The document describes a method for correlating thermodynamic modeling and molecular simulations to predict liquid-liquid equilibrium in ternary polymer mixtures. The method uses a modified double lattice theory thermodynamic model combined with molecular simulations to determine interaction energy parameters. Dummy atoms are used to represent polymer segments in the simulations. The method is applied to four real ternary polymer systems, with the energy parameters directly used in the thermodynamic model. The results show good agreement with experimental observations using one or no adjustable parameters.

1. Fluid Phase Equilibria 307 (2011) 202–207
Contents lists available at ScienceDirect
Fluid Phase Equilibria
journal homepage: www.elsevier.com/locate/fluid
Correlation of thermodynamic modeling and molecular simulations for
liquid–liquid equilibrium of ternary polymer mixtures based on a
phenomenological scaling method
Suk Yung Oh, Young Chan Bae∗
Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Republic of Korea
a r t i c l e i n f o
Article history:
Received 27 September 2010
Received in revised form 27 April 2011
Accepted 30 April 2011
Available online 30 May 2011
Keywords:
Molecular thermodynamics
Molecular simulation
Ternary polymer mixtures
a b s t r a c t
In our previous study [S.Y. Oh, Y.C. Bae, J. Phys. Chem. B 114 (2010) 8948–8953], we presented a new
method to predict liquid–liquid equilibria in ternary simple liquid mixtures by using a combination of a
thermodynamic model and molecular simulations. As a continuation of that effort, we extend our previ-
ously developed method to ternary polymer systems. In the simulations, we used the dummy atoms to
calculate the pair interaction energy values between the polymer segments and the solvent molecules.
Furthermore, a thermodynamic model scaling concept is introduced to consider the chain length depen-
dence of the energy parameters. This method was applied to ternary mixtures incorporating low to high
molecular weight polymers. The method presented here well described the experimental observations
using one or no adjustable parameters.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Phase behaviors and thermodynamic properties of polymer
mixtures have proven to be of noticeable interest from both exper-
imental and theoretical points of view due to their importance for
the design and optimization of the polymer production plants. In
many cases, most commercial polymeric materials produced are
a mixture of multiple components; however, compared to binary
systems, experimental and theoretical considerations on multi-
component polymer systems have been limited. It is necessary to
have a proper model and method able to describe the behaviors
of the multi-component mixtures in the widest range of operating
conditions.
The most widely known theoretical treatment of polymer solu-
tions is the lattice model of Flory and Huggins [1,2]. Based on the
Flory–Huggins (FH) model, Scott [3] has treated the system sol-
vent + non-solvent + polymer with the assumption of “single liquid
approximation” and has obtained valuable qualitative results. A
similar approach was given by Tompa [4], which has considered
the effect of the chain length on the phase diagram. Following
these classical works, numerous studies have been done on the
applicability and the modification of the FH theory during the last
few decades [5–10]. Recently, Hu and co-workers [11–13] reported
∗ Corresponding author. Tel.: +82 2 2298 0529; fax: +82 2 2296 0568.
E-mail address: ycbae@hanyang.ac.kr (Y.C. Bae).
URL: http://www.labmtl.com (Y.C. Bae).
Monte Carlo simulation results of liquid–liquid equilibria (LLE) for
ternary chain molecule systems and developed a new lattice model
for multi-component systems. Equation of state (EOS) models,
based on theories such as the lattice fluid (LF) theory or statistical
associating fluid theory (SAFT) have also successfully interpreted
the phase equilibrium of the ternary polymer systems at times
[14–18].
Almost all of the thermodynamic models are capable of calculat-
ing the phase equilibria for ternary systems by using binary model
parameters obtained from experimental data for the three sub-
stituent binaries. Accurate predictions are often possible for ternary
vapor–liquid equilibria (VLE) with binary data alone, but are hardly
obtained for ternary LLE [19]. Furthermore, in common cases where
constituent binary data are not available, many fitting parameters
may have to be used to correlate the phenomena. Due to the lim-
ited data on the polymer mixtures, phase equilibrium calculations
of ternary polymer mixtures are more difficult than ordinary liquid
mixtures.
