On the molecular mechanism of thermal diffusion in liquids

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On the molecular mechanism ofthermal diffusion in liquidsBjørn Hafskjold a , Tamio Ikeshoji b & Signe Kjelstrup Ratkjea

a Department of Physical Chemistry, University ofTrondheim-Norwegian Institute of Technology, N-7034,Trondheim, Norwayb National Institute for Advanced Interdisciplinary Research,Higashi 1-1-4, Tsukuba, Ibaraki, 305, JapanPublished online: 23 Aug 2006.

To cite this article: Bjørn Hafskjold , Tamio Ikeshoji & Signe Kjelstrup Ratkje (1993) On themolecular mechanism of thermal diffusion in liquids, Molecular Physics: An InternationalJournal at the Interface Between Chemistry and Physics, 80:6, 1389-1412

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MOLECULAR PHYSICS, 1993, VOL. 80, No. 6, 1389-1412

On the molecular mechanism of thermal diffusion in liquids

By BJORN H A F S K J O L D t , T A M I O IKESHOJI~ , and S IGNE K J E L S T R U P R A T K J E t

# Department of Physical Chemistry, University of Trondheim-Norwegian Institute of Technology, N-7034 Trondheim, Norway

:~ National Institute for Advanced Interdisciplinary Research, Higashi 1-1-4, Tsukuba, Ibaraki 305, Japan

(Received 31 March 1993; accepted 21 April 1993)

A recently developed non-equilibrium molecular dynamics algorithm for heat conduction is used to compute the thermal conductivity, thermal diffusion factor, and heat of transfer in binary Lennard-Jones mixtures. An internal energy flux is established with local source and sink terms for kinetic energy.

Simulations of isotope mixtures covering a range of densities and mass ratios show that the lighter component prefers the hot side of the system at stationary state. This implies a positive thermal diffusion factor in the definition we have adopted here. The molecular basis for the Soret effect is studied by analysing the energy flux through the system. In all cases we found that there is a difference in the relative contributions when we compare the hot and cold sides of the system. The contribution from the lighter component is predominantly flux of kinetic energy, and this contribution increases from the cold to the hot side. The contribution from the heavier component is predominantly energy transfer through molecular interactions, and it increases from the hot to the cold side. This explains why the thermal diffusion factor is positive; heat is conducted more effectively through the system if the lighter component is enriched at the hot side. Even for very large heat fluxes, we find a linear or almost linear temperature profile through the system, and a constant thermal conductivity. The entropy production per unit volume and unit time increases from the hot to the cold side.

1. Introduction

Mass transfer by thermal diffusion (the Soret effect) in liquid mixtures is norm- ally a small effect compared with ordinary mass diffusion. The Soret effect may be quantified by the thermal diffusion factor, which is the relative concentration gradient divided by the relative temperature gradient. Typical values for the thermal diffusion factor range from 0"3 to 1.0 for aqueous solutions of sugars [1]. Similar values have been reported for mixtures of hydrogen in noble gases [2] and hydrocarbon mixtures [3, 4]. Larger effects have been demonstrated theoretically for components with large mass differences [5], which may explain why hot surfaces in air heaters stay clean from dust particles [6]. The thermal diffusion factor tends to infinity at the mixture's critical point, and values between l0 and 100 have been measured in the critical region [7]. The Soret effect may be important in separation technology even in cases where it is small. One early application of the effect was the separation of isotopes [8].

F rom a fundamental point of view, perhaps the most interesting property of the thermal diffusion factor is its sign. The sign has been found to depend on the

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1390 B. Hafskjold et al.

composition for some binary mixtures [9]. Kinetic theory predicts that the sign of the thermal diffusion factor depends on the fluid density, temperature, and component mass ratio in addition to the pair potential parameters of the system [5]. The reason why and how these parameters affect the sign of the thermal diffusion factor is not understood. If it were, it could lead to improved separation technologies for binary mixtures, e.g., by changing external conditions of a system, and new separation techniques might be developed.

The size of the thermal diffusion factor may be important in electrolysis. When concentration gradients develop during electrolysis, the coupling between heat and mass transport leads to a temperature gradient. This is the Dufour effect, which may be seen as the reciprocal of the Soret effect [10]. It is superimposed on the Peltier effect. The contribution from the Dufour effect to the local heat production at the electrodes may be substantial [11].

General knowledge of the coupling between heat and mass transport may thus become important for the engineering design of industrial processes [12], as concluded also by Ratkje and coworkers [13, 14].

Molecular dynamics (MD) simulation has proved to be a powerful tool in studies of transport mechanisms in liquids (see e.g., Haile [15]). Several MD algorithms have been developed and most of the recent work employs non-equilibrium MD (NEMD) [16, 17]. Specific algorithms have also been developed for coupled heat and mass transport [18-21]. These studies have focused on the method developments, and the present work and a recent paper by Kincaid et al. [22] represent the first systematic studies covering a range of parameter values. One of the strong features of NEMD is the metod's ability to determine a system's transport properties at the molecular scale. From the molecular transport properties, phenomenological coefficients as given by irreversible thermodynamics can be computed. This can give information about the relationships between the phenomenological coefficients and molecular properties, which a thermodynamic experiment alone cannot give. This relationship is our primary interest; the main objective of this paper is to contribute to the answer of the fundamental question: what is the cause of thermal diffusion?

A second objective of the present work is related to methodological questions. We will discuss some formal relations between the macroscopic and microscopic descriptions of heat transport in mixtures in the theoretical part of the paper, section 2. The concept of mass transfer is clear in the macroscopic as well as the microscopic level because it includes only mechanical, absolute quantities. The concept of heat transport in mixtures is less clear because (1) several alternative sets of flux-force relationships may be defined (see e.g., de Groot and Mazur [23]), (2) heat may be operationally defined at the macroscopic level, but is not easily given by a mechanical analogue at the microscopic level, and (3) heat and energy are relative quantities, which depend on a chosen reference state. The choice of a frame of reference is required for a complete specification of the fluxes and forces used. We present in section 2 the expressions for the macroscopic and microscopic heat and energy fluxes, respectively, and show how they are related.

In the calculations we employ a newly developed NEMD algorithm [24] to coupled heat and mass transport in a binary Lennard-Jones mixture. The algorithm, which is briefly described in section 3, represents an experimental situation with a heat source and sink to generate a flux of internal energy and heat through the system. The algorithm is therefore called the heat exchange algorithm.

