Coarse-graining and phase behavior of model star polymer–colloid mixtures insolvents of varying qualityArash Nikoubashman, Nathan A. Mahynski, Barbara Capone, Athanassios Z. Panagiotopoulos, and ChristosN. Likos Citation: The Journal of Chemical Physics 143, 243108 (2015); doi: 10.1063/1.4931410 View online: http://dx.doi.org/10.1063/1.4931410 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Accurate coarse-grained models for mixtures of colloids and linear polymers under good-solvent conditions J. Chem. Phys. 141, 244905 (2014); 10.1063/1.4904392 Surface adsorption of colloidal brushes at good solvents conditions J. Chem. Phys. 137, 104703 (2012); 10.1063/1.4752195 A novel algorithm for creating coarse-grained, density dependent implicit solvent models J. Chem. Phys. 128, 154115 (2008); 10.1063/1.2899729 Grand canonical Monte Carlo simulation of a model colloid–polymer mixture: Coexistence line, criticalbehavior, and interfacial tension J. Chem. Phys. 121, 3253 (2004); 10.1063/1.1773771 Phase behavior of mixtures of colloidal platelets and nonadsorbing polymers J. Chem. Phys. 117, 9947 (2002); 10.1063/1.1518007

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THE JOURNAL OF CHEMICAL PHYSICS 143, 243108 (2015)

Coarse-graining and phase behavior of model star polymer–colloid mixturesin solvents of varying quality

Arash Nikoubashman,1,2,a) Nathan A. Mahynski,1 Barbara Capone,3Athanassios Z. Panagiotopoulos,1 and Christos N. Likos3,b)1Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA2Institute of Physics, Johannes Gutenberg University of Mainz, Staudingerweg 7, 55128 Mainz, Germany3Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

(Received 30 July 2015; accepted 9 September 2015; published online 23 September 2015)

We study the effective interactions and phase behavior of star polymer–colloid mixtures throughtheory and Monte Carlo simulations. We extend previous theoretical approaches for calculatingthe effective star–colloid pair potential to take into account attractive contributions, which becomesignificant at worsening solvent conditions. In order to assess the validity of our simulation andtheory, we compute the effective interactions via virtual move parallel tempering Monte Carlosimulations using a microscopic bead-spring model for the star polymer and achieve excellentagreement. Finally, we perform grand canonical Monte Carlo simulations of the coarse-grainedsystems to study the effect of solvent quality on the phase behavior. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4931410]

I. INTRODUCTION

Mixtures of hard colloidal particles and soft, penetrablecolloids have attracted a great deal of interest in the last twodecades.1–4 The vast majority of the investigations have beenbased on the prototypical Asakura-Oosawa model,5 in whichpolymers are modeled as ideal particles that are excluded fromthe colloids within a spherical region whose size is set by thesum of the radii of the colloids and the polymers. However,more realistic models of polymers have also been investi-gated,6–11 leading to better agreement between experimentalresults and the Asakura-Oosawa model. A great deal of inter-est in colloid-polymer mixtures has focused around questionsrelated to the interplay between macroscopic phase separationand the glass and gelation lines induced by the combinationof hard-sphere repulsions and the polymer-induced depletionattractions.12–14

Considerably less is known about mixtures of soft colloidswith other additives. The prototype of deformable colloids withtunable softness is star polymers.15,16 These macromoleculesare made of f homopolymer arms attached to a common cen-tral core; their penetrability depends both on the functionalityf and on the interaction of the homopolymer arms with thesolvent.17 Star polymer solutions present a rich phase diagramthat, depending on f and on the density of stars in solution,spans from fluid to solid phases.18–20 Moreover, their mixtureswith linear chains21 or with smaller stars22 display very richequilibrium phase diagrams as well as a variety of dynamicallyarrested states, which are manifestations of the deformabilityof the polymer-based colloids.

