Tutorial Mesodyn

J Mol Model (2011) 17:607–619
DOI 10.1007/s00894-010-0755-z

ORIGINAL PAPER

Modeling and analysis of the compatibility
of polystyrene/poly(methyl methacrylate) blends
with four inducing effects
Dan Mu & Jian-Quan Li & Yi-Han Zhou

Received: 8 February 2010 / Accepted: 1 April 2010 / Published online: 5 June 2010
# Springer-Verlag 2010

Abstract The compatibility of polystyrene (PS) and poly Introduction
(methyl methacrylate) (PMMA) blends was studied over a
wide range of compositions at 383, 413 and 443 K, Polymer blend systems play an important role in the plastic
respectively, by atomistic and mesoscopic modeling. All industry because they can be tuned to have a better
the calculated Flory–Huggins interaction parameters combination of physical properties than the corresponding
showed positive values; furthermore, they were all above homopolymers. The process of physical blending is much
the critical Flory–Huggins interaction parameter value, easier to perform than chemical blending (e.g., grafting),
which means that the PS/PMMA blends were immiscible. and the former is also more economical. As a general rule,
Both the addition of a block copolymer and the introduction the aim of blending is to obtain a product with new or better
of a shear field influenced the phase morphologies of the properties. Because of the different physical properties of
blends, while the degree of influence depended on the candidate polymers, it is not always easy to obtain
compositions of the blends. The study of PS/PMMA blends completely miscible polymer mixtures. When the compo-
doped with nanoparticles showed that the mesoscopic phase nents of polymers can form energetically favorable inter-
was influenced by not only the properties of the nano- actions, such as hydrogen bonding, dipole interactions, etc.,
particles, such as their size, number and number density, it is possible to obtain a miscible blend. For immiscible
but also the compositions of the blends. The effect of the blends, adding interferents is a common method of
surface roughness of the planes on the phase separation of changing the microscopic morphology. This improves the
the blends was also studied. blend miscibility and thus enhances its properties.
Polystyrene (PS) and poly(methyl methacrylate)
Keywords PS/PMMA blend . Inducing effect . Phase (PMMA) are conventional model systems in polymer
morphology . Nanoparticle science. PS/PMMA blends are a well-known immiscible
combination [1-8], and their bulk and surface phase
separation has been observed [9, 10]. Immiscible blends
D. Mu (*) are known to have properties combining those of their
Department of Chemistry, Zaozhuang University,
component polymers, and also to have segregated structures
Shandong 277160, China
e-mail: mudanjlu1980@yahoo.com.cn with domains that are predominantly formed from the
individual homopolymers. It has been shown that changing
J.-Q. Li the relative homopolymer proportions in such blends can
Physics and Electronic Engineering Department,
change the domain structure, surface morphology, and even
Zaozhuang University,
Shandong 277160, China the phase morphology [11-13].
Some contradictory results can be found in the literature
Y.-H. Zhou regarding the miscibilities of certain polymers and the
National Analytical Research Center of Electrochemistry
possibility of predicting miscibility from surface tension or
and Spectroscopy, Changchun Institute of Applied Chemistry,
Chinese Academy of Sciences, from parameters derived from it. These contradictions can
Changchun, China be attributed to several reasons. Firstly, the structures and