In section 4, we analyse the internal energy flux in terms of the relative contri-

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Thermal diffusion in liquids 1391

butions from each component and the flux of kinetic energy, potential energy, and the flux due to energy transfer between the molecules. The energy flux induces a mass flux and a concentration gradient in the mixture. Under the constraint of a constant energy flux, a stationary state with zero mass flux is reached, and a balance between the temperature gradient and the gradient in chemical potential is established. In order to build systematic knowledge we shall start our work with isotope mixtures and thereby eliminate the effects on the results of different molecular sizes and potential depths. At stationary state, the thermal diffusion factor and the thermal conductivity can be computed, but not the separate Soret and Dufour effects. The relative contributions to the internal energy flux can also be computed, and they give insight into the mechanism of heat transport, as discussed in the final parts of the paper.

Results for a simulated Kr /Ar mixture will be used for a comparison with previous work, and we discuss the features of thermal diffusion in some real mixtures on the basis of the isotope study. Predictions for general cases are attempted. In future work we aim to eliminate some of the restrictions introduced here.

2. Heat and mass flux in a binary mixture

As a starting point for our discussion, we have chosen the entropy production per unit volume and unit time a [23]:

= - : u . V , ( l ) k = l

which is a sum of products of conjugate fluxes and forces. The fluxes are the internal energy flux Ju and the mass flux Jk of each species k. In the expressions for the forces, V T is the temperature gradient, ~k is the partial specific Gibbs energy of component k, and F k is an external force acting on species k (per unit mass). Several sets of fluxes and forces give the same result for tr. The variables within one set should be independent in order to have the Onsager reciprocal relations fulfilled, which is why we have eliminated one mass flux (Jn) by making use of E" k=lJk = 0 and introduced the difference # k - #~. Neither irreversible thermodynamics nor N E M D simulations give other prescriptions for how the sets shall be chosen, but the experimental or computat ional conditions cause some sets to be more convenient than others. It is not even clear if one can distinguish between fluxes and forces on formal grounds in irreversible thermodynamics without some support from a molecular theory [16]. The derivation of equation (1) includes the usual concepts of fluxes and forces on phenomenological grounds, however, and we shall adopt these in our N E M D work.

The entropy production is independent of the frame of reference. The internal energy flux J v and the mass flux "]k used here refer to a barycentric frame of reference. Some of the contributions to the internal energy flux depend on a reference state for the potential energy and, in our N E M D simulations, we have chosen this to be particles infinitely distant apart (see equation (4) below). At the thermodynamic level, each #k depends on a chosen reference state, typically pure component k. I f the same reference state is chosen for all components, e.g., pure component for n and infinite dilution of k in n for all the other components, the reference state contributions to #k -- #~ would cancel.

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The heat flux is related to the internal energy flux by n-1

Jq = JU -- Z ( h k -- hn)Jk, (2) k=l

where hk is the partial specific enthalpy. For an ideM mixture, hk is independent of the composition and equal to the specific enthalpy of pure component k. In this paper, we shall consider an isotope mixture with molecular masses rnk. We then have hk = rnn hn/rnk and, therefore,

n-l (mn--k~kk 1)Jk. Jq = ~U -- hn ~ (3)

Furthermore, at the stationary state we consider in this work (Jk = 0) we obtain as a general result Jv = Jq. Whereas Jv in general depends on a reference state for the potential energy or #k, Jq does not. Moreover, in the absence of viscous forces, Jq is independent of the frame of reference [25].

Different definitions of the fluxes and forces have led to inconsistent terminology in the literature, de Groot and Mazur [23] use the term 'heat flux' and the symbol Jq for the flux that we here call Ju, and the symbol J'q, which they call 'reduced heat flux', for our Jq. Haase [25] uses 'heat current density' and 'energy flux density' and the symbol J~ for our Jv and 'heat current density' and the symbol JQ for our Jq. Forland et al. [10] use 'total heat flux' and the symbol J~ for our Ju and 'measurable heat flux' and the symbol Jq for our Jq. Pierleoni and Ciccotti [26] use the term 'heat current' for Jq at the macroscopic level and the same term for Jr: at the microscopic level. We shall use the term 'internal energy flux' for Ju because it is a flux of the internal energy, and 'heat flux' for Jq. Note that Jv is defined through the derivation that leads to equation (1) [23] and Jq is defined by a transformation of the conjugate fluxes and forces (equation (2)).

The flux-force relationships cannot be established directly at the molecular level. The fluxes and forces may be computed, however, and relations established indirectly. The energy flux in a control volume (CV) is given in MD as [16]

Jv-----pi~cv [ lmi (v i - v )2+~bi] (v i - v ) -~ j~_ l [ (v i - v ) .F i j ] r i j , (4)

where V is the size of the control volume, m i and vi are the mass and velocity, respectively, of particle i, v is the barycentric velocity of the system, ~bi is the potential energy of particle i in the field of all the other particles, Fij is the force acting on i due to j, and rij is the vector from the position of i to the position o f j . This is a local, instantaneous analogue to the macroscopic internal energy flux. We assume that it relates to the macroscopic flux in the same way as other statistical mechanical quantities relate to the thermodynamic properties. In particular, we assume that the thermodynamic limit is adequately represented by a relatively small-sized system.

The presence of the quantity ~bi in equation (4) shows that the internal energy flux depends on a reference state, which in this case is infinite volume (all particles infinitely distant apart). If we shift the reference energy by an amount q~0, say, this would lead to an additional term

J g = ~ Z ~3i (5) i c C V

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in the energy flux. At a stationary state with Jk = 0, the time average of the sum vanishes, and Jv becomes independent of the reference state, as it should.

Equation (4) suggests that the internal energy flux may be interpreted in terms of three contributions. The first and second terms are the kinetic energy and potential energy, respectively, carried by a moving particle. The third term is the energy transferred due to motion of a particle in the field of the other particles. The sign in front of the third term is a consequence of the sign conventions for rij and FU; energy is transferred from i to j if i moves towards j against a repulsive force from j. For a hard sphere system, this term would be non-zero only when particles collide, and it is therefore often referred to as a collision term. We shall use the name interrnolecular energy transfer for the third term, to include energy transfer by soft interactions.

The potential energy flux and the intermolecular energy transfer are not sym- metric in i a n d j because vi is generally different from vj. The factor 1/2 is therefore n o r a symmetry factor as in a total energy summation over all particle pairs. Equation (4) must be used with care because it is not generally valid if the control volume is small compared with the range of the force. To see this, consider three layers and let i and j occupy layers 1 and 2, respectively, with the third layer in between. We assume that i and j are within the range of the force. The energy transfer from i to j will then pass through layer 3, and thus contribute to the energy flux in that layer. We must therefore assign a certain part of the energy transfer to each layer, corresponding to the distance of the transfer within that layer. This will modify the factor 1/2. At a stationary state, and if the cross-sectional area to the flux is constant, we find that the geometrical factor is 1/2 on average. Only in this case can equation (4) be used as its stands.