Modern molecular thermodynamic methods combined with
molecular simulation have been able to overcome the difficulties
originating from the above-mentioned problems [20–22]. In our
recent paper [20], we presented a new method to predict LLE in
ternary liquid mixtures by using a combination of a thermody-
namic model and molecular simulations. A modified double lattice
(MDL) theory [23,24] is adopted to express the Helmholtz function
of mixing, and the Dreiding force field [25] is used in simulations.
In prediction of the phase diagrams, model parameters are simply
obtained from the molecular simulations so correlating work is not
0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.fluid.2011.04.026

2. S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 203
Table 1
Ternary systems investigated in this study [29].
System T (K) (1) (2) (3)
A 223.15 258.15 298.15 PEG 200 n-Heptane Dichloro methane
B 323.15 PS 233,000 2-Butanone 1-Propanol
C 323.15 PS 233,000 2-Butanone Methanol
D 298 PS 300,000 Ethyl acetate Methanol
required. In this study, extending the method to ternary polymer
mixtures is presented by revising a previously developed method.
Dummy atoms are introduced to represent the polymer segments
in simulations, and chain length dependence is accounted for in the
energy parameters to scale down that of the monomer–solvent pair.
Four real ternary polymer systems are investigated by the method
presented here.
2. Thermodynamic model
The MDL theory is chosen as a thermodynamic model because
this model provided good predictions and more satisfactory results
than those given by the FH theory or lattice cluster theory (LCT)
[26–28], according to our previous work [20].
The expression of the multi-component lattice model starts
with a simple cubic lattice (coordination number, z = 6) containing
Nr =
K
i=1
Niri sites that are composed of K components. Each
component i fills the ri lattice sites completely by Ni molecules. Fol-
lowing our previous work [20], the Helmholtz function of mixing
is given by
mixA
NrkT
=
K
i=1
i
ri
ln i +
Cˇ
2
K
i=1
K
j=1
i j
1
ri
−
1
rj
2
+
K
i=1
K
j=1
aij i j −
⎛
⎝
K
i=1
K
j=1
a∗
ij i j
⎞
⎠
2
(1)
where kT have their usual meaning, and i(= Niri/Nr) is the volume
fraction of species i. aij and a∗ are a generalized function of the
reduced interchange energy ˜εij, chain lengths ri and rj as.
aij =
˜εij
2
2 − i
ri
+
j
rj
+
1
ri
+
1
rj
(2)
a∗
ij
= ˜εij
C
4
(3)
where universal constants, Cˇ and C are 0.1415 and 1.7986,
respectively [23]. The reduced interchange energy between i and j
segments, ˜εij, is given by
˜εij =
(εii + εjj − 2εij)
kT
(4)
where εij is the interaction energy between segments i and j. The
first two terms of Eq. (1) are the entropy contribution, and the last
terms of Eq. (1) are the energy contribution.
The chemical potential of component i is necessary for calculat-
ing the binodal line.
i
kT
=
∂( mixA/NrkT)
∂Ni T,V,Nj /= i
(5)
A coexistence curve can be determined from the following con-
ditions
i
= i
(i = 1, 2, . . . , K) (6)
where i is the change in chemical potential upon isothermally
transferring component i from the pure state to the mixture. Super-
scripts and denote two phases at equilibrium. For calculation of
the phase equilibrium, the critical condition is given by
A ≡
A 11 A 12 · · · A 1K
A 21 A 22 · · · A 2K
.
.
.
A K1 A K2 · · · A KK
= 0 (7)
and
A ≡
∂ A
∂ 1
∂ A
∂ 2
· · ·
∂ A
∂ K
A 21 A 22 · · · A 2K
.
.
.