608 J Mol Model (2011) 17:607–619

the properties of partially miscible polymer blends depend blending materials and planes or doped nanoparticles.
on the processing conditions used, which may influence Furthermore, in the study of the compatibility of PEO/PMMA
conclusions about polymer compatibility when they are blends in our former work, the MesoDyn simulation method
deduced from macroscopic properties or from the structure was successfully applied to detect the phase morphologies of
of the blend. Secondly, methods that can be used to predict PEO/PMMA blends. From a theoretical point of view, these
miscibility are limited [14, 15]. For example, various results clarified several conflicting conclusions of different
experimental difficulties can be encountered during the experiments [26]. Therefore, the MesoDyn method has been
determination of the Flory–Huggins interaction parameter, shown to be reliable when dealing with PS/PMMA blends.
χ, which can be used to characterize the strength of the The mesoscale simulation process used in this paper was
interaction between the polymers. χ can be determined by carried out with the MesoDyn module in the Material Studio
various methods, such as inverse gas chromatography, commercial software provided by Accelrys.
solvent diffusion, or the dependence of composition on
mechanical properties, but all of these methods have their
drawbacks. Finally, when blend systems have two calori- Simulation details, results and discussions
metric Tg values, it does not necessarily indicate that their
components are immiscible [16]. Fortunately, computer This paper consists of two parts: calculations performed via
calculations and simulations have become available as (1) molecular dynamic (MD) simulations and (2) mesoscopic
efficient methods of solving these experimental problems. simulations. The intermediate parameter connecting the
This paper is divided into two main parts. The first part microscale and the mesoscale is the Flory–Huggins param-
is about the calculation of typical thermodynamic parameter, χ. MD simulation is used to accurately study bulk
eters, such as the solubility parameter, of the studied polymer properties when this is difficult to do experimental-
systems via a microscopic calculation method, and com- ly. Although MD simulation is widely used nowadays, it is
paring those calculated parameters with experimental data difficult to detect the phase separation of blends via a
in order to maintain a high level of accuracy in further microscopic method, such as MD simulation. Therefore, it is
calculations and simulations; the second part describes necessary to combine it with a mesoscopic method, such as
mesoscopic dynamics simulations of several kinds of MesoDyn simulation, to perform a full study of polymer
inducing effects that can change the phase morphologies blends from the microscale to the mesoscale.
of PS/PMMA blends. Mesoscale structures are of the
utmost importance during the production processes of Atomistic molecular simulation
many materials, such as polymer blends, block copolymer
systems, surfactant aggregates in detergent materials, latex Model and simulation
particles, and drug delivery systems.
Mesoscopic dynamics models have been receiving increas- The selection of the representative chain length (Nmon) was
ing attention recently, as they form a bridge between our primary task. This simulation process was carried out at
microscale and macroscale properties [17-20]. Mesoscopic room temperature (298 K) on an SGI workstation by
dynamics (MesoDyn) [21-23] and dissipative particle dy- following the MD procedure in the commercial Cerius2
namics (DPD) [24, 25] methods both treat the polymer chain package program provided by Accelrys.
at a mesoscopic level by grouping atoms together and then The simulation processes (including the construction of the
coarse-graining them to be persistent-length polymer chains. models) were the same for both PS and PMMA, so we will use
These two mesoscopic modeling methods increase the scale an atomistic chain length N of 10 for PS as an example here
of the simulation by several orders of magnitude compared to describe the simulation process in detail. Initially, the
with atomistic simulation methods. MesoDyn utilizes dy- proposed polymer chain structure was generated using the
namic mean-field density functional theory (DFT), in which rotational isomeric state (RIS) model of Flory [27], which
the dynamics of phase separation can be described by describes the conformations of the unperturbed chains. Three
Langevin-type equations to investigate polymer diffusion. PS polymer chains were then generated following the same
Thermodynamic forces are found via mean-field DFT, using principle, each with ten repeating units; they were then put
the Gaussian chain as a model. The most important into a simulation box with a density of 1.05, which is the
molecular parameter is the “incompatibility parameter” χN, density of bulk PS [28]. Minimization was then done using
where χ is the Flory–Huggins interaction parameter and N is the conjugate gradient method (CGM) with a convergence
the degree of polymerization. level of 0.1 kcal mol−1 Å−1 until the energy reached a
However, owing to the soft interaction potential of the minimum. This minimized configuration was refined by MD
DPD method, it is not suitable to apply the DPD method in our simulation for 30 ns under constant temperature and density
work, especially when dealing with the interactions between (NVT ensemble). A time step of 1 fs was used to ensure the

J Mol Model (2011) 17:607–619 609

stability of simulation. The Nosé–Hoover [30] thermostat in
a canonical ensemble was used to create the canonical
molecular dynamics trajectories, and the COMPASS force
field was chosen to deal with such a condensed state system
[31-33]. The spline treatment of van der Waals (VDW)
interactions was chosen, with a spline window of between 11
and 12 Å. The Ewald summation [34], an accurate method of
calculating long-range interactions, was applied in order to
calculate long-range electrostatic interactions. Next, the
refined polymer was annealed in the temperature range from
383 K (above 373 K [28], the Tg of PS) to 443 K (below
513 K [28], the melting temperature Tm of PS) in steps of
10 K for 50 cycles. The simulation time for each annealing
cycle was set to 2 ns, and the system was minimized at the
beginning of each annealing cycle. Because such oligomers
always have strong end–end interactions, a system consisting Fig. 1 The solubility parameter for different PS chain lengths
of oligomers is different from a real polymeric system. Thus,
an MD simulation was executed with the NPT (with constant
temperature and pressure) ensemble for 20 ns; furthermore, numbers of repeating units of PMMA. These become
MD simulations with a repeating ensemble cycle from NVT constant from N=60 onwards, and they are 6.40% smaller
to NPT were carried out three times with a simulation time of than the experimental results. This may result from the
20 ns for each cycle to reduce the end–end interactions presence of a bulky CH3 group in PMMA, which also
sufficiently. Finally, an MD simulation was executed with restricts free rotation around the C–C bond of its backbone.
the NVT ensemble for 30 ns at 298 K to collect data. In The fact that the calculated relaxation times were
general, about 200 ns was needed in total for each simulated shorter than the experimental results presented a rather
case. During the whole simulation process, the selection of difficult obstacle to overcome in the simulations. How-
the appropriate force field was a crucial task for obtaining ever, taking N=60 to represent the whole polymer chain
accurate information about the structures and dynamics of for PMMA appeared a reasonable approach. Hence, the
the blends. The COMPASS force field has been shown to be minimum representative polymer chain lengths of PS and
a suitable choice for calculating solubility parameters, PMMA, Nmon(PS) and Nmon(PMMA), were chosen to be
especially for polymeric systems [35]. 30 and 60, respectively. Figure 3 shows a flowchart that
illustrates the process described above.
Results and discussion The blend systems were binary, with PS (density =
1.05 gcm−3) and PMMA (density = 1.188 g cm−3) as the
The cohesive energy density (CED) of each pure compo- components. The glass transition temperature of PS is
nent was used to calculate the solubility parameter, δ,
according to the equation below:
rffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi Ecoh
d ¼ CED ¼ ;
V

