Many body effects on the phase separation and structure of dense polymer-particle melts Lisa M. Hall and Kenneth S. Schweizer Citation: The Journal of Chemical Physics 128, 234901 (2008); doi: 10.1063/1.2938379 View online: http://dx.doi.org/10.1063/1.2938379 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Grafted nanoparticles as soft patchy colloids: Self-assembly versus phase separation J. Chem. Phys. 142, 074901 (2015); 10.1063/1.4908044 The effect of polymer-induced attraction on dynamical arrests of polymer composites with bimodal particle size distributions J. Rheol. 57, 1669 (2013); 10.1122/1.4822254 Polymer-mediated spatial organization of nanoparticles in dense melts: Transferability and an effective one- component approach J. Chem. Phys. 133, 144905 (2010); 10.1063/1.3501358 Many-body interactions and coarse-grained simulations of structure of nanoparticle-polymer melt mixtures J. Chem. Phys. 133, 144904 (2010); 10.1063/1.3484940 Effect of shear on nanoparticle dispersion in polymer melts: A coarse-grained molecular dynamics study J. Chem. Phys. 132, 024901 (2010); 10.1063/1.3277671 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 65.39.15.37 On: Sun, 15 Mar 2015 10:30:49
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Many body effects on the phase separation and structure of dense polymer-particlemeltsLisa M. Hall and Kenneth S. Schweizer Citation: The Journal of Chemical Physics 128, 234901 (2008); doi: 10.1063/1.2938379 View online: http://dx.doi.org/10.1063/1.2938379 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Grafted nanoparticles as soft patchy colloids: Self-assembly versus phase separation J. Chem. Phys. 142, 074901 (2015); 10.1063/1.4908044 The effect of polymer-induced attraction on dynamical arrests of polymer composites with bimodal particle sizedistributions J. Rheol. 57, 1669 (2013); 10.1122/1.4822254 Polymer-mediated spatial organization of nanoparticles in dense melts: Transferability and an effective one-component approach J. Chem. Phys. 133, 144905 (2010); 10.1063/1.3501358 Many-body interactions and coarse-grained simulations of structure of nanoparticle-polymer melt mixtures J. Chem. Phys. 133, 144904 (2010); 10.1063/1.3484940 Effect of shear on nanoparticle dispersion in polymer melts: A coarse-grained molecular dynamics study J. Chem. Phys. 132, 024901 (2010); 10.1063/1.3277671
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Many body effects on the phase separation and structure of densepolymer-particle melts
Lisa M. Hall and Kenneth S. Schweizera�
Department of Chemical & Biomolecular Engineering, Department of Materials Science, and FrederickSeitz Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana,Illinois 61801, USA
�Received 2 April 2008; accepted 8 May 2008; published online 16 June 2008�
Micron-sized particles, or “fillers,” have long been usedas additives to enhance the material properties of polymermelts and crosslinked rubbers.1 With the vigorous progressof nanoscience over the last decade, the emphasis has turnedto nanosized fillers where the high surface-to-volume ratiocan greatly increase the amount of matrix polymer influ-enced by hard particles even at low volume fractions.2–5
Many experimental studies have documented that smallamounts of nanofillers can significantly modify the compos-ite glass transition and mechanical properties.6–9 Changingparticle size, grafting polymer on the fillers, and modifyingsurface chemistry all can significantly affect polymer nano-composite �PNC� properties. A central issue is the degree ofspatial dispersion of the nanoparticles in the polymer matrix.To realize the full potential of PNCs for specific applications,design rules which link particle chemistry and size andmonomer-filler interfacial interactions to microstructure andmiscibility are required.
Monte Carlo,10–12 molecular dynamics,13–17 and dissipa-tive particle dynamics18 computer simulations have been re-cently employed to study model PNCs, and multiple insightshave been gleaned. However, the relatively large disparity in
length scales, and the relevant high total packing fraction,renders simulations expensive and difficult to equilibrate.Prior theoretical studies have employed density functionaltheory but only in the dilute one and two particle limits.19
The polymer reference interaction site model �PRISM�integral equation theory is a microscopic statistical mechani-cal approach originally developed to describe polymer solu-tions, melts, blends, and block copolymers.20 Early workwith this theory for polymer-particle mixtures focused onrelatively small and dilute fillers under athermal solutionconditions.21 Recently, PRISM theory has been extended andsystematically applied to model PNCs to allow a broad ex-ploration of the relevant parameter space for densemelts.22–28 A major focus of this work was structure in thedilute one and two particle limits. At this level PRISM theoryhas been shown to agree well with simulations for variouspair correlation functions.19,23,27 The critical role of thestrength and range of the monomer-filler interfacial attractionin controlling whether polymers mediate entropic depletionattraction, steric stabilization, or bridging has beenestablished.24–26 Since polymer entropy is maximized awayfrom the filler surface, at low interfacial attraction strength�pn �units of thermal energy, kBT� there is a strong depletionattraction between particles. At intermediate �pn, each nano-particle is stabilized by a bound polymer layer resulting in aa�Electronic mail: [email protected].
THE JOURNAL OF CHEMICAL PHYSICS 128, 234901 �2008�
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repulsive potential of mean force and homogeneous disper-sion. Further increasing �pn promotes the sharing of boundpolymer layers by particles, which can trigger an enthalpicbridging network-like type of phase separation. In all cases,at nonzero filler loadings the monomer-scale liquid-likepacking is suppressed by the addition of nanoparticles, andthe mixture bulk modulus is reduced. Theoretical predictionsfor the small angle scattering signature of bound polymerlayers appear to be consistent with experiments on a silica-polystyrene nanocomposite.29
A significant limitation of all prior PRISM studies is thatcomputational difficulties prevented determination of thespinodal phase diagram beyond the low filler volume frac-tion regime.25 The present work removes this limitation byimplementing a more efficient numerical algorithm to solvethe coupled nonlinear integral equations. This technical ad-vance allows the exploration of new physics and results inqualitatively new insights concerning structure and phase be-havior. For example, the identification of critical points andconstruction of full spinodal curves, from the dilute filler todilute polymer limits, are achieved for the first time. Thedilute polymer regime is relevant to colloid and nanoparticlescience where polymers are used as low concentrationadditives.30 Under certain conditions many body particle cor-relation effects are important, or even dominant, for under-standing miscibility. In some cases, spectacular differencescompared to the virial analysis25 are found. For example, forspecific regimes of the interfacial cohesion strength andrange, a mixture with a positive filler second virial coeffi-cient and repulsive dilute filler limit potential of mean forcecan undergo a collective bridging driven phase separation.
Section II summarizes the PNC model and PRISMtheory and presents virial coefficient calculations. The nu-merical solution method is briefly mentioned, with more de-tails given in the Appendix. The spinodal phase boundariesfor various particle sizes and interfacial attraction ranges aredetermined in Sec. III. Section IV discusses structure as afunction of filler volume fraction for systems that are highlysensitive to many body correlation effects. Special attentionis paid to the length-scale-dependent structural changes nearphase separation. Section V discusses the possibility ofkinetic aggregation and gelation triggered by deep contact orbridging minima in the nanoparticle potential of mean forcethat can emerge prior to phase separation. The conclusionsare given in Sec. VI.
