Velocity dependence of friction

ChemicalPhysics 152 (1991) 221-228 North-Holland

Velocity dependence of friction S.-B. Zhu, Sutjit Sin&

J. Lee ’ and G.W. Robinson

SubPicosecond and Quantum Radiation Laboratory. P.O. Box 4260, Texas Tech University, Lubbock, TX 79409. USA Received 30 April 1990

Here, we carry out a set of nonequilibrium molecular dynamics calculations for a bath of host particles containing one small test particle. The test particle is under the influence of a fluctuating external force which imparts a constant terminal velocity along one particular direction. We are thus able to investigate the velocity dependence of the microscopic friction. If the velocity of the test particle is comparable to or exceeds the average thermal velocity of the host particles, nonlinear effects arise that cause a breakdown of Stokes’ law. The friction obtained can be successfully tit to a function derived from a simple model.

1. Introduction Microscopic friction plays a central role in applications of the Langevin equation [ 1 ] to the examination of stochastic variables in dynamical processes. The Kramers [ 21 formulation, or its modifications [ 3-61, for solving barrier crossing problems provides an important example. Brownian motion theory [ 7 ] was developed on the basis of hydrodynamic concepts in which the force caused by the friction is a linear function of both the velocity v and the radius a of the Brownian particle. However, it is important to recognize that Stokes’ relation [ 8 ] f= 6n~av

(1)

between the friction force fand the viscosity coeffrcient rl of the medium is valid only for streamline flow past a particle [ 9 1. Macroscopic friction laws have undergone a number of generalizations [ lo- 121. These include consideration of non-Markovian effects [ lo], mass effects [ 111, and cage effects [ 12 1. The behavior of a small (both in mass and size) hard sphere [ 13 ] or Lennard-Jones [ 141 test particle in the dense fluid phase has been numerically studied by the use of equilibrium molecular dynamics methods. Large deviations of the transport properties from the Enskog ’ Present address: Spectral Sciences, Inc., 111 South Bedford

theory of uncorrelated motion have been found, indicating that highly collective effects must be involved [ 151. Eq. ( 1) becomes inapplicable if the velocity of the particle becomes very large. This situation may occur in ultrafast dynamical processes even at the microscopic level. At least two complications are introduced. First, the friction is mainly determined by the loculstructure and properties of the nearby fluid (the “cage”), which, in general, are changed from those in the bulk fluid. These differences between cage and bulk fluid are sensitively dependent upon the dynamical behavior of the test particle. In particular, if the velocity of the test particle becomes comparable to or exceeds that of the host particles, local compression of the fluid [ 16 ] may occur, creating a nonlinearity in the microscopic friction. A second type of complication is caused by the comparatively slow response of the surrounding host particles to changes in state of a rapidly moving low-mass test particle. This brings in non-Markovian memory effects, which give rise to a frequency dependence of the friction [ 17 1. Most literature has laid an emphasis on this frequency dependence [ 3,18 1, while much less attention has been paid to the nonlinearity of the friction [ 11,19,20]. It is therefore the main purpose of the present paper to discuss the velocity dependence of the microscopic friction in condensed phases as a consequence of nonlinear effects.

Street, Burlington, MA 01803, USA. 0301-0104/91/$03.50

0 1991 - Elsevier Science Publishers B.V. (North-Holland)

S.-B. Zhu et al. / Velocity dependence offriction

222

2. Background 2.1. Theory Consider the one-dimensional stochastic motion of a test particle acted on by a time dependent external force A (t ). It is assumed that this process can be described by a phenomenological Langevin equation

$/l(t)-

ji(

u,t-7)

v(z) dT+R(u,t)

