J. Phys. Chem. B 2006, 110, 17207-17211
17207
Breakdown of the Stokes-Einstein Relationship: Role of Interactions in the Size Dependence of Self-Diffusivity Manju Sharma† and S. Yashonath*,†,‡,§ Solid State and Structural Chemistry Unit and Center for Condensed Matter Theory, Indian Institute of Science, Bangalore-560012, India ReceiVed: July 11, 2006
Einstein and others derived the reciprocal dependence of the self-diffusivity D on the solute radius ru for large solutes based on kinetic theory. We examine here (a) the range of ru over which Stokes-Einstein (SE) dependence is valid and (b) the precise dependence for small solutes outside of the SE regime. We show through molecular dynamics simulations that there are two distinct regimes for smaller solutes: (i) the interaction or Levitation effect (LE) regime for solutes of intermediate size and (ii) the D ∝ 1/ru2 for still smaller solutes. We show that as the solute-solvent size ratio decreases, the breakdown in the Stokes-Einstein relationship leading to the LE regime has its origin in dispersion interaction between the solute and the solvent. These results explain reports of enhanced solute diffusion in solvents existing in the literature seen for small solutes for which no explanation exists.
1. Introduction The theory of Brownian motion was first proposed by Einstein nearly a hundred years ago through a series of papers.1-4 This pioneering contribution has since expanded into a subject area of research and has been applied to diverse areas such as biology, chemistry, physics, and materials science. In the theory of Brownian motion, the particle or solute performing the Brownian motion is assumed to be significantly larger in radius than the solvent. Being a large solute particle, it suffers collisions with the solvent molecules simultaneously from different directions. On the basis of kinetic theory, Einstein derived an expression for the self-diffusion coefficient and showed that it goes as the inverse of the radius of the suspended particle, ru:
D)
RT 1 NA 6πηru
(1)
where R is the gas constant, T is the temperature, NA is the number of molecules in a gram-mole or the Avogadro number, and η is the viscosity of the fluid. Einstein suggested eq 1 as a method to determine the real size of the atom or molecule.1,2 It has been used to determine the sizes of the molecules over the years. The above relationship was also obtained by Sutherland5 around the same time using similar arguments and later Chandrasekar6 showed that a similar relationship can be derived by the application of the Langevin equation to the case of a Brownian particle. An idea of the importance of the above relationship can be formed by the number of contributions investigating the range of validity of the Stokes-Einstein relationship when viscosity, mass, size of the solute, nature of the interaction (hard sphere, soft sphere, Lennard-Jones, square well, etc.), and density are * Addresss correspondence to this author. † Solid State and Structural Chemistry Unit. ‡ Center for Condensed Matter Theory. § Also at the Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore.
varied. The focus of some of these studies has been largely on the inverse relationship between D and η. van Gunsteren and co-workers,7 in a molecular dynamics (MD) study, found that the product of viscosity (η) with the diffusion coefficient (D) of pure water is not independent of the mass of water. This suggests a deviation from Stokes’ law, which predicts that the product is independent of mass. Alder et al.8 found that the inverse relationship between selfdiffusivity and viscosity is valid throughout the density range they studied for a hard sphere fluid. In a detailed study, Kivelson et al.9 investigated the validity of the Stokes-Einstein relationship for solutes of mass and size larger than the solvent. Their study also investigated the situation when the solute mass is comparable to the solvent mass; many previous theoretical studies have found that the Stokes-Einstein relationship does not hold under this situation, although most of the experimental studies do suggest that the Stokes-Einstein relation is valid.10-13 Alder et al.14 studied the mass and size dependence of the solute in a binary mixture of hard sphere fluids at high densities. They found deviations from Enskog theory for smaller solute sizes but found the results in agreement with experiment. Alder and Alley15 studied a one-component fluid of hard disks by molecular dynamics. They found long-time correlations in the displacement distributions. Berner and Kivelson16 found that the Stokes-Einstein relationship between D and η is valid over 5 orders of magnitude of variation of η. In an extensive investigation, Willeke17 reported MD simulation of binary mixtures interacting through Lennard-Jones potential and studied the validity of the Stokes-Einstein relationship over a wide range of mass and size ratios of solute to solvent. One of their conclusions, among others, is that for σuu/σvv < 1 the StokesEinstein relationship is not valid for 1/16 < mu/mv < 50. Lamanna et al.18 have studied the dependence of D on η in protein solutions and thereby examine the validity of the Stokes-Einstein relationship as a function of concentration of the solution. Yamamoto and Onuki19 showed that the diffusivity is heterogeneous on time scales less than the stress relaxation time of a tagged particle. Masters and Keyes20 investigated