II. MODEL, THEORY, AND VIRIAL COEFFICIENTS
The mixture model and specific version of PRISMtheory employed are identical to prior studies24–26 and areonly briefly summarized. The homopolymer is an ideal freelyjointed chain �FJC� composed of N hard spheres of diameterd and a bond length between adjacent segments of � /d=4 /3 that is typical of the persistence length of flexiblepolymers.20,31 Nanoparticles are hard spheres of diameter D.Intermolecular potentials are pair decomposable, and at thesite level the excluded volume is additive. In general, non-contact potentials between sites of type i and j, Uij�r�, can bepresent. The total packing fraction is fixed at �t=0.4 which
results in a typical experimental value of the polymer meltdimensionless isothermal compressibility.20 The numberdensities of polymer segments �denoted by the subscript p�and nanoparticles �denoted by the subscript n� are �p
= �1−���t / �d3� /6� and �n=��t / �D3� /6�, respectively,where � is the nanoparticle volume fraction. Unless notedotherwise, all lengths are in units of the monomer diameter,and all energies are in units of the thermal energy.
PRISM theory for this model PNC, with chain end ef-fects preaveraged, is then given by three coupled integralequations for the site-site intermolecular pair correlationfunctions.20,24 Each species j is described by an intramolecu-lar probability distribution function or structure factor inFourier space, � j�k�. The particle �n�k�=1, and the FJCstructure factor is20
�p�k� = �1 − f2 − 2N−1f + 2N−1fN+1�/�1 − f�2, �1�
where f =sin�kl� / �kl�. The site-site intermolecular pair corre-lation function g, is related to the site-site intermoleculardirect correlation function C by the generalized Ornstein–Zernike or Chandler–Andersen matrix equations20,32
H> �k� = �> �k�C> �k���> �k� + H> �k�� , �2�
where �> �k� is a diagonal matrix of elements �i�i�k��ij, C> �k�contains elements Cij�k�, and H> �k� contains elements�i� jhij�k�, where hij�r�=gij�r�−1. Beyond contact, the appro-priate closure approximations for PNCs are23–26 the site-sitePercus–Yevick closure for polymer-polymer �p-p� andpolymer-nanoparticle �p-n� correlations, and the hypernettedchain �HNC� closure for nanoparticle-nanoparticle �n-n�correlations,
Cij�r� = �eUij�r� − 1��1 + hij�r� − Cij�r�� , �3�
Cnn�r� = − Unn�r� + hnn�r� − ln gnn�r� , �4�
where =1 /kBT. Inside the hard core distance of closest ap-proach, hij�r�=−1.
The chemical nature of the polymer and nanoparticle isencoded in the functions Uij�r�. We focus on the case ofpurely hard core monomer-monomer and filler-filler interac-tions. A two-parameter interfacial attraction betweenmonomers and nanoparticles of an exponential form isemployed,24
Upn�r� = − �pne−�r−rc�/d, r � rc, �5�
where rc= �d+D� /2 and is the spatial range in units ofmonomer diameter. The exponential attraction is similar inshape to the potential between a Lennard–Jones unit and acolloid.22,25,33
Prior work utilized the iterative Picard method to nu-merically solve the PRISM equations.23–26,34 This method iseasy to implement, but becomes extremely slow at high fillervolume fractions and near spinodal phase separation.26 Welargely circumvent this difficulty in the present work by em-ploying an inexact Newton method35–37 as discussed in theAppendix.
234901-2 L. M. Hall and K. S. Schweizer J. Chem. Phys. 128, 234901 �2008�
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Solution of the PRISM equations yields all real spacesite-site pair correlation functions gij�r�. Of special interest isthe filler potential of mean force �PMF� given in units of thethermal energy by
Wnn�r� = − ln�gnn�r�� . �6�
The nanoparticle second virial coefficient B2 follows in thedilute two particle limit ��→0� as
B2
B2HS=
− 3hnn�k → 0�4�D3 , �7�
where B2HS is the vacuum hard sphere value. Polymer ad-sorption mediated repulsions increase B2, while depletion orbridging induced attractions decrease B2.24 The lowest levelvirial description predicts phase separation only if B2�0,and the spinodal curve is simply25,38
�S = − �8B2
B2HS�t�−1
. �8�
The polymer and nanoparticle collective partial structurefactors are given by
where I is the identity matrix and S� is the dimensionalizedversion of S, given by Sii� =�iSii. Macroscopic spinodal phaseseparation corresponds to the simultaneous divergence atk=0 of all partial structure factors.
All calculations are performed for a degree of polymer-ization of N=100. The effect of chain length on structure andphase behavior is relatively weak since the total volume frac-tion is high.25,26 This aspect will be explicitly addressed in afuture study that considers the silica-polyethylene oxidePNC.39 The key variables are thus the size asymmetry ratioD /d �chosen here to be 5, 10, 15�, filler volume fraction0 � 1, and the strength �in units of the thermal energy�and range of �in units of the monomer diameter� the interfa-cial cohesion. A wide range of strengths, 0 �pn 4, andthree values of the range, =0.25,0.5,1.0, are studied aspreviously motivated.24,25
A major focus of the subsequent sections is on theD /d=10 system. The normalized nanoparticle second virialcoefficient for this case, as a function of interfacial attractionstrength and for three spatial ranges, is shown in Fig. 1. Forall cases B2 is extremely large and negative at small attrac-tion strengths, transitions to a spatial range-sensitive regimeat intermediate strengths where it is positive and larger thanits bare hard sphere value �B2 /B2HS�1� due to adsorptionand bound layer formation, and then again becomes stronglyattractive due to bridging mediated polymer-particle complexformation.25 At low and intermediate attraction strengths thetrends are monotonic in spatial range: Longer range interfa-cial cohesion results in more positive �or less negative� sec-ond virial coefficients. However, the breadth in attractionstrength of the repulsive B2 regime, and the onset of strongbridging and a negative B2 at a high degree of cohesion, is anonmonotonic function of range. This results in the multiple“curve crossings” seen in Fig. 1 at high �pn.
III. SPINODAL PHASE DIAGRAM
A. Numerical procedure and general results
The spinodal phase boundaries, plotted in two differentformats �energy or temperature�, are given in Figs. 2 and 3.As discussed in the Appendix, the data points were deter-mined by incrementally changing � �closed symbols� or �pn
�open symbols�, using the previously converged solution asan initial guess at the next � or �pn. In the � vs �pn repre-sentation this corresponds to a vertical or horizontal trajec-tory in the phase diagram, respectively. As the spinodal de-mixing boundary is approached, a large zero wavevectorpeak emerges in the collective structure factors and conver-gence becomes very difficult. Each point in the figures rep-resents the last � or �pn which could be converged, incre-menting by 0.0001 in �pn, or in the fourth significant figurein �. The smoothness of the curves, closeness of the threepoints at �pn=0 for =0.25, 0.5, and 1.0 in Fig. 3 �whichrepresent the same system converged separately�, and thefact that the open and closed symbols align well, all attest tothe accuracy of the numerical method of estimating the lo-cation of the spinodal instability. Comparison with estimatesbased on extrapolation of the k=0 structure factors is givenin the Appendix, and the agreement is good.
Results over the entire range of volume fraction areshown in Figs. 2 and 3. Most experimental studies of PNCsare for ��0.3. However, the high volume fraction part ofthe phase diagram is relevant to colloidal or nanoparticlesystems with dilute polymer additives.30 Moreover, two re-cent experimental studies have examined miscible PNCs toexceptionally high loadings. Mackay et al. studied mixtures
FIG. 1. �Color� Filler second virial coefficient normalized by its hard spherevalue as a function of monomer-particle attraction strength in units of thethermal energy for D /d=10 and three values of spatial range.