,

0 where the mass of the test particle has been absorbed into the other variables, and C( v,t) denotes the memory kernel, which here is assumed to be a function of both the time and the test particle velocity ZLConcomitantly, the random force R(u,t) must also depend on v. With certain specific forms of the various functions given, the Laplace transform of eq. (2) can be determined. Such results indicate of course that the frequency dependence and the velocity dependence are mixed in the solutions to this equation. There are two factors that may affect the value of [(v,t). The finite interaction time caused by a relatively large velocity of the test particle always tends to decrease the microscopic friction. This non-Markovian effect has been extensively studied [ 3,17 1. In addition to this first factor, an ultrafast motion, where the velocity of the test particle exceeds the mean thermal velocity of the host particles, may be affected by a nonlinear behavior, resulting in local turbulence and shock waves. If this occurs, there could be a considerable increase in the microscopic friction [ 161. Such nonlinear effects are a principal topic of gas dynamics but have rarely been taken into account in stochastic processes in the condensed phase. Thus, the two factors, both of which increase in importance with increasing test particle velocity, influence the effective microscopic friction in opposite ways. The first decreases it, the second increases it [21], and the combination tends to produce an extremely complex picture for the ultrafast dynamics. This microscopic friction relates to an effective viscosity introduced in a previous paper [ 22 1. The mathematical prescription for dealing generally with a real system, where both non-Markovian effects and nonlinear effects in the velocity are pres-

ent, is not at hand. However, in this paper we use a simplified formalism. If the external force A(t) is artificially adjusted so that it exactly balances the resultant force from the heat bath at every instant of time, the test particle loses its accelerations and moves with a constant terminal velocity vo. To obtain a fast uniform motion, a large average force is required, and to maintain constant uo, the artificial force must fluctuate with time near this average value. This is possible to achieve in a computer experiment. When the terminal velocity is achieved, the friction might then be considered to depend parametrically on vo. In other words, at each vo, one could have a different memory kernel #‘. This is indeed what our final results will show. This friction function applies so long as the velocity of the test particle does not change during the experiment. Taking a time average of eq. ( 2 ) over the time span r, and keeping in mind that dvldt vanishes at each instant of time, we have, r ;

s 0

r A(vo,t)

dt+

s 0

I dt

r c(u,,,r)

I 0

dr-;

I 0

R(v,,t)

dt. (3)

Since the time variations in both A ( uo,t) and R ( vo,t) depend only on the random fluctuations in the bath, the former should average to a constant (A ( vo) ) and the latter should average to zero if r is sufficiently long compared with the relaxation times of the system. In this case, =voRvo)

7

(4)

where [ 23 ] r

s

&o,=)_li; (r-t)

C(v,,t) dt.

0

If the terminal velocity is measured and the friction is known, the averaged applied force can be calculated. On the other hand, if the averaged applied force and terminal velocity are known then the velocity dependent friction c( vo) can be determined. The +I’Both non-Markovian processes embedded in the time dependence of C(v,t) and nonlinear effects embedded in the velocity dependence of [( v,t) may enter the arena of this problem. A further discussion of this is presented later in the paper.

S.-B. Zhu et al. / Velocity dependence offriction

latter will constitute the method employed in this paper. It is also precisely the situation encountered in Stokes’ Law [8] laboratory determinations of friction (viscosity) when a sphere falls through a medium under the influence of the fixed gravitational force, as briefly described in the next section. 2.2. Experiment Consider a spherical body that falls in a viscous medium with an increasing velocity until the gravitational force is just balanced by the frictional resistance. The constant terminal velocity v. achieved creates the situation in a falling-sphere viscometer. In fact, using the Stokes’ equation ( 1 ), it is possible to obtain the viscosity of the medium by measuring u. [ 241 #2.With an added external electric field, this sedimentation method can also be applied to the measurement of the elementary electronic charge

[261. It would of course be very difficult to measure microscopic friction in an ultrafast dynamical process by the direct use of the above mentioned laboratory method. Nevertheless, the scheme can easily be effectuated through a molecular dynamics approach with the aid of computer simulations.