Very recently, attention was turned to mixtures of starpolymers and smaller hard colloidal particles.23–25 The addi-

a)Electronic mail: anikouba@uni-mainz.deb)Electronic mail: christos.likos@univie.ac.at

tion of hard colloids to such solutions allows one to tunethe location of the phase transitions in the phase diagram aswell as to control the location of the line of dynamic arrestof the stars and its interplay with the demixing binodal. Fora good solvent, the factors setting the properties of the star-colloid mixtures are the density of additives in solution andthe size ratio q = Rg/Rc between the average radius of gyra-tion of the star polymer Rg and the colloid radius Rc. Thisdimensionless quantity defines two distinct regions: the firstone is the so-called protein limit q & 1, in which the radiusof gyration of the stars is bigger than the colloidal radius.Here, the degrees of freedom of the macromolecules play animportant role and many colloidal particles can interact at thesame time with the same star polymer thus setting space formany body interactions.26–28 Recent combined experimentaland theoretical studies have revealed that the addition of smallcolloidal depletants to a solution of star polymers has theability to drastically change the dynamical scenario inducing,upon increasing density of the depletant, several rheologicalstates, including phase separation and fluid like suspension.Moreover, the combined theoretical and experimental studiesproven the accuracy of a theoretical approach that is able to pre-dict the phase behavior of the complex solution.23,25,29 When,on the other hand, the radius of the colloids is bigger thanthe radius of the star polymers, the solution is in the so-calledcolloidal limit (q . 1). In this regime, the main equilibriumproperties can often be rationalized by means of the Asuakura-Oosawa picture of depletion.30–34

Most of the computational studies on mixtures of stars andrigid colloids have been carried out at athermal solvent condi-tions for both components, where all interactions involved arepurely entropic in nature.23,24 However, changes in solventquality affect both the softness and penetrability of the stars.17

Hence, their interaction with the additives in solution and theproperties of mixtures of hard and soft colloids are expected to

0021-9606/2015/143(24)/243108/8/$30.00 143, 243108-1 © 2015 AIP Publishing LLC

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243108-2 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

be strongly dependent on the interactions between stars andsolvent, thus opening a new path to control the equilibriumproperties of the assemblies making temperature a key param-eter in influencing phase separation.

Simulations on mixtures of homopolymer chains and col-loids in various solvent qualities show parallels between phasediagrams when the interaction between the polymers and thesolvent worsens and when the size of the chains is reduced.11

A selective change of the solvent quality for the star polymer,thereby creating an intramolecular-intermolecular imbalancein the interactions of the monomers constituting the star poly-mer and in the star–colloid interaction, is a possible path toexperimentally control phase separation behavior in mixturesof soft and hard colloids.

The goal of the present manuscript is to analyze the effectsof worsening solvent quality for the colloidal particles arisingfrom changing temperature in colloid-polymer mixtures. Thelarge number of interaction sites makes it computationallyinfeasible to study the phase behavior of star–colloid mixturesusing microscopic models, which retain the molecular de-tails of the polymers. Therefore, we develop a coarse-graineddescription of the system, where we integrate over all thepolymer’s internal degrees of freedom and map it to a singleultrasoft interaction site. We then use this model in conjunc-tion with grand canonical Monte Carlo (GCMC) simulationsand histogram reweighing techniques to determine the phasebehavior of the mixtures.

The manuscript is structured as follows: we present ourtheoretical model for the effective star–colloid pair potential inSection II, and then compare our theory to simulation data inSection III. In Section IV, we investigate the effect of size ratioand solvent quality on the phase behavior of these systems. InSection V, we summarize and draw our conclusions.

II. THEORETICAL APPROACH

The morphology of each star polymer is determined byits functionality f and its arm length P, while the rigid col-loids are characterized by their radius Rc. The most commonphysical origin of attraction between two colloidal particlesis the dispersion force between fluctuating dipoles. Followingstandard arguments,35 one can show that starting from atomicdipoles and using superposition to construct mesoscopic parti-cles of volumes V1 and V2, made of material elements (denotedby the indices 1 and 2) and immersed in a solvent (denotedby the index 3), the resulting dispersion forces Udisp(r) atinterparticle separation r have the general form,35,36

Udisp(r) ∼ −A132V1V2

r6 , (1)

with the Hamaker constant A132 > 0, which depends on theparticular materials and the solvent as

A132 ≈(

A11 −

A33

) (A22 −

A33

). (2)

In Eq. (2), the quantity Aii, i = 1,2,3 denotes the Hamakerconstant between two material elements of type i in vacuum.Accordingly, the dispersion forces can be completely switchedoff, ideally, when the two materials and the solvent are iden-tical, in the so-called athermal solvent limit. In practice, if a

polymer (chosen now to be denoted by index 1) is dissolvedin a solvent such that A11 � A33, i.e., if the dispersion forcesare weak in comparison to the strong steric repulsions, goodsolvent conditions are achieved and the statistics of the chainare characterized by the properties of a self-avoiding randomwalk.