10.1021/jp064364a CCC: $33.50 © 2006 American Chemical Society Published on Web 08/10/2006
17208 J. Phys. Chem. B, Vol. 110, No. 34, 2006 solute motion as a function of the size and mass ratio. They found an increase in the self-diffusivity over and above the Stokes-Einstein value for light solutes when the solute size was about 1/4 of the solvent size. There have been a number of studies attempting to understand the D-η relationship in supercooled liquids. Bagchi21 has shown that near the glass transition temperature the self-diffusivity exhibits a change from the viscosity-dependent, Stokes-Einstein to an activated, hopping motion nearly independent of viscosity. Subsequently, Tarjus and Kivelson22 showed that there is a decoupling of the translational motion from viscosity below the supercooled transition in a one-component system. Similar findings have been reported by Bhattacharya and Bagchi23 but for a two-component system in the supercooled regime. They23 have carried out mode coupling theory calculations of solutesolvent mixtures and found that the small solutes have selfdiffusivities higher than that given by the Stokes-Einstein relationship. They have also carried out studies with a range of solute sizes by choosing solutes smaller as well as larger than the solvent size, as a function of solute-solvent size ratio. Jung et al.24 also investigated through MD simulation the decoupling and breakdown of the Stokes-Einstein relationship in supercooled liquids. In a study based on computer simulation and mode coupling theory of the solute-solvent system, Srinivas et al.25 showed that varying the solute-solvent interaction from attractive to repulsive can change the diffusion coefficient of the solute nonlinearly. Ravichandran and Bagchi26 have investigated the anisotropy associated with diffusion for Gay-Berne ellipsoids in a sea of Lennard-Jones spheres as a function of density. Viot, Tarjus, and Kivelson27 examined the breakdown of the Stokes-Einstein relationship for translational diffusion in supercooled liquids at low temperatures. In a density functional theoretical treatment of a tagged particle in a viscous liquid, Bagchi28 examined the validity of the inverse relationship between the orientational correlation time and the self-diffusivity of a tagged particle suggested by the Debye-Stokes-Einstein relationship. Phillies et al.29 reported deviations from the Stokes-Einstein dependence of self-diffusivity on viscosity in a light-scattering study of hydroxypropylcellulose:water mixture. Nigra and Evans30 have constructed the Stokes-Einstein relationship between the viscosity and the self-diffusivity for a fluid interacting through a modified form of the square-well potential. Ould-Kaddour and Barrat31 as well as Easteal and Woolf32 have carried out computer simulation investigations of solutesolvent mixtures to understand the variation of diffusion coefficient of the solute with size. They found that selfdiffusivity is higher than the SE value. More recently, Noworyta et al.33 observed that small solutes in water also showed higher diffusivities than given by the Stokes-Einstein expression. Recently, Liu, Goree, and Vaulina34 have shown in two dimensions and for particles interacting via Yukawa pair potential that the relationship between self-diffusivity and viscosity can break down near the disordering transition. We examine here the range of ru over which eq 1 holds and show that this relationship breaks down under certain conditions. We also obtain the dependence of D on ru when the StokesEinstein relationship breaks down. There are three regimes with differing dependence of D on ru. These have been obtained through molecular dynamics simulations. We show that the three regimes seen here are a function of the ratio of the solute to solvent radius. The present results provide the underlying reason as well as systematically explain the previous reports of enhanced self-diffusivity exhibiting deviations from the SE
Sharma and Yashonath TABLE 1: Lennard-Jones Interaction Parameters for the Solvent and the Solute type of interaction
, kJ/mol
σ, Å
vv uv uu
0.25 1.0 0.99
4.1 1.0-4.7 0.3-4.0
relationship by four different groups, namely, Willeke,17 Bhattacharya and Bagchi,23 Ould-Kaddour and Barrat,31 and Noworyta et al.33 2. Methods We study a binary mixture of monatomic spherical solute and solvent particles. All particles interact with other particles through simple (6-12) Lennard-Jones potential:
φRβ ) 4Rβ
[( ) ( ) ] σRβ rRβ
12
-
σRβ rRβ
6
R, β ) u, v
(2)
The total potential energy, Utot, is the sum of solvent-solvent (Uvv), solvent-solute (Uvu), and solute-solute (Uuu) intermolecular interaction energies.