To determine the minimum representative polymer chain
length of the blend components, Nmon, cases with different
chain lengths, N=10, 20, 30, 40, 50, 60, 70 and 80 for both
PS and PMMA, were selected in order to calculate the δ
value of each component. Figure 1 displays the dependence
of δ of PS on the number of repeating units of PS. It can be
seen that a nearly constant value is approached when the
number of repeating units reaches 30, and the values are
8.32% smaller than the experimental data. This may result
from the presence of a bulky C6H5 group in PS, which
restricts free rotation around the C–C bond of its backbone.
Figure 2 shows the calculated δ values for different Fig. 2 The solubility parameter for different PMMA chain lengths

610 J Mol Model (2011) 17:607–619

chains were randomly distributed in the simulation box, the
model was assumed to be a well-mixed model and accepted
for further simulations; otherwise, it was discarded, and a
new blending model was built until the component chains
contacted each other sufficiently. Figure 4 shows a typical
1/1 blend. Because of the end–end interactions of short
polymer chains, the initial amorphous structure of the blend
must first be minimized. The subsequent simulation process
was the same as that used to calculate δ values for the pure
components, and the simulation temperatures were also
383, 413 and 443 K.
The tendency for binary polymer chains to mix depends on
the cohesive energies of the blends and their pure components.
Both are applied to calculate the χ parameters of blends with
different compositions. The χ parameter is defined as

ΔEmix
χ¼ Vmon ;
RT
where Vmon is the unit volume of one monomer. Here, Vmon
equals 165.54 cm3 mol−1 for blends of PS/PMMA. ΔEmix for
a binary mixture is defined as

Ecoh Ecoh Ecoh
ΔEmix ¼ f1 þ f2 À ;
V pure1 V pure2 V mix

where f is the volume fraction, and the subscripts “pure”
and “mix” denote the cohesive energy densities of the pure
components and the binary mixture, respectively. χ is shown
as a function of the PS volume fraction in Fig. 5. Several
features of these data are impressive. Firstly, the three χ
curves vary in almost the same way at different temperatures,
which means that temperature was not an efficient factor for
changing the degree of phase separation. This is a totally
different conclusion from that obtained for PEO/PMMA
blends, which are miscible [26]. Secondly, the χ value shows
a decreasing trend as the volume fraction of PS increases,
except for the 1/4 blend case, where it shows a small rising
Fig. 3 The procedure used to select Nmon for PS trend. Then the 4/1, 6/1 and 8/1 blend cases also show another
decreasing trend in χ. These two decreasing trends are termed
T1 and T2, respectively. Considering the absolute data on
373 K, so PS is prone to crystallize at a low temperature.
slope variation for T1 and T2, where T1 T2, it is supposed
PMMA starts to soften at the glass transition temperature
that the degree of phase separation would intensify with
(Tg =378 K), and does not substantially depolymerize until
increasing PS content when the blends have a higher content
453 K [36]. Therefore, the blending simulation temper-
of PS. In addition, the 1/4 and 4/1 blend cases produced
atures adopted were 383, 413 and 443 K.
outstandingly high χ values for both the T1 and the T2 lines,
PS/PMMA blends with different weight percentages of
which means that it is possible that extremely distinct phase
PS were put in a three-dimensional (3D) periodic boundary
separation was present in these two blend cases.
simulation box with the corresponding density. Table 1
A positive χ value alone cannot prove that the blends are
describes the blending models in detail, including the molar
immiscible; it is necessary to calculate the critical value of
ratios of PS to PMMA, the chain number per unit cell, the
χ via the equation below:
compositions by weight, and the densities. The blend
properties calculated in subsequent studies depend on how 2
well the components are blended. Therefore, when the 1 1 1
ð χ AB Þcritical ¼ pffiffiffiffiffiffiffi þ pffiffiffiffiffiffi ;
blending model was generated, if the PS and the PMMA 2 mA mB