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of polystyrene linear chains and crosslinked nanoparticles upto 80 wt % in filler.40 Anderson and Zukoski have homoge-neously mixed 44 nm silica particles in polyethylene oxideto a filler packing fraction approaching that of random closepacking.41
B. Specific trends of phase behavior
The spinodal phase boundaries for D /d=5, 10, and 15 at=0.5 are given in Fig. 2. For �pn�1, demixing occurs atvery low volume fractions driven by entropic depletioneffects30,42,43 that favor contact aggregation of fillers.24,25 Inthe other extreme, corresponding to mixing a small amountof polymer in the hard sphere fluid �high ��, the depletionattraction is much weaker. This results in miscibility even inthe athermal, pure entropy limit beyond some volume frac-tion; in the absence of interfacial attraction, roughly 10%�25%� polymer dissolves in the mixture at D /d=15 �5�. Atboth low and high volume fractions, increasing filler sizeresults in a wider immiscibility region.
Miscibility increases sharply as �pn grows beyond�0.1–0.25 in both the high and low volume fraction regionsof the phase diagram. Although no attempt was made to findthe exact location of the critical points, Fig. 2�b� indicatesthe depletion critical point is at �c�0.1 for all three particlesizes. Increasing filler volume fraction beyond �c results inincreased miscibility near the depletion �high temperature�phase boundary. A miscibility window at intermediate valuesof interfacial attraction or temperature exists due to polymeradsorption and the formation of thermodynamically stablepolymer layers. In the representation of Fig. 2�a�, the breadthof the miscibility window is not strongly dependent on sizeasymmetry ratio D /d. Further increasing �pn promotes thesharing of bound layers between particles.24,25 This results ina bridging network, or polymer-particle complex, type ofphase separation beginning at �pn�0.8, 0.9, and 1.1 forD /d=15, 10, and 5, respectively. The bridging critical pointis at a very high �c�0.95 and is not strongly dependent onparticle size. The spinodal curve decreases slowly, compared
FIG. 2. �Color� �a� Spinodal phase diagram for =0.5 and three values of D /d. Numerical data points were determined by approaching the spinodal instabilityvia incrementing � �closed symbols� and �pn �open symbols�. Though some symbols appear close to the �=0 or 1 axes, all systems are, of course, misciblein these limits. Lines are a guide to the eye. �b� Alternate representation of the same spinodal phase boundary in terms of reduced temperature. Thehomogeneous miscible and phase separated depletion and bridging regimes are indicated.
FIG. 3. �Color� Spinodal phase diagram for D /d=10 and three values ofspatial range. The phase boundaries were determined in the same manner asin Fig. 2. Solid lines are a guide to the eye. Dashed lines show the spinodalpredicted by the virial calculation �Ref. 25� of Eq. �8�; endpoints of thedashed lines are spatial range labels.
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to its depletion side analog, with increasing interfacial attrac-tion strength. Overall, miscibility at both high and low �pn
moderately decreases with increasing particle size, thoughthe shape of the phase boundaries remains similar.
In the temperature-volume fraction representation of Fig.2�b�, the two critical points are reminiscent of lower andupper critical solution temperatures �LCST and UCST� ob-served in some polymer blends and solutions.44,45 The hightemperature phase boundary shifts to lower temperatures andbroadens as particle size is increased, as expected for adepletion-like mechanism.42,43 The critical temperature of thelow �pn �high temperature� boundary is at �3.2, 4.0, and 7.7for D /d=15, 10, and 5, respectively. A nearly inverse rela-tionship between D /d and this critical temperature is found�not plotted�. In contrast, the low temperature bridging sideof the phase diagram is relatively insensitive to D /d. For allsize asymmetry ratios, the bridging phase boundary tempera-ture increases roughly linearly with volume fraction up to thecritical point at �c�0.95, after which it sharply drops.
The analogy of PNC phase behavior to polymer solu-tions and blends that display a UCST and LCST is at leastpartially superficial. The low temperature UCST of a poly-mer mixture is driven by an unfavorable enthalpic interaction�or positive Flory � parameter� where the system separatesinto polymer rich and solvent rich domains.45 On the otherhand, the PNC analog is driven by favorable �attractive�polymer-particle interactions via the formation of a particle-polymer network or complex. The high temperature LCSTphase transition of polymer mixtures is not well understoodat the molecular level, but is generically associated with a �parameter that has negative enthalpic and positive entropiccontributions.44 Its PNC analog is the polymer-mediateddepletion attraction between particles which results in an un-favorable entropy of mixing modified by a favorable en-thalpy of polymer adsorption.
Despite the different physics involved, it is interesting tocompare the shapes of the PNC demixing boundaries withthe LCST/UCST analogs for polymer mixtures. The shape ofthe PNC high temperature LCST curve actually resembles anupside down version of the highly asymmetric in composi-tion polymer-solvent UCST phase boundary where the nano-particle plays the role of the polymer species. For UCSTpolymer solutions miscibility decreases �higher Tc� with in-creasing macromolecule size, while the PNC analog ofUCST-like miscibility is also reduced with increasing fillersize. On the other hand, the low temperature PNC phaseboundary follows a roughly linear dependence on composi-tion, a functional form that is much more akin to the LCSTdemixing boundaries of polymer mixtures.44
Figure 3 presents the phase diagrams for fixed D /d=10and three values of the spatial range of the interfacial cohe-sion. For weak polymer-particle attractions, the demixing re-gion is monotonically broadened as the attraction range de-creases. Such a trend is expected from the classic analysis ofdepletion attraction.42 The depletion critical point is at �pn
�0.11, 0.25, and 0.47 for =1.0, 0.5, and 0.25, respectively.This trend is understandable since miscibility requires favor-able interfacial cohesion, which at fixed contact strength isweaker for shorter range attractions. We note that the product
of the interfacial attraction strength and spatial range at thedepletion critical point is nearly constant, �pn�0.11–0.13.We do not have a rigorous explanation for this interestingnumerical result. However, the idea that bound layer forma-tion and miscibility requires a threshold degree of interfacialcohesion seems plausible. The simplest estimate of the latteris in the spirit of a mean field Flory �-parameter given by theintegrated strength of the interfacial attraction,
�I − rc
�
dr��pne−�r−rc�/�d� � − �pnD2d , �10�
where rc= �D+d� /2 and D�d has been assumed in the finalproportionality. This quantity is an elementary measure ofthe enthalpy gained by monomer adsorption on a filler par-ticle. It competes against polymer-induced entropic depletionattraction which favors filler contact aggregation. At the sim-plest Asakura–Oosawa �AO� level42,43 the depletion attrac-tion strength scales as kBT�D /d� and has a spatial range setby the small monomer diameter d. Hence, the depletionanalog of Eq. �10� is
�AO − D
�
dr�WAO�r� � − kBTD
d�D2d� � − kBTD3. �11�
Bound layer formation and miscibility are expected to occurwhen the depletion and adsorption driving forces are compa-rable leading to the qualitative criterion �pnD2d�kBTD3, orin dimensionless energy units �pn� �D /d�. This simple ar-gument provides a physical interpretation of our findings thatthe product �pn is nearly constant at the depletion criticalpoint if D /d is fixed. It also rationalizes our numerical find-ing that the critical temperature Tc scales roughly with theinverse size asymmetry ratio d /D.
