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Collective Structure and Dynamics in Dense Colloid-Rod Polymer Suspensions Y.-L. Chen and K. S. Schweizer* Department of Chemical Engineering, Department of Materials Science and Engineering, and Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801 Received April 1, 2002. In Final Form: June 26, 2002 Microscopic liquid-state theories are employed to study the structure, free volume, and dynamical properties of rod polymer-colloid suspensions with an emphasis on the high particle density regime. Depletion effects result in strong local clustering of the colloids and polymers in the one-phase region, and more so with increasing colloid-to-rod size asymmetry ratio. The colloidal collective cage order at high densities is a nonmonotonic function of rod concentration reflecting competing physical effects. A dynamic consequence is a strong modification of the colloidal glass transition volume fraction. Far from the glass transition, depletion attraction between colloids can suppress (enhance) self-diffusion (shear viscosity) by modest factors of ∼2-4 and lead to a violation of the Stokes-Einstein relation with increasing rod polymer concentrations. Polyelectrolytes are also studied using a simple nonadditive excluded volume model. The additional polymer-polymer repulsions result in suppression of depletion-driven fluid-fluid phase separation and the tendency of the rods to preferentially segregate near the colloids. Consequences of the latter effect include a more dramatic modification of local colloidal structure and glass formation, the emergence of a statistical adsorption or “haloing” phenomenon absent for neutral rods, and a strong collective organization of the charged polymers on multiple length scales at high rod concentrations.
I. Introduction Polymer-colloid suspensions are ubiquitous in diverse fields of science and engineering.1-5 In recent years, there has been a resurgence of interest in the fundamental physical behavior of such complex mixtures.6-19 The most common spherical “colloids” (micelles, proteins, microgels, quantum dots, nanoparticles, and so forth) and polymers (synthetic or biological) can vary in global size from nanometers to microns. The phase behavior, viscoelasticity, flocculation, gelation, and vitrification of polymercolloid suspensions play important roles in industrial processes ranging from production of car paints and motor * Corresponding author. E-mail:
[email protected]. Fax: (217) 333-2736. Telephone: (217) 333-6440. (1) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: 1983. (2) Vincent, B. Adv. Colloid Interface Sci. 1974, 4, 193. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, UK, 1989. (4) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: Oxford, UK, 1999. (5) Lewis, J. A. J. Am. Ceram. Soc. 2000, 83, 3241. (6) Fuchs, M.; Schweizer, K. S. J. Phys.: Condens. Matter 2002, 14, R239. (7) Gast, A. P.; Russel, W. B. Phys. Today 1998, 51, 24. (8) Poon, W. C.; Pusey, P. N.; Lekkerkerker, H. N. W. Phys. World 1996, 9, 27. (9) van Blaaderen, A. MRS Bull. 1998, 23, 39. (10) Dijkstra, M.; van Roij, R.; Evans, R. J. Chem. Phys. 2000, 113, 4799. (11) Eisenreigler, E. J. Chem. Phys. 2000, 113, 5090. (12) Eisenreigler, E. Phys. Rev. E 1997, 55, 3116. (13) Bohlius, P. G.; Louis, A. A.; Hansen, J. P.; Meijer, E. J. Chem. Phys. 2001, 114, 4296. (14) Tunier, R.; Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 2001, 113, 10768. (15) Schmidt, M. Phys. Rev. E 2000, 62, 3799. (16) Brugger, M. B.; Lekkerkerker, H. N. W. Macromolecules 2000, 33, 5532. (17) Vliegenhart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 1999, 111, 4153. (18) Adams, M.; Dogic, Z.; Keller, S. L.; Fraden, S. Nature 1998, 393, 349. (19) Bolhius, P.; Frenkel, D. J. Chem. Phys. 1994, 101, 9869.
oils to protein crystallization and self-assembly of functional nanoparticle materials.3-9 A problem of major importance in particle technology is the consequences of so-called “depletion attractions” between colloids induced by polymer additives.1-9 The classic theoretical work in this area is of a statistical thermodynamic nature and focuses on equilibrium phase behavior.3,20-23 Major simplifications are generally invoked regarding the description of the polymer component of both a model and statistical mechanics nature. For example, effective one-component colloid models have been developed, where the (small) polymer only enters implicitly via a low density pair decomposable depletion potential.20-22 More recently, two-component mean field theories have been constructed for the free energy in the dilute, small polymer limit.16,17,23 Structural studies have generally been restricted to the elementary question of the potential of mean force between a pair of large colloids dissolved in a polymer solution.11,20,21,24-28 Recently, the development and application of a microscopic statistical mechanical approach to the equilibrium structure, thermodynamics, and phase behavior of such mixtures have been pursued.6,28-31 This approach is based (20) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (21) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (22) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (23) Lekkerkerker, H. N. W.; Poon, W. C.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Europhys. Lett. 1992, 20, 559. (24) Mao, Y.; Cates, M. E.; Lekkerkerker, H. N. W. J. Chem. Phys. 1997, 106, 3721. (25) Joanny, J. F.; Leibler, L.; deGennes, P. G. J. Polym. Sci., Polym. Phys. Ed. 1979, 17, 1073. (26) Lin, K. H.; Crocker, J. C.; Zeri, A.; Yodh, A. G. Phys. Rev. Lett. 2001, 87, 088301. (27) Verma, R.; Crocker, J. C.; Lubensky, T. C.; Yodh, A. G. Phys. Rev. Lett. 1998, 81, 4004. (28) Chatterjee, A. P.; Schweizer, K. S. J. Chem. Phys. 1998, 109, 10464-10477. (29) Chatterjee, A. P.; Schweizer, K. S. Macromolecules 1999, 32, 923. (30) Fuchs, M.; Schweizer, K. S. Europhys. Lett. 2000, 51, 621.
