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Molecular collisions and self-diffusion inliquidsD.E. O'Reilly aa Argonne National Laboratory, Argonne, Illinois, 60439

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To cite this article: D.E. O'Reilly (1975): Molecular collisions and self-diffusion in liquids, MolecularPhysics: An International Journal at the Interface Between Chemistry and Physics, 29:4, 1189-1196

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MOLECULAR PHYSICS, 1975, VOL. 29, No. 4, 1189-1196

Molecular coll is ions and self-diffusion in l iquidst

by D. E. O'REILLY Argonne National Laboratory, Argonne, Illinois 60439

(Received 2 July 1974)

The velocity autocorrelation function and coefficient of self-diffusion are derived for a hard-sphere fluid. An earlier quasi-lattice model of self-diffusion in liquids is re-examined and a less restrictive expression for the self-diffusion coefficient is derived. The coefficient of self-diffusion of a light or heavy isotope in the normal liquid is considered both for hard spheres and in the quasi-lattice approximation. An experiment is proposed to clarify the nature of the isotope effect in self-diffusion.

1. INTRODUCTION In a recent paper [1 a] a quasi-lattice model was formulated for the calculation

of the velocity autocorrelation function and self-diffusion coefficients of liquids. Subsequent numerical calculations on a variety of liquids [1 b, 2] yielded reasonable agreement between the observed and calculated pre-exponential factors (Do). The model predicts essentially an Arrhenius behaviour for the self-diffusion coefficient which is observed [2] for a variety of van der Waals and polar liquids over wide temperature ranges. I t is the purpose of the present paper to investigate more thoroughly some of the assumptions inherent in the model; in particular the precise meaning of a ' hard ' collision which was defined as a collision which results, on the average, in a zero dot product between the molecular velocity just before the collision and immediately after the collision. Other types of molecular collisions, which will be referred to as ' so f t ' collisions in the following, were neglected. This assumption receives some support from the molecular dynamics calculations [3] of Einwohner and Alder on mole- cular square-well systems in which they concluded that the effect of soft collisions between hardcore collisions can be neglected in evaluating the consequences of molecular collisions. This conclusion substantiates the van der Waals hypothesis which asserts that only hard-core collisions are really important and the effect of soft collisions in the attractive part of the potential can be neglected.

In the present work the velocity autocorrelation function for a hard-sphere fluid is derived and a self-diffusion coefficient is calculated which agrees with earlier [4] calculations which employed certain assumptions which are not used here. The quasi-lattice model is then improved with the aid of the expression for the encounter rate given by Einwohner and Mder [3] and the results of the hard-sphere fluid calculation.

t Based on work performed under the auspices of the U.S. Atomic Energy Commission.

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1190 D.E . O'Reilly

Of considerable interest in the present work is the effect of isotopic substitu- tion on the coefficient of self-diffusion in liquids. It has been demonstrated experimentally [5] with liquid benzene that the self-diffusion coefficients of heavy tracer benzene molecules diffusing in ordinary benzene are virtually unaltered from that of ordinary benzene. Likewise molecular dynamics cal- culations [6] on light and heavy argon atoms diffusing in ordinary argon indicate that the self-diffusion coefficients of these atoms are nearly the same as for normal argon atoms. In the present work this effect is considered and the effect of isotopic substitution for hard spheres is evaluated. Experiments are proposed to evaluate experimentally the influence of isotopic substitution on self-diffusion in liquids.

2 . H A R D - S P H E R E F L U I D

First we consider the dynamics of a binary collision between hard spheres of equal diameter but different masses which are denoted by m 1 and m 2. The final velocity of particle 1 (vx') is related to the initial velocities (v s and v2) by the following equation [7]:

??12 1712 V s - - - - P X [ (V s - V 2 ) X P] - - (V s - V 2 ) • P P

m I q- m~ m 1 q- m~

m ~ v I q- m ~ v ~ + , (1)

m I -.t-- m,~

where P is a unit vector in the direction of rs2-- r s - r~; r s and r 2 are the radial vectors of particle 1 and particle 2. Equation (1) results from the conservation of linear momentum and kinetic energy. It readily follows from equation (1) that