In previous experimental studies of star polymer–colloidmixtures, the rigid colloids were composed of exactly thesame material as the star polymers, resulting in A11 = A22,so that dispersion forces between all components could beignored, and only short-range, repulsive interactions neededto be taken into account in the modeling of the mixture;25

in fact, this choice of material for the colloidal particles wasmade precisely for the purpose of suppressing the dispersionforces between colloids and star polymers. More generally,colloidal particles can be made of some other material, so thatsignificant dispersion forces between star polymers and col-loids can appear, according to Eq. (2). This being the case, thenalso dispersion forces between colloidal particles are expectedto appear. However, as Eq. (1) shows, the overall dispersionattraction between colloids also scales with the product of theparticles’ volume. Accordingly, the ratio of the colloid-colloiddispersion attraction to its star-colloid counterpart will scaleas q−3. Since we are interested here in the q > 1 case (in fact,we are dealing mostly with the region q > 2), the colloid-colloid dispersion attractions are expected to be one order ofmagnitude weaker than the star-colloid ones, and therefore,we ignore them. Accordingly, for two particles separated bydistance D, the star-star effective interaction U11(D) is givenby its well-established form for good solvent conditions,15,25

and the colloid-colloid effective interaction U22(D) is modeledas a hard-sphere potential. We thus need to modify only thestar-colloid effective interaction U12(D).

In order to derive an expression for the (dispersion) attrac-tion between a colloid and a star polymer, let us first consider asolid colloid and a point-like monomer at distance R from thecolloid’s center, which coincides with the origin of the systemof coordinates. Denoting as φmc(r) the attractive contributionto the interaction between a monomer and a molecule of thecolloid, the attraction between the colloidal particle and themonomer, Φatt(R), can be expressed as

Φatt(R) =

ρc(r)φmc(|r − R|)dr, (3)

where ρc(r) denotes the density profile of the colloidal particle,assumed to be spherosymmetric. Eq. (3) allows for a “deepcoarse-graining”: if the interaction potential φmc(r) betweenthe constituent matter of the colloid and the monomer (e.g.,polystyrene) is known, one can use Eq. (3) in conjunctionwith the density profile of the colloid to compute Φatt(R).However, explicit knowledge of φmc(r) is not necessary fordetermining the effective colloid-monomer interaction, as willbe demonstrated.

Let us now extend our considerations from a single mono-mer to an entire star polymer, with its anchor point separatedfrom the colloid by the vector D. Without loss of generality,we separate the effective star–colloid interaction, U12(D), intotwo contributions, a purely repulsive one due to steric effects,Urep(D), and an attractive one due to the mismatch in solvent

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243108-3 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

FIG. 1. Schematic representation of a colloid in the density cloud of a starpolymer at a distance of D.

quality, Uatt(D). Previously, the steric contribution Urep(D) hasbeen determined analytically by considering the osmotic pres-sure exerted by the star on the surface of the colloid,23,37 andthus, we will concentrate here on deriving an expression forUatt(D).

To develop an analytic expression for Uatt(D), we firstmodel the star polymer by its inhomogeneous monomer distri-bution ρ

(0)s (r), as shown schematically in Fig. 1. Here, we use

the expression for an isolated, unperturbed star as derived inRef. 38. It is conceivable that this approximation breaks downas D → 0, because the star polymer might deform and wraparound the colloid, leading to an anisotropic density distri-bution, which is not captured in our theory. We surmise thatthis effect should become more pronounced with decreasingq. However, the excellent agreement between the theory andsimulations (see Section III) provides an a posteriori justifica-tion for our approximation, even for size ratios q ≈ 1. With thegeometry depicted in Fig. 1, the density profile of the star isgiven by

ρs(r2) = α(D)ρ(0)s (|r2 − D|)Θ(|r2| − Rc). (4)

In Eq. (4), the step functionΘ takes into account that the mono-mers of the star are excluded from a region around the colloiddue to steric hindrance, and α(D) is a numerical coefficient,which guarantees that the integrals of ρs and ρ

(0)s are identical.