10.1021/la020309r CCC: $22.00 © 2002 American Chemical Society Published on Web 09/07/2002
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on a fusion and generalization of liquid-state integral equation theory of spheres32 and polymers (polymer reference interaction site model, or PRISM, theory33-39). The theory is, in principle, applicable for all polymer/colloid size asymmetry ratios and species concentrations. It has the advantage of predicting not only thermodynamic properties and fluid-fluid demixing but also structural correlations of all species over all length scales.6,30 Most of the PRISM work considers hard spheres mixed with flexible random coil polymers under strictly athermal solution conditions. Multiple comparisons with experiments for local structure, scattering patterns, thermodynamic properties, and fluid-fluid phase separation have demonstrated the theory to be quite accurate.6,30,31,40-43 Novel predictions for the role of polymer/particle size asymmetry on fluid-fluid phase separation boundaries have been made and recently verified experimentally.40 The PRISM approach has been very recently applied to treat rigid rod polymers mixed with hard spheres in the limit when one of the species is dilute.44 On the basis of the simplest Percus-Yevick (PY) closure approximations, the depletion potentials, second virial coefficients, and insertion chemical potentials have been studied. Such dilute conditions are in many ways the most demanding for liquid-state theory, since the strength of polymerinduced depletion attractions between large colloids can be much larger than the thermal energy.6,31,45 The goal of the present paper is to initiate study of the structure of true two-component rod-particle mixtures in the high particle volume fraction/relatively low polymer concentration regime. This is a situation relevant, for example, to ceramic and colloid processing, assembly of photonic band gap materials, and fabrication of porous/ membrane materials.3,5,7-9 Here, the strengths of depletion attractions are modest in the one-phase homogeneous region,6,31 which optimizes the accuracy of PRISM-PY. The influences of size asymmetry (particle diameter, D ) 2R, relative to rod length, L) and polymer concentration on physical clustering and collective structure are investigated. We also address the role of polymer-polymer interactions, which can vary because of solvent quality changes or charging (polyelectrolytes), resulting in effectively nonadditive excluded volume interactions that can significantly modify thermodynamic and structural properties. The structural information is combined with the simplest versions of modern microscopic statistical dynamical theories to determine the impact of depletion forces on transport coefficients in the homogeneous phase and also the colloidal glass transition. The paper is organized as follows. The molecular models (31) Fuchs, M.; Schweizer, K. S. Phys. Rev. E 2001, 64, 021514. (32) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986. (33) Schweizer, K. S.; Curro, J. G. Adv. Chem. Phys. 1997, 98, 1 and references within. (34) Schweizer, K. S.; Curro, J. G. Adv. Polym. Sci. 1994, 116, 319. (35) Schweizer, K. S.; Curro, J. G. Phys. Rev. Lett. 1987, 58, 246. (36) Curro, J. G.; Schweizer, K. S. J. Chem. Phys. 1987, 87, 1842. (37) Schweizer, K. S.; Curro, J. G. Macromolecules 1988, 21, 3070. (38) Chandler, D. Studies in Statistical Mechanics; North-Holland: Amsterdam, 1982; Vol. VIII, 274. (39) Chandler, D.; Andersen, H. C. J. Chem. Phys. 1972, 57, 1930. (40) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2002, 116, 2201. (41) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. Langmuir 2002, 18, 1082. (42) Kulkarni, A. M.; Chatterjee, A. P.; Schweizer, K. S.; Zukoski, C. F. Phys. Rev. Lett. 1999, 83, 4554. (43) Kulkarni, A. M.; Chatterjee, A. P.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2000, 113, 9863. (44) Chen, Y. L.; Schweizer, K. S. J. Chem. Phys., 2002, 117, 1351. (45) Dickman, R.; Attard, P.; Simonian, V. J. Chem. Phys. 1997, 107, 205.
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employed and PRISM-PY theory are described in section II. Section III presents structural correlation predictions in both real and Fourier space for ideal needles and model charged rods. The evolution of a length scale dependent “free volume” in the suspensions as a function of rod concentration is also studied. The influence of rod additives on the colloidal glass transition is addressed in section IV. The modification of suspension shear viscosity and colloid self-diffusion by depletion effects in the homogeneous regime far from the glass transition is also studied in section IV. The paper concludes in section V with a brief discussion and summary. II. Equilibrium Theory We are interested in ternary mixtures of polymer rods, spherical particles, and small molecule solvent. To simplify the problem, we treated the solvent as a background continuum (vacuum) whose chemical nature only influences the effective pair interactions in a binary mixture of rods and particles. A. Formulation and Molecular Models. The PRISM theory33,34 is a generalization of the small molecule RISM theory of Chandler and Andersen38,39 and is based on an interaction site representation of molecules and a generalization of the matrix Ornstein-Zernike equation.33,34,38,39 The latter relates the total intermolecular site-site pair correlation functions as a function of scalar site separation, hij(r) ) gij(r) - 1, the intermolecular sitesite direct correlation functions, Cij(r), and the intramolecular structure factors, ω ˆ i(k), where the subscripts i and j denote the mixture species, r denotes the real space separation, and k denotes the wavevector. In Fourier space, the PRISM matrix integral equations for molecules composed of identical interaction sites (on average) are
hˆ ij(k) ) ω ˆ i(k)[C ˆ ij(k) ω ˆ j(k) +
∑l Cˆ il(k)Flhˆ lj(k)]
(1)
where explicit chain end effects have been ignored.33-37 The collective partial structure factors are given by
ˆ i(k)δij + FiFjhˆ ij(k) S ˆ ij(k) ) Fiω
(2)
In numerical work, the diagonal partial structure factors are nondimensionalized by taking out a factor of site density Fi. We consider hard sphere colloids of diameter D, ω ˆ s(k) ) 1, at a reduced density, FsD3, or volume fraction, φs ) (π/6)FsD3. An interaction site model of a rigid rod is adopted, which has the virtue of being naturally generalizable to treat chain semiflexibility and site-site intermolecular attractive or repulsive tail potentials. For simplicity, we consider the continuum version of a tangent bead rod of N sites with bond length l defined by the limiting procedure N f ∞, l f 0, L ) Nl. This results in a continuous rod with an intramolecular structure factor given by
ω ˜ p(k) )
1 ω ˆ (k) ) 2 N p
∫01(1 - t)
sin(tkL) dt tkL
(3)
A smooth spherocylinder can be modeled by assigning a hard core diameter, d, to each “site”, as is appropriate for athermal good solvent conditions. The infinitely thin “needle” model follows in the d f 0 limit and crudely describes ideal or Θ-like solvent conditions, where the