The mass flux of component k is

1 Jk = --~ iec~vukmi('oi__ -- V). (6)

The partial specific enthalpy may be computed by MD via a series of equilibrium computations at different compositions or by a perturbational MD simulation [27].

The temperature gradient is in N E M D given by computations of the local temperature according to

1 N 3 N k B T = "2 i~v_v, mi(vi - v) 2, (7)

where N is the number of particles in the control volume and kB is Boltzmann's constant.

We consider now a binary mixture of neutral, spherical molecules with no viscous flow. The external force is the gravity or some other force that does not discriminate between the components, so that F k equals F for all k. The internal energy flux and mass flux that appear in equation (1) are then [23]:

V T Ju = -Luu--T- T - L u I V ( ~ - ~ - ~ )

L , , V T L V { # l - # 2 ' x J' = - k - - - T - ) '

(8 a)

(8b)

where the L's are phenomenological coefficients. This 'energy set' is most convenient in N E M D simulations because both fluxes are purely mechanical. The alternative

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'heat set' defined by equation (9),

V T 1 Jq = - L q q T2 Lql - ~ V T ( # I -- #2) (9 a)

L V T L 1 J1 = - l q ~ - - - 11 "~VT( ~I -- #2), (9b)

is most convenient under experimental conditions, because the heat change is directly measurable [10]. Relationships between the various sets can be found in textbooks by de Groot and Mazur [23], Forland et al. [10], and others. In equation (9), the subscript T on the gradient denotes isothermal conditions. The coefficients L11 are identical in equations (8b) and (9b). The mass flux Jl vanishes at stationary state, which makes Ju = Jq. If there is a pressure gradient in the system that for some reason does not lead to viscous flow, it will contribute only to the chemical potential gradient.

We shall explore systematically these flux-force relationships for a binary iso- topic mixture. The particle diameters are then equal for the two components, and so are the potential depths; only the particle masses are different.

The thermal conductivity for a binary mixture may be defined in several ways [23]; we shall use the definition at J1 = 0 derived by solving equation (9b) for VT(#l -- #2) and substituting this into (9a):

--~1 (Lqq Lq lL lq~ . (10) AJI=0 = _ _ L11 f

This definition will be used to analyse our molecular dynamics results. In a binary system at uniform pressure, the Gibbs-Duhem equation leads to

V : r ( # 1 - # 2 ) = 1 0 # l V w l (11) w2 Owl

where wk is the weight fraction of component k. For an ideal mixture, such as a mixture of isotopes, the gradient in the partial specific Gibbs energy at zero pressure gradient simplifies to

kBT V T ( # l -- #2) = WlW2(mlw 2 q_ m2wl ) Vwl. (12)

This makes the isotope system a convenient object for NEMD simulations because it avoids the Gibbs energy altogether.

The mass flux in equation (9b) may be expressed alternatively as [23]

J1 : - p w l W 2 D T V T - pDVwl, (13)

where p is the mass density, D T is the thermal diffusion coefficient,

t l q (14) D T -- p w l w 2 T 2 '

and D is the diffusion coefficient,

D- - Lll 0#l (15) pw2TOwl

The thermal diffusion and diffusion coefficients relate to the particular choice of frame of reference and thermodynamic forces that we have used: the barycentric frame of reference and the forces given as gradients in equation (9).

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The thermal diffusion factor o12 is defined at a stationary state as

- V l n T ]J,=0 wlw=\VTJ*,=o' (16)

which is readily computed by NEMD. The ratio DT/D is the Soret coefficient. The thermal diffusion factor is independent of the frame of reference.

The heat of transfer, defined by

�9 = ,

q \JlJvr=o

is related to the thermal diffusion factor by

q* Lq I 0~1 = L11 = WIOtl2Ow1

(17)

(18)

if we assume that the Onsager reciprocal relation Lql : Llq is valid. For an ideal mixture, this reduces to

q, kBT (19) -- Ot12" ml w 2 -}- m2w 1

Our interest in the heat of transfer is related to the question of macroscopic heat transport; it will tell us the relative contribution to the heat flux from the mass flux according to

Jq : -Aj ,=0VT + q*Jl. (20)

3. Details of the system and the computations

3.1. Pair potential and reduced variables

A binary mixture of isotopes was modelled by a 'switched' Lennard-Jones potential,

�9 r [o-ij\12 / O ' i j \ 6 ] uij(r ) : I-~{ij[~V) --~--~) ]Sij(r), (21)

where So(r ) is a switching function used to truncate the potential in a smooth way around a cut-off distance rc/aO, which was arbitrarily set equal to 2"0. We used the following switching function:

1 Si j ( r ) - / r ~,24 , (22)

1+ { }-- krcy

where the exponent 24 was chosen for computational convenience. The various reduced quantities are defined in table 1. In the computer code, all

variables are used in 'MD units' (denoted by superscript ~ ). All results reported here are given in 'Lennard-Jones units' (denoted by *). The conversion between MD units and Lennard-Jones units follows from the two basic relations,

N 3 NL3x {a.l'~3 pC p* = -V ~ - L_~L>,L~ / l \- t7x J ~--- Pt2(~ ) 3 _ (Lx) 3 (23)

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1396 B. Hafsk jo ld et al.

Table 1. Definitions of reduced quantities in Lennard-Jones and MD units.

Definition in Quantity Lennard-Jones units Definition in MD units

. N 3 p[] N 3 Density P = "~f f l l = ~ L x = p* (Lx) 3

Temperature T* -- kBT T u _ k,. . .T_ T* ~11 s

Pressure P* = P a~l P[] = P L~ = P*(L~) 3 ~11 s

m m Mass rn* = - - m [ ] - - m *

ml ml

T i m e t*=' (s 1/2 tu t ((11"~ 1/2 t*

r r r* Length r* = - - r [] . . . .