2m2 v ? - v l - - P P. ( 2 )

ms + m2

Likewise, for particle 2 one obtains

2ml - - (vs- PP. (3) m s + m~

Equations (2) and (3) state that the difference between the initial and final velocity of particle 1 (or particle 2) lies along the vector P. We now take the dot product of equation (2) with v s and of equation (3) with v 2 and add the resulting equations. This yields the following expression:

V s " V1 t VS 2 V 2 " V2 t V2 2 2 + - • ( ( v l - ( 4 )

m 2 m z m 1 m 1 m 1 -b m 2

Next we calculate the average over a Maxwell-Boltzmann distribution of mole- cular velocities of both sides of equation (4). The right-hand side of equation (4) yields just the thermal average value of the square of the component of the relative velocity V along P. This is readily calculated by standard kinetic theory techniques [8] (Appendix) to yield

2kT ( v . , (5)

/z

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Self-diffusion in liquids 1191

where/x is the reduced mass (l~=mlms/ml+ms) and the bar in equation (5) designates a thermal average. Thus equation (4) yields

v I �9 V l t V s " v s ' 6 k T 4 k T - - + . . . . . . ( 6 )

ms ml mlm 2 mlms

Let us first consider the case that m z = m s = m . obtain from equation (6) the following result:

kT �9 V l v - - Vl

m

Then V 1 " V l ' = V s " VS t and w e

and

2m 2 kT V 1 �9 V 1' - - - - (7)

m 1 + m~ m I

2m I kT vs" v ( - - - (8)

n,/1 .~- m s ms

Let us follow particle 1 through successive collisions with particles of mass m s after the first for which the thermal average value of the dot product v 1 �9 v 1 is equal to 2ms/3(ml+ rnz) times the thermal average value of vl s. Let Vl" be the velocity of particle 1 immediately after the second collision. Consider the average value of (Vl". Vl' ) (v l ' . Vl). Writing out this expression in terms of the polar coordinates of vz" and v 1 relative to v x' which is chosen as the polar axis, it follows that

(V1 tt" VI ')(V1 t " V1) = (Vln)S(Vlt)S(~)I) 2 COS 01 COS 0 s

= (Vln)S(~I ' )S(Vl) s COS 01S

where 01 iS the angle between vz" and VI' , 0 s is the angle between V 1' and vl, and Ozs is the angle between v 1 and vl". With the approximation that (molecular chaos approximation)

COS 01cOS 0 2 ~ COS 01cOS 0 2 (10)

it follows that ( V l ~ Vl) i s equal to ~(me/ml+ms)S(kT/ml) and each successive collision of particle 1 with particles of mass 2 reduces the dot product of initial (vl) and final velocity by the factor 2ms/3(m 1+ m2). We now may construct the velocity autocorrelation function of a hard-sphere fluid. The probabili ty Wn('r) of n collisions after a t ime r from r = 0 is given by the Poisson distribution [9] (Markovian process)

,(.)- wn(- )= ~. ~ee exp ( - - / r e ) , (11)

where re is the mean t ime between collisions. The velocity autocorrelation function ~b(,)= (vl( , ) �9 vl(0))/(v~ s) is then clearly given by

(9)

That is, the average value of the dot product between v I and v z' is one-third of vz s=3kT/m. For m 1 #m~ it follows that

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1192 D.E . O'Reilly

n=0 ,~=o ni ~e exp ( -T / re ) , (12)

where An is the amplitude of Vl(m �9 vl/vl z after n collisions and [ = 1 - (An/An_l) = 4m~/3(ml+mz). The sum in equation (12) is readily performed to yield, for a hard-sphere fluid, the following result (particle 1):

~bl(~- ) = exp ( - ['r/We). (13)

With the aid of the well-known relation [10]

D=�89 S <v(.) �9 v(0)) dr (14) 0

one obtains

kT re D = - - - - (15)

rnl [

The mean time between collisions is given by [3]

( zrkr~ 1/2 ~'e-a =4aap \ - ~ j g(a), (16)

where a is the hard-sphere diameter, g(o) is the value of the equilibrium radial distribution function at R = a, and p is the number density of the fluid. The result for D is as follows:

3 f k r ~ ''2 D = 8--~ ~k2-~/ g(a) -1. (17)

If mt=m 2, equation (17) reduces to

D = 8 ~ \~rmj g(a)-i " (18)

Equation (18) is equivalent to expressions derived in references [4] for the coefficient of self-diffusion of a hard-sphere fluid. In reference [4 a] it was assumed that ~b(r) has an exponential decay whim in reference [4 b~ the ' plateau time ' was taken to be infinitesimally short. Equation (17) demonstrates that for a hard-sphere fluid D is inversely proportional to the square root of the reduced mass t~ of the foreign sphere of mass m 1 and the host fluid spheres of mass m 2.