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rods can pass through each other but not the colloidal particles.13-17,19,46,47 Globally isotropic orientations for the rod particles are assumed. In three dimensions, this holds for all concentrations of ideal needles. For finite thickness spherocylinders, the rod concentration must be below the isotropicnematic (I-N) phase transition. For L . d large aspect ratio rods, Onsager theory48,49 predicts that (in the absence of particles) a purely nematic phase emerges at a rod site number density of Fp,NI ≈ 4.25/Ld2. When the rod site density is below Fp,NI, the isotropic rods can strongly repel above the dilute-semidilute crossover concentration Fp* ≈ N/Vp ) 3N/(4πRg3) ) (18x2)/(πdL2), which is generally much lower than the I-N phase transition for rods with large aspect ratios. In terms of a dimensionless rod molecule density (in units of the dilute-semidilute threshold), the I-N transition is estimated as cp,NIVp ) (4.25π/(18x3))(L/d) ≈ 0.43(L/d), where cp is the rod molecule number density. For L/d < 100 or so, greater than two-body (second virial) interactions are important in the isotropic phase.48,49 If d , D and L, the approximation of a finite thickness rod by a needle is geometrically plausible. We also consider a hybrid “charged needle” model, which crudely mimics the presence of Coulomb interactions. It is defined as a rigid rod of zero thickness when it interacts with particles, but it interacts with other rods with a nonzero thickness d. This choice of intermolecular excluded volume potentials mimics Coulomb repulsions between rods at an effective diameter level in the spirit of Onsager’s analysis. Accounting for the electrostatic interactions in this manner ignores the fact that the ionic strength of the solution would increase as the density of charged rods increases, and screening effects would make the rods effectively “thinner”. However, in the presence of a sufficient amount of added salt, the Debye-Hu¨ckel screening length could be kept roughly constant for low rod concentrations and/or charge densities. We close the integral equations with the site-site Percus-Yevick (PY) approximation, which has been shown to be quite accurate for the pure components.32-39 For the athermal mixture, the site-site PY closure is given by33,39
gij(r) ) 0
r < Rij
Cij(r) ) 0
r > Rij
B. Thermodynamic Properties and Phase Separation. Thermodynamic properties are computed using standard statistical mechanical relations and the compressibility route. Spinodal liquid-liquid phase separation follows from the condition of simultaneous divergence at zero wavevector of all partial structure factors, S ˆ ij(k)0) f ∞. Using eqs 1 and 2, this condition can be explicitly written as33,34,50-52
1 - cpC ˜ pp(0) - FsC ˆ ss(0) + cpFs[C ˜ pp(0) C ˆ ss(0) C ˜ ps2(0)] ) 0 (5) where C ˜ pp ) N2C ˆ pp, C ˜ ps ) NC ˆ ps, and cp ) Fp/N. Equation 5 is rigorously equivalent to the vanishing of the determinant of the matrix of second derivatives of the free energy. Detailed results for the predicted spinodals are given elsewhere;51,52 in the present work, we only indicate their location in specific applications. For mixtures of charged needles and hard spheres, polymer-colloid excluded volume interactions favor miscibility, since the rods can reduce interpolymer repulsions by statistically accumulating near the neutral spheres. Hence, as expected from prior studies using the PY closure in the purely athermal limit,6,30,32,45 no spinodal phase separation is found for the systems examined. In the opposite limit of nonadditivity, the ideal needles experience excluded volume interaction only with particles, which leads to stronger induced depletion attraction between spheres. In this case, spinodal phase separation is predicted by PRISM-PY.51,52 We also compute the excess chemical potential, βδµ, for inserting a probe sphere of diameter d* into a colloidneedle mixture. This quantity is a measure of the fractional equilibrium “free volume” in the suspension, defined as R ≡ Vfree/V ) exp(-βδµ). The needle occupies no spacefilling volume, and for dynamics applications, we are interested in studying how the free volume associated with the colloids is modified by the physical clustering driven by depletion attractions. This quantity can be computed using the standard “thermodynamic charging” formula solely on the basis of knowledge of C ˆ fs(k)0) in the needle-particle suspension:28-31,51,52
βδµ ) (4)
where Rij is the distance of closest approach between interaction sites of type i and type j. Since ideal needles can pass through each other, Cpp(r) ) 0 for all r. The ideal needle model is a crude representation of rods in Θ solvents below the dilute-semidilute crossover concentration. It has been the subject of several computer simulation studies because of its technical/computational simplicity,19,46,47 and it has the advantage of removing from the problem the molecular parameter L/d. For charged rods, the hypernetted-chain (HNC) closure is often employed.32 However, our crude mimicking of Coulombic forces by a nonadditive purely excluded volume model justifies the use of the PY closure as a first approximation.32 In all cases, the integral equations are solved numerically using a standard Picard iteration procedure.32-34 (46) Yaman, K.; Jeppesen, C.; Marques, C. M. Europhys. Lett. 1998, 42, 221. (47) Vroege, G. J.; Lekkerkerker, H. N. W. Rep. Prog. Phys. 1992, 55, 1241. (48) Doi, M.; Edwards, S. F. Theory of Polymer Dynamics; Oxford Press: Oxford, 1986. (49) Eppenga, R.; Frenkel, D. Phys. Rev. A 1985, 31, 1776. (50) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774.
∫0F Cˆ fs(k)0;Fp;F′s) dF′s s
(6)
Here the subscript “f” refers to the probe sphere, β ) 1/(kBT), and Cfs is the probe sphere-colloid direct correlation function. The latter follows from solving a single linear integral equation for a probe sphere dissolved in a particle fluid characterized by S ˆ ss(k) of the needle-particle mixture.
ˆ fs(k) S ˆ ss(k) hˆ fs(k) ) C gfs(r) ) 0,
r < (d* + D)/2
Cfs(r) ) 0,
r > (d* + D)/2
(7)
Before proceeding to applications, we make a few comments on the reliability of the site-site PY closure approximation. Recent PRISM-PY studies of mixtures consisting of rods and hard spheres when one of the species is dilute find that, for the L > D “nanoparticle” regime, the errors incurred by the PY closure appear to be remarkably small for depletion potentials and many (51) Chen, Y. L. M.S. Thesis, University of Illinois, Urbana, IL, unpublished work, 2001. (52) Chen, Y. L.; Schweizer, K. S. In preparation.
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thermodynamic properties.44,51 But as the L < D colloid regime is entered, the errors for some properties such as the depletion potential increase and the entropic attraction is quantitatively underpredicted. A deep understanding of the limitations of the local nature of the PY approximation has been recently developed for random coilsphere systems.6,30,31 The primary issue is the neglect of the restriction of polymer conformational (orientational for rigid rods) fluctuations due to the presence of hard particles inherent to the local nature of the PY approximation. Recent work has proposed and successfully applied modified PY (m-PY) closures for the polymercolloid direct correlations which include in a thermodynamically self-consistent manner nonlocal entropic repulsion effects.6,30,31 However, this approach is mathematically and numerically rather involved, and its formulation for macromolecules that are not random coils is an open problem. Thus, in this first exploratory study of rodsphere mixtures, the simpler PY closure for polymercolloid direct correlation is used. Our emphasis on high density sphere suspensions guarantees the depletion potentials to be relatively weak compared to kBT in the homogeneous fluid regime, which optimizes the accuracy of the PY approximation for both polymer-colloid and colloid-colloid direct correlations.28,29,31,45
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Figure 1. Site-site pair correlation functions for an L/D ) 0.5 ideal needle-sphere mixture at state points in proximity and far away from the spinodal and at low and high sphere densities. The solid and dashed lines are φs ) 0.0367 and 0.508 and cp/cp* ) 0, respectively; the dot-dashed and dotted lines are at (φs ) 0.0367, cp/cp* ) 20 ≈ 0.99cp+/cp*) and (φs ) 0.508, cp/cp* ) 1.3 ≈ 0.99cp+/cp*), respectively, where cp+/cp* is the reduced needle number density at the spinodal boundary.