O'll Lx L*x

Veloc i t y v* v(ml~ 1/2 V [] = V ( / ' n l~ 1/2 \ e l l / \ e l l , /

u u Energy U* = - U • = - = U*

s s

Energy flux J~r = Ju m' ~ 1/2 3 (m"~ '/2 e l l \ s J ~ = J U \e l l . ] = Ju(L~)

Mass flux Jk* J k ~ k e n / ml \e l l / . . . . . . Js (Lx)

(ml'~ 1/2 )~[] = )~ L2x (m._~_l'~l/2= .~*(L*) 2 Thermal conductivity A* = .-~ kB \e l 1 / kB kq 1 ,]

Heat of transfer q** = q, ml q, [] = q, ml = q** s g l l

Entropy production a = OBk-- \e l l f = a k-- \ e l l /

and

T* - ka T _ T n. (24) s

By this defini t ion, Lx ~ = 1 and rr~l = 1. The t h e r m o d y n a m i c states and po ten t i a l pa rame te r s used in this work are l isted

in table 2. Unless otherwise s ta ted in table 2, the molecu la r d iamete r ra t io and the po ten t ia l dep th ra t io were uni ty and the overal l system t empera tu re was T* = 2.0. This is a supercr i t ical s tate for the i so tope mixture . The densi ty, t empera tu re , and mass ra t io for the K r / A r mix ture (runs 18 and 19) were set in acco rdance with previous studies o f this mixture [18,20,21]. This is a l iquid state ju s t above the tr iple point , co r r e spond ing to T = 116 K.

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Table 2. Parameter values used in this study. The mixture was equimolar in all cases. Unless otherwise stated, the molecular diameter ratio and the potential depth ratio were 1.0, the overall system temperature was 2.0, the number of particles was 500, and the MD cell was cubic.

Run No. p* m l/m2 J~j Comments

1 0.4 2 0-108

2 0-4 5 0-108

3 0.4 10 0-108

4 0.6 2 0-282

5 0.6 5 0.282

6 0.6 10 0.282

7 0.8 2 0.342

8 0-8 5 0-342

9 0.8 10 0.342

10 0.4 10 0

11 0.4 10 0.215

12 0.4 10 0-323

13 0.4 10 0-431

14 0-4 1 0.108

15 0.4 10 0-3

16 0.4 1 0.3

17 0.4 10 0.3

18 0.8671 2.098 0.361

19 0.8671 2.098 0.361

20 0'4 10 0"3

N = 1000 L / L x = Lz/Lx = 0 .5

N = 1000 L , / / . ~ = C z / L x = 0-5

N = 4000

N = 4000

T* = 0"692, olt/o22 = 1"067 s ~-- 1"394,

1 500 000 time steps T* = 0'692, erl~/o22 = 1 "067

Ell/~22 = 1"394, r c = 2"5~ij With Lennard-Jones/spline potential

The system consists o f N par t ic les in a box o f vo lume V = LxLyLz with per iod ic b o u n d a r y condi t ions . W e used N = 500 unless otherwise s ta ted in table 2. There were N/2 part ic les o f each o f the two species. In mos t cases, the box was cubic, but runs 14 and 15 were made with Ly/Lx = L: /Lx = 0.5 in o rde r to examine the wall effects descr ibed below. The system was d iv ided into 32 layers o f equal thickness and pe rpend icu la r to the x axis, to enable c o m p u t a t i o n s o f the local densi ty, t empera ture , compos i t ion , energy flux, energy, and o ther quant i t ies o f interest .

3.2. The heat exchange algorithm

The energy flux th rough the system was genera ted as follows: One layer at each end o f the box ( label led H in figure 1) was used for the add i t i on o f kinetic energy to the system. A region o f two layers in the centre o f the box, label led C in figure 1, was used for the remova l o f energy. As a consequence, the system deve loped hot and cold regions, separa ted by middle regions ( label led M in figure 1) with a t empera tu re

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gradient and an internal energy flux. At each time step, a specified amount of kinetic energy A U was added to the hot region by scaling and shifting the velocity of each particle. The shifting was done in order to conserve zero momentum for the total system. If the velocity of particle i before energy addition was v[, then the velocity after energy addition was

v i = (1 + a)v; + O, (25)

where

[t-- a E m i v ; (26)

E mi

The summation is over all the particles in the hot region. A specified value of A U leads to a quadratic equation in a :

1 ~2 = - - - - 'Oi ]. AU 2 ~ mi[((1 + a ) v [ + / 3 ) 2 (27)

All particles in the hot region

The same procedure was used to scale and shift the velocities in the cold region, except that the specified amount of energy was removed at each time step.

If the hot (cold) region contains less than two particles at a given time, or if all the particles have identical velocities, the velocity scaling with the zero momentum

Figure 1.

C M H M M ~/~ .~ C

|

+a

!

Unit cell

X

Layout of the cubic system showing the periodic boundary conditions and the hot (H), cold (C), and middle (M) regions.

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requirement cannot be fulfilled, and the procedure will not be valid. This was not observed in any of the runs reported here, however.

Instead of imposing an energy flux on the system, we could equally well have thermostatted the hot and cold layers. This would fix the temperature gradient and lead to a stationary state similar to that discussed in this paper.

At a stationary state, the x component of the internal energy flux must equal half the rate at which energy is added and removed in the hot and cold regions, respectively,

~'sign (x )AU } Yv = I, 2-~tL---~ ' 0, 0 (28)

where sign (x) is + for 0 < x < Lx/2 and - for Lx/2 < x < Lx, and 8t is the time step. This was used as a test of internal consistency of the algorithm.

The barycentric velocity of the total system is zero if the numerical algorithm conserves the initial zero momentum. Density fluctuations may, however, cause the barycentric velocity of a local control volume to be non-zero, and we need to determine the basis for the time and space average for v in equations (4-7). We also need to interpret Tas defined by local, mechanical quantities in equation (7). We have in this work taken v to be zero in each control volume on the basis of computed average values during long runs and the fact that we work at a stationary state. This implies that T is interpreted as the thermodynamic temperature.

3.3. The computations

Most of the computat ions were made on IBM RS6000/320H, 350, and 550 workstations. Preliminary computat ions were also made on a Cray X-MP. The 500 particle code ran at 8 time steps per CPU second on the IBM 350, and the 1000 particle code ran at 4 times steps per CPU second.

Each run included at least 1 000 000 time steps of length 6t* = 0-002, starting from a randomized configuration or a previous run. The first 200000 time steps were discarded due to their transient behaviour. A temperature drift was usually observed at the beginning of each run. I f the drift exceeded a certain limit during the first 10 000 time steps, or if the initial temperature was not as specified, the velocities were scaled (subject to conservation of total momentum) to the set temperature. The internal energy flux was computed at each 20th time step, and subaverages were taken over 500 instantaneous flux calculations, i.e., at 10 000 time step intervals, and dumped to file for later analysis.

After some preliminary runs with the leap-frog algorithm, it was abandoned due to a substantial drift in the total energy (11% over 1 000 000 time steps). The Verlet velocity algorithm [28] was used for the production runs. The energy conservation was then found to depend on the internal energy flux. A comparison for 5 runs at p* --- 0-4 and ml/m 2 = 10 (runs 3 and 10-13) showed that the highest internal energy flux gave a 4% reduction in total energy over 1 000 000 time steps, which was the worst case. This is 1/5 of the energy drift that MacGowan and Evans [ 18] considered acceptable in their work.

