Determination of Statistically Reliable Transport...

D. Keffer, Department of Chemical Engineering, University of Tennessee, Knoxville

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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

1512 Middle Drive Knoxville, TN 37996-2200

[email protected]

†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


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1. Introduction

Equilibrium molecular dynamics simulations have been used to generate self-diffusivities

and transport diffusivities of bulk mixtures for decades [1, 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

2.1. 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


3

∑=β

β

•αβ

αµ∇

=−

cN

1

~T

Lj , (1)

where αj is the diffusive mass flux of component a, Nc is the number of components in the

system, T is the absolute temperature, βµ~ is the specific chemical potential of component β, and

•αβL is the phenomenological coefficient relating the flux of α to the driving force of β.

Alternatively, one can rigorously derive Eq. (1) using modern Nonequilibrium Thermodynamics.

(See Appendix A.)

In Eq. (1) 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

( ) ( ) ττ+ℑ⋅ℑ= ∫∞

•β

•α

•αβ dtt

dVk1L

0B , (2)

where d is the dimensionality of the system, V is the volume, kB is Boltzmann’s constant, t is the

time variable over which the ensemble is averaged, τ is the observation time, and

( ) ( ) ( )[ ]∑α

=

•α

•α −=ℑ

N

1ii tvtvmt . (3)


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Here mα is the mass of a molecule of component α, Nα is the number of molecules of component

α, iv is the velocity of the ith particle, and •v is the velocity of the frame of reference, in this

case the center of mass. Eqs. (1), (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

•βα

•αβ = LL . (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

0jcN

1=∑

=αα . (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:

0LcN

1=∑

=α

•αβ . (6)

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]:


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( ) ( ) ( ) ( )[ ] ( ) ( )[ ]τ

−τ+−τ+=ττ+

∞→τ

∞

∫ 2tBtBtAtA

limdtdtdBt

dtdA

0. (7)

This allows us to compute the phenomenological coefficient from either the velocities or the

positions,

( ) ( )( ) ( ) ( )( )[ ]

( ) ( )( ) ( ) ( )( )[ ]τ

−−τ+−τ+

⋅−−τ+−τ+

=∑

∑β

=

••β

α

=

••α

∞→τ

•αβ 2

trtrtrtrm

trtrtrtrm

limdVk

1L

N

1iii

N

1iii

, (8)

where ir is the position of the ith 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:

•ββ

•βα

•αβ

•αα =−=−= LLLL . (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

ββ

•ββ

αα

•αα

βα

=µ∇−=µ∇=− j~T

Lw1~

TL

w1j . (10)


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Equation (10) 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

ββαα

=∇ρ−=∇ρ=− jwDwDj , (11)

where ρ is the mass density, D is the binary transport diffusivity (sometimes called the mutual

diffusivity), and wα 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. (10) and (11) we arrive at an expression for the diffusivity

in terms of the remaining phenomenological coefficient:

p,T

w

~

TL

w1D

∂µ∂

ρ=

α

α•αβ

β. (12)

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

** JxcDxcDJ ββαα =∇−=∇=− , (13)

where *Jα is the molar flux of component α, c is the concentration, and xα is the mole fraction.

It can be shown that the transport diffusivities that appears in Eqs. (11) and (13) are the same. If

one computes the phenomenological coefficients relative to the molar-averaged velocity, then

one has


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( ) ( )( ) ( ) ( )( )[ ]

( ) ( )( ) ( ) ( )( )[ ]τ

−−τ+−τ+

⋅−−τ+−τ+

=∑

∑β

=

α

=

∞→ταβ 2

trtrtrtr

trtrtrtr

limdVk

1L

N

1i

*i

*i

N

1i

*i

*i

* , (14)

where ( )tr * 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

p,T

*

xTL

cx1D

∂µ∂

=α

ααβ

β, (15)

where µα is the molar chemical potential. We will call D as defined by Eq. (15) the LIT

transport diffusivity.

2.2. The Darken Equation

In 1948, while studying binary alloys, Darken derived an approximate relationship

between the self-diffusivities and the transport diffusivities in an isothermal binary system [4].

