J. Phys. Chem. 1982, 86, 3047-3049
3047
Some Comments on the Kirkwood and Rice-Aiinatt Transport Theories F. P. Ricci and D. Rocca’ Istltuto di Flsice “0.M r m i ” dell Unlversite dl Rome. P.k A M Mor0 2, 00185 Rome, Meter& del C A R . (Recelvd: Febnvlry 5, 1982; I n Final F m : M r c h 11. 1982)
Ita&, and buppo Nazlonak, di Sbvtlwa delle
We report an unambiguous check of the Kirkwood and Rice-Allnatt equations for diffusion in simple liquids over a large range in temperature and at various densities using molecular dynamics experiments. We show the inadequacy of these theories in predicting the dependence of D on density and temperature. Introduction Among the various theories on transport processes in dense fluids those proposed by Kirkwood’ and Rice and Allnatt2 (in the following abbreviated as K and RA) have great importance since, starting from basic principles, they give explicit expressions for the transport coefficients which can be numerically evaluated. The validity of these theories requires the fulfillment of a few hypotheses which are by no means obvious. The major one is that the time integral of the force autocorrelation function (see eq 2 of next section) must reach a plateau value; it is now possible, by means of computer experiments, to obtain directly such an autocorrelation function. In the case of a simple liquid (argon simulated with BPB potential) two such experiments are reported in the literature,% one near the triple point and another near the critical point. In both cases the numerical results indicate that the plateau-value hypothesis is wrong. Moreover, the friction coefficient at high density turns out to be negative. The situation is however not too clear since many authorsM have found that the final expressions (see eq 4, 6, and 7) given by the K and RA theories, yield transport coefficients which are almost in quantitative agreement with the experiments. The lack of complete agreement was ascribed to uncertainties in the knowledge of the radial distribution function, g(r),and of have the interatomic potential, V ( r ) . Many therefore suggested that a revised kinetic theory of classical liquids based on the K and RA ideas is a worthwhile proposition. However, the comparison between theory and experiment has been carried out only in a limited density and temperature range, i.e., near the melting point. Therefore, we think it necessary to reconsider this comparison in a much wider range to avoid fortuitous cancellations in the errors. Moreover, it would be desirable to make the comparison in a case where there are no ambiguities with respect to the choice of g(r) and of V(r). To accomplish these objectives we decided to check the K and RA equations (see eq 4,6, and 7) for the self-diffusion, D, in the case of the Lennard-Jones (L-J) fluid for which extensive molecular dynamics experiment^"^ in a large temperature range and at various densities are available. (1)J. G. Kirkwood, J. Chem. Phys., 14, 180 (1946). (2)S.A.Rice and A. R. h t t , J.Chem. Phys., 34,2144,2156(1961). (3)(a) R. A.Fisher and R. 0. Watts, A u t . J. Phys., 26,21(1972);(b) S. I. Smedley and L. V. Woodcock, J. Chem. SOC.,Faraday Trans. 2,70, 955 (1974). (4)For a review see: (a) S. A. Rice and P. Gray, ‘The Statistical Mechanics of Simple Liquids”,Interscience,New York, 1965; (b) S.A. Rice, J. P. Boom, and H. T. Davie in “SimpleDense Fluids”,H. L. Frish and Z. W. Salzburg, E&., Academic Press, New York, 1968; (c) J. S. Ku and K. D. Luks, J.Phys. Chem., 76, 2133 (1972). (5) (a) B. Cleaver, 5. I. Smedley, and P. N. Spencer, J. Chem. SOC., Faraday Trans. 1,68,1720(1972);(b) B.Cleaver and T. Herdlicka, ibid., 72, issi (1976). (6) L. Palleschi, S. Sacchetta, and F. P. Ricci, Mol. Phys., 42, 961 (1981). (7) (a) L.Verlet, Phys. Rev., 165,201 (1968);(b) D.Levesque and L. Verlet, Phys. Reu. A, 2, 2514 (1970).