3. Computer simulations Computer simulations have the advantage of controlling experimental conditions that cannot be controlled in a real laboratory situation. In the present work, we perform a set of nonequilibrium molecular dynamics experiments on a test particle, whose mass m and size u, are small compared with the host particle mass M and size a,. This test particle is given a constant terminal velocity v. along the xdirection by the application of an external field. Specifically, we choose mjMz0.26 and a,,,/~~= 0.579, with a,,,,+,=f (a,+ trM). The system density p is 1.3028~7~~. As mentioned earlier, this steady motion can be realized for a microscopic test particle by imposing the constraint that at every instant of time the artificial external force on the particle just balances the solvent ** This

simple experiment is also briefly discussed in numerous text books of physical chemistry. See, e.g., ref. [ 251.

223

force. No external forces are exerted along they- and z-directions. Therefore, motions of the particle along these two directions are random in the zeroth order of approximation that neglects coupling with the orderly motion along the x-direction. By assuming that the average value of the stochastic forces along the xdirection vanishes over a very long time period, the time average of the applied artificial force needed to achieve the constant velocity v. thus equals, as indicated by eq. (4), the effective frictional force vo[( tr, ) on the test particle. To minimize the boundary effects with an affordable computational effort we consider a reasonably large sample which contains one test particle and 499 host particles. Standard periodic boundary conditions are employed to extend the size of the box. The host particles can be separated into two groups. Those whose distances from the test particle are less than 2. lo,,., are referred to as “cage particles”, while the others constitute a “heat bath”. The “heat bath” is assumed to be placed in a thermostat [ 27 ] having a temperature of 0.9174rM, where eM is the LennardJones interaction parameter of a host particle. Thus, velocities of the “heat bath” particles are adjusted at each time step so that they maintain this constant temperature. The test particle interaction parameter is chosen such that e,=O.3664~, and the standard combining law, emM=FM, is assumed. Also, the transport momentum of the entire heat bath is restrained to zero by adjusting the net velocities of the whole system to zero at each time step. The equations of motion are solved by the use of a modified version of the Verlet algorithm [ 27,281. These simulations are carried out on the CRAY YMP/832 located at the Pittsburgh Supercomputing Center. Over 20 CPU hours are required for the entire set of calculations.

4. Results By varying (A ( vo) ) , different values of v. can be achieved. It will be convenient to define a reduced velocity Pas the ratio of the terminal velocity v. of the test particle to the one-dimensional mean thermal velocity ,/m of the host particles. For the systems to be studied here, the one-dimensional mean thermal velocity Vof the test particle will not exceed

224

S.-B. Zhu et al. / Velocity dependence offriction

2. We can now symbolize the average microscopic frictional force from eq. (4) as&( V) = I’c( V) . This frictional force can be determined for different Vs, the resulting dependence being illustrated in fig. 1. It can be quite accurately fit to a simple function

fo=z 1 +#f’

(5)

2000 I

i

where y represents the linear friction coefficient and ,ygives a measure of the nonlinear effects. As seen in the figure, the friction force tends to saturate as the velocity becomes large, while, in the range of low reduced velocities ( V-SK1 ),fo is linear with V. The latter is the limit met in the Brownian motion problem. In general, the dependence of the friction on the velocity will be jointly determined by the nonlinearity and non-Markovian effects. To gain more intuition about this dependence, we plot in fig. 2 the mean square random solvent force (MSF= (R *) ) acting on the test particle as a function of the reduced velocity V. Note that in this problem ( RZ ) = (A)2 - ( A2). It is known from previous studies [ 2 1,291 that the mean square solvent force measures the initial value of the memory kernel, c( uo,t= 0). It is seen that this quantity remains nearly constant within the computational uncertainty of these runs. In barrier crossing problems [ 30-331, a constant MSF would correspond to a regime where the initial value of the memory kernel does not sensitively depend upon the barrier height. However, the functionality [ 34,35 ] of the memory kernel, according to eq. ( 5 ), must still depend on the dynamical behavior of .,._.