Utot ) Uvv + Uvu + Uuu
(3)
The Lennard-Jones potential parameters and σ are different for the solute and solvent and we have not employed the Lorentz-Berthelot combination rules for interactions between the solute and the solvent.35,36 The solute diameter (assumed to be the same as the LennardJones parameter σ) has been varied to obtain the dependence of self-diffusivity on solute diameter. We have not included the electrostatic interactions to keep the system simple. Interactions between the solute and the solvent atoms were computed by using the following rule:35-37 σuv ) σuu + 0.7 Å. Note that the diameter of the solute, σuu, is 2ru. 2.1. Molecular Dynamics Simulations. All calculations have been carried out in the microcanonical ensemble using the Verlet leapfrog algorithm within the molecular dynamics method using DLPOLY.38 Cubic boundary conditions have been used. Integration of coordinates of both the solvent and the solute have been carried out initially with scaling of the velocities until equilibration was attained. Subsequently, runs were carried out without scaling of the velocities during which the averages were computed from the positions and their derivatives. 3. Computational Details The system we have simulated consists of 550 particles with 500 solvents (of mass 80 amu) and 50 solutes (of mass 200 amu) in a cubic simulation cell of length 33.3 Å. We also carried out runs with 40 amu instead of 200 amu for the solute mass and the results are similar. This corresponds to a reduced number density, F* ) 0.933. Simulations have been carried out at T* ) 1.663. The density chosen is higher than the triple point density, which is typically around F* ) 0.7. An equilibration period of 2 ns was followed by a 6 ns production run. Energy conservation obtained was better than 5 × 10-5. A time step of 15 fs has been used for all solute sizes with the exception of 0.3 Å for which 10 fs was used. The storage interval of 1 ps has been used for most properties and averaged over 6 ns. For intermediate scattering functions and velocity autocorrelation functions, a storage interval of 100 fs was used and properties averaged over only 700 ps.
Role of Interactions on Self-Diffusivity in Solution
Figure 1. Time evolution of the mean square displacement for different solute-solvent size ratios.
Figure 2. Plot of self-diffusivity, D, vs κ ) σuu/σvv at T* ) 1.663. Filled traingles indicate simulations with solute-solvent dispersion interactions; + are points without dispersion interaction between solute and solvent. Dashed and dotted lines are fits to 1/σuu2 and 1/σuu, respectively. Insets: (a) Plot of self-diffusivity against reciprocal of the square of the solute diameter in the small solute size limit. (b) Plot of the self-diffusivity against reciprocal of the solute diameter in the large solute size limit.