( )αββαα

α +

∂∂

= ,self,selfp,T

DxDxxlnaln

D , (16)

where aα 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:


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[ ]

α

α

=

••

∞→τα

∑ +−τ+−τ+

τ=

N

)t(r)t(r)t(r)t(r1lim

d21D

N

1i

2 ii

,self . (17)

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. (17) 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

( )βααβα

ααββα +

∂∂

+−= ,off,offp,T

,tranDarkenLIT DxDxxlnaln

Dxx2DD , (18)

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, 10]. These statements need careful interpretation. One can show that the three


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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. [11], which give trajectories in the

isobaric-isothermal ensemble. Additional parameters of the simulation are given in Table 1. 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 (104 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. (15)) and the Darken

Equation (Eq. (16)) for the transport diffusivity can be related to each other via

p,TBp,T x

lnTk

xxlnaln

∂

µ∂=

∂∂

α

αα

α

α . (19)

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 [12]. 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


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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. (19)

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. (19) [13]. We have previously shown that this equation of state agrees extremely

well with the pressure of the simulations [14].

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. (17)). 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 1 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 least-

squares 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. (14)). 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 τmax. The parameter, τmax, is

varied in this work. For the transport diffusivity based on the MSD correlation function, we


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numerically regressed from observation times of τmin to a value of τmax. Both parameters, τmin

and τmax, are varied in this work. In order to minimize the number of parameters, τmax 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 1, 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-hand-

side of Eq. (19 ) has a minimum value of 0.812 at 20% methane. The maximum value was 1.0 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


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is 1.2%. 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 self-

diffusivities 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 τmin

and τmax, 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 τmax.


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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 τmin only because we do not

vary it over the same broad range as τmax.

In Figure 5, we plot the mean diffusivity and its standard deviation as functions of τmax,

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 τmax. We do not vary τmin 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 τmax. Jolly and Bearman [1] 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 τmax. They report excellent agreement between the two

methods, but the criteria for selection of different values of τmax are not reported. They are able

to obtain very small standard deviations because they apparently chose a very small τmax, about

1.8 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 τmax lie in a region where the transport diffusivity is a strong function of τmax.

Thus, these small values of τmax 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 τmin and τmax 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.


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If one naively uses the same prescription as that stated for the self-diffusivity, namely that τmin

and τmax 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 τmax. The value of τmin 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 1 or 2 ps. The value of τmin 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 0.12 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 τmax across

the plateau in order to obtain a number which is independent of τmax.

Using this algorithm, we plot in Figure 7 the transport diffusivity from LIT using both

MSD and VCF Methods. The value of τmin is 1 ps and we have averaged the diffusivity over the

result obtained across a range of τmax from 0.02 to 0.12 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 11%. The average standard deviation

across all compositions for the LIT transport diffusivity via VCF is 17%. This is compared to

1.2% 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 τmax, where, at least

for these simulations, the transport diffusivity is a strong function of τmax.


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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 5.1% for mixtures of methane and ethane at 350 K and 98 bar.

It is possible, using very short observation times (τmax = 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

τmax, 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


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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,

10]. 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.


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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 cross-

correlations 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 184-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


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support from the UT Engineering Fundamentals Division.


19

Thermodynamic Parameters total number of molecules 10000 temperature (K) 350 K pressure (bar) 97.99 Chemical Identity Properties intermolecular potential Lennard-Jones

4CHσ (Å) 3.822

6H2Cσ (Å) 4.418

4CHε (K) 137

6H2Cε (K) 230

4CHM (grams/mole) 16.042

6H2CM (grams/mole) 30.068

long-range cut-off distance (Å) 15 Numerical Integration Parameters integration algorithm Gear fifth-order predictor corrector [15, 16] time step (fs) 2 number of equilibration steps 100,000 number of data production steps 1,000,000 Auxiliary Parameters sampling interval for thermodynamic properties (fs)

2

sampling interval for self-diffusivity (fs) 2000 sampling interval for transport diffusivity (fs) 20 temperature-controlling frequency (fs-1) 10-5 pressure-controlling frequency (fs-1) 10-5 Diffusivity Parameters minimum elapsed time for self-diffusivity (fs) 500,000 maximum elapsed time for self-diffusivity (fs) 1,000,000 minimum elapsed time for transport diffusivity (fs)

τmin

maximum elapsed time for transport diffusivity (fs)

τmax

Table 1. Simulation parameters.