I

Reduced

Velocity

Fig. I _Microscopic friction force fo( v) versus reduced velocity V. Circles:computerexperimentalresults;solid curve:best tit from eq. (5) with y=56.11 andX=0.4317.

?3:5 Reduced

Velocity

Fig. 2. Mean square forces (in reduced units) acting on the test particle.

the test particle. Unfortunately, in the present work, we are not able to provide much detailed information on the functionality of the memory kernel, since the fluctuation-dissipation relation is not necessarily valid for these nonequilibrium systems [ 36-381, and, besides this, only a limited range of velocities could be studied. This brings up another limitation of this work: the velocity range of the test particle can span only moderate reduced velocities in order to guarantee the complete relaxation of host particles during the time that the test particle passes through the sample box. For V=2, the maximum velocity that was used, the passage time of the test particle through the distance of one box length equals 3.792 where r= ~,&M/E~) ‘12.This is long compared with the relaxation time of the host particles. In fact, considering the random motion of the test particle along the yand z-directions and the fact that the test particle does not enter the sample box at the same point as its last passage, we could have allowed the test particle to move with a somewhat larger constant velocity than implied by the above conditions. In any case, the limit VG 2 should give reliable results with respect to this issue. To make this section more definitive, we can illustrate the results obtained by choosing as an example a specific analytical form for the velocity dependent friction in eq. (2). A simple choice would use the exponential memory kernel, C( VJ) = a exp( - Bt) in reduced velocity space, where the parameters cy and /3 may be functions of the reduced terminal velocity V. Using eq. (4), one then finds,

225

S.-B. Zhu et al. / Velocity dependence offriction

(6)

fo(V=~~(VlB(V.

We note that the actual form of cy( V), the memory kernel at t=O, is given by the curve in fig. 2, which we will assume for all practical purposes to be a constant oo. Assuming that B can be expressed as a series expansion, /.?( V)=/30( l+bl

V+baV-Z+...)

)

from eq. (5) we find

This means that r=c~~//?~ and x=6,. The friction function saturates with increasing V as shown in fig. 1. This construction then provides an explicit form for c( u,t), which can then presumably be used in time dependent problems. Certainly, this analysis is a bit oversimplified because of the limitation V< 2. Higher powers of V do not therefore enter our considerations. Thus, many different (and more realistic) functional forms of the friction kernel would be consistent with the results found here. However, we believe this diversion is illustrative of the point we are trying to make, namely that the velocity dependence and the time dependence in these types of problems are mixed. In this particular case, the relaxation of the memory kernel becomes faster as the test particle velocity increases. This causes the problem to appear more Markovian at higher test particle velocities. In some problems too a spatial dependence is present [ 39,401. Thus, an analytical unravelling by various transformations of all these various dependencies in real ultrafast dynamical friction problems would seem to be a near hopeless task. The concept of an effective friction [ 22 ] is thus all the more inviting to experimentalists.

5. A simple friction model In order to describe more physically the motion of a test particle in viscous flow, we now consider an ultrasimple model introduced by H. Eyring and described on page 627 of ref. [ 4 11. Imagine that the liquid has a lattice-like structure and that the molecules are confined to their individual cells by their neighbors. To move to a neighboring vacant site, the test

particle must pass through a “bottleneck” region of high potential energy. The maximum of this potential energy function can be assumed to lie midway between the original and final state lattice sites and rep resents an “activated state” [ 42,431. If there is an external force on the test particle, the potential energy of the “bottleneck” becomes distorted. Let &be the frequency of jumping in the “forward direction” along the field direction, and let k, be the frequency corresponding to the “reverse direction”. Then the net velocity of the test particle with respect to the lattice sites in the direction of the applied force can be expressed as (8)

2

v=b(h-k,)

where b denotes the distance between lattice sites. The frequencies can be evaluated using the Eyring theory of transport phenomena [ 42,43 ] kf=koe

k,-b

+blZkeT

(9)

,

e-b/2keT

(10)

