Non-equilibrium molecular dynamics algorithm for the calculation of ...

MOLECULARPHYSICS, 1991, VOL. 72, NO. 4, 893-898

Non-equilibrium molecular dynamics algorithm for the calculation of thermal diffusion in simple fluid mixtures By D. J. EVANS and P. T. C U M M I N G S t Research School of Chemistry, Australian National University, P.O. Box 4, Canberra, ACT 2601, Australia (Received 2 August 1990; accepted 18 October 1990)

We describe a non-equilibrium molecular dynamics algorithm for thermal conductivity/diffusion in binary simple fluid mixtures. The external field drives both a heat current and a diffusive current permitting calculation of the linear response coefficients LQQ and Llo which are required for the calculation of the thermal conductivity and the thermal diffusion (Sorer effect). The algorithm is momentum-preserving and has the interesting property it does not satisfy the adiabatic incompressibility of phase space. However, it does generate the correct response. 1.

Introduction

An interesting problem in predicting the transport properties of fluids from a molecular viewpoint is computing the mutual diffusion coefficient in fluid mixtures. Both equilibrium molecular dynamics methods [1-4] and non-equilibrium molecular dynamics (NEMD) methods [5, 6] have been developed for this task. In the NEMD case, a corresponding N E M D algorithm has been developed for computing the thermal conductivity [5, 6]. In the two NEMD algorithms (thermal conductivity and mutual diffusion), external fields (FQ and Fn, respectively) drive an energy current density and a diffusive current density (JQ and j l respectively and defined in greater detail below). However, the external field in the thermal conductivity algorithm also drives a non-zero diffusive current density j l , which has the property that j l , __, LIQFQ as

FQ --, 0

(1)

and the external field in the mutual diffusion algorithm also drives a non-zero energy current density J~ which has the property that J~LQ,

Fn

as

Fn--'O.

(2)

The phenomenological crosscoefficients L,Q and L,Q describe, respectively, the thermal diffusion (Soret effect) and diffusion thermoeffect (Dufour effect). Thus, we refer to the NEMD thermal conductivity algorithm as the thermal conductivity/diffusion algorithm and the NEMD mutual diffusion algorithm as the diffusion thermoeffect algorithm. As pointed out by Evans and MacGowan [6], the NEMD algorithm for thermal conductivity/diffusion used by MacGowan and Evans [5] involved the use of inconsistent microscopic definitions of the energy current density vector (see Section 2 below). This inconsistency does not affect the thermal conductivity and the mutual diffusion t Permanent Address: Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, VA 22903-2442, USA. 0026-8976/91 $3.00 9 199! Taylor & Francis Ltd

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D . J . Evans and P. T. Cummings

coefficient. However, it does lead to incorrect values of the phenomenological coefficient L~e. The diffusion thermoeffect algorithm given in [5] is correct so that the value of LQ~ obtained by using equation (2) and the correct definition for the energy current density is exact. The basic problem with the MacGowan-Evans thermal diffusion algorithm [5] is that MacGowan and Evans introduced a new definition of the heat flux vector which differed from both the Irving-Kirkwood and BearmanKirkwood definitions. This means that the relation between the consequently redefined MacGowan-Evans phenomenological transport coefficients and the conventional phenomenological coefficients is nontrivial, especially in relation to L1Q. One possible route to L~Q is via the Onsager reciprocal relation (ORR) [7-9] L1Q = LQI.

(3)

However, it would be desirable to have an independent route to LtQ so that the O R R could be tested by simulation. This requires developing a correct form of the N E M D thermal conductivity/diffusion algorithm. This is the subject of this paper. In Section 2, we define the important quantities (such as the various forms of the energy current density) and write down the N E M D thermal conductivity/diffusion algorithm. Several important properties of the algorithm are described. We give our conclusions in Section 3. 2.

Thermal conductivity/diffusion algorithm

We begin by introducing the notation adopted in this paper. For the binary mixtures under consideration here, the species index for any quantity is indicated by a superscripted Greek letter. The number of the molecule is indexed by a subscripted Roman letter. The number of species v molecules is N v, v = 1, 2. The total number of molecules is 2

N=

~ N v. v=l

The mass of molecule i of species v is m~ = m v since all species v molecules have the same mass. The position of molecule i of species v is q~ and its momentum is p~. Thus, the centre of mass velocity u is given by

u = ~. ~.p,/~, N~m," = ~. N~,,/ ~. N'rn'. v=l

i~l

i=1

(4)

vfl

~ffil i = l

It is useful to define the tensor quantity S~ by S~ =

~ m"

\ my

-- u

I +

~*(~b~l . . . . . ~=t j=l 7,j -,j ,~,

(5)

where * indicates that the sum does not include the term j = i, # = v, I is the unit tensor (I U = 6q, where 6ij is the Kronecker delta), q~,~and F~~ are respectively the pair potential and force between molecule i of species v and molecule j of species #, and q~ = q~ - q~. We define the related tensor quantity $ by 2

S

=

Nv

~, E Sr v=l

(6)

i--I

The field-free Hamiltonian, H0, is given by Ho = v=l

~ p • p~ 1 2m v + -2 v = ] i~" =1

i=l

, /~=1

.,.