20

References: [1] D. L. Jolly, R. J. Bearman, Molecular dynamics simulation of the mutual and self-diffusion coefficients in Lennard-Jones liquid mixtures, Mol. Phys. 41 (1980) 137-147. [2] M. Schoen, C. Hoheisel, The mutual diffusion coefficient D12 in binary liquid model mixtures. Molecular dynamics calculations based on Lennard-Jones (12-6) potentials. I. The method of determination, Mol. Phys. 52 (1984) 33-56. [3] G. S. Heffelfinger, F. van Swol, Diffusion in Lennard-Jones fluids using dual control-volume grand-canonical molecular dynamics simulation (DCV-GCMD), J. Chem. Phys. 100 (1994) 7548-7552. [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. 175 (1948) 184-201. [5] P. C. Carman, Self-diffusion and interdiffusion in complex-forming binary systems, J. Phys. Chem. 71 (1967) 2565-2572. [6] W. A. Steele in H. J. M. Hanley (Ed.), Time correlation functions, Transport Phenomena in Fluids. Dekker, New York, 1969. [7] K. E. Gubbins in K. Singer (Ed.), Thermal transport coefficients for dense fluids, A Specialist Periodical Report. Statistical Mechanics. Vol. 1, Burlington House, London, 1973. [8] D. D. Fitts, Nonequilibrium Thermodynamics: A Phenomenological Theory of Irreversible Processes in Fluid Systems. McGraw Hill, New York, 1962. [9] J. M. Haile, Molecular Dynamics Simulation. John Wiley & Sons, Inc., New York, 1992. [10] J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids. Academic Press, London, 1986. [11] S. Melchionna, G. Ciccotti, B. L. Holian, Hoover NPT dynamics for systems varying in size and shape, Mol. Phys. 78 (1993) 533-544. [12] B. Widom, Potential distribution theory and the statistical mechanics of fluids, J. Phys. Chem. 86 (1982) 869-872. [13] J. J. Nicolas, K. E. Gubbins, W. B. Streett,D. J. Tildesley, Equation of state for the Lennard-Jones fluid, Mol. Phys. 37 (1979) 1429-2454. [14] D. Keffer, P. Adhangale, The composition dependence of self and transport diffusivities from molecular dynamics simulations, submitted to Chemical Eng. J. (2003)


21

[15] C. W. Gear, The Numerical Integration of Ordinary Differential Equations of Various Orders. Argonne National Laboratory, ANL-7126, 1966. [16] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1971. [17] A. N. Beris, B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure. Oxford Science Publications, New York, 1994. [18] C. Truesdell, Rational Thermodynamics. Springer-Verlag, New York, 1984. [19] B. D. Coleman, C. Truesdell, On the reciprocal relations of onsager, J. Chem. Phys. 33 (1960) 28-31. [20] J. Wei, Irreversible thermodynamics in engineering, Ind. Eng. Chem. 58 (1966) 55-60. [21] J. Wei, J. C. Zahner, Comment on the general reciprocity relations of van Rysselberghe, J. Chem. Phys. 43 (1965) 3421. [22] H. B. Callen, Thermodynamics and an Introduction to Thermostatics. John Wiley and Sons, New York, 1985. [23] L. Onsager, Reciprocal relations in irreversible processes, part 1, Phys. Rev. 37 (1931) 405-426. [24] L. Onsager, Reciprocal relations in irreversible processes, part 2, Phys. Rev. 38 (1931) 2265-2279. [25] M. Grmela, H. C. Öttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56 (1997) 6620-6632. [26] H. C. Öttinger, M. Grmela, Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E 56 (1997) 6633-6655. [27] H. C. Öttinger, General projection operator formalism for the dynamics and thermodynamics of complex fluids, Phys. Rev. E 57 (1998) 1416-1420. [28] B. J. Edwards, An analysis of single and double generator formalisms for the macroscopic description of complex fluids, J. Non-Equilib. Thermodyn. 23 (1998) 301-333. [29] J. J. de Pablo, H. C. Öttinger, An atomistic approach to general equation for the nonequilibrium reversible-irreversible coupling, J. Non-Newtonian Fluid Mech. 96 (2001) 137-162.


22

Figure Captions Figure 1. 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. (14)) as

functions of observation time. Figure 4. Mean and standard deviation of the LIT transport diffusivity via MSD correlation

functions as functions of τmin and τmax. Figure 5. Mean and standard deviation of the LIT transport diffusivity via the velocity

correlation function as functions of τmax. Figure 6. Mean and standard deviation of the LIT transport diffusivity via the MSD and VCF

Methods as functions of τmax. 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.