The bridging critical points are at �c�0.95 and �pn
�0.4, 0.9, and 1.7 for =1.0, 0.5, and 0.25, respectively.Increasing the interfacial cohesion range strongly shifts thebridging critical temperature to higher values. Interestingly,we again find that at fixed D /d the product of the interfacialattraction strength and spatial range at the critical point isnearly constant, �pn�0.40–0.45. Note that this numericalvalue is roughly a factor of 4 larger than its depletion criticalpoint analog. The underlying physics is presumably some-what different since the filler volume fraction is extremelyhigh at the bridging critical point and the polymer concen-tration is low. Depletion is not qualitatively important sinceFigs. 2 and 3 show that at such a high filler volume fractionthere is no entropic depletion driven demixing. Rather, thekey physical process is that polymers must localize to someextent, and hence lose entropy, in order to form tight bridgesbetween nanoparticles. The enthalpic driving force for thelatter is still controlled to zeroth order by adsorption andhence Eq. �10� is relevant. The entropic cost will scale as thethermal energy. Balancing these two contributions again sug-gests that for fixed D /d the product �pn is roughly constant,in agreement with our numerical calculations. The depen-dence of the bridging critical point on D /d at fixed spatialrange of interfacial cohesion is more subtle. Figure 2 showsthat, at fixed attraction range, bridging occurs at higher co-hesion strength �lower temperature� as the fillers become
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smaller. But this is a much weaker effect than the depen-dence of the depletion critical temperature on D /d. Qualita-tively, the trend seems intuitive since smaller fillers have ahigher radius of curvature and hence bridging via short rangeinterfacial attractions will be less efficient.
The shape of the bridging spinodal curves in Fig. 3 de-pend in a complex manner on attraction range. At intermedi-ate cohesion strength and/or high filler volume fractions, alonger range interfacial attraction monotonically results inless miscibility. However, as the strength of the attractiongets larger, the spinodal curves of different ranges cross.Hence, competing effects must be operative. At the highestvalues of �pn, the spinodal curves of the shortest and longestrange attraction systems are nearly identical, but the interme-diate range attraction mixture is far more miscible. Thissubtle and surprising result is more evidence for the presenceof competing effects associated with attraction strength andrange under strong bridging conditions.25
C. Comparison with virial analysis
The decrease in miscibility with particle size in Figs. 2and 3 is expected from the prior virial based spinodal analy-sis based on Eq. �8� since the nanoparticle PMF in the dilutelimit scales linearly with D /d.24,25 However, the virial analy-sis must fail at high enough filler volume fraction, cannotpredict critical points, and can strongly overpredict miscibil-ity on the bridging side of the spinodal. These trends areevident in Fig. 3 which compares the full many body calcu-lation of demixing curves with their �nearly vertical� virialanalogs based on Eq. �8�. The virial results �dashed lines�approximate well the full calculation on the depletion side ofthe phase diagram up to the critical point and also reproducethe correct ordering of curves at very high �pn�3 where � issmall. However, the bridging spinodals are far less vertical
than their virial based analogs. Moreover, phase separation atthe virial level requires a negative second virial coefficient, aconstraint the full calculations reveal is not necessary. Sys-tems with positive second virial coefficients �Fig. 1� and arepulsive PMF can undergo demixing because of many bodyeffects. Moreover, the =0.25 spinodal curve crosses its =0.5 analog, thereby making the =0.25 system the leastmiscible in the intermediate �pn regime according to thevirial analysis, though it is the most miscible system whenmany-body effects are taken into account. Clearly, the con-sequences of many body effects are sensitive to the range ofthe interfacial attraction, as expected on qualitative physicalgrounds. The virial calculation always overpredicts miscibil-ity �except above the critical point� relative to the full calcu-lation, with the most dramatic differences at intermediate �pn
for =0.5 and 1.0.In the remainder of the paper we focus primarily on the
D /d=10 system and consider the structural correlations andthermodynamic properties. Emphasis is on state points wherethe second virial coefficient is positive, and hence demixingis driven entirely by filler many body correlation effects.
D. Filler PMF
The PMF provides physical insight into the phase dia-gram trends. Examples for D /d=10 in the dilute limit aregiven in Figs. 4 and 5 for =0.25 and 1.0 and a wide rangeof attraction strengths corresponding to depletion, steric sta-bilization, and bridging behavior. Figure 4 shows that thePMF for =0.25 and �pn=0.4 is similar to that for =1.0and �pn=0.1. This explains why depletion phase separation issimilar for various short �compared to nanoparticle diameter�attraction ranges. The shifted value of the spinodal �pn is dueto the greater overall interfacial cohesion at larger attraction
FIG. 4. �Color� Nanoparticle potential of mean force in the dilute two particle limit for D /d=10 and the four indicated values of attraction strength for �a�=0.25 and �b� =1.0.
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range. One notable difference, though, is the oscillatory na-ture of the PMF, and the longer range repulsive tail for the=1 system.
The bridging behavior is qualitatively different for short�=0.25� and long �=1.0� range attraction systems. Uponincreasing �pn from 1 to 3 with a short attraction range, theprimary bridging minimum deepens but remains located at aparticle-particle separation of approximately one monomerdiameter, and other bridging minima are comparatively small�Fig. 4�a��. The bridging minima are so sharp at high �pn thatthe PMF is akin to that of a depletion system shifted in the rcoordinate by 1d. In contrast, the PMF of the =1 mixture inFig. 4�b� is almost purely repulsive at �pn=1 �PMF minimumof −0.36�. In addition, increasing �pn results in the emer-gence of multiple, rather deep bridging minima, and an over-all lowering of the PMF that extends to an interfiller separa-tion of �8 monomer diameters. These completely differentPMF features for the short and long range interfacial cohe-sion systems demonstrate it is coincidental that the =0.25and 1.0 spinodal curves in Fig 3 are similar at very high �pn
in the low volume fraction region. Moreover, these differ-ences rationalize the divergence of the spinodal curves fromone another as nanoparticle loading increases substantially.The analogous PMF’s at =0.5 �not shown� are similar tothe =1.0 results except the global minima occur at shorterdistance and, at �pn=3, are not as deep. This explains thegreater miscibility for the =0.5 mixture compared to boththe =0.25 and 1.0 systems at very high �pn as seen inFig. 3.
The above structural differences, as well as qualitativelydifferent dependences on volume fraction of the PMF, ac-count for the complex cohesion range dependence of thespinodal curves in Fig. 3. Figure 5 compares the dilute limit
PMF to its �=0.1 analog for =0.25, 0.5, and 1.0 at fixed�pn=2, where the normalized second virial coefficients are0.53, 1.32, and 1.68, respectively. At �pn=2 all these systemsphase separate with increasing filler volume fraction; the=1 mixture has the lowest spinodal boundary, followed bythe =0.5, and then =0.25 systems. However, at the viriallevel all are completely miscible at all volume fractions!Each mixture does phase separate due to many body effectsdespite a positive B2. Moreover, the systems with larger posi-tive B2 phase separate first; specifically, demixing occurs at�=0.674, 0.355, and 0.178 for =0.25, 0.5, and 1.0, respec-tively. Increasing filler volume fraction from 0 to 0.1 deepensthe bridging minima for the =1.0 and =0.5 mixtures, aneffect which is much stronger when =1.0, resulting in thissystem demixing at the lowest volume fraction. On the otherhand, the local attractive minimum of the PMF for the short-est range attraction changes little �becomes slightly weakerwith increasing ��, and this mixture is the most miscible.The underlying physics seems to be simple geometry:Many body effects are weakest for the shortest rangepolymer-particle attraction.
In summary, when phase separation is induced at lowvolume fraction by a very strong depletion �small �pn� orbridging �very high �pn� attraction, many body effects areweak and the virial analysis25 is in rather good agreementwith the full calculations. A weaker bridging attraction al-lows the system to reach a higher volume fraction beforephase separation, thereby increasing the influence of manybody effects. These many body effects are much more sig-nificant for the longer range bridging cases of =0.5 and 1.0,as physically expected. This results in a decreased miscibilitywith increasing cohesion range at moderate to high �pn,opposite to the trend predicted from a second order virialcalculation.