Equation (18) has also been inferred by Dymond and Alder [4 c] for a hard-sphere system. As is clear from the approximation of equations (10) and (11), in the derivation of equation (18) we have neglected correlations between collisions and, as is stated in reference [4 c], the error produced by this assump- tion is significant only at high density, where it amounted to about 30 per cent in D. The deviations of ~b(t) from exponential behaviour are small, even at the highest densities studied [4 d].

3. MOLECULAR LIQUIDS

The results of molecular dynamics calculations on liquid argon [11] and water [12] demonstrate that the velocity autocorrelation functions are markedly non-exponential in contrast to that of a hard-sphere fluid. Rahman [11 b, c]

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Self-diffusion in liquids 1193

analysed r into ' s h p ' (S~(t)) and 'rattling and R~(t) are defined as follows:

and

' (R,(t)) parts of r S,(t)

m N

S,(t)-- 3kTN i~I__ viii(O)v,~l(t),

m N R~(t) = 3kTN i x-1 (v~Q(O)viQ(t) + vl~l(O)vl'l(t))

(19)

(20)

r = s,(t) + R,(t), (21) where ~t refers to the direction of rl(r) - ri(0), i.e. the direction of ' eventual atomic displacement' [11 b] in time r. ~:~, ~71 and ~i form a Cartesian coordinate system for particle 1 and N is the total number of particles in the system. S~(t) was further [11 b, e] decomposed into S~+(t) and S~-(t), where S+(t) is the part of S~(t) arising from particles with v~i(O)> 0 and S~-(t) is the part of S~(t) with v~(0)<0. The function S~-(t)+R~(t) for ~ > 0 oscillates rapidly and yields [1 a] only a small contribution to the self-diffusion coefficient. S+(r) is a more slowly decaying function with a negative tail and contributes by far the largest part to the coefficient of self-diffusion.

In the following we will construct r on the basis of a cell (quasi-lattice) model for the liquid. Each molecule is regarded as being in a potential well produced by neighbouring molecules. In addition, fluctuations in density of the liquid will occur on a microscopic scale which we will ascribe [1 a] to voids of molecular dimensions. If a molecule attains sufficient velocity it may surmount the potential energy barrier between a normal ' lattice ' site and a neighbouring void and in this way diffuse in a manner that is vaguely similar to the mechanism of vacancy diffusion in a solid [13, 14]. In contrast to a solid at low tempera- tures, in a liquid there are frequent ' h a r d ' collisions which will reduce the contribution of the quasi-vacancy diffusion and enhance the rattling contribution to the coefficient of self-diffusion in comparison to the respective contributions in a solid [13]. We will define what is meant by a hard collision later.

For the quasi-vacancy diffusion to occur it is necessary for the particle to have sufficient mean square velocity to go over the average potential barrier which we designate by ~0. The probability mean square velocity [1 a] with velocity greater than (2g0/ml) 1/2 we designate by (v2)g o. Let the mean time between hard molecular collisions be re'. The probability mean square vel- ocity of a molecule in traversing the energy barrier will be (v~)r this quan-

tity must be multiplied by the probability of a vacancy (Pv) at a ' lattice' site subtended by the direction of v to form the ensemble average of qg(t) = (v(0) �9 v(t)) during a diffusive step. The time dependence of ~b(t) is clearly given by exp ( - f t / r e ' ) in analogy with the results of w 2. Hence the sliding part of r may be readily seen to be given by

~b s(t) = (v2)Qpv exp ( - It~re'). (22)

We have constructed Cs(t) in equation (22) by evaluation of the thermal average rather than the time average in equation (14). In writing equation (22) we have neglected the contribution of ' s o f t ' collisions to ~bs(t), i.e. we have adopted the van der Waals hypothesis [3].