J Mol Model (2011) 17:607–619 611

Table 1 Details of the
PS/PMMA blend cases carried Number System Content wt% of PMMA p(gcm−3)
out via MD simulation
1 Pure PMMA 1PMMA chain 100 1.1880
2 1/6 1PS, 6PMMA 92.02 1.1772
3 1/4 1PS, 4PMMA 88.49 1.1724
4 1/3 1PS, 3PMMA 85.22 1.1681
5 1/2 1PS, 2PMMA 79.36 1.1604
6 1/1 1PS, 1PMMA 65.78 1.1429
7 2/1 2PS, 1PMMA 49.01 1.1220
8 3/1 3PS, 1PMMA 39.06 1.1099
9 4/1 4PS, 1PMMA 32.46 1.1021
10 6/1 6PS, 1PMMA 24.27 1.0925
11 8/1 8PS, 1PMMA 19.38 1.0868
12 Pure PS 1PS chain 0 1.0500

where mA and mB represent the actual number of repeating mixing ΔHm, could be used to characterize the miscibility
units of PS and PMMA, respectively. When the calculated of polymer blends [37].
and the critical values of χ are based on the same reference However, it is also necessary to calculate thermodynamic
volume, χAB)critical can be used to estimate the compatibil- parameters such as entropy, enthalpy and Gibbs free energy
ity of the blends. The blends are miscible if (χAB)calculated (G) to illustrate the phenomenon of polymer blending. The
(χAB)critical. In this case, the Gibbs free energy, ΔGm, is entropy change upon mixing long polymers is so small that
negative. In contrast, when (χAB)calculated ≫ (χAB)critical, the the mixing enthalpy becomes a key influence on the degree
blends are totally immiscible; the phases will even separate. of mixing. The entropy change upon mixing, ΔSm, must be
When the difference between χAB)calculated and χAB)critical is considered in addition to the enthalpy of mixing. Hence, it
small or even zero, the blends will be partially or even was important to compute ΔHm in order to study the
totally miscible. The value of χAB)critical in our studied PS/ miscibilities of the PS/PMMA blends using the following
PMMA blends was 0.0486, which was much smaller than equation [37]:
the calculated χ values. Therefore, we can see that the PS/
PMMA blends were immiscible at 383, 413 and 443 K. hn oi2
ΔHm ¼ Vm ðCEDÞ0:5 ÀðCEDÞ0:5 fPSfPMMA;
As long as the calculated χAB and the corresponding PS PMMA

critical values are based on the same reference volume,
comparing the calculated χAB with critical values will where Vm is the total volume of the blend. The CED values
provide useful information about the degree of miscibility used to compute ΔHm were obtained from the data in the
of the blends. Therefore, χAB, rather than the enthalpy of MD simulations.

Fig. 4 Amorphous cell from the
1/1 PS/PMMA blend case.
Red, O; gray, C; white, H

612 J Mol Model (2011) 17:607–619

they exhibit the strongest attractive or repulsive interactions.
Because of the low entropy of mixing, polymeric blends
are mostly incompatible, and their phases will separate
under the appropriate conditions. Therefore, it is necessary
to carry out further research from a dynamic point of
view at the mesoscopic level to study how the phase
separations of PS/PMMA blends are influenced by certain
inducing effects.