III. Liquid Structure The depletion effect can result in large structural changes. In this section, we examine the real space intermolecular pair correlation functions gpp(r), gss(r), and gps(r), and the collective concentration fluctuation structure factors S ˆ ss(k) and Spp(k) over all length scales for a range of polymer/colloid size asymmetries and densities and for the ideal and charged needle models. A. Ideal Needles. Two mixtures are studied with size ratios L/D ) 0.5 and 2 (or, equivalently, Rg/R ∼ 0.29 and 1.16). The intermolecular structure is examined at low and very high sphere densities with the needle concentration far below and near the spinodal phase boundary. The latter is reported nondimensionalized by the dilutesemidilute crossover concentration Fp/Fp* ) cp/cp*. The most local measure of physical clustering of the spheres by needle addition is characterized by the contact value gss(D), which is found in all cases to increase linearly with needle concentration up to the spinodal. Stronger intersphere attraction implies a higher local collision rate between spheres. Figure 1 shows examples of the three site-site pair correlation functions for the L/D ) 0.5 mixture at low and high sphere volume fractions. The contact value gss(D) increases strongly with needle density. The relative increases as the spinodal is approached are larger for the more compressible lower sphere volume fraction. Increasing polymer concentration also enhances the site-site needle-needle pair correlation gpp(r), and the effects are much more dramatic for the dense mixture. Local contacts between needles and colloids are quantified by gps(r) and are reduced with increasing rod concentration because of the depletion effect. Oscillatory features in the rod-sphere and rod-rod correlations emerge at high particle densities because of imprinting of the colloidal short-range order on the spatial distribution of needles. However, as the needle concentration increases, the oscillatory feature is diminished because of depletion-driven colloid clustering and a cage coherence disruption process discussed below. The collective local colloidal structure depends in a complex manner on needle addition. Figure 2 shows an example of the evolution of the colloidal structure factor
Figure 2. Colloid structure factor for L/D ) 0.5 ideal needles at a colloid volume fraction of 0.508 and cp/cp* ) 0, 0.3, 0.5, 1, and 1.3.
for L ) D/2 at a very high density representative of a suspension compressed beyond its pure hard sphere equilibrium fluid-crystal transition. The peak wavevector increases monotonically with polymer concentration, corresponding to a depletion-driven enhancement of the average local density of the colloidal cage. The behavior at small wavevectors and local length scales is summarized in Figure 3 for both values of L/D. The magnitude of the peak in S ˆ ss(k) on local length scales (k* ≈ 7/D) is a measure of the tightness and degree of coherence of the colloidal “cage”. From Figures 2 and 3, one sees that, for L ) D/2, S ˆ ss(k*) displays an interesting nonmonotonic dependence on needle concentration. A small amount of polymer rods can partially “melt” or disrupt the initial highly ordered local collective packing of the spheres. But, with increasing rod concentration, this trend is reversed. One interpretation is that as fluid-fluid phase separation is approached, pretransitional concentration fluctuations emerge, resulting in a densification of the cage with an increase in the
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Figure 3. (left panel) Colloidal local cage order parameter as a function of reduced rod concentration. The circles represent “charged” needles with aspect ratio L/d ) 20 at φs ) 0.513, the squares are ideal needles at φs ) 0.513, the crosses are ideal needles, and the plus signs are charged needles at φs ) 0.35. The lines through the points are guides to the eye. Shaded (empty) symbols represent systems with size ratio L/D ) 2 (0.5). For the high volume fraction L/D ) 0.5 charged needle system, Sss(k*) ≈ 4.94 at cp/cp* ) 5. (right panel) Analogous results for the zero-wavevector colloidal structure factor for φs ) 0.513.
local order parameter S ˆ ss(k*). These qualitative trends are also seen in random coil-colloid suspensions,6,30 but there are quantitative differences due to the different geometry of rigid rods. The dramatic behavior described above is found only when the mixture initially contains high volume fractions of spheres, and the nonmonotonicity severely weakens for L ) 2D (and eventually vanishes as L/D further increases) even though the dimensionless particle osmotic compressibility structure factor, S ˆ ss(k ) 0), indicates that the system is approaching the spinodal boundary (Figure 3). This behavior in the L > D regime is consistent with the intuitive van der Waals idea38,39 that the relatively slowly varying depletion attraction induced by long rods does not perturb local hard sphere packing correlations. Results (not shown) for gpp(r) and gps(r) show trends qualitatively similar to, but quantitatively weaker than, that observed in the L/D ) 0.5 case of Figure 1. Very little changes in gss(r) are found at low needle densities for L ) 2D. Enhancement in the sphere-sphere contact value can still be observed when large amounts of needles are added, although the effect is much smaller compared to that in the L/D ) 0.5 case. The cage disruption effect at high colloid volume fractions also changes significantly at moderate volume fractions. An example for the φs ) 0.35 case is shown in Figure 3. Nonmonotonic response is again found, but now, cage suppression is only a weak effect and is replaced by significant enhancement of the local order parameter with increasing polymer concentration. The collective structure factor of the needles S ˆ pp(k) shows features on several length scales for high volume fraction suspensions (Figure 4). At low needle densities (far below the spinodal boundary), S ˆ pp(k) is independent of k at small wavevectors (k , k* or r . D), and hence, the polymers fill space homogeneously, although their osmotic compressibility (S ˆ pp(k)0)) is enhanced as cp/cp* increases. At high needle densities, S ˆ pp(k) scales as k-2 at small wavevectors, indicating the proximity of the spinodal boundary and long-range rod-rod concentration fluctuations. Oscillations in S ˆ pp(k) on a length scale of order D emerge with increasing needle concentration, which represents an “imprinting” of colloidal local order on the
Chen and Schweizer
Figure 4. Needle collective structure factors in the ideal needle-sphere system of L/D ) 0.5, φs ) 0.514.
