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Dynamics of Tracer Particles in Gel-like Media† Douglas C. Viehman‡ and Kenneth S. Schweizer* Department of Chemical and Biomolecular Engineering, Department of Materials Science, and Frederick Seitz Materials Research Laboratory, UniVersity of Illinois, 1304 West Green Street, Urbana, Illinois 61801 ReceiVed: July 9, 2008; ReVised Manuscript ReceiVed: September 23, 2008
Our previously proposed theory of kinetic arrest and activated barrier hopping in binary mixtures of hard and sticky spheres is applied to the problem of repulsive particle tracer diffusion. For a dynamically frozen matrix, a tracer kinetic arrest diagram is determined using a simplified version of ideal mode coupling theory. The matrix particles cluster more with increasing degree of attraction, resulting in extra free volume for tracer motion that shifts the onset of localization to higher volume fractions. At the nonergodicity transition, the tracer localization length is roughly its diameter and decreases exponentially with increasing matrix volume fraction. If the matrix is formed via thermoreversible gelation of a fluid, then the lifetime of the kinetically arrested state is a critical question. The naïve mode coupling theory localization boundary for a pure sticky particle fluid is computed and compared to the tracer localization boundary under dynamically frozen matrix conditions. Above a threshold degree of attraction, the matrix can be treated as effectively static for times less than the mean activated barrier hopping time, which grows strongly with increasing matrix particle stickiness. I. Introduction The structure, equilibrium phase behavior, and nonequilibrium states of nanoparticle and colloid suspensions are of fundamental scientific and engineering interest.1,2 For (effective) onecomponent fluids of attractive particles, major theoretical progress for predicting the emergence of kinetically driven gels and glasses has been achieved based on the ideal mode coupling theory (MCT),3-6 a simplified version known as “naïve” MCT (NMCT),7-9 and a microscopic nonlinear stochastic Langevin equation (NLE) theory that goes beyond MCT to treat activated barrier hopping and non-Gaussian dynamic heterogeneity phenomena.8,10-13 Very recently, we generalized the NMCT and NLE approaches to binary mixtures of spherical particles.14 The NMCT has been compared to its more sophisticated ideal MCT analog for mixtures. A basic issue is colocalization of both species versus a partial transition where only one species localizes and the other freely diffuses. The binary mixture NMCT nonergodicity transition signals a crossover to activated barrier hopping dynamics on a two-dimensional nonequilibrium free energy surface.14 This dynamical “landscape” has a rich structure with barrier locations and heights that are sensitive to the relative magnitude of the cooperative displacements of the sticky and repulsive particles, strength of the attractive intermolecular forces, mixture composition, and total fluid packing fraction. The present paper applies our recent theory to study the dynamics of a single repulsive tracer particle for two distinct situations. First, we consider a matrix composed of attractive spheres that are dynamically frozen to create a gel-like medium. This problem is relevant to the flow of nanoparticles and colloids in porous media. The primary goal is to understand the role of matrix concentration and structure on tracer particle localization. The second situation corresponds to treating the sticky particle matrix as a fluid that can undergo a kinetic gel transition. Here †
Part of the “Karl Freed Festschrift”. * To whom correspondence should be addressed. E-mail: kschweiz@ illinois.edu. ‡ Present address: School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332.