IV. STRUCTURAL CHANGES APPROACHING PHASESEPARATION
Very recent work26 has presented an initial explorationof the structure of model PNCs for filler volume fractions upto ��0.4. This limited study was performed without knowl-edge of where the true spinodal boundaries were located.Moreover, systematic exploration of how length scale depen-dent properties, such as the polymer and filler osmotic com-pressibilities and microphase separation “bound polymerlayer” peaks, depend on system parameters was notattempted. In this section we study these issues using themore powerful numerical algorithm.
Our focus is on the most dramatic situation where manybody effects alone are responsible for demixing in thebridging region on the phase diagram. This motivates thefollowing choice of system parameters: =0.5, �pn=0.9, andD /d=10 in Figs. 6–9. As seen in Fig. 2�a�, this mixture issquarely in the miscibility gap. Moreover, Fig. 1 shows thisPNC has a strongly repulsive second virial coefficient,B2 /B2HS�1.6, and hence thermodynamically stable boundpolymer layers exist which result in steric stabilization in thedilute nanoparticle limit. Phase separation does occur due tomany body effects, but only at a very high volume fraction
FIG. 5. �Color� Nanoparticle potential of mean force in the dilute twoparticle limit �dashed lines� and at �=0.1 �solid lines� for D /d=10,�pn=2, and three values of interfacial cohesion range.
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close to the bridging critical point at ��0.93, thereby al-lowing an exceptionally wide range of volume fractions andphysical behavior to be explored. Moreover, this model sys-tem is relevant to the recent experiments of Anderson and
Zukoski who have studied silica-polyethylene oxide nano-composites using x-ray scattering which are miscible fromdilute to ultrahigh filler volume fractions.41
A. Real space correlations
The PMF for the above system as a function of volumefraction, including the dilute and hard sphere limits, is shownin Fig. 6. Nearly symmetric monomer-scale oscillations existin the dilute two particle limit. As filler loading increases, themonomer-scale oscillations diminish, and a broader oscilla-tory feature emerges on the nanoparticle diameter plus itsbound polymer layer length scale. This acts to strengthen thebridging attraction until very high volume fractions nearphase separation, beyond which the attraction weakensslightly. The nanoparticle repulsion at contact induced by thebound polymer layer decreases with increasing volume frac-tion. As �→0.9, the particles experience a complex effec-tive attraction with local minima at both contact and twomonomer diameter separations, which extends to a fillersurface-to-surface separation of roughly four monomer diam-eters. Phase separation is induced at ��0.93 driven bybridging. Eventually, the system resolubilizes. The pure hardsphere PMF is also shown in Fig. 6, and is of a grosslydifferent form than that near the critical point.
The corresponding monomer-particle radial distributionfunctions are presented in Fig. 7. While the strength ofmonomer-particle correlation at contact decreases with nano-particle loading, an excess of polymer �bound layer� is stillpresent around the filler, even as phase separation is ap-proached. Oscillatory monomer-scale order decreases, andnanoparticle plus polymer layer-scale order grows, with in-
FIG. 6. �Color� Nanoparticle potential of mean force for various volumefractions and fixed =0.5, �pn=0.9, and D /d=10. The �=1 curve is thepure hard sphere result at a total packing fraction of 0.4, computed using theHNC closure.
FIG. 7. �Color� Monomer-filler pair correlation function for various fillervolume fractions for =0.5, �pn=0.9, and D /d=10. The �=0, 0.25, 0.5,0.75, and 0.9 lines are labeled by a square, diamond, triangle, x, and circle,respectively, and have contact values gpn�rc�=4.1, 3.5, 3.0, 2.7, and 3.2,respectively. The inset shows the nonrandom part of the pair correlationfunction, hpn�r�, weighted by the surface area factor �r /rc�2.
FIG. 8. �Color� Monomer-monomer pair correlation function for variousfiller volume fractions and =0.5, �pn=0.9, and D /d=10. The inset showsthe nonrandom part of the pair correlation function, hpn�r�, weighted by thesurface area factor �r /d�2.
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creasing volume fraction. The longer range oscillations aredramatically enhanced in amplitude and spatial extent �manyfiller diameters� as spinodal phase separation is approached.This feature is most clearly seen in the surface area weightednonrandom part of the monomer-particle pair correlationfunction shown in the inset of Fig. 7.
Figure 8 presents the corresponding monomer-monomerpair correlation functions. Local monomer-scale order de-creases, and filler-scale order increases, as � approaches thephase separation value. The various structural features areoverlaid on the typical radius-of-gyration �Rg� scale “corre-lation hole.” Overall, the dramatic reduction in contact valueprimarily reflects the smaller number of polymers present asfiller volume fraction increases at fixed total packing frac-tion. The peak that emerges at r�10d�D is a signature ofadsorption and bridging. The corresponding surface areaweighted representation is shown in the inset of Fig. 8. Thefiller-scale order imprinted on the monomer interchain pack-ing correlations is clearly evident and becomes increasinglystrong and spatially long range as the spinodal boundary isapproached.
B. Collective structure factors
Fourier space partial collective density fluctuations areexperimentally measurable using selective labeling scatteringmethods. Results for the same system studied above areshown in Fig. 9. Consider first the nanoparticle Snn�k� inFig. 9�a�. The wide angle cage peak at kD�2� quantifies thespatial coherence of local filler packing. As volume fractionincreases towards 0.9, the cage peak intensity dramaticallyincreases and shifts towards larger wavevector indicatingtighter packing. The nanoparticle osmotic compressibilitySnn�0� first decreases with filler volume fraction as wouldoccur for hard spheres in a vacuum, but then shows an up-
turn at �=0.9 as phase separation is approached. Though at�=0.9 demixing is imminent, the cage peak continues toshow significantly more order due to polymer-mediatednanoparticle clustering and enhanced local densification rela-tive to pure hard spheres; for example, the peak intensitiesare 2.4 and 1.9 at 2� /kD of 0.96 and 0.93 for the �=0.9system and hard spheres of the same total packing fraction,respectively. This increased wide angle scattering peak,coupled with the presence of a maximum at k=0, impliesthat polymer bridging induces local packing order at thesame time that long wavelength concentration fluctuationsintensify.
The polymer bound layer can be thought of as a mi-crophase, distinct from the relatively unperturbed polymerfar from the filler particle.26 The spatial structure of thebound layer correlates with that of the nanoparticles, givingrise to the filler-scale microphase peak in the polymer struc-ture factor in Fig. 9�b�. Similar to the cage peak in Snn�k�, themicrophase order in Spp�k� persists as a k→0 upturn devel-ops at high volume fractions. However, unlike the trend inSnn�k�, the polymer microphase peak intensity initially growssignificantly with filler volume fraction and then decreasesweakly as � increases from 0.5 to 0.9. The ordering lengthscale decreases monotonically, 2� /kd=12, 11, and 9.9�2� /kD=1.2, 1.1, and 0.99� at �=0.5, 0.75 and 0.9, respec-tively. A key difference in the interpretation of Snn�k� andSpp�k� is that the nanoparticle length scale peak in Spp�k� hasbeen imprinted on the bound polymer layer. At some highenough nanoparticle volume fraction, essentially all poly-mers are associated with one or more fillers. Further increas-ing volume fraction reduces the total amount of polymer andbound layer, thereby explaining the decrease in imprintedorder at very high nanoparticle loading. Note that with grow-ing � the polymer osmotic compressibility Spp�k=0� first
FIG. 9. �Color� �a� Nanoparticle collective structure factor for various volume fractions for =0.5, �pn=0.9, and D /d=10. The �=1 curve corresponds to purehard sphere result at a total packing fraction of 0.4. �b� Small angle regime of the monomer-monomer collective structure factor for the same systems.