4. Results and discussion

4.1. The method

The internal energy flux was computed from equation (4) for each layer as well as

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Ho~

0 . 5 0

x 0 . 2 5

> ,

L.,

r- 0 .00

0 c

Q .4- c - 0 . 2 5

C o l d

~ T r

Hof

I T m

.o x - d l r e c t l o n �9 y - d l r e c t l o n

- 0 . 5 0 ~ - - • 0 . 0 0 .2 0 . 4 0 .6 0 . 8 1 . 0

x/L

Figure 2. The internal energy flux in x and y directions for run 13 (p* = 0-4, T* = 2-0, ml/m2 = 10.0, and J~ = 0.431), computed with equation (4) in each layer. The x component is positive in the left half of the cell, and negative in the right half due to the symmetry. The x and y components of the macroscopic internal energy flux, computed from equation (26), are shown as the solid lines.

f rom equat ion (28) for the middle regions. At a s tat ionary state, the two methods should give identical results. A compar i son is shown for run 13 in figure 2. Each data point represents an average over 40 000 instantaneous flux computa t ions for each layer, which contained on average 15.6 particles.

The x componen t o f the local internal energy flux computed f rom equat ion (4) agrees well with the imposed flux, except for the underest imation o f the flux near the

Figure 3.

Hot Cold

0 . 5 i i i i J u e Run n o .

0 . 4 3 1 13 0 . 4

0.3 0 . 3 2 3 12

0 . 2 0 . 2 1 5 1 1

5,

0.1 0 . 1 0 8 3

0 . 0 0 . 0 0 . 1 0 . 2 0 . 5 0 .4 . 0 . 5

x / L

The internal energy flux for four different values of J~, shown as function of x D. The flux is an average of the data for the two half cells.

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3 . 0

2 . 5

2 . 0

.=

~* 1.5 g D.

E 0

~" 1 .0

0 . 5

I I I

. ~ ' l ~ T e r n p e r o t u r e g o p

" - 4 .

" ' ~ ' g " D - 0 0 �9 - ~ . , o

O O ~ O O O

O O O

, 0 . 6

0 . 0 ; i i i 0 . 0 0 .1 0 . 2 0 . 3 0 . 4

x/L

o_

0 0 0

r

1.0 0 . 5

0 . 4 O.B

0.3 ~

0 .6 0 . 2

0.1 0 .4

0.0 0 . 5

a .

E O o

r O

-o-

O L.

CB B O

Figure 4. e, Temperature, O, density, and �9 mole fraction of component 1 (heavy component) as functions o f x • for run 13 (p* = 0"4, T* = 2.0, m]/rn 2 = 10.0, and Jb = 0.431). The dashed line is a fit to the 12 central points of the middle region, and it is drawn to show the deviation from the linear temperature profile in the hot and cold layers, and in the adjacent points of the middle region.

hot and cold regions. In the centre o f the middle regions, the difference between the local and imposed internal energy fluxes in x direction is within the statistical uncertainties in the averages, given in figure 2 as one standard deviation of the average. The y component is zero on average, as it should be.

I I I I

1 .0

X

q , .

~ 1 .0 L O C

W

0 . 5

0 . 0 0 . 0 0.1 0 .2 0 . 3 0 . 4 0 . 5

x / L

Figure 5. Total energy flux computed with a non-cubic 1000 particle system (run 15) and a cubic 4000 particle system (run 17) at p" = 0.4, T* = 2.0, and mffrn 2 = 10-0. These two systems are twice as long in the x direction as the 500 particle system. The energy flux is normalized to the macroscopic value .It} = 0-3.

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1402 B. Hafskjold e t al.

There may be signs of oscillatory behaviour of the x component of the flux, possibly due to the small size of the system. The thickness of each middle region is 4"71all at p* = 0-4. The difference between the local and imposed fluxes in the middle region layers closest to the hot and cold regions may also be significant. The reason for this difference is not clear; one possibility is lack of local equilibrium and too large a perturbation of the system by the velocity scaling in the hot and cold layers. A reduction in the imposed energy flux should then reduce the effect, but results for the total internal energy flux for runs 3, 11, 12, and 13 (figure 3) show that this is not the case. (From here on, we will use the symmetry of the system and show quantities that are averages over the two half cells. We will also leave out the error bars in the flux results.) Another possibility is that the hot and cold layers act as walls that affect the structure of the system and the local energy flux. The density profile shown by open circles in figure 4 for the highest internal energy flux has, however, only very weak oscillations. In fact, it is very much smoother here than in simulations with hard walls [28]. It seems as if the hot and cold layers do not have an impact on the structure of the system other than that given by the density and temperature dependence at equilibrium.

If there is a wall effect or an incomplete energy dissipation, the effect should be reduced if the system size is increased or the MD cell is elongated in x direction. We therefore increased the system size from 500 to 1000 and 4000 particles for an elongated and a cubic cell, respectively. The elongated cell had L y / L x = L z / L x =

0.5. Each system was divided into 32 layers of equal thickness A x D = 1/32, which corresponds to A x* ~ 0.6 in both cases, each layer containing on average 31.25 and 125 particles for N = 1000 and 4000, respectively. The thickness of each middle region was therefore 9"43~r ll, or twice the thickness of the 500 particle cubic cell. Stationary state results from these runs are shown in figure 5. The effect on the internal energy flux close to the hot and cold layers is drastically reduced or absent for the larger systems. These larger systems were sufficiently slow to permit a study of their transient behaviour when going from a one-component system to a binary mixture by changing the mass ratio, but we shall not analyse this observation here.

The mass flux Jk was zero within statistical uncertainties at a stationary state. The stationary state temperature profile is shown by the filled circles in figure 4

for run 13. The temperature gradient is constant in the centre of the middle region, which implies that the thermal conductivity is constant. The temperatures in the hot and cold regions are not on the same linear profile as that in the middle region, but rather somewhat higher and lower, respectively. The temperature gap appears to be proportional to the energy flux. Combined with the observation that the energy flux is smaller than expected close to the hot and cold layers, the temperature gap is not consistent with local equilibrium in the system, unless the thermal conductivity is much smaller here than in the rest of the system. The temperature profiles in the larger systems discussed above were found to be much smoother (although not linear close to the hot region) than for the 500 particle system. We therefore concluded that there is lack of l o c a l equilibrium in the hot and cold regions. At present we shall concentrate our discussion on the linear region, and leave the question of local equilibrium to a future discussion.

