Post on 29-Apr-2023
HIGHLIGHT
Determining the Phase Behavior of Nanoparticle-FilledBinary Blends
GANG HE,1 VALERIY V. GINZBURG,2 ANNA C. BALAZS11Chemical Engineering Department, University of Pittsburgh, Pittsburgh,Pennsylvania 15261
2The Dow Chemical Company, Building 1702, Midland, Michigan 48674
Received 4 January 2006; accepted 16 May 2006DOI: 10.1002/polb.20887Published online in Wiley InterScience (www.interscience.wiley.com).
Correspondence to: A. C. Balazs (E-mail: balazs1@engr.pitt.edu)
ABSTRACT: Nanoparticle addi-
tives provide a means of impart-
ing the desired electrical, optical,
or mechanical properties to a
polymeric matrix. The difficulty
faced in creating these composites
is determining the optimal condi-
tions for forming a thermody-
namically stable mixture, where
the particles will not phase sepa-
rate from the matrix material.
This challenge is even more
daunting when the polymeric ma-
trix is itself a multicomponent
mixture, as is often the case in
advanced materials. Ideally, the
nanoparticles would not only con-
tribute the needed physical prop-
erties, but also stabilize the mix-
ture so that the entire system
forms a single-phase system. In
this study, we use a free energy
expression for a binary blend that
contains nanoparticles and take
the interaction parameters between
the different species to be inde-
pendent variables. Thus, the par-
ticles can have distinct enthalpic
interactions with each of the poly-
meric components. Using this ex-
pression, we determine the condi-
tions under which the mixture
forms a stable, single-phase mate-
rial. In particular, we isolate how
variations in the system’s parame-
ters (e.g., polymer composition,
particle volume fraction, particle
size, interaction energies) affect
the phase diagrams. The findings
provide guidelines for creating ef-
fective formulations and can allow
researchers to understand how
choices made in the nature of the
components affect the overall mac-
roscopic properties. VVC 2006 Wiley
Periodicals, Inc. J Polym Sci Part B: Polym
Phys 44: 2389–2403, 2006
Keywords: nanocomposites;
nanoparticles; theory
2389
INTRODUCTION
One of the challenges in formulating new polymeric
mixtures is predicting how additives will affect the
overall phase behavior and performance of the mate-
rial. This is particularly difficult when the additives are
solid nanoparticles, which are typically introduced to
impart or enhance a range of physical properties (e.g.,
optical, electrical, and mechanical). To provide guide-
lines for creating the desired mixtures, researchers
have developed theoretical models for the equilibrium
and dynamic behavior of mixtures of diblock copoly-
mers and nanoparticles,1–8 as well as models for the
structural evolution of particle-filled binary blends.9–16
However, there have as yet been few systematic theo-
retical studies on the thermodynamic properties of mix-
Valeriy V. Ginzburg is a Research Specialist at The Dow Chemical
Company in Midland, Michigan. He received his B.S. in Physics
and Ph.D. in Polymer Physics at the Moscow Institute of Physics
and Technology (Russia). He subsequently worked as a postdoctoral
Research Associate at the University of Colorado (1993–1997) and the
University of Pittsburgh (1998–2000), before joining the Corporate
R&D at The Dow Chemical Company in early 2001. His research inter-
ests include polymer, colloid, and nanocomposite thermodynamics,
mechanical properties of blends and composites, polymer crystals and
liquid crystals.
Anna C. Balazs is the Distinguished Professor of Chemical Engineering
and the Robert von der Luft professor at the University of Pittsburgh.
She is also a Senior Visiting Fellow at the Oxford Center for Advanced
Materials and Composites at Oxford University, UK. She received her
B.A. in Physics at Bryn Mawr College and her Ph.D. in Materials Sci-
ence at the Massachusetts Institute of Technology. After postdoctoral
work in the Polymer Science Department at the University of Massa-
chusetts, Amherst, she joined the faculty at the University of Pittsburgh
in 1987. Her research involves developing theoretical and computa-
tional models to capture the behavior of polymer blends, nanocompo-
sites, and multicomponent fluids. Balazs served as the Chair of the Di-
vision of Polymer Physics of the APS (2000–2001), Co-Chair of the
Spring MRS meeting (2000), and currently serves on the APS Public
Policy Committee. In addition, she is a Fellow of the APS. She was
Chair of the Polymers West Gordon Conference in 1999, and in 2003
received the Maurice Huggins Award.
ANNA C. BALAZS
VALERIY V. GINZBURG
Gang He is a Technology Development Engineer at Timbre Technol-
ogy Inc. He received his B.A. in Physics at Sichuan University, P.R.C.
and Ph.D. in Physics at the Johns Hopkins University. After postdoc-
toral work in the Department of Mechanical Engineering at the Massa-
chusetts Institute of Technology and the Department of Chemical Engi-
neering at the University of Pittsburgh, he joined Timbre Technology
Inc. in 2005. His current interest involves applying electromagnetic
simulation and optimization methods to Critical Dimension metrology
for semiconductor process controls.GANG HE
2390 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
tures of nanoparticles and binary polymer blends.
Thus, there are few guidelines that could elucidate
how the size, volume fraction, or chemical nature of
the particles affect the phase behavior of the system.
