satisfy
8 Schoen
where δU is given in Eqs. (2), (3) and δü by
and the tilde refers to environmental variables. Viewing the environment as vir-
tual pistons, displacements of the boundary between them and the lamella
satisfy the equation = —δsα. Moreover, because the supersystem is mate-
rially closed, = —δN. From these two conditions and Eqs. (12)-(14), the
equilibrium conditions
are deduced. Now suppose the lamella is subject to thermal, mechanical, and
chemical reservoirs that maintain temperature, normal stress, and chemical po-
tential fixed at the values , and µ Assume also that the “complementary”
strains A, R, αxlx, and αyly are kept fixed. Then one has, from Eqs. (12) and
(14)]
Because of Eqs. (9), (10), and (15) this is equivalent to δ 0; that is, when
the lamella is at equilibrium at fixed T, µ, A, R, Tzz, αxlx, and αyly, is mini-
mum.
3. Derjaguin’s Approximation
To make contact with the SFA experiment one has to realize that the confining
surfaces are only locally parallel. Because of the macroscopic curvature of the
substrate surfaces, Tzz becomes a local quantity which varies with the vertical
distance sz = sz(x,y) between the substrate surfaces (see Fig. 2). Since the
sphere-plane arrangement (see Sec. II Al) is immersed in bulk fluid at pressure
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-26-320.jpg)
![Pbulk the total force exerted on the sphere by the film in the z direction can be
expressed as
9
Structure and Phase Behavior of Soft Condensed Matter
FIG. 2 Side view of film confined between a sphere of macroscopic radius R and a pla-
nar substrate surface. The shortest distance between two points located on the surface of
the sphere and of the substrate, respectively, is denoted by h (from Ref. 48).
which depends on the (bulk) thermodynamic state specified by T and µ. This
solvation, or depletion, force plays a vital role in the context of binary mixtures
of colloidal particles of different sizes [34] (this volume, chapter by M.
Schmidt). Because of their practical importance for colloid-polymer mixtures
[35], depletion forces in binary hard-sphere mixtures have recently received a
lot of attention and have been studied by a range of methods, including integral
equations based upon sophisticated hypernetted chain closure approximations
[36-41], density functional theory [42,43], virial expansion [44], and computer
simulation [45-47].
To evaluate the integral in Eq. (17), it is convenient to transform from carte-
sian to cylindrical coordinates (see Fig. 2) to obtain
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-27-320.jpg)
![where the arguments µ, and T have been dropped to simplify notation and the
far right side follows from (see Fig. 2) [48]. In Eq. (18)
10 Schoen
which follows from Eq. (11) (fixed R, αxlx, αyly) and a similar expression for
the bulk reser!voir
where V is the bulk volume. In Eq. (19) the excess grand potential Θex
:= Θ —
Θbulk is also introduced. Assuming V = Asz, the far right side of Eq. (19) ob-
tains because the bulk phase is isotropic. Furthermore, note that f(sz(p)) van-
ishes in the limit sz because of [49]
so that f(sz) may be interpreted as the excess normal pressure exerted on the
sphere by the fluid. In Eq. (19), F(h) still depends on the curvature of the sub-
strate surfaces through R. Experimentally, one is normally concerned with
measuring F(h)/R rather than the solvation force itself [27], because for macro-
scopically curved substrate surfaces this ratio is independent of R. This can be
rationalized by realizing that Tzz(sz) + Pbulk vanishes on a microscopic length
scale much smaller than R. The upper integration limit in Eq. (19) may then be
taken to infinity to give
because Θex
vanishes in the limit sz according to the definition in Eq.
(19). In Eq. (22) we introduce θex
(h) as the excess grand potential per unit area
of a fluid confined between two planar substrate surfaces separated by a dis-
tance h. The far right side of Eq. (22) is known as the Derjaguin approximation
(see Eq. (6) in Ref. 29). As pointed out recently by Götzelmann et al. [43], the
Derjaguin approximation is exact in the limit of a macroscopic sphere (i.e., if R
), which is the only case of interest here. A rigorous proof can be found in
the appendix of Ref. 50. A similar “Derjaguin approximation” for shear forces
exerted on curved substrates has recently been proposed by Klein and Ku-
macheva [51].
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-28-320.jpg)
![Eq. (22) is a key expression because it links the quantity F(h)/R that can be
determined directly in SFA experiments to the local stress Tzz available from
computer simulations (see Sec. IV A1). It is also interesting that differentiating
Eq. (22) yields
11
Structure and Phase Behavior of Soft Condensed Matter
Eq. (23) is particularly useful because it relates a derivative of experimentally
accessible data directly to the stress exerted locally on the macro-scopically
curved substrates at the point (0,0, sz = h) (see Fig. 2, Secs. IV A2, IV A3).
B. Symmetry and Homogeneity of Thermodynamic Potentials
An important issue in the thermodynamics of confined fluids concerns their
symmetry which is lower than that of a corresponding homogeneous bulk phase
because of the presence of the substrate and its inherent atomic structure [52].
The substrate may also be nonplanar (see Sec. IV C) or may consist of more
than one chemical species so that it is heterogeneous on a nanoscopic length
scale (see Sec. VB 3). The reduced symmetry of the confined phase led us to re-
place the usual compressional-work term —Pbulk V in the bulk analogue of Eq.
(2) by individual stresses and strains. The appearance of shear contributions
also reflects the reduced symmetry of confined phases.
1. Atomically Smooth Substrates
The simplest situation is one in which a planar substrate lacks any crystal-lo-
graphic structure. Then the confined fluid is homogeneous and isotropic in
transverse (x,y) directions. All off-diagonal elements of T vanish, Txx = Tyy =
, and Eq. (5) simplifies to
By symmetry, ⱖ f(A) at fixed T, µ, and sz. Hence, under these conditions one
can formally integrate Eq. (24) to obtain
taking the zero of 傼 to correspond to zero interfacial area. From Eqs. (6), (10),
and (25) one gets
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-29-320.jpg)
![which is the analogue of the bulk relation Θ = —Pbulk V. From Eq. (9) it is
straightforward to realize that
12 Schoen
is a nontrivial quantity (because in general ⬆ Tzz), whereas its bulk ana-
logue vanishes trivially because = Tzz = Pbulk on account of the higher
symmetry of bulk phases reflected by Eq. (21) [52]. From Eqs. (10), (24), and
(25), the Gibbs-Duhem equation
follows immediately.
2. The Two-dimensional Ideal Gas in an External Potential
While the smooth substrate considered in the preceding section is sufficiently
realistic for many applications, the crystallographic structure of the substrate
needs to be taken into account for more realistic models. The essential compli-
cations due to lack of transverse symmetry can be delineated by the following
two-dimensional structured-wall model: an ideal gas confined in a periodic
square-well potential field (see Fig. 3). The two-dimensional lamella remains
rectangular with variable dimensions sx and sy and is therefore not subject to
shear stresses. The boundaries of the lamella coinciding with the x and y axes
are anchored. From Eqs. (2) and (10) one has
FIG. 3 Schematic of the two-dimensional square-well potential u(x) of depth e, width
d, and period l (from Ref. 48).
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-30-320.jpg)
![for the free energy of the ideal gas under these premises. From standard text-
book considerations one also knows the statistical-physical expression [53]
13
Structure and Phase Behavior of Soft Condensed Matter
where β = 1/kBT (kB is Boltzmann’s constant). The canonical partition function
can be written more explicitly as — qN
/N! where the atomic partition
function is given by
where 1 is the single-atom configurational integral, and A is the thermal de
Broglie wavelength. The far right side of Eq. (31) follows immediately because
the potential energy of a molecule in the present two-dimensional ideal gas does
not depend on its y coordinate (see Fig. 3).
The configuration integral depends on sx in a piecewise fashion. For sx in
the nth period of the potential, that is for (n — 1)l sx ⭐ nl (n 僆 ), one ob-
tains
where
and
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-31-320.jpg)
![From Eqs. (29)—(31) one has
14 Schoen
With the help of Eq. (32) the first two expressions can be written explicitly as
and
where Pbulk = β-1
exp(β/µ)/A2
is the pressure of the two-dimensional ideal bulk
gas in thermodynamic equilibrium with the confined fluid.
Fig. 4 displays plots of -Txx and -Tyy versus sx. From these it is clear that
both stresses are functions of the size of the lamella. The most significant con-
sequence of this is that, unlike Eq. (24), Eq. (29) cannot be integrated at fixed T,
µ, and sy in general to yield an expression analogous to Eq. (25) without addi-
tional equations of state, that is Txx = Txx(sx), Tyy = Tyy(sx). In other words, a
Gibbs-Duhem equation corresponding to Eq. (28) does not obtain for the pres-
ent two-dimensional structured-wall model. The same conclusion holds for
more realistic three-dimensional structured-wall models [54]. The lack of a
Gibbs-Duhem equation for general thermodynamic transformations is a direct
consequence of the additional reduction of the confined fluid’s symmetry
caused by the discrete atomic structure of the substrate (see Sec. II B 1).
3. Coarse-grained Thermodynamics
While a Gibbs-Duhem equation does not exist for general transformations dsα
d , a specialized (i.e., “coarse-grained”) Gibbs-Duhem equation
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-32-320.jpg)
![n nⴕ = n ⬆ m (n, m integer), T is constant provided n and nⴕ are sufficiently
large for intensive properties to be independent of the (microscopic) size of the
lamella. Under these conditions Eq. (37) can be integrated to get
16 Schoen
Eq. (39) may be differentiated subsequently to give
Equating the expressions for dΘ given in Eqs. (37) and (40) and rearranging
terms yields the coarse-grained Gibbs-Duhem equation
which permits one to define the (transverse) isothermal compressibility k
where A = n2
a as detailed in Ref. 55. Note that a similar definition is prevented
for general transformations dsa dsⴕα according to the discussion in Sec. II
B2.
C. Statistical Physics
1. Stress-Strain Ensembles for Open and Closed Systems
To achieve a description of confined soft condensed matter at the molecular
level one has to resort to the principles of statistical physics. To make contact
with, say, SFA experiments it is convenient to introduce statistical physical en-
sembles depending explicitly on a suitable set of stresses and strains. For sim-
plicity, the lamella is treated quantum mechanically, following the procedure
originated by Schrödinger [56] and extended by Hill [53] and McQuarrie [57],
so that its energy states are formally discrete. The energy eigenvalues Ej(N,
A,R,sZ,αxlx,αyly) are implicit functions of the number of fluid molecules, ex-
tent and shape of the lamella, and the registry of the substrates, which control
the external field acting on the fluid molecules. Index j signifies the collection
of quantum numbers necessary to determine the eigenstate uniquely. The en-
semble comprises an astronomical number N of systems each in the same
macroscopic state, which, as an example, is taken to be specified by the set {T,
µ, A, R, αxlx,αyly} of ensemble parameters. Since the ensemble is isolated, it
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-34-320.jpg)
![satisfies the following constraints:
17
Structure and Phase Behavior of Soft Condensed Matter
where njNsz is the number of systems having N molecules between substrates
separated by sz and occupying eigenstate j. It is assumed that the isolated en-
semble has fixed total energy , fixed total number of molecules , and fixed
total volume . The total number of ways of realizing a given distribution n =
{njNsz} over the allowed “superstates” characterized by triplets (j,N,sz) is W(n)
= !/%j %N %sz
n
jNs! Since the number of systems is extremely large, the
most probable distribution, denoted by n*, overwhelms all others. It is found by
maximizing W(n) subject to the constraints [see Eqs. (43)]. The result for the
probability of a system’s occupying superstate (j, N, sz) is
where the partition function
and the set of Lagrangian multipliers {λ1,λ2, λ3} are determined through
equivalence of thermodynamic and statistical expressions as follows.
The statistical expression for the internal energy is simply = !jNsz
PjNsz Ej, from which its exact differential follows as
Ej can be obtained from Eq. (44) so that dEj can be replaced. One
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-35-320.jpg)
![eventually obtains
18 Schoen
At the molecular level one may interpret (uEj/uA)N,sz,R,αxlxαylyas the interfa-
cial tension of the system in superstate (j,N,sz). Similar meaning can be attached
to the other partial derivatives of Ej appearing in Eq. (47). Invoking also the
principle of conservation of probabilityjNsdPjNSz = 0), Eq. (47) can be recast
as
Following the lengthy argument by Hill [53], one identifies
Then a comparison of the microscopic Eq. (48) with its macroscopic counter-
part Eq. (5) allows one to identify the Lagrangian multipliers as
Therefore, from Eq. (44) one obtains
Upon substitution into Eq. (49), this yields
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-36-320.jpg)
![where the far right side is obtained by comparing the statistical expression in
Eq. (52) with the thermodynamic Eqs. (9) and (10). By exactly the same ap-
proach one can also derive statistical-physical expressions for other mixed
stress-strain ensembles [58,59]. Finally, from Eq. (45) one has for the partition
function
19
Structure and Phase Behavior of Soft Condensed Matter
where :=j
ex
P(βEj) is the canonical-ensemble partition function and =
exp(—βΘ) is its counterpart in the grand canonical ensemble. Since this chapter
is exclusively concerned with classical systems, is replaced by its classical
analogue
for the special case of spherically-symmetric molecules where is the config-
uration integral, Λ := (h2
β/2πm)1/2
is the thermal de Broglie wavelength (h is
Planck’s constant and m the molecular mass). The limiting expression class
can be derived from the quantum mechanical within the framework of the
Kirkwood and Wigner theory [53]. In the classical limit one has to replace the
quantum mechanical PjNSz by the analogous probability density distribution
where U(rN
) is the configurational energy of the system and Xclass is the classi-
cal counterpart of X obtained by replacing in Eq. (53) by class (see also
this volume, chapter by Nielaba).
2. Correlation Functions
Since we shall also be interested in analyzing the confined fluid’s microscopic
structure it is worthwhile to introduce some useful structural correlation func-
tions at this point. The simplest of these is related to the instantaneous number
density operator
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-37-320.jpg)
![where δ is the Dirac delta function. The mean value of π is given by
20 Schoen
where P[1]
(ri = r) is the probability of the center of mass of molecule i being at
r regardless of the positions of the other molecules (and regardless of orienta-
tion, see Sec. IV A 3). Since the molecules are equivalent, P[1]
is independent of
i and the summation on i in Eq. (57) can be performed explicitly to yield
In general, π[1]
(r) is a function of the vector position of the point of observation
r. However, if one is concerned mainly with the inhomogeneity of the confined
fluid in the normal (z) direction, the average over the interfacial area is ade-
quate. Averaging yields
to which we shall henceforth refer simply as the local density.
The translational microscopic structure of the confined fluid is partially re-
vealed by correlations in the number density operator, given by
where the “self-term” gives no new information beyond the mean density.
Again invoking the equivalence of fluid molecules, we recognize the cross-term
in Eq. (60) as the pair distribution function
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-38-320.jpg)
![which is related to the mean local density through the pair correlation function
by
21
Structure and Phase Behavior of Soft Condensed Matter
In general, g[2]
is a six-dimensional function of the position of reference (x,y,z)
and observed (xⴕ,yⴕ,zⴕ) molecules. However, to be consistent with the approxi-
mation for the local density [see Eq. (59)], we take g[2]
to be a function only of
the normal coordinate (z) of the reference molecule and the cylindrical coordi-
nates π12 and z12 of the observed molecule (2) relative to the reference mole-
cule (1), where the distance vector between the two r12 = π12 + z12êz and êz is
the unit vector in the z direction (see Sec. IV B).
III. MONTE CARLO SIMULATIONS
A key problem in the equilibrium statistical-physical description of condensed
matter concerns the computation of macroscopic properties Omacro like, for ex-
ample, internal energy, pressure, or magnetization in terms of an ensemble aver-
age O of a suitably defined microscopic representation 0(rN
) (see Sec. IV A 1
and VA 1 for relevant examples). To perform the ensemble average one has to
realize that configurations rN
:= {r1, r2,.., rN} generally differ energetically so
that a certain probability of occurrence is associated with each configuration.
Therefore, to compute the correct value O , 0(r) needs to be multiplied by the
relevant probability density function f(rN
;X), where X is a set of thermodynamic
state variables (for example, T, µ, and a combination of stresses and strains).
Analytically, the computation of ensemble averages along this route is a for-
midable task, even if microscopically small representations of the system of in-
terest are considered, because f(rN
; X) is generally a very complicated function
of the spatial arrangement of the N molecules. However, with the advent of
large-scale computers some forty years ago the key problem in statistical
physics became tractable, at least numerically, by means of computer simula-
tions. In a computer simulation the evolution of a microscopically small sample
of the macroscopic system is determined by computing trajectories of each mol-
ecule for a microscopic period of observation. An advantage of this approach is
the treatment of the microscopic sample in essentially a first-principles fashion;
the only significant assumption concerns the choice of an interaction potential
[25]. Because of the power of modern supercomputers which can literally han-
dle hundreds of millions of floating point operations per second, computer sim-
ulations are nowadays viewed as “a third branch complementary to the… two
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-39-320.jpg)
![traditional approaches” [60]: theory and experiment.
There are basically two different computer simulation techniques known as
molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular
trajectories are computed by solving an equation of motion for equilibrium or
nonequilibrium situations. Since the MD time scale is a physical one, this
method permits investigations of time-dependent phenomena like, for example,
transport processes [25,61-63]. In MC, on the other hand, trajectories are gener-
ated by a (biased) random walk in configuration space and, therefore, do not per
se permit investigations of processes on a physical time scale (with the “dynam-
ics” of spin lattices as an exception [64]). However, MC has the advantage that
it can easily be applied to virtually all statistical-physical ensembles, which is of
particular interest in the context of this chapter. On account of limitations of
space and because excellent texts exist for the MD method [25,61-63,65], the
present discussion will be restricted to the MC technique with particular empha-
sis on mixed stress-strain ensembles.
If one wishes to compute O numerically by means of MC one immediately
realizes that this requires the a priori unknown function f(rN
; X) according to
which the random walk in configuration space has to be carried out. However, if
the random walk is carried out in a biased way as a Markov process it turns out
that only the ratio f(rN
m+1; X)/f(rN
m; X) is relevant to generate a new configura-
tion (m + 1) from a given old one (m). The subsequent discussion will show that
f (rN
m+1; X)/f(rN
m; X) is computationally accessible. Because this scheme gen-
erates configurations with the correct probability of occurrence, Omacro can be
computed via the simple expression
22 Schoen
where the prime is attached as a reminder that the summation is restricted to
configurations generated according to their importance (importance sampling)
and Mmax should be large enou gh [usually, Mmax (106
-109
) is sufficient,
depending on the particular physical situation and quantity of interest]. How-
ever, before turning to practical aspects of MC, a brief introduction to Markov
processes seems worthwhile because it rarely appears in the literature.
A. Stochastic Processes
Let y(t) be a random process, that is a process incompletely determined at any
given time t. The random process can be described by a set of probability distri-
butions {Pn} where, for example, P2 (y1 t1,y2t2) dy1 dy2 is the probability of
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-40-320.jpg)
![finding y1 in the interval [y1,y1 + dy1] at t = t1 and in the interval [y2, y2+ dy2]
at
another time t = t2. Thus the set {Pn} forms a hierarchy of probability distri-
butions describing y(t) in greater detail the larger n is.