Based on this analysis, we conclude that the heat exchange algorithm also works well for binary mixtures of Lennard-Jones particles. The lack of local equilibrium extends at most one layer (thickness Ax* ~ 0.3) into the middle regions for the 500 particle system. These layers may be discarded in the stationary state calculations,

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leaving a region of some 4trll in thickness for computation of local fluxes and other variables. The larger systems provide a wider region and better statistics at the cost of increased computer time which, even for a 4000 particle system, is not exorbitant.

The use of fully periodic boundary conditions is a nice feature for simulations of ionic systems and other systems with long range forces, which will be the topic of a future paper.

4.2. Comparison with previous MD simulations

MacGowan and Evans [18] and Paolini and Ciccotti [20] have reported values for all four L coefficients appearing in equation (9) for an equimolar Kr/Ar mixture at the same state as we used in run 18. From their data, we computed values for the thermal conductivity and the thermal diffusion factor. In order to compute the heat of transfer from equation (18), we need data for Otq/Ow 1. Gardner et al. [27] have reported partial molar quantities, including #l and #2, as functions of composition. It is clear that the mixture is very nearly ideal. The enthalpy of mixing is significantly different from zero ( -20 4- 5 J mo1-1 for an equimolar mixture), but the data for the chemical potentials are too scattered to give a non-zero confidence value for O#l/OWl. We have therefore assumed the mixture to be ideal, and the heat of transfer was computed from equation (19). Comparison of the data is shown in table 3, which also includes the thermal conductivity computed by Gardner et al. [27]. Our result for the thermal conductivity agrees with MacGowan and Evans within the statistical uncertainties, but the disagreements with the other authors are significant. Our thermal diffusion factor and heat of transfer agree with the literature values within the statistical uncertainties. Our statistical uncertainties are substantially smaller than for the previous results. This could be due to our larger system, but it is also clear that the algorithm can produce precise data. Comparison with previous results is somewhat difficult because the potential models differ in their cut-offs, and the numbers of particles also differ. We therefore made an additional run with the same parameters as in run 18, except that the switching function was centred at 2"5crij instead of 2"Ocrij. This gave the results in parentheses in table 3.

Two independently written versions of the NEMD code were used for pure argon at T* = 0-72 and p* = 0.8444 in order to reduce the risk of programming errors. The thermal conductivity obtained with the two codes agreed within statistical uncertainties (A* = 6.81 4- 0.05 with the present code, N --- 1000, and A* = 6.74 with N = 1024 [24]). These results are also in agreement with literature data for A* [30, 31].

The results obtained for the thermal diffusion factor using the heat exchange and the Verlet velocity algorithms were also compared with results obtained by Kincaid et al. [22] by the 'fuzzy wall' leap-frog algorithm at p*---0-4. Agreement within statistical errors was obtained. This is shown later in figure 7. Again, there is a slight difference in the pair potentials used by Kincaid et al. (the Lennard-Jones/ spline potential introduced by Holian and Evans [32]) and us, but a run with the Lennard-Jones/spline potential at p*= 0.4, T * = 2.0, and m l / m 2 = 10 (run 20) verified that the resulting difference in the thermal diffusion factors is smaller than the statistical uncertainty. Comparison of runs 3, and 11-13 (with the switched Lennard-Jones potential) and run 20 (with the Lennard-Jones/spline potential) shows no difference in thermal conductivity, thermal diffusion factor, and heat of transfer. This does not support the opinion that thermal diffusion is more sensitive than other properties to changes in the molecular characteristics [3]. By comparison,

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Table 3. Comparison with literature values for thermal conductivity, thermal diffusion factor, and heat of transfer for an equimolar Kr/Ar Lennard-Jones mixture.

Property Our result Literature values

A* 5"77 • 0"03 a 5"90 4- 0.08 ~ 6.2+0.1 a

(5"77 4- 0-05) b 6.28 e

1 "78 -4- 0 - 0 7 a 1.6 4- 0-5 c cq2 (1'5 4- 0"1) 6 2.1 -4- 0.4 a

** 1'91 + 0"08 a 1-7 -4- 0 . 6 ~ q (1.4 -4- 0-1) b 2.2 4- 0-5 a

a Results from run 18, r c = 2"Oaij. b Results from run 20, r c = 2"5aij.

MacGowan and Evans [18], N = 108 and 256, r~ = 2"5aij. aPaolini and Ciccotti [20], N = 256, force switched off between 3'0all and 3"33a11. e Gardner et al. [27], N = 108, r c = 2"5~11.

there is a significant difference in the two systems compression factors: Z = 1.098 • 0.001 for the switched Lennard-Jones system, and Z = 1.283 • for the Lennard-Jones/spl ine system.

4.3. Stationary state results for isotopes

The linear region o f the temperature profile for each run was used to compute the thermal conductivi ty according to equat ion (20). Similarly, the linear par t o f the composi t ion profile was used to compute the thermal diffusion factor f rom equat ion (16) and the heat o f transfer f rom equat ion (19). The results are given in table 4, where the uncertainties are given as one s tandard deviation. The thermal conductivity for p* = 0.4 and m l / m 2 = 10 was plotted as funct ion o f the internal energy flux in figure 6 to extrapolate to zero flux. N o clear dependence on the energy flux was found over the range studied, but the statistical uncertainties in A* decrease with increasing flux. Too large an energy flux will increase the tendency to phase segre- gation if the the rmodynamic state is close to a phase transit ion [24]. Fo r these reasons, we chose an internal energy flux o f the order o f 0.3 for mos t o f the production runs. A similar trend is shown for the thermal diffusion factor in the same figure. The heat o f transfer is p ropor t iona l to oq2 at constant composi t ion, temperature, and mass ratio for these isotope mixtures. At this part icular state, q** = l lcq2.

The thermal conductivity, thermal diffusion factor, and heat o f transfer were plotted as functions o f mass ratio ml/m2 in figure 7, and as functions o f density in figure 8. The main observations f rom these figures are that the thermal conductivity is a weak function o f the mass ratio and a strong function o f the density. The thermal diffusion factor and the heat o f transfer, on the other hand, depend strongly on the mass ratio as well as on the density.