To address the aforementioned issue, Ginzburg
recently carried out a thermodynamic analysis to deter-
mine the influence of nanoparticles on the miscibility of
blends of A and B homopolymers.17 The enthalpic inter-
actions in the system are characterized by the Flory–
Huggins parameters, vAB, vAP, and vBP, where the first
of these terms describes the interaction between the
homopolymers and the latter two describe the interaction
between the respective polymer and the particle. The lat-
ter study focused on the specific case where vAB ¼ vBPand vAP ¼ 0, implying that the nanoparticles are essen-
tially A-like and, thus, are preferentially wet by the A
domains. It was found that the addition of these A-likenanoparticles can either promote or hinder the mixing of
polymers, depending on the particle radius RP and the
polymer degree of polymerization N.Herein, we extend the previous work to investigate the
most general case, where all three enthalpic interaction
parameters vAB, vAP, and vBP are considered as independ-
ent variables. In this manner, we can establish how the
chemical nature of the particles affects the phase behavior
of the composites. By coating the surface, researchers can
readily modify the chemistry of the particles and, thus,
the enthalpic interactions between these solids and the
polymeric components. Understanding how to tailor the
nature of the coating to control the thermodynamic prop-
erties of the mixture is vital for creating the desired for-
mulations. Since the particle size and the chains’ degree
of polymerization influence the phase behavior at given
values of the three v parameters, we also examine the
phase space as a function of RP and N. The predictions
from these studies allow researchers to understand how
choices made in the nature of the components affect the
overall macroscopic properties.
In what follows, we first describe the theoretical model
and method for determining the stability of the polymer/
particle mixtures. In Results and Discussion, we present
sample phase diagrams that illustrate the effect of the inde-
pendent variables. In Conclusions, we summarize the
observed trends and discuss the salient means of control-
ling the phase behavior of these complex mixtures.
THE MODEL
The system under consideration is composed of homopoly-
mers, A and B, whose respective degrees of polymerization
are given by NA and NB. Here, we only consider the case
where NA ¼ NB ¼ N. In the binary mixture, the volume
fractions of the A and B components are / and (1 � /),
respectively. To this polymer blend, we add a volume frac-
tion w of nanoparticles of radius RP. The addition of the
particles changes the total volume fraction of A and B pol-
ymers to (1 � w) / and (1 � w) (1 � /), respectively. We
define the parameter a ¼ vPq0, where vP ¼ 4pR3P/3 is the
volume of one particle, and N/q0 : N 4pr30/3 is the vol-
ume occupied by a single homopolymer chain, with r0being the radius of a monomer.
The free energy of the above ternary mixture can be
written as
F ¼ ð1�wÞ/N
lnð1�wÞ/
N
� �þ ð1�wÞð1�/Þ
N
� lnð1�wÞð1�/Þ
N
� �þ vABð1�wÞ2/ð1�/Þ
þwa
lnwþ pRP
r0
� �4w� 3w2
ð1�wÞ2" #
þ ð1�wÞwa
pRp
r0
� �3R2
P
2Nr20þ vAP
r0RP
� �
�wð1�wÞ/þ vBPr0RP
� �wð1�wÞð1�/Þ: ð1Þ
In the above equation, the first three terms represent the
Flory–Huggins mixing free energy for the A and B compo-
nents. The fourth term (enclosed in square brackets)
describes the contributions of the particles and consists of
the ideal and nonideal entropic interactions of the particles.
We use the Carnahan-Starling equation of state18 to
describe the nonideal contribution to this free energy. The
addition of the Carnahan-Starling term allows us to explic-
itly take into account the excluded volume interactions
between the finite-sized particles. In particular, the term
yields the entropy loss due to the steric interactions, pre-
venting the particles from overlapping.19 The appearance
of a particle size dependent prefactor, p RP
r0
� �, and its func-
tional form was discussed in Ref. 17. We take the same
functional form of p(x) ¼ max (tanh (x � 1),0) in this
calculation. This choice of p(x) interpolates between two
extreme cases. At RP ¼ r0 (x ¼ 1), this term tends to 0;
in this manner, the nonideal term from the Carnahan-
Starling equation does not contribute to the equation. This
ensures consistent treatments of particles and monomers
on homopolymer chains, since they have the same size
and the nonideal term has been ignored for monomers.
The other extreme case corresponds to larger particles
wherein p RP
r0
� �must approach 1 to correctly model the
nonideal ‘‘entropic free energy’’ of hard spheres given by
the Carnahan-Starling equation of state.18 Notice that
p(x) increases monotonically toward 1 as x increases.The term immediately following the Carnahan-Starling
term (the fifth term) describes the effective ‘‘depletion inter-
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2391
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
action’’ between the particles and the polymer.1 In particular,
the particles dispersed in the melt acquire an effective ‘‘sur-
face tension’’ since the area occupied by the a particle is
blocked for the nearby chains. The derivation of this term is
given in Refs. 1 and 17. The term can be derived by consider-
ing the stretching of the polymer chains in the vicinity of a
particle. On average, each polymer chain acquires a stretch-
ing free energy of3R2
P
2Nr20
every time a chain encounters a par-
ticle. Summing up these contributions in a unit volume,
we obtain the fifth term in eq 1. Note that this entropic
contribution is independent of the chemical nature of the
chain; it only depends on chain length and particle size.
The remaining terms in the free energy describe the
enthalpic interactions between the particles and each of
the polymers. In the enthalpic interaction terms, a pre-
factor r0RP
appears because the total surface area of the
particles per unit volume at a given particle concentra-
tion is proportional to the surface-to-volume ratio of
particles. At the same time, when RP ¼ r0, the defini-
tion of v as a monomer–monomer interaction requires
that the prefactor be equal to 1.