the matrix particles can be localized in a transient sense, but on a long enough time scale, structural relaxation occurs due to rare activated barrier hopping motions. The primary question addressed is the effective time scale over which the attractive particle fluid can be treated as a dynamically frozen matrix, and its implications for treating the gel matrix as an effectively arrested porous medium for tracer diffusion. Section II specializes our general theory14 to the tracer problem. Predictions for the tracer kinetic arrest transition and localization length as a function of matrix volume fraction and strength of attraction are given in Section III. In Section IV the gel matrix kinetic arrest diagram, and the mean time scale, below which it can be treated as dynamically arrested, are computed. The paper concludes with a brief discussion. II. Theory and Model The details of our theory for binary mixtures of hard and sticky spheres of identical diameter (D) have been presented elsewhere.14 The tracer problem corresponds to the infinite dilution limit of a single hard sphere moving in a matrix of attractive particles. The structure of the latter is described at the pair correlation function level using equilibrium integral equation theory. Since the tracer is present at infinite dilution, its presence does not modify the statistical structure of the matrix. Consequently, subtle issues associated with treating the system in a quenched-annealed manner15 do not enter. A. Naïve Mode Coupling Theory. For a dynamically frozen matrix of particles of volume fraction φ, the force-force time correlation function experienced by the tracer follows immediately as a special case of the full mixture result14 and is given by
〈ft(0) · ft(t)〉 )
1 3β2
dq 2 2 q Ctm(q)FmSmm(q)Γs(q, t) ∫ (2π) 3
(1)
where Fm is the matrix particle number density, φ ) πFmD3/6, Ctm(q) is the Fourier transform of the tracer-matrix direct
10.1021/jp8060784 CCC: $40.75 2008 American Chemical Society Published on Web 11/12/2008
Dynamics of Tracer Particles in Gel-like Media
J. Phys. Chem. B, Vol. 112, No. 50, 2008 16111
correlation function, Smm(q) ) [1 - FmCmm(q)]-1 is the dimensionless matrix collective structure factor, q is the wavevector, and β is the inverse thermal energy. The quantity Fmq2C2tm(q)Smm(q) is proportional to a Fourier-resolved effective mean-square force on the tracer exerted by its surroundings,8,10 and Γs(q,t) ) 〈eiq · [r(t)-r(0)]〉 is the tracer dynamic structure factor. Ergodic behavior corresponds to Γs(q,t) f 0 as t f ∞, but this quantity is nonzero at long times for a localized tracer. Within NMCT, the nonergodicity parameter, or Debye-Waller factor, is of an Einstein oscillator (harmonic solid) form7,8
Γs(q, t f ∞) ) exp(-q2rL2 /6)
(2)
where rL is the particle localization length. A self-consistent equation for the localization length can be derived in several ways.7,8,10 The simplest approach follows from the Einstein solid description where β〈ft(0) · ft(t f ∞)〉 is a spring constant, and an effective equipartition relation applies.
β〈ft(0) · ft(t f ∞)〉rL2 /2 ) 3kBT/2
(3)
Combining eqs 1-3, the equation for the tracer localization length is given by
rL-2 )
1 9
dq 2 F q2Ctm (q)Smm(q)e-(q r /6) ∫ (2π) 3 m 2 2 L
(4)
Ideal kinetic arrest corresponds to a finite rL value solution of eq 4, and the delocalized regime corresponds to rL ) ∞. If the matrix is treated as a fluid, then its self-consistent NMCT ideal localization equation is given by -2 rL,m )
1 9
dq 2 F q2Cmm (q)Smm(q)e-(q r ∫ (2π) 3 m
2 2 /6)(1+S-1 (q)) L,m mm
(5)
where all correlation functions are for the pure matrix fluid. B. Nonlinear Generalized Langevin Equation. For fluids, the NMCT ideal nonergodicity transition signals a crossover to transient localization and activated hopping dynamics.8,10 A nonlinear Langevin equation (NLE) theory for single particle motion has been developed to describe this regime based on a local equilibrium idea and quantification of dynamical caging constraints in an average manner via pair correlation functions.10 The NLE in the high friction overdamped limit is given by