In accordance with equat ion (4), the total energy flux may be analysed in terms o f the kinetic and potential energy fluxes and the intermolecular energy transfer. Selected results for m l / m 2 = 10 and p* = 0.4 and 0"8 are shown on relative scales in figure 9. Three features emerge: (1) the potential energy flux is small in all cases, and it varies little over the middle region; (2) the ratio between the intermolecular energy transfer and the kinetic energy flux increases with increasing density (at p* = 0.8 the

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Table 4. Results for the thermal conductivity, thermal diffusion factor, and heat of transfer, a

Run No. p* ml/m2 A* at2 q**

1 0-4 2 2"20 + 0"07 0"5 4- 0-1 1"4 4- 0-4 2 0"4 5 2-48 + 0"07 0"9 4- 0" 1 5"5 + 0-8 3 0-4 10 2"42 4- 0"07 1 '2 4- 0'1 13 ::k 1 4 0"6 2 3-82 4- 0"07 0"8 4- 0-1 2'5 + 0'4 5 0'6 5 3"92 4- 0-06 1-7 + 0-1 10'0 -t- 0.7 6 0'6 10 4"15 + 0"08 1-8 4- 0"1 20 4- 1 7 0"8 2 7'3 4- 0-2 1"8 4- 0"2 5"4 4- 0"7 8 0"8 5 7.2 4- 0"2 2-0 4- 0-2 12 4- 1 9 0-8 10 7-7 4- 0"2 2"8 4- 0"2 31 4- 2

10 0"4 10 N/A N/A N/A 11 0"4 10 2-554-0"04 1'404-0'07 15"44-0"8 12 0-4 10 2-47 4- 0-02 1-24 + 0"05 13'6 4- 0"6 13 0"4 10 2-504-0"02 1"21 • 13'34-0"4 14 0-4 1 1'93 4- 0"08 N/A N/A 15 0'4 10 2"71 4-0-02 1-204-0"03 13'24-0.3 16 0"4 1 2-04 4- 0-03 N/A N/A 17 0"4 10 2"81 +0"02 1'194-0'03 13-1 4-0"3 18 0-8671 2-098 5"77 4- 0-03 1"78 • 0"07 1-91 4- 0-08 19 0-8671 2-098 5"774-0"05 1'55:0'1 1-44-0-1 20 0"4 10 2-47 4- 0"02 1"24 4- 0"04 13-6 4- 0"4

a N/A, not available.

3 . 0 i ~ I I 3 . 0

Figure 6.

>. 2 . 5

>

= 2 . 0 C 0

"~ 1 . 5 E L.

e"

~ 1.0

2.5 o Q

C 2 . 0 o

m ,,i

" o 1 . 5

0

E i . g

1.o ~

0 , 5 t t i i 0 , 5 0 . 0 0 . 1 0 . 2 0 , 3 0 . 4 0 , 5

Ju

e , Thermal conductivity and ~7, thermal diffusion factor shown as functions of the internal energy flux for p* = 0.4, T* = 2-0, and mj/m2 = 10.0.

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0

0

" r

4O

I 20

] 0

0 '~ g

6

4

2

0

I I ! I I

I

V - - V ~ v

V V' ' ' ~

f_o o

I I I I I

2 4 6 8 10

m l / / m 2

12

Figure 7. The thermal diffusion factor (lower graph), thermal conductivity (central graph), and heat of transfer (upper graph) as functions of mass ratio for three densities v, p* = 0.8; v , p* = 0.6; and o, p* = 0.4; T* = 2-0. The thermal diffusion factor is also compared with results by Kincaid et al. [22] (p* = 0.4), [], obtained with the fuzzy wall algorithm and the Lennard-Jones/spline potential.

intermolecular energy transfer accounts for some 80% of the internal energy flux); and (3) the same ratio increases form the hot to the cold region of the system. We have not yet found an explanation for the first feature. The last two features are consistent with an increasing collision rate with increasing density. We expect that these features also characterize the heat flux in a real binary isotope mixture, because the energy flux is equivalent to the thermodynamically measurable heat flux at a stationary state.

For an understanding of the Soret effect, it is interesting to see how the two species contribute to the total internal: energy flux. We therefore regrouped the local flux (equation (4)) as

J u ~ J~:,~ + Ju,2 (29)

where the k component contribution was determined as the sum over type k particles:

y#i

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t -

O i .

.4-

Q "1-

"10 t -

O r

i . g r

1- -

40

30

20

10

0

10

8

6

4

2

0

I I I I I

f

,,..._..-.-----'-"~'~-'-'-" .----

J ~ 4

o 3

~ 2

L ID

,-- 0 0.3

I I I I I

0.4 0.5 0 .6 0 .7 0 .8 0 .9

P* Figure 8. The thermal diffusion factor (lower graph), thermal conductivity (central graph),

and heat of transfer (upper graph) as functions of reduced density for three mass ratios: m]/m 2 =V, 10; V, 5; e, 2; T* = 2.0.

The two contributions are shown for p* -~ 0.4 in figure 10 for ml/m2 = 2 and 10. At ml/mz = 2, both species carry about equal amounts of energy. At ml/m2 = 10, the lighter component carries a substantially higher part than the heavier component and there is a clear increase in the ratio Jt~,l/Ju. 2 from the hot to the cold side of the middle region. A concentration gradient, which places the heavier component in the cold region and the lighter component in the hot region, will optimize the heat transport properties of the total system. Division of the internal energy flux into kinetic energy flux, potential energy flux, and intermolecular energy transfer for each of the components shows that the most significant changes from the hot to the cold side are a decrease in the kinetic energy flux for the lighter component and an increase in the intermolecular energy transfer for the heavier component. The lighter component becomes more effective by migrating towards the high temperature side, leaving the heavier component for the cold side where it can be most effective.

According to the second law of thermodynamics, the system's entropy production is positive when there is a heat flux through it. For a fixed heat flux, the system will eventually reach a stationary state of minimum entropy production, i.e.,

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I I I I

�9 S ' . , , o . . . .

0 . 5 (Run no. 9)

o

~ 0.0

g

p'=o.4 (Run no. 3 )

0.5

0 . 0 ~ - I I I

0.0 0,1 0.2 0.3 0,4 0,5

x/L Figure 9. �9 Total energy flux, O, kinetic energy flux, O, potential energy flux, and v , inter-

molecular energy transfer for two densities at T* = 2.0 and ml/m 2 = 10.0. The energy flux is normalized to the macroscopic value, which was J~ = 0.108 for/9* = 0.4, and J~ = 0.342 and/9* = 0.8.

maximum thermal conductivity, which is given locally by

V T o r = - J ~ : . T 2 . (31)

Figures 3 and 4 show that both Jt~ and V T are constant, and the entropy production increases towards the cold region of the system as 1 / T 2. The entropy production does not, however, say which stationary state this is. In fact, the answer to this question must be related to the system's molecular interactions, and cannot be found from thermodynamics alone.

4.4. Real mixtures

The energy flux contributions for the equimolar Kr /Ar mixture are shown in figure 11. The pattern obtained is similar to that for isotope mixtures. The potential energy contribution does not play any significant role, the kinetic energy contribution is also small. The intermolecular energy transfer dominates the picture com- pletely. The variations across the system are the same as described above. The small decrease in the kinetic energy is compensated for by a similar small increase in the intermolecular energy transfer. This increase means that Kr is taking over the

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1 . 0

0 . 5

+- o

.l,- :1 .o 0 . 0 i .