The simplest random process is completely stochastic so that one may write,
for example, P2(y1t1,y2t2) = P1(y1t1)P1(y2t2). However, here we are con-
cerned with a slightly more complex process known as the Markov process,
characterized by
23
Structure and Phase Behavior of Soft Condensed Matter
where K1 (yxtx y2t2) is the conditional probability of finding y in the interval
[y2,y2 + dy2] at t = t2 provided y =y1 at an earlier time t = tx (tx t2). Some
important properties are the following:
1. Normalization, that is K1(y1t1y2t2) dy2 = 1.
2. P1(y1t1)K1(y1t1y2t2)dy1 = P1(y2t2)
3 Perhaps most importantly, a Markov process has a “one-step memory”; that
is, to find y in the interval [yn,yn + dyn] at t = tn depends only on the realiza-
tion y = yn-1 at the immediately preceding time t = tn
-
1 but is independent of
all earlier realizations {ymtm}, 1 ⱕ m ⱕ n - 2. Mathematically, this can be
cast as Kn-1(y1t1,…,yn-1 tn-1 |yntn) K1 (yn-1tn-1|yntn).
4 Stationarity, that is P1(y1 t1) = P1 (y1) and P2(y1t1,y2t2)= P2(y1,y2;t2-t1).
Consider now
and assume that y(t) is a Markov process. Then
Because of Eqs. (64) and (66), Eq. (65) can be rewritten as
Alternatively, because of property (4), this last expression can be cast as
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-41-320.jpg)
![“simple” molecules having only (three) translational degrees of freedom. Under
these premises one has (see Sec. IIC 1] [66]
where ,…, N so that the inte-
25
Structure and Phase Behavior of Soft Condensed Matter
gration is carried out over the unit-cube volume . The summation over sz [see
Eq. (53)] has been replaced by an integral, and the dimensionless quantity B is
defined by
The MC method can be implemented by a modification of the classic Me-
tropolis scheme [25,67]. The Markov chain is generated by a three-step se-
quence. The first step is identical to the classic Metropolis algorithm: a
randomly selected molecule i is displaced within a small cube of side length 2δr
centered on its original position
where 1 = (1,1,1) and ξ is a vector whose three components are pseudorandom
numbers distributed uniformly on the interval [0,1]. During the MC run δr is
adjusted so that 40-60% of the attempted displacements are accepted. With the
identification one obtains
from Eqs. (55) and (74) because Nm = Nm+1 and szm+1 = sz,m, where
9U is the change in configurational energy associated
with the process An efficient way to compute ∆U is detailed below
in Sec. III.C.
In the second step it is decided with equal probability whether to remove (AN =
-1) a randomly chosen molecule or to create (∆N = +1) a new one at a randomly
chosen point in the system (see also Sec. III D). From Eqs. (55) and (74) the
transition probabilities for addition (“+”) and subtraction (“-“) are given by
where
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-43-320.jpg)
![Since only one molecule is added to (or removed from) the system, Um is sim-
ply the interaction of the added (or removed) molecule with the remaining ones.
If one attempts to add a new molecule, N is the number of molecules after addi-
tion, otherwise it is the number of molecules prior to removal. If a cutoff for the
interaction potential is employed, long-range corrections to Um must be taken
into account because of the density change of m 1/Asz. Analytic expressions for
these corrections can be found in the appendix of Ref. 33.
In the third and final step the substrate separation is changed according to
26 Schoen
and the coordinates of fluid molecules are scaled via zm+1 = zmsz,m+l/sZ,m.
Because N is held constant the transition probability associated with this step is
where
∆sz := sz,m+1 - sz,m and the same comments concerning corrections to ∆U
apply as in step 2. On each pass through the three-step sequence the number of
attempts in steps 1, 2, and 3 is chosen to be N, N, and 1, respectively, in order to
realize a comparable degree of events in each of the steps. Because the third
step moves all N molecules at once, and the first two affect only one molecule
at a time, the sought balance is roughly achieved. The algorithm described here
can easily be amended by additional steps if, for example, one is interested in
situations in which the shear stress(es) is (are) also among the controlled param-
eters so that αx (and αy) may vary too [58,59]. Applying the analysis of Wood
[68] to each step of the algorithm separately, one can verify that the resulting
transition probabilities indeed comply with the requirements of a Markov
process as stated in Eq. (74).
C. The Taylor-expansion Algorithm for “Simple”
Fluids
According to Allen and Tildesley, the standard recipe to evaluate ∆U in step one
of the algorithm described in Sec. III B involves “computing the energy of atom
i with all the other atoms before and after the move (see p. 159 of Ref. 25, ital-
ics by the present author) as far as “simple” fluids are concerned. The evalua-
tion of ∆U can be made more efficient in this case by realizing that for
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-44-320.jpg)
![short-range interactions U can be split into three contributions U = U1 + U2 +
U3 corresponding to three different spatial zones, where U1 is the configura-
tional energy between atom i and N1 neighboring molecules located in a pri-
mary zone immediately surrounding i. Similarly, U2 refers to interactions
between i and N2 molecules in a secondary zone adjacent to the primary zone
and, last but not least, U3 refers to interactions between i and the remaining
molecules in an outermost tertiary zone whose upper limit is identical with the
potential cutoff by which the computational burden is reduced already in con-
ventional implementations. Savings of computer time depend on the sizes of the
three zones (i.e., the values of N1, N2, N3) and different degrees of sophistica-
tion with which the three terms are treated. It turns out that a sphere of radius r1
centered on ri can be associated with the primary zone. The secondary zone can
be a spherical shell of thickness ∆r = r2 - r1 (bulk fluid) or a cylindrical shell of
the same thickness but infinite height (confined fluid, slit geometry). If r2 is
sufficiently large one may assume ∆U3 = U3(ri,m+1) - U3(ri,m) b 0 because
[see Eq. (77)] is small compared with typical distances corresponding to ter-
tiary-zone interactions. Thus, N3 interactions are entirely neglected during the
course of the simulation. For the secondary zone one assumes that ∆U2 =
U2(ri,m+1) — U2(ri,m) is not entirely negligible but small enough to be ap-
27
Structure and Phase Behavior of Soft Condensed Matter
proximated by a Taylor expansion
truncated after the first nonvanishing term, where F2 is the total force exerted
on i in the initial configuration m by the N2 atoms in the secondary zone. For
the primary zone no simplifying assumptions can be made because U1 will
strongly depend on δr. Thus, on the basis of these assumptions, ∆U in Eq. (78)
can be written explicitly as
It is clear that Eq. (85) is numerically reliable provided δr is sufficiently small.
However, a detailed investigation in Ref. 69 reveals that δr can be as large as
some ten percent of the diameter of a fluid molecule. Likewise, r1 should not be
smaller than, say, the distance at which the radial pair correlation function has
its first minimum (corresponding to the nearest-neighbor shell). Under these
conditions, and if combined with a neighbor list technique, savings in computer
time of up to 40% over conventional implementations are measured for the first
(canonical) step of the algorithm detailed in Sec. III B. These are achieved be-
cause, for pairwise interactions, only N1 + N2 contributions need to be com-
puted here before i is moved (U1 and F2), and only N1 contributions need to be
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-45-320.jpg)
![evaluated after i is displaced by δr (U1) where some efficiency is inevitably lost
because the computation of forces is numerically more demanding than the
computation of energies. A much larger number of 2(N1 + N2 + N3) such terms
must be evaluated conventionally.
D. Orientationally Biased Creation of Molecules
If fluid molecules have rotational degrees of freedom the algorithm outlined in
Sec. III B must be modified in various respects. First, in addition to random dis-
placements in step 1, the molecule is rotated randomly by a small angle incre-
ment ∆ ⑀ (—max, max) around an axis chosen at random from the three
axes of the Cartesian coordinate system. A rotation attempt is again accepted
with the probability given in Eq. (78) where now ∆U refers to the change in
configurational energy associated with the rotation attempt. During the course
of a simulation max is adjusted to preserve an overall acceptance ratio of 40-
60%.
For a removal attempt a molecule is selected irrespective of its orientation.
To enhance the efficiency of addition attempts in cases where the system pos-
sesses a high degree of orientational order, the orientation of the molecule to be
added is selected in a biased way from a distribution function. For a system of
linear molecules this distribution, say, g( ), depends on the unit vector Û
parallel to the molecule’s symmetry axis (the so-called microscopic director
[70,71]) and on the macroscopic director which is a measure of the average ori-
entation in the entire sample [72]. The distribution g can be chosen in various
ways, depending on the physical nature of the fluid (see below). However,
g( ) must be normalized to one [73,74]. In other words, an addition is at-
tempted with a preferred orientation of the molecule determined by the macro-
scopic director of the entire simulation cell. The position of the center of
mass of the molecule is again chosen randomly. According to the principle of
detailed balance the probability for a realization of an addition attempt is given
28 Schoen
by [73]
whereas
for a removal attempt, where rⴞ is given in Eq. (80). However, B has to be re-
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-46-320.jpg)
![placed by Bⴕ
denned analogously as (linear molecules only)
where I is the moment of inertia. Quantities Û+ and Û- are the microscopic di-
rectors of the film molecule to be added to or removed from the film. Biased ad-
dition is indispensable if, for example, the confined fluid is a liquid crystal in
thermodynamic equilibrium with a nematic bulk phase (see Sec. IV A 3). In this
case g( ) is identified as the orientational distribution function of film mole-
cules, which is computed as a histogram averaged over all configurations pre-
ceding the actual one; if the thermodynamic state of the bulk liquid crystal is
isotropic, g( ) = 1 is a suitable choice. One realizes from Eqs. (86), (87) that
in this case one recovers Eq. (79).
IV. MICROSCOPIC STRUCTURE
A. Planar Substrates
1. Molecular Expressions for the Normal Stress
Within the framework of Monte Carlo simulations, the relation between meas-
urable quantities and the microscopic structure of confined phases can now be
examined. An example of such a measurable quantity is the solvation force
F(h)/2π R (see Sec. IIA 1). From a theoretical perspective and according to the
discussion in Sec. IIA 3 its investigation requires the stress Tzz(sz) exerted nor-
mally by a confined fluid on planar substrates [see Eqs. (19) and (22)]. Using
29
Structure and Phase Behavior of Soft Condensed Matter
Eqs. (11) and (53) one can derive a molecular expression for Tzz from
for “simple” fluids, where the shorthand notation is introduced to represent the
weighted sum over N. Depending on how the partial derivative of is worked
out, two mathematically different but physically equivalent expressions for Tzz
obtain. The first of these is obtained by following the procedure of Hill [53] and
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-47-320.jpg)
![transforming variables according to zt $ i- = zisz-1
(i = 1,…,N) to obtain
30 Schoen
assuming that
where uff is the fluid-fluid interaction potential (corresponding, for example, to
the Lennard-Jones (12,6) potential) and the fluid-substrate interaction depends
on the distance between a fluid molecule and a substrate atom. Scaling in Eq.
(90) affects UFF through the argument
and UFS because of for a fluid molecule located at ri, and a
wall atom at [see Eqs. (1)]. According to the product rule, the differentiation
in Eq. (90) yields three terms grouped as Tzz = TZZ,FF + TZZ,FS where
and
In Eqs. (93) and (94) Wzz is an element of Clausius’s virial matrix [53] and ul is
the derivative of u with respect to its argument. Therefore Eqs. (93) and (94) are
termed “virial” expressions henceforth.
A different expression for Tzz can be obtained directly from Eq. (89) without
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-48-320.jpg)
![transforming coordinates. It is then convenient to recast the
31
Structure and Phase Behavior of Soft Condensed Matter
configuration integral as [75]
Applying Leibniz’s rule for the differentiation of a parameter integral [76], it
follows from Eqs. (89) and (95) that
because U becomes infinitely large if z1 = sz on account of the divergence of
Ufs at this point. In Eq. (96), g2 is defined analogously to g1 in Eq. (95) and the
above argument may be repeated N -1 times, to obtain finally
where is the total force exerted by the film on substrate
k. Equation (97) is therfore termed the “force” expression henceforth. The far
right side of Eq. (97) is a statement of mechanical stability of the confined fluid.
Virial and force expressions provide a useful test of internal consistency of a
computer simulation and should agree within a few percent (see, for example,
Table II in [77]).
2. Stratification
To illustrate the relationship between the microscopic structure and experimen-
tally accessible information, we compute pseudo-experimental solvation-force
curves F(h)/R [see Eq. (22)] as they would be determined in SFA experiments
from computer-simulation data for Tzz [see Eqs. (93), (94), (97)]. Numerical
values indicated by an asterisk are given in the customary dimensionless (i.e.,
reduced) units (see [33,75,78] for definitions in various model systems). Results
are correlated with the microscopic structure of a thin film confined between
plane parallel substrates separated by a distance sz = h. Here the focus is specif-
ically on a “simple” fluid in which the interaction between a pair of film mole-
cules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A
confined “simple” fluid serves as a suitable model for approximately spherical
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-49-320.jpg)
![OMCTS molecules confined between mica surfaces, which is perhaps the most
thoroughly investigated system in SFA experiments [27,30]. Because OMCTS
is chemically inert and electrically neutral, the influence of charges on the mica
surfaces can safely be ignored.
Plots of f(sz) and F(h)/R versus sz and h, respectively, are shown in Fig. 5.
The oscillatory decay of both quantities is a direct consequence of the oscilla-
tory dependence of Tzz on sz, which has also been investigated by integral equa-
tion approaches of varying sophistication [36,85-89]. As can be seen in Fig. 5,
zeros of f(sz) correspond to successive extrema of F(h)/R because of Eq. (22).
In actual SFA experiments the only portions of the F(h)/R curve generally ac-
cessible are those where d[F(h)/R]/dh 0. Regions where d[F(h)/R]/dh 0 are
inaccessible, because θeX
(H) would decrease upon compression of the film [see
Eq. (23)]. However, structural changes accompanying the variation of F(h)/R in
the accessible regimes are rather obscure, as can be inferred from Fig. 6 where
plots of the local density are presented. On account of Eq. (23) accessible por-
tions of the pseudo-experimental curve can be related to the local stress at the
point (0,0, sz = h) of minimum distance between the surfaces of the macro-
scopic sphere and the planar substrate (see Fig. 2). By correlating the local
stress Tzz(h) with the film’s local structure at (0,0, A) via π[1]
(z;sz = h), one can
establish a direct correspondence between pseudo-experimental data (i.e.,
32 Schoen
FIG. 5 The excess pressure f(sz) (, dashed line) and the solvation force per radius
F(h)/R (full line) as functions of sz and h, respectively, for a confined fluid composed of
“simple” molecules (from Ref. 48)
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-50-320.jpg)
![F(h)/R) and local microscopic structure of the film.
Plots of a sequence of local densities π[1]
(z; sz = h) in Fig. 6 over the range
2.60 ⱕ h* ⱕ 4.00 illustrate this correlation. In an actual SFA experiment 2.59 ⱕ
h*ⱕ 3.06 and 3.53 ⱕ h*ⱕ 4.00 are accessible portions of the solvation-force
33
Structure and Phase Behavior of Soft Condensed Matter
FIG. 6 The local density π[1]
(z;sz) as a function of z/sz for a “simple” fluid confined by
planar substrates, (a) sz*:=2.60 (䊏), sz*=2.80 (*), sz*=3.00 (+); (b) sz*=3.20 (䊏),
sz*=3.40 (*), sz*=3.55 (+); (c) sz*=3.80 (䊏), sz*=4.00 (*) (from Ref. 48).
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-51-320.jpg)
![curve whereas 3.06 h* 3.53 demarcates the inaccessible range because here
d[F(h)]/dh 0. Plots in Figs. 6(a) and 6(c) show that in the experimentally ac-
cessible regions the film consists locally of two and three strata, respectively.
For h* = 2.60 the film is locally compressed (F(h) 0) whereas it is stretched
for h* = 3.00 (F(h) 0). Under compression the film appears to be less strati-
fied, as is reflected by smaller heights of less well separated peaks of π[1]
(z; sz
= h) compared with the other two curves in Fig. 6(a). For h* = 2.80, F(h) 0
and Tzz(sz = h) has almost assumed a minimum value, indicating that for this
particular value of h film molecules are locally accommodated most satisfacto-
rily between the surfaces of the macroscopic sphere and the planar substrate. It
is therefore not surprising that peaks in π[1]
(z;sz = h) are taller for h* = 2.80
compared with the two neighboring values of h [see Fig. 6(a)].
In the next accessible region (3.53 ⱖ h* ⱖ 4.00) the film consists of three
molecular strata for which the most pronounced structure is observed for h*
3.80, corresponding to a point at which F(h)/R nearly vanishes [see Fig. 6(c)].
As before [see Fig. 6(a)] this is reflected by the peak height in the contact strata
(i.e., the strata closest to the substrate) whereas inner portions of the film remain
largely unaffected. Plots of π[1]
(z;sz = h) in the inaccessible regime in Fig. 6(b)
show that here the film undergoes a local reorganization characterized by the
vanishing (appearance) of a whole stratum. The reorganization is gradual, as
one can see in the plot of π[1]
(z; sz = h) for h* = 3.4, where two shoulders ap-
34 Schoen
FIG. 6 Continued.
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-52-320.jpg)
![pear at z/sz ±0.1.
Stratification, as illustrated by the plots in Fig. 6, is due to constraints on the
packing of molecules next to the wall and is therefore largely determined by the
repulsive part of the intermolecular potential [55]. It is observed even in the ab-
sence of intermolecular attractions, such as in the case of a hard-sphere fluid
confined between planar hard walls [42,90-92]. For this system Evans et al. [93]
demonstrated that, as a consequence of the damped oscillatory character of the
local density in the vicinity of the walls, Tzz is a damped oscillatory function of
sz, if sz is of the order of a few molecular diameters, which is confirmed by Fig.
5.
For one-dimensional confined hard-rod [49,94,95] and Tonks-Takahashi flu-
ids [49,96,97] the close relationship between stratification and the oscillatory
decay of Tzz(sz) has been demonstrated analytically. On the basis of a density-
functional approach Iwamatsu [98] has recently analyzed the solvation force in
various experimental systems. In the context of dielectric media the analogue of
the solvation force between planar walls [f(sz) in the current notation] is known
as the Casimir force. It arises because the walls modify the spectrum of electro-
magnetic fluctuations between them such that the vacuum energy of the electro-
magnetic field becomes size- and shape-dependent [99].