d ∂ -ζs rm F (r ) + δf ) 0, dt ∂rm eff m 〈δf(0)δf(t)〉 ) 2kBTζsδ(t) (6) where rm(t) denotes the scalar displacement of a tagged particle from its initial position, ζs ) kBT/Ds is the short-time tracer friction constant, δf(t) is the corresponding white noise random force, and Feff(rm) is an effective or nonequilibrium “free energy” given by
βFeff(rm) ) -3 ln(rm) -
∫
2 b FmCmm(q)Smm(q) -(q2rm2/6)(1+Smm -1 (q)) dq e (7) -1 3 (2π) 1 + Smm(q)
The first term is a translational entropy contribution that favors the delocalized state, whereas the second term quantifies the localizing contribution due to intermolecular forces. Beyond the NMCT ideal nonergodicity transition, the effective free energy acquires a barrier of height FB and a local minimum (localized state). If the noise term in eq 5 is dropped, activated barrier hopping is “turned off”, and the NMCT ideal kinetic arrest theory for fluids is recovered.8,10 The dimensionless mean first passage, or activated barrier crossing, time is8,16
ζs /ζ0 FB/kBT τ ) e τ0 √K0KB
(8)
where τ0 ) ζ0D2/kBT is the suspension Brownian time, ζ0 the dilute Stokes-Einstein friction constant, and K0 and KB are the absolute magnitudes of the localization well and barrier curvatures in units of kBT/D2, respectively. Following prior work,8 the matrix short-time friction constant is determined based on a binary collision model17 corresponding to ζs ) ζ0gmm(D) where gmm(D) is the contact value of the pair correlation function. C. Model. The tracer and matrix spheres have identical diameters and interact via a hard-core repulsion. An attractive interaction between the matrix particles is also present14
( D a- r ),
υ(r) ) -ε exp
r>D
(9)
where ε is the attraction strength at contact in units of kBT, and a is the spatial range. Note that the tracer interacts with the matrix only via an excluded volume interaction; however, the attraction between matrix particles modifies matrix structure and strongly affects the tracer dynamics. All numerical results presented below are for a ) 0.02D, which is typical of van der Waals attractions for nanoparticles or colloids and is relevant to recent experimental studies of “biphasic mixtures”.14,18,19 As in our prior work,14 the pair correlations and structure factors are determined using the Ornstein-Zernike integral equation theory with the Percus-Yevick closure.20 The integral equations are numerically solved using the Picard algorithm. III. Localization Transition in a Frozen Matrix A. Background. The classic Lorentz problem of a single point particle moving in an array of disordered and immobile hard obstacles (oVerlapping randomly placed spheres) has been studied using simulation21 and mode-coupling-like theories.22 A strict localization transition (zero long time diffusion constant) occurs due to a continuum percolation transition where the void space available for tracer motion becomes disconnected into finite clusters. Recent simulations21 in three dimensions have found the percolation and localization transition occurs at an obstacle volume fraction of φc ) 0.44, which is ∼10% higher than the MCT prediction22 of φc ) 3/8 ) 0.375. We note that ideal MCT always predicts a kinetic nonergodicity transition whether the matrix is frozen or a fluid. For the latter, this is an artifact of the approximate factorization of multipoint dynamic correlation functions, and the ideal MCT transition signals a dynamical crossover to activated transport. However, in a frozen medium a true localization transition does occur for geometric continuum percolation reasons.
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Figure 1. Tracer kinetic arrest diagram for dynamically frozen matrices in the representation of matrix volume fraction versus attraction strength in units of the thermal energy. The dash-dot curve is a fit to a sigmoidal function: φc ) 0.259 + 0.208(1 + exp(4.5 - ε))- 1. The region inside the lower dashed curve indicates where equilibrium spinodal phase separation occurs.