Mesoscopic simulation

PS/PMMA blends with a block copolymer and a shear field

To map the representative polymer chains onto Gaussian
chains, the bead number (Nmeso) and the unit chain length
Fig. 5 Flory–Huggins interaction parameter χ as a function of the PS
volume fraction for binary blends of PS/PMMA at 383, 413 and
of the Gaussian chain (ameso) are estimated by
443 K, respectively
N mon
Nmeso ¼ ;
C1
To calculate ΔGm, the change in entropy during mixing, ameso ¼ amon C1;
ΔScomb was needed, and this was calculated using the
m
combinatorial entropy term for binary systems via Flory– where Nmon is the number of repeating units in the polymer
Huggins theory [38] for two-component systems, as given by chain. amon is the reference unit monomer length, and C∞ is
the characteristic ratio of a chain of infinite length. Nmon
ΔScomb ¼ ÀR½# A ln # A þ # B ln # B Š:
m was determined by the representative chain lengths of PS
and PMMA, i.e., Nmon =30 for PS (MW =3200) and Nmon =
Thus, by using the thermodynamic formula 60 for PMMA (MW =6000), while C∞(PS)=9.85 and C∞
(PMMA)=8.65 were found in the Polymer Handbook [29].
ΔGm ¼ ΔHm À T ΔScomb ;
m The connection between the microscale and the mesoscale
it was possible to calculate the Gibbs free energy of mixing. is as follows:
Both ΔGm and ΔHm exhibited similar trends at 413 K over
the entire composition range, as shown in Fig. 6. The vÀ1 ij ¼ # AB RT ;
extreme values of these two curves both appeared in the 1/1
blend case. It is presumed that when the molar proportions where the parameter χAB is calculated by atomistic
of PS and PMMA are the same, such as in the 1/1 blend, simulation for each blend composition at different temper-
atures. R is the molar gas constant, 8.314 J mol−1 K−1, and
T is the simulation temperature. The calculated χ values for
various compositions were all positive at 383, 413 and
443 K. Nmeso and ν−1εij are the input parameters used in
MesoDyn simulations to describe the number of Gaussian
beads and the interaction between beads. In the 1/1 blend
case, the Nmeso values for PS and PMMA were 3 and 6,
respectively.
In order to stabilize the numerical calculation, the time
step for MesoDyn calculation, τ, was chosen to be 100 ns,
and the total simulation time was 1000 μs for each blend
case. A default noise parameter value of 75.002 was used
for numerical speed and stability. The grid dimensions
adopted were 32×32×32 nm. Ten mesoscopic simulations
were carried out with different PS/PMMA blend composi-
tions, 1/1, 1/2, 1/3, 1/4, 1/6, 2/1, 3/1, 4/1, 6/1 and 8/1 in
Fig. 6 Changes in enthalpy ΔHmix (solid curve) and Gibbs free molar ratio, which roughly covered the whole composition
energy ΔGmix (dotted curve) with volume fraction of PS range.

J Mol Model (2011) 17:607–619 613

Results and discussion

The order parameter, denoted P, is defined as the average
difference between the local density squared and the overall
density squared, as given by the equation
Z
1 Â Ã
Pi ¼ h2 ðrÞ À h2 dr;
i i
V V

where ηi is the dimensionless density (volume fraction) of
species i. The larger the value of the order parameter, the
stronger the phase separation. A decrease in P indicates
better compatibility or miscibility, and the polymer phases
mix more randomly.
It should be pointed out that the P data calculated by
mesoscopic simulation can only be compared for the same Fig. 7 V ROP value versus volume fraction of PS for ten blend cases
blend composition. Therefore, plain PS/PMMA blends with 26 block copolymers added. The zero point on the Y axis was set
were used as reference cases, and their order parameter as the reference line. When the data are above this line, it means that
the block copolymer has an enhancing effect on the PS/PMMA blend;
data were labeled “A;” in addition, the order parameter data when the data are below this line, it has a deteriorating effect on the
of PS/PMMA blends of the same compositions as for the blend. Isodensity surface pictures of a 6/1 (PS/PMMA) blend at 443,
“A” blends but with inducing effects, such as being doped 413 and 383 K are displayed from top to bottom in the chart,
with a block copolymer or nanoparticles, or having planes respectively. Red, PS; green, PMMA
introduced, were labeled “B.” The value of (B − A)/A is
termed the variation rate of order parameters (V ROP for value of the 3/1 blend case also turned from positive to
short). By comparing the V ROP values, we can detect negative. Based on this information, and using the 1/1
which factor is the most effective at changing the phase blend case as a reference, the V ROP values of pairs of
morphology. blend cases (such as the 1/3 and 3/1 blend cases) turned
In order to study the influence of the addition of a block from positive to negative as the temperature was gradually
copolymer on the phase morphology, a block copolymer increased. Following this principle, when the temperature
made of PS and PMMA with a volume fraction of 10% was was raised above 443 K, the V ROP values of the 1/4 and 4/
added to each blend. Generally speaking, when block 1 blend cases would be expected to become negative. The
copolymers of the same composition are added to an three pictures inserted into Fig. 7 are the phase morphol-
immiscible blend, the compatibility of the blend will be ogies of the 6/1 blend case with block copolymers added at
enhanced. However, in our study, two opposite results were 443, 413 and 383 K (top to bottom, respectively).
obtained when the block copolymer was added: an In order to investigate the influence of the shear field on
enhancement of or a deterioration in compatibility were the microscopic phase morphology, the shear rate was set to
both observed. 0.001 ns−1 and 0.002 ns−1 as examples of low and high
Several striking features of these data can be seen in shear effects. Simulations were then performed on the basis
Fig. 7. First, the V ROP values of blends with less PS (the of plain PS/PMMA blends with P data used as a reference.
1/1, 1/2, 1/3, 1/4, 1/6, 2/1 and 3/1 blend cases) fluctuated It can be seen from Fig. 8 that low shear stress can have a
around zero, and this trend was most obvious for the 1/1, 1/ significant inducing effect on PS-rich blends, such as in the
3 and 1/6 blend cases, which means that adding a block 4/1 and 6/1 blend cases. Phase separation can clearly be
copolymer to the blends with less PS did not result in an seen in the inserted pictures. However, shear stress had no
obvious inducing effect. However, obvious enhancing obvious effect on the 8/1 blend case. When the shear rate
effects were seen for the blends with more PS (the 4/1, 6/ was increased to 0.002 ns−1, all of the V ROP values for the
1 and 8/1 blend cases). This result can be attributed to the ten different blend cases were high, as shown in Fig. 9,
difference in the self-diffusion coefficients between PS and which means that high shear stress plays an important role
PMMA and the difference in their self-compatibility in changing the microscopic phase morphology, especially
properties. Second, the V ROP values of the 1/2, 1/1 and in PS-rich blends. In addition, the V ROP value of the 8/1
2/1 blend cases were negative at 383 K; when the blend case was relatively high. Furthermore, the phase
temperature was increased to 413 K, the V ROP value of morphologies of all the blends exhibited obvious phase
the 1/3 blend case turned from positive to negative. When separation. In a word, the higher the shear rate, the more
the temperature was raised again, to 443 K, the V ROP obvious the phase separation.