~..

, (Iqq l)

(7)

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N E M D algorithm f o r thermal conductivity

In terms of these quantities, the Irving-Kirkwood energy current density is given by JQ V

=

L v=l i=1

~kmy

-- "

~ Si

(8)

where V is the volume of the system. We can define the diffusive current density J~ of species v by Nv

J~V

=

~ p~ -

N~m~u

=

N~m~(u ~ -

(9)

u),

i=l

where 1 NV N V m v ~ v~

uv -

(lO)

i=l

is the center of mass velocity of species v molecules. An alternative definition of the energy current density is given by Bearman and Kirkwood [10] as 2

j3K

=

jQ _

~

~V[hV + 89

_

.)2],

( l l)

v=l

where h v is the specific enthalpy of species v in the co-moving frame. Since there is no known microscopic expression for h v, it is essentially impossible to develop an algorithm to drive this energy current density. MacGowan and Evans [5] proposed a third energy current density ~ME V

=

i

v=l i=l

\ m ~ -- "~

" my

--

I +

Z

#=l j=l

(~,'~l

.,~ _,j

,j (12)

This amounts to replacing u in the Irving-Kirkwood tensor by u v. The advantage of JoME is that a momentum-preserving N E M D algorithm satisfying adiabatic incompressibility of phase space (see below) could be constructed whose adiabatic dissipative flux was v J ~ E ' F Q . As pointed out by Evans and MacGowan [6], unfortunately, this definition of the heat flux is not free of all diffusive contributions to the energy flux and therefore leads to phenomenological transport coefficients which are different from those to Bearman and Kirkwood. In this paper, we introduce an N E M D algorithm for thermal conductivity[ diffusion in binary mixtures. It works with the experimentally more accessible IrvingKirkwood definition of the heat flux. However, consistent with the findings of MacGowan and Evans [5], it does not appear possible to simultaneously satisfy the requisite periodic boundary conditions and momentum conservation required for computer simulations and the condition known as the adiabatic incompressibility of phase space (AIF). Our new algorithm violates AIF. That is, the quantity A ~ 0 where A is defined by 0 a = ~-~./"

(13)

In this equation,/" is the phase space vector containing the positions and momenta of all the molecules. In general, one of the goals in developing N E M D algorithms has been satisfaction of AIF. This is not a necessary condition for the algorithm to be useful. For example, the N E M D algorithm for thermal conductivity in pure fluids

896

D . J . Evans and P. T. Cummings

developed by Gillan and Dixon [11] does not satisfy AIF but it does obey the response theorem of MacGowan and Evans [12] - VJ" F e =

(14)

I:Io-- k B T A

except for terms of order 1IN. In fact, the violation of AIF by the Gillan and Dixon algorithm was only marginal (i.e. of order I/N). It also marginally violates momentum conservation. (Note that equation (14) differs from the corresponding equation in MacGowan and Evans [5] by a minus sign. The form of the response theorem given in equation (14) is consistent with other publications concerning the response theorem by Evans and co-workers.) In equation (14), Fe is the external field applied to the system in order to drive the current density J, ka is Boltzmann's constant, T is the absolute temperature and H0 = dHo/dt is the time derivative of the field-free Hamiltonian (i.e., the energy of the system without any external field terms). Satisfaction of equation (14) means that the algorithm has the correct response. The algorithm given below will be shown to be momentum preserving and to satisfy equation (14) with the correct Irving- Kirkwood energy current density. We assign a colour cv to each molecule of species v, v = 1, 2 such that the colour constraint 2

Z NVcV =

0

(t5)

v=l

is satisfied. Then the equations of motion we propose are given by dq~

p~

dt

m v

dt

-

F[ +

(16)

S~--~S + ~S +

kaTcVl

" F e,

(17)

where F / i s the force on molecule i of species v, 2

N~~

#=1 j=l

and where FQ is the external field for driving the heat flux. We now describe several properties of this algorithm. First, the algorithm is momentum conserving since

v=l

NVdPY E =o.