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increases due to the fillers acting as “holes,” then decreasessince fillers pack more tightly and their bound layers inter-fere, and then again increases as the systemapproaches phase separation due to depletion or bridging.
C. Osmotic compressibilities
Figure 10 presents detailed calculations of the particleand polymer inverse dimensionless osmotic compressibilitiesas a function of volume fraction �vertical path on the phasediagram� for the same very weak bridging system as above��pn=0.9�, and four other systems of the same =0.5 andD /d=10 but under depletion ��pn=0.249�, fully miscible��pn=0.5�, and strong bridging ��pn=1.5 and 2.5� conditions.The pure hard sphere fluid result is also shown where �corresponds to the hard sphere packing fraction divided by0.4, or the equivalent PNC volume fraction if the polymer isreplaced by vacuum.
Consider first the nanoparticle results in Fig. 10�a�. Pre-dictably, increasing filler loading initially decreases the nano-particle compressibility for the fully miscible case. However,Snn�0� is a nonmonotonic function of volume fraction. Quali-tatively this is because the presence of the polymer matrixenhances the structural ordering of the fillers, so their com-pressibility is lower than that of the analogous hard spheresystem in which the polymer is replaced by vacuum. But atvery high volume fractions the effect of polymer diminishessince its concentration becomes small, and the filler com-pressibility increases to meet the hard sphere line at �=1.The Snn�0� of the bridging systems ��pn=0.9, 1.5, and 2.5�also initially decreases with volume fraction until phaseseparation is approached, at which point the compressibilityincreases sharply since 1 /Snn�0�→0 at the spinodal bound-
ary. The depletion system shows a continuously increasingcompressibility since it phase separates at a very low volumefraction. Finally, note the reduced osmotic compressibility ofmany of the filled systems relative to the analogous hardsphere fluid. This again reflects the presence of bound poly-mer layers which provide steric stabilization that can bequalitatively thought of as increasing the nanoparticle sizeand hence the effective volume fraction.
The corresponding inverse of the polymer dimensionlessosmotic compressibility, 1 /Spp�k→0�, is given in Fig. 10�b�.The behavior is even more complex than that of the nano-particle analog, suggesting additional competing effects. At�=0 all lines meet the pure polymer limit of 1 /Spp�0�=5.9�off scale�. The addition of a small amount of particles intro-duces randomly distributed holes to the relatively unper-turbed polymer matrix which drastically increases the poly-mer compressibility �concentration fluctuations� in all cases.Interestingly, for those systems not approaching phase sepa-ration, the polymer hardens �Spp�0� decreases� as nanopar-ticle volume fraction increases beyond �0.2. This may becaused by an increasing amount of bound polymer, which ismore structured than the surrounding matrix and can over-compensate for the softening effect of adding more holes tothe mixture. Of course, the fact that 1 /Spp�0�→0 at phaseseparation eventually reverses this trend and the compress-ibility increases sharply. However, even the �pn=0.5 misciblesystem shows increasing polymer compressibility for��0.7–1. At these high volume fractions, it appears that somuch of the polymer is adsorbed to the particle that thebound layer hardening effect is diminished, while the com-peting effect of additional particles acting as empty space inthe polymer is again more relevant.
FIG. 10. �Color� �a� Inverse of the dimensionless filler osmotic compressibility as a function of volume fraction for various interfacial cohesion strengths,=0.5, and D /d=10. The black solid curve corresponds to pure hard spheres at a total packing fraction of 0.4�. �b� Analogous results for the inverse of thedimensionless polymer osmotic compressibility.
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D. Local filler cage and polymer microphase orderparameters
The amplitude and location �k*� of the nanoparticle cagepeak of Snn�k� is shown in Fig. 11�a� for the same five sys-tems studied in Sec. IV C. Fillers that experience adepletion-like attraction are ordered on a shorter length scale�larger k*� than hard spheres, while miscible/weak bridgingnanoparticles that carry a bound layer are organized on alonger length scale. Note that the k* of the two strong bridg-ing systems increases above the hard sphere value as phaseseparation is approached via increasing volume fraction. Al-though the particles maintain a bound layer, the bridgingattraction just after the bound layer �“telebridging”24� resultsin interfiller correlation at smaller separations than for hardspheres in a vacuum.
The degree of nanoparticle cage coherence Snn�k*� in-creases with volume fraction above that of hard spheres forall systems. The miscible PNC ��pn=0.5� cage peak intensitygoes through a very shallow minimum at high �. Interest-ingly, the very weak bridging system ��pn=0.9� remains sig-nificantly more locally ordered than hard spheres, with nosign of a decrease in cage peak intensity towards the hardsphere limit even at ��0.9. With only 8% polymer and nearphase separation, Snn�k*�=2.5 for the �pn=0.9 system, muchlarger than the value of 1.7 for its analog where the polymeris replaced by vacuum. In comparison, the maximum of thestructure factor is �2.85 at solidification ���0.5� for aLennard–Jones or hard sphere fluid.46 Note that for all thesystems in Fig. 11�a� the polymer mediates enhanced fillerorder locally, i.e., Snn�k*� is above its analogous pure hardsphere fluid value. But on long �k=0� length scales the op-posite is true near demixing boundaries in the sense that
Snn�k=0� far exceeds the analogous hard sphere value corre-sponding to greatly enhanced compressibility.
The intensity and location of the filler-induced mi-crophase polymer peak of Spp�k� for the above miscible andbridging mixtures are shown in Fig. 11�b�. Results are notshown for the �pn=0.249 depletion-like system since there islittle bound polymer, and hence an upturn of Spp�k� at lowwavevector exists but there is no microphase peak. Thistrend further supports our assertion that the microphase peakis a signature of the bound layer. There is no microphasepeak at very low and high volume fractions, and for the�pn=2.5 system at most volume fractions. Under theseconditions, the nonzero wavevector peak becomes only ashoulder of the k=0 peak. Increasing nanoparticle loadingenhances the degree of filler-imprinted polymer order atsmall volume fractions due to the increasing amount ofbound layer. Notably, the increase in Spp�k*� with volumefraction for the �pn=1.5 mixture plateaus before increasingagain as phase separation is approached. This plateau behav-ior reflects competing physics, and may be explained by anincreasing amount of shared polymer layer, and thereforedecreasing total bound polymer, under strong bridging con-ditions. The final increase near phase separation may reflectan interference �overlap� of the microphase peak and thegrowing, broad k=0 macrophase separation peak. The mis-cible system Spp�k*� decreases with volume fraction after itsmaximum of 29 at �=0.62, as does the �pn=0.9 mixtureafter its maximum of 25 at �=0.75. At these very high vol-ume fractions almost all of the polymer is associated withfiller. Hence, adding more nanoparticles does not increasethe amount of bound polymer as it does at lower volume
FIG. 11. �Color� �a� Amplitude of the nanoparticle collective cage peak as a function of volume fraction for various interfacial strengths, =0.5, and D /d=10. The smooth black curve �no data points� corresponds to the pure hard sphere fluid result at a total packing fraction of 0.4�. Inset shows the peak location.�b� Analogous results for the microphase peak intensity of the collective polymer structure factor. Inset shows the peak location.
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fraction, but rather only serves to decrease the total amountof polymer and degree of ensemble-averaged polymer orderas quantified by Spp�k�.
Figure 11�b� also shows that increasing �pn at fixednanoparticle volume fraction generally results in a decreaseof the polymer microphase peak intensity. This trend likelyarises from higher interfacial cohesion promoting the sharingof polymer in tighter configurations, decreasing the totalamount of bound layer. However, at very high volume frac-tions the �pn=0.5 and 0.9 curves cross. As discussed earlier,the amount of bound layer is likely saturated, and a higherSpp�k*� at �pn=0.9 may reflect the greater spatial order rela-tive to the miscible mixture, rather than a larger amount ofbound layer.