This factor can be calculated a priori from Eq. (4) and is shownfor various colloid sizes in Fig. 2. Here, it is clearly visible thatα(D) increases with decreasing q and as D → 0, since the starpolymer is increasingly eclipsed by the colloid.

FIG. 2. Correction term α as a function of star–colloid separation D forvarious size ratios q.

We can now calculate the overall attractive contributionbetween the colloid and the star polymer, given the micro-scopic interaction φmc(|r1 − r2|) and the density profiles ρc andρs. Taking advantage of the radial symmetry of the system andemploying Eq. (3) then yield

Uatt(D)α(D) =

ρ(0)s (|r2 − D|)Θ(|r2| − Rc)Φatt(r2)dr2. (5)

Note that Eq. (5) does not depend on the microscopic inter-action φmc(r) any more, but on the attractive contributionof the whole colloid-monomer interaction, Φatt(r), instead,which is a known input parameter from the simulations. Forcomputational purposes, it is convenient that the ratio U (0)

att (D)≡ Uatt(D)/α(D) takes the form of a convolution between theunperturbed star density ρ

(0)s (r) and the modified potential

Ψ(r) ≡ Θ(r − Rc)Φatt(r), leading to

U (0)att (k) = ρ

(0)s (k)Ψ(k), (6)

where symbols with a tilde (∼) indicate Fourier transformedquantities. The quantity ρ

(0)s (k) is known from Ref. 38, whereas

Ψ(k) is also known from the modeling of the attractions, seeSection III. Thus, we can compute the ratio Uatt(D)/α(D) bytaking the inverse Fourier transformation of Eq. (6). To modelmore realistically the exclusion of the monomers from theinterior of the colloid, which enters into the calculation of thefunction Ψ, we replace the sharp step function Θ(r) with asmoother Fermi-Dirac function subtracted from unity,

Θ(r) → 1 − 1exp(r/ξ) + 1

. (7)

The steepness of the cutoff of Θ(r) can be controlled throughthe parameter ξ, and we found that ξ = 0.1 leads to a reason-ably sharp and smooth boundary.

Finally, we augment the considerations of the attractionsby one additional factor: aiming at modeling of various condi-tions of solvent quality (usually controlled in experiments bya change of temperature), we introduce a scaling factor λ≥ 0, which multiplies the attractive potentialΦatt(r) and conse-quently, via Eq. (5), also Uatt(D),Uatt(D) → λUatt(D)

= α(D)

ρ(0)s (|r2 − D|)Θ(|r2| − Rc)λΦatt(r2)dr2. (8)

Choosing, e.g., λ = 0 allows us to switch off all attractions. Ac-cordingly, the total effective potential between a star polymerand a rigid colloid, U12(D), is a λ-dependent quantity given by

U12(D) = Urep(D) + λUatt(D). (9)

The repulsive contribution Urep(D) diverges as the colloid sur-face approaches the star anchor,23,37 and therefore, it is pref-erable to plot U12(D) against the center-to-surface separationz = D − Rc. In Figure 3, the star-colloid potential U12(z) forf = 18, q = 1.85, and λ = 0.1 is shown, together with theindividual contributions from the right-hand side of Eq. (9),in which the attractive term has been modeled according toEq. (12) to follow. It is clearly visible that the total potential isdominated by Urep(D) when the colloid surface approaches theanchor point of the star polymer. As the colloid moves furtherout from the core region, Uatt(D) becomes more important,

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243108-4 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

FIG. 3. Effective interaction between a star polymer and a rigid colloid vs.the distance between star anchor and colloid surface. Data are shown forfunctionality f = 18, size ratio q = 1.85, and interaction strength λ = 0.1.

causing a long-ranged attraction in U12(D), well beyond thestar’s radius of gyration. In fact, already a relatively smallmismatch in solvent quality is sufficient to cause a sizableattraction between the star and the colloid. This effect is espe-cially pronounced when the colloid surface is located in thecorona of the star.

From a microscopic point of view, this behavior can beunderstood as follows: due to the solvent mismatch for thecolloid, it is energetically favorable to shield it from the solventby wrapping the star’s arms around it. This reasoning suggeststhat the attractive contribution should become larger as thecolloid is approaching the star center, since the monomer den-sity increases significantly as D → 0 (see Fig. 2). This trendis confirmed by our calculations as shown by the evolution ofUatt(D) in Fig. 3. At the same time however, the high monomerdensity in the core region hinders the colloid from penetratingit due to excluded volume effects.

