Determining the phase behavior of nanoparticle-filled binary blends

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: [email protected])

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


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-



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



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.



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.



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.



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.



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).



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.



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.



(/ 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.



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.

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Determining the phase behavior of nanoparticle-filled binary blends

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