3. Orientational Effects
In the other model system the film consists of (soft) ellipsoidal Gay-Berne mol-
ecules [78,100-102]. Depending on the thermodynamic state, bulk phases con-
sisting of Gay-Berne molecules can be either isotropic or nematic [103]. The
Gay-Berne fluid is therefore a suitable model for liquid crystals, which are cur-
rently intensively studied in SFA experiments [104-110] because of their impor-
tance in such diverse fields as, say, display technology [111] and lubrication
[112]. Here the only case considered is that of a confined Gay-Berne fluid in
thermodynamic equilibrium with a nematic bulk phase [102,113]. Parameters of
the fluid-substrate intermolecular potential parameters are chosen so that a
homeotropic anchoring of fluid molecules to the substrate surface is favored
(i.e., fluid molecules in the vicinity of the substrate surface are preferentially or-
dered normal to the surface). The “nematic” Gay-Berne film between
homeotropically anchoring substrates may be viewed as a rough model for a
film of (dimers of) 4´-n-octyl-4-cyanobiphenyl (8CB) molecules confined be-
tween hydrophobic substrate surfaces consisting of mica coated with dihexade-
cyldimethyl ammonium acetate (DHDAA) monolayers [107].
As in the case of a “simple”-fluid film, the normal component of the stress
tensor is a damped oscillatory function of substrate separation (see Fig. 7). Over
35
Structure and Phase Behavior of Soft Condensed Matter
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-53-320.jpg)
![the range 4.0 ⱖ sz ⱖ 16.75,f(sz) exhibits four maxima separated by a distance
∆sz b 3.2, which is slightly smaller than the large diameter of a film molecule
[102]. In analogy with results for confined “simple” fluids (see Sec. IV A 2), it
is plausible to associate oscillations in f(sz) with the formation of molecular
strata parallel with the walls. Fig. 7 also shows that f(sz) oscillates around zero
in the limit of large sz [see Eqs. (19), (21)] as it should [49].
However, f(sz) also exhibits shoulders at characteristic values of sz separated
by the same distance ∆s*z 3.2 as the maxima. Portions of f(sz) between
neighboring minima (i.e., s*z 6.80, 6.80 ⱖ s*z ⱖ 10.00, 10.00 ⱖ s*z ⱖ
13.20, and 13.20 ⱖ s*z ⱖ 16.40) are remarkably similar. In order to correlate
the microscopic structure of the confined film with features of f(sz), it is con-
36 Schoen
FIG. 7 Same as Fig. 5, but for a “nematic” Gay-Berne film confined between
homeotropically anchoring substrates (from Ref. 48).
venient to label these portions as “decrease,” “increase,” and “shoulder” zones
and to introduce the density-alignment distribution defined by [78,101]
where (z,u2
z;sz)dzdu2
z is the probability of finding a film molecule at position
z with orientation uz, which is the cosine of the angle θ between the micro-
scopic director Û and the z axis. The argument u2
z of the probability density f(z,
u2
z; sz) reflects the nonpolarity of Gay-Berne molecules (i.e., the equivalence of
Û and —Û). By definition, u2
z = 1 if Û is orthogonal to the plane of a wall and
Copyright © Marcel Dekker 2000](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-54-320.jpg)
![found excellent. The price of these pans is moderate, and they last a long
time—manufacturers, Messrs. Deane and Dray.[36]
SERIES OF SMALL RECEIPTS FOR A SQUAD, OUTPOST,
OR PICKET OF MEN,
Which may be increased in proportion of companies.
Camp Receipts for the Army in the East.
(From the Times of the 22nd January, 1855.)[37]
No. 15. Camp Soup.—Put half a pound of salt pork in a saucepan, two
ounces of rice, two pints and a half of cold water, and, when boiling, let
simmer another hour, stirring once or twice; break in six ounces of biscuit,
let soak ten minutes; it is then ready, adding one teaspoonful of sugar, and a
quarter one of pepper, if handy.
No. 16. Beef Soup.—Proceed as above, boil an hour longer, adding a pint
more water.
Note.—Those who can obtain any of the following vegetables will find
them a great improvement to the above soups:—Add four ounces of either
onions, carrots, celery, turnips, leeks, greens, cabbage, or potatoes,
previously well washed or peeled, or any of these mixed to make up four
ounces, putting them in the pot with the meat.
I have used the green tops of leeks and the leaf of celery as well as the
stem, and found, that for stewing they are preferable to the white part for
flavour. The meat being generally salted with rock salt, it ought to be well
scraped and washed, or even soaked in water a few hours if convenient; but
if the last cannot be done, and the meat is therefore too salt, which would
spoil the broth, parboil it for twenty minutes in water, before using for soup,
taking care to throw this water away.
No. 17.—For fresh beef proceed, as far as the cooking goes, as for salt
beef, adding a teaspoonful of salt to the water.
No. 18. Pea Soup.—Put in your pot half a pound of salt pork, half a pint
of peas, three pints of water, one teaspoonful of sugar, half one of pepper,
four ounces of vegetables, cut in slices, if to be had; boil gently two hours,](https://image.slidesharecdn.com/5251364-250618142541-2b705d60/85/Computational-Methods-In-Surface-And-Colloid-Science-Borwko-72-320.jpg)
Computational Methods In Surface And Colloid Science Borwko
- 1. Computational Methods In Surface And Colloid Science Borwko download https://ebookbell.com/product/computational-methods-in-surface- and-colloid-science-borwko-10502728 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Computational Methods In Organometallic Catalysis From Elementary Reactions To Mechanisms Yu Lan https://ebookbell.com/product/computational-methods-in-organometallic- catalysis-from-elementary-reactions-to-mechanisms-yu-lan-46084622 Computational Methods In Engineering S P Venkateshan Prasanna Swaminathan https://ebookbell.com/product/computational-methods-in-engineering-s- p-venkateshan-prasanna-swaminathan-50401588 Computational Methods In Drug Discovery And Repurposing For Cancer Therapy Ganji Purnachandra Nagaraju https://ebookbell.com/product/computational-methods-in-drug-discovery- and-repurposing-for-cancer-therapy-ganji-purnachandra- nagaraju-51335908 Computational Methods In Psychiatry Gopi Battineni Mamta Mittal https://ebookbell.com/product/computational-methods-in-psychiatry- gopi-battineni-mamta-mittal-53997480
- 3. Computational Methods In Engineering Finite Difference Finite Volume Finite Element And Dual Mesh Control Domain Methods J N Reddy https://ebookbell.com/product/computational-methods-in-engineering- finite-difference-finite-volume-finite-element-and-dual-mesh-control- domain-methods-j-n-reddy-55754940 Computational Methods In Lanthanide And Actinide Chemistry Michael Dolg https://ebookbell.com/product/computational-methods-in-lanthanide-and- actinide-chemistry-michael-dolg-56362562 Computational Methods In Fluid Mechanics A Handbook Gabriel Alozondo https://ebookbell.com/product/computational-methods-in-fluid- mechanics-a-handbook-gabriel-alozondo-56363670 Computational Methods In Elasticity And Plasticity Solids And Porous Media 1st Edition A Anandarajah Auth https://ebookbell.com/product/computational-methods-in-elasticity-and- plasticity-solids-and-porous-media-1st-edition-a-anandarajah- auth-2006300 Computational Methods In Engineering Science Proceedings Of Enhancement And Promotion Of Computational Methods In Engineering And Science X Aug 2123 2006 Sanya China 1st Edition Zhenhan Yao https://ebookbell.com/product/computational-methods-in-engineering- science-proceedings-of-enhancement-and-promotion-of-computational- methods-in-engineering-and-science-x-aug-2123-2006-sanya-china-1st- edition-zhenhan-yao-2016302
- 5. COMPUTATIONAL METHODS IN SURFACE AND COLLOID SCIENCE Copyright © Marcel Dekker 2000
- 6. DANIEL BLANKSCHTEIN Department of Chemical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts S. KARABORNI Shell International Petroleum Company Limited London, England LISA B. QUENCER The Dow Chemical Company Midland, Michigan JOHN F. SCAMEHORN Institute for Applied Surfactant Research University of Oklahoma Norman, Oklahoma P. SOMASUNDARAN Henry Krumb School of Mines Columbia University New York, New York Rochester, New York ERIC W. KALER Department of Chemical Engineering University of Delaware Newark, Delaware CLARENCE MILLER Department of Chemical Engineering Rice University Houston, Texas DON RUBINGH The Procter & Gamble Company Cincinnati, Ohio BEREND SMIT Shell International Oil Products B. V. Amsterdam, The Netherlands JOHN TEXTER Strider Research Corporation SURFACTANT SCIENCE SERIES FOUNDING EDITOR MARTIN J. SCHICK 1918–1998 SERIES EDITOR ARTHUR T. HUBBARD Santa Barbara Science Project Santa Barbara, California ADVISORY BOARD Copyright © Marcel Dekker 2000
- 7. 1. Nonionic Surfactants, edited by Martin J. Schick(see also Volumes 19, 23, and 60) 2. Solvent Properties of Surfactant Solutions, edited by Kozo Shinoda(see Volume 55) 3. Surfactant Biodegradation, R. D. Swisher (seeVolume 18) 4. Cationic Surfactants, edited by Eric Jungermann(see also Volumes 34, 37, and 53) 5. Detergency: Theory and Test Methods (in three parts), edited by W. G. Cutler and R. C. Davis(see also Volume 20) 6. Emulsions and Emulsion Technology (in three parts), edited by Kenneth J. Lissant 7. Anionic Surfactants (in two parts), edited by Warner M. Linfield(see Volume 56) 8. Anionic Surfactants: Chemical Analysis, edited by John Cross 9. Stabilization of Colloidal Dispersions by Polymer Adsorption, Tatsuo Sato and Richard Ruch 10. Anionic Surfactants: Biochemistry, Toxicology, Dermatology, edited by Christian Gloxhuber(see Volume 43) 11. Anionic Surfactants: Physical Chemistry of Surfactant Action, edited by E. H. Lu- cassen-Reynders 12. Amphoteric Surfactants, edited by B. R. Bluestein and Clifford L Hilton(see Vol- ume 59) 13. Demulsification: Industrial Applications, Kenneth J. Lissant 14. Surfactants in Textile Processing, Arved Datyner 15. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applica- tions, edited by Ayao Kitahara and Akira Watanabe 16. Surfactants in Cosmetics, edited by Martin M. Rieger(see Volume 68) 17. Interfacial Phenomena: Equilibrium and Dynamic Effects, Clarence A. Miller and P. Neogi 18. Surfactant Biodegradation: Second Edition, Revised and Expanded, R. D. Swisher 19. Nonionic Surfactants: Chemical Analysis, edited by John Cross 20. Detergency: Theory and Technology, edited by W. Gale Cutler and Erik Kissa 21. Interfacial Phenomena in Apolar Media, edited by Hans-Friedrich Eicke and Ge- offrey D. Paifitt 22. Surfactant Solutions: New Methods of Investigation, edited by Raoul Zana 23. Nonionic Surfactants: Physical Chemistry, edited by Martin J. Schick 24. Microemulsion Systems, edited by Henri L. Rosano and Marc Clausse 25. Biosurfactants and Biotechnology, edited by Nairn Kosaric, W. L. Cairns, and Neil C. C. Gray 26. Surfactants in Emerging Technologies, edited by Milton J. Rosen 27. Reagents in Mineral Technology, edited by P. Somasundaran and Brij M. Moudgil 28. Surfactants in Chemical/Process Engineering, edited by Darsh T. Wasan, Martin E. Ginn, and Dinesh O. Shah 29. Thin Liquid Films, edited by I. B. Ivanov 30. Microemulsions and Related Systems: Formulation, Solvency, and Physical Prop- erties, edited by Maurice Bourrel and Robert S. Schechter 31. Crystallization and Polymorphism of Fats and Fatty Acids, edited by Nissim Garti and Kiyotaka Sato 32. Interfacial Phenomena in Coal Technology, edited by Gregory D. Botsaris and Yuli M. Glazman Copyright © Marcel Dekker 2000
- 8. 33. Surfactant-Based Separation Processes, edited by John F. Scamehorn and Jef- frey H. Harwell 34. Cationic Surfactants: Organic Chemistry, edited by James M. Richmond 35. Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske 36. Interfacial Phenomena in Petroleum Recovery, edited by Norman R. Morrow 37. Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh and Paul M. Holland 38. Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Grätzel and K. Kalyanasundaram 39. Interfacial Phenomena in Biological Systems, edited by Max Bender 40. Analysis of Surfactants, Thomas M. Schmitt 41. Light Scattering by Liquid Surfaces and Complementary Techniques, edited by Dominique Langevin 42. Polymeric Surfactants, Irja Piirma 43. Anionic Surfactants: Biochemistry, Toxicology, Dermatology. Second Edition, Re- vised and Expanded, edited by Christian Gloxhuber and Klaus Künstler 44. Organized Solutions: Surfactants in Science and Technology, edited by Stig E. Friberg and Björn Lindman 45. Defoaming: Theory and Industrial Applications, edited by P. R. Garrett 46. Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe 47. Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Do- biáš 48. Biosurfactants: Production • Properties • Applications, edited by Nairn Ko saric 49. Wettability, edited by John C. Berg 50. Fluorinated Surfactants: Synthesis • Properties • Applications, Erik Kissa 51. Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J. Pugh and Lennart Bergström 52. Technological Applications of Dispersions, edited by Robert B. McKay 53. Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J. Singer 54. Surfactants in Agrochemicals, Tharwat F. Tadros 55. Solubilization in Surfactant Aggregates, edited by Sherril D. Christian and John F. Scamehorn 56. Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache 57. Foams: Theory, Measurements, and Applications, edited by Robert K Prud’— homme and Saad A. Khan 58. The Preparation of Dispersions in Liquids, H. N. Stein 59. Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax 60. Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn M. Nace 61. Emulsions and Emulsion Stability, edited byJohan Sjöblom 62. Vesicles, edited by Morton Rosoff 63. Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K Spelt 64. Surfactants in Solution, edited by Arun K. Chattopadhyay and K L. Mittal 65. Detergents in the Environment, edited by Milan Johann Schwuger 66. Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda 67. Liquid Detergents, edited by Kuo-Yann Lai Copyright © Marcel Dekker 2000
- 9. 68. Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Mar- tin M. Rieger and Linda D. Rhein 69. Enzymes in Detergency, edited by Jan H. van Ee, Onno Misset, and Erik J. Baas 70. Structure—Performance Relationships in Surfactants, edited by Kunio Esumi and Minoru Ueno 71. Powdered Detergents, edited by Michael S. Showell 72. Nonionic Surfactants: Organic Chemistry, edited by Nico M. van Os 73. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and Ex- panded, edited by John Cross 74. Novel Surfactants: Preparation, Applications, and Biodegradability, edited by Kris- ter Holmberg 75. Biopolymers at Interfaces, edited by Martin Malmsten 76. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applica- tions, Second Edition, Revised and Expanded, edited by Hiroyuki Oh-shima and Kunio Furusawa 77. Polymer-Surfactant Systems, edited by JanC. T. Kwak 78. Surfaces of Nanoparticles and Porous Materials, edited by James A. Schwarz and Cristian I. Contescu 79. Surface Chemistry and Electrochemistry of Membranes, edited by Torben Smith S0rensen 80. Interfacial Phenomena in Chromatography, edited by Emile Pefferkorn 81. Solid-Liquid Dispersions, Bohusiav Dobiäš, Xueping Qiu, and Wolfgang von Ry- binski 82. Handbook of Detergents, editor in chief: Uri Zoller Part A: Properties, edited by Guy Broze 83. Modern Characterization Methods of Surfactant Systems, edited by Bernard P. Binks 84. Dispersions: Characterization, Testing, and Measurement, Erik Kissa 85. Interfacial Forces and Fields: Theory and Applications, edited by Jyh-Ping Hsu 86. Silicone Surfactants, edited by Randal M. Hill 87. Surface Characterization Methods: Principles, Techniques, and Applications, ed- ited by Andrew J. Milling 88. Interfacial Dynamics, edited by Nikola Kallay 89. Computational Methods in Surface and Colloid Science, edited by Magorzata Borówko ADDITIONAL VOLUMES IN PREPARATION Adsorption on Silica Surfaces, edited by Eugene Papirer Fine Particles: Synthesis, Characterization, and Mechanisms of Growth, edited by Tadao Sugimoto Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer and Harald Luders Copyright © Marcel Dekker 2000
- 10. COMPUTATIONAL METHODS IN SURFACE AND COLLOID SCIENCE edited by Małgorzata Borówko Maria Curie-Skłodowska University Lublin, Poland MARCEL DEKKER, INC. NEW YORK.hBASEL Copyright © Marcel Dekker 2000
- 11. ISBN: 0-8247-0323-5 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more infor- mation, write to Special Sales/Professional Marketing at the headquarters address above. Copyright © 2000 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the pub- lisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA Copyright © Marcel Dekker 2000
- 12. Preface Interfacial systems are frequently encountered in a large variety of phenomena in biology and industry. A few examples that come to mind are adsorption, catalysis, corrosion, flotation, osmosis, and colloidal stability. In particular, surface films are very interesting from a cognitive point of view. Surface science has a long history. For many years, natural philosophers were curious about interfacial phenomena be- cause it was quite clear that matter near surface differs in its properties from the same matter in bulk. Decades of patient analysis and laboratory experiments gave only an approximate picture of a situation at the interface, which follows from a great complexity of investigated systems. However, much of the progress in science consists of asking old questions in new, more penetrating, and more wide-ranging ways. One of the scientific advances that shaped history during the 20th century is the revolution in computer technology. It has given a strong impetus to the development of mathematical modelling of physical processes. The powerful new tools are ve- hemently accelerating the pace of interfacial research. We can easily carry out cal- culations that no one had previously imagined. Computer simulations have already had quite impressive achievements in surface science, so it seems timely to write a monograph summarizing the results. The existing books cover the simple, rather than the advanced, theo-retical ap- proaches to interfacial systems. This volume should fill this gap in the literature. It is the purpose of this volume to serve as a comprehensive reference source on theory Copyright © Marcel Dekker 2000
- 13. and simulations of various interfacial systems. Furthermore, it shows the power of statistical thermodynamics that offers a reliable framework for an explanation of in- terfacial phenomena. This book is intended primarily for scientists engaged in the- oretical physics and chemistry. It should also be a useful guide for all researchers and graduate students dealing with surface and colloid science. The book is divided into 18 chapters written by different experts on various as- pects. In many areas of contemporary science, one is confronted with the problem of theoretical descriptions of adsorption on solids. This problem is discussed in the first part of the volume. The majority of inter-facial systems may be considered as fluids in confinement. Therefore, the first chapter is devoted to the behavior of con- fined soft condensed matter. Because quantum mechanics is a paradigm for micro- scopic physics, quan-tum effects in adsorption at surfaces are considered (Chapter 2). The theory of simple and chemically reacting nonuniform fluids is discussed in Chapters 3 and 4. In Chapters 5 and 6, the current state of theory of adsorption on energetically and geometrically heterogeneous surfaces, and in random porous media, is presented. Recent molecular computer-simulation studies of water and aqueous electrolyte solutions in confined geometries are reviewed in Chapter 7. In Chapter 8, the Monte Carlo simulation of surface chemical reactions is discussed within a broad context of integrated studies combining the efforts of different dis- ciplines. Theoretical approaches to the kinetic of adsorption, desorption, and reac- tions on surfaces are reviewed in Chapter 9. Chapters 10 through 14 examine the systems containing the polymer molecules. Computer simulations are natural tools in polymer science. This volume gives an overview of polymer simulations in the dense phase and the survey of existing coarse-grained models of living polymers used in computer experiments (Chapters 10 and 11). The properties of polymer chains adsorbed on hard surfaces are dis- cussed in the framework of dynamic Monte Carlo simulations (Chapter 12). The systems involving surfactants and ordering in microemulsions are described in Chapters 13 and 14. Chapters 15 through 17 are devoted to mathematical modeling of particular sys- tems, namely colloidal suspensions, fluids in contact with semi-permeable mem- branes, and electrical double layers. Finally, Chapter 18 summarizes recent studies on crystal growth process. I hope that this book will be useful for everyone whose professional activity is connected with surface science. I would like to thank A. Hubbard for the idea of a volume on computer simula- tions in surface science and S. Sokołowski for fruitful discussions and encourage- ment. I thank the authors who contributed the various chapters. Finally, R. Zagórski is acknowledged for his constant assistance. Malgorzata Borówko Copyright © Marcel Dekker 2000
- 14. Contents Preface iii Contributors vii 1. Structure and Phase Behavior of Confined Soft Condensed Matter 1 Martin Schoen 2. Quantum Effects in Adsorption at Surfaces 77 Peter Nielaba 3. Integral Equations in the Theory of Simple Fluids 135 Douglas Henderson, Stefan Sokołowski, and Malgorzata Borówko 4. Nonuniform Associating Fluids 167 Malgorzata Borówko, Stefan Sokołowski, and Orest Pizio 5. Computer Simulations and Theory of Adsorption on Energetically and Geometrically Heterogeneous Surfaces 245 Andrzej Patrykiejew and Malgorzata Borówko 6. Adsorption in Random Porous Media 293 Orest Pizio 7. Water and Solutions at Interfaces: Computer Simulations on the Molecular Level 347 Eckhard Spohr Copyright © Marcel Dekker 2000