These arguments indicate that both Urep(D) and Uatt(D)should increase with decreasing q, since, on the one hand, thecolloid takes up more volume as it approaches the star center,increasing the steric repulsion Urep(D). On the other hand, thecontact area between the star polymer and the colloid grows,leading to an increase of Uatt(D). In Figure 4, we show thetotal effective interaction U12(D) between a star polymer anda colloid for various size ratios q at fixed f = 18 and λ = 0.1,

FIG. 4. The star-colloid effective potential U12(z) against the surface-to-center separation z for various size ratios q at fixed functionality f = 18 andinteraction strength λ = 0.1.

indicating that the attractive contribution Uatt(D) increases inmagnitude faster than Urep(D) with colloid size.

III. SIMULATIONS AND COMPARISON TO THEORY

In order to validate the effective potential from theory, wecomputed the effective potential between a single rigid colloidand an isolated star polymer via Monte Carlo (MC) simula-tions. We used a bead-spring model for the polymer, whereeach arm was comprised of P = 50 monomers with radius Rm.This choice falls in the scaling limit for the potentials andmodels we used.23 The bonded interactions are given by thefinitely extensible nonlinear elastic (FENE) potential,39

ΦFENE(r) =

− k2

(R0

σ

)2

ln1 −

(rR0

)2(r ≤ R0),

∞ (r > R0),(10)

with σ = 2Rm, the spring constant k, and maximum exten-sion R0. We used the standard Kremer-Grest parameters k= 30 ε/σ2 and R0 = 1.5σ,40 which prevent unphysical bondcrossings. Non-bonded interactions between the monomersare modeled through the purely repulsive Weeks-Chandler-Andersen (WCA) potential,41 to mimic good solvent condi-tions for the star polymer,

Φmm(r) =

4ε(σ

r

)12−

(σ

r

)6+ ε (r ≤ 21/6σ),

0 (r > 21/6σ).(11)

The excluded volume interactions between the monomers andthe colloid, Φrep(r), are also modeled through a functionalform identical to that of Eq. (11) but with the change of scaleσ → σmc = Rc + Rm. More accurate pair potentials have beenderived when Rc ≫ Rm,42 but we decided to keep the samemonomer-colloid interaction potential for all size ratios q forthe sake of consistency. This approach is justified by the excel-lent agreement between the MC simulations and theory previ-ously established for the athermal case,23 where a similar rangeof size ratios was investigated. In the case of an implicit sol-vent, thermal effects can be introduced into our microscopicmodel by adding an attractive contribution to the monomer-colloid interaction,17

Φatt(r) =

−ε (r ≤ 21/6σmc),4ε

(σmc

r

)12−

(σmc

r

)6

(r > 21/6σmc).(12)

In agreement with the considerations put forth in Section II,we consider also a scaled version of the monomer-colloidattraction by multiplying Eq. (12) with a factor λ ≥ 0. Notethat for λ = 1, the total monomer-colloid interaction, Φmc(r)= Φrep(r) + λΦatt(r), becomes the standard Lennard-Jonespotential. For purposes of computational efficiency, we trun-cated the monomer-colloid potential at rcut = 2.5σmc andshifted it so that Φatt(rcut) = 0.

In order to compute the effective star–colloid potentialfrom the MC simulations, we make use of the relation betweenthe Boltzmann factor and the correlation function g12(D) atvanishing density,43

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243108-5 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

U12(D) = −kBT ln [g12(D)] . (13)

In practice however, it is difficult to retrieve a sufficient num-ber of samples for high-functionality stars, especially forsmall distances between the star center and the surface of thecolloid. Hence, it is necessary to employ biasing techniques44

to improve the sampling in those regions. To this end, weemployed the virtual move parallel tempering Monte Carlo(VMPT-MC) technique,45 which is a combination of the adap-tive parallel tempering algorithm46 and the waste recyclingmethod developed by Frenkel.47 In this work, we did not swapconfigurations of the system obtained at different temperatures,but rather considered a single temperature kBT/ε = 1.

The MC simulations consisted solely of translationalmoves for the monomers and the colloid, and we performed500 translation attempts for each monomer before attemptinga colloidal move. Hence, each MC step consisted of 500 f P + 1individual moves, and we ran our simulations for at least105 MC steps to ensure that the computed potentials haveconverged to their final profile.