A K1 A K2 · · · A KK
= 0 (8)
where A ij is defined by
A ij =
∂2( mixA/NrkT)
∂ i∂ j T,V
(i, j = 1, 2, . . . , K) (9)
3. Molecular simulation
Thermodynamic modeling was further applied to the LLEs of real
ternary polymer systems chosen from the LLE data handbook [29],
Table 1. Pairwise interaction energy parameters were determined
by a molecular simulation technique developed by Blanco et al.,
[30,31] available in the Blends module of the commercial software
Materials Studio (version 4.4) from Accelrys. The procedure used in
this study involves the following stages. First, the structures of the
molecules were constructed using the sketching tools in the materi-
als visualizer. The repeat units of polymers, which are polyethylene
glycol (PEG) and polystyrene (PS), and solvent molecules, were built
in this investigation. Only one molecule of each kind was used in
the Blends calculation. After the construction of the molecules,
the geometry was optimized by energy minimization, and then
the pairwise interaction energy values were calculated by using
a Monte Carlo approach that includes constraints arising from the
excluded volume constraint method [30,31]. This method allows
for the generation of a large number of relative orientations of the
two molecules, thereby allowing for a more reliable estimate of the
average interaction energy in binary mixtures. The pairwise inter-
action is calculated for a specified number of times and a probability
distribution, P(εij) is generated. An explicit temperature depen-
dence of εij is incorporated by averaging the binding energies using
the Boltzmann method:
εij T
=
dεijP(εij)εij exp(−εij/kT)
dεijP(εij) exp(−εij/kT)
(10)
From the simulation results, attractions for pairs were given
as a negative pair potential because the interaction energy is
defined as the energy required to separate two bodies to an infinite

3. 204 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207
Table 2
Energy parameters obtained for the ternary systems using molecular simulation.
System ε11/k (K) ε22/k (K) ε33/k(K) ε12/k (K) ε13/k (K) ε23/k (K)
A 136.36 545.39 424.29 250.65 245.39 482.99
B 747.49 399.79 281.29 521.38 422.89 334.45
C 747.49 399.79 152.76 521.38 292.10 242.45
D 747.49 434.94 152.76 525.06 292.10 254.55
n
X
X
O
O
ethyl acetate
a b
polystyrene
Fig. 1. Examples of the constructed structures in the simulations, either the whole
molecule in the case of solvent or the monomer repeat unit in the case of polymer:
(a) ethyl acetate, (b) polystyrene.
intermolecular separation. When applying our model, we take
an opposite direction of the simulated energy value because the
energy definitions of the two methods (our model and Blends) are
equal to an amount, but opposite in sign. The results are listed in
Table 2. The obtained energy parameters (εii and εij) are further
directly used in Eq. (4) or (12) of the thermodynamic model. In
the simulations, the average interaction energy was calculated for
the generated N configurations. As N is increased, a better average
interaction energy is obtained. Here 107 pair configurations were
generated during the simulations.
Fig. 1 shows examples of the structures in the simulations, either
the whole molecule in the case of the solvent (Fig. 1a) or the
monomer repeat unit in the case of the polymer (Fig. 1b). Dummy
atoms are introduced to consider the effects of polymer connec-
tivity, which are indicated by a letter “X” in Fig. 1b. Two dummy
atoms were located at the head and tail positions of the polymer
segment, making some positions of the polymer segment inacces-
sible to solvent molecules. Therefore, any configurations violating
this constraint would be rejected. To obtain the optimum struc-
ture of the redefined polymer segment, these two dummy atoms
represent –CH3 in the united atom approximation. The interaction
energies associated with the dummy atoms were not calculated
although the dummy atoms were considered to be methyl groups.
The repeat units further than one CH3 group away do not hinder the
solvent positions, but using a single dummy atom at both ends of
the polymer is sufficient to mimic a real polymer segment in these
simulations [30].
The Dreiding force field [25] was used in these simulations,
which is a simple generic force field based on simple hybridiza-
tion rules rather than on specific combinations of atoms. This force
field is useful for predicting structures and dynamics of organic, bio-
logical, and main-group inorganic molecules. Energetically, the van
der Waals interactions are captured by the Lennard–Jones poten-
tial, and electrostatic interactions are defined by atomic monopoles
and a screened Coulombic term. An explicit Lennard–Jones 12–10
potential represents hydrogen bonding.
1.000.750.500.250.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
(3)
(2)
298.15K
253.15K
223.15K
(1)
Fig. 2. Phase diagram of the ternary liquid–liquid equilibria of a PEG 200 (1) -n-
heptane (2) -dichloro methane (3) system at three different temperatures (223.15 K,
253.15 K and 298.15 K). Open symbols are experimental data [29], and the lines are
calculated by the model.
4. Results and discussion
4.1. Ternary LLE phase diagrams in polymer systems
In ternary systems listed in Table 1, the polymer molecule is
regarded as component 1 and the smaller solvent molecule of the
two solvents as component 3. We setr3 = 1 for the smallest molecule
and the chain length parameters of the components 1 and 2 are
given by the classical method as
ri =
viMWi
v3MW3
(i = 1, 2) (11)
where vi and MWi are specific volume and molecular weight of an
ith component, respectively. In the lattice models, size parameter,
ri, is more of volume ratio than weight ratio because the size is
related to the occupied sites of the lattice. Therefore, Eq. (11) is
more reasonable than weight ratio when using the lattice thermo-
dynamics.