Here ko represents the frequency in the absence of the external force and b(A) /2 is the work done by the external force (A ) to move the test particle a distance b/2 into the activated state. This work represents the change of the free energy of activation in the two directions. Substitution of eqs. (9) and (10) into eq. (8) yields v=2bk,sinh(b(A)/2kBT). Solving for (A) (A) =yln

(11)

in eq. ( 11 ), one obtains &+J(v/2bb)2+

1. (12)

1

If the force (A) is such that the test particle reaches a steady state, the net velocity v becomes the terminal velocity vo. Note that only in the limit where the net velocity is much smaller than the jump velocity in the absence of the external force, i.e. when u*: 2bb, are v and (A) linearly related. Otherwise, the effective viscosity (proportional to (A) /v) is dependent on the force itself and the flow is non-Newtonian [ 441. The implication is that when the external force is comparable to or exceeds the average stochastic forces, nonlinear effects are expected to set in. Thus, eq. ( 12 ) and eq. ( 5 ) provide fully independent as-

S.-B. Zhu et al. / Velocity dependenceoffriction

226

sessments of the nonlinear velocity dependent microscopic friction. Fig. 3 shows that the data obtained in section 4, which were seen in fig. 1 to be successfully lit to a simple empirical function, can also be fit to eq. ( 12 ) with b=0.0557aM and k,,=5.094d-i. The apparently small value of the fitting parameter b, the “distance between lattice sites”, is caused partly by shortcomings of the Eyring model and partly by the fact that b is an effective parameter for the test particle in the fluid host bath. It evidences the importance of correlated motion in this problem [ 15 1. It is evident from fig. 4 that eq. (5) and eq. ( 12) are almost identical in the range L’S 5, which certainly encompasses the actual range of V used in the present computations. Therefore, throughout a rather wide range of reduced velocities, the microscopic friction can be expressed by the simple functions de-

V

too

I

0.50

p

1

1 .oo

Reduced

s

r

7

1.50

I

2.00

8

2.50

Velocity

Fig. 3. Same as for fig. 1 except that the solid curve is now replaced by eq. (12) with b=O.O557u,,,and k0=5.094Jm.

120

(

I

96-

C .-0

72_

2.00

4.00

Reduced

6.00

8.00

Velocity

Fig. 4. A comparison of eq. (5) (dashed curve) and eq. ( 12) (solid curve) for V= 0- 10.

scribed by either of these equations. While eq. (5) has the advantage of simplicity, eq. ( 12) provides a certain amount of physical insight into the origin of the nonlinearity.

6. Conclusions By comparing curves depicted in figs. 1 and 2, we conclude that the dynamical processes for a test particle system studied in the range of low reduced velocities are approximately linear. They are also likely to be fully Markovian providing the size of the test particle is considerably larger than that of a host particle. In this range the hydrodynamic concept and the Brownian motion theory provide an adequate description for these relatively slow motions. However, as the reduced velocity approaches and then exceeds unity, nonlinear and non-Markovian effects become of prime importance. Frequency dependence describes this behavior, but the frequency to be used is dependent not only on static parameters, such as barrier curvature, but also on dynamic parameters such as test particle velocity. In order to supply a full picture for ultrafast dynamical processes, it is thus necessary to take into account these nonlinear velocity effects. The above two factors were seen to affect the microscopic friction in opposite ways, the first decreasing the friction, the latter increasing it. In fact, the results here have indicated that for these systems, when nonlinear effects are present, at high velocities the system becomes more Markovian than at low velocities. As a consequence of the nonlinear effects discovered here, to describe a stochastic process in inhomogeneous systems involving small (both in mass and size) “test particles” dissolved in a bath of large host particles, a nonlinear Langevin or Fokker-Planck equation must be introduced [ 19,20,45,46]. This means that in such a case the standard Grote-Hynes theory [ 3 ] describing barrier crossing rates in the nonMarkovian, but linear, regime gives an incomplete picture. Also studied in this paper is the velocity dependence of the mean square forces acting on a test particle. This provides a measure of the initial value of the memory kernel. This was found not to vary significantly with velocity over the range covered. Such