- 15. 8. Surface Chemical Reactions 387 Ezequiel Vicente Albano 9. Theoretical Approaches to the Kinetics of Adsorption, Desorption, and Reactions at Surfaces 439 H. J. Kreuzer and Stephen H. Payne 10. Computer Simulations of Dense Polymers 481 Kurt Kremer and Florian Müller-Plathe 11. Computer Simulations of Living Polymers and Giant Micelles 509 Audrey Milchev 12. Conformational and Dynamic Properties of Polymer Chains Adsorbed on Hard Surfaces 555 Audrey Milchev 13. Systems Involving Surfactants 631 Friederike Schmid 14. Ordering in Microemulsions 685 Robert Holyst, Alina Ciach, and Wojciech T. Góźdź 15. Simulations of Systems with Colloidal Particles 745 Matthias Schmidt 16. Fluids in Contact with Semi-permeable Membranes 775 Sohail Murad and Jack G. Powles 17. Double Layer Theory: A New Point of View 799 Janusz Stafiej and Jean Badiali 18. Crystal Growth and Solidification 851 Heiner Müller-Krumbhaar and Yukio Saito Index 933 Copyright © Marcel Dekker 2000
- 16. Contributors Ezequiel VicenteAlbano, Ph.D. Instituto de Investigaciones Fisicoquímcas Teóricas y Aplicadas, Universidad National de La Plata, La Plata, Argentina Jean Badiali, Ph.D. Structure et Réactivité des Systémes Interfaciaux, Uni- versité P. et M. Curie, Paris, France Magorzata Borówko, Ph.D. Department for the Modelling of Physico- Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland Alina Ciach, Ph.D. Institute of Physical Chemistry, Polish Academy of Sci- ences, Warsaw, Poland Wojciech T. Góźdź, Ph.D. Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland Douglas Henderson, Prof. Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah Robert Hołyst, Ph.D. Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland Kurt Kremer, Ph.D. Max-Planck-Institut für Polymerforschung, Mainz, Germany Copyright © Marcel Dekker 2000
- 17. H. J. Kreuzer, Dr.rer.nat., F.R.S.C. Department of Physics, Dalhousie Uni- versity, Halifax, Nova Scotia, Canada Andrey Milchev, Ph.D., Dr.Sci.Habil. Institute for Physical Chemistry, Bulgarian Academy of Sciences, Sofia, Bulgaria Florian Müller-Plathe, Ph.D. Max-Planck-Institut für Polymerfor-schung, Mainz, Germany Heiner Müller-Krumbhaar, Prof. Dr. Institut für Festkörperforschung, Forschungszentrum Jülich, Jülich GMBH, Germany Sohail Murad, Ph.D. Department of Chemical Engineering, University of Illinois at Chicago, Chicago, Illinois Peter Nielaba, Prof. Dr. Department of Physics, University of Konstanz, Konstanz, Germany Andrzej Patrykiejew, Ph.D. Department for the Modelling of Physico- Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland Stephen H. Payne Department of Physics, Dalhousie University, Halifax, Nova Scotia, Canada Orest Pizio, Ph.D. Instituto de Quimica de la Universidad Nacional Au- tonoma de México, Coyoacán, México Jack G. Powles, Ph.D., D.es.Sc. Physics Laboratory, University of Kent, Canterbury, Kent, England Yukio Saito, Ph.D. Department of Physics, Keio University, Yokohama, Japan Friederike Schmid, Dr.rer.nat. Max-Planck-Institut für Polymerfor- schung, Mainz, Germany Matthias Schmidt, Dr.rer.nat. Institut für Theoretische Physik II, Hein- rich-Heine-Universität Düsseldorf, Düsseldorf, Germany Copyright © Marcel Dekker 2000
- 18. Martin Schoen, Dr.rer.nat. Fachbereich Physik - Theoretische Physik, Ber- gische Universität Wuppertal, Wuppertal, Germany Stefan Sokołowski, Ph.D. Department for the Modelling of Physico-Chem- ical Processes, Maria Curie-Skłodowska University, Lublin, Poland Eckhard Spohr, Ph.D. Department of Theoretical Chemistry, University of Ulm, Ulm, Germany Janusz Stafiej, Ph.D. Department of Electrode Processes, Institute of Phys- ical Chemistry, Polish Academy of Sciences, Warsaw, Poland Copyright © Marcel Dekker 2000
- 19. 1 Structure and Phase Behavior of Confined Soft Condensed Matter MARTIN SCHOEN Fachbereich Physik—Theoretische Physik, Bergische Universität Wuppertal, Wuppertal, Germany I. Introduction 2 II. Equilibrium Theory of Confined Phases 3 A. Thermodynamics 3 B. Symmetry and homogeneity of thermodynamic potentials 11 C. Statistical physics 16 III. Monte Carlo Simulations 21 A. Stochastic processes 22 B. Implementation of stress-strain ensembles for open and closed systems 24 C. The Taylor-expansion algorithm for “simple” fluids 26 D. Orientationally biased creation of molecules 28 IV. Microscopic Structure 29 A. Planar substrates 29 B. The transverse structure of confined fluids 41 C. Nonplanar substrates 45 V. Phase Transitions 49 A. Shear-induced phase transitions in confined fluids 49 B. Liquid-gas equilibria in confined systems 56 References 66 1 Copyright © Marcel Dekker 2000
- 20. I. INTRODUCTION In many areas of contemporary science and technology one is confronted with the problem of miniaturizing parts of the system of interest in order to control processes on very short length and time scales [1]. For example, to study the ki- netics of certain chemical reactions, reactants have to be mixed at a sufficiently high speed. By miniaturizing a continuous-flow mixer, Knight et al. have re- cently shown that nanoliters can be mixed within microseconds, thus permitting one to study fast reaction kinetics on time scales unattainable with conventional mixing technology [2]. The importance of designing and constructing micro- scopic machines gave rise to a new field in applied science and engineering known as “microfabrication technology” or “microengineering” [3]. A central problem in the operation of such small mechanical machines is posed by fric- tion between movable machine parts and wear. Lubricants consisting of, say, or- ganic fluids can be employed to reduce these ultimately destructive phenomena. Their functioning depends to a large extent on the nature of the interaction be- tween the fluid and the solid substrate it lubricates [4,5]. In the case of micro- machines the lubricant may become a thin confined film of a thickness of only one or two molecular layers. The impact of such severe confinement is perhaps best illustrated by the dramatic increase of the shear viscosity in a hexadecane film of a thickness of two molecular layers, which may exceed the bulk shear viscosity by four orders of magnitude [6]. Understanding the effect of confinement on the phase behavior and materi- als properties of fluids is therefore timely and important from both a fundamen- tal scientific and an applied technological perspective. This is particularly so because the fabrication and characterization of confining substrates with pre- scribed chemical or geometrical structures on a nano-to micrometer length scale can nowadays be accomplished in the laboratory with high precision and by a variety of techniques. For example, by means of various lithographic methods [3,7] or wet chemical etching [8] the surfaces of solid substrates can be en- dowed with well-defined nanoscopic lateral structures. In yet another method the substrate is chemically patterned by elastomer stamps and, in certain cases, subsequent chemical etching [9-12]. The development of a host of scanning probe devices such as the atomic force microscope (AFM) [13-17] and the surface forces apparatus (SFA) [18- 22], on the other hand, enables experimentalists to study almost routinely the behavior of soft condensed matter confined by such substrates to spaces of mo- lecular dimensions. However, under conditions of severe confinement a direct study of the relation between material properties and the microscopic structure of confined phases still remains an experimental challenge. 2 Schoen Copyright © Marcel Dekker 2000
- 21. Computer simulations, on the other hand, are ideally suited to address this particular question from a theoretical perspective. Generally speaking, computer simulations permit one to pursue the motion of atoms or molecules in space and time. Since the only significant assumption concerns the choice of interaction potentials, the behavior of condensed matter can be investigated essentially in a first-principles fashion. At each step of the simulation one has instantaneous ac- cess to coordinates and momenta of all molecules. Thus, by applying the laws of statistical physics, one can determine the thermomechanical properties of condensed matter as well as its underlying microscopic structure. In many cases the insight gained by computer simulations was and is unattainable by any other theoretical means. Perhaps the most prominent and earliest example in this re- gard concerns the prediction of solid-fluid phase transitions in hard-sphere flu- ids at high packing fraction [23]. However, because of limitations of computer time and memory required to treat dense many-particle systems, computer simulations are usually restricted to microscopic length and time scales (with hard-sphere fluids, which may be viewed as a model for colloidal suspensions [24] (this volume, chapter by M Schmidt), and Brownian dynamics [25] as two prominent exceptions). This lim- itation can be particularly troublesome in investigations of, say, critical phe- nomena where the correlation length may easily exceed the microscopic size of the simulation cell. In confinement, on the other hand, a phase may be physi- cally bound to microscopically small volumes in one or more dimensions by the presence of solid substrates so that computer simulations almost become a natu- ral theoretical tool of investigation by which experimental methods can be com- plemented. It is then not surprising that the study of confined phases by simulational techniques is still flourishing [26], illustrated here for one particu- lar aspect, namely the relation between microscopic structure and phase transi- tions in confined fluids. In Sec. II an introduction to equilibrium theory of confined phases will be given. Sec. III s devoted to formal and technical aspects of computer simulations. In Sec. IV the microscopic structure of confined phases will be analyzed for a number of different systems. The chapter con- cludes in Sec. V with a description of phase transitions that are unique to phases in confined geometry. II. EQUILIBRIUM THEORY OF CONFINED PHASES A. Thermodynamics 1. Experiments with the Surface Forces Apparatus The force exerted by a thin fluid film on a solid substrate can be measured with nearly molecular precision in the SFA [27]. In the SFA a thin film is confined 3 Structure and Phase Behavior of Soft Condensed Matter Copyright © Marcel Dekker 2000
- 22. between the surfaces of two cylinders arranged such that their axes are at right angles [27]. In an alternative setup the fluid is confined between the surface of a macroscopic sphere and a planar substrate [28]. However, crossed-cylinder and sphere-plane configurations can be mapped onto each other by differential-geo- metrical arguments [29]. The surface of each macroscopic object is covered by a thin mica sheet with a silver backing, which permits one to measure the sepa- ration h between the surfaces by optical interferometry [27]. The radii are macroscopic so that the surfaces may be taken as parallel on a molecular length scale around the point of minimum distance. In addition, they are locally planar, since mica can be prepared with atomic smoothness over molecularly large areas. This setup is immersed in a bulk reservoir of the same fluid of which the film consists. Thus, at thermodynamic equilibrium temperature T and chemical potential µ are equal in both subsystems (i.e., film and bulk reservoir). By ap- plying an external force in the direction normal to both substrate surfaces, the thickness of the film can be altered either by expelling molecules from it or by imbibing them from the reservoir until thermodynamic equilibrium is reestab- lished, that is, until the force exerted by the film on the surfaces equals the ap- plied normal force. Plotting this force per radius R, F/R, as a function of h yields a damped oscillatory curve in many cases (see, for instance, Fig. 1 in Ref. [30]). In another mode of operation of the SFA a confined fluid can be exposed to a shear strain by attaching a movable stage to the upper substrate (i.e., wall) via a spring characterized by its spring constant k [6,31,32] and moving this stage at some constant velocity in, say, the x direction parallel to the film-wall interface. Experimentally it is observed that the upper wall first “sticks” to the film, as it were, because the upper wall remains stationary. From the known spring con- stant and the measured elongation of the spring, the shear stress sustained by the film can be determined. Beyond a critical shear strain (i.e., at the so-called “yield point” corresponding to the maximum shear stress sustained by the film) the shear stress declines abruptly and the upper wall “slips” across the surface of the film. If the stage moves at a sufficiently low speed the walls eventually come to rest again until the critical shear stress is once again attained so that the stick-slip cycle repeats itself periodically. This stick-slip cycle, observed for all types of film compounds ranging from long-chain (e.g., hexadecane) to spheroidal [e.g., octamethylcyclotetra-siloxane (OMCTS)] hydrocarbons [21], has been attributed by Gee et al. [30] to the for- mation of solid-like films that pin the walls together (region of sticking) and must be made to flow plastically in order for the walls to slip. This suggests that the structure of the walls induces the formation of a solid film when the walls are properly registered and that this film “melts” when the walls are moved out of the correct registry. As was first demonstrated in Ref. 33, such solid films 4 Schoen Copyright © Marcel Dekker 2000
- 23. may, in fact, form in “simple” fluids between commensurate walls on account of a template effect imposed on the film by the discrete (i.e., atomically struc- tured) walls. However, noting that the stick-slip phenomenon is general, in that it is observed in every liquid investigated, and that the yield stress may exhibit hysteresis, Granick [21] has argued that mere confinement may so slow me- chanical relaxation of the film that flow must be activated on a time scale com- parable with that of the experiment. This more general mechanism does not necessarily involve solid films which can be formed only if the (solid-like) structure of the film and that of the walls possess a minimum geometrical com- patibility. 2. The Fluid Lamella For a theoretical analysis of SFA experiments it is prudent to start from a some- what oversimplified model in which a fluid is confined by two parallel sub- strates in the z direction (see Fig. 1). To eliminate edge effects, the substrates are assumed to extend to infinity in the x and y directions. The system in the thermodynamic sense is taken to be a lamella of the fluid bounded by the sub- strate surfaces and by segments of the (imaginary) planes x = 0, x = sx, y = 0, and y = sy. Since the lamella is only a virtual construct it is convenient to asso- ciate with it the computational cell in later practical applications (see Secs. IV, 5 Structure and Phase Behavior of Soft Condensed Matter FIG. 1 Schematic of two atomically structured, parallel surface planes (from Ref. 134). Copyright © Marcel Dekker 2000
- 24. V). It is assumed that the lower substrate is stationary in the laboratory coordi- nate frame, whose origin is at 0, and that the substrates are identical and rigid. The crystallographic structure of the substrate is described by a rectangular unit cell having transverse dimensions lx x ly. In general, each substrate consists of a large number of planes of atoms parallel with the x-y plane. The plane at the film-substrate interface is called the surface plane. It is taken to be contained in the x-y plane. The distance between the surface planes is sz. To specify the transverse alignment of the substrates, registry parameters αx and αy are intro- duced. Coordinates of a given atom (2) in the upper surface plane (z = sz) are related to its counterpart (1) in the lower surface plane (z = 0) by 6 Schoen Thus the extensive variables characterizing the lamellar system are entropy S, number of fluid molecules N, sx, sy, sz, αxlx, and αyly. Gibbs’s fundamental relation governing an infinitesimal, reversible transfor- mation can be written where the mechanical work can be expressed as The primes denote restricted summations over Cartesian components (α, β = x,y,z), dsα is a displacement in the α direction, Aα is the area of the α-directed face of the lamella, and Tαβ is the average of the β-component of the stress ap- plied to Aα. Note that if the force exerted by the lamella on Aα points outward, Tαβ 0. Thus, dWmech is the mechanical work done by the system on the sur- roundings. Terms involving diagonal and off-diagonal elements of the stress tensor T in Eq. (3) respectively represent the work of compressing and shearing the lamella. Note that because the substrates are rigid they cannot be com- pressed or sheared. This is the reason for the absence of the four off-diagonal contributions involving Txz, Tyz, Txy, and Tyx. To introduce area A/ Az as an independent variable, the transformation Copyright © Marcel Dekker 2000
- 25. is introduced. In terms of these new variables Eq. (2) becomes 7 Structure and Phase Behavior of Soft Condensed Matter where the interfacial tensions and are defined by Note that the definition of R is arbitrary. However, the present choice seems simplest and has a transparent physical interpretation. The work done by the system in an infinitesimal reversible transformation at constant S, N, A, sz, αxlx, and αyly is given by because dsy = -sysx -1 dsx. It is then clear that the fourth term in Eq. (5) is the net work done by the lamella as its shape (R = sx/sy) is changed at fixed area. To recast the thermodynamic description in terms of independent variables that can be controlled in actual laboratory experiments (i.e., T, µ, and the set of strains or their conjugate stresses), it is sensible to introduce certain auxiliary thermodynamic potentials via Legendre transformations. This chapter is prima- rily concerned with where the grand potential is given by and F is the free energy. The exact differential of the grand potential follows as where Eqs. (5) and (10) have also been employed. Other relevant potentials can be obtained by suitable Legendre transformations of F or Θ with respect to, say, Tzz, Tzx, or Tzy (see Sec. VA1). Conditions for thermodynamic equilibrium of the lamella can be derived by considering the lamella plus its environment as an isolated supersystem. As- suming the entropy of the supersystem to be fixed, one knows that the internal Copyright © Marcel Dekker 2000
- 26. energy must be minimum in a state of thermodynamic equilibrium. In mathe- matical terms, an infinitesimal virtual transformation that would take the system from this state must satisfy 8 Schoen where δU is given in Eqs. (2), (3) and δü by and the tilde refers to environmental variables. Viewing the environment as vir- tual pistons, displacements of the boundary between them and the lamella satisfy the equation = —δsα. Moreover, because the supersystem is mate- rially closed, = —δN. From these two conditions and Eqs. (12)-(14), the equilibrium conditions are deduced. Now suppose the lamella is subject to thermal, mechanical, and chemical reservoirs that maintain temperature, normal stress, and chemical po- tential fixed at the values , and µ Assume also that the “complementary” strains A, R, αxlx, and αyly are kept fixed. Then one has, from Eqs. (12) and (14)] Because of Eqs. (9), (10), and (15) this is equivalent to δ 0; that is, when the lamella is at equilibrium at fixed T, µ, A, R, Tzz, αxlx, and αyly, is mini- mum. 3. Derjaguin’s Approximation To make contact with the SFA experiment one has to realize that the confining surfaces are only locally parallel. Because of the macroscopic curvature of the substrate surfaces, Tzz becomes a local quantity which varies with the vertical distance sz = sz(x,y) between the substrate surfaces (see Fig. 2). Since the sphere-plane arrangement (see Sec. II Al) is immersed in bulk fluid at pressure Copyright © Marcel Dekker 2000
- 27. Pbulk the total force exerted on the sphere by the film in the z direction can be expressed as 9 Structure and Phase Behavior of Soft Condensed Matter FIG. 2 Side view of film confined between a sphere of macroscopic radius R and a pla- nar substrate surface. The shortest distance between two points located on the surface of the sphere and of the substrate, respectively, is denoted by h (from Ref. 48). which depends on the (bulk) thermodynamic state specified by T and µ. This solvation, or depletion, force plays a vital role in the context of binary mixtures of colloidal particles of different sizes [34] (this volume, chapter by M. Schmidt). Because of their practical importance for colloid-polymer mixtures [35], depletion forces in binary hard-sphere mixtures have recently received a lot of attention and have been studied by a range of methods, including integral equations based upon sophisticated hypernetted chain closure approximations [36-41], density functional theory [42,43], virial expansion [44], and computer simulation [45-47]. To evaluate the integral in Eq. (17), it is convenient to transform from carte- sian to cylindrical coordinates (see Fig. 2) to obtain Copyright © Marcel Dekker 2000
- 28. where the arguments µ, and T have been dropped to simplify notation and the far right side follows from (see Fig. 2) [48]. In Eq. (18) 10 Schoen which follows from Eq. (11) (fixed R, αxlx, αyly) and a similar expression for the bulk reser!voir where V is the bulk volume. In Eq. (19) the excess grand potential Θex := Θ — Θbulk is also introduced. Assuming V = Asz, the far right side of Eq. (19) ob- tains because the bulk phase is isotropic. Furthermore, note that f(sz(p)) van- ishes in the limit sz because of [49] so that f(sz) may be interpreted as the excess normal pressure exerted on the sphere by the fluid. In Eq. (19), F(h) still depends on the curvature of the sub- strate surfaces through R. Experimentally, one is normally concerned with measuring F(h)/R rather than the solvation force itself [27], because for macro- scopically curved substrate surfaces this ratio is independent of R. This can be rationalized by realizing that Tzz(sz) + Pbulk vanishes on a microscopic length scale much smaller than R. The upper integration limit in Eq. (19) may then be taken to infinity to give because Θex vanishes in the limit sz according to the definition in Eq. (19). In Eq. (22) we introduce θex (h) as the excess grand potential per unit area of a fluid confined between two planar substrate surfaces separated by a dis- tance h. The far right side of Eq. (22) is known as the Derjaguin approximation (see Eq. (6) in Ref. 29). As pointed out recently by Götzelmann et al. [43], the Derjaguin approximation is exact in the limit of a macroscopic sphere (i.e., if R ), which is the only case of interest here. A rigorous proof can be found in the appendix of Ref. 50. A similar “Derjaguin approximation” for shear forces exerted on curved substrates has recently been proposed by Klein and Ku- macheva [51]. Copyright © Marcel Dekker 2000