Application to low molecular weight polymer systems is first
considered by using system A of Table 1. Table 3 provides the model
parameters used in this system. The predicted results of our model
and the experimental cloud point data of PEG 200 solutions at
different temperatures (223.15 K, 253.15 K and 298.15 K) [29] are
shown in Fig. 2. These systems show Type 1 phase behavior of the
Treybal classification [32], i.e. in a ternary system that has only
Table 3
Model parameters of the ternary PEG solutions (note: ˜εij = (εii + εjj − 2εij)/kT).
System T (K) r1 r2 r3 ˜ε12 ˜ε13 ˜ε23
A 223.15 2.772 2.288 1 0.809 0.313 0.017
258.15 0.699 0.271 0.014
298.15 0.605 0.234 0.012

4. S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 205
Table 4
Model parameters of the ternary PS solutions (note: ˜εij = (εii + εjj − 2εij − εr )/kT).
System r1 r2 r3 εr/k ˜ε12 ˜ε13 ˜ε23
B 2966.4 1 1 61.4 0.323 0.566 0.038
C 5483.9 1 1 87.3 0.323 0.978 0.209
D 7060.8 1 1 104.3 0.444 1.061 0.264
one separate region. Due to the higher values of r1, r2, and ˜ε12,
PEG and n-heptane produce a pair of partially miscible liquids. As
depicted in Fig. 2, the mixing region of the phase diagram increases
slightly as the temperature increases from 223.15 to 298.15 K. The
model parameters ri and εij are regarded as temperature indepen-
dent in this work, but slightly decreasing ˜εij induces a larger mixing
region. Reasonable agreement between the model and experimen-
tal data without the use of adjustable parameters confirms the
successful application of our proposed method to the ternary poly-
mer systems. Accurate predictions were already expected for low
molecular weight polymer mixtures because their behaviors are
not significantly different from those of single liquid mixtures.
For high molecular weight polymer mixtures, prediction is not
easy so an additional concept is introduced to interpret unusual
phenomena. To model the experimental observations appropri-
ately, it turned out to be necessary to introduce a chain length
dependence of the energy parameters. For polymer mixtures, local
correlations between unlike segments are amplified due to the
combined effects of large chain length and self avoidance, leading
to a reformulation of the segment-segment interaction parame-
ter [33]. For that purpose, Eq. (4) was reformulated by analogy to
the scaling concept given by de Gennes [34] for the chain length
dependence of the FH interaction parameter, as follows:
˜εij =
(εii + εjj − 2εij − εr)
kT
(12)
where εr is the reduced interaction energy of chain length depen-
dence and was obtained in the light of the ternary LLE data. The
values of εr are expected to increase as r1 increases. Thus, it might
be better interpreted by using the functional form of εr, as follows:
εr
k
= ˛(r1 − 1)ˇ
(13)
Similar functional approaches are given by other literatures
[35–37].
Systems B, C, and D of PS in Table 1 [29] were considered to inves-
tigate the ternary mixtures of high molecular weight polymers. We
set r2 = r3 = 1 for solvent molecules because the chain length of the
polymer is very high (r1 r2, r3). Only small differences within 0.1
percent result from r2 being exactly 1 instead of being calculated
using Eq. (11). To incorporate the chain length dependence of the
polymer, the reduced interchange energies of the pairs of polymer
and solvent (1–2 and 1–3) are calculated by Eq. (12), and those of
the solvents pair (2–3) are calculated by Eq. (4). For maintaining
simplicity of the method, εr for 1–2 and 1–3 are identical for the
same system.
Table 4 provides the model parameters of the three ternary PS
systems and, using these parameters, the calculated results were
compared with experimental data as shown in Figs. 3–5. The calcu-
lated curve fit fairly well to experimental values using only one
scaling parameter (εr). All three PS solutions have one separate
region and a relatively high value of the reduced interchange energy
(˜ε13) confirmed that the components 1 and 3 (PS and alcohol) pro-
duce a partly miscible pair. The values of εr are increased reasonably
as r1 increases. Fig. 6 shows the reduced energy parameter versus
the polymer chain length parameter. Eq. (13) with ˛ = 0.45 and
ˇ = 0.61 gives a good approximation of the chain length depen-
dence. As such the present method could be used as a predictive
tool in high polymer mixtures using the scaling function with Eq.