V. POTENTIAL OF MEAN FORCE AT PHASESEPARATION AND KINETIC AGGREGATION
Equilibrium phase separation, whether depletion-like orbridging mediated, requires the presence of an attractivecomponent in the filler PMF. An interesting question is whatis the strength of the most cohesive nanoparticle configura-tion �minimum of Wnn�r�� at the spinodal boundary for vari-ous filler sizes and ranges of interfacial attraction? Besidesits intrinsic interest, such information is likely important indetermining whether equilibrium phase separation is realizedin practice versus being preempted by a nonequilibrium tran-sition to an amorphous glass, gel, or aggregated cluster state.A rule of thumb for the latter is when the maximum attrac-tion of the particle PMF is �−3 to −4kBT.30,47 Such an esti-mate has support from dynamic mode coupling theory.48,49
We use it here to qualitatively address the question of equili-bration versus gelation of the model PNC’s studied in prior
sections. For PNCs, nonequilibrium states composed of clus-ters of fillers in contact, or finite polymer-particle complexes,could occur at relatively low nanoparticle volume fractionsbelow a percolation threshold. At higher filler volume frac-tions a space-spanning contact depletion-like or bridging gelcan form.
A. PMF depth along spinodal boundaries
The depth of the global minimum of the PMF at thespinodal phase separation boundary plotted versus the corre-sponding volume fraction is given in Fig. 12. Each pointfrom the calculations in Figs. 2 and 3 corresponds to a pointin Figs. 12�a� and 12�b�, respectively. As volume fractionincreases below the critical point ��0.1 for depletion and0.95 for bridging�, the spinodal �pn is increasing; above thecritical point the spinodal �pn decreases with filler volumefraction. Intuitively, a stronger particle attraction causesphase separation at lower volume fraction. As � becomesvery small, the depth of the PMF minimum at phase separa-tion increases sharply, as an infinite attraction would theo-retically be required to induce phase separation as �→0.The PMF minimum on the bridging side of the phase dia-gram is significantly smaller than that on the depletion side.Because the bridging phenomenon is longer range and mul-tiple attractive wells can simultaneously exist, it has a greatereffect at lower nanoparticle volume fraction, or equivalentlydoes not require as deep a PMF minimum to phase separateat a given volume fraction. However, at very high � on thebridging side of the spinodal, the global minimum of thePMF occurs at contact due to the small amount of polymeravailable for bridging. Therefore it is not surprising that the
FIG. 12. �Color� Value of the global minimum in the nanoparticle potential of mean force at the spinodal phase separation boundary for �a� =0.5, D /d=5, 10, and 15 systems and �b� D /d=10, =0.25, 0.5, and 1.0 systems. Open symbols correspond to the high �pn �bridging� side of the phase diagrams inFigs. 2 and 3, and closed symbols correspond to the low �pn �depletion� side.
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PMF minimum required for phase separation becomes simi-lar for the depletion and bridging sides of the phase diagramat very high nanoparticle volume fractions.
We now consider the more detailed trends in Fig. 12. Atfixed interfacial cohesion range, larger nanoparticles havedeeper PMF minima at the spinodal. The origin of this trendis not entirely obvious due to the presence of competingeffects. Specifically, the strength of the attractive features ofthe PMF does increase roughly linearly with D /d,24 but thedemixing volume fraction also decreases with D /d. A repre-sentative example is on the depletion side at �=0.1 �near thecritical point�, where the PMF minimum at phase separationis −4.5, −5.8, and −6.6kBT for D /d=5, 10, and 15, respec-tively. On the bridging side of the phase diagram at �=0.3,the PMF minimum is −1.4, −2.1, and −2.8kBT for the sameD /d values. However, the dependence of the value of thePMF minimum on filler volume fraction is qualitativelysimilar on both sides of the phase diagram for all threevalues of particle diameter.
The dependence of the PMF minimum at phase separa-tion on the range of the interfacial cohesion is illustrated inFig. 12�b�. The behavior is even more complex. Qualita-tively, the depletion-like attraction in the PMF does notchange its effective spatial range, and is similar in shape,though shifted in �pn, as increases �see Fig. 4�. Hence, anearly identical value of the PMF minimum can inducephase separation at a particular volume fraction regardless ofthe cohesion spatial range. At =0.25, the change in thePMF minimum with volume fraction is very similar to thatfor depletion, shifted to a weaker minimum by a factor ofroughly the square of �D+d� /d. As seen in Fig. 4 and 5, thebridging attraction at =0.25 is sharp, similar to the deple-tion attraction but shifted in the r coordinate by 1d. Increas-ing serves to flatten and decrease the depth of the PMFminimum as seen in Fig. 12�b�, except at ��0.5 where the=0.5 and 1.0 results tend to overlap. This behavior is un-derstandable since the increasing range of the bridging at-traction enhances its consequences at a given volumefraction.
B. Nonequilibrium aggregation and gelation
The results in Fig. 12 suggest that, depending on systemvariables, the PMF minimum may be deep enough to inducekinetic gelation or cluster aggregation before thermodynamicphase separation is reached. For the sake of qualitative dis-cussion, we consider that a PMF attraction of −4kBT anddeeper results in effectively irreversible nanoparticle stickingevents on the experimental time scale.
For the depletion systems in Fig. 12 which phase sepa-rate with increasing volume fraction, the polymer-inducedminimum in the PMF is always deeper than −4kBT. Becausethe depletion critical point is at a low ��0.1, below thecritical point disconnected clusters of particles, reflecting ar-rested phase separation, are likely favored over a gelnetwork. Many real PNCs do in fact have dispersed clustermorphologies as observed by microscopy experiments.2,4,29
On the other hand, the results in Fig. 12 also suggestpolymer-mediated bridging interactions are irreversible only
at very low volume fractions ��0.05� for D /d=5 and 10 at=0.5 and for D /d=10 at =1.0, and only up to ��0.08for D /d=15 and =0.5. For these situations, aggregates offinite polymer-particle networks may form instead of a per-colated gel. However, decreasing to 0.25 at D /d=10 re-sults in the bridging minimum becoming deeper than −4kBTup to a much higher volume fraction of ��0.23, and apercolated polymer-particle network is more likely.
There is also the possibility of polymers mediating aglass �or jamming-like� transition of fillers via repulsiveforces between nanoparticles carrying bound layers. For barehard sphere colloid suspensions, glassy dynamics is experi-mentally observed when the volume fraction is �0.575.50
Our calculations were performed at a total packing fractionof only 0.4, for which pure hard spheres have a cage peakintensity of only �1.9. However, as seen in Fig. 11�a�, themiscible PNC can have much larger cage peaks, �2.5, avalue attained by hard spheres only at the elevated volumefraction of �0.48. Of course, this is still well below the onsetof glassy dynamics, but an increase of the total packing frac-tion employed in the calculation may indeed result39 in theprediction of polymer bound layer mediated vitrification ashas been experimentally observed.41
VI. SUMMARY AND DISCUSSION
We have employed microscopic integral equation theoryto study the real and Fourier space structure, many bodyeffects, and phase behavior of dense mixtures of hard spheresand adsorbing homopolymers. The effects of particle size�D /d=5,10,15�, and the strength and range of interfacialattraction, have been systematically investigated over the en-tire range of nanoparticle volume fraction for the first time.Special attention has been paid to the structural changes nearphase separation.