- 29. Eq. (22) is a key expression because it links the quantity F(h)/R that can be determined directly in SFA experiments to the local stress Tzz available from computer simulations (see Sec. IV A1). It is also interesting that differentiating Eq. (22) yields 11 Structure and Phase Behavior of Soft Condensed Matter Eq. (23) is particularly useful because it relates a derivative of experimentally accessible data directly to the stress exerted locally on the macro-scopically curved substrates at the point (0,0, sz = h) (see Fig. 2, Secs. IV A2, IV A3). B. Symmetry and Homogeneity of Thermodynamic Potentials An important issue in the thermodynamics of confined fluids concerns their symmetry which is lower than that of a corresponding homogeneous bulk phase because of the presence of the substrate and its inherent atomic structure [52]. The substrate may also be nonplanar (see Sec. IV C) or may consist of more than one chemical species so that it is heterogeneous on a nanoscopic length scale (see Sec. VB 3). The reduced symmetry of the confined phase led us to re- place the usual compressional-work term —Pbulk V in the bulk analogue of Eq. (2) by individual stresses and strains. The appearance of shear contributions also reflects the reduced symmetry of confined phases. 1. Atomically Smooth Substrates The simplest situation is one in which a planar substrate lacks any crystal-lo- graphic structure. Then the confined fluid is homogeneous and isotropic in transverse (x,y) directions. All off-diagonal elements of T vanish, Txx = Tyy = , and Eq. (5) simplifies to By symmetry, ⱖ f(A) at fixed T, µ, and sz. Hence, under these conditions one can formally integrate Eq. (24) to obtain taking the zero of 傼 to correspond to zero interfacial area. From Eqs. (6), (10), and (25) one gets Copyright © Marcel Dekker 2000
- 30. which is the analogue of the bulk relation Θ = —Pbulk V. From Eq. (9) it is straightforward to realize that 12 Schoen is a nontrivial quantity (because in general ⬆ Tzz), whereas its bulk ana- logue vanishes trivially because = Tzz = Pbulk on account of the higher symmetry of bulk phases reflected by Eq. (21) [52]. From Eqs. (10), (24), and (25), the Gibbs-Duhem equation follows immediately. 2. The Two-dimensional Ideal Gas in an External Potential While the smooth substrate considered in the preceding section is sufficiently realistic for many applications, the crystallographic structure of the substrate needs to be taken into account for more realistic models. The essential compli- cations due to lack of transverse symmetry can be delineated by the following two-dimensional structured-wall model: an ideal gas confined in a periodic square-well potential field (see Fig. 3). The two-dimensional lamella remains rectangular with variable dimensions sx and sy and is therefore not subject to shear stresses. The boundaries of the lamella coinciding with the x and y axes are anchored. From Eqs. (2) and (10) one has FIG. 3 Schematic of the two-dimensional square-well potential u(x) of depth e, width d, and period l (from Ref. 48). Copyright © Marcel Dekker 2000
- 31. for the free energy of the ideal gas under these premises. From standard text- book considerations one also knows the statistical-physical expression [53] 13 Structure and Phase Behavior of Soft Condensed Matter where β = 1/kBT (kB is Boltzmann’s constant). The canonical partition function can be written more explicitly as — qN /N! where the atomic partition function is given by where 1 is the single-atom configurational integral, and A is the thermal de Broglie wavelength. The far right side of Eq. (31) follows immediately because the potential energy of a molecule in the present two-dimensional ideal gas does not depend on its y coordinate (see Fig. 3). The configuration integral depends on sx in a piecewise fashion. For sx in the nth period of the potential, that is for (n — 1)l sx ⭐ nl (n 僆 ), one ob- tains where and Copyright © Marcel Dekker 2000
- 32. From Eqs. (29)—(31) one has 14 Schoen With the help of Eq. (32) the first two expressions can be written explicitly as and where Pbulk = β-1 exp(β/µ)/A2 is the pressure of the two-dimensional ideal bulk gas in thermodynamic equilibrium with the confined fluid. Fig. 4 displays plots of -Txx and -Tyy versus sx. From these it is clear that both stresses are functions of the size of the lamella. The most significant con- sequence of this is that, unlike Eq. (24), Eq. (29) cannot be integrated at fixed T, µ, and sy in general to yield an expression analogous to Eq. (25) without addi- tional equations of state, that is Txx = Txx(sx), Tyy = Tyy(sx). In other words, a Gibbs-Duhem equation corresponding to Eq. (28) does not obtain for the pres- ent two-dimensional structured-wall model. The same conclusion holds for more realistic three-dimensional structured-wall models [54]. The lack of a Gibbs-Duhem equation for general thermodynamic transformations is a direct consequence of the additional reduction of the confined fluid’s symmetry caused by the discrete atomic structure of the substrate (see Sec. II B 1). 3. Coarse-grained Thermodynamics While a Gibbs-Duhem equation does not exist for general transformations dsα d , a specialized (i.e., “coarse-grained”) Gibbs-Duhem equation Copyright © Marcel Dekker 2000
- 33. may be derived for cases in which the transverse dimensions of the lamella are changed only discretely, that is, in such a way that the surface plane at the fluid- wall interface of the lamella always comprises an integer number n of unit cells in both x and y directions so that 15 Structure and Phase Behavior of Soft Condensed Matter FIG. 4 Plots of -Txx (—) and - Tyy (—) versus sx for the ideal gas confined to the two- dimensional periodic square-well potential depicted in Fig. 3. Distance is measured in units of the period l; stress in units of the pressure of the bulk ideal gas at the given T and µ (d/l = 0.20) (from Ref. 54). Thus, the exchange of work between the lamella and its surroundings is effected on a coarse-grained length scale defined in units of {lx,ly}. Eliminating sx and sy in Eq. (11) in favor of n gives where work contributions due to shear and deformations of the shape of the lamella are neglected for simplicity. In Eq. (37), α := lxly is the unit-cell area and is the “mean” stress applied transversely on the n ⫻ n lamella. If T, µ, and sz are fixed, Txx and Tyy are periodic in sx and sy, having periods lx and ly, re- spectively. Thus, for the restricted class of transformations Copyright © Marcel Dekker 2000
- 34. n nⴕ = n ⬆ m (n, m integer), T is constant provided n and nⴕ are sufficiently large for intensive properties to be independent of the (microscopic) size of the lamella. Under these conditions Eq. (37) can be integrated to get 16 Schoen Eq. (39) may be differentiated subsequently to give Equating the expressions for dΘ given in Eqs. (37) and (40) and rearranging terms yields the coarse-grained Gibbs-Duhem equation which permits one to define the (transverse) isothermal compressibility k where A = n2 a as detailed in Ref. 55. Note that a similar definition is prevented for general transformations dsa dsⴕα according to the discussion in Sec. II B2. C. Statistical Physics 1. Stress-Strain Ensembles for Open and Closed Systems To achieve a description of confined soft condensed matter at the molecular level one has to resort to the principles of statistical physics. To make contact with, say, SFA experiments it is convenient to introduce statistical physical en- sembles depending explicitly on a suitable set of stresses and strains. For sim- plicity, the lamella is treated quantum mechanically, following the procedure originated by Schrödinger [56] and extended by Hill [53] and McQuarrie [57], so that its energy states are formally discrete. The energy eigenvalues Ej(N, A,R,sZ,αxlx,αyly) are implicit functions of the number of fluid molecules, ex- tent and shape of the lamella, and the registry of the substrates, which control the external field acting on the fluid molecules. Index j signifies the collection of quantum numbers necessary to determine the eigenstate uniquely. The en- semble comprises an astronomical number N of systems each in the same macroscopic state, which, as an example, is taken to be specified by the set {T, µ, A, R, αxlx,αyly} of ensemble parameters. Since the ensemble is isolated, it Copyright © Marcel Dekker 2000
- 35. satisfies the following constraints: 17 Structure and Phase Behavior of Soft Condensed Matter where njNsz is the number of systems having N molecules between substrates separated by sz and occupying eigenstate j. It is assumed that the isolated en- semble has fixed total energy , fixed total number of molecules , and fixed total volume . The total number of ways of realizing a given distribution n = {njNsz} over the allowed “superstates” characterized by triplets (j,N,sz) is W(n) = !/%j %N %sz n jNs! Since the number of systems is extremely large, the most probable distribution, denoted by n*, overwhelms all others. It is found by maximizing W(n) subject to the constraints [see Eqs. (43)]. The result for the probability of a system’s occupying superstate (j, N, sz) is where the partition function and the set of Lagrangian multipliers {λ1,λ2, λ3} are determined through equivalence of thermodynamic and statistical expressions as follows. The statistical expression for the internal energy is simply = !jNsz PjNsz Ej, from which its exact differential follows as Ej can be obtained from Eq. (44) so that dEj can be replaced. One Copyright © Marcel Dekker 2000
- 36. eventually obtains 18 Schoen At the molecular level one may interpret (uEj/uA)N,sz,R,αxlxαylyas the interfa- cial tension of the system in superstate (j,N,sz). Similar meaning can be attached to the other partial derivatives of Ej appearing in Eq. (47). Invoking also the principle of conservation of probabilityjNsdPjNSz = 0), Eq. (47) can be recast as Following the lengthy argument by Hill [53], one identifies Then a comparison of the microscopic Eq. (48) with its macroscopic counter- part Eq. (5) allows one to identify the Lagrangian multipliers as Therefore, from Eq. (44) one obtains Upon substitution into Eq. (49), this yields Copyright © Marcel Dekker 2000
- 37. where the far right side is obtained by comparing the statistical expression in Eq. (52) with the thermodynamic Eqs. (9) and (10). By exactly the same ap- proach one can also derive statistical-physical expressions for other mixed stress-strain ensembles [58,59]. Finally, from Eq. (45) one has for the partition function 19 Structure and Phase Behavior of Soft Condensed Matter where :=j ex P(βEj) is the canonical-ensemble partition function and = exp(—βΘ) is its counterpart in the grand canonical ensemble. Since this chapter is exclusively concerned with classical systems, is replaced by its classical analogue for the special case of spherically-symmetric molecules where is the config- uration integral, Λ := (h2 β/2πm)1/2 is the thermal de Broglie wavelength (h is Planck’s constant and m the molecular mass). The limiting expression class can be derived from the quantum mechanical within the framework of the Kirkwood and Wigner theory [53]. In the classical limit one has to replace the quantum mechanical PjNSz by the analogous probability density distribution where U(rN ) is the configurational energy of the system and Xclass is the classi- cal counterpart of X obtained by replacing in Eq. (53) by class (see also this volume, chapter by Nielaba). 2. Correlation Functions Since we shall also be interested in analyzing the confined fluid’s microscopic structure it is worthwhile to introduce some useful structural correlation func- tions at this point. The simplest of these is related to the instantaneous number density operator Copyright © Marcel Dekker 2000
- 38. where δ is the Dirac delta function. The mean value of π is given by 20 Schoen where P[1] (ri = r) is the probability of the center of mass of molecule i being at r regardless of the positions of the other molecules (and regardless of orienta- tion, see Sec. IV A 3). Since the molecules are equivalent, P[1] is independent of i and the summation on i in Eq. (57) can be performed explicitly to yield In general, π[1] (r) is a function of the vector position of the point of observation r. However, if one is concerned mainly with the inhomogeneity of the confined fluid in the normal (z) direction, the average over the interfacial area is ade- quate. Averaging yields to which we shall henceforth refer simply as the local density. The translational microscopic structure of the confined fluid is partially re- vealed by correlations in the number density operator, given by where the “self-term” gives no new information beyond the mean density. Again invoking the equivalence of fluid molecules, we recognize the cross-term in Eq. (60) as the pair distribution function Copyright © Marcel Dekker 2000
- 39. which is related to the mean local density through the pair correlation function by 21 Structure and Phase Behavior of Soft Condensed Matter In general, g[2] is a six-dimensional function of the position of reference (x,y,z) and observed (xⴕ,yⴕ,zⴕ) molecules. However, to be consistent with the approxi- mation for the local density [see Eq. (59)], we take g[2] to be a function only of the normal coordinate (z) of the reference molecule and the cylindrical coordi- nates π12 and z12 of the observed molecule (2) relative to the reference mole- cule (1), where the distance vector between the two r12 = π12 + z12êz and êz is the unit vector in the z direction (see Sec. IV B). III. MONTE CARLO SIMULATIONS A key problem in the equilibrium statistical-physical description of condensed matter concerns the computation of macroscopic properties Omacro like, for ex- ample, internal energy, pressure, or magnetization in terms of an ensemble aver- age O of a suitably defined microscopic representation 0(rN ) (see Sec. IV A 1 and VA 1 for relevant examples). To perform the ensemble average one has to realize that configurations rN := {r1, r2,.., rN} generally differ energetically so that a certain probability of occurrence is associated with each configuration. Therefore, to compute the correct value O , 0(r) needs to be multiplied by the relevant probability density function f(rN ;X), where X is a set of thermodynamic state variables (for example, T, µ, and a combination of stresses and strains). Analytically, the computation of ensemble averages along this route is a for- midable task, even if microscopically small representations of the system of in- terest are considered, because f(rN ; X) is generally a very complicated function of the spatial arrangement of the N molecules. However, with the advent of large-scale computers some forty years ago the key problem in statistical physics became tractable, at least numerically, by means of computer simula- tions. In a computer simulation the evolution of a microscopically small sample of the macroscopic system is determined by computing trajectories of each mol- ecule for a microscopic period of observation. An advantage of this approach is the treatment of the microscopic sample in essentially a first-principles fashion; the only significant assumption concerns the choice of an interaction potential [25]. Because of the power of modern supercomputers which can literally han- dle hundreds of millions of floating point operations per second, computer sim- ulations are nowadays viewed as “a third branch complementary to the… two Copyright © Marcel Dekker 2000
- 40. traditional approaches” [60]: theory and experiment. There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are gener- ated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the “dynam- ics” of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular empha- sis on mixed stress-strain ensembles. If one wishes to compute O numerically by means of MC one immediately realizes that this requires the a priori unknown function f(rN ; X) according to which the random walk in configuration space has to be carried out. However, if the random walk is carried out in a biased way as a Markov process it turns out that only the ratio f(rN m+1; X)/f(rN m; X) is relevant to generate a new configura- tion (m + 1) from a given old one (m). The subsequent discussion will show that f (rN m+1; X)/f(rN m; X) is computationally accessible. Because this scheme gen- erates configurations with the correct probability of occurrence, Omacro can be computed via the simple expression 22 Schoen where the prime is attached as a reminder that the summation is restricted to configurations generated according to their importance (importance sampling) and Mmax should be large enou gh [usually, Mmax (106 -109 ) is sufficient, depending on the particular physical situation and quantity of interest]. How- ever, before turning to practical aspects of MC, a brief introduction to Markov processes seems worthwhile because it rarely appears in the literature. A. Stochastic Processes Let y(t) be a random process, that is a process incompletely determined at any given time t. The random process can be described by a set of probability distri- butions {Pn} where, for example, P2 (y1 t1,y2t2) dy1 dy2 is the probability of Copyright © Marcel Dekker 2000
- 41. finding y1 in the interval [y1,y1 + dy1] at t = t1 and in the interval [y2, y2+ dy2] at another time t = t2. Thus the set {Pn} forms a hierarchy of probability distri- butions describing y(t) in greater detail the larger n is. The simplest random process is completely stochastic so that one may write, for example, P2(y1t1,y2t2) = P1(y1t1)P1(y2t2). However, here we are con- cerned with a slightly more complex process known as the Markov process, characterized by 23 Structure and Phase Behavior of Soft Condensed Matter where K1 (yxtx y2t2) is the conditional probability of finding y in the interval [y2,y2 + dy2] at t = t2 provided y =y1 at an earlier time t = tx (tx t2). Some important properties are the following: 1. Normalization, that is K1(y1t1y2t2) dy2 = 1. 2. P1(y1t1)K1(y1t1y2t2)dy1 = P1(y2t2) 3 Perhaps most importantly, a Markov process has a “one-step memory”; that is, to find y in the interval [yn,yn + dyn] at t = tn depends only on the realiza- tion y = yn-1 at the immediately preceding time t = tn - 1 but is independent of all earlier realizations {ymtm}, 1 ⱕ m ⱕ n - 2. Mathematically, this can be cast as Kn-1(y1t1,…,yn-1 tn-1 |yntn) K1 (yn-1tn-1|yntn). 4 Stationarity, that is P1(y1 t1) = P1 (y1) and P2(y1t1,y2t2)= P2(y1,y2;t2-t1). Consider now and assume that y(t) is a Markov process. Then Because of Eqs. (64) and (66), Eq. (65) can be rewritten as Alternatively, because of property (4), this last expression can be cast as Copyright © Marcel Dekker 2000
- 42. where and . Equation (68) is known as the Chap- man-Kolmogoroff equation. Suppose a small characteristic time interval πC exists such that yn-1 changes without strongly affecting K1(yn-2yn;t) so that the latter may be expanded in a Taylor series as 24 Schoen From Eqs. (68) and (69) one gets To proceed it is convenient to define the transition probability per time interval πC as which satisfies With Eqs. (71), (72), and one may multiply both sides of Eq. (70) by P1(yn-2tn-2) and integrate over dyn-2 to obtain (see properties 1, 2, and 4) For stationary situations and Eq. (73) is then satisfied by where is the probability for the transition n - 1 $ n. Equation (74) reflects microscopic reversibility and is a special formulation of the principle of de- tailed balance. B. Implementation of Stress-Strain Ensembles for Open and Closed Systems Consider now, as an illustration, a confined fluid in material and thermal contact with a bulk reservoir and under fixed normal stress Tzz. For simplicity we as- sume the substrates to be in fixed registry αx = αy = 0 and the fluid to consist of Copyright © Marcel Dekker 2000
- 43. “simple” molecules having only (three) translational degrees of freedom. Under these premises one has (see Sec. IIC 1] [66] where ,…, N so that the inte- 25 Structure and Phase Behavior of Soft Condensed Matter gration is carried out over the unit-cube volume . The summation over sz [see Eq. (53)] has been replaced by an integral, and the dimensionless quantity B is defined by The MC method can be implemented by a modification of the classic Me- tropolis scheme [25,67]. The Markov chain is generated by a three-step se- quence. The first step is identical to the classic Metropolis algorithm: a randomly selected molecule i is displaced within a small cube of side length 2δr centered on its original position where 1 = (1,1,1) and ξ is a vector whose three components are pseudorandom numbers distributed uniformly on the interval [0,1]. During the MC run δr is adjusted so that 40-60% of the attempted displacements are accepted. With the identification one obtains from Eqs. (55) and (74) because Nm = Nm+1 and szm+1 = sz,m, where 9U is the change in configurational energy associated with the process An efficient way to compute ∆U is detailed below in Sec. III.C. In the second step it is decided with equal probability whether to remove (AN = -1) a randomly chosen molecule or to create (∆N = +1) a new one at a randomly chosen point in the system (see also Sec. III D). From Eqs. (55) and (74) the transition probabilities for addition (“+”) and subtraction (“-“) are given by where Copyright © Marcel Dekker 2000