The above equations apply to spherical particles and
linear chains. For particles with different shapes, one
must modify both the equation of state for the particles
and the form of the depletion interaction term. We note
that this calculation has been carried out for the case
of rod-like particles and diblock copolymers4 using our
hybrid ‘‘SCF/DFT’’ model, which combines the self-con-
sistent field (SCF) theory with density functional theory
(DFT).2,3,5,6
The homogeneous phase can be stable only if the
free energy assumes a positive definite, quadratic form
with respect to long wavelength density variations near
the uniform state. This implies that the following three
conditions must all be satisfied to guarantee the stabil-
ity (or metastability) of the homogeneous phase; viola-
tion of any of these equations lead to phase separation.
The conditions are
1:@2F
@/2> 0
2:@2F
@w2> 0
3: J ¼ det
�����@F2
@/2@F2
@/@w
@F2
@/@w@F2
@w2
�����¼ @F2
@/2
� �@F2
@w2
� �� @F2
@/@w
� �2> 0
ð2ÞAs mentioned earlier, in previous studies,17 it was as-
sumed that vAB ¼ vBP and vAP ¼ 0. This assumption
describes the scenario where the particles are inherently
A-like or coated by an A layer. On the other hand, in
many practical applications, all three interaction parame-
ters, vAB, vAP, and vBP, can be independent of each
other. In this paper, we study the phase behavior of the
most general case, where vAB, vBP, and vAP all vary in-
dependently. Through these calculations, we can investi-
gate how to exploit surface modification as a means of
controlling the morphology of mixtures that comprise a
given polymer composition and particles of a specified size.
From the first condition in eq 2, we have,
vAB <1
2N/ð1� /Þð1� wÞ ð3Þ
In other words, the system will be unstable when vABis sufficiently large (i.e. greater than the right hand side
of the expression). At the same time, since (1 � w) < 1,
this value is always greater than the critical point in
the spinodal curve for the pure, unfilled homopolymer
blends, vcAB ¼ 12N/ð1�/Þ. Thus, properly chosen particles
could potentially act as compatibilizers within the tem-
perature range defined by these two values of vAB (i.e.
the value specified in eq 3 and vcAB), given that the
other two conditions are also satisfied. The second con-
dition can be easily transformed into a restriction on
/vAP þ (1 � /)vBP,
/vAP þ ð1�/Þ vBP <1
2
RP
r0
� �ð2vAB/ð1� /Þ þGðwÞÞ
� AðN;RP; vAB;w;/Þ
GðwÞ ¼ 1
awþ 1
Nð1� wÞ þ pRP
r0
� ��
8� 2w
að1� wÞ4 �3R2
P
aNr20
" #ð4Þ
It will become clear why we choose to write the condition
in the above format after we review the third condition.
The third condition can be transformed into the fol-
lowing expression:
/vAPþð1�/ÞvBP <AðN;RP;vAB;w;/Þ
�2r0RP
� �ð1�2wÞ2 �2vABð1�wÞ2þ 1�w
N/ð1�/Þ� ��1
� 1
2ðvAP�vBPÞ�
1
2ð1�2wÞ�
�
RP
r02vABð1�wÞð1�2/Þþ 1
Nln
/1�/
� ��2�AðN;RP;vAB;w;/Þ�BðN;RP;vAB;wÞ�
1
2ðvAP�vBPÞ�CðN;RP;vAB;w;/Þ
� �2ð5Þ
It is now obvious that satisfying Conditions 1 and 3
automatically guarantees that Condition 2 is also met.
The fact that Conditions 1 (or 2) and 3 imply Condi-
2392 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
tion 2 (or 1) can also be seen easily from eq 2, since
the product in the first term of Condition 3 has to be
greater than 0; so one of the two factors being greater
than 0 automatically implies that the second factor is
also greater than 0.
Using the above-mentioned equations, we now gen-
erate phase diagrams to determine how the stability of
the homogeneous phase is affected by the system
parameters, vAB, vAP, vBP, Rp, and N, as well as the
particle volume fraction w and polymer blend compo-
sition /. It would be difficult to plot a complete, mul-
tidimensional phase diagram in terms of all these
parameters; we can, however, analyze how the size of
the stability region depends on a number of factors. The
studies described in the following text are not meant to
be exhaustive, but rather to illustrate how small changes
in the parameter space can lead to significant changes in
the thermodynamic stability of the mixture.
RESULTS AND DISCUSSION
In the following discussion, we restrict our discussion
to the cases wherein vAP � 0 and vBP � 0. For con-
venience, we introduce the variables x ¼ /vAP þ (1
� /) vBP and y ¼ (vAP � vBP)/2. The variable x is a
measure of the average enthalpic ‘‘field’’ acting on the
particles, due to the homopolymers. The variable y is a
measure of the relative affinity between the particles
and the different homopolymers; when y is negative,
the particles prefer the A phase, while a positive yindicates a preferential interaction between the par-
ticles and the B phase.
In the coordinate system of x ¼ /vAP þ (1 � /)vBPand y ¼ (vAP � vBP)/2, the curve that separates the
stable (metastable) from the unstable region is de-
fined by the function x ¼ A(N, RP, vAB, w, /) �B(N, RP, vAB, w) [y � C(N, RP, vAB, w, /)]
2 (see eq 5),
which represents a parabola with its central axis being
parallel to the x ¼ /vAP þ (1 � /)vBP axis (see the
plots labeled (a) in the appropriate figures). Only sys-
tems with (x, y) to the left of the parabola are stable.