The interaction parameters used in the present work are listed in Table 1. 4. Results and Discussions 4.1. Three Distinct Regimes of Dependence of D on Solute Diameter. In Figure 1 we show the time evolution of the mean square displacement (msd) for different solute sizes. The absence of oscillations and the linear variation with time of the msd suggests good statistics. With increase in solute size there is a decrease in the slope of the msd curve. However, surprisingly, we see that for sizes of σuu between 0.7 and 0.9 Å (0.17 e κ e 0.22) there is an increase in the slope with increase in κ. The variation of self-diffusivity with the ratio of the solute to solvent Lennard-Jones diameter κ ) σuu/σvv obtained from the msd plot (see Figure 1) is shown in Figure 2. We have computed D for 0.17 e κ e 0.98 corresponding to solute sizes from 0.7 to 4.0 Å. Three distinct regimes of dependence of D on κ can be seen. In the limit of the large solute size, in the approximate range 0.5 e κ e 1.0 we see that, as σuu decreases, D increases gradually. In the inset we expand this region along with the fit of D ∝ 1/σuu to the MD points. The curve fits well
J. Phys. Chem. B, Vol. 110, No. 34, 2006 17209 to these points as indicated by the inset to Figure 2b. Below κ ) 0.5, the curve begins to deviate from the fit suggesting that the Stokes-Einstein (SE) relationship is not valid for solutes of still smaller sizes. We note that in the original derivation Einstein had made the assumption that the solute is larger than the solvent.1-3,39 We note that this estimate of 0.5 for κ below which the dependence of D on 1/σuu begins to deviate from the SE is only approximate. The present results suggest that the reciprocal relationship between self-diffusivity and the solute diameter is valid up to quite small solute sizes; solutes half the size of the solvent still follow the Stokes-Einstein relationship. Below κ < 0.5, self-diffusivity begins to increase steeply with the solute diameter, σuu. It reaches a maximum at κ ≈ 0.22. Note that since we have determined D values for solute diameter at an interval of 0.1 Å, the precise value at which the maximum occurs can only be determined with this accuracy. More precise determination is possible provided simulations are carried out at smaller intervals of σuu and the error bar on the self-diffusivity can be reduced to a value less than the change in D on going from one size to the next. Previously, we have observed such a maximum for guests diffusing within porous solids such as zeolites40,41 as well as binary liquid mixtures.37 Briefly, the reason for this maximum is due to the following: normally the medium in which the diffusant exists exerts strong attractive forces (due to dispersion interaction between the diffusant and the medium). However, when the size of the diffusant is comparable to the void size in which it exists, there is a mutual cancellation of the forces exerted by the medium on the diffusant leading to the diffusant essentially levitating. The effect responsible for the maximum in self-diffusivity is therefore known as the Levitation Effect (LE). In these studies, it was found that such a maximum occurs irrespective of the nature of the geometrical or topological structure of the zeolite and the nature of the guest suggesting that the maximum is a universal behavior of all guests in all types of porous solids. Further, the only condition for the existence of the maximum in self-diffusivity is the presence of dispersion or van der Waals interaction between the diffusant and the medium in which it is present. This has been demonstrated unambiguously by us.40,41 In previous studies, the interaction between the guests as well as the guest and the host porous solids has been modeled in terms of simple (6-12) Lennard-Jones potential. No electrostatic or polarization interactions were present. It has been demonstrated in one of the earlier studies that the ratio that influences the LE is the levitation parameter γ defined as the ratio of the length at which the interaction between the solute and the solvent is optimum to the Void diameter. Since the void diameter is proportional to the solvent diameter, κ is γ times a constant. Also shown in Figure 2 is the nature of D-κ dependence in the absence of the dispersion term. Note that the principal change from the previous curve is the absence of the diffusivity maximum at intermediate sizes of the solute. As stated above, the only condition for the existence of the maximum in selfdiffusivity with guest size as was shown by us in the context of guest-zeolite systems41 is the presence of dispersion interactions. Our results here demonstrate that similar behavior can be seen in the context of solute-solvent systems. This maximum arises from the levitation effect (LE). We shall therefore refer to this regime where the maximum in self-diffusivity is seen as the interaction regime (IR) or levitation effect (LE) regime since dispersion interactions are essential to the existence of this regime. LE refers to the minimum in the friction or the force on the guest exerted by the host or more generally the medium through which the guest is diffusing that occurs when the size
17210 J. Phys. Chem. B, Vol. 110, No. 34, 2006
Figure 3. Velocity autocorrelation function for the solute particles in the three regimes: 0.7 (from the SS regime), 0.9 (from the LE regime) and 3.0, 4.0 Å (both from the SE regime).