In view of the small contribution of R~(t)+ S~-(t) to the coefficient of self- diffusion in liquid argon we will not consider the rattling contribution but direct

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1194 D . E . O'Reilly

our attention to ~bs(t) as given by equation (22). The ' encounte r ' rate (per molecule) P between two molecules i and j at a separation R = Irij[ has been given by Einwohner and Alder [3] and is as follows:

F=epR~ ( ~----T) I /2 (g(R) )AV, (23)

where <g(R)>AV is the average value of the pair-distribution function averaged over all relative positions r,j such that Ir,jl=-R. An ' e n c o u n t e r ' is defined as the location of two molecule i, j at a specific separation rij and with r,j �9 vtt < 0. For R < a, an encounter will be a hard collision. For all hard collisions we may replace the factor R e by a2 to good approximation and we define the time *c' as follows using equation (23):

ze' = 4p cr z ( (g(R))av )o, (24)

where ((g(R))Av)0 is the thermal average value of <g(R)>Av for hard collisions. Hard collisions (R < a) will closely approximate collisions between hard spheres because of the rapid dependence of the pair potential between molecules on R. The relative kinetic energy between molecules i a n d j undergoing a hard collision is equal to �89 2) and will be completely converted to potential energy when the distance of closest approach occurs. Let the potential of mean force be V(R), then g(R) is given by definition as [10]

g(R) = exp ( - V(R)/kT). (25)

The thermal average value of �89 e) will be accompanied by a mean potential energy that will be proportional to the minimum energy E in the pair-potential energy between molecules. Hence at closest approach between molecules i a n d j during a hard collision, it follows by conservation of energy that

( V(R)> 0 = �89 e) - a~, (26)

where ~ is a constant which we expect is not greatly different from unity and <V(R)> 0 is the mean value of V(R) at closest approach between molecules i and j. Let %' be the value of V(R) such that

( (g(R))Av)0 = exp ( - %'/kT). (27)

Using the value of (v2)g given in reference [1 a], equation (14) and [=4me/ 3(m1+ me) one obtains the sliding contribution to D as follows:

1 l,e Ds=4~pa-----~\kT] \.kT]

x [ '+akT(l+~)]exp(s/k)

x exp {-(~+w)/kT}, (28)

where r #0- %', s=so+sw-So', s0 = - ( O e o / O T ) p , Sw = -(Ow/OT)e, and S'o= - (0%'/aT)p. p v = e x p ( - w / k T ) where w is the mean work required to form a ' v a c a n c y ' in the liquid. As with hard spheres, equation (28) states that the self-diffusion coefficient is inversely proportional to the square root of the

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Sell-diffusion in liquids 1195

reduced mass/z. However, % and %' will depend on/n I and m s in addition, so that a simple dependence of D on/z -1/s is not to be expected from equation (28).

To apply equation (28) to real liquids we will use the experimental value of e+w to calculate Do, the pre-exponential factor in equation (28). w may be evaluated by analysis [1, 2] of experimental density data, isothermal compressi- bility data, the scaled particle theory of liquids [15] and by the difference [2] of activation energies for diffusion and intramolecular relaxation. Generally these four sources of w are consistent and hence the quantity e may be evaluated. By evaluation [14] of equation (26), %' may be estimated and hence r determined. From the expression [14] for c0', s 0' may be evaluated; Sw may be calculated from the methods of analysing w and go may be estimated [1 b]. Hence D o may be calculated from equation (28) and compared with experiment. This procedure has been carried out for liquid sodium [14] and the agreement with experiment is reasonably good; the calculated D O (1-6 x 10 -3 cm 2 sec -1) is some- what higher than the experimental value (1.1 x 10 -3 cm 2 see-I).

It would be of interest to ascertain the dependence of D on m I and m z experi- mentally by determining the pre-exponential factor Do. An ideal system for such experiments would appear to be liquid hydrogen and deuterium. The ratio mz/ml=2 for a small amount of H z diffusing in liquid D2 and the reduced

_ 4 mass Ix-g, so that a rather large difference in D O for H s in H z and Ds in H e would be expected. Quantum-mechanical effects may be present, but possibly these may be taken into account.