With the aid of the relationships vAP ¼ x þ (1 � /)yand vBP ¼ x � /y, the curves in (a) can be transformed
and expressed directly in terms of vAP and vBP. Theseplots are labeled (b) in the corresponding figures. In
the following subsections, we discuss how each of the
parameters, vAB, w, /, RP, and N, affect the stability
region in the phase diagram.
Dependence of Phase Diagram on xAB
We first examine how the stability region in the (vAP,vBP) plane is affected by variations in the value of
vAB. As discussed earlier, to facilitate the analysis, we
first plot the shape of the stable region in the coordi-
nate system of (x ¼ /vAP þ (1 � /)vBP, y ¼ (vAP– vBP/2.0), and then transform the data back to the coor-
dinate system of (vAP, vBP). With respect to vAB, thisparameter must satisfy Condition 1 (eq 3), which is
determined solely by the composition of the homopoly-
mer blends and the volume fraction w of particles, and
does not depend on the details of the particles (size,
preferential interactions with the two homopolymers).
For any vAB in the range determined by Condition 1,
Condition 3 dictates that the region of stability lies to
the left of the parabolas in Figures 1(a) and 2(a). In
plots (a) and (b), the blue, green, and black curves are
for vAB equal to 0.0002, 0.012, and 0.024, respectively.
The other relevant parameters are fixed at w ¼ 0.18,
RP ¼ 10r0 and N ¼ 100. Figure 1 is for a symmetric
mixture with / ¼ 0.5, while Figure 2 is for an asym-
metric mixture with / ¼ 0.35. By setting w ¼ 0 in eq
3, we see that for the pure binary blend with N ¼ 100,
the system becomes thermodynamically unstable for
vAB � 0.02 when / ¼ 0.5, and for vAB � 0.022 when
/ ¼ 0.35. On the hand, at w ¼ 0.18, Condition 1 dic-
tates that the particle-filled system is thermodynami-
cally unstable for vAB � 0.0244 when the particles are
added to a symmetric mixture (/ ¼ 0.5); when the
particles are added to a blend where / ¼ 0.35, Condi-
tion 1 indicates that the composite will become ther-
modynamically unstable when vAB � 0.0268. Thus, as
noted earlier, for the correct choice of particle interac-
tions and size, the solid fillers can stabilize an other-
wise immiscible blend.
Figures 1 and 2 illustrate the latter point, showing
that the addition of solid, nanoscopic spheres can pro-
mote the formation of a thermodynamically stable mix-
ture. In addition, these figures are particularly useful
for pinpointing how the relative affinity between the
particles and homopolymers affects the phase behavior
of the polymer/particle mixture. Figures 1(a) and 2(a)
reveal that the width of the defining parabolas de-
creases as vAB is increased. This implies that at larger
vAB, the stability of the system becomes more sensitive
to the preferential wetting interactions between the
polymers and particles. In other words, the allowed
range of (vAP � vBP)/2 in the stable region is much
smaller for larger vAB. For the symmetric case [Fig.
1(a)], the curve is centered at (vAP � vBP)/2 ¼ 0,
implying that at larger vAB, the interactions between
the particles and each of the polymers need to be com-
parable in value. Consequently, particles with strong
preferential interactions will destabilize the symmetric
mixture at higher values of vAB.This behavior is in contrast with that observed for
an asymmetric mixture. In Figure 2(a), the central
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2393
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
axis of the stable region is shifted along the y ¼ (vAP� vBP)/2 axis as vAB is increased. Systems containing
neutral particles (vAP ¼ vBP ¼ 0) can become unstable
when vAB is large, while systems containing particles
that display preferential interactions can be stabilized.
It is instructive to consider some general features of
Figures 1(b) and 2(b), which mark the region of phase
stability in the first quadrant of the (vAP, vBP) plane
(where vAP � 0 and vBP � 0). For example, consider
the case where vAB ¼ 0.012 and note that the stable
region of the phase map encompasses points where
both vAP and vBP are greater than vAB. The role played
by the particles in this scenario is similar to that
played by the ‘‘compatibilizing’’ homopolymer C in
A/B/C ternary blends, where all three components are
incompatible, but the incompatibility between A and C
and between B and C is greater than that between A
and B.20–22 In effect, the relatively larger unfavorable
interactions with C promote a greater intermixing of
the A and B and, for the appropriate volume fractions
of C and v parameters, can lead to an overall thermo-
dynamically stable mixture. Another interesting exam-
ple is provided by the case where vAB is greater than
vAP and vBP. In this case, the particles play a more tra-
ditional compatibilizing role, reducing the unfavorable
interactions between the A and B polymers.
Figure 2. Similar to Figure 1, except that / ¼ 0.35, and the respective values of vAB for
the three curves in (a) and (b) are as follows: 0.0002 (blue), 0.013 (green), and 0.027
(black).
Figure 1. (a). Phase map in the (x ¼ /vAP þ (1 � /)vBP, y ¼ (vAP � vBP)/2) plane. Themap indicates where the mixture is stable, given that w ¼ 0.18, / ¼ 0.5, RP ¼ 10r0, and N¼ 100. The blue, green, and black curves correspond to vAB ¼ 0.0002, 0.012, and 0.024,
respectively. (b) Phase map in the (vAP, vBP) coordinate system.