spatial organization of needles. At very large wavevectors (k . k*, kL . 1, or in real space r , D, L), the intramolecular structure of the needle (ω ˜ p(k)) is probed, and S ˆ pp(k) scales as 1/k, as a direct consequence of the rigid rod architecture. B. Charged Needles. Figure 3 also presents results for two charged needle cases (both with L/d ) 20) with the same D/L ratios as those of the ideal needles. Note that the rod is still “thin”, since its thickness is 10 or 40 times smaller than the particle diameter. In the absence of colloids, the isotropic-nematic phase transition is estimated to occur at cp/cp* ∼ 10. One sees from Figure 3 that the dimensionless osmotic compressibility initially increases with rod concentration, as expected on the basis of a depletion attraction effect. But, the increases of S ˆ ss(0) are significantly smaller than those for the neutral ideal needle even at low polymer concentrations and tend to saturate or even weakly decrease as the rod concentration is further increased and rod-rod repulsions become important. The strong influence of excluded volume nonadditivity for charged needles is also reflected in the colloidal order. At low polymer concentrations, the disruption of local colloidal structure at high volume fractions caused by adding small amounts of charged needles is very similar to that of the ideal needle case, since rod-rod repulsion is not important. However, with increasing charged needle concentration, phase separation is not observed, in contrast to the case of the ideal needle mixture. Indeed, the charged needles can reduce their mutual repulsion by preferentially segregating (in a statistical sense) toward the spheres. This results in a further decrease of the local order parameter of the colloids, S ˆ ss(k*). In effect, this is an “antidepletion” phenomenon, where for entropic reasons the polymers reorganize as if there were a rod-sphere attraction. As seen in Figure 3, the “cage disruption” process does not continue indefinitely. This is because eventually the available volume for organizing rods near the colloids will be completely used. Hence, there is a transition from needle-induced local cage melting to enhanced cage ordering around cp/cp* ∼ 2 for the L ) D/2 case and at a modestly larger value for L ) 2D. The origin of the nonmonotonic behavior of the local cage order may be
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Figure 5. Needle-needle structure factors in the charged needle-sphere system of L/d ) 20, L/D ) 0.5, φs ) 0.513.
Figure 7. Colloidal potential of mean force for (a) charged and (b) ideal needles with L ) D/2 for the various indicated rod concentrations.
Figure 6. Needle-sphere site-site pair correlation function for systems of dense (φs ) 0.513) hard spheres with L/D ) 0.5 and charged needles with aspect ratio L/d ) 20.
physically distinct from the neutral needle analogue and strongly coupled to the very high total mixture volume fraction. Indications for intensified polymer organization can be found by examining S ˆ pp(k) in Figure 5, which differs dramatically from the ideal needle behavior (Figure 4) beginning at cp/cp* ∼ 1. This demonstrates explicitly the crucial influence of inter-rod excluded volume interactions in determining the spatial correlations of charged needles. Intense imprinting of colloidal correlations on rod organization on the ∼D length scale emerge. On intermediate d < r < L length scales, rod collective fluctuations are strongly suppressed with increasing polymer concentration. This reflects the rod-rod repulsions and formation of a “physical mesh” in semidilute solution. At even larger wavevectors, the charged needles also tend to organize on the needle diameter length scale (k** ≈ 7/d) because of very local packing effects, as indicated by a broad peak in S ˆ pp(k) for high needle densities (cp/cp* g 2). This suggests the rods are forced to pack very tightly because of a high total mixture volume fraction, η ) φs + φp ) (π/6)FsD3 + (π/4)FpLd2 ≈ 0.55. A local signature of these strong collective packing effects can be observed in the real space gps(r), shown in Figure 6. Enhancements in the local colloid-needle site contacts are found as rod density increases. Such a trend is opposite of the behavior of ideal needles (Figure 1) and is suggestive of an excluded volume driven rod-sphere
attraction akin to a “local adsorption” of needles onto spheres. The differences between the consequences of adding ideal versus charged needles on the real space colloid-colloid correlations at high volume fractions are most clearly seen by examining the particle-particle potential of mean force (PMF), shown in Figure 7. In both cases, polymer addition results in an enhanced contact value (more attractive PMF). However, the change of the repulsive barrier feature is qualitatively different, since it increases in amplitude and sharpness as more ideal needles are added, but decreases in amplitude and broadens with increasing charged needle concentration. These differences between charged and neutral needles are found (not shown) to be qualitatively the same for the φs ) 0.35 case. At the moderate colloid volume fraction of φs ) 0.35, the local colloidal order (S ˆ ss(k*)) again shows a nonmonotonic response with increasing polymer concentration (Figure 3) but in a much weaker manner than at high colloid density and in a distinctly different fashion than that for ideal needles. Close examination (not shown) of the colloidal structure factors reveals that the peak wavevector, k*, is also a nonmonotonic function of charged needle density, first shifting to higher values (by ∼5% at cp/cp* ≈ 2) and then reversing. This nonmonotonic shift is also found to occur for high colloid volume fractions at lower charged needle densities (by ∼2% at cp/cp* ≈ 1 for φs ) 0.513). This trend is not found for ideal needles. It suggests that the initial introduction of needles (ideal or charged) induces tighter packing of the spheres. However, while more ideal needles induce even more tighter spheresphere packing, increased charged needle concentration results in the needles preferentially segregating toward the colloids, and the attendant needle-needle repulsion results in an increased average particle-particle separation. The charged needle structure factor S ˆ pp(k) behaves the same as in the case of ideal needles at large wavevectors.
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Figure 9. Free volume distribution corresponding to the results of Figure 8. The curves are two-parameter fits of the numerical data to a Gaussian function, as described in the text. Figure 8. Fractional free volume for inserting a probe sphere of diameter d* into an L/D ) 2, φs ) 0.514 ideal needle-colloid mixture as a function of reduced rod concentration and variable probe size.