Our interest is a finite diameter tracer sphere moving in matrices composed of nonoverlapping (generally sticky) hard spheres with correlated structure. For the most elementary problem of a single hard sphere moving in a quenched matrix with hard sphere fluid correlations, and if the matrix and tracer particles have the same diameter, then early MCT theory23 analysis predicted a localization transition at φc ≈ 0.18-0.22, depending on details of the approximations. A more recent MCT15 of tracer localization in a quenched hard sphere matrix found φc ≈ 0.17 in the dilute tracer limit. It is interesting to compare these MCT-based results with recent simulation studies by Yethiraj and co-workers24,25 of a mobile sphere in a quenched matrix of identical sized hard spheres. The latter was constructed in several distinct manners including: (i) a “random” matrix of immobilized spheres that are randomly and sequentially inserted into the simulation cell and quenched at the position of insertion, and (ii) a “templated” matrix created using an equilibrated hard sphere fluid configuration at φ ) 0.45 followed by quenching the particles in place and randomly removing matrix spheres until the desired matrix volume fraction is obtained. The simulations determined a threshold matrix volume fraction, φperc, beyond which there is no percolating pore volume and a tracer is localized. The simulations found φperc ) 0.24 and 0.32 for the random and templated matrices, respectively.24 These studies emphasize the importance of many body topological effects that cannot be captured by any theory, including ours, which considers structure, and hence dynamical constraints, only at the pair level. We are not aware of prior theoretical or simulation studies of the tracer localization problem in matrices of attractive spheres, which is the primary focus of our work. Given localization in frozen matrices is intimately related to continuum percolation, we expect activation barriers will be infinite beyond the MCT transition for this system. Hence, in this section we consider only the NMCT predictions for the kinetic arrest phenomenon and the associated tracer localization length. B. Kinetic Arrest Diagram. Figure 1 shows the NMCT results in the format of a critical matrix volume fraction versus attraction strength. For ε ) 0, the dynamically frozen matrix is a set of steric obstacles and a tracer localization transition is predicted at φc ≈ 0.258; as expected, this volume fraction is much smaller than φc ≈ 0.432 for a one-component fluid of mobile hard spheres.8 Our value of φc ≈ 0.258 is modestly larger than prior MCT-like theoretical calculations15,23 and falls in betweenthecriticalvolumefractionsdeterminedfromsimulation24,25 based on the two frozen medium protocols discussed above. As the attraction between matrix particles is increased from zero, Figure 1 shows the volume fraction required for tracer
Viehman and Schweizer
Figure 2. Tracer localization length at the kinetic arrest boundary as a function of matrix attraction strength. Inset: same results plotted as a function of volume fraction.
localization monotonically increases in a sigmoidal manner. The increase is slow for weak attractions with the critical φc growing from 0.258 to ∼0.3 as ε/kBT increases from 0 to 3. Between ε/kBT ) 3 and 7, φc grows more rapidly. However, at very high attractions a limiting saturation behavior is attained, indicative of a “ground state” packing structure of the sticky particle matrix. The results in Figure 1 can be well-fit to a sigmoidal curve where φc changes most rapidly at 4.5 kBT and saturates at about 0.467. The kinetic arrest diagram can also be read in a “horizontal” manner, that is, fixing φ and increasing matrix attraction strength. In this case, the tracer undergoes a delocalization transition at a critical attraction strength that increases with volume fraction. This trend emphasizes that as the strength of the attraction grows the sticky matrix spheres more strongly cluster, which opens up additional space or “free volume” for tracer motion, thereby delaying localization. The matrix of attractive spheres can also undergo an equilibrium demixing phase transition as indicated by the fluid-fluid (F + F) spinodal curve in Figure 1. Based upon approximate theories of demixing and dynamics employed, spinodal phase separation occurs in the parameter regime where the tracer is delocalized and does not directly interfere with the kinetic arrest transition. More generally, spinodal demixing is characterized by unbounded growth of the long wavelength density fluctuations, which play an essentially negligible role in determining the small-scale localization process.9,26 C. Localization Lengths. The main panel of Figure 2 shows the tracer localization length along the kinetic arrest boundary falls in the range of 0.8-1.2D and follows a sigmoidal dependence on ε. These localization lengths are much larger than at the NMCT glass transition of a fluid of mobile hard spheres8 where rL ≈ 0.2D. The physical reasons are the lower matrix volume fractions and the dynamically frozen nature of the matrix. Moreover, despite the trend that the critical volume fraction grows with increasing attraction strength, the tracer localization length also increases, contrary to typical onecomponent glasses and gels.3-6,8-10 This trend again reflects enhanced matrix particle clustering with increasing ε, which opens up more free volume for tracer motion. The inset of Figure 2 documents the nearly linear growth of the tracer localization length with volume fraction. Beyond the kinetic arrest boundaries, Figure 3 shows that tracer localization lengths sharply decrease with increasing volume fraction. The general form of the decrease is quite similar for different matrix attraction strengths if the volume fraction is measured as the excess relative to its critical (φc) NMCT value. The form of the volume fraction dependence is nearly exponential, with a common slope, as shown in the inset. Such trends have also been previously found for one-component
Dynamics of Tracer Particles in Gel-like Media
Figure 3. Tracer localization length as a function of excess matrix volume fraction beyond the kinetic arrest transition. Circles, upsidedown triangles, and squares represent matrices with ε ) 1, 3, and 5 kBT, respectively. Inset: log-linear plot of the same results.