of the guest is comparable to the size of the void or neck through which it is diffusing. For a more detailed discussion, see refs 37, 41, and 42. As the solute size decreases further below 0.9 Å or κ ) 0.22 we see that D decreases for the first time instead of an increase! This is the only range where D decreases with decrease in σuu. By κ ) 0.17 (σuu ) 0.7 Å), the self-diffusivity has reached a minimum and on decrease of σuu below this value, selfdiffusivity begins to increase again. A fit to the MD points between 0.3 and 0.7 Å or 0.07 e κ e 0.17 to the curve D ∝ 1/σuu2 is shown in the inset. It is seen that the fit reproduces the molecular dynamics data well. This regime is expected on the basis of kinetic theory and exists in diverse systems such as guest-porous solids, liquid mixtures, etc.37,40,41 We term this regime as the small solute (SS) regime. Several studies on the dependence of the self-diffusivity as a function of the ratio of the solute to solvent size report deviation from Stokes-Einstein relation. One such study by Rasaiah and co-workers33 reports that Li0 has a higher D than predicted by SE expression in the MD study of uncharged solutes in water. The ratio of the solute size to the void size in water at the rather low density investigated (0.35 gm/cc) is sufficiently small that it lies in either the LE regime or the SS regime (where the dependence is 1/σuu2 on solute size rather than 1/σuu). We also note that in a detailed computer simulation study of the solutesolvent system by Ould-Kaddour and Barrat,31 the authors found that the solutes have a higher diffusivity than the SE prediction for some of the solute-solvent ratios between κ ) 0.66 and 0.066 of the size of the solvent. The points in this range correspond to the LE and SS regimes discussed in Figure 2. Here the LE regime is seen for the range 0.17 < σuu/σvv < 0.48 approximately. Thus, the enhanced diffusivity seen by Rasaiah and co-workers33 and Kaddour and Barrat31 is explained in terms of the solute-solvent interaction that is responsible for the maximum in self-diffusivity.Mode coupling theory treatment of solute-solvent systems for small solutes has also been investigated by Bhattacharyya and Bagchi.23 They also find a higher diffusivity for small solutes when the solute/solvent ratios are smaller. Thus all three studies in purely spherical solvents as well as in water suggest a deviation from SE relation for small solutes. They all suggest a higher diffusivity than predicted by SE. Our results here are in agreement with these. Further they provide the underlying reason for the different dependences and the precise nature of the variation of D with
Sharma and Yashonath
Figure 4. Intermediate scattering function for the solute particles in the three regimes: κ ) 0.17 (0.7 Å, from the SS regime), 0.22 (0.9 Å, from the LE regime), and 0.49 and 0.73 (3.0 Å, 4.0 Å, both from the SE regime).
solute-to-solvent ratio. Noworyta found that neutral Li atom diffusing within liquid water at 298 K and supercritical water at 683 K exhibits an enhanced self-diffusivity much higher than expected from SE expression.33 Recently, we have37 shown that a size-dependent maximum in D also exists in simple liquid mixtures interacting via Lennard-Jones consisting of one component of larger size and another of relatively smaller size. The latter is varied in diameter and one observes that D exhibits a maximum. This confirms that the maximum exists in all types of systems such as guesthost, liquid mixtures, etc. and the only condition in which it does not exist is the absence of dispersion and/or electrostatic interactions. However, since dispersion interactions are the subtlest of all molecular interactions and exist even in rare gases, there appears to be no system where the LE cannot be seen. To obtain greater insight into the nature of motion of the solute in the solvent, we show a plot of the velocity autocorrelation function (vacf) for four different solute sizes (see Figure 3). Of these, κ ) 0.17 (0.7 Å) is from the SS regime and κ ) 0.22 (0.9 Å) is from the LE regime while κ ) 0.73 and 