At small wavevectors, S ˆ pp(k) is independent of k at all system densities, since there is no macrophase separation. Addition of charged needles to a dense sphere fluid has qualitatively similar effects on the colloid long wavelength fluctuations for both L/D size ratios examined. The dimensionless osmotic compressibility of the spheres, S ˆ ss(0), increases with the addition of a small amount of needles but decreases at larger concentrations (Figure 3). The crossover point occurs around the same reduced rod concentration at which the nonmonotonic behavior of the sphere local structure factor is observed. In contrast to the case of the sphere structure, the effects of adding more charged needles on the collective long wavelength needle fluctuations depend on the L/D size asymmetry ratio. For L/D ) 2 (not shown), S ˆ pp(k)0) decreases monotonically with higher needle density by roughly a factor of 3 when cp/cp* ) 0f3. In contrast, for L/D ) 0.5, we find that S ˆ pp(k)0) increases monotonically with higher needle density by a factor of ∼12 when cp/cp* ) 0f3. Hence, at cp/cp* ) 3, mixtures with longer charged needles are more than an order of magnitude less compressible than those in the L ) D/2 case. This large difference is surprising. A physical interpretation might be that when the needles are longer than the spheres they tend to segregate less toward the colloids (reduced “haloing”) and thus at high volume fractions experience more mutual repulsions. C. Local Free Volume. To further probe the influences of polymer additives and depletion attraction on colloid spatial organization at very high densities, we have computed the “free volume” for inserting a probe sphere of (variable) diameter d* into the L ) D/2, φs ) 0.514 colloid-needle suspension using eqs 6 and 7. The results, normalized by the corresponding value in the rod-free fluid, are presented in Figure 8. The dependence on reduced rod concentration up to nearly the spinodal is shown for probe diameters varying from D/10 to D. This is the size range expected to be relevant to colloid self-diffusion and structural relaxation.3,4 The available free volume increases monotonically with rod concentration because of depletion-induced clustering of the colloids. This effect is enhanced as the length scale associated with the free volume increases toward the
colloid diameter. Nonmonotonic variations with rod density are not found, in contrast with the collective local order parameter S ˆ ss(k*). This is perhaps not surprising, since the present definition of “free volume” corresponds to a purely k ) 0 thermodynamic property versus S ˆ ss(k*) which is a measure of local cage coherence. It is often suggested that a critical quantity for transport and structural relaxation is the distribution of free volume.3-4,53 One can envision many definitions for this quantity, but for simplicity, we consider
P(d*) ) A exp(-βδµ(d*))
(8)
where A is a normalization constant determined from integration over all possible values of free volume, Vf ) π(d*)3/6. An “average” 〈Vf〉 follows from integration over the free volume of the product VfP(Vf). In Figure 9, we plot (d*/D)2e-βδµ(d*), which is the relevant distribution of differential free volume. Clearly, the mean free volume increases with rod concentration, as does the breadth of the distribution, indicating enhanced “heterogeneity” of the free volume. The numerical results are fit to a twoparameter (location of maximum and variance) Gaussian function; the overall amplitude has been shifted to agree with the numerical data. Interestingly, the accuracy of the Gaussian fit worsens monotonically with increasing rod concentration, primarily in the large free volume tail region. IV. Dynamics In this section, we combine our equilibrium correlation function results with modern statistical dynamic theories that are well described in the literature. We adopt an effective one-component fluid model for purposes of the dynamical analysis. Such a simplification is reasonable when the second component (polymer rods) is at low concentration, is relatively small, and moves quickly relative to the colloids.3,4 These are the primary conditions we have worked under in this paper and are especially appropriate for “ideal needles” which can pass through each. Thus, the rods serve only to influence the colloidal equilibrium structure and are implicitly assumed to remain in equilibrium on the colloidal dynamics time scales of interest. We also ignore solvent mediated hydrodynamic interactions,3,4 not only because they are (53) Debenedetto, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, 1996.
Dense Colloid-Rod Polymer Suspensions
very difficult to include but also because it has been recently shown that many colloidal dynamics phenomena can be reasonably well understood over a (surprisingly) wide range of concentrations, both far from and approaching the glass transition without an explicit accounting for many particle hydrodynamics.54-56 We consider primarily ideal needles, since one of our goals is to estimate how much depletion attractions can modify one-phase colloidal dynamics before fluid-fluid phase separation is encountered. The L/D ratio cases examined are consistent with the situation where fluidfluid phase transition preempts the fluid-crystal transition.3-4,7-9,22,23 A. Depletion Effects on the Colloidal Glass Transition. In the absence of polymers, hard sphere colloids undergo a glass transition at a volume fraction of57 φs ≈ 0.57. This phenomenon is well described theoretically by the idealized self-consistent mode coupling theory (MCT), which predicts a kinetic glass transition when the local cage constraints due to purely repulsive forces become strong enough to induce particle localization.55,56 The approximate MCT glass transition occurs at φs ≈ 0.515 and S ˆ ss(k*) ∼ 3.45 (on the basis of PY hard sphere input). In the previous section, we found that the addition of small amounts of ideal needles disrupts the local structure of highly ordered hard sphere fluids, thereby reducing the cage constraints. Hence, as suggested by others,58-60 if the hard sphere fluid is initially glassy, then adding a small amount of needles could suppress local order and return the hard sphere fluid to the liquid state. Such a phenomenon has been experimentally observed61 in model hard sphere colloid fluids mixed with small flexible polymers of Rg/R ∼ 0.08. The very recent combined experiment, computer simulation, and MCT study of polymer-colloid mixtures has emphatically verified the basic theoretical ideas.62 To quantify this “glass melting” effect using the PRISMPY structural input, we require a criterion for the glass transition. We adopt the simplest condition for glass formation of S ˆ ss(k*) = 3.45, which is expected from MCT to yield a reliable result compared to the full numerical calculations as long as the vitrification process is driven primarily by excluded volume or “jamming” considerations.58-60,62 A similar correlation between the equilibrium fluid-crystal transition and S ˆ ss(k*) ) 2.85 is known as the “Hansen-Verlet rule” and is quite successful, since crystallization is also dominated by local collective packing considerations.32 Examples of the “glass melting” behavior are shown in Figure 10 in the form of a nonequilibrium phase diagram. As expected on the basis of the results in Figure 3, major stabilization of the fluid phase is found for the smaller needle L ) D/2 mixture for small rod additions. Note that the random close packing of hard spheres63 occurs at a (54) Cohen, E. G. D.; Verberg, R.; deSchepper, I. M. Physica A 1998, 251, 251. (55) Gotze, W.; Sjogren, L. Rep. Prog. Phys. 1992, 55, 241. (56) Fuchs, M. Transp. Theory Stat. Phys. 1995, 24, 855. (57) Pusey, P. N. In Liquids, Freezing and the Glass Transition; Hansen, J. P., Levesque, D., Zinn-Justin, J., Eds.; North-Holland: Amsterdam, 1991. (58) Bergenholtz, J.; Fuchs, M. Phys. Rev. E 1999, 59, 5706. (59) Fabbian, L.; Gotze, W.; Sciortino, F.; Thiery, P. T. Phys. Rev. E 1999, 59, R1347. (60) Dawson, K.; Foffi, G.; Fuchs, M.; Gotze, W.; Sperl, M.; Tartaglia, P.; Voightmann, T.; Zaccarelli, E. Phys. Rev. E 2000, 63, 011401. (61) Illett, S.; Orrock, A.; Poon, W. C.; Pusey, P. N. Phys. Rev. E 1995, 51, 1344. (62) Pham, K. N.; Puertas, A. M.; Bergenholtz, J.; Egelhaaf, S. U.; Moussaid, A.; Pusey, P. N.; Schofield, A. B.; Cates, M. E.; Fuchs, M.; Poon, W. C. K. Science 2002, 296, 104. (63) Bennett, C. H. J. Appl. Phys. 1972, 43, 2727.