In our theoretical description, we used the monomer den-sity profile and the radius of gyration, Rg, of an isolated star tocompute the effective potential U12(D). In order to assess thequality of this approximation, we first measured Rg for systemswith and without a colloidal particle. We were not able tofind any appreciable effect, even for systems with rather largecolloids (q ≈ 1) and thermal interactions (λ = 0.1), which pro-vides an a posteriori justification for our assumption.

Figure 5 shows a comparison between the results fromtheory and simulations for various λ values at f = 18 and q= 1.85, highlighting the excellent agreement between the two.As in previous studies,23,37 the center-to-surface separation ofthe simulation data had to be shifted by rcore � 0.2 Rg to coin-cide with the effective potentials from theory. This mismatchis due to microscopic effects, which are still relevant despitethe chains being long enough to have reached the scaling limitof the stars. Indeed, in the star center, there is a dense corewhose size scales as rcore � σ

f , whereas the gyration radius

of the whole star scales as Rg � σ f 1/5Pν, with ν � 0.6.48

Using f = 18 and P = 50, we find indeed rcore/Rg � 0.2. Inthe simulations, the chains are stretched near the center insteadof having the statistics of a semi-dilute polymer solution. Thecore region is still a significant portion of the whole star, which

FIG. 5. Solid lines show the effective star–colloid interactions U12 fromtheory, while symbols show the results from our VMPT-MC simulations. Datashown for functionality f = 18 and size ratio q = 1.85.

FIG. 6. Fraction of monomers S(z) within the monomer-colloid interactionrange for size ratio q = 1.85. Symbols correspond to simulation data, whilethe line shows the result from theory.

will of course drop out of sight as P ≫ 1. Since in theory thecore is already vanishingly small, it has to be subtracted fromthe simulation data to have a fair comparison with theory.

As an additional test of the agreement between our theoryand simulations, we compared the fraction of monomers S(D)within the interaction range of the monomer-colloid inter-action. This quantity is readily available in our theoreticaldescription by integrating ρs for a spherical shell with thick-ness rcut around the colloid. In the simulations, S(D) can becomputed through

S(D) = 4πf P

rcut

0gmc(r,D)r2dr, (14)

where gmc(r,D) is the radial distribution function between themonomers and the colloid at a fixed star-colloid separationD. Figure 6 shows S(z) for both theory and simulation. Thetheory slightly underestimates S(z) in the vicinity of the starcenter, which is likely due to the fact that we redistribute themass contained in ρ

(0)s cut out by the colloid homogeneously

across the star’s density profile [see Eq. (4)]. In reality, themonomers are displaced more locally because of the bondedinteractions. Furthermore, it appears that S(z) is independentof λ, indicating that the attraction between the monomers andthe colloid does not cause a significant deformation of the starpolymer, over the range of λ investigated.

IV. PHASE BEHAVIOR

Having developed the set of interactions Ui j(D) amongthe components i, j = 1,2 of the mixture (index 1 denotingthe stars and index 2 the rigid colloids), we can now pro-ceed to an investigation of the mixing behavior of the system,depending on size ratio q and attraction strength λ. To thisend, we performed GCMC simulations on these star–colloidmixtures using this coarse-grained pair potential description,while neglecting triplet interactions among stars, because theyare known to only contribute a relatively small correction totheir pairwise counterparts.49 The colloid-colloid interactionswere modeled using a hard sphere potential, which diverges atcontact. All simulations were conducted in a cubic simulationbox. We chose the edge length L such that for all combinationsof chemical potentials considered, the box contained up to afew hundred Nc colloids, and a similar amount of Ns stars. The

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243108-6 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

colloid packing fraction is defined as ϕc = 4πNcR3c/(3L3) and

the star packing fraction as ϕs = 4πNsR3g/(3L3); the overlap

concentration of stars thus occurs at ϕ∗s = 1.We carefully checked our data for finite-size effects by

varying L at selected state points. Moves in the MC simulationsincluded particle insertion, deletion, and translations, whichwere conducted with equal probability and accepted accordingto the Metropolis rule. Each run was equilibrated for 107 MCsteps, followed by a production period of equal length, duringwhich the particle number densities were recorded every 500steps. These measurements were then combined to producetwo-dimensional histograms of star polymer and colloid densi-ties.