0.98 (of solute sizes 3.0 and 4.0 Å) are from the SE regime. The motion of the solute from the SS regime exhibits a noticeable backscattering as indicated by the value of vacf (ca. -0.1) at around t ) 4 ps. This suggests that the solute diffusion does encounter collisional effects leading to reversal of the initial velocity. The vacf of a solute from IR or interaction regime or LE regime with κ ) 0.22 is seen to have little or no backscattering. This is surprising since it is larger in size than the 0.7 Å solute and is therefore expected to have more collisions. However, this surprising behavior is seen to arise from the relatively smoother potential energy landscape as has been discussed by us previously.37,41 The lower backscattering leads to an increase in D. The vacf of the SE regime (with κ ) 0.73 and 0.98) show significant backscattering and oscillations. The value of vacf is -0.25 and the rapid oscillations suggest the increased difficulty encountered by the large solute in its motion. A solute in the SE regime is unable to have facile movement as expected and performs significant vibrational motion before diffusive motion is seen. We show in Figure 4 the decay of the intermediate scattering function for different solute sizes (0.7, 0.9, 2.0, and 3.0 Å corresponding to κ values of 0.17 (SS), 0.22 (LE), 0.49 (SE)
Role of Interactions on Self-Diffusivity in Solution and 0.73 (SE)). Although the single-exponential fit to κ ) 0.73 in Figure 4.1 appears to fit well, on expansion of the figure it can be seen that the fit is poor at initial times close to zero. Biexponential fit provides a good fit throughout for this κ. Thus, except for 0.9 Å solute (or κ ) 0.22 from the IR or LE regime), all other sizes show relatively poorer agreement with the single exponential fit. Only the LE regime particle has a good fit over the whole range. These results are in good agreement with previous studies by us37,41 and suggest that carefully planned neutron scattering measurements may be able to distinguish between the different regimes by obtaining the intermediate scattering function. 5. Conclusions The results presented here suggest that the Stokes-Einstein relationship breaks down for relatively small solute diameter. The SE regime is seen to be valid until the ratio of solute to solvent reaches about 0.5. A further decrease in the ratio is followed by a maximum in D arising from the levitation effect. The exact value of κ at which the transition from the SE to the LE regime occurs may depend on the nature of the interaction between the solute and solvent particles. We also note that the transition occurs only if the dispersion interaction exists. We have not investigated the effect of the presence of electrostatic interactions here. Further decrease in κ leads to transition from the LE to the SS regime where D ∝ 1/σuu2. It is clear that the important quantity that determines the range over which each regime is valid is the ratio of the size of the solute to the solvent. More precisely, even this ratio is not the correct quantity but it is the ratio of the solute size to the neck diameter of the voids present within the solvent.37,41 However, since the neck diameters are proportional to the size of the particles (for face centered close-packed solids the neck radius is 0.155 times the radius of the solvent) and since the neck radius is not easily estimated, we could use the solvent size in place of the neck diameter. In summary, we see that the SE regime exists for γ e σuu/ σvv ) 0.48. The LE regime is observed for the range 0.17 e γ < 0.48. The SS regime is seen for really small solute sizes: γ e 0.17. We note that the onset of the LE regime is over (0.17, 0.48) size ratio. As already discussed, in a close packed solid, the neck diameter is 0.155R, where R is the radius of the packing spheres. However, in a liquid this is likely to be larger. Thus, we see that the lower limit is close to this value. The maximum in self-diffusivity is seen when the LE condition of mutual cancellation of forces40,41 occurs (that is the solute diameter is comparable to the neck diameter). The upper limit of the LE regime corresponds to the maximum size of the solute that can diffuse without encountering too many collisions with the solvent, which in turn depends on the structure of the voids and necks in the liquid. This points to the need for a systematic study of the void structure in dense liquids and solids. We note that some of enhanced diffusivities reported in the literature but as yet unexplained can be accounted for by the present study. More detailed study of the three regimes is under progress. Recognizing that three distinct regimes exist in a liquid