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
(3)
(2)
(1)
Fig. 3. Phase diagram of the ternary liquid–liquid equilibria of a PS 233,000 (1) -2-
butanone (2) -1-propanol (3) system at 323.15 K. Open symbols are experimental
data [29], and the line is calculated by the model.
(13) as an universal function of each polymer. However, it is very
difficult to accurately predict the phase equilibrium of high polymer
mixtures on account of the various limitations in polymer materi-
als.
4.2. Significance and limitation
The validity of using the molecular simulation to predict the
phase equilibrium depends on the theoretical model it is based
1.000.750.500.250.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
(3)
(2)
(1)
Fig. 4. Phase diagram of the ternary liquid–liquid equilibria of a PS 233,000 (1) -
2-butanone (2) -methanol (3) system at 323.15 K. Open symbols are experimental
data [29], and the line is calculated by the model.

5. 206 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207
1.000.750.500.250.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
(3)
(2)
(1)
Fig. 5. Phase diagram of the ternary liquid–liquid equilibria of a PS 300,000 (1) -
ethyl acetate (2) -methanol (3) system at 298 K. Open symbols are experimental
data [29], and the line is calculated by the model.
upon and the simulation technique applied. MDL theory, of course,
has many obstacles for this type of calculation. Our thermodynamic
model is based on an incompressible lattice, so the volume change
of mixing is not considered. Flexibilities and structural effects of
the polymer chains are also neglected; it assumed that all polymers
have freely flexible chains. Lastly, polydispersity of the polymer is
neglected, and all polymers are regarded as monodisperse. These
limitations in dealing with polymers may lead to errors in phase
equilibrium calculations. In simulations, choosing a proper force
field for a given chemical is important, because calculated values
are strongly dependent on the force field applied. The suitability of
the Dreiding force field was confirmed again in this work for the
method combined with a thermodynamic model.
The methodology outlined here can well predict the ternary LLE
of mixtures of simple liquids or oligomers, but one scaling parame-
ter is introduced for accurate prediction of high polymer mixtures.
This shows the possibility of calculating the ternary phase behavior
of polymer mixtures where constituent binary data are not avail-
able, because the model parameters of any given chemicals can
1000050000
0
50
100
150
εr
/Κ
r1
(polymer chain length parameter)
Fig. 6. Chain length dependence of reduced energy parameter. Open symbols are
the reduced energy parameters of the systems B, C, and D. The line is calculated by
Eq. (13).
be achieved in such a simple manner. Though it has restrictions
on prediction, qualitative study or relative comparisons are always
possible. For binary polymer mixtures, Fan et al. [30] and Patnaik
et al. [22] showed similar studies using molecular simulations with
the FH model. They provided a detailed understanding of the mis-
cibility of the polymer mixtures qualitatively, but had difficulties
in predicting the phase behavior quantitatively. The ultimate goal
of applying the method to the real systems was successful from the
viewpoint of both quantitative and qualitative prediction. Despite
the use of an inevitable scaling parameter, this work provides a
new method to predict the ternary polymer mixtures and describes
quantitatively the phase behavior of polymer with mixed-solvent
systems fairly well in spite of its simplicity.
5. Conclusions
A novel approach to calculations of LLE of ternary polymer mix-
tures is introduced by combination of a thermodynamic model and
molecular simulations. MDL theory and the blend module with
a Dreiding force field are adopted as a means to the method. In
the calculation of high polymer mixed-solvent systems, a certain
degree of scaling on the energy parameters of polymer-solvent
pairs is necessary in order to match the experimental observations.
Good agreement was obtained between the experimental data and
the calculated results with using one or no adjustable parameters.
Despite a few limitations, the present method can serve as a basis
to develop more efficient methods for practical applications.
Acknowledgments
This work was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF) grant
funded from the Ministry of Education, Science and Technology
(MEST) of Korea for the Center for Next Generation Dye-sensitized
Solar Cells (No. 2011-0001055).
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