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Figure 5. Comparison of the matrix fluid ideal kinetic arrest boundary (triangles) and the tracer localization boundary for dynamically frozen matrices (circles). The intersection of these two curves defines a crossover attraction strength, which occurs at ε ) εc ≈ 3.2 kBT, beyond which the matrix kinetically arrests (in the ideal MCT sense) before the tracer localizes under the frozen matrix idealization. The equilibrium spinodal phase separation curve is also indicated (squares). All smooth curves through the points are guides to the eye except the dash-dot curve through the circles, which is the function described in the caption of Figure 1.
IV. Dynamical Matrix Effects
Figure 4. (a) Tracer-matrix radial distribution functions, gtm(r), for φ ) 0.35, and matrix attraction strengths of (in order of decreasing contact value) ε ) 0, 2, 4, and 5 kBT. (b) Corresponding matrix radial distribution functions, gmm(r). Note that gmm(r ) 2D) grows with increasing attraction strength. The inset shows the contact value in a log-linear format; the smooth curve is a guide to the eye.
glassy hard sphere fluids8 and molecular analogs26 and have been understood based on an analytic analysis.27 D. Structural Origin of Localization. The trends in Figures 1 and 2 can be qualitatively understood from examining the radial distribution functions for various attraction strengths at fixed volume fraction. As a representative example, Figure 4a shows tracer-matrix pair correlation functions for φ ) 0.35. As attraction strength increases, the amplitude of gtm(r) in the contact region monotonically decreases. The amplitude of the tracer-matrix local pair correlation function inversely correlates with the amount of free space the tracer experiences in the matrix. Consequently, the trends in Figure 4a reflect the enhanced clustering of matrix particles, which is explicitly documented in Figure 4b via calculations of the matrix-matrix pair correlation function. Figure 4b shows a large increase of contact value with increasing matrix attraction strength. At high attractions, gmm(r) also has peaks at r ) 2D that signify enhanced matrix clustering beyond contact. These structural trends of enhanced matrix contacts, and suppressed matrix-tracer contacts, with increasing matrix attraction strength at fixed volume fraction provide an intuitive understanding of the results in Figures 1 and 2.
We now consider a different situation where the matrix is a fluid that can be kinetically arrested on a finite time scale. Here, the glass or thermoreversible gel-like structure is, in principle, transient, and the ideal MCT transition signals a crossover to activated barrier hopping dynamics. If the mean hopping time (a simple measure of the lifetime of the static matrix constraints experienced by a tracer particle) is long enough relative to the relevant experimental time scale, then treating the matrix as effectively frozen in a dynamic sense is a reasonable approximation. The ideal NMCT kinetic arrest boundary for the pure matrix fluid is shown in Figure 5. With increasing (relatively) small attraction strength, the arrest boundary shifts to higher volume fractions corresponding to the well-known re-entrant “glass melting” effect.2,3,5,6,9 However, beyond an attraction strength on the order of two thermal energy units, the attraction strength becomes sufficient to induce physical gelation via the formation of “bonds”. The critical ideal gelation volume fraction then strongly decreases with increasing attraction strength. Results are only shown for volume fractions above ∼0.1, where it is at least plausible that a percolated network exists. It is significant to note that Figure 5 also shows that, based on the approximate theoretical methods employed, the ideal gelation boundary does not cross the equilibrium spinodal (liquid-gas-like) demixing curve for this system characterized by a very short-range attraction (a ) 0.02D). Figure 5 also addresses the question of whether treating the fluid matrix as an effectively frozen medium is sensible for describing tracer motion. One sees that for attraction strengths less than ∼3 kBT the answer is no, that is, the matrix is not even kinetically arrested based on MCT. However, for ε > εc ≈ 3.2 kBT, tracer localization is predicted to occur after (at a higher volume fraction) the matrix dynamically freezes in the ideal MCT sense. The matrix kinetic arrest predicted by MCT is, in principle, destroyed by activated barrier hopping, which can possibly restore ergodicity on the relevant experimental time scale. As a semiquantitative measure of the lifetime of the frozen matrix idealization, we have computed the mean barrier hopping time of matrix particles using the NLE and Kramers theories. Results are shown in Figure 6. The hopping time grows very rapidly, in a stronger than exponential fashion, with attraction strength; it can become astronomically large, for example, ∼1020 and
16114 J. Phys. Chem. B, Vol. 112, No. 50, 2008
Figure 6. Logarithm (base 10) of the mean barrier hopping time, in units of the Brownian time, of the matrix particles as a function of attraction strength evaluated along the tracer localization boundary in Figure 1. Results are shown only for attraction strengths above the crossover value (ε > εc ≈ 3.2) corresponding to systems where the matrix becomes an ideal kinetically arrested gel at a volume fraction below the critical value φc required for tracer localization as computed based on literally frozen matrix conditions.