Indeed, the molecular dynamics experiments yield both
D and g(r) values for a well-defined intermolecular potential. Comparison between Theories and Experiments The main Kirkwood idea’ was to write the equation of motion for the single molecule in the form of a Langevin equation
dPi/dt = Xi - {iPi/m
+ Gi
(1)
where Pi is the momentum of the ith molecule, Xiis the external force acting on it, Gi is the fluctuating intermolecular force, and { is the friction coefficient. Gi and {are expressed in terms of the intermolecular forces acting on the molecule, Fi,and its ensemble averages. Particularly 1 { = - JT(Fi(0) Fi(t)) dt (2) 3kT o One hypothesis, necessary for the validity of K theory, is that in eq 2, as a function of T, a plateau value for the integral must exist, so that the integral does not depend on T. If this has to make sense, T must be very short on a macroscopic scale (Le., T u Using eq 5 they obtain
f~ = (8/3)(*kT/M)'/2~g(u)u2
(6) Flgure 1. D' vs. T' at p' = 0.85: (a) DoexDtl, (b) D o K (c) , D*w.
where g(u) is the value of g(r) for r = u (7)
4c
so that in this case eq 3 can be written
D = kT/(cH -k ls)
(8)
To evaluate (s RA made the same approximation used by K in calculating 5; confined only to the soft force field, whereas in the Kirkwood case the integrals are evaluated by using the whole intermolecular potential (repulsive and attractive parts). As indicated at the end of the Introduction, we will check the K and RA equations as follows: (1) In the case of K theory we calculate D (DK) from eq 3 using eq 4, where g(r) is taken from a molecular dynamics experiment. obWe can compare DK with the experimental D (Dexptl) tained by the same molecular dynamics experiment. (2) In the case of RA theory we calculate D ( D M )from eq 8 using eq 6 and 7 where g(r) is the same as in the K case. Following the Rice suggestion, u is the Lennard-Jones parameter which appears in eq 5. Then we compare DM with De,,. For the sake of generality, we use reduced quantities, indicated by the asterisk, i.e. D* = ( D / u ) ( m / t ) 1 / 2 P = f u / ( m t ) ' J 2 p* = pu3/m T* = k T / t
3c
2c
10
Flgure 2. D' vs. T' at P* = 0.65. The symbols have the same meaning as in Figure 1.
For example, if we refer to a L-J argon-type fluid, Le., u = 3.405 X cm and t = 1.65 X erg, and we express D , l, p, and T in cgs units, we have D* 1860D (3.25 X lo''){ p* = 0 . 5 9 5 ~ T* = T/120 From g(r) and V(r) taken from ref 7 we calculate { and therefore DK and DRA(in which g(u) = g(r=u); the integrals in eq 4 and 7 are evaluated numerically. Therefore DK and DRAhave an indetermination smaller than a few percent. The De,ptl values are taken from the same molecular dynamics calculations' and they are given with an error on the order of 5%. In Table I we report D*K, D*RA,and D*,& for reduced densities of 0.85, 0.75, and 0.65 and for 0.72 5 P 5 3.67. We have considered only these three densities since they are the only ones for which the molecular dynamics calculations give both g(r) and D. In Figures 1 and 2 we report D* = D * ( P ) along isochores at p* = 0.85 and 0.65, respectively; in Figures 3 and 4 we report D* = D*(p*)along isotherms for P = 1.00 and 2.20, respectively. Discussion Looking at Table I and Figures 1-4, we note the following general trend (1) D- has a much stronger density and temperature dependence than both DK and DRA. (2) De,, is always higher than DK and DRAexcept for low
0.65
0.75
0.85
Flgwe 3. D' vs. p' at T' = 1.00: (0)D*exDtl, ( 0 )D a K (A) , DSRA.