2394 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
To more fully grasp the effect of increasing vAB in
the above-mentioned systems, we again focus on the
first quadrant of the (vAP, vBP) plane and determine
how the area of the stability region in this quadrant
depends on the value of vAB. As noted earlier, RP/r0¼ 10 and w ¼ 0.18, with one curve for / ¼ 0.5 and
another for / ¼ 0.35. As shown in Figure 3, for both
cases, the area of the stable region increases slowly at
smaller values of vAB and then decreases sharply at
larger values of vAB (where the two polymers begin to
phase separate). At very large vAB, the system is local-
ized deep within the spinodal region for the polymer
blend, and thus, the area diminishes to 0. Within this
region, one cannot tune the particle–polymer interac-
tion to compatibilize the blend.
We can analyze a special case alluded to earlier in
more detail. For the special case of neutral particles (vAP¼ vBP ¼ 0), we can determine the value of vAB that
separates the stable and unstable mixtures by solving the
equation C(N, RP, vAB, w, /)2 � A(N, RP, vAB, w, /)/
B(N, RP, vAB, w) ¼ 0. The solution as a function of /provides the spinodal curve for systems of neutral par-
ticles and is plotted in Figure 4 for two different particle
concentrations and sizes. For comparison, the spinodal
curve for pure homopolymer blends is also plotted in the
same figure (see black curve). For the cases examined,
the smaller particles (RP/r0 ¼ 3) act as compatibilizers
of the binary blends since these curves (in red) lie above
the spinodal curve for the pure binary blend. However,
the larger particles (RP/r0 ¼ 10) can act either as a com-
patibilizer for symmetric or relatively less asymmetric
mixtures, or as a ‘‘demixer’’ for highly asymmetric mix-
tures (see blue curves).
Dependence of Phase Diagram onParticle Loading (ww)
With the aid of similar plots, we can analyze how the
shapes of the stable regions in the (vAP, vBP) plane
vary with respect to other compositional parameters. In
particular, we determine the behavior of the system at
three different values of the volume fraction w of filler
particles. We focus on the following three cases: vAB¼ 0.01, 0.02, and 0.0205. The other parameters are
fixed at RP ¼ 10r0, / ¼ 0.5, and N ¼ 100. (Recall that
for the pure symmetric blend at this value of N, vcAB¼ 0.02.) It is particularly noteworthy that for vAB¼ 0.01, the width of the stability region in the (vAP,
Figure 3. The size (area) of stable region in the first quadrant of the (vAP, vBP) plane as a
function of vAB for different values of /. Other parameters in the plot are RP/r0 ¼ 10, w ¼ 0.18,
N ¼ 100. The blue and red curves in the plot correspond to / ¼ 0.5 and 0.35, respectively. In
both cases, the area decreases at large vAB, implying that the system is much more sensitive to
the interactions between the particles and the two homopolymers.
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2395
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
vBP) coordinate frame does not vary systematically
with variations in w. For example, in Figure 5(b), the
width is larger for both small and large w and reaches
a minimum at some intermediate value of w. For vAB
¼ 0.02, however, the trend shifts so that the width of
the stability region systematically increases with in-
creasing w. The image in Figure 6(b), where vAB¼ 0.0205, is very similar to the corresponding curve
Figure 4. Spinodal curve for pure homopolymer blends (black) and systems filled with a w¼ 0.18 volume fraction of particles (top red, RP ¼ 3r0, top blue, RP ¼ 10r0), or a w ¼ 0.1 vol-
ume fraction of particles (lower red, RP ¼ 3r0; lower blue, RP ¼ 10r0). All particles are ‘‘neu-
tral,’’ meaning that vAP ¼ vBP ¼ 0. As shown in the figure, the compatibilizing effect of the fil-
ler particles deteriorates as the particle size becomes larger, especially for highly asymmetrical
mixtures.
Figure 5. Phase maps for various particle volume fractions, w. In plots (a) and (b), w¼ 0.18, 0.1, and 0.05 for the blue, green, and black curves, respectively. Other parameters are N¼ 100, RP ¼ 10r0, / ¼ 0.5, and vAB ¼ 0.01.
2396 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
for vAB ¼ 0.02 (not shown), but with the increase in
vAB, the stability region in the (vAP, vBP) becomes nar-
rower for all three of the selected w values.
We can gain additional insight into the behavior of
these filled systems by considering not only the width
but also the entire size (area) of the stable region. In Fig-
ure 7, we plot this area for the first quadrant of the (vAP,vBP) plane as a function of w; the blue, red, and green
curves correspond to vAB ¼ 0.01, 0.02, and 0.0205,
respectively. For vAB < 0.02, the plot reaches a mini-
mum at some intermediate value of w and diverges as wtends to 0. For vAB > 0.02, there is a region of small w
Figure 6. Similar to Figure 5, but with vAB ¼ 0.0205. In plots (a) and (b), w ¼ 0.18, 0.1 and
0.05 for the blue, green and black curves, respectively. Other parameters are N ¼ 100, RP
¼ 10r0, and / ¼ 0.5.
Figure 7. The size (area) of the stable region in the (vAP > 0, vBP > 0) plane as a function
of w for different values of vAB. The blue, red, and green curves in the plot correspond to vAB¼ 0.01, 0.02, and 0.0205, respectively. Other parameters are N ¼ 100, RP ¼ 10r0, and / ¼ 0.5.