- 44. Since only one molecule is added to (or removed from) the system, Um is sim- ply the interaction of the added (or removed) molecule with the remaining ones. If one attempts to add a new molecule, N is the number of molecules after addi- tion, otherwise it is the number of molecules prior to removal. If a cutoff for the interaction potential is employed, long-range corrections to Um must be taken into account because of the density change of m 1/Asz. Analytic expressions for these corrections can be found in the appendix of Ref. 33. In the third and final step the substrate separation is changed according to 26 Schoen and the coordinates of fluid molecules are scaled via zm+1 = zmsz,m+l/sZ,m. Because N is held constant the transition probability associated with this step is where ∆sz := sz,m+1 - sz,m and the same comments concerning corrections to ∆U apply as in step 2. On each pass through the three-step sequence the number of attempts in steps 1, 2, and 3 is chosen to be N, N, and 1, respectively, in order to realize a comparable degree of events in each of the steps. Because the third step moves all N molecules at once, and the first two affect only one molecule at a time, the sought balance is roughly achieved. The algorithm described here can easily be amended by additional steps if, for example, one is interested in situations in which the shear stress(es) is (are) also among the controlled param- eters so that αx (and αy) may vary too [58,59]. Applying the analysis of Wood [68] to each step of the algorithm separately, one can verify that the resulting transition probabilities indeed comply with the requirements of a Markov process as stated in Eq. (74). C. The Taylor-expansion Algorithm for “Simple” Fluids According to Allen and Tildesley, the standard recipe to evaluate ∆U in step one of the algorithm described in Sec. III B involves “computing the energy of atom i with all the other atoms before and after the move (see p. 159 of Ref. 25, ital- ics by the present author) as far as “simple” fluids are concerned. The evalua- tion of ∆U can be made more efficient in this case by realizing that for Copyright © Marcel Dekker 2000
- 45. short-range interactions U can be split into three contributions U = U1 + U2 + U3 corresponding to three different spatial zones, where U1 is the configura- tional energy between atom i and N1 neighboring molecules located in a pri- mary zone immediately surrounding i. Similarly, U2 refers to interactions between i and N2 molecules in a secondary zone adjacent to the primary zone and, last but not least, U3 refers to interactions between i and the remaining molecules in an outermost tertiary zone whose upper limit is identical with the potential cutoff by which the computational burden is reduced already in con- ventional implementations. Savings of computer time depend on the sizes of the three zones (i.e., the values of N1, N2, N3) and different degrees of sophistica- tion with which the three terms are treated. It turns out that a sphere of radius r1 centered on ri can be associated with the primary zone. The secondary zone can be a spherical shell of thickness ∆r = r2 - r1 (bulk fluid) or a cylindrical shell of the same thickness but infinite height (confined fluid, slit geometry). If r2 is sufficiently large one may assume ∆U3 = U3(ri,m+1) - U3(ri,m) b 0 because [see Eq. (77)] is small compared with typical distances corresponding to ter- tiary-zone interactions. Thus, N3 interactions are entirely neglected during the course of the simulation. For the secondary zone one assumes that ∆U2 = U2(ri,m+1) — U2(ri,m) is not entirely negligible but small enough to be ap- 27 Structure and Phase Behavior of Soft Condensed Matter proximated by a Taylor expansion truncated after the first nonvanishing term, where F2 is the total force exerted on i in the initial configuration m by the N2 atoms in the secondary zone. For the primary zone no simplifying assumptions can be made because U1 will strongly depend on δr. Thus, on the basis of these assumptions, ∆U in Eq. (78) can be written explicitly as It is clear that Eq. (85) is numerically reliable provided δr is sufficiently small. However, a detailed investigation in Ref. 69 reveals that δr can be as large as some ten percent of the diameter of a fluid molecule. Likewise, r1 should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. III B. These are achieved be- cause, for pairwise interactions, only N1 + N2 contributions need to be com- puted here before i is moved (U1 and F2), and only N1 contributions need to be Copyright © Marcel Dekker 2000
- 46. evaluated after i is displaced by δr (U1) where some efficiency is inevitably lost because the computation of forces is numerically more demanding than the computation of energies. A much larger number of 2(N1 + N2 + N3) such terms must be evaluated conventionally. D. Orientationally Biased Creation of Molecules If fluid molecules have rotational degrees of freedom the algorithm outlined in Sec. III B must be modified in various respects. First, in addition to random dis- placements in step 1, the molecule is rotated randomly by a small angle incre- ment ∆ ⑀ (—max, max) around an axis chosen at random from the three axes of the Cartesian coordinate system. A rotation attempt is again accepted with the probability given in Eq. (78) where now ∆U refers to the change in configurational energy associated with the rotation attempt. During the course of a simulation max is adjusted to preserve an overall acceptance ratio of 40- 60%. For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system pos- sesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g( ), depends on the unit vector Û parallel to the molecule’s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director which is a measure of the average ori- entation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g( ) must be normalized to one [73,74]. In other words, an addition is at- tempted with a preferred orientation of the molecule determined by the macro- scopic director of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given 28 Schoen by [73] whereas for a removal attempt, where rⴞ is given in Eq. (80). However, B has to be re- Copyright © Marcel Dekker 2000
- 47. placed by Bⴕ denned analogously as (linear molecules only) where I is the moment of inertia. Quantities Û+ and Û- are the microscopic di- rectors of the film molecule to be added to or removed from the film. Biased ad- dition is indispensable if, for example, the confined fluid is a liquid crystal in thermodynamic equilibrium with a nematic bulk phase (see Sec. IV A 3). In this case g( ) is identified as the orientational distribution function of film mole- cules, which is computed as a histogram averaged over all configurations pre- ceding the actual one; if the thermodynamic state of the bulk liquid crystal is isotropic, g( ) = 1 is a suitable choice. One realizes from Eqs. (86), (87) that in this case one recovers Eq. (79). IV. MICROSCOPIC STRUCTURE A. Planar Substrates 1. Molecular Expressions for the Normal Stress Within the framework of Monte Carlo simulations, the relation between meas- urable quantities and the microscopic structure of confined phases can now be examined. An example of such a measurable quantity is the solvation force F(h)/2π R (see Sec. IIA 1). From a theoretical perspective and according to the discussion in Sec. IIA 3 its investigation requires the stress Tzz(sz) exerted nor- mally by a confined fluid on planar substrates [see Eqs. (19) and (22)]. Using 29 Structure and Phase Behavior of Soft Condensed Matter Eqs. (11) and (53) one can derive a molecular expression for Tzz from for “simple” fluids, where the shorthand notation is introduced to represent the weighted sum over N. Depending on how the partial derivative of is worked out, two mathematically different but physically equivalent expressions for Tzz obtain. The first of these is obtained by following the procedure of Hill [53] and Copyright © Marcel Dekker 2000
- 48. transforming variables according to zt $ i- = zisz-1 (i = 1,…,N) to obtain 30 Schoen assuming that where uff is the fluid-fluid interaction potential (corresponding, for example, to the Lennard-Jones (12,6) potential) and the fluid-substrate interaction depends on the distance between a fluid molecule and a substrate atom. Scaling in Eq. (90) affects UFF through the argument and UFS because of for a fluid molecule located at ri, and a wall atom at [see Eqs. (1)]. According to the product rule, the differentiation in Eq. (90) yields three terms grouped as Tzz = TZZ,FF + TZZ,FS where and In Eqs. (93) and (94) Wzz is an element of Clausius’s virial matrix [53] and ul is the derivative of u with respect to its argument. Therefore Eqs. (93) and (94) are termed “virial” expressions henceforth. A different expression for Tzz can be obtained directly from Eq. (89) without Copyright © Marcel Dekker 2000
- 49. transforming coordinates. It is then convenient to recast the 31 Structure and Phase Behavior of Soft Condensed Matter configuration integral as [75] Applying Leibniz’s rule for the differentiation of a parameter integral [76], it follows from Eqs. (89) and (95) that because U becomes infinitely large if z1 = sz on account of the divergence of Ufs at this point. In Eq. (96), g2 is defined analogously to g1 in Eq. (95) and the above argument may be repeated N -1 times, to obtain finally where is the total force exerted by the film on substrate k. Equation (97) is therfore termed the “force” expression henceforth. The far right side of Eq. (97) is a statement of mechanical stability of the confined fluid. Virial and force expressions provide a useful test of internal consistency of a computer simulation and should agree within a few percent (see, for example, Table II in [77]). 2. Stratification To illustrate the relationship between the microscopic structure and experimen- tally accessible information, we compute pseudo-experimental solvation-force curves F(h)/R [see Eq. (22)] as they would be determined in SFA experiments from computer-simulation data for Tzz [see Eqs. (93), (94), (97)]. Numerical values indicated by an asterisk are given in the customary dimensionless (i.e., reduced) units (see [33,75,78] for definitions in various model systems). Results are correlated with the microscopic structure of a thin film confined between plane parallel substrates separated by a distance sz = h. Here the focus is specif- ically on a “simple” fluid in which the interaction between a pair of film mole- cules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A confined “simple” fluid serves as a suitable model for approximately spherical Copyright © Marcel Dekker 2000
- 50. OMCTS molecules confined between mica surfaces, which is perhaps the most thoroughly investigated system in SFA experiments [27,30]. Because OMCTS is chemically inert and electrically neutral, the influence of charges on the mica surfaces can safely be ignored. Plots of f(sz) and F(h)/R versus sz and h, respectively, are shown in Fig. 5. The oscillatory decay of both quantities is a direct consequence of the oscilla- tory dependence of Tzz on sz, which has also been investigated by integral equa- tion approaches of varying sophistication [36,85-89]. As can be seen in Fig. 5, zeros of f(sz) correspond to successive extrema of F(h)/R because of Eq. (22). In actual SFA experiments the only portions of the F(h)/R curve generally ac- cessible are those where d[F(h)/R]/dh 0. Regions where d[F(h)/R]/dh 0 are inaccessible, because θeX (H) would decrease upon compression of the film [see Eq. (23)]. However, structural changes accompanying the variation of F(h)/R in the accessible regimes are rather obscure, as can be inferred from Fig. 6 where plots of the local density are presented. On account of Eq. (23) accessible por- tions of the pseudo-experimental curve can be related to the local stress at the point (0,0, sz = h) of minimum distance between the surfaces of the macro- scopic sphere and the planar substrate (see Fig. 2). By correlating the local stress Tzz(h) with the film’s local structure at (0,0, A) via π[1] (z;sz = h), one can establish a direct correspondence between pseudo-experimental data (i.e., 32 Schoen FIG. 5 The excess pressure f(sz) (, dashed line) and the solvation force per radius F(h)/R (full line) as functions of sz and h, respectively, for a confined fluid composed of “simple” molecules (from Ref. 48) Copyright © Marcel Dekker 2000
- 51. F(h)/R) and local microscopic structure of the film. Plots of a sequence of local densities π[1] (z; sz = h) in Fig. 6 over the range 2.60 ⱕ h* ⱕ 4.00 illustrate this correlation. In an actual SFA experiment 2.59 ⱕ h*ⱕ 3.06 and 3.53 ⱕ h*ⱕ 4.00 are accessible portions of the solvation-force 33 Structure and Phase Behavior of Soft Condensed Matter FIG. 6 The local density π[1] (z;sz) as a function of z/sz for a “simple” fluid confined by planar substrates, (a) sz*:=2.60 (䊏), sz*=2.80 (*), sz*=3.00 (+); (b) sz*=3.20 (䊏), sz*=3.40 (*), sz*=3.55 (+); (c) sz*=3.80 (䊏), sz*=4.00 (*) (from Ref. 48). Copyright © Marcel Dekker 2000
- 52. curve whereas 3.06 h* 3.53 demarcates the inaccessible range because here d[F(h)]/dh 0. Plots in Figs. 6(a) and 6(c) show that in the experimentally ac- cessible regions the film consists locally of two and three strata, respectively. For h* = 2.60 the film is locally compressed (F(h) 0) whereas it is stretched for h* = 3.00 (F(h) 0). Under compression the film appears to be less strati- fied, as is reflected by smaller heights of less well separated peaks of π[1] (z; sz = h) compared with the other two curves in Fig. 6(a). For h* = 2.80, F(h) 0 and Tzz(sz = h) has almost assumed a minimum value, indicating that for this particular value of h film molecules are locally accommodated most satisfacto- rily between the surfaces of the macroscopic sphere and the planar substrate. It is therefore not surprising that peaks in π[1] (z;sz = h) are taller for h* = 2.80 compared with the two neighboring values of h [see Fig. 6(a)]. In the next accessible region (3.53 ⱖ h* ⱖ 4.00) the film consists of three molecular strata for which the most pronounced structure is observed for h* 3.80, corresponding to a point at which F(h)/R nearly vanishes [see Fig. 6(c)]. As before [see Fig. 6(a)] this is reflected by the peak height in the contact strata (i.e., the strata closest to the substrate) whereas inner portions of the film remain largely unaffected. Plots of π[1] (z;sz = h) in the inaccessible regime in Fig. 6(b) show that here the film undergoes a local reorganization characterized by the vanishing (appearance) of a whole stratum. The reorganization is gradual, as one can see in the plot of π[1] (z; sz = h) for h* = 3.4, where two shoulders ap- 34 Schoen FIG. 6 Continued. Copyright © Marcel Dekker 2000
- 53. pear at z/sz ±0.1. Stratification, as illustrated by the plots in Fig. 6, is due to constraints on the packing of molecules next to the wall and is therefore largely determined by the repulsive part of the intermolecular potential [55]. It is observed even in the ab- sence of intermolecular attractions, such as in the case of a hard-sphere fluid confined between planar hard walls [42,90-92]. For this system Evans et al. [93] demonstrated that, as a consequence of the damped oscillatory character of the local density in the vicinity of the walls, Tzz is a damped oscillatory function of sz, if sz is of the order of a few molecular diameters, which is confirmed by Fig. 5. For one-dimensional confined hard-rod [49,94,95] and Tonks-Takahashi flu- ids [49,96,97] the close relationship between stratification and the oscillatory decay of Tzz(sz) has been demonstrated analytically. On the basis of a density- functional approach Iwamatsu [98] has recently analyzed the solvation force in various experimental systems. In the context of dielectric media the analogue of the solvation force between planar walls [f(sz) in the current notation] is known as the Casimir force. It arises because the walls modify the spectrum of electro- magnetic fluctuations between them such that the vacuum energy of the electro- magnetic field becomes size- and shape-dependent [99]. 3. Orientational Effects In the other model system the film consists of (soft) ellipsoidal Gay-Berne mol- ecules [78,100-102]. Depending on the thermodynamic state, bulk phases con- sisting of Gay-Berne molecules can be either isotropic or nematic [103]. The Gay-Berne fluid is therefore a suitable model for liquid crystals, which are cur- rently intensively studied in SFA experiments [104-110] because of their impor- tance in such diverse fields as, say, display technology [111] and lubrication [112]. Here the only case considered is that of a confined Gay-Berne fluid in thermodynamic equilibrium with a nematic bulk phase [102,113]. Parameters of the fluid-substrate intermolecular potential parameters are chosen so that a homeotropic anchoring of fluid molecules to the substrate surface is favored (i.e., fluid molecules in the vicinity of the substrate surface are preferentially or- dered normal to the surface). The “nematic” Gay-Berne film between homeotropically anchoring substrates may be viewed as a rough model for a film of (dimers of) 4´-n-octyl-4-cyanobiphenyl (8CB) molecules confined be- tween hydrophobic substrate surfaces consisting of mica coated with dihexade- cyldimethyl ammonium acetate (DHDAA) monolayers [107]. As in the case of a “simple”-fluid film, the normal component of the stress tensor is a damped oscillatory function of substrate separation (see Fig. 7). Over 35 Structure and Phase Behavior of Soft Condensed Matter Copyright © Marcel Dekker 2000
- 54. the range 4.0 ⱖ sz ⱖ 16.75,f(sz) exhibits four maxima separated by a distance ∆sz b 3.2, which is slightly smaller than the large diameter of a film molecule [102]. In analogy with results for confined “simple” fluids (see Sec. IV A 2), it is plausible to associate oscillations in f(sz) with the formation of molecular strata parallel with the walls. Fig. 7 also shows that f(sz) oscillates around zero in the limit of large sz [see Eqs. (19), (21)] as it should [49]. However, f(sz) also exhibits shoulders at characteristic values of sz separated by the same distance ∆s*z 3.2 as the maxima. Portions of f(sz) between neighboring minima (i.e., s*z 6.80, 6.80 ⱖ s*z ⱖ 10.00, 10.00 ⱖ s*z ⱖ 13.20, and 13.20 ⱖ s*z ⱖ 16.40) are remarkably similar. In order to correlate the microscopic structure of the confined film with features of f(sz), it is con- 36 Schoen FIG. 7 Same as Fig. 5, but for a “nematic” Gay-Berne film confined between homeotropically anchoring substrates (from Ref. 48). venient to label these portions as “decrease,” “increase,” and “shoulder” zones and to introduce the density-alignment distribution defined by [78,101] where (z,u2 z;sz)dzdu2 z is the probability of finding a film molecule at position z with orientation uz, which is the cosine of the angle θ between the micro- scopic director Û and the z axis. The argument u2 z of the probability density f(z, u2 z; sz) reflects the nonpolarity of Gay-Berne molecules (i.e., the equivalence of Û and —Û). By definition, u2 z = 1 if Û is orthogonal to the plane of a wall and Copyright © Marcel Dekker 2000
- 55. Other documents randomly have different content
- 56. In which no eggs or milk are required: important in the Crimea or the field. Put on the fire, in a moderate-sized saucepan, 12 pints of water; when boiling, add to it 1lb. of rice or 16 tablespoonsful, 4oz. of brown sugar or 4 tablespoonsful, 1 large teaspoonful of salt, and the rind of a lemon thinly pealed; boil gently for half an hour, then strain all the water from the rice, keeping it as dry as possible. The rice-water is then ready for drinking, either warm or cold. The juice of a lemon may be introduced, which will make it more palatable and refreshing. THE PUDDING. Add to the rice 3oz. of sugar, 4 tablespoonsful of flour, half a teaspoonful of pounded cinnamon; stir it on the fire carefully for five or ten minutes; put it in a tin or a pie-dish, and bake. By boiling the rice a quarter of an hour longer, it will be very good to eat without baking. Cinnamon may be omitted. No. 23A.—Batter Pudding. Break two fresh eggs in a basin, beat them well, add one tablespoonful and a half of flour, which beat up with your eggs with a fork until no lumps remain; add a gill of milk, a teaspoonful of salt, butter a teacup or a basin, pour in your mixture, put some water in a stew-pan, enough to immerge half way up the cup or basin in water; when boiling put in your cup or basin and boil twenty minutes, or till your pudding is well set; pass a knife to loosen it, turn out on a plate, pour pounded sugar and a pat of fresh butter over, and serve. A little lemon, cinnamon, or a drop of any essence may be introduced. A little light melted butter, sherry, and sugar, may be poured over. If required more delicate, add a little less flour. It may be served plain. No. 24.—Bread and Butter Pudding. Butter a tart-dish well, and sprinkle some currants all round it, then lay in a few slices of bread and butter; boil one pint of milk, pour it on two eggs
- 57. well whipped, and then on the bread and butter; bake it in a hot oven for half an hour. Currants may be omitted. No. 25.—Bread Pudding. Boil one pint of milk, with a piece of cinnamon and lemon-peel; pour it on two ounces of bread-crumbs; then add two eggs, half an ounce of currants, and a little sugar: steam it in a buttered mould for one hour. No. 26.—Custard Pudding. Boil one pint of milk, with a small piece of lemon-peel and half a bay- leaf, for three minutes; then pour these on to three eggs, mix it with one ounce of sugar well together, and pour it into a buttered mould: steam it twenty-five minutes in a stew-pan with some water (see No. 115), turn out on a plate and serve. No. 27.—Rich Rice Pudding. Put in ½lb. of rice in a stew-pan, washed, 3 pints of milk, 1 pint of water, 3oz. of sugar, 1 lemon peel, 1oz. of fresh butter; boil gently half an hour, or until the rice is tender; add 4 eggs, well beaten, mix well, and bake quickly for half an hour, and serve: it may be steamed if preferred. No. 28—Stewed Macaroni. Put in a stewpan 2 quarts of water, half a tablespoonful of salt, 2oz. of butter; set on the fire; when boiling, add 1lb. of macaroni, broken up rather small; when boiled very soft, throw off the water; mix well into the macaroni a tablespoonful of flour, add enough milk to make it of the consistency of thin melted butter; boil gently twenty minutes; add in a tablespoonful of either brown or white sugar, or honey, and serve. A little cinnamon, nutmeg, lemon-peel, or orange-flower water may be introduced to impart a flavour; stir quick. A gill of milk or cream may now be thrown in three minutes before serving. Nothing can be more light and nutritious than macaroni done this way. If no milk, use water.