We subsequently employed histogram reweighting tech-niques50–52 to combine histograms from runs at many differentimposed star polymer and colloid chemical potentials, µp andµc, respectively. This allows us to build up the underlyingentropy surface, or density of states, which can be reweightedto produce the partition function of the binary mixture at anychosen external condition. From the partition function, theprobability of observing a certain system with a given densityof polymers and colloids may be obtained at any µp and µc.This probability distribution can then be mapped onto theknown three-dimensional Ising distribution at the critical point,accounting for mixing of the two density fields,53 allowing usto accurately identify the critical point in these fluids when thedistributions are equal within statistical and simulation error.Subsequently, the fluid–fluid binodal curves were identified byusing the equal peak area criterion.

Figure 7 shows the coexistence lines and the critical pointsfor fixed star functionality f = 18 and various size ratios q. Atdensities above the binodal, the system phase separates intotwo fluid phases, one polymer-rich and colloid-lean (“colloidalgas” phase), while the other is polymer-lean and colloid-rich(“colloidal liquid” phase). We observed phase separation for allinvestigated size ratios q, a finding which is in contrast to previ-ous estimates23 using a Barker-Henderson mapping onto effec-tive hard spheres.54 This metric essentially reflects the changein volume, ∆Vmix, when a single star and single colloid aremixed together and was previously used under the assumptionthat a bulk system is expected to phase separate when ∆Vmix< 0, whereas the system is expected to be stable against demix-ing when ∆Vmix > 0. This line of argument had previously led

FIG. 7. Binodal curves of athermal star polymer–colloid mixtures for varioussize ratios q at fixed functionality f = 18.

to the conclusion that for any f , there exists a threshold q abovewhich the mixture will not phase separate.23 This finding stoodin contrast to other GCMC simulations which suggested thatfluid-fluid phase separation of these mixtures persists deep intothe protein limit.24

Although the approach based on the non-additivity param-eter does capture some essential features of the mixture (e.g.,the fact that as q ≫ 1 the cross-interaction becomes very softand penetrable), it is quantitatively inaccurate in predictingwhen the system would be stable against demixing. One reasonfor this seems to be the fact that the underlying interactionsU11(D) and U12(D) are too soft and feature tails that are toolong-ranged to be mapped onto effective hard spheres in areliable way. Furthermore, this non-additivity parameter isbased solely on pairwise interactions and is not a rigorousrepresentation of the behavior of a bulk system, as evidencedby Fig. 7 which clearly demonstrates that phase separationindeed persists in the protein limit for a star ( f = 18) which hadbeen predicted to yield only a stable single phase at q & 1.25.

The binodal shifts to lower ϕc and higher ϕs as q isincreased. These trends are in qualitative agreement withprevious microscopic MC simulations of mixtures of rigid col-loids and low-functionality stars on a lattice.24 The predictedcoexistence densities for both the colloids and stars are roughlya factor of two lower in our coarse-grained simulations, thoughthis discrepancy is not unreasonable considering the presentmodel is not on a lattice, nor does it contain any explicit detailsabout the conformational changes of a star’s corona whenencroached upon by multiple colloids and their resulting three-body interaction.

Let us now turn our attention to the effect of solvent qualityon the phase behavior by systematically increasing λ. This isof particular importance since experimentally it is very hardto tune the star-colloid interactions to purely steric repulsions,and therefore, one needs to know how strong the effect of resid-ual attractions on the demixing binodals might be. Moreover,it has been demonstrated that for high-functionality stars, theaddition of colloids brings about a melting of the glass formedby the star polymers, with the glass line ending at the demixingbinodal resulting in arrested phase separation.25 The evolutionof the (repulsive) glass line upon addition of colloids is notexpected to depend drastically on the star-colloid interactions,since it is mostly dictated by short-range, caging effects. Thebinodal, on the other hand, can be shifted by large amountsonce attractions are present, and their interplay can be used tocontrol the location of structurally arrested states in the phasediagram.