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. (1). 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 (1) 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. [17-21] 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. (1). Only then can one claim that Eq. (1) 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. (1): some of the principles of LIT that are claimed as being universal are actually

only valid under very special circumstances [17-21]. In summary, we must verify two issues for

the present article: first, we must verify that Eq. (1) 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. (1). 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, )c,c,s(u 21 , where s is the entropy

density and 1c , 2c are the molar concentrations of each component [17]. Alternatively, one can


24

also express the entropy density as a potential function: )c,c,u(s 21 . From Equilibrium

Thermodynamics, it is known that

2211i

iccsTup,

Tcs,

T1

us

µ+µ++−=µ

−=∂∂

=∂∂ , (A.1)

where p is the thermodynamic pressure expressed in terms of density variables u , 1c , 2c , and

functions s , T , 1µ , 2µ [17]. With a properly expressed equilibrium potential function,

)c,c,u(s 21 , 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

∑= ∂

∂∂∂

=∂∂ 3

1i

i

i tX

Xs

ts , (A.2)

where ( )T21 c,c,uX ≡ . In this expression, the first entity on the right-hand side is identified as a

thermodynamic force or affinity, iΛ , and the second as an associated nonequilibrium flux, iJ [17,

22]:

∑=

Λ=∂∂ 3

1iii J

ts . (A.3)

The fluxes appearing in the above expression are functions of all affinities and all intrinsic

system parameters: ( )21321i ,,T,,,J µµΛΛΛ . Expanding each iJ in a power series, and

subsequently neglecting all non-linear terms close to equilibrium, one obtains linear

relationships between the fluxes and affinities,

∑=

Λ′=3

1jjiji LJ , (A.4)


25

where ( )21ij ,,TL µµ′ is the phenomenological coefficient relating the appropriate affinity to the

corresponding flux [17, 22].

For the phenomenological coefficients defined above, it has been shown, with

reservations expressed [17, 18], that the Onsager Reciprocal Relations hold [23, 24]:

jiij α=α . (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

ii Xs

∂∂=Λ , 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 [17]. 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. (1) 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 )c,c,u(s 21 , and thus Eq. (A.1) still applies at every position x. The production rate of


26

entropy density is also still described by Eq. (A.2); however, the flux iJ is no longer expressed

by tXi

∂∂ . Indeed, this latter quantity then takes on the form

ii J

tX

⋅−∇=∂

∂ , (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 iX 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

0JXs

ts 2

1ii

i=⋅∇

∂∂

+∂∂ ∑

= . (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

i

2

1i iJ

Xs

ts

⋅

∂∂

∇=∂∂ ∑

= . (A.8)

One can now recover the form of Eq. (A.3) provided that

∂∂

∇=Λi

i Xs ; (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,


27

j2

1jiji LJ Λ′′= ∑

= , (A.10)

assuming that the fluid is isotropic so that no spatial dependence is included in ijL ′′ .

Equation (A.10) 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 [17-21].

Indeed, up until the late 1990s, 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.10):

jiij LL ′′=′′ . (A.11)

Hence one satisfies the second objective by noting that Eq. (A.11) applies to Eq. (A.10) under

the auspice of Eq. (A.9). For the purposes of the present article, this amounts to using as the

proper equation set

2221212

2121111

LT1L

T1J

LT1L

T1J

µ∇′′−µ∇′′−=

µ∇′′−µ∇′′−= , (A.12)


28

with 2112 LL ′′=′′ and constant temperature.


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. (15) as

αα∞→τα

α

ααα τ

∂∂

= f1limxlnaln

xdNx21D

p,Tb,tran , (B.1)

where

( ) ( )( ) ( ) ( )( )[ ]

( ) ( )( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) ( )( )[ ]∑∑

∑

α

=

α

=

α

=αα

−−τ+−τ+⋅−−τ+−τ+=

−−τ+−τ+=

N

1i

N

1j

*j

*j

*i

*i

2N

1i

*i

*i

trtrtrtrtrtrtrtr

trtrtrtrf

. (B.2)

One can rewrite the self-diffusivity from Eq. (17) as

α∞→τα

α τ= f1lim

dN21D ,self , (B.3)

where

[ ]∑α

=α +−τ+−τ+=

N

1i

2 *i

*i )t(r)t(r)t(r)t(rf . (B.4)

The quantity fα contains only the diagonal (i=j) elements of fαα. We can include all the off-

diagonal elements in another function,

( ) ( )( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) ( )( )[ ]∑∑α

=

α

≠=

αα −−τ+−τ+⋅−−τ+−τ+=N

1i

N

ij1j

*j

*j

*i

*i

* trtrtrtrtrtrtrtrf (B.5)

This new function satisfies the relationship

*fff ααααα += . (B.6)