temperatures (T*I1.17 for DK and T* C 0.83 for D M ) and the highest density; the disagreement between the experimental and theoretical values of D reaches a factor of 2 practically in the whole range of temperature (0.90 I P I3.67) at the lowest density (p* = 0.65). (3) DK and DM have almost the same temperature and density dependence. The only difference is that DK is almost 10-15% higher than DRA. It is also interesting to note that DK N Derptl at p* = 0.85 and P N 1.17 while DRAN DeXptl at p* = 0.85 and T* N 0.83. These agreements are only fortuitous; they are due to the different density and temperature dependence and happen to occur in the liquid range. Therefore, it is clear that, when we compare experiment and theoretical expressions in a limited temperature and density range, we
Kirkwood and Rice-Allnatt Transport Theories
The Journal of Physical Chemistty, Vol. 86, No. 15, 1982 3040
TABLE I: Comparison between the Molecular Dynamics Data for D and the Theoretical Predictions from Kirkwood and Rice-Allnatt Theories‘ (AD*/ (AD*/
T* S* S*H f*S 107D*K 1O2D*RA 1OzD*exptl D*expU)K D*exptl)RA 2.888 24.3 14.3 13.8 11.9 10.3 16.7 -0.29 -0.39 8.3 12.5 -0.22 -0.34 2.202 22.5 12.4 14.2 9.8 6.6 19.5 8.4 14.7 6.5 5.5 -0.01 -0.16 1.273 t 0.04 7.5 14.8 5.1 -0.13 5.8 18.7 6.0 1.127 4.3 +0.16 5.7 14.9 5.0 4.3 17.6 0.88 ~0.06 t0.23 3.7 4.9 14.8 4.6 4.0 17.0 0.786 t 0.30 t0.13 3.3 4.3 16.7 4.3 14.8 3.8 0.719 2.8 14.1 3.9 + 0.48 +0.25 2.9 4.3 15.3 0.658 1.2 -0.42 -0.48 13.6 23.3 10.9 12.5 20.9 0.75 2.845 -0.31 -0.39 11.2 6.2 13.0 6.8 16.8 7.8 1.304 -0.27 6.0 -0.36 9.3 5.0 13.0 15.8 6.8 1.070 -0.31 -0.24 5.1 3.4 12.9 5.6 7.4 14.8 0.827 -0.57 41.4 19.6 -0.55 9.6 10.9 17.9 18.6 0.65 3.669 11.6 -0.52 -0.48 10.6 22.0 6.1 11.2 15.8 1.827 -0.52 -0.47 9.4 10.4 19.5 5.5 11.4 12.3 1.584 -0.49 -0.44 13.7 3.5 11.4 13.6 7.0 7.6 1.036 -0.49 -0.44 6.3 12.3 3.0 11.4 13.2 6.8 0.900 a AD* = D* - D*axpa, D*K = T*/f*, D*RA = T * / ( ~ * H t S*S). D values are taken from ref 7. In the case of small differences in temperature, we use the interpolation forinula for Dexptl proposed in eq 49 of ref 7. P*
0.85
tractive parts of the potential,“ or taking into account the correlation function between hard and soft forces, seems to us hopeless. In fact, the cross correlations between hard and soft forcesgtend to increase the {value, whereas from Figures 1-4 { should decrease to meet the experimental value; an attempt to change the hard-core values to r < Q only has the effect to increase D predicted by RA toward the limit of the K prediction, which, as we have seen, is not enough.
10 -*A N
25
20
15.
e ’I 0.65
Flguro 4. D’ vs. p’ at T’ In Figure 3.
0.75
I
0.85
= 1.80. The symbols are the same as
can deduce wrong conclusions about the validity of the theory. In conclusion, from Table I and Figures 1-4 we uul state that the Kirkwood equation does not agree with the experiments. If besides this disagreement we considler also the failure of the plateau-value hypothesis as pointed out by the authors of ref 3, we must conclude that, at lsast for simple fluids, the K kinetic theory of transport prlocesses must be abondoned. The same conclusion for the RA approach can be drawn. Any attempt to improve the friction constant formalism choosing different ways to separate the repulsive ,and at-
Conclusions In this report we have shown unambiguously that both the Kirkwood and the Rice-Allnatt equations for D are unable to give good estimates of the self-diffusion coefficient in a Lennard-Jones fluid. The main difference is in the density and temperature dependence, which is much lower than in the case of “experimental” data. The fact that some claimed, in the case of simple fluids, a moderate agreement between theories and experimental data in a limited range of p and Tis meaningless since this agreement is due to a fortuitous choice of temperature and density. Indeed, the theoretical values meet the experimental ones only near the melting point just where the previous comparisons were made. What remains an open question is why the RA theory was able to predict the ionic mobility in liquid Kr as was reporteda for measurements along the coexistence curve. It is worth noting that in this case the temperature range is large enough to exclude a fortuitous agreement. The agreement is probably due to the fact that the ion size is much larger than the Kr atom; therefore, the hypothesis that there exists a plateau value for eq 3 is more reliable. Computer simulation for ion dynamics in a L-J fluid should be very helpful to understand this point. (9)J. A. Palyvos and H.T.Davis, J. Phys. Chem., 71, 439 (1967).