Representative results quantifying the effects of star-colloid attractions on the demixing binodals are shown inFigure 8. It is evident that the coexistence lines drastically shiftto higher polymer- and colloid-concentrations as the solventquality is decreased. With increasing λ, the effective cross-interaction between the colloid and polymer becomes energeti-cally more favorable, offsetting the entropic penalty for the twospecies to overlap. This entropic penalty, better known as thedepletion interaction, drives phase separation in the athermalcase. The system becomes stable at higher densities as thisdriving force is reduced, shifting the binodal as shown in thefigure. The increased propensity of stars and colloids to be

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243108-7 Nikoubashman et al. J. Chem. Phys. 143, 243108 (2015)

FIG. 8. Binodal curves of thermal star polymer–colloid mixtures for variousinteraction strengths λ at fixed functionality f = 18 and size ratio q = 1.85.

in the neighborhood of one another suppresses the colloid-induced depletion between the stars and stabilizes the mixture.Accordingly, a larger region of miscibility between the twocomponents opens up, in which the ergodic-to-nonergodictransition of the star polymers upon addition of colloids canbe investigated before it interferes with the binodal line of thedemixing transition.

Although the values of λ that bring about this change inthe location of the binodals may seem quite small, the effectivecross-interaction between the colloid and star polymer is verysensitive to the value of λ. To illustrate this, and to rationalizeour findings, we consider the evolution of the reduced secondvirial coefficient B∗2 of the cross interaction, which quantifiesthe deviation from ideal gas behavior due to pairwise interac-tions, relative to that of hard spheres,

B∗2 =3

�Rc + Rg

�3 ∞

0{1 − exp [−βU12 (D)]} D2dD.

(15)

As B∗2 approaches zero, the stars and colloids have effectivelyno excluded volume and behave essentially like ideal gas parti-cles with respect to each other at the two-body level. WhenB∗2 = 1, the stars and colloids interact as if they were effectivelyhard spheres with radii Rg and Rc, respectively. We wouldlike to remark that B∗2 can exceed unity in our systems, sincethe star’s corona extends well beyond Rg. Regardless, as B∗2decreases, the excluded volume which drives the depletion-based attraction between the two species is reduced, whichstabilizes the mixture against phase separation. We have foundB∗2 = 1.4, 1.2, 0.89, and 0.55 as we merely increased λ from0.00 to 0.03, which is consistent with the phase behavior shownin Figure 8.

V. CONCLUSIONS

We analyzed the impact that the addition of small, hardcolloids has on the behavior of suspensions of large star poly-mers. For this purpose, we employed a previously derivedcoarse-graining approach for the steric interactions betweenmonomers and colloids, and we enhanced it with a mean-fieldapproach to take into account attractions of varying strengththat can develop between the two components as an effect of

varying solvent quality. Our approach is valid for the range ofgood solvent conditions for the polymer where the statisticsof the chain still follow the rules of a self-avoiding randomwalk; the solvent effects primarily manifest themselves byincreasing the attraction between the star polymers and therigid colloids. The validity of our coarse-grained models wasconfirmed via monomer-resolved simulations. To complete thebridging of the length scales all the way to the macroscopicone, we conducted grand canonical Monte Carlo simulationsbased on the coarse-grained model, to investigate the phasebehavior of these mixtures for various size ratios and solventqualities.

Our investigations have brought forward two significantnovel aspects: First, the evolution of the phase diagram asa function of size ratio for athermal solvents, demonstratingthat demixing persists for much more asymmetric mixturesthan previously thought. Second, we were able to demon-strate that even weak attractions between star polymers andcolloidal particles can have drastic effects on the location ofthe demixing binodals, which, in comparison to the athermalcase, expose a broader thermodynamically stable region ofthe phase diagram. More detailed investigations are called for,both theoretically and experimentally, to look into the locationof the structural arrest line and the possibility of novel glassystates for these thermal soft/hard colloidal mixtures. Thesequestions will the subject of future investigations.

ACKNOWLEDGMENTS

We acknowledge financial support for this work fromthe Princeton Center for Complex Materials (PCCM), a U.S.National Science Foundation Materials Research Science andEngineering Center (Grant Nos. DMR-0819860 and DMR-1420541). N.A.M. received support from the U.S. NationalScience Foundation (Grant No. CBET-1403049). B.C. ack-nowledges support by the Austrian Academy of Sciences(ÖAW) through an APART-Fellowship, No. 11723. Addi-tional support by the ETN-COLLDENSE (H2020-MCSA-ITN-2014, Grant No. 642774) is gratefully acknowledged.

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