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Figure 10. PRISM-MCT predictions of the glass melting effect induced by ideal needles. Open (shaded) circles represent the fluid-glass boundary for systems with size ratio L/D ) 0.5 (2). The solid (dot-dashed) line is the spinodal boundary for L/D ) 0.5 (2). The analogous L/D ) 0.5 charged needle result is also shown as open squares.
volume fraction of ∼0.64. Hence, experimentally, an increase of colloid volume fraction above the glass transition of only ∼10% is possible. The magnitude of the maximum shift of the glass volume fraction in Figure 10 is ∼6%, thus demonstrating it as a strong effect. The depletion-driven fluid stabilization trend reverses well before the fluid-fluid spinodal is encountered, and the glass volume fraction can (slightly) exceed its pure hard sphere value at very high rod concentrations. It is interesting to compare the influence of macromolecular architecture on the “glass melting” phenomenon for small polymers. Prior PRISM calculations6 for Rg/R ≈ 0.3 and random coils yield an ∼2% enhancement of the colloid vitrification volume fraction at a reduced polymer concentration of cp/cp* ≈ 0.02 compared to the present ideal needles 2% shift at cp/cp* ≈ 0.2. Dimensional analysis yields the corresponding number of polymer molecules per colloid of φs-1(cp/cp*)(R/Rg)3, which is ∼1.5 for the ideal coils and ∼15 for rigid rods. More rods are required because of their lower effective (fractal) dimensionality. However, in absolute polymer density units (g/cc), the rod polymers are far more efficient at disrupting the colloidal cage. This can be seen by using the standard connection49 between the dilute-semidilute crossover and the polymer radiusof-gyration, whence c*rod/c*coil ∼ (2/N)3/2, where N is the degree of polymerization. Thus, even for a modest value of N ∼ 100, the required rod monomer concentration relative to its coil analogue is crod/ccoil ∼ 10(2/100)3/2 ∼ 0.0282, corresponding to a monomer number density ∼35 times smaller. As a caveat, we note that very recent MCT work58,59,62 has discovered a crossover with increasing cp/cp* from a glass transition to a “gelation” type transition when Rg/R , 1. The gel state is distinguished from the glass by localization of colloids on a shorter length scale controlled by the range of the attractive depletion potential and not the ∼0.1D length scale characteristic of vibrational amplitudes in a hard sphere glass. We have not addressed this gelation phenomenon here. In the longer needle system (L/D ) 2), the density at which the fluid-glass transition occurs increases only slightly when a small amount of needles are added (cp/cp* < 1) and a weak nonmonotonicity with increasing polymer concentration is barely detectable. This near insensitivity
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of the glass transition to polymer additives when L > D is expected on the basis of the results in Figure 3 and the general van der Waals argument that slowly varying attractions cannot significantly modify local packing in a dense liquid. What is perhaps surprising is the sharpness in the change of the glass melting effect from L < D to L > D. Finally, analogous results for the L ) D/2 charged needle studied in section IIIB are also shown in Figure 10. At low polymer concentrations, there is very little difference compared to the ideal needle behavior except for perhaps a small tendency for the charged rod to be less efficient at stabilizing the fluid phase. However, at larger polymer concentrations, because of the lack of fluid-fluid phase separation, the charged needles induce a more dramatic “glass melting” behavior that eventually also reverses, displaying the same nonmonotonic form as found for ideal needles. This provides support for the idea that the nonmonotonicity is not fundamentally due to long wavelength concentration fluctuations that are precursors of phase separation. B. Transport Coefficients in the Fluid Regime. Far from the glass transition (φs < 0.5), the transport coefficients of model hard sphere colloids have been recently well described by a microscopic theory of Cohen et al.54 This is a simplified mode-coupling approach that accounts for binary collisions and collective caging in a non-selfconsistent “mean field” fashion. The critical structural information is the colloid-colloid pair correlation function at contact gss(D;φ) (which quantifies the binary collision rate) and the static structure factor S ˆ ss(k). Given the Stokes-Einstein self-diffusion coefficient of a single, infinitely dilute colloid (D0), the zero frequency shear viscosity and translational self-diffusion constant are given by54
ηN(φ) ) η0gss(D;φ) + kBT ∞
∫ dk 120π2 0
{[ k2
]
∂S ˆ ss(k;φ)/∂k S ˆ ss(k;φ)
2
Ds(φ) )
[
DSS(φ) 1+
DSS(φ)D2 36πφ
∫0∞dkSˆ
}
1 DS(k;φ)
k2[S ˆ ss(k;φ) - 1]2
(9)
]
-1
S S ss(k;φ)[D (k;φ) +DS (φ)]
(10) DSS(φ) ) D0/gss(D;φ) DS(k;φ) )
D0d(k) gss(D;φ) S ˆ ss(k;φ)
(11) (12)
where d(k) ) [1j0(kD) + 2j2(kD)]-1, with jl(kD) being the spherical Bessel function of order l, DSS(DS) is the socalled short time (collective) self-diffusion coefficient, Ds is the long time self-diffusion coefficient, ηN is the shear viscosity associated with the intermolecular part of the collective stress tensor, and η0 follows from the singlesphere Stokes-Einstein equation 6πRD0η0 ) kBT. Despite the neglect of explicit hydrodynamic interactions, these equations describe remarkably well the experimental transport data over the entire range of colloid densities up to φs ≈ 0.5, at which point the viscosity (self-diffusion constant) has increased (decreased) by nearly 2 (1) orders of magnitude relative to its dilute limit.54 We apply this approach to the needle-colloid mixtures using the PRISM-PY results for the required equilibrium
Figure 11. Normalized self-diffusion coefficient for hard spheres in the ideal needle-sphere mixture with size ratio L/D ) 0.5 for cp/cp* ) 0 (solid line), 3 (square), 5 (triangle), 8 (diamond), and 10 (×). The normalization is with respect to the dilute sphere diffusion coefficient, which is found to be Ds(φs)0)/ D0 ) 1/(1 + 1.133cp/cp*). The inset shows the normalized viscosity for the same system normalized with respect to the dilute sphere viscosity, given by ηs(φs)0)/η0 ) (1 + 1.133cp/cp*).