- 58. No. 29.—Macaroni Pudding. Put 2 pints of water to boil, add to it 2oz. of macaroni, broken in small pieces; boil till tender, drain off the water and add half a tablespoonful of flour, 2oz. of white sugar, a quarter of a pint of milk, and boil together for ten minutes; beat an egg up, pour it to the other ingredients, a nut of butter; mix well and bake, or steam. It can be served plain, and may be flavoured with either cinnamon, lemon, or other essences, as orange flower-water, vanilla, c. No. 30.—Sago Pudding. Put in a pan 4oz. of sago, 2oz. of sugar, half a lemon peel or a little cinnamon, a small pat of fresh butter, if handy, half a pint of milk; boil for a few minutes, or until rather thick, stirring all the while; beat up 2 eggs and mix quickly with the same; it is then ready for either baking or steaming, or may be served plain. No. 31.—Tapioca Pudding. Put in a pan 2oz. of tapioca, 1½ pint of milk, 1oz. of white or brown sugar, a little salt, set on the fire, boil gently for fifteen minutes, or until the tapioca is tender, stirring now and then to prevent its sticking to the bottom, or burning; then add two eggs well beaten; steam or bake, and serve. It will take about twenty minutes steaming, or a quarter of an hour baking slightly. Flavour with either lemon, cinnamon, or any other essence. No. 32.—Boiled Rice semi-curried, for the premonitory symptoms of Diarrhœa. Put 1 quart of water in a pot or saucepan; when boiling, wash ½lb. of rice and throw it into the water; boil fast for ten minutes; drain your rice in a colander, put it back in the saucepan, which you have slightly greased with butter; let it swell slowly near the fire, or in a slow oven till tender; each grain will then be light and well separated. Add to the above a small tablespoonful of aromatic sauce, called “Soyer’s Relish or Sultana Sauce,” with a quarter of a teaspoonful of curry
- 59. powder; mix together with a fork lightly, and serve. This quantity will be sufficient for two or three people, according to the prescriptions of the attending physician. No. 33.—Figs and Apple Beverage. Have 2 quarts of water boiling, into which throw 6 dry figs previously opened, and 2 apples, cut into six or eight slices each; let the whole boil together twenty minutes, then pour them into a basin to cool; pass through a sieve; drain the figs, which will be good to eat with a little sugar or jam. No. 34.—Stewed French Plums. Put 12 large or 18 small-size French plums, soak them for half an hour, put in a stew-pan with a spoonful of brown sugar, a gill of water, a little cinnamon, and some thin rind of lemon; let them stew gently twenty minutes, then put them in a basin till cold with a little of the juice. A small glass of either port, sherry, or claret is a very good addition. The syrup is excellent. No. 35.—French Herb Broth. This is a very favourite beverage in France, as well with people in health as with invalids, especially in spring, when the herbs are young and green. Put a quart of water to boil, having previously prepared about 40 leaves of sorrel, a cabbage lettuce, and 10 sprigs of chervil, the whole well washed; when the water is boiling, throw in the herbs, with the addition of a teaspoonful of salt, and ½oz. of fresh butter; cover the saucepan close, and let simmer a few minutes, then strain it through, a sieve or colander. This is to be drunk cold, especially in the spring of the year, after the change from winter. I generally drink about a quart per day for a week at that time; but if for sick people it must be made less strong of herbs, and taken a little warm. To prove that it is wholesome, we have only to refer to the instinct which teaches dogs to eat grass at that season of the year. I do not pretend to say that it would suit persons in every malady, because the doctors are to decide
- 60. upon the food and beverage of their patients, and study its changes as well as change their medicines; but I repeat that this is most useful and refreshing for the blood. No. 36.—Browning for Soups, etc. Put ½lb. of moist sugar into an iron pan and melt it over a moderate fire till quite black, stirring it continually, which will take about twenty-five minutes: it must colour by degrees, as too sudden a heat will make it bitter; then add 2 quarts of water, and in ten minutes the sugar will be dissolved. You may then bottle it for use. It will keep good for a month, and will always be found very useful. No. 37.—Toast-and-Water. Cut a piece of crusty bread, about a ¼lb. in weight, place it upon a toasting-fork, and hold it about six inches from the fire; turn it often, and keep moving it gently until of a light-yellow colour, then place it nearer the fire, and when of a good brown chocolate colour, put it in a jug and pour over 3 pints of boiling water; cover the jug until cold, then strain it into a clean jug, and it is ready for use. Never leave the toast in it, for in summer it would cause fermentation in a short time. I would almost venture to say that such toast-and-water as I have described, though so very simple, is the only way toast-water should be made, and that it would keep good a considerable time in bottles. Baked Apple Toast-and-Water.—A piece of apple, slowly toasted till it gets quite black, and added to the above, makes a very nice and refreshing drink for invalids. Apple Rice Water.—Half a pound of rice, boiled in the above until in pulp, passed through a colander, and drunk when cold. All kinds of fruit may be done the same way. Figs and French plums are excellent; also raisins. A little ginger, if approved of, may be used. Apple Barley Water.—A quarter of a pound of pearl barley instead of toast added to the above, and boil for one hour, is also a very nice drink.
- 61. Citronade.—Put a gallon of water on to boil, cut up one pound of apples, each one into quarters, two lemons in thin slices, put them in the water, and boil them until they can be pulped, pass the liquor through a colander, boil it up again with half a pound of brown sugar, skim, and bottle for use, taking care not to cork the bottle, and keep it in a cool place. For Spring Drink.—Rhubarb, in the same quantities, and done in the same way as apples, adding more sugar, is very cooling. Also green gooseberries. For Summer Drink.—One pound of red currants, bruised with some raspberry, half a pound of sugar added to a gallon of cold water, well stirred, and allowed to settle. The juice of a lemon. Mulberry.—The same, adding a little lemon-peel. A little cream of tartar or citric acid added to these renders them more cooling in summer and spring. Plain Lemonade.—Cut in very thin slices three lemons, put them in a basin, add half a pound of sugar, either white or brown; bruise all together, add a gallon of water, and stir well. It is then ready. French Plum Water.—Boil 3 pints of water; add in 6 or 8 dried plums previously split, 2 or 3 slices of lemon, a spoonful of honey or sugar; boil half an hour and serve. For Fig, Date, and Raisin Water, proceed as above, adding the juice of half a lemon to any of the above. If for fig water, use 6 figs. Any quantity of the above fruits may be used with advantage in rice, barley, or arrowroot water. Effervescent Beverages. Raspberry Water.—Put 2 tablespoonfuls of vinegar into a large glass, pour in half a pint of water; mix well. Pine-apple Syrup.—Three tablespoonfuls to a pint. Currant Syrup.—Proceed the same. Syrup of Orgeat.—The same. Orange-Flower Water.—The same, adding an ounce of lump sugar, is a most soothing drink, and is to be procured at Verrey’s, in Regent Street, or Kuntz’s, opposite Verrey’s. Put two tablespoonfuls to a glass of water. It is
- 62. also extremely good with either Soda, Seltzer, or Vichy Water, the last of which is to be obtained at the depôt, Margaret-street, Cavendish-square.
- 63. ARMY RECEIPTS. SOYER’S FIELD AND BARRACK COOKERY FOR THE ARMY. N.B.—These receipts are also applicable for barracks, in camp, or while on the march, by the use of Soyer’s New Field Stove, now adopted by the military authorities. These receipts answer equally as well for the navy. Each stove will consume not more than from 12 to 15lbs. of fuel, and allowing 20 stoves to a regiment, the consumption would be 300lbs. per thousand men. The allowance per man is, I believe, 3½lbs. each, which gives a total of 3500lbs. per thousand men. The economy of fuel would consequently be 3200 lbs. per regiment daily. Coal will burn with the same advantage. Salt beef, pork, Irish stew, stewed beef, tea, coffee, cocoa, c., can be prepared in these stoves, and with the same economy. They can also be fitted with an apparatus for baking, roasting, and steaming.
- 64. No. 1.—Soyer’s Receipt to Cook Salt Meat for Fifty Men. Head-Quarters, Crimea, 12th May, 1856. 1. Put 50 lbs. of meat in the boiler. 2. Fill with water, and let soak all night. 3. Next morning wash the meat well. 4. Fill with fresh water, and boil gently three hours, and serve. Skim off the fat, which, when cold, is an excellent substitute for butter. For salt pork proceed as above, or boil half beef and half pork—the pieces of beef may be smaller than the pork, requiring a little longer time doing. Dumplings, No. 21, may be added to either pork or beef in proportion; and when pork is properly soaked, the liquor will make a very good soup. The large yellow peas as used by the navy, may be introduced; it is important to have them, as they are a great improvement. When properly soaked, French haricot beans and lentils may also be used to advantage. By the addition of 5 pounds of split peas, half a pound of brown sugar, 2 tablespoonfuls of pepper, 10 onions; simmer gently till in pulp, remove the fat and serve; broken biscuit may be introduced. This will make an excellent mess. No. 1A.—How to soak and plain-boil the Rations of Salt Beef and Pork, on Land or at Sea. To each pound of meat allow about a pint of water. Do not have the pieces above 3 or 4 lbs. in weight. Let it soak for 7 or 8 hours, or all night if possible. Wash each piece well with your hand in order to extract as much salt as possible. It is then ready for cooking. If less time be allowed, cut the pieces smaller and proceed the same, or parboil the meat for 20 minutes in the above quantity of water, which throw off and add fresh. Meat may be soaked in sea water, but by all means boiled in fresh when possible. I should advise, at sea, to have a perforated iron box made, large enough to contain half a ton or more of meat, which box will ascend and descend by pulleys; have also a frame made on which the box might rest when lowered overboard, the meat being placed outside the ship on a level with the water, the night before using; the water beating against the meat through
- 65. the perforations will extract all the salt. Meat may be soaked in sea water, but by all means washed. No. 2.—Soyer’s Army Soup for Fifty Men. Head-Quarters, 12th May, 1856. 1. Put in the boiler 60 pints, 7½ gallons, or 5½ camp kettles of water. 2. Add to it 50lbs. of meat, either beef or mutton. 3. The rations of preserved or fresh vegetables. 4. Ten small tablespoonfuls of salt. 5. Simmer three hours, and serve. P.S.—When rice is issued put it in when boiling. Three pounds will be sufficient. About eight pounds of fresh vegetables. Or four squares from a cake of preserved ditto. A tablespoonful of pepper, if handy. Skim off the fat, which, when cold, is an excellent substitute for butter. No. 2A.—Salt Pork with Mashed Peas, for One Hundred Men. Put in two stoves 50lbs. of pork each, divide 24lbs. in four pudding- cloths, rather loosely tied; putting to boil at the same time as your pork, let all boil gently till done, say about two hours; take out the pudding and peas, put all meat in one caldron, remove the liquor from the other pan, turning back the peas in it, add two teaspoonfuls of pepper, a pound of the fat, and with the wooden spatula smash the peas, and serve both. The addition of about half a pound of flour and two quarts of liquor, boiled ten minutes, makes a great improvement. Six sliced onions, fried and added to it, makes it very delicate. No. 3.—Stewed Salt Beef and Pork. For a Company of One Hundred Men, or a Regiment of One Thousand Men.
- 66. Head-Quarters, 12th June, 1855. Put in a boiler, of well-soaked beef 30lbs., cut in pieces of a quarter of a pound each. 20lbs. of pork. 1½lb. of sugar. 8lbs. of onions, sliced, 25 quarts of water. 4lbs. of rice. Simmer gently for three hours, skim the fat off the top, and serve. Note.—How to soak the meat for the above mess.—Put 50lbs. of meat in each boiler, having filled them with water, and let soak all night; and prior to using it, wash it and squeeze with your hands, to extract the salt. In case the meat is still too salt, boil it for twenty minutes, throw away the water, and put fresh to your stew. By closely following the above receipt you will have an excellent dish. No. 4.—Soyer’s Food for One Hundred Men, using Two Stoves. Head-Quarters, Crimea. Cut or chop 50lbs. of fresh beef in pieces of about a ¼lb. each; put in the boiler, with 10 tablespoonfuls of salt, two ditto of pepper, four ditto of sugar, onions 7lbs. cut in slices: light the fire now, and then stir the meat with a spatula, let it stew from 20 to 30 minutes, or till it forms a thick gravy, then add a pound and a half of flour; mix well together, put in the boiler 18 quarts of water, stir well for a minute or two, regulate the stove to a moderate heat, and let simmer for about two hours. Mutton, pork, or veal, can be stewed in a similar manner, but will take half an hour less cooking. Note.—A pound of rice may be added with great advantage, ditto plain dumplings, ditto potatoes, as well as mixed vegetables. For a regiment of 1000 men use 20 stoves. No. 5.—Plain Irish Stew for Fifty Men. Cut 50lbs. of mutton into pieces of a quarter of a pound each, put them in the pan, add 8lbs. of large onions, 12lbs. of whole potatoes, 8 tablespoonfuls of salt, 3 tablespoonfuls of pepper; cover all with water,
- 67. giving about half a pint to each pound; then light the fire; one hour and a half of gentle ebullition will make a most excellent stew; mash some of the potatoes to thicken the gravy, and serve. Fresh beef, veal, or pork, will also make a good stew. Beef takes two hours doing. Dumplings may be added half an hour before done. No. 6.—To Cook for a Regiment of a Thousand Men. Head-Quarters, Crimea, 20th June, 1855. Place twenty stoves in a row, in the open air or under cover. Put 30 quarts of water in each boiler, 50lbs. of ration meat, 4 squares from a cake of dried vegetables—or, if fresh mixed vegetables are issued, 12lbs. weight—10 small tablespoonfuls of salt, 1 ditto of pepper, light the fire, simmer gently from two hours to two hours and a half, skim the fat from the top, and serve. It will require only four cooks per regiment, the provision and water being carried to the kitchen by fatigue-parties; the kitchen being central, instead of the kitchen going to each company, each company sends two men to the kitchen with a pole to carry the meat. No. 7.—Salt Pork and Puddings with Cabbage and Potatoes. Put 25lbs. of salt pork in each boiler, with the other 50lbs. from which you have extracted the large bones, cut in dice, and made into puddings; when on the boil, put five puddings in each, boil rather fast for two hours. You have peeled 12lbs. of potatoes and put in a net in each caldron; put also 2 winter cabbages in nets, three-quarters of an hour before your pudding is done; divide the pork, pudding, and cabbage in proportion, or let fifty of the men have pudding that day and meat the other; remove the fat, and serve. The liquor will make very good soup by adding peas or rice, as No. 1a. For the pudding-paste put one quarter of a pound of dripping, or beef or mutton suet, to every pound of flour you use; roll your paste for each half an inch thick, put a pudding-cloth in a basin, flour round, lay in your paste, add your meat in proportion; season with pepper and a minced onion; close your pudding in a cloth, and boil.