614 J Mol Model (2011) 17:607–619

Column-shaped nanoparticles were applied as an exam-
ple. In order to determine the factor that exerts the most
influence on phase separation, seven different kinds of
nanoparticles were made for the doping study. In Table 2,
the number of nanoparticles in each layer (Np), the radius of
each nanoparticle (rp), the height of each nanoparticle (hp),
the number of layers (NL) and the total number of
nanoparticles doped (Ntp) are listed in detail. Among these
cases, the 4-3-4-2 case (with four nanoparticles in each of
two layers, a nanoparticle radius of 3 nm, and a nano-
particle height of 4 nm) was used as a reference, and all the
other cases were derived from this. For example, adding
only one more nanoparticle in the center resulted in the 4-3-
4-3 case; increasing the number of layers to four but not
changing any other settings yielded the 4-3-4-4 case;
Fig. 8 V ROP data based on plain blends versus the volume fraction
of PS. Isodensity surface pictures of a 4/1 (PS/PMMA) blend induced doubling the nanoparticle density of each layer produced
by a shear rate of 0.001 ns−1 are also shown, as are isodensity surface the 8-3-4-2 case; doubling the nanoparticle height yielded
pictures of a 4/1 (PS/PMMA) blend at 383, 443 and 413 K (top to the 4-3-8-2 case; adding one more nanoparticle to the middle
bottom in the chart, respectively). Red, PS; green, PMMA of the simulation box and increasing the number of layers to
three on the basis of the 4-3-8-2 case produced the 4-3-8-3
PS/PMMA blends doped with nanoparticles case; and doubling the nanoparticle radius on the basis of the
4-3-8-2 case gave the 4-6-8-2 case. The parameters chosen
The pairs of blend cases, such as the 1/3, 3/1 and the 1/2, 2/ were the same as those used in the previous mesoscopic
1 blend pairs, were then used as examples to study the simulations; the χ value was that obtained at 400 K, and
inducing effect of doping with nanoparticles. In addition, the total simulation time was 10 ms in each case. The simu-
the 1/3 (with a PS volume fraction of 16.21%) and 1/2 lation models were the plain PS/PMMA blends shown in
(with a PS volume fraction of 22.49%) blend cases were Table 1.
used to represent compositions rich in PMMA, while the 2/
1 (with a PS volume fraction of 53.72%) blend case was Result and discussion
used to represent compositions that contain roughly equal
amounts of PS and PMMA, and the 3/1 (PS volume Figure 10 shows the V ROP values obtained for various
fraction of 63.52%) blend case was chosen to represent kinds of nanoparticle doping situations. A few trends can be
compositions rich in PS. seen in these cases. First, the trends seen in these four
pictures are almost the same. Second, since V ROP1/3 V
ROP1/2 V ROP2/1 V ROP3/1, it can be deduced that the
degree of phase separation intensifies with increasing
PMMA content when the blend is rich in PMMA. Third,
since V ROP4-3-4-3 V ROP4-3-4-2 and V ROP4-3-8-3 V
ROP4-3-8-2, the order of the PS/PMMA blend increases
when the number of nanoparticles of the same size is