S.-B. Zhu et al. / Velocity dependence offriction

studies, when extended to systems with greater nonlinearity, should supply important information on the applicable range of conventional linear approximations in the theoretical evaluation of chemical reaction rates. One final comment should be made. Though the computations presented in this paper are based on nonequilibrium (steady state) dynamics, the general conclusions that we have obtained for the microscopic friction are independent of the way in which they were derived. Because of this friction form, which is nonlinear in the test particle velocity, it is easy to see that the fluctuation-dissipation relation [ 36-381 breaks down and non-Maxwellian velocity distributions occur [ 45-481 #3. Particles having different velocities will simply undergo a different type of dynamics. Since this dynamics varies on a timescale which is so fast that the heavy host particles cannot respond, no amount of equilibration time will smooth out the deviations from Maxwellian. Because only a finite number of cage particles contact the test particle, the distribution function of the system initially out of equilibrium will never approach its equilibrium form [ 501.

Acknowledgements Financial support at the SPQR Laboratory has been shared by the Robert A. Welch Foundation (D-0005, 50% and D-1094, 5Oh), the National Science Foundation (CHE86 1138 1, 28%) and the State of Texas Advanced Research Program ( 1306, 17%). Computer time was furnished by the Pittsburgh Supercomputing Center. 113This work on the breakdown

of the Maxwell velocity distribution for a quasidissociative diatomic molecule [48] has been criticized by Keirstead and Wilson [49]. However, results in refs. [ 471 and new results [ 5 1] on the quasidissociative diatomic molecule indicate that the Maxwell velocity distribution is either not easily reached or is never reached in systems where a low-mass test particle subjected to large force fields is immersed in a heavy particle heat bath.

References [ 1] P. Langevin, Compt. Rend. Acad. Sci. (Paris), 530.

146 ( 1908)

227

[2] H.A. Kramers, Physica 7 (1940) 284. [ 31 R.F. Grate and J.T. Hynes, J. Chem. Phys. 73 ( 1980) 1392. [4] B. Carmeli and A. Nitzan, Phys. Rev. Letters 49 ( 1982) 423. [ 51 P. Hanggi, Phys. Rev. A 26 (1982) 2996. [6] E. Pollak, H. Grabert and P. Hang& J. Chem. Phys. 91 (1989) 4073. [ 71 A. Einstein, Ann. Physik 17 (1905) 549. [S] G.G. Stokes, Trans. Cambridge Philos. Sot. 9(8) (1851). [ 9 ] ET. Gucker and R.L. Seifert, Physical Chemistry (Norton, New York, 1966). [IO] J.T. Hynes, J. Chem. Phys. 57 (1972) 5612. [ 111 J.T. Hynes, J. Chem. Phys. 62 ( 1975) 2972. [ 121 J.T. Hynes, R. Kapral and M. Weinberg, J. Chem. Phys. 70 (1979) 1456. [ 13 ] B.J. Alder, W.E. Alley and J.H. Dymond, J. Chem. Phys. 6 I (1974) 1415. 114 K Toukubo, K. Nakanishi and N. Watanabe, J. Chem. Phys. 67 (1977) 4162. 115 B.J. Alder, D.M. Gass and T.E. Wainwright, J. Chem. Phys. 53 (1970) 3813. and E.M. Lifshitz, Fluid Mechanics 1’6 ‘IL.D. Landau (Pergamon, London, 1959). 117 H. Mori, Progr. Theoret. Phys. (Kyoto) 28 (1962) 763. [ 181 J.L. Skinner and P.G. Wolynes, J. Chem. Phys. 69 ( 1978) 2143. [ 191 J.T. Hynes, R. Kapral and M. Weinberg, Physica A 81 (1975) 485. [20] J.T. Hynes, R. Kapral and M. Weinberg, Physica A 81 (1975) 509. [ 211 S.-B. Zhu, J. Lee and G.W. Robinson, J. Chem. Phys. 88 (1988) 7088. [22] J. Lee, S.-B. Zhu and G.W. Robinson, J. Phys. Chem. 91 (1987) 4273. [ 23 ] M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, with formulas, graphs and mathematical tables (U.S. Government Printing Office, Washington, 1972), eq. (25.4.58). [ 241 R.S. Bradley, The phenomena of fluid motions (AddisonWesley, Reading MA, 1967 ) . [25] E.A. Moelwyn-Hughes, Physical Chemistry, 2nd Ed. ( Pergamon Press, Oxford, 196 I ) p. 7 14. [26] J.D. Stranathan, The Particles of Modem Physics (Blakiston, Philadelphia, 1954). [27] W.G. Hoover,Ann. Rev. Phys. Chem. 34 (1983) 103. [ 28 ] W.C. Swope, H.C. Andersen, P.H. Berens and K.R. Wilson, J. Chem. Phys. 76 (1982) 637. (291 R. Kubo, Rep. Prog. Phys. 29( 1) (1966) 255. [ 301 G. Ciccotti, D. Frenkel and I.R. McDonald, eds., Simulation of Liquids and Solids. Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics, (North-Holland, Amsterdam, 1987). [31] G. Ciccotti and W.G. Hoover, eds., Molecular-Dynamics Simulation of Statistical-Mechanical Systems (NorthHolland, Amsterdam, 1986). [32] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids (Clarendon, Oxford, 1987).