A P P E N D I X

Thermal average o[ (V . ~)2 The probability of a collision between a molecule of mass m I and a molecule

of mass rn 1 per unit volume per unit time such that V lies between V and V + d V and 0 lies between 0 and 0 + dO is proportional to [8]

exp ( - ~IXV2) V z sin 0 cos 0 dO dV,

where 0 is the angle between V and P. The thermal average of (V. p)2 is obtained as follows:

exp ( - } ~ V 2) V 5 dV S2sin O cos z 0 dO (V . e)~= o o

~r12 oo

exp ( - ~ / x V 2) V z dV ~ sin 0 cos 0 dO o 0

The integrals over 0 yield a factor of �89 To evaluate the integrals over V, let E = }/xV 2. Then,

( y . p ) ~ = ( ~ ) ( ~ ) i e x p ( - e / k T ) e s d e - ---~ - -~ . . . . . . .

(--~2) ~o e X p ( - ' / k T " d "

2kT tx

as given in equation (5).

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1196 Sel[-diffusion in liquids

REFERENCES

[1] (a) O'REmLY, D. E., 1971, J. chem. Phys., 55, 2876; 1973, Ibid., 58, 1272 (erratum); (b) 1972, Ibid., 56, 2490.

[2] (a) O'IL8ILLY, D. E., and PETERSON, E. M., 1972, J. chem. Phys., 56, 2262. (b) O'REILLY, D. E., PETERSON, E. M., and SCHEIE, C. E., 1973, J. chem. Phys., 58, 4072. (c) O'REILLY, D. E., PETERSON, E. M., SCttEIE, C. E., and SEYFARTH, E., 1973, J. chem. Phys., 59, 3576. (d) O'REILLY, D. E., PETERSON, E. M., and ScrIEm, C. E., 1974,J. chem. Phys., 60, 1603.

[3] EINWOHNER, T., and ALDER, B. J., 1968, J. chem. Phys., 49, 1458. [4] (a) LONatmT-HIGGINS, H. C., and POPLE, J. A., 1956, J. chem. Phys., 25, 884. (b)

H~LEAND, E., 1961, Physics Fluids, 4, 681. (c) DYMOND, J. H., and ALDER, B. J., 1966, J. chem. Phys., 45, 2061. (d) ALDER, B. J., and WAINWRIGHT, T. E., 1967, Phys. Rev. Lett., 18, 988.

[5] ALLEN, G. G., and DUNLOP, P. J., 1973, Phys. Rev. Lett., 30, 316. [6] RArIMAN, A. (private communication). [7] HAUSER, W., 1965, Introduction to the Principles of Mechanics (Addison-Wesley), p. 242. [8] FOWLER, R., and GUGGENHEIM, E. A., 1965, Statistical Thermodynamics (Cambridge

University Press), p. 491. [9] FELLER, W., 1950, An Introduction to Probability Theory and its Applications (Wiley),

Chap. VI. [10] EGELSTAFF, P. A., 1967, Introduction to the Liquid State (Academic Press), Chap.

V, p. 134. [11] (a) RAHMAN, A., 1964, Phys. Rev., 136, A405; (b) 1966, Reactor Physics in the Reson-

ance and Thermal Regions, edited by A. G. Goodjohn and G. C. Ponvraning (M.I.T. Press), p. 123. (c) 1966, J. chem. Phys., 45, 2585. (d) VERLET, L., 1967, Phys. Rev., 159, 98; 1968, Ibid., 165, 201.

[12] (a) RAHMAN, A., and STILLINGER, F. H., 1971, J. chem. Phys., 55, 3336. (b) STILLIN- GEe, F. H., and RAHMAN, A., 1972, J. chem. Phys., 57, 1281.

[13] O'REILLY, D. E., and PETEaSON, E. M., 1972, J. chem. Phys., 56, 5536. [14] O'REtLLY, D. E., 1974, J. chem. Phys., 78, 2275. [15] REtss, H., FRISCH, H. L., HEL~AND, E., and LEBOWtTZ, J. L., 1960, J. chem. Phys.,

32, 119.

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