Table 2 Mesoscopic simulations of PS/PMMA blends doped with
nanoparticles

Simulation Case Np rp (nm) hp (nm) NL Ntp

1 4-3-4-2 4 3 4 2 8
2 4-3-4-3 4 3 4 3 9
3 4-3-4-4 4 3 4 4 16
4 4-3-8-2 4 3 8 2 8
Fig. 9 V ROP data with the shear effect versus the volume fraction of
PS. Isodensity surface pictures of an 8/1 (PS/PMMA) blend induced 5 4-3-8-3 4 3 8 3 9
by a shear rate of 0.002 ns−1 are also shown, as are isodensity surface 6 4-6-8-2 4 6 8 2 8
pictures of an 8/1 (PS/PMMA) blend at 443, 413 and 383 K (top to 7 8-3-4-2 8 3 4 2 16
bottom in the chart, respectively). Red, PS; green, PMMA

J Mol Model (2011) 17:607–619 615

Fig. 10 V ROP data for the 1/2, 2/1 and the 1/3, 3/1 blend pairs doped with nanoparticles at 383, 413 and 443 K

increased. Fourth, since V ROP8-3-4-2 V ROP4-3-4-4, the From the third, fourth and fifth results discussed above
order of the PS/PMMA blend can be increased by (V ROP4-6-8-2 ≫ V ROP8-3-4-2 V ROP4-3-4-4 V ROP4-3-8-3
increasing the density of the nanoparticles of the same V ROP4-3-8-2 VROP4-3-4-3 V ROP4-3-4-2), we can see
size. Fifth, it can be seen from the extremely high value of that the factors investigated had the following order of
V ROP4-6-8-2 that the nanoparticle size is the factor that influence on the phase separation: nanoparticle size
exerts the most influence on the phase morphology. nanoparticle density number of doped nanoparticles.

Table 3 Mesoscopic simulations
of PS/PMMA blends induced “ci” type Row number (X-axis) Row number (Y-axis) Radius (nm)
by “ci” and “co” planes ci-444 4 4 4
ci-882 8 8 2
“co” type E/Tg (X-axis) E/Tg (Y-axis) Height (nm)
co-444 2/4 2/4 4
co-448 2/4 2/4 8
co-4412 2/4 2/4 12
co-4432 2/4 2/4 32
co-884 4/8 4/8 4
co-888 4/8 4/8 8
co-8812 4/8 4/8 12
g
This is the abbreviation for co-8832 4/8 4/8 32
“effective/total row number.”

616 J Mol Model (2011) 17:607–619

Table 4 Mesoscopic simulations
of PS/PMMA blends induced “gra” type Row number (X-axis) Height (nm) Sfin X-axis
by “gra” and “rg” planes
gra-444 4 4,3,2,1 N
gra-888 8 8,7,6,5,4,3,2,1 N
gra-2(444) 8 4,3,2,1,1,2,3,4 Y
gra-2(448) 8 8,4,2,1,1,2,4,8 Y
gra-2(888) 16 8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8 Y
“rg” type Layer number Height (nm) Sf in the X- and Y-axes
rg-442 2 2,4 Y
f
This is the abbreviation for rg-884 4 8,4,2,1 Y
“symmetry;” “N” means “no” and rg-16168 8 16,14,12,10,8,6,4,2 Y
“Y” means “yes”