input. An extra uncontrolled assumption is the continued applicability of the dynamic theory in the presence of polymer-mediated attractions, as was also adopted in the MCT analysis of the glass transition. Physically, this corresponds to the idea that depletion attractions enter dynamically only by modifying repulsive force binary collisions and collective cage structure and that explicit frictional consequences are of secondary importance. The accuracy of this simplification is not known a priori. Results are shown in Figure 11 for the L/D ) 0.5 mixture in a format normalized by the dilute sphere limit value. Each curve terminates at the spinodal demixing condition. As expected, for all cases the self-diffusion constant decreases monotonically as the needle density increases because of polymer-induced physical clustering of the colloids that strongly enhances their collision rate and hence friction. For a given needle concentration, the reduced colloidal mobility decreases by nearly an order of magnitude with increasing sphere volume fraction, and it decreases increasingly rapidly as needle concentration increases. At fixed colloid density, increasing polymer concentration results in a factor ∼2-4 lowering of the reduced diffusion constant. The corresponding shear viscosity results are shown in the inset. For the cases studied, viscosity enhancements by a modest factor of j2 are found before the spinodal is encountered. The viscosity appears to be less affected by polymer additives than selfdiffusion. The origin of this trend is the different dependences of Ds and ηN on the length scale dependent collective structural correlations in eqs 9 and 10 and the length scale dependent structural consequences of depletion attractions. Analogous calculations have been performed for the charged needle model. We refrain from presenting more figures and only summarize our findings. At high (low) colloid volume fraction φs ) 0.5 (0.052), the normalized shear viscosity increases by a factor of ∼3 (6) as the reduced rod volume fraction increases from 0 to 5. The reduced self-diffusion constant decreases by a factor of ∼3 (4) over the analogous range of polymer concentration. The stronger dynamical consequences of polymer additions for the lower colloid volume fraction system are a reflection
Dense Colloid-Rod Polymer Suspensions
Figure 12. Stokes-Einstein plot in a format normalized by the polymer-free value as a function of colloid volume fraction for various reduced ideal needle concentrations cp/cp* ) 0 (solid line), 1 (circle), 3 (square), 5 (triangle), 8 (diamond), and 10 (dot). The curves terminate at the fluid-fluid spinodal phase separation volume fraction.
of the increased efficiency of the depletion effect to induce local colloid clustering (and hence enhanced collision rate) when the suspension is more dilute. Figure 12 presents a normalized “Stokes-Einstein” (SE) plot for ideal needles that quantifies the extent to which the dilute limit SE relation between self-diffusion constant and viscosity holds with increasing colloid volume fraction and polymer additives. For the pure hard sphere suspension, the SE relation is rather well obeyed to within 20% over the entire volume fraction window. However, the addition of polymer results in increasingly larger deviations due to the prediction that translational diffusion is more affected by depletion forces and colloid restructuring than by the shear viscosity. We hope these calculations motivate future experimental tests. V. Conclusion and Discussion We have presented a microscopic liquid-state theory study of the structure and dynamical properties of rod polymer-colloid suspensions with an emphasis on the high particle density regime. Ideal needles and charged needles of size asymmetry ratios of L/D ) 0.5 and 2 have been examined to address the role of variable rod-rod forces and solvent quality. Our primary findings for ideal needles are as follows. (i) Needles induce strong local clustering of the colloids and polymers in the one-phase region and more so with increasing size asymmetry ratio D/L. However, local contacts and colloid imprinted oscillations in the needle-sphere pair correlations are reduced with increasing polymer concentration because of pretransitional concentration fluctuations and colloidal cage disruption. (ii) For L ) D/2, the colloid collective cage
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order at high densities is a nonmonotonic function of rod concentration reflecting competing physical effects. A dynamic consequence is a strong and nonmonotonic modification of the colloidal glass transition volume fraction. Far from the glass transition, depletion attraction between colloids can suppress (enhance) self-diffusion (shear viscosity) by a factor of up to ∼4 (2), and the StokesEinstein relation is predicted to be increasingly violated with increasing rod concentration. (iii) For the L ) 2D case, all the effects in point (ii) are much smaller, consistent with the longer spatial range of depletion attractions and elementary van der Waals fluid concepts. (iv) Local free volume on the 0.1-1 colloid diameter length scale increases, and the corresponding distribution significantly broadens and acquires a non-Gaussian tail, with increasing needle concentration. The nonadditive excluded volume aspect inherent to charged needles results in the suppression of depletiondriven fluid-fluid phase separation and the tendency of the polymers to preferentially segregate near the spheres. The primary consequences of this different physics are as follows. (i) Depletion-induced enhancements of long wavelength colloid and polymer concentration fluctuations (osmotic compressibility) are much smaller and can even be suppressed relative to the pure hard sphere behavior at high charged needle concentrations. (ii) The influence of charged rod additions on local colloidal structure and glass formation is more dramatic than the ideal needle case. (iii) The real space polymer-colloid pair correlation function exhibits enhanced local clustering features as a function of needle concentration. This is opposite to the behavior of ideal needles and reflects an entropy-driven statistical adsorption or “haloing” phenomenon. (iv) At high rod concentrations, the L ) D/2 charged needles show strong collective organization on both the colloid diameter and rod thickness length scales. Many of our results should be testable by computer simulation, light, neutron, and X-ray scattering, tracer diffusion, and viscometry experiments. The phase separation of ideal needle-sphere mixtures, and a more thorough study of the role of size asymmetry (including the nanoparticle regime), will be presented elsewhere.34 Improvement of the theory beyond the PY approximation to include nonlocal conformational entropic repulsions between rods and spheres is presently under study and is likely important for proper description of demixing in purely athermal (good solvent) suspensions.30,31 The combination of the rigid rod studies with recent progress for flexible chains will allow the synthetic and biologically important class of semiflexible polymers mixed with spherical objects to be systematically addressed. Acknowledgment. We gratefully acknowledge many useful discussions and correspondence with Matthias Fuchs. Stimulating discussions with A. P. Chatterjee, J. A. Lewis, and C. F. Zukoski are also acknowledged. This work was supported by the Department of Energy through the Frederick Seitz Materials Research Laboratory via Grant No. DEFG02-91ER45439. LA020309R