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2397
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
where the plot tends to 0, implying that a minimum vol-
ume fraction of particles are needed to stabilize the mix-
ture. The vAB ¼ 0.02 example is the limiting case for
this system. The addition of nanoparticles with the
appropriate values of vAP and vBP can shift the system
from the critical point to a stable phase. In particular, for
vAB ¼ 0.02, the area of the stable region approaches a
finite value as w tends to 0, although the minimum is
still obtained at some intermediate value of w.
Dependence of Phase Diagram on BlendComposition (//)
We also examined how the shape of the stable region
in the (vAP, vBP) plane depends on the composition of
the homopolymer blends. We considered the following
values for /: 0.5, 0.65, and 0.80. To these blends, we
added a particle volume fraction of w ¼ 0.18. The
other parameters are held fixed at RP ¼ 10r0 and
N ¼ 100. Figure 8 shows the data for vAB ¼ 0.01, and
Figure 9 reveals the plots for vAB ¼ 0.025. (We also
considered the case of vAB ¼ 0.0244; however, the
plots appear very similar to those for vAB ¼ 0.025
and, thus, are not shown.) Figures 8 and 9 both indi-
cate that for fixed particle conditions, the stability
region can be modified by altering the composition
of the blend. Such plots are particularly useful, since
modifying the particles or just the coatings on a
particle (to tailor the v’s) is synthetically more chal-
Figure 8. Phase maps for various blend compositions. In plots (a) and (b), / ¼ 0.8, 0.65,
and 0.5 for blue, green, and black curves, respectively. Other parameters are N ¼ 100, RP
¼ 10r0, and vAB ¼ 0.01.
Figure 9. Similar to Figure 8, except that vAB ¼ 0.025. Notice that for relatively symmetrical
blends, the system cannot be stabilized by adding particles at this volume fraction (w ¼ 0.018)
and value of vAB (no black curve for / ¼ 0.5).
2398 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
lenging than simply adding more or less polymer to
the formulation.
By comparing Figures 8 and 9, we further see that
the stability region depends not only on / but also quite
significantly on the value of vAB. Note that in Figure 9
(for vAB ¼ 0.025), there is no curve for / ¼ 0.5 be-
cause there is no stable region for symmetric mixtures
with w ¼ 0.18 at this value of vAB. As noted earlier, for
w ¼ 0.18 and symmetric blends, we can use eq 3 to
obtain an upper limit or the most stringent constraint on
the value of vAB that will yield a stable particle-filled
blend; for any particle size and values of vAP and vBP,this constraint must be met to yield a thermodynamically
stable system. Specifically, we find that the limiting
value is vAB ¼ 1/(2N(0.25)(1 � w)) ¼ 0.0244. In other
words, symmetric blends cannot be stabilized by the
addition of particles when vAB � 0.0244. Since the
product of (/(1 � /)) reaches a minimum at / ¼ 0.5,
we can see from eq 3 that at vAB < 0.0244, blends can
be stabilized by the appropriate nanoparticles.
In Figure 10, we show the size of the stable region in
the first quadrant of the (vAP, vBP) plane as a function of
/ for the three vAB values mentioned earlier. The blue,
red, and green curves correspond to vAB ¼ 0.01, 0.0244,
and 0.025, respectively. Notice that the size of the stabil-
ity region is the smallest for symmetric mixtures. For
sufficiently large vAB, the figure clearly shows that sym-
metric and relatively less asymmetric systems cannot be
stabilized by tuning the enthalpic interactions between
particles and the two homopolymers of the systems con-
sidered in the plot (i.e. 0.2 � / � 0.8).
Dependence of Phase Diagram on Particleand Polymer Sizes (RP and N)
In Figure 11, we show the effect of varying the parti-
cle size RP for / ¼ 0.5 with N ¼ 100, w ¼ 0.18, and
vAB ¼ 0.02. (To facilitate comparison with Fig. 13, the
data in Fig. 11 are multiplied by N.) It can clearly be
seen that increasing the particle size from RP/r0 ¼ 5 to
RP/r0 ¼ 10 has a significant effect on the phase behav-
ior of the filled mixture. This point is also illustrated
in Figure 12, which shows that the overall stability
region in the (vAP, vBP) plane decreases monotonically
as the particle radius increases (the data is for both /¼ 0.5 and / ¼ 0.35). The translational entropy of
the particles decreases as the particle size increases,
Figure 10. The size of stable region in the (vAP > 0, vBP > 0) plane as a function of / for
various values of vAB. The blue, red, and green curves in the plot correspond to vAB ¼ 0.01,
0.0244, and 0.025 respectively. Other parameters in the plot are RP/r0 ¼ 10, w ¼ 0.18, N ¼ 100.
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2399
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
thereby diminishing the free energy of mixing. With
respect to design rules for creating stable formulations,
the observations indicate that for fixed compositions
and interaction energies, the mixture can potentially be
compatibilized by reducing the size of the particle
additives.
In Figure 13, we show the effect of changing the
homopolymer chain length N in a symmetric mixture
Figure 11. Phase maps for various particle sizes. The blue and green curves in plots (a) and
(b) correspond to particle sizes of 5 and 10, respectively. Other parameters are N ¼ 100, w¼ 0.18, / ¼ 0.5, and vAB ¼ 0.02.