Furthermore, all of the PS/PMMA blend cases showed the vigorous phase separation on rougher planes. The
same trends as those described above with nanoparticle values of V ROPco-884 and V ROPco-888 were almost
doping. the same at 383, 413 and 443 K. The values of
V ROPco-8812 at 383 and 413 K were also the same, but
PS/PMMA blends induced by various planes the value of V ROPco-8812 at 443 K jumped to 13.3,
which was one order of magnitude higher than its
To investigate which planes induce the most intense phase values at 383 and 413 K. On a much rougher surface,
separation in PS/PMMA blends, four different types of such as a co-8832 type of plane, V ROPco-8832,443K
inducing planes were designed as substrates using our own V ROPco-8832,413K V ROPco-8832,383K, which reveals
program. These four types were denoted “ci,” “co,” “gra,” that PS/PMMA blends undergo more intense phase
and “rg.” The “ci” series of planes used half-spheres with separation at higher temperatures. The surfaces of the
different radii as a mask that simulated different degrees of co-4xx series of planes were rougher than those of the co-
surface roughness. The “co” series had equally spaced 8xx series of planes. The V ROP values of the co-4xx
cubic columns as a mask. The columns had different sizes series were higher than those of the co-8xx series at
and heights to simulate different degrees of surface 443 K, which is also consistent with the conclusion that
roughness. The “gra” series were planes with different more intense phase separation occurs on rougher planes.
widths to simulate different degrees of surface roughness. (2) For the “ci” planes, V ROPci-882 V ROPci-444 at the
The mask was generated by gradually increasing the same temperature. There was no obvious difference
column height across the plane, so that it resembled stairs between the V ROPci-882 values at 383 and 443 K, and
viewed side on. In addition, monodirectional asymmetric there was no obvious difference between the V ROPci-
planes, such as gra-444 and gra-888, and monodirectional 444 values at 383 and 443 K either. Furthermore, the V
symmetric planes, such as gra-2(444), gra-2(448) and gra-2 ROPci-882 and V ROPci-444 values at 413 K were both
(888), were considered. The “rg” series were bidirectional higher than their corresponding values at lower
symmetric planes originating from monodirectional sym- (383 K) or higher (443 K) temperatures. This shows
metric planes, as used in the “gra” series. Tables 3 and 4 that temperature is the most influential factor on the
describe these four types of planes used in the simulations, phase separation of PS/PMMA blends induced by “ci”
and Fig. 11 shows the eighteen planes pictorially. planes, but also that the V ROP values vary irregularly
with temperature. Both the temperature and the degree
Results and discussion of surface roughness contribute to the results.
(3) For the “rg” planes, the same order of V ROPrg-16168
Figure 12 shows the V ROP values of PS/PMMA blends V ROPrg-884 V ROPrg-442 was seen at three temper-
induced by the four types of planes mentioned above; atures, which shows that the degree of surface roughness
eighteen isodensity surface pictures are shown for the 1/3 is an important factor for inducing phase separation,
blend case at 413 K as examples at the top of each chart. especially for rougher planes with high step gradients,
The following trends can be discerned from these figures: such as rg-16168. Nearly the same value of V ROPrg-442
was observed at 383 and 443 K—the same situation as
(1) For the “co” planes, V ROPco-8832 V ROPco-8812 V seen for V ROPrg-884—so it can be presumed that for
ROPco-888 V ROPco-884 and V ROPco-4432 V the smoother planes, such as rg-884 and rg-442,
ROPco-4412 V ROPco-448 V ROPco-444, which temperature is more important than the degree of
shows that the PS/PMMA blends underwent more surface roughness in inducing phase separation.

J Mol Model (2011) 17:607–619 617

Fig. 11 The four types of plane used in the simulations: “ci,” “rg,” “gra,” and “co”

(4) For the “gr” planes, the V ROP values for gr-444 were surface roughness plays a small role in inducing phase
lower than those for other “gr” planes at 383, 413 and separation for “gr” planes. In contrast, it can be deduced
443 K, and all of these planes presented similar values at from the extremely high value of V ROP at 413 K that
383 and 443 K; in addition, the other “gr” planes except temperature is the main inducing factor for these planes.
for gr-444 presented nearly the same V ROP values at (5) For all of the “ci,” “rg,” and “gr” planes, the V ROP
383, 413 and 443 K. This shows that the degree of values at 413 K were much higher than those at 383

618 J Mol Model (2011) 17:607–619

Fig. 12 V ROP data for the 1/3 blend case at 383, 413 and 443 K, respectively. Isodensity surface pictures are shown for PS/PMMA blends
induced by different types of planes at 413 K. Red, PS; green, PMMA

and 443 K; furthermore, the V ROP values at 383 and and spread; they could only adjust themselves in the “small
443 K were nearly the same for each plane, which cubic spaces” formed by neighboring column masks.
reveals that using a higher (443 K) or lower (383 K) Furthermore, the big difference between the diffusion
temperature had nearly the same effect on phase abilities of PS and PMMA was also a barrier to movement
separation, and that there was a “critical” temperature and adjustment. Because of the two reasons mentioned
between 443 and 383 K (such as 413 K) that could above, the degrees of phase separation differed widely
induce the most vigorous phase separation. The between the first three types of plane and the “co” planes.
influence of each “ci,” “rg,” or “gr” plane on phase
separation mainly depended on this critical tempera-
ture, not on the degree of surface roughness, in Conclusion
contrast to what was seen for the “co” planes.
The two most important processes during the MD simula-
The “ci,” “rg,” and “gr” planes were different from the tion are the annealing process after minimization and the
“co” planes in that the first three types of plane had MD simulations with three NVT and NPT ensemble cycles.
continuous mask distributions on their surfaces, which The strong polymer end–end interactions, which evidently
allowed the PS/PMMA blends to move and spread more influence the simulation results, can only be reduced by
freely until they could adjust themselves to reach stable performing the above two processes. Based on the
phase separation. In contrast, the “co” planes did not have calculated δ values for different numbers of repeating units
continuous mask distributions on their surfaces, which of PS and PMMA, their representative chain lengths were
meant that the polymer did not have enough room to move found to be 30 and 60, respectively. By comparing the

J Mol Model (2011) 17:607–619 619

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