At low interfacial cohesion strength an entropic deple-tion attraction, modified by weak polymer adsorption, in-duces phase separation. The critical point occurs at �10%filler loading, roughly independent of size asymmetry ratioD /d and attraction range. Increasing the cohesion range ordecreasing the size asymmetry ratio improves miscibility, ingood accord with a second order virial analysis valid25 at lowvolume fractions. At high interfacial attraction strengths anetwork or polymer-particle complex type of phase separa-tion occurs via the formation of bridges and sharing of boundpolymer layers between nanoparticles. Prior virial basedcalculations25 correctly predict less miscibility with increas-ing D /d, and the dependence of spinodal curves on understrong bridging conditions where demixing occurs at verylow filler volume fraction. However, the full many body cal-culations show that increasing cohesion range reduces mis-cibility at higher volume fraction, opposite to the trend pre-dicted by the virial analysis. Moreover, the shape of thebridging spinodals is qualitatively different, with a criticalpoint at �95% filler volume fraction. The bridging spinodalcurves for different interfacial ranges cross, indicating a non-monotonic dependence of miscibility on cohesion range dueto many body effects. The latter intensify with increasingnanoparticle-filler attraction range, and phase separation oc-
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curs even for systems with a positive filler second virialcoefficent B2.
To highlight the dramatic many body effects, a detailedstudy of the structure of a D /d=10 PNC was performedunder intermediate cohesion strength conditions where thesystem is stable �repulsive B2� in the dilute filler limit. Atextremely high nanoparticle volume fractions phase separa-tion does occur driven by many body bridging. As filler vol-ume fraction increases from zero, the wide angle cage peakof the collective nanoparticle structure factor monotonicallygrows and significantly exceeds its pure hard sphere analogdue to the presence of repelling bound polymer layers. How-ever, the zero wavelength filler concentration fluctuationsfirst decrease significantly due to bound layer packingeffects, and then increase as the ultrahigh volume fractiondemixing boundary is approached. The corresponding collec-tive polymer structure factor displays an osmotic compress-ibility that is also nonmonotonic, first increasing with nano-particle additions and then decreasing, before finallyincreasing again near phase separation. The intensity of thefiller-mediated bound polymer layer microphase peak ini-tially increases dramatically with loading, then saturates, andultimately decreases as the particle volume fraction exceeds�75%.
The role of the strength of interfacial attraction on col-lective structure and long wavelength concentration fluctua-tions, from the weak depletion dominated regime through thestrong bridging regime, was studied for a D /d=10 PNC overa wide range of filler volume fractions. Complex, nonmono-tonic behaviors of both the polymer and nanoparticleosmotic compressibilities were found. Their origin is thepresence of competing processes associated with particlecluster formation via depletion and polymer-mediated bridg-ing which are precursors of phase separation, and boundlayer formation which promotes steric stabilization and goodfiller dispersion.
The attractive minimum in the nanoparticle potential ofmean force at phase separation is predicted to be deeper than4kBT under some circumstances, suggesting nonequilibriumgelation or aggregate formation can preempt equilibrium de-mixing. For the systems examined here, the absolute value ofthe depletion contact attraction in the filler PMF is alwayslarger than 4kBT at the spinodal boundary, suggesting frus-tration of macroscopic phase separation in favor of the for-mation of aggregates or contact gels. Polymer-induced bridg-ing of nanoparticles can also lead to kintically trappedconfigurations when phase separation occurs at low fillervolume fractions and for short interfacial cohesion of=0.25 up to ��0.23.
There is also the question of whether the polymers canbe trapped in nonequilibrium states due to their adsorptionon nanoparticle surfaces. This issue has not been consideredhere, but our calculations �Fig. 7� of the interfacialmonomer-nanoparticle pair correlation functions gpn�r�, canprovide some insight by examining the strength of the asso-ciated potential of mean force, −ln�gpn�r��. If this quantity issignificantly attractive at contact then the monomer desorp-tion events required for equilibration may not occur on theexperimental time scale. Of course, explicitly dynamical
considerations are also important, such as whether the poly-mers are short �unentangled� or long �entangled�. For thesame chemical system, it may well be possible that equili-bration is possible for unentangled PNCs but not theirentangled analogs, as suggested by recent studies of thesilica-PEO nanocomposite.41
The present theory can be generalized to treat nonspheri-cal fillers, and the more powerful computational method hasallowed exploratory studies of rodlike, platelike, andnanocube fillers composed of a discrete number of elemen-tary units. This work is relevant to the increasing experimen-tal interest in nonspherical fillers such as carbon nanotubesand exfoliated clays.3 The effect of particle dimensionalityon nanocomposite phase behavior and structure will be dis-cussed in a future publication.39 Finally, efforts to use thetheory to describe PNCs characterized by chemical heteroge-neity of the filler and/or polymer are planned.
ACKNOWLEDGMENTS
This work was supported by the Division of MaterialsSciences and Engineering, U.S. Department of Energy undercontract with UT-Battelle, LLC. We thank Ben Andersonand Chip Zukoski for many informative and stimulatingdiscussions.
APPENDIX: NUMERICAL ALGORITHM
Newton’s method can be used to numerically solve inte-gral equations. However, the full Newton’s method requirescalculation of very large matrices as each function is repre-sented by many points. Special mathematical methods havebeen developed corresponding to a modified Newton’smethod which render solution of the Ornstein–Zernike inte-gral equations tractable even for a large number of points.35
Implementation of these methods can be quite complex. Weemploy the inexact Newton’s method algorithm based onsoftware available in the public domain.36 The KINSol pro-gram, based on NKSOL, was chosen because it is written inthe computer language C. Details of this program can befound in the user documentation,37 and in a recent paperdescribing the SUNDIALS software suite.51 A multilevelmethod,52 or the use of wavelets,53 may allow even fastersolution of the generalized PRISM Ornstein–Zernike equa-tions. However, the KINSol program is relatively easy to useand so drastically increased the ease of convergence over thePicard method that further improvements were not exploredat this time. Generally, the solution from a previously con-verged system at a slightly different � or �pn is used as theinitial guess in the iterative procedure. Nevertheless, somesystems do not converge even if a guess from a very similarsystem is used. As this nonconvergence is approached uponincrementing � or �pn, the k→0 values of the structurefactors increase sharply, indicating the system is nearingspinodal phase separation.
Since PRISM theory is an equilibrium approach, in prin-ciple physically reliable results inside a spinodal demixingregion do not exist. Hence, when strong precursors to phaseseparation are observed upon changing � or �pn by verysmall increments, the last system that can be converged is
234901-14 L. M. Hall and K. S. Schweizer J. Chem. Phys. 128, 234901 �2008�
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reported as an indication of the spinodal phase boundary.54
We have tested the consistency of this estimate of the loca-tion of the spinodal phase boundary by alternatively deter-mining a demixing point from extrapolation of 1 /Spp�k=0�and 1 /Snn�k=0� to zero. As a representative example of theoutcome of this procedure, we report results based on a lin-ear extrapolation using the last two points of each of theD /d=10 and =0.5 mixture curves in Fig. 10. The lastconverged volume fraction for the �pn=0.249 system is�=0.044, while extrapolation to zero of 1 /Snn�k=0�and 1 /Spp�k=0� yields spinodal values �=0.049 and 0.048,respectively. This 10% deviation is the largest we typicallyencounter. For the �pn=0.9 mixture, the last converged�=0.927, while the corresponding values based on extrapo-lation are �=0.935 and 0.934, respectively. We find the threeestimates of spinodal volume fraction at �pn=1.5 and 2.5 areidentical to two significant digits. Since the various estimatesof the spinodal are generally so similar, the last volumefraction converged is employed as the spinodal points inFig. 2 and 3.
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