Figure 12. The size of stable region in the plane of (vAP > 0, vBP > 0), as a function of RP/
r0 for different values of /. The blue and green curves in the plot correspond to / ¼ 0.5 and
0.35, respectively. Other parameters in the plot are vAB ¼ 0.02, w ¼ 0.18, N ¼ 100.
2400 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
(/ ¼ 0.5), filled with a volume fraction of w ¼ 0.18
of particles of size RP/r0 ¼ 10. The value of NvAB for
the three curves shown in plots (a) and (b) are held
constant at 2.0. The blue, green, and black curves cor-
respond to N ¼ 100, 1000, and 2000 respectively. As
shown in the plot, at constant NvAB, shorter chains
tend to have a larger region in the (vAP, vBP) phase
space where the system can be stabilized. This is
because we hold NvAB constant, and the absolute value
of the difference in vAB from its critical values is thus
larger for shorter polymer chains; so the system is fur-
ther away from a critical mixture.
Comparison of Theoretical Predictions withAvailable Experimental Studies
Before concluding, we comment that to date, there have
been few systematic experimental studies on particle-
filled binary blends; thus, it is difficult to make a direct,
quantitative comparison between the above-mentioned
theoretical predictions and experiments. We can, how-
ever, make a connection to the available experimental
studies and point out the qualitative agreement between
the latter findings and the trends obtained from these cal-
culations. In the case of A-like particles, the previous
theoretical predictions17 showed good qualitative agree-
ment with experimental studies on a PVA/PMMA blend
filled with fumed silica.16 In particular, the theory repro-
duced the experimental observation that at low PMMA
content, the particles have an adverse effect on polymer
miscibility, but for high PMMA content, they can ac-
tually compatibilize the two polymers.16 In general
terms, Figures 8 and 9 herein show analogous behavior;
that is, at a fixed particle volume fraction, one can tailor
the blend composition to yield a thermodynamically sta-
ble composite. Note that Figure 9, in particular, indicates
that a / ¼ 0.5 mixture is not stable at the fixed value of
vAB and w, while the / ¼ 0.65 and / ¼ 0.80 mixtures
are stable, supporting the observation that particles can
have an adverse effect at a small value of /, but an ad-
vantageous affect at larger /.Studies have also been carried out on styrene-acrylo-
nitrile (SAN)/ethylene-propylene diene monomer rubber
(EPDM) blends that contained surface-treated calcium
carbonate fillers.23 Although the pure blend is immiscible,
the researchers found that surface modification of the filler
allowed them to control the filler–polymer interactions
and, consequently, create a miscible mixture. In particular,
in this system, the interfacial tension between each homo-
polymer and filler was lower than that between the two
homopolymers.23 Consequently, one could argue that the
particles acted as compatibilizers, driving the system to
form a thermodynamically stable mixture. Such behavior
is indeed seen in Figures 1(b) and 2(b); for the cases
where vAB is greater than both vAP and vBP, the addition
of the particles can lead to a miscible composite.
More recently, Composto and coworkers examined
thin films of binary blends and nanoparticles24 and
found that interfacially active nanoparticles can stabi-
lize the structure of polymer blend films and prevent
film rupture. We again note that when vAB is greater
than both vAP and vBP, the particles can act as compa-
tibilizers, causing the entire system to be miscible [see
Figs. 1(b) and 2(b)], whereas the pure binary blend
would phase separate. This finding could potentially
explain the latter experimental observations on the par-
ticle-filled films.
Figure 13. Phase map for various values of the homopolymer chain length N. Other parame-
ters are RP/r0 ¼ 10, NvAB ¼ 2, w ¼ 0.18, and / ¼ 0.5.
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2401
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
Mackay et al. have recently conducted experiments
on mixtures of nanoparticles and a single homopoly-
mer.25 They find that when Rg is less than Rp, the
nanoparticle/homopolymer mixture is thermodynami-
cally stable, while the mixture phase separates in the
opposite limit. The plots in Figure 12 show qualitative
agreement with the latter argument. Finally, it is note-
worthy that Krishnamoorti and coworkers examined
the effect of clay particles on the miscibility of binary
blends26; the clay particles, however, have a high as-
pect ratio and one would have to modify the above
theory to obtain predictions on the thermodynamic
behavior of blends and nonspherical nanoparticles.
CONCLUSIONS
We investigated the phase behavior of a composite that
comprises a binary AB homopolymer blend and spheri-
cal nanoparticles. We considered the most general case
where all three enthalpic interaction parameters, vAB,vAP, and vBP, can vary independently. To carry out this
study, we extended the expression for the free energy
utilized by Ginzburg to probe the phase behavior of bi-
nary blends containing A-like particles.17 Herein, we
determined how the various parameters, vAB, w, /, N,and RP, affect the general shape of the stability region
in the particle–polymer interaction space (vAP, vBP).By plotting the data in terms of the (vAP, vBP) coordi-nates, we could readily visualize how changes in the
above-mentioned variables affect the range of particles
that could be used to create a stable composite.
We note that the model presented herein does
encompass certain limitations. In particular, we treat
the behavior of the system in a mean-field fashion,
neglecting the effects of fluctuations. In addition, we
have commented earlier that the particles can be
coated by a polymeric layer to tailor their enthalpic
interactions with the components of the blend. We do
not, however, take into account the chain-like charac-
teristics of a polymeric coating, but rather, model its
presence through an appropriately chosen value of the
v parameter. If the chains that make up the coating are
relatively short, this approximation is valid; however,
if they are relatively long, we should take into account
the configurational entropy of these anchored moieties
within the total free energy expression for the system.