1060 in elementary Brownian time units (τ0) for ε/kBT ) 5 and 6, respectively (not plotted). Indeed, beyond ε ≈ 4.5 kBT, the hopping time is more than 13 orders of magnitude longer than τ0. For particles of diameter D ≈ 0.1-1 µm, the Brownian time is on the order of 10-2 to 10 s, respectively. Hence, for attraction strengths only modestly larger than εc, the matrix will appear as a dynamically frozen gel on typical experimental time scales, resulting in an effectively localized tracer. Of course, the precise critical value of attraction strength beyond which the matrix will appear as dynamically arrested does depend on tracer particle diameter (τ0 ∝ D3) and attraction range, but these dependencies are of second order importance relative to the influence of the bond strength variable, ε. V. Summary and Discussion We applied our theory for glass and gel-like dynamics in binary sphere mixtures14 to the problem of tracer diffusion in a dynamically frozen matrix of attractive particles. A kinetic arrest diagram has been determined that predicts increasing matrix attraction delays the onset of localization to higher volume fractions. The physical reason is enhanced matrix particle clustering, which results in additional free volume for tracer motion. The tracer localization length is roughly its diameter at the arrest transition and decreases exponentially with increasing volume fraction. If the matrix is formed via physical gelation of a fluid, then the lifetime of the kinetically arrested state is a critical question. The NMCT localization boundary for sticky particles has been presented and compared to the tracer localization boundary for dynamically frozen matrices. For the model studied, an ideal matrix gel transition occurs for attraction strengths ε > 3.2 kBT. However, the thermally activated barrier hopping process still occurs and can potentially restore ergodicity on the relevant time scale. The matrix can then be plausibly treated, to a first approximation, as effectively static only for times less than the mean activated barrier hopping time. Calculations reveal that this mean hopping time rapidly grows as a stronger-thanexponential function of matrix attraction strength. As a consequence, the dynamically frozen matrix idealization is effectively realized for attractions strengths in excess of roughly 4.0-4.5 kBT. An interesting future direction is to treat the fully coupled dynamics of the tracer and mobile matrix particles via stochastic
Viehman and Schweizer trajectory solution of the two nonlinear Langevin equations.11 The effective free energy experienced by the tracer will depend not only on tracer displacement as in the frozen matrix limit, but also on matrix particle stochastic dynamical displacements, which will induce an explicit fluctuating time-dependence to the caging force felt by the tracer. The origin of the latter is not only large amplitude barrier hopping motions, but also smaller scale non-Fickian diffusion of the matrix particles. Solution of the theory at this level will allow the calculation of all single particle dynamical properties of the tracer and matrix particles, including the self-diffusion constant, mean-square