i=1

This follows straightforwardly from the definition of the $ tensors and the colour constraint, equation (15). Hence, once the simulation is initialized in the centre of mass frame (the frame in which u = 0) it remains in the center of mass frame. Second, the simulation obeys the response theorem of MacGowan and Evans, equation (14). To see this, we first compute the time derivative of the field free Hamiltonian, equation (7), given by dH~ dt

~" ~ ~=~ i=l

~$

+ ~S

+ k.TcVl

" Fe

(19)

897

N E M D algorithm for thermal conductivity

In the centre of mass frame, = . Qv v=l i = l - ~

and Nv

v

=

E ,,': i=1

An additional, general property of the diffusive current densities is that $~ + j2 = 0. Using these three results, equation (19) can be written as dHo dt -

[

1(1 IQ-

1)j

I

l(c I m'

m'

~2)J']V'F

+k,T(~

C2)jl. S

0.

(20)

We impose a further condition on the colour charges, viz. ~

1

1

ml

m2

--

cI

c2

ml

mX.

(21)

This, combined with the color constraint, equation (15), leads to the following values for c ~ and cz cI =

!amlm2N2 ml N~ + m2 N2 ,

c2 =

#m~m2N~ miNi + m2 N2 .

(22)

Using this equation, equation (20) becomes dn0 = dt

[JQ + #k, T J ' ] V . FQ.

(23)

The next step is to compute A given by equation (13). It is straightforward to show that a

=

~tdlV

9

Fo_ + O(1).

(24)

Finally, using equations (23) and (24), we find that

dH0 dt

kB TA = JQ V . FQ,

(25)

which we see agrees with equation (14) when J is identified with - J Q and Fe with Fe. Hence we see that the algorithm defined by equations (16) and (17) satisfies the response theorem. 3.

Conclusions

In this paper, we have introduced a new NEMD algorithm for thermal conductivity/diffusion which is momentum preserving and satisfies the response theorem of MacGowan and Evans for systems in which AIF does not hold. This is the first practical algorithm which violates AIF to order N and order Fe. In view of

898

D . J . Evans and P. T. Cummings

the importance of phase space volume in the theory of dissipative dynamical systems [13], it will be interesting to study the ramifications of this development. Combined with the diffusion thermoeffect algorithm given in [5], this algorithm permits the calculation of the thermal conductivity and mutual diffusion coefficient in binary mixtures, and also permits independent calculation of the crosscoefficients Lie and Lel. Simulations employing these algorithms are presently underway. The extension of the algorithm to multicomponent mixtures is currently under investigation. P.T.C. acknowledges the financial support of this research by the National Science Foundation and ICI Films (Industry/University Collaborative Research Grant CBT-8801213) and Comalco Research of Australia and the support of the National Science Foundation through the US/Australia Cooperative Science Program (Grant INT-8913457) for the provision of travel funds to visit the Australian National University where this research was performed. P.T.C. is also indebted to the Department of Chemical Engineering at the University of Massachusetts for its support during the final stages of this research through its Distinguished Visiting Scholar Program. References [1] JACUCCI,G., and MCDONALD,I. R., 1975, Physica A, 80, 607. [2] JOLLY,D. L., and BEARMAN,R. J., 1980, Molec. Phys., 41, 137. [3] SCHOEN,M., and HOHEISEL,C., 1984, Molec. Phys., 52, 33. [4] SCHOEN,M., and HOHEISEL,C., 1984, Molec. Phys., 52, 1042. [5] MAcGOWAN,D., and EVANS,D. J., 1986, Phys. Rev. A, 34, 2133. [6] EVANS,D. J., and MAcGOwAN,D. M., 1987, Phys. Rev. A, 36, 948. [7] DEGROOT,S. R., and MAZUR,P., 1984, Nonequilibrium Thermodynamics (Dover). [8] FITTS,D. D., 1962, Nonequilibrium Thermodynamics (McGraw-Hill), [9] HANLEY,H. J. M., editor, 1969, Transport Phenomena in Fluids (Dekker). [10] BEARMAN,R. J., and KIRKWOOD,J. G., 1958, J. chem. Phys., 28, 136. [ll] GILLAN,M. J., and DIXON, M., 1983, J. Phys. C, 16, 869. [12] MAcGOwAN,D., and EVANS,D. J., 1986, Phys. Lett. A, ll7, 414. [13] SCHUSTER,H. G., 1989, Deterministic Chaos: An Introduction, second edition (V. H. H. Verlgsgesellschaft, Weinheim).