In these studies, both vAP and vBP are taken to be
� 0; that is, we focused on the first quadrant of the
(vAP, vBP) phase space. This region is particularly
interesting since the values of the v parameters imply
that the particles could have an enthalpically unfavora-
ble interaction with the respective homopolymers.
Nonetheless, we find that the mixture forms a stable
composite for a broad range of system parameters. This
is in fact one of the challenges faced by researchers in
formulating effective mixtures; the design space is enor-
mous, encompassing choices in the values of vAP, vBP,vAB, w, /, N, and RP. The phase diagrams we calculated
here are useful in indicating the limiting values of the
relevant parameters that will still yield a stable system.
It is, however, typically the case that one is con-
strained to use a particular set of polymers and par-
ticles since they impart the desired macroscopic prop-
erties. Consequently, the values of vAB, vAP, and vBPare fixed. In this scenario, our findings on the role of
w, /, and RP are of particular benefit. Specifically, the
results shown in Figures 6 and 7 reveal that there is
commonly an optimal value of w, which yields the
largest stability region. Furthermore, for vAB ¼ 0.0205
in Figure 7, we observed that the size of the stability
region tends to 0 for particle volume fractions below a
critical value; this highlights the fact that frequently wmust be above a minimum amount to stabilize the sys-
tem. It is also worth pointing to the findings in Figure 10,
which indicate the difficulty of creating stable systems
with symmetric or nearly symmetric mixtures of homo-
polymers. In other words, the stability region can be
tailored by considering more asymmetric mixtures of
the A and B homopolymers. Finally, as seen from Fig-
ure 12, reducing the size of the particles also helps in
broadening the stability region for a given set of vparameters. In more general terms, the findings pre-
sented herein can provide important guidance in creat-
ing new functional composite materials.
G. He and A. C. Balazs gratefully acknowledge finan-cial support from the NSF.
REFERENCES AND NOTES
1. Huh, J.; Ginzburg, V. V.; Balazs, A. C. Macromole-cules 2000, 33, 8085.
2. Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.;Balazs, A. C. Science 2001, 292, 2469.
3. Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.;Balazs, A. C. Macromolecules 2002, 35, 1060.
4. Shou, Z.; Buxton, G. A.; Balazs, A. C. ComposInterfaces 2003, 10, 343.
5. Lee, J. Y.; Thompson, R. B.; Jasnow, D.; Balazs,A. C. Macromolecules 2002, 35, 4855.
6. Lee, J. Y.; Balazs, A. C.; Thompson, R. B.; Hill,R. M. Macromolecules 2004, 37, 3536.
7. Buxton, G. A.; Balazs, A. C. Phys Rev E 2003, 67,031802.
8. He, G.; Balazs, A. C. J Comput Theor Nanotech-nol 2005, 2, 99.
2402 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 44 (2006)
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb
9. Ginzburg, V. V.; Qiu, F.; Paniconi, M.; Peng, G.; Jas-now, D.; Balazs, A. C. Phys Rev Letts 1999, 82, 4026.
10. Ginzburg, V. V.; Peng, G.; Qiu, F.; Jasnow, D.;Balazs, A. C. Phys Rev E 1999, 60, 4352.
11. Lee, B.; Douglas, J. F.; Glotzer, S. C. Phys Rev E1999, 60, 5812.
12. Balazs, A. C.; Ginzburg, V. V.; Qiu, F.; Peng, G.;Jasnow, D. J Phys Chem B 2000, 104, 3411.
13. Zhu, Y. J.; Ma, Y. Q. Phys Rev E 2003, 67, 041503.14. Laradji, M.; MacNevin, G. J Chem Phys 2003,
119, 2275.15. Laradji, M. J Chem Phys 2004, 120, 9330.16. Lipatov, Y. S.; Nesterov, A. E.; Ignatova, T. D.;
Nesterov, D. A. Polymer 2002, 43, 875.17. Ginzburg, V. V. Macromolecules 2005, 38, 2362.18. Carnahan, N. F.; Starling, K. E. J Chem Phys
1969, 51, 635.
19. Matsen, M. W.; Barrett, C. J Chem Phys 1998,109, 4108.
20. ten Brinke, G.; Karasz, F. E.; MacKnight, W. J.Macromolecules 1983, 16, 1827.
21. Huang, C.; Olvera de la Cruz, M.; Swift, B. W.Macromolecules 1995, 28, 7996.
22. Huang, C.; Olvera de la Cruz, M. Phys Rev E1996, 53, 812.
23. Murgic, Z. H.; et al. Macromol Mater Eng 2002,287, 684.
24. Chung, H.; Ohno, K.; Fukuda, T.; Composto, R. J.Nano Letts 2005, 5, 1878.
25. Mackay, M. E.; Tuteja, A.; Duxbury, P. M.;Hawker, C. J.; Van Horn, B.; Guan, Z.; Chen, G.;Krishnan, R. S. Science 2006, 311, 1740.
26. Yurekli, K.; Karim, A.; Amis, E. J.; Krishnamoorti,R. Macromolecules 2003, 36, 7256.
BEHAVIOR OF NANOPARTICLE-FILLED BINARY BLENDS 2403
Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb