Determination of Statistically Reliable Transport Diffusivities from Molecular Dynamics Simulation

Transcription

1 Determination of Statistically Reliable Transport Diffusivities from Molecular Dynamics Simulation submitted to: Journal of Non-Newtonian Fluid Mechanics David J. Keffer, Brian J. Edwards, and Parag Adhangale Department of Chemical Engineering The University of Tennessee 52 Middle Drive Knoxville, TN Author to whom correspondence should be addressed. Abstract Using molecular dynamics simulations we determine the composition dependence of the self-diffusivity and transport diffusivity of a methane/ethane mixture at high pressure. We compute the transport diffusivity in two ways. First, the transport diffusivity is generated from the simulated self-diffusivities using an approximation known as the Darken Equation. Second, the transport diffusivity is generated from the simulated phenomenological coefficients, based upon linear irreversible thermodynamics. We discuss the relative advantages of the two methods in terms of (i) accuracy and (ii) computational demands of the approach. We find that the Darken Equation gives values of the transport diffusivity within 6% of the more rigorous approach and is subject to substantially less statistical error with less computational effort. We find that the mean and standard deviation of the transport diffusivity obtained from linear irreversible thermodynamics are strong functions of the implementation of the infinite-time limit required in the evaluation. We suggest and implement an algorithm for statistically reliable transport diffusivities from molecular dynamics simulations. Keywords: Transport; Diffusivity; Molecular dynamics; Simulation

2 . Introduction Equilibrium molecular dynamics simulations have been used to generate self-diffusivities and transport diffusivities of bulk mixtures for decades [, 2]. Non-equilibrium simulations have also been used to obtain diffusivities [3]. Our purpose in this work is to compare on the basis of (i) numerical accuracy and (ii) computational effort two different techniques for obtaining the transport diffusivity from equilibrium molecular dynamics simulations. The first technique is founded in Linear Irreversible Thermodynamics (LIT). It is a rigorous method that relates the transport diffusivity to phenomenological coefficients, which can be obtained from the time dependence of correlation functions of position and velocities of the simulated system. The second method is based on the Darken Equation, an approximate but derivable relationship, which allows one to calculate the transport diffusivity directly from the self-diffusivities [4]. While the Darken Equation is not rigorous and has taken severe criticism in the literature [5], we will show (i) that for some systems it is a reasonable approximation and (ii) that when obtaining transport diffusivities from molecular dynamics simulation it has substantial statistical and computational advantages over the more rigorous LIT approach. 2. Background 2.. Linear irreversible thermodynamics As a starting point, we begin with Linear Irreversible Thermodynamics. One can choose to begin with the generalized expression for mass flux of component α [6, 7], which under isothermal conditions reduces to 2

3 Nc L αβ j = α µ ~ β, () T β= where j is the diffusive mass flux of component a, N α c is the number of components in the system, T is the absolute temperature, µ ~ β is the specific chemical potential of component β, and L αβ is the phenomenological coefficient relating the flux of α to the driving force of β. Alternatively, one can rigorously derive Eq. () using modern Nonequilibrium Thermodynamics. (See Appendix A.) In Eq. () we include a superscripted bullet on the phenomenological coefficient to remind the reader that these coefficients implicitly demand that three items be specified. First, one must specify the nature of the flux; in this case it is a mass flux of component α. Second, one must specify the driving forces; in this case they are specific chemical potentials. Third, one must specify a frame of reference; in this case we choose the center of mass. These phenomenological coefficients, once determined, can be used, generally speaking, only under these three conditions, so it is important to state them. The phenomenological coefficients are related to a correlation function via L αβ = dvk B 0 I α () t I ( t + τ) dτ β, (2) where d is the dimensionality of the system, V is the volume, k B is Boltzmann s constant, t is the time variable over which the ensemble is averaged, τ is the observation time, and I α N α [ i ] () = m v () t v () t t. (3) α i= 3

4 Here m α is the mass of a molecule of component α, N α is the number of molecules of component α, v i is the velocity of the i th particle, and v is the velocity of the frame of reference, in this case the center of mass. Eqs. (), (2), and (3) along with the specifications denoted by the bullet completely define the phenomenological coefficients in a manner that allows them to be calculated from molecular dynamics simulations. The phenomenological coefficients that appear above are not all independent. They are related by three types of constraints. First, Onsager s reciprocity requires that L αβ = L βα. (4) The choice of reference frame creates an additional stipulation on the phenomenological coefficients. For the specifications made above, this constraint is of the form N c j α α= = 0. (5) This constraint and the fact that either the driving forces (chemical potentials) are independent away from mechanical equilibrium or that they are related by the Gibbs-Duhem Equation at equilibrium [8], one arrives at constraints of the form: N c L αβ = 0. (6) α= The particular form of the constraint is dependent on the choice of flux, driving force, and frame of reference. The Green-Kubo Integral given in Eq. (2) can be rewritten in the equivalent form of a long-time limit of a displacement correlation [9]: 4

5 [ A( t + τ) A( t) ][ B( t + τ) B( t) ] da db ( t + τ) () t dτ = lim dt dt τ 2τ 0. (7) This allows us to compute the phenomenological coefficient from either the velocities or the positions, Nα mα i i= Nβ mβ i i= L αβ = lim dvk τ 2 τ [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] i i, (8) where r i is the position of the i th particle and r is the position of the chosen reference frame. Frequently, one does not see the frame of reference position included in this expression. However, if one is to obey the symmetry relations of Eqs. (4) and (6), one must include the frame of reference. The only exception is if the frame of reference is not a function of time. (If the frame of reference were the center of mass, this would be fixed in a microcanonical simulation, due to conservation of momentum. However, the center of mass is not fixed in a canonical ensemble, where the thermostats do not conserve momenta.) If we limit ourselves to isothermal diffusion in a binary mixture, then a consequence of Eqs. (4) and (6) is that there is only one independent phenomenological coefficient: αα L αβ = L βα = ββ L = L. (9) One can prove analytically, via substitution, that Eq. (8) satisfies the symmetry of Eq. (9). Using Eq. (9) and the Gibbs-Duhem Relation, we can rewrite the mass fluxes as L αα L ββ j = µ ~ α α = µ ~ β = j. (0) w β β T w α T 5

6 Equation (0) is a form of Fick s Law but it is not a useful form for engineers, who more frequently work with a different form of Fick s Law j = ρd w α α = ρd w β = j, () β where ρ is the mass density, D is the binary transport diffusivity (sometimes called the mutual diffusivity), and w α is the mass fraction of component α. Again, this version of Fick s Law requires three specifications; the flux is a mass flux of α, the driving force is the gradient of the mass fraction, and the frame of reference is the mass-averaged velocity. If we equate the fluxes in Eqs. (0) and () we arrive at an expression for the diffusivity in terms of the remaining phenomenological coefficient: D = ρ w β L αβ T ~ µ w α α T,p. (2) One can also perform this derivation on a molar basis. If one takes as specifications that the flux is a molar flux, the driving force is the molar chemical potential, and the frame of reference is the molar-averaged velocity, then one writes Fick s Law as where Jα = cd xα = cd xβ = Jβ, (3) J α is the molar flux of component α, c is the concentration, and x α is the mole fraction. It can be shown that the transport diffusivities that appears in Eqs. () and (3) are the same. If one computes the phenomenological coefficients relative to the molar-averaged velocity, then one has 6

7 Lαβ = dvk lim τ Nα i= Nβ i= [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] i [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] i 2τ i i, (4) where r () t is the position of the center of moles of the system. Under these specifications the same symmetry of Eq. (9) applies to the phenomenological coefficients superscripted with a star. In this case, the diffusivity is related to the phenomenological coefficient via D = cx β L αβ µ α, (5) T x α T,p where µ α is the molar chemical potential. We will call D as defined by Eq. (5) the LIT transport diffusivity The Darken Equation In 948, while studying binary alloys, Darken derived an approximate relationship between the self-diffusivities and the transport diffusivities in an isothermal binary system [4]. ( x D + x D ) lnaα D = α self, β β self, α, (6) ln xα T,p where a α is the activity of component α. This expression, while approximate, has been widely used because it is easier to obtain self-diffusivities than transport diffusivities, both experimentally and from molecular-level simulations. One can see that self-diffusivities are easier to obtain from simulations by writing the Einstein Relation for the self-diffusivity of component α: 7

8 [ r (t + τ) r (t + τ) r (t) r (t)] Nα 2 i i + D = α = lim i self,. (7) 2d τ τ Nα The significant difference between the correlation function required for the phenomenological coefficient (Eq. (8)) and that required for the self-diffusivity is that the latter is a single-particle correlation function. Thus one averages over particles within the simulation. Equation (8) is a correlation of the center of mass (or center of moles). Thus, at each time step in a simulation of N molecules, one is generating N independent pieces of information for the self-diffusivity but only one piece of information for the transport diffusivity. We should also point out that the definition of the self-diffusivity in Eq. (7) is equally valid for component α in a pure fluid as well as in a mixture. The numerical values will differ for the two cases, as they should since the self-diffusivity is certainly a function of thermodynamic state. We have derived a rigorous relationship between the LIT and Darken transport diffusivities. (See Appendix B.) The result is DLIT ( x D x D ) lnaα = DDarken 2xα xβdtran, αβ + β off, α + α off, β, (8) ln xα T,p where the three additional terms from left to right represent respectively the correlation between α and β particles, the correlations (excluding self-correlations) between α particles, and the correlations (excluding self-correlations) between β particles. In the literature, one finds statements that the Darken Equation describes a system in which cross-correlations are negligible [2, 0]. These statements need careful interpretation. One can show that the three 8

9 additional terms do not vanish separately in any system. On the contrary, only the prescribed linear combination of correlation terms vanishes. (See Appendix B.) 3. Molecular dynamics Simulation In this work, we perform molecular dynamics simulations of a binary mixture of methane and ethane in the isobaric-isothermal ensemble, using a time-tested, home-made FORTRAN with MPI code. The methane and ethane are single-center pseudo atoms. We use the Hoover thermostat and barostat formulation of Melchionna et al. [], which give trajectories in the isobaric-isothermal ensemble. Additional parameters of the simulation are given in Table. We intentionally provide very detailed information regarding the simulation parameters because we are attempting to answer a question of statistical accuracy, which can be affected by the choices of many of these parameters. The simulations are larger (0 4 molecules) and longer (2 ns) than are typically used to generate diffusivities because we are interested in answering questions that can be obscured by statistical noise. The thermodynamic factors that appear in the LIT formulation (Eq. (5)) and the Darken Equation (Eq. (6)) for the transport diffusivity can be related to each other via lna α ln x α T,p xα lnµ kbt α = x. (9) α T,p As such, we require the same factor for both formulations. In theory, this partial derivative can be evaluated from molecular-level simulation. In the isobaric-isothermal ensemble, one could use the Widom Particle Insertion Method to evaluate the chemical potential [2]. One could then run simulations varying composition slightly and use a finite-difference rule to obtain an approximation of the derivative. In practice, this method is subject to statistical error. In this 9

10 work, we are interested in the error due to the dynamic contribution to the transport diffusivity, arising from the correlation functions given above. In order to isolate this error, we avoid introducing error due to the thermodynamic factor by using an analytical expression for Eq. (9) rather than obtaining it from simulation. Using the Lennard-Jones Equation of State with mixing rules, we obtain an analytical expression for the activity and the partial derivative of the activity in Eq. (9) [3]. We have previously shown that this equation of state agrees extremely well with the pressure of the simulations [4]. We will see shortly that it is necessary to make a very clear statement of a few of the details concerning the evaluation of the diffusivities. We computed the self-diffusivities using the mean square displacements (MSD) (Eq. (7)). We saved positions (without periodic boundary conditions, of course) every 2 ps. For our standard runs of 2 ns, we regressed the slope of the MSD versus time plots, over observations times from 0.5 to ns. This formula of averaging over observation times ranging from a quarter to a half of the total simulation duration has proven to strike a good balance between having a sufficiently long observation time to capture the long-time limit behavior and sufficiently short observation time to avoid dwindling statistical accuracy as the observation time approaches the simulation duration. The linear leastsquares regression does not require that the intercept pass through the origin, since the short-time behavior is nonlinear. We computed the transport diffusivities from both the Green-Kubo Integral of the Velocity Correlation Vunction (VCF) (Eq. (2)) and from MSD (Eq. (4)). We saved the minimum necessary data every 20 fs. For the transport diffusivity based on the VCF, we numerically integrated from observations times of 0 to a value of τ max. The parameter, τ max, is varied in this work. For the transport diffusivity based on the MSD correlation function, we 0

11 numerically regressed from observation times of τ min to a value of τ max. Both parameters, τ min and τ max, are varied in this work. In order to minimize the number of parameters, τ max is the same for MSD and VCF methods. 4. Results and discussion We simulated mixtures of methane and ethane at 350 K and a pressure of 98 bar across the entire range of composition. Using the methods described above, we obtained plots of the MSD versus observation time. In Figure, we show one example of such a plot for methane in a 50/50 mole percent mixture. The main point of this plot is to illustrate the high degree of linearity in the curve and the small degree of variation between the x, y, and z components of the diffusivity, which should be the same since the system is isotropic. This figure demonstrates that we have no difficulty computing self-diffusivities. We do not encounter statistical problems because the self-diffusivity is computed from a single-particle correlation function, which allows for adequate averaging over all particles. We also point out that the mean square displacement measured corresponds to an average displacement of 200 Å. In Figure 2, we plot the self-diffusivities and the transport diffusivity from the Darken Equation as functions of composition. In this plot, the thermodynamic factor on the left-handside of Eq. (9 ) has a minimum value of 0.82 at 20% methane. The maximum value was.0 at the pure components. The error bars are one standard deviation and are computed from five independent repetitions of the simulation (at each composition) using different seeds for the random number generator responsible for determining initial positions and velocities. The average relative standard deviation for both self-diffusivities and the Darken transport diffusivity

12 is.2%. If the only purpose is to generate self-diffusivities, however, there is no point in performing repetitions; a single simulation suffices. We now turn to calculating the transport diffusivity from LIT. We used the same simulations to generate the LIT results as was used to generate the Darken results. In Figure 3, we show one example of the MSD of methane in a 50/50 mole percent mixture. In sharp contrast to the MSD for the self-diffusivity, the MSD for the transport diffusivity is not strictly linear. Moreover, the x, y, and z components of the MSD do not overlap. This figure demonstrates that the exact procedure with which one can generate statistically reliable selfdiffusivities should not be used to calculate transport properties. Again, the difference lies in the fact that the self-diffusivity is based on a single particle correlation function whereas the transport diffusivity is based on the system correlation function. We also note that the net displacement illustrated in Figure 3 is smaller than the size of a particle. The velocity correlation function (not shown) contains analogous noise; it does not provide a solution to the problem at hand. While the MSD data in Figure 3 are noisy, we should point out that the symmetry of the four phenomenological coefficients (Eq. (9)) is maintained to machine precision for each x, y, and z component at all instants in time. Therefore, there can be no reduction in noise by averaging over the various forms of the phenomenological coefficients. We should also point out that this symmetry is kept if and only if one includes the frame of reference velocity in the VCF and the frame of reference position in the MSD correlation function. In Figure 4, we plot the mean diffusivity and its standard deviation as a function of τ min and τ max, the limits on the observation time over which the least squares regression is performed. We see that the mean and standard deviation of the diffusivity vary sharply as functions of τ max. 2

13 In other words, the value of the transport diffusivity one obtains from this analysis depends upon the choice of observation times. We see a weaker dependence on τ min only because we do not vary it over the same broad range as τ max. In Figure 5, we plot the mean diffusivity and its standard deviation as functions of τ max, the upper limit of integration for the VCF. We see again that the mean and standard deviation of the diffusivity vary sharply as functions of τ max. We do not vary τ min in this case since the lower limit of integration is fixed at zero. Early work, which established the calculation of LIT transport diffusivities, does not discuss the importance of the choice of τ max. Jolly and Bearman [] report standard deviations of their transport diffusivity of 3.8%. Schoen and Hoheisel [2] report standard deviations of their transport diffusivity less than 3%. In evaluating the VCF and MSD Methods, Schoen and Hoheisel chose different values of τ max. They report excellent agreement between the two methods, but the criteria for selection of different values of τ max are not reported. They are able to obtain very small standard deviations because they apparently chose a very small τ max, about.8 ps for VCF and 2.4 ps for MSD. It is unclear whether this puts them in the long-time limit. We will show however that for the simulations under consideration in the present work that such small estimates of τ max lie in a region where the transport diffusivity is a strong function of τ max. Thus, these small values of τ max do not provide definitive transport diffusivities. It is the purpose of this current work to suggest a more rigorous procedure for the unambiguous determination of the transport diffusivity. The crucial problem is how to select τ min and τ max such that one is in the long-time limit, but has not yet reached the region of dwindling statistical accuracy. At this stage, we have not identified a simple rule of thumb that works most of the time, as we have for the self-diffusivity. 3

14 If one naively uses the same prescription as that stated for the self-diffusivity, namely that τ min and τ max are a quarter and a half of the simulation duration respectively, then one is averaging in the region of high statistical error. Even if the simulations are repeated ten times (which we did), one still obtains a standard deviation that is 85% of the mean value for the transport diffusivity. As an alternative procedure, we recommend generating diffusivities for both the MSD and VCF Methods as functions of τ max. The value of τ min used only in the MSD Method should be kept large enough to exclude short-time nonlinear behavior in Figure 3, which for this system is about or 2 ps. The value of τ min will increase with a decrease in temperature or a decrease in density. Curves like those displayed in Figures 4 and 5 should be examined for plateaus. We find a common plateau in the MSD and VCF Methods from observation times of 0.02 to 0.2 ns in Figure 6, which is a close-up of data shown in Figures 4 and 5. The standard deviation is relatively constant over this plateau as well. We then average the diffusivity over τ max across the plateau in order to obtain a number which is independent of τ max. Using this algorithm, we plot in Figure 7 the transport diffusivity from LIT using both MSD and VCF Methods. The value of τ min is ps and we have averaged the diffusivity over the result obtained across a range of τ max from 0.02 to 0.2 ns. The error bars represent one standard deviation based on eight repetitions of the simulation. The average standard deviation across all compositions for the LIT transport diffusivity via MSD is %. The average standard deviation across all compositions for the LIT transport diffusivity via VCF is 7%. This is compared to.2% for the Darken Equation. These standard deviations are larger than those obtained in early studies, but we again point out that those studies used very small values of τ max, where, at least for these simulations, the transport diffusivity is a strong function of τ max. 4

15 We observe qualitative similarities between the LIT transport diffusivity obtained from MSD and VCF Methods. Both share the same deviations from linearity. Whether this deviation is real or is rather an artifact of the simulations due to too few repetitions is not known for certain. The error bars for all nine compositions overlap the Darken Equation, making the linear behavior a statistical possibility. Moreover, we did not always observe the same trend in deviations when examining the plots for each of the eight individual sets of simulations before averaging. Regardless, the Darken Equation appears to be a very good approximation. The average difference between the transport diffusivities from LIT (MSD) and the Darken Equation is 5.% for mixtures of methane and ethane at 350 K and 98 bar. It is possible, using very short observation times (τ max = 2.5 ps), to obtain standard deviations of less than 3%, as Schoen and Hoheisel report [2]. However, the mean values of the diffusivity are on average 70% lower than the Darken Equation. Additionally, with this small τ max, we observe a decrease in diffusivity with increasing methane mole fraction, which is aphysical. The results of this work may be surprising because the calculation of the transport diffusivity via molecular dynamics simulation has been considered an established procedure for over twenty years. Nevertheless, employing careful simulation and theory, we have demonstrated that the choice of observation time has a profound effect on the transport diffusivity. We have suggested an arbitrary but unambiguous method for determination of the transport diffusivity that gives reasonable standard deviations, reasonable agreement between MSD and VCF Methods, and reasonable agreement between LIT and the Darken Equation. One of the purposes of this paper was to determine the validity of the Darken Equation by comparing it with the more rigorous formalism of LIT. On this point, we can say only that 5

16 the method of determination of the transport diffusivity via LIT is not statistically accurate enough to make a definitive statement on the relative applicability of the Darken Equation. Within statistical error, the two methods yield equivalent results for this system. Earlier work indicates that LIT should yield a transport diffusivity 5% greater than the Darken Equation [2, 0]. We cannot confirm this observation. Using the Darken Equation has two advantages over LIT: (i) it requires only a single simulation to generate a statistically reliable transport diffusivity and (ii) this reduction in statistical noise provides a smooth and monotonic dependence of the transport diffusivity as a function of fluid composition. 5. Conclusions In this work, we have performed molecular dynamics simulations of mixtures of methane and ethane at 350 K and 98 bar. We have used linear irreversible thermodynamics (LIT) and the Darken Equation to obtain the transport diffusivity. We find that using LIT to obtain the transport diffusivity is subject to a strong dependence on the choice of observation times over which the diffusivity is calculated. Because there is no absolute rule for the choice of observation time, the mean value and standard deviation of the transport diffusivity are difficult to obtain definitively. Previous work in this area, which is considered to have made the calculation of the transport diffusivity from LIT a standard procedure, does not discuss this issue. We present an algorithm for obtaining statistically reliable transport diffusivities, which, while arbitrary, is nonetheless methodical and based on a balance of capturing the long-time limit of the dynamic correlation functions, while minimizing statistical errors associated with long observation times. 6

17 We find, within the statistical limits of our study, that for the system examined here LIT and the Darken Equation yield equivalent mean values of the transport diffusivities. In terms of computational effort, the Darken Equation requires a single simulation, whereas LIT requires multiple repetitions. Holding temperature and pressure constant, we find that the transport diffusivity increases with increasing mole fraction of methane, as one would expect. Identification and application of reference frame in the MSD correlation function and in the Green-Kubo Integral over the VCF is essential, if one wishes to maintain the symmetry properties (such as the Onsager reciprocal relations) in the phenomenological coefficients. Finally, the idea that crosscorrelations must be zero in order for the Darken Equation to apply must be interpreted carefully; a linear combination of the cross-correlations collectively, not individually, must go to zero. We are currently pursuing transport diffusivities via nonequilibrium molecular dynamics simulations to compare on the basis of computational efficiency and numerical accuracy against the results of LIT and the Darken Equation presented here. Acknowledgements Access to a 84-node IBM RS/6000 SP at Oak Ridge National Laboratory was made possible through the UT/ORNL Computational Science Initiative. P.A. would like to acknowledge 7

18 support from the UT Engineering Fundamentals Division. 8

19 Thermodynamic Parameters total number of molecules 0000 temperature (K) 350 K pressure (bar) Chemical Identity Properties intermolecular potential Lennard-Jones σ (Å) CH4 σ (Å) 4.48 C2H6 ε (K) 37 CH4 ε (K) 230 C2H6 M (grams/mole) CH4 M (grams/mole) C2H long-range cut-off distance (Å) 5 Numerical Integration Parameters integration algorithm Gear fifth-order predictor corrector [5, 6] time step (fs) 2 number of equilibration steps 00,000 number of data production steps,000,000 Auxiliary Parameters sampling interval for thermodynamic properties (fs) sampling interval for self-diffusivity (fs) 2000 sampling interval for transport diffusivity (fs) 20 temperature-controlling frequency (fs - ) 0-5 pressure-controlling frequency (fs - ) 0-5 Diffusivity Parameters minimum elapsed time for self-diffusivity (fs) 500,000 maximum elapsed time for self-diffusivity (fs),000,000 minimum elapsed time for transport diffusivity (fs) maximum elapsed time for transport diffusivity (fs) Table. Simulation parameters. 2 τ min τ max 9

20 References: [] D. L. Jolly, R. J. Bearman, Molecular dynamics simulation of the mutual and self-diffusion coefficients in Lennard-Jones liquid mixtures, Mol. Phys. 4 (980) [2] M. Schoen, C. Hoheisel, The mutual diffusion coefficient D2 in binary liquid model mixtures. Molecular dynamics calculations based on Lennard-Jones (2-6) potentials. I. The method of determination, Mol. Phys. 52 (984) [3] G. S. Heffelfinger, F. van Swol, Diffusion in Lennard-Jones fluids using dual controlvolume grand-canonical molecular dynamics simulation (DCV-GCMD), J. Chem. Phys. 00 (994) [4] L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. of the Amer. Inst. of Mining and Metall. Eng. 75 (948) [5] P. C. Carman, Self-diffusion and interdiffusion in complex-forming binary systems, J. Phys. Chem. 7 (967) [6] W. A. Steele in H. J. M. Hanley (Ed.), Time correlation functions, Transport Phenomena in Fluids. Dekker, New York, 969. [7] K. E. Gubbins in K. Singer (Ed.), Thermal transport coefficients for dense fluids, A Specialist Periodical Report. Statistical Mechanics. Vol., Burlington House, London, 973. [8] D. D. Fitts, Nonequilibrium Thermodynamics: A Phenomenological Theory of Irreversible Processes in Fluid Systems. McGraw Hill, New York, 962. [9] J. M. Haile, Molecular Dynamics Simulation. John Wiley & Sons, Inc., New York, 992. [0] J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids. Academic Press, London, 986. [] S. Melchionna, G. Ciccotti, B. L. Holian, Hoover NPT dynamics for systems varying in size and shape, Mol. Phys. 78 (993) [2] B. Widom, Potential distribution theory and the statistical mechanics of fluids, J. Phys. Chem. 86 (982) [3] J. J. Nicolas, K. E. Gubbins, W. B. Streett,D. J. Tildesley, Equation of state for the Lennard-Jones fluid, Mol. Phys. 37 (979) [4] D. Keffer, P. Adhangale, The composition dependence of self and transport diffusivities from molecular dynamics simulations, submitted to Chemical Eng. J. (2003) 20

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22 Figure Captions Figure. Mean square displacement of methane as a function of observation time in a 50/50 mole percent mixture of methane and ethane at 350 K and 98 bar. Figure 2. Self-diffusivities of methane and ethane and the transport diffusivity via the Darken Equation as functions of composition. In most cases the error bars are smaller than the markers. Figure 3. Mean square displacement of the molar-averaged methane position (Eq. (4)) as functions of observation time. Figure 4. Mean and standard deviation of the LIT transport diffusivity via MSD correlation functions as functions of τ min and τ max. Figure 5. Mean and standard deviation of the LIT transport diffusivity via the velocity correlation function as functions of τ max. Figure 6. Mean and standard deviation of the LIT transport diffusivity via the MSD and VCF Methods as functions of τ max. This is a close-up of the short-time behavior shown in Figs. 4 and 5. Figure 7. LIT transport diffusivities from MSD and VCF Methods as functions of composition. 22

23 Appendix A. A derivation of the mass flux equation To test the validity of the Darken Equation for transport diffusivities using molecular simulations, it is absolutely necessary to have an unambiguous starting point, as expressed by Eq. (). Furthermore, it is also necessary to take great care when applying physical principles to this starting point so that no erroneous information is included in the subsequent analysis. Equation () must be guaranteed to be compatible with the theory of Linear Irreversible Thermodynamics (LIT) for any such subsequent analysis to be applicable; however, it has been well documented that LIT has often been misused in applications such as those discussed in the main body of this article see Refs. [7-2] for detailed explanations of this phenomenon. To guarantee that LIT is applied properly to the problem at hand, one must choose the proper starting point for the development of Eq. (). Only then can one claim that Eq. () is a valid equation set to which one can apply the principles of LIT, such as the Onsager Reciprocal Relations. Subsequently, one must also ensure that the oft-misused principles of LIT actually do apply to Eq. (): some of the principles of LIT that are claimed as being universal are actually only valid under very special circumstances [7-2]. In summary, we must verify two issues for the present article: first, we must verify that Eq. () is a proper flux/force equation set in the theory of LIT, and second, that the required principles of LIT needed for the purposes of this article are applicable to Eq. (). As shown below, although these two issues seem intuitively to be essentially the same, they are, in fact, quite distinct. Consider the two-component system under investigation in this article. In terms of volumetric properties, the internal energy density, u (a thermodynamic potential function), may be expressed in terms of the system density variables, u(s,c,c 2 ), where s is the entropy density and c, c 2 are the molar concentrations of each component [7]. Alternatively, one can 23

24 also express the entropy density as a potential function: s(u,c,c 2 ). From Equilibrium Thermodynamics, it is known that s = u T, s ci µ = i T, p = u + st + cµ + c2µ 2, (A.) where p is the thermodynamic pressure expressed in terms of density variables u, c, c 2, and functions s, T, µ, µ 2 [7]. With a properly expressed equilibrium potential function, s(u,c,c 2 ), one is then in a position to apply LIT to the problem at hand, close to, but not at, thermodynamic equilibrium. Taking the density variables as functions of an Eulerian spatial coordinate, x, and the time, t, an evolution equation for the production rate of entropy density is calculated as s = t 3 i= s X i X t i, (A.2) X u,c where ( ) T, c 2 thermodynamic force or affinity, 22]:. In this expression, the first entity on the right-hand side is identified as a Λ i, and the second as an associated nonequilibrium flux, J i [7, s = t 3 i= Λ i J i. (A.3) The fluxes appearing in the above expression are functions of all affinities and all intrinsic system parameters: J ( Λ, Λ, Λ,T, µ µ ) i 2 3, 2. Expanding each J i in a power series, and subsequently neglecting all non-linear terms close to equilibrium, one obtains linear relationships between the fluxes and affinities, 3 J i = L ij Λ j, (A.4) j= 24

25 ij, 2 is the phenomenological coefficient relating the appropriate affinity to the where L ( T, µ µ ) corresponding flux [7, 22]. For the phenomenological coefficients defined above, it has been shown, with reservations expressed [7, 8], that the Onsager Reciprocal Relations hold [23, 24]: α ij = α ji. (A.5) These famous relationships state the symmetry of material interactions between affinities and fluxes. However, Onsager s derivation of this result employed only affinities of the form Λ i = s X i, referred to as relaxational affinities; i.e., those affinities arising from processes conducted entirely within a given fluid particle, viewed as a complete thermodynamic subsystem at an internal state of equilibrium [7]. Thus everything discussed so far applies to systems undergoing relaxational processes only. For examining transport processes, such as mutual diffusion between two components of a binary mixture at constant temperature and pressure, relaxational affinities are not relevant. The standard analysis presented above must be replaced with one appropriate to the case at hand, involving transport affinities; i.e., those affinities arising from processes involving the physical movement of a quantity of energy from one thermodynamics subsystem (cf. fluid particle) to another. Consequently, for the problem under consideration in this article, one must begin the derivation of Eq. () anew, using the preceding analysis as a general outline. Consider a system experiencing only transport, as opposed to relaxational, processes. The thermodynamical equilibrium state of each subsystem is still described by the potential function s(u,c,c 2 ), and thus Eq. (A.) still applies at every position x. The production rate of 25

26 entropy density is also still described by Eq. (A.2); however, the flux J i is no longer expressed by X i. Indeed, this latter quantity then takes on the form t Xi t = Ji, (A.6) where the nabla operator denotes the spatial gradient of it operant and the flux is now a spatial vector field. For a discrete number of subsystems, the nabla operator becomes a delta operator, thus making explicit that the variable X i is being transferred from one subsystem to another. In a continuous system, this concept is expressed mathematically by the gradient operator. Substituting Eq. (A.6) into (A.2), one obtains s + t 2 i= s Ji X i = 0. (A.7) Notethat the summation is now over only two components, since no relaxational processes are occurring (since temperature and pressure are taken as constants). Integrating this expression by parts over all fluid particles (i.e., over all positions x) using no-penetration boundary conditions, and then setting the integrand to vanish yields s = t 2 i= s Ji X i. (A.8) One can now recover the form of Eq. (A.3) provided that Λi s = Xi ; (A.9) in words, if the transport affinity is defined as the gradient of the partial derivative of the entropy density with respect to the variable of interest. Consequently, Eq. (A.4) transfers directly to transport phenomena, 26

27 J 2 = i L ij j= Λ j, (A.0) assuming that the fluid is isotropic so that no spatial dependence is included in L ij. Equation (A.0) satisfies the first objective, which was to determine a relationship between fluxes and affinities that is compatible with LIT. Unfortunately, in this case, the derivation of the Reciprocal Relations of Onsager does not apply to transport affinities [7-2]. Indeed, up until the late 990s, no rigorous derivation of the Onsager relations for transport processes was presented. This situation has changed, however, since more mathematically structured treatments of nonequilibrium thermodynamics on multiple length and time scales have been formulated in the past several years [25-29]. Without excessive detail that can be found in the references cited immediately above, the crux of the matter is that the symmetry requirements imposed on irreversible transport processes by the newly discovered mathematical structure dictate satisfaction of Onsager-type reciprocal relations for transport, as well as relaxational, processes. These symmetry requirements arise on a coarse-grained level of description through projection operation of atomistic information. As a consequence, Onsager-type reciprocal relations may be applied to the phenomenological coefficients appearing in Eq. (A.0): L ij = L ji. (A.) Hence one satisfies the second objective by noting that Eq. (A.) applies to Eq. (A.0) under the auspice of Eq. (A.9). For the purposes of the present article, this amounts to using as the proper equation set J J 2 = L µ T = L 2 µ T L2 µ T L 22 µ T 2 2, (A.2) 27