. i - c o o

~ 1 . 0

G

0 . 5

0.0

0.0

I I I I

m,//m2=2 (Run no. 1)

ml/rn1=lO (Run no. 5)

I I I I

0.1 0.2 0.3 0.4 0.5

x/L Figure 10. V, Total energy flux and the contributions from O, component 1 and e , compo-

nent 2 for two mass ratios at T* = 2-0 and p* = 0.4. The energy flux is normalized to the macroscopic value, which was It) = 0.108.

leading role as the heat carrier as the temperature goes down, with a slight component separation as a consequence of this.

The main difference between isotope mixtures and the mixture of argon and krypton is that a difference in size of the molecules and the potential depth may also influence the results, especially at high densities. From the analogue to the isotope system at p* = 0.8 (figure 9), we expect, however, that this influence is a secondary effect to the mass ratio effect.

Ecenarro et al. [3] have reported experimental oq2 values for liquid hydrocarbon mixtures that confirms the main observation from our study; the lighter component migrates towards the hot side of the system. Their numerical values are similar to ours. In a study of binary mixtures of H2 with noble gases, Trangove and Dunlop [2] reported values for al2 approximately equal to 0.3 and slightly decreasing with increasing mass ratio. The order of magnitude agrees with our results, but the isotope results increase with increasing mass ratio. The observed trend must therefore be due to features of the pair potential, such as differences in the molecular diameter and potential depth, and not the mass difference.

We have calculated the heat of transfer from equation (19) for the Kr /Ar mixture, assuming that the mixture is ideal. The result is given in table 3. Our value lies within the uncertainty of the results of MacGowan and Evans [18] and Paolini and Ciccotti [20], so it seems clear that we should expect a value of q** close to 1.9 for

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o

- i . Q

L t - O U

g

Q

1 .0

0 . 5

0.0

1 . 0

0 . 5

0.0

0.0

I I l I

(o) . . . . . . . v - < . ' , .

\ \ / 4 / * Total ene rgy f l u x ~ /! . In te rm. en. t r a n s f e r 7 o Kinet ic energy f l u x \ \ r �9 Poten t ia l ene rgy f l u x

(b)

/ \

* Componen t 2 (Ar)

I I I I

0.1 0 .2 0 .3 0.4 0 .5

x/L

Figure 11. (a) The total energy flux, the kinetic energy flux, the potential energy flux, and the intermolecular energy transfer, and (b) the total energy flux and the contributions from each component for the Kr/Ar mixture at T* = 0-692, p* = 0-8671, and ml/m2 = 2-098. The energy flux is normalized to the macroscopic value, which was J~ = 0.361.

an equimolar Kr/Ar mixture under these conditions. This value corresponds to 2.7 kJ m o n a, which is twice the system's kinetic energy, 3RT/2 (1.4 kJmol-1).

The heat of transfer has frequently been related to the activation energy of diffusion [33], a model that does not explain experimental data for salt mixtures and aqueous electrolyte solutions [11]. Activation energies for diffusion computed from MD results for Kr /Ar mixtures at various temperatures [27] are found to be of the same order of magnitude as the heat of transfer, but the uncertainty of the activation energy is very large, because there are only a few temperature points. The activation energy also shows an abnormally high temperature dependency. Therefore, we cannot find a basis for the similarity between the heat of transfer and the activation energy for interdiffusion for the Kr/Ar mixture, and we believe that this similarity is not generally valid.

4.5. Predictions

Do our results provide any clue as to why sign changes in a12 have been observed? We have so far found only positive thermal diffusion factors, and the effect of variations in overall composition has not been studied yet. A change in sign of the thermal diffusion factor should follow from a change in the role of the

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participating components. This could occur if the lighter particles carried a relatively larger portion of the energy flux in the colder region, e.g., if the lighter particles also had a deeper potential. It would be very interesting to run a simulation for a moderate mass ratio and s163 < 1 to see if this were the case. In any case, our isotope results show that the kinetic energy flux and intermolecular energy transfer of the lighter and heavier species, respectively, are the key mechanisms responsible for thermal diffusion in isotope mixtures. We can only speculate that this will also be the case for compositions other than the 50/50 mixture we have studied here. It is difficult to predict how ot12 will depend on the composition, but experimental data [2] suggest that at2 decreases with increasing content of the heavier component. A possible explanation is that a smaller overall amount of the lighter component will give less optimal heat conduction when it migrates to the hot side. This trend is confirmed by some unpublished results obtained with the fuzzy wall algorithm described by Kincaid et al. [22, 34].

5. Conclusion

Heat conduction in isotope mixtures has been studied at different densities and mass ratios. This study shows a clear picture of how heat is transported in binary systems; the contribution from the lighter component is predominantly flux of kinetic energy, and this contribution increases from the cold to the hot side. The contribution from the heavier component is predominantly intermolecular energy transfer through molecular interactions, and it increases from the hot to the cold side. This explains why the thermal diffusion factor is positive; heat is conducted more effectively through the system if the lighter component migrates towards the hot side. Even for very large heat fluxes, we find a linear or almost linear temperature profile through the system, and a constant thermal conductivity. The entropy production per unit volume and unit time increases from the hot to the cold side.

The formal basis for a comparison between the phenomenological equations for coupled heat and mass transport and the molecular expressions for the internal energy and mass fluxes have been critically reviewed. At a stationary state with vanishing diffusive mass fluxes, the molecular internal energy flux is equal to the phenomenological, measurable heat flux. The relationship between the molecular and thermodynamic fluxes requires information about the partial specific enthalpies if the mass fluxes are non-zero.

The heat exchange algorithm for non-equilibrium molecular dynamics simulations has been implemented for computation of thermal conductivity, thermal diffusion factor, and heat of transfer for binary Lennard-Jones mixtures. The method has comparable efficiency with other methods for computation of the thermal diffusion factor and thermal conductivity, and precise values of these transport coefficients may be obtained. The method may be extended to ionic fluids and other fluids with long-range forces.

This work was supported by The Norwegian Research Council for Science and the Humanities (NAVF) with travel grants to T.I. (Grants no. 432.91/031 and 432.91/065) and computer time on the Cray X-MP in Trondheim. The Department of Petroleum Technology and Applied Geophysics is acknowledged for providing computer time on an IBM RS6000/550 workstation. We also thank John M. Kincaid for several useful discussions.

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On the molecular mechanism of thermal diffusion in liquids