displacement, non-Gaussian parameters, wavevector-dependent incoherent dynamic structure factor, and other properties sensitive to dynamical fluctuations.13 The model can also be extended to investigate the consequences of attractive interactions between tracer and matrix particles, particle size differences, and matrices and/or tracers composed of nonspherical objects.26 Acknowledgment. This work was supported by the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award Number DMR-0642573. We thank Grzegorz Szamel for insightful comments on the original version of the manuscript. References and Notes (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1991. (2) Poon, W. C. K. J. Phys.: Condens. Matter 2002, 14, R859. (3) (a) Dawson, K. A. Curr. Opin. Colloid Interface Sci. 2002, 7, 218. (b) Sciortino, F. Nat. Mat. 2002, 1, 145. (4) (a) Go¨tze, W.; Sjo¨gren, L. Rep. Prog. Phys. 1992, 55, 241. (b) Go¨tze, W. J. Phys.: Condens. Matter 1999, 11, A1. (5) Fabbian, L.; Go¨tze, W.; Sciortino, F.; Tartaglia, P.; Thiery, F. Phys. ReV. E 1999, 59, R1347. (6) (a) Bergenholtz, J.; Fuchs, M. Phys. ReV. E 1999, 59, 5706. (b) Bergenholtz, J.; Poon, W. C. K.; Fuchs, M. Langmuir 2003, 19, 4493. (c) Dawson, K.; Foffi, G.; Fuchs, M.; Go¨tze, W.; Sciortino, F.; Sperl, M.; Tartaglia, P.; Voigtmann, Th.; Zaccarelli, E. Phys. ReV. E 2000, 63, 011401. (7) Kirkpatrick, T. R.; Wolynes, P. G. Phys. ReV. A 1987, 35, 3072. (8) Schweizer, K. S.; Saltzman, E. J. J. Chem. Phys. 2003, 119, 1181. (9) Chen, Y.-L.; Schweizer, K. S. J. Chem. Phys. 2004, 120, 7212. (10) Schweizer, K. S. J. Chem. Phys. 2005, 123, 244501. (11) (a) Saltzman, E. J.; Schweizer, K. S. Phys. ReV. E 2006, 74, 061501. (b) Saltzman, E. J.; Schweizer, K. S. J. Chem. Phys. 2006, 125, 044509. (12) Chen, Y. L.; Kobelev, V.; Schweizer, K. S. Phys. ReV. E, 2005, 71, 041405. (13) Schweizer, K. S. Curr. Opin. Colloid Interface Sci. 2007, 12, 297. (14) Viehman, D. C.; Schweizer, K. S. J. Chem. Phys. 2008, 128, 084509. (15) (a) Krakoviack, V. Phys. ReV. Lett. 2005, 94, 065703. (b) Krakoviack, V. Phys. ReV. E 2007, 75, 031503. (16) Hanggi, P.; Talkner, P.; Borkovec, M. ReV. Mod. Phys. 1990, 62, 251. (17) (a) Cohen, E. G. D.; Verberg, R.; de Schepper, I. M. Phys. A 1998, 251, 251. (b) Verberg, R.; de Schepper, I. M.; Cohen, E. G. D. Phys. ReV. E 1997, 55, 3143. (18) (a) Plunkett, K. N.; Mohraz, A.; Haasch, R. T.; Lewis, J. A.; Moore, J. S. J. Am. Chem. Soc. 2005, 127, 14574. (b) Mohraz, A.; Weeks, E. R.; Lewis, J. A. Phys.ReV. E, 2008, 77, 060403R. (19) Lewis, J. A. AdV. Funct. Mater. 2006, 16, 2193. (20) Hansen, J. P.; McDonald I. R. Theory of Simple Liquids; Academic: London, 1986. (21) Hofling, T.; Franosch, T.; Frey, E. Phys. ReV. Lett. 2006, 96, 165901. (22) Gotze, W.; Leutheusser, E.; Yip, S. Phys. ReV. A 1981, 23, 2634. (23) Leutheusser, E. Phys. ReV. A 1983, 28, 2510. (24) Chang, R.; Yethiraj, A. Phys. ReV. E 2004, 69, 051101. (25) Sung, B. J.; Yethiraj, A. J. Chem. Phys. 2008, 128, 054702. (26) (a) Yatsenko, G.; Schweizer, K. S. J. Chem. Phys. 2007, 126, 014505. (b) Yatsenko, G.; Schweizer, K. S. Phys. ReV. E 2007, 76, 041506. (27) Schweizer, K. S.; Yatsenko, G. J. Chem. Phys. 2007, 127, 164505.
JP8060784