28 with L 2 = L 2 and constant temperature. 28

29 Appendix B. A derivation relating the Darken transport diffusivity to the transport diffusivity from Linear Irreversible Thermodynamics. One can rewrite the LIT transport diffusivity from Eq. (5) as lnaα Dtran, αα = lim fαα, (B.) 2dNxαxb ln xα τ τ T,p where Nα fαα = i= Nα Nα = i i= j= [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] i 2 [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] r ( t + τ) r ( t + τ) One can rewrite the self-diffusivity from Eq. (7) as where D, α = lim fα 2dNα τ τ i i [( ) ( r () t r () t )] j j. (B.2) self, (B.3) N α [ ri (t + τ) r (t + τ) ri(t) r (t)] α = + i= 2 f. (B.4) The quantity f α contains only the diagonal (i=j) elements of f αα. We can include all the offdiagonal elements in another function, N α Nα αα = i= j= j i [( r ( t + τ) r ( t + τ) ) ( r () t r () t )] r ( t + τ) r ( t + τ) i i [( ) ( r () t r () t )] f (B.5) This new function satisfies the relationship f αα = f f. (B.6) αα α + We can define an off-diagonal diffusivity as j j 29

30 Doff, α = lim fαα. (B.7) 2dNα τ τ so that we can rewrite the LIT transport diffusivity as lnaα tran, αα = self, α + off, α xβ ln xα T,p ( D D ) D. (B.8) If we switch indices, we have an equivalent form of the transport diffusivity, lna D. (B.9) ( D D ) β tran, ββ = β + self, off, β xα ln xβ T,p The thermodynamic partial derivative in Eqs. (B.8) and (B.9) can be shown to be identical. From the constraints on the phenomenological coefficients, we know that we have the same symmetry in the transport diffusivities, namely D. (B.0) LIT = D tran, αα = D tran, αβ = D tran, βα = D tran, ββ We can take a linear combination of Eqs. (B.8) and (B.9) to give a new expression for the transport diffusivity, D. (B.) LIT = xβ D tran, αα + xαd tran, ββ Substitution of Eqs. (B.8) and (B.9) into Eq. (B.) yields ( D + D + D + D ) lnaα LIT = self, α off, α self, β off, β ln xα T,p D, (B.2) and further manipulation yields: DLIT lna α = ln xα T,p ( xβdself, α + xαdself, β ) + ( xβdoff, α + xαdoff, β ) + ( x D + x D ) + ( x D + x D ) α self, α α off, α β self, β β. (B.3) off, β 30

31 The first term in parentheses gives rise to the Darken Equation. The last two terms in parentheses are directly related to the transport diffusivities in Eq. (B.). DLIT = DDarken lna ( ) α D + D + ( x D x D ) + xα xβ tran, αα tran, ββ β off, α + ln xα T,p α off, β. (B.4) Using the symmetry in Eq. (B.0), we can rewrite this expression as DLIT ( x D x D ) lnaα = DDarken 2xα xβdtran, αβ + β off, α + α off, β ln xα T,p. (B.5) Thus, we see very clearly that the Darken Equation contains the contributions to the transport diffusivity excluding correlations between α and β particles (in the D tran, αβ term), all non-self α-α correlations (in the D off, α term), and all non-self β-β correlations (in the D off, β term). The thermodynamic factor in front of the off-diagonal elements has been absorbed already into D Darken and D tran, αβ. We also see that, because Dtran, αβ = DLIT, the correlations will never vanish individually. Agreement with the Darken Equation can only come from a system where the three correlation terms collectively vanish. 3

32 4.0E+04 mean square displacement (Å 2 ) 3.5E E E+04 x y z 2.0E+04.5E observation time (ns) Figure. 32

33 3.0E-07 self-diffusivity (Me) 2.5E-07 self-diffusivity (Et) transport diffusivity (Darken) 2.0E-07 diffusivity (m 2 /s).5e-07.0e E E mole fraction methane Figure 2. 33

34 3.5E E+00 mean square displacement (Å 2 ) 2.5E E+00.5E+00.0E x y z 5.0E-0 0.0E observation time (ns) Figure 3. 34

35 .4E-07 τ min = 0 ps & ps.2e-07.0e-07 τ min = 0 ps means diffusivity (m 2 /sec) 8.0E E-08 standard deviations 4.0E-08 τ min = 0 ps 2.0E-08 τ min = 0 ps & ps 0.0E maximum observation time (ns) Figure 4. 35

36 .6E-07.4E-07.2E-07 mean standard deviation.0e-07 diffusivity (m 2 /sec) 8.0E E E E E E E maximum observation time (ns) Figure 5. 36

37 .6E-07.4E-07 means MSD.2E-07 VCF diffusivity (m 2 /sec).0e E E-08 standard deviations VCF 4.0E E-08 MSD 0.0E maximum observation time (ns) Figure 6. 37

38 2.3E-07 2.E-07 LIT (MSD) LIT (VCF) Darken.9E-07.7E-07 diffusivity (m 2 /s).5e-07.3e-07.e E E E mole fraction methane Figure 7. 38