228

S-B

Zhu et al / Vekmty dependence offrnctlon

[ 331 .l.A McCammon and S.C. Harvey, Dynamtcs of Proteins

and Nucleic Acids (Cambridge Umv. Press, Cambridge, 1987) [ 341 S.-B. Zhu, J. Lee, G.W. Robmson and S.H. Lin, Chem. Phys. Letters, 148 (1988) 164. (351 S.-B. Zhu, J. Lee, G.W. Robmson and S.H. Lm, J. Chem. Phys. 90 (1989) 6335 [ 361 M.W. Evans, J. Chem. Phys. 76 ( 1982) 5480. [ 371 M. Ferrano, P. Grigolini, M. Leoncmi, L Pardi and A. Tam, Mol. Phys. 53 (1984) 1251. [ 38 ] S.-B. Zhu, J. Lee and G.W. Robinson, J. Opt. Sot. Am. B 6 (1989) 250 [39] B. Carmeh and A Nitzan, Phys. Rev. Letters 51 ( 1983) 233;Phys.Rev A29 (1984) 1481. [ 401 Smut Smgh, R. Krishnan and G.W. Robinson, Chem. Phys. Letters 175 ( 1990) 338. [41] J 0. Hirschfelder, C.F. Curt~ss and R.B. Bard, Molecular Theory of Gases and Liquids (Wiley, New York, 1954 )

[42] H. Eyrmg, J Chem. Phys. 4 (1936) 283 [43] S. Glasstone, K.J. Laidler and H. Eynng, Theory of Rate Processes (McGraw-Hill, New York, 194 1). [44] M Remer, Deformation and Flow (Lewis, London, 1949) [45 ] S -B Zhu, S. Singh and G W Robmson, Phys Rev A 40 (1989) 1109. [46] S.-B. Zhu, Phys. Rev A 42 (1990) 3374. [47] S.-B. Zhu, J. Lee and G.W. Robmson, Chem Phys. Letters 163(1989)328,169(1990)355;170(1990)368. [48] S.-B Zhu and G.W. Robmson, J Phys. Chem. 93 (1989) 164 [ 49) W.P. Keustead and K.R. Wilson, J. Phys Chem. 94 ( 1990) 918 [ 501 I. Oppenheim, Progr. Theoret Phys (Kyoto) Suppl. 99 (1989) 369 [ 5 1 ] S.-B. Zhu and G W Robmson, J Phys. Chem , m press.