- 68. This receipt is more applicable to barrack and public institutions than a camp. Fresh meat of any kind may be done the same, and boiled with either salt pork or beef. No. 8.—Turkish Pilaff for One Hundred Men. Put in the caldron 2lbs. of fat, which you have saved from salt pork, add to it 4lbs. of peeled and sliced onions; let them fry in the fat for about ten minutes; add in then 12lbs. of rice, cover the rice over with water, the rice being submerged two inches, add to it 7 tablespoonfuls of salt, and 1 of pepper; let simmer gently for about an hour, stirring it with a spatula occasionally to prevent it burning, but when commencing to boil, a very little fire ought to be kept under. Each grain ought to be swollen to the full size of rice, and separate. In the other stove put fat and onions the same quantity with the same seasoning; cut the flesh of the mutton, veal, pork, or beef from the bone, cut in dice of about 2oz. each, put in the pan with the fat and onions, set it going with a very sharp fire, having put in 2 quarts of water: steam gently, stirring occasionally for about half an hour, till forming rather a rich thick gravy. When both the rice and meat are done, take half the rice and mix with the meat, and then the remainder of the meat and rice, and serve. Save the bones for soup for the following day. Salt pork or beef, well soaked, may be used—omitting the salt. Any kind of vegetables may be frizzled with the onions. No. 9.—Baking and Roasting with the Field Stove. By the removal of the caldron, and the application of a false bottom put over the fire, bread bakes extremely well in the oven, as well as meat, potatoes, puddings, c. Bread might be baked in oven at every available opportunity, at a trifling cost of fuel. The last experiment I made with one was a piece of beef weighing about 25lbs., a large Yorkshire pudding, and about 10lbs. of potatoes, the whole doing at considerably under one pennyworth of fuel, being a mixture of coal and coke; the whole was done to perfection, and of a nice brown colour. Any kind of meat would, of course, roast the same. Baking in fixed Oven.—In barracks, or large institutions, where an oven is handy, I would recommend that a long iron trough be made, four feet in
- 69. length, with a two-story movable grating in it, the meat on the top of the upper one giving a nice elevation to get the heat from the roof, and the potatoes on the grating under, and a Yorkshire pudding, at the bottom. Four or five pieces of meat may be done on one trough. If no pudding is made, add a quart more water. No. 10.—French Beef Soup, or Pot-au-feu, Camp Fashion. For the ordinary Canteen-Pan. Put in the canteen saucepan 6lbs. of beef, cut in two or three pieces, bones included, ¾lb. of plain mixed vegetables, as onions, carrots, turnips, celery, leeks, or such of these as can be obtained, or 3oz. of preserved in cakes, as now given to the troops; 3 teaspoonfuls of salt, 1 ditto of pepper, 1 ditto of sugar, if handy; 8 pints of water, let it boil gently three hours, remove some of the fat, and serve. The addition of 1½lb. of bread cut into slices or 1lb. of broken biscuits, well soaked, in the broth, will make a very nutritious soup; skimming is not required. No. 11.—Semi-Frying, Camp Fashion, Chops, Steaks, and all Kinds Meat of. If it is difficult to broil to perfection, it is considerably more so to cook meat of any kind in a frying-pan. Place your pan on the fire for a minute or so, wipe it very clean; when the pan is very hot, add in it either fat or butter, but the fat from salt and ration meat is preferable; the fat will immediately get very hot; then add the meat you are going to cook, turn it several times to have it equally done; season to each pound a small teaspoonful of salt, quarter that of pepper, and serve. Any sauce or maître d’hôtel butter may be added. A few fried onions in the remaining fat, with the addition of a little flour to the onion, a quarter of a pint of water, two tablespoonfuls of vinegar, a few chopped pickles or piccalilly, will be very relishing. No. 11A.—Tea for Eighty Men, Which often constitutes a whole Company.
- 70. One boiler will, with ease, make tea for eighty men, allowing a pint each man. Put forty quarts of water to boil, place the rations of tea in a fine net, very loose, or in a large perforated ball; give one minute to boil, take out the fire, if too much, shut down the cover; in ten minutes it is ready to serve. No. 12.—Coffee a la Zouave for a Mess of Ten Soldiers, As I have taught many how to make it in the camp, the canteen saucepan holding 10 pints. Put 9 pints of water into a canteen saucepan on the fire; when boiling add 7½ oz. of coffee, which forms the ration, mix them well together with a spoon or a piece of wood, leave on the fire for a few minutes longer, or until just beginning to boil. Take it off and pour in 1 pint of cold water, let the whole remain for ten minutes or a little longer. The dregs of the coffee will fall to the bottom, and your coffee will be clear. Pour it from one vessel to the other, leaving the dregs at the bottom, add your ration sugar or 2 teaspoonfuls to the pint; if any milk is to be had make 2 pints of coffee less; add that quantity of milk to your coffee, the former may be boiled previously, and serve. This is a very good way for making coffee even in any family, especially a numerous one, using 1 oz. to the quart if required stronger. For a company of eighty men use the field-stove and four times the quantity of ingredients. No. 13.—Coffee, Turkish Fashion. When the water is just on the boil add the coffee and sugar, mix well as above, give just a boil and serve. The grouts of coffee will in a few seconds fall to the bottom of the cups. The Turks wisely leave it there, I would advise every one in camp to do the same. No. 14.—Cocoa for Eighty Men. Break eighty portions of ration cocoa in rather small pieces, put them in the boiler, with five or six pints of water, light the fire, stir the cocoa round till melted, and forming a pulp not too thick, preventing any lumps forming, add to it the remaining water, hot or cold; add the ration sugar, and when
- 71. just boiling, it is ready for serving. If short of cocoa in campaigning, put about sixty rations, and when in pulp, add half a pound of flour or arrowroot. Easy and excellent way of Cooking in Earthen Pans. A very favourite and plain dish amongst the convalescent and orderlies at Scutari was the following:— Soyer’s Baking Stewing Pan, the drawing of which I extract from my “Shilling Cookery.” The simplicity of the process, and the economical system of cooking which may be produced in it, induced me to introduce it here. Each pan is capable of cooking for fifteen men, and no matter how hard may be the meat, or small the cutting, or poor the quality,—while fresh it would always make an excellent dish. Proceed as follows:—Cut any part of either beef (cheek or tail), veal, mutton, or pork, in fact any hard part of the animal, in 4oz. slices; have ready for each 4 or 5 onions and 4 or 5 pounds of potatoes cut in slices; put a layer of potatoes at the bottom of the pan, then a layer of meat, season to each pound 1 teaspoonful of salt, quarter 1 of pepper, and some onion you have already minced; then lay in layers of meat and potatoes alternately till full; put in 2 pints of water, lay on the lid, close the bar, lock the pot, bake two hours, and serve. Soyer’s Baking Stewing Pan. Remove some of the fat from the top, if too much; a few dumplings, as No. 21, in it will also be found excellent. By adding over each layer a little flour it makes a thick rich sauce. Half fresh meat and salt ditto will also be
- 72. found excellent. The price of these pans is moderate, and they last a long time—manufacturers, Messrs. Deane and Dray.[36] SERIES OF SMALL RECEIPTS FOR A SQUAD, OUTPOST, OR PICKET OF MEN, Which may be increased in proportion of companies. Camp Receipts for the Army in the East. (From the Times of the 22nd January, 1855.)[37] No. 15. Camp Soup.—Put half a pound of salt pork in a saucepan, two ounces of rice, two pints and a half of cold water, and, when boiling, let simmer another hour, stirring once or twice; break in six ounces of biscuit, let soak ten minutes; it is then ready, adding one teaspoonful of sugar, and a quarter one of pepper, if handy. No. 16. Beef Soup.—Proceed as above, boil an hour longer, adding a pint more water. Note.—Those who can obtain any of the following vegetables will find them a great improvement to the above soups:—Add four ounces of either onions, carrots, celery, turnips, leeks, greens, cabbage, or potatoes, previously well washed or peeled, or any of these mixed to make up four ounces, putting them in the pot with the meat. I have used the green tops of leeks and the leaf of celery as well as the stem, and found, that for stewing they are preferable to the white part for flavour. The meat being generally salted with rock salt, it ought to be well scraped and washed, or even soaked in water a few hours if convenient; but if the last cannot be done, and the meat is therefore too salt, which would spoil the broth, parboil it for twenty minutes in water, before using for soup, taking care to throw this water away. No. 17.—For fresh beef proceed, as far as the cooking goes, as for salt beef, adding a teaspoonful of salt to the water. No. 18. Pea Soup.—Put in your pot half a pound of salt pork, half a pint of peas, three pints of water, one teaspoonful of sugar, half one of pepper, four ounces of vegetables, cut in slices, if to be had; boil gently two hours,
- 73. or until the peas are tender, as some require boiling longer than others—and serve. No. 19. Stewed Fresh Beef and Rice.—Put an ounce of fat in a pot, cut half a pound of meat in large dice, add a teaspoonful of salt, half one of sugar, an onion sliced; put on the fire to stew for fifteen minutes, stirring occasionally, then add two ounces of rice, a pint of water; stew gently till done, and serve. Any savoury herb will improve the flavour. Fresh pork, veal, or mutton, may be done the same way, and half a pound of potatoes used instead of the rice, and as rations are served out for three days, the whole of the provisions may be cooked at once, as it will keep for some days this time of the year, and is easily warmed up again. N.B. For a regular canteen pan triple the quantity. No. 20.—Receipts for the Frying-pan. Those who are fortunate enough to possess a frying-pan will find the following receipts very useful:—Cut in small dice half a pound of solid meat, keeping the bones for soup; put your pan, which should be quite clean, on the fire; when hot through, add an ounce of fat, melt it and put in the meat, season with half a teaspoonful of salt; fry for ten minutes, stirring now and then; add a teaspoonful of flour, mix all well, put in half a pint of water, let simmer for fifteen minutes, pour over a biscuit previously soaked, and serve. The addition of a little pepper and sugar, if handy, is an improvement, as is also a pinch of cayenne, curry-powder, or spice; sauces and pickles used in small quantities would be very relishing; these are articles which will keep for any length of time. As fresh meat is not easily obtained, any of the cold salt meat may be dressed as above, omitting the salt, and only requires warming; or, for a change, boil the meat plainly, or with greens, or cabbage, or dumplings, as for beef; then the next day cut what is left in small dice— say four ounces—put in a pan an ounce of fat; when very hot, pour in the following:—Mix in a basin a tablespoonful of flour, moisten with water to form the consistency of thick melted butter, then pour it in the pan, letting it remain for one or two minutes, or until set; put in the meat, shake the pan to loosen it, turn it over, let it remain a few minutes longer, and serve.
- 74. To cook bacon, chops, steaks, slices of any kind of meat, salt or fresh sausages, black puddings, c. Make the pan very hot, having wiped it clean, add in fat, dripping, butter, or oil, about an ounce of either; put in the meat, turn three or four times, and season with salt and pepper. A few minutes will do it. If the meat is salt, it must be well soaked previously. No. 21.—Suet Dumplings. Take half a pound of flour, half a teaspoonful of salt, a quarter teaspoonful of pepper, a quarter of a pound of chopped fat pork or beef suet, eight tablespoonfuls of water, mixed well together. It will form a thick paste, and when formed, divide it into six or eight pieces, which roll in flour, and boil with the meat for twenty minutes to half an hour. Little chopped onion or aromatic herbs will give it a flavour. A plainer way, when Fat is not to be obtained.—Put the same quantity of flour and seasoning in a little more water, and make it softer, and divide it into sixteen pieces; boil about ten minutes. Serve round the meat. One plain pudding may be made of the above, also peas and rice pudding thus:—One pound of peas well tied in a cloth, or rice ditto with the beef. It will form a good pudding. The following ingredients may be added: a little salt, sugar, pepper, chopped onions, aromatic herbs, and two ounces of chopped fat will make these puddings palatable and delicate. BILL OF FARE FOR LONDON SUPPERS. In introducing the subjoined Bill of Fare, applicable to the London suppers, I must here repeat that which I have previously mentioned, that my idea is far from replacing the dishes now so much in vogue both at the “Albion,” Simpson’s in the Strand, Evans’ Cider Cellars, and such-like places; but now and then a couple of dishes taken from these receipts cannot fail to prove agreeable to the partakers, without much interfering with the regular routine of the nightly business of such establishments. No. 1.—Plain Mutton Chops and Rump Steaks. Though almost anybody can boast of being able to cook a plain steak or a chop, very few can say they can do them to perfection. First of all, to obtain this important point, either the mutton or beef ought to be kept till
- 75. properly set, according to season; secondly, the chop especially is more preferable when cut and beat, some time before cooking, so as to set the meat and prevent its shrinking; it at all times requires a sharp fire (the broiling City fires may be taken as an example, and the continual red heat of the gridiron); lay your gridiron over a sharp fire, two minutes after lay on your chop or steak, turn three or four times; when half done, season highly with salt and pepper, and when done, serve immediately, on a very hot dish. Ten minutes will do a steak of 1½lb., and about six minutes a chop. No. 2.—Rumpsteak and Potatoes. Of all steaks, rumpsteaks are far more preferable than any other, not excepting the fillet of beef, as the meat in England is so rich, while in France they eat only the fillet of beef—that being the only eatable steak of a French ox. Have your steak cut as even as possible, nearly an inch thick, and weighing from about 1½lbs. to 2lbs.; broil it sharply as described above, season when properly done, lay it on a very hot dish, put on 2oz. or more of maître d’hôtel butter (No. 2), turn it three or four times on the dish quickly, when a most delicious gravy will be formed, then place about a pound of fried potatoes round it, and serve. For smaller steaks, for cooking be guided by size. Ditto with Anchovy Butter, of which use 2oz. in lieu of the maître d’hôtel butter, and omit the potatoes. Same with Pimento Butter. Same with Shalot Butter, well rubbing the dish prior to putting the steak on it. No. 3.—Mutton and Lamb Cutlets a la Bouchere. The word à la bouchere, in English, means the butcher’s wife’s plain fashion, and at one time had only the merit of economy; but a real gourmet, the illustrious Cambaceres, who lived in the time of the first Empire, being served with this dish at a little country inn, while travelling, discovered the correctness of the proverb that “the nearer the bone the sweeter the meat,” and on returning to Paris introduced it to the fashionable circle, and for a long period this exquisite cotelettes d’agneaux de maison, or house-lamb
- 76. cutlet, and the dainty and justly celebrated cutlets de presalé, were figuring on all the banqueting tables of the Paris gourmets in perfect negligé, being dressed in the following unceremonious manner:—Take either a neck of lamb or mutton, neither too fat nor too lean, chop the cutlets about six inches in length, cutting them as usual, leaving a bone in each; flatten them with the chopper, not trimming them at all, season them highly with salt and pepper, broil them very quick, and serve hot. Lamb, mutton, and veal cutlets may be done the same. For plain cutlets with fried potatoes, cut them either à la bouchere or trim them, and proceed as for rumpsteak. Ditto for Cutlets à la maître d’hôtel. For relishing sauce, see List of Sauces. No. 4—Lamb and Mutton Cutlets, semi-Bouchere. Cut your cutlets from the neck, one inch thick; beat them flat with a chopper without trimming them, roll them in flour, butter over; season with salt, pepper, a little chopped shalot; broil on a sharp fire, turn three or four times, and serve. No. 5.—Relishing Steak. (Mutton, Veal, Pork, Chops and Cutlets, Fowls, Pigeons, Grilled Bones, Kidneys, c.) Chop fine a tablespoonful of green pickled chillies: mix with two pats of butter, a little mustard, a spoonful of grated horseradish; have a nice thick steak, spread the steak on both sides with the above, season with half a teaspoonful of salt, put on a gridiron on a sharp fire, turn three or four times; put on a hot dish with the juice of half a lemon and two teaspoonfuls of walnut ketchup, and serve. If glaze is handy, spread a little over the steak. Mutton, lamb, veal, pork, chops and cutlets may be done the same; as well as kidneys; also grilled fowls, pigeons—the latter may be egged and bread-crumbed. Proceed the same for cooking according to size. Any of the above may be half done before rubbing in the Chili butter.
- 77. No. 6.—Fillet de Bœuf, Parisian Fashion. Cut a piece of the fillet of beef crosswise, including some fat, the thickness of an inch; beat it slightly flat with a chopper, set on a gridiron, put it on a very sharp fire, turn it two or three times; when half done, season with a quarter of a teaspoonful of salt, quarter that of pepper, put on a hot plate, rub over with an ounce of maître d’hôtel butter (as No. 2); serve up with fried potatoes. Mutton chops, veal chops, and lamb chops may be dressed similar. No. 7.—Fillet of Beef, semi-Chateaubriant. Cut it double the thickness of the above, butter lightly over, set on the gridiron on a slowish fire, turn several times; when half done, place it nearer the fire; season with half a teaspoonful of salt, quarter one of pepper, a little cayenne, and serve with sauce à la Mussulman (as No. 17). Maître d’hôtel butter or anchovy butter may be used instead; serve fried chipped potatoes round. No. 8.—Chops, semi-Provençal, or Marseilles Fashion. When the chop is half broiled, scrape half a clove of garlic and rub over on both sides of the chop; serve with the juice of a lemon. For semi- Provençal, the clove of garlic is cut in two, and the flat part is placed at the end of a fork and rubbed on the chop. No. 9.—Chop or Steak a la Sultana. Add a tablespoonful of Sultana sauce in a dish to each pound of meat; place in a dish and serve; when the steak is done, turn it in it three or four times, and it will make a most delicious gravy. No. 10.—Mushroom Kidney Sandwich. Broil 3 plain kidneys à la Brochette to keep them flat. Broil also 6 large mushroom heads; well season with salt and pepper (cayenne if approved of). A few minutes will do them; then rub a little fresh butter inside the
- 78. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com