J. Phys. Chem. B, Vol. 110, No. 34, 2006 17211 interacting through van der Waals interactions is important. It might help resolve many of the anomalies that exist in diverse systems in nature. The present results offer an explanation for the enhanced self-diffusivity seen by several groups previously: Willeke,17 Masters and Keyes,20 Bhattacharya and Bagchi,23 and Noworyta et al.33 Acknowledgment. The authors wish to thank the Department of Science and Technology, New Delhi for financial support in carrying out this work. The authors also acknowledge CSIR, New Delhi for a research fellowship to M.S. and partial financial support from CSIR, New Delhi (grant no. 80(0060)/ 06/EMR-II). References and Notes (1) Einstein, A. Ann. Phys. 1906, 19, 371. (2) Einstein, A. Ann. Phys. 1906, 19, 289. (3) Einstein, A. Ann. Phys. 1911, 34, 591. (4) Einstein, A. InVestigations on the Theory of the Brownian MoVement; Dover Publications, Inc.: Mineola, NY, 1956. (5) Sutherland, W. Philos. Mag. 1905, 9, 781. (6) Chandrashekar, S. ReV. Mod. Phys. 1943, 15, 1. (7) Walser, R.; Mark, A. E.; van Gunsteren, W. F. Chem. Phys. Lett. 1999, 303, 583. (8) Alder, B. J.; Gass, M.; Wainwright, T. E. J. Chem. Phys. 1970, 53, 3813. (9) Kivelson, D.; Jensen, S. K.; Ahn, M.-K. J. Chem. Phys. 1973, 58, 428. (10) Stearn, A. E.; Irish, E. M.; Eyring, H. J. Phys. Chem. 1940, 44, 981. (11) Gray, P. Mol. Phys. 1963, 7, 235. (12) Onsanger, L. Ann. N.Y. Acad. Sci. 1945, 46, 241. (13) McCall, D. W.; Douglass, D. C. J. Phys. Chem. 1967, 71, 987. (14) Alder, B.; Alley, W. E. J. Chem. Phys. 1974, 61, 1415. (15) Alder, B.; Alley, W. E. J. Stat. Phys. 1978, 19, 341. (16) Bernert, B.; Kivelson, D. J. Phys. Chem. 1979, 83, 1401. (17) Willeke, M. Mol. Phys. 2003, 101, 1123. (18) Lamanna, R.; Delmelle, M.; Cannistraro, S. Phys. ReV. E 1994, 49, 5878. (19) Yamamoto, R.; Onuki, A. Phys. ReV. Lett. 1998, 81, 4915. (20) Masters, A. J.; Keyes, T. Phys. ReV. A 1983, 27, 2603. (21) Bagchi, B. J. Chem. Phys. 1994, 101, 9946. (22) Tarjus, G.; Kivelson, D. J. Chem. Phys. 1995, 103, 3071. (23) Bhattacharyya, S.; Bagchi, B. J. Chem. Phys. 1997, 106, 1757. (24) Jung, Y.; Garrahan, J. P.; Chandler, D. Phys. ReV. E 2004, 69, 061205. (25) Goundla, S.; Bhattacharyya, S.; Bagchi, B. J. Chem. Phys. 1999, 110, 4477. (26) Ravichandran, S.; Bagchi, B. J. Chem. Phys. 1999, 111, 7505. (27) Viot, P.; Tarjus, G.; Kivelson, D. J. Chem. Phys. 2000, 112, 10368. (28) Bagchi, B. J. Chem. Phys. 2001, 115, 2207. (29) Phillies, G. D. J.; O’Connell, R.; Whitford, P.; Streletzky, K. A. J. Chem. Phys. 2003, 119, 9903. (30) Nigra, P.; Evans, G. J. Chem. Phys. 2005, 122, 244508. (31) Ould-Kaddour, F.; Barrat, J.-L. Phy. ReV. A 1992, 45, 2308. (32) Easteal, A.; Woolf, L.; Jolly, D. Physica 1983, 121A, 286. (33) Noworyta, J. P.; Koneshan, S.; Rasaiah, J. J. Am. Chem. Soc. 2000, 122, 11194. (34) Liu, B.; Goree, J.; Vaulina, O. Phys. ReV. Lett. 2006, 96, 015005. (35) Parrinello, M.; Rahman, A.; Vashishtha, P. Phys. ReV. Lett. 1983, 50, 1073. (36) Vashishtha, P.; Rahman, A. Phys. ReV. Lett. 1978, 40, 1337. (37) Ghorai, P. K.; Yashonath, S. J. Phys. Chem. B 2005, 109, 5824. (38) Dlpoly-2.13 reference manual, version-2.13; Smith, W.; Forester, T.; CCLRC, Daresbury Laboratory: Daresbury, Warrington WA4 4AD, England, 2001. (39) Einstein, A. Ann. Phys. 1905, 17, 549. (40) Yashonath, S.; Santikary, P. J. Chem. Phys. 1994, 100, 4013. (41) Yashonath, S.; Santikary, P. J. Phys. Chem. 1994, 98, 6368. (42) Ghorai, P. K.; Yashonath, S. J. Phys. Chem. B 2005, 109, 3979.