We can define an “off-diagonal diffusivity” as


30

*,off f1lim

dN21D αα

∞→ταα τ

= . (B.7)

so that we can rewrite the LIT transport diffusivity as

( )ααα

α

βαα +

∂∂

= ,off,selfp,T

,tran DDxlnaln

x1D . (B.8)

If we switch indices, we have an equivalent form of the transport diffusivity,

( )βββ

β

αββ +

∂

∂= ,off,self

p,T,tran DD

xlnaln

x1D . (B.9)

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

βββααβαα =−=−== ,tran,tran,tran,tranLIT DDDDD . (B.10)

We can take a linear combination of Eqs. (B.8) and (B.9) to give a new expression for the

transport diffusivity,

ββαααβ += ,tran,tranLIT DxDxD . (B.11)

Substitution of Eqs. (B.8) and (B.9) into Eq. (B.11) yields

( )ββααα

α +++

∂∂

= ,off,self,off,selfp,T

LIT DDDDxlnaln

D , (B.12)

and further manipulation yields:

( ) ( )

( ) ( )

++++

+++

∂∂

=ββββαααα

βααββααβ

α

α

,off,self,off,self

,off,off,self,self

p,TLIT DxDxDxDx

DxDxDxDx

xlnaln

D . (B.13)


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.1).

( ) ( )βααβα

αββααβα +

∂∂

+++= ,off,offp,T

,tran,tranDarkenLIT DxDxxlnaln

DDxxDD . (B.14)

Using the symmetry in Eq. (B.10), we can rewrite this expression as

( )βααβα

ααββα +

∂∂

+−= ,off,offp,T

,tranDarkenLIT DxDxxlnaln

Dxx2DD . (B.15)

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 αβ,tranD term), all non-self

α-α correlations (in the α,offD term), and all non-self β-β correlations (in the β,offD term). The

thermodynamic factor in front of the off-diagonal elements has been absorbed already into

DarkenD and αβ,tranD . We also see that, because LIT,tran DD −=αβ , the correlations will never

vanish individually. Agreement with the Darken Equation can only come from a system where

the three correlation terms collectively vanish.


32

1.5E+04

2.0E+04

2.5E+04

3.0E+04

3.5E+04

4.0E+04

0.5 0.6 0.7 0.8 0.9 1.0observation time (ns)

mea

n sq

uare

dis

plac

emen

t (Å

2 ) xyz

Figure 1.


33

0.0E+00

5.0E-08

1.0E-07

1.5E-07

2.0E-07

2.5E-07

3.0E-07

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0mole fraction methane

diffu

sivi

ty (m

2 /s)

self-diffusivity (Me)

self-diffusivity (Et)

transport diffusivity (Darken)

Figure 2.


34

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

3.0E+00

3.5E+00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0observation time (ns)

mea

n sq

uare

dis

plac

emen

t (Å

2 )

xyz

0.00

0.03

0 000 0 005 0 0100.0 0.01

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

3.0E+00

3.5E+00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0observation time (ns)

mea

n sq

uare

dis

plac

emen

t (Å

2 )

xyz

0.00

0.03

0 000 0 005 0 0100.0 0.01

Figure 3.


35

0.0E+00

2.0E-08

4.0E-08

6.0E-08

8.0E-08

1.0E-07

1.2E-07

1.4E-07

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0maximum observation time (ns)

diffu

sivi

ty (m

2 /sec

)

standard deviations

means

τmin = 0 ps & 1 ps

τmin = 0 ps & 1 ps

τmin = 10 ps

τmin = 10 ps

Figure 4.


36

-4.0E-08

-2.0E-08

0.0E+00

2.0E-08

4.0E-08

6.0E-08

8.0E-08

1.0E-07

1.2E-07

1.4E-07

1.6E-07


diffu

sivi

ty (m

2 /sec

)

meanstandard deviation

Figure 5.


37

0.0E+00

2.0E-08

4.0E-08

6.0E-08

8.0E-08

1.0E-07

1.2E-07

1.4E-07

1.6E-07


diffu

sivi

ty (m

2 /sec

)

standard deviations

means

MSD

MSD

VCF

VCF

Figure 6.


38

5.0E-08

7.0E-08

9.0E-08

1.1E-07

1.3E-07

1.5E-07

1.7E-07

1.9E-07

2.1E-07

2.3E-07

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0mole fraction methane

diffu

sivi

ty (m

2 /s)

LIT (MSD)LIT (VCF)Darken

Figure 7.

Determination of Statistically Reliable Transport...

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