Velocity and angular distributions of molecules emitted from a liquid surface

21 February 1997

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 266 (1997) 161-168

Velocity and angular distributions of molecules emitted from a liquid surface L.F. Phillips Chemistry Department, University of Canterbury, Christchurch, New Zealand Received 13 August 1996; in final form 6 January 1997

Abstract

Evaporation from the surface of a liquid is considered from the viewpoints of the kinetic theory of gases, capillary-wave theory, and statistical reaction-rate theory. The velocity vectors of molecules emitted from a planar surface are concentrated around the normal to the surface and their velocity distribution is non-Maxwellian. The abnormal velocity distribution and high mean kinetic energy of molecules emitted from a liquid surface provide a mechanism for coupling of heat and matter fluxes in the gas within a few mean paths of the surface.

1. Introduction

A well-known difficulty with current theoretical treatments of evaporation and condensation is that abnormal gas-phase temperature and composition profiles are predicted to exist between two liquid surfaces held at different temperatures. The slope of the calculated temperature profile is often found to be in the direction opposite to the applied temperature difference, under which conditions the vapour close to the surface from which evaporation is taking place is predicted to be supersaturated. Bedeaux et al. [1] term this result a remarkable phenomenon; Koffman et al. [2] consider it so unacceptable as to cast doubts on the fundamental theory. There are two aspects of current theoretical treatments which might give rise to this difficulty. One is the general assumption [2-5] of an unperturbed Maxwell distribution of velocities for molecules evaporating from a liquid surface, an assumption which we shall show to be incorrect. The other is the omission of a term for

coupling of macroscopic heat and matter fluxes through the liquid surface via a heat of transport which involves the latent heat of vaporization [6]. This coupling is absent from theoretical treatments based on solving the Boltzmann equation, and has also been omitted from most treatments using irreversible thermodynamics [1]. One approach to predicting the velocity distribution for molecules evaporating from a surface is to suppose that the molecules have a Maxwellian distribution of velocities prior to evaporation, albeit with a very small mean free path, and that the molecules which escape from the surface have a velocity distribution which can be obtained by shifting the zero of the kinetic energy scale of the original distribution by an amount equal to the height of the barrier provided by the heat of vaporization, and then discarding molecules for which the resulting kinetic energy is negative. When an exponential curve is truncated in this way it remains the same exponential curve, so the velocity distribution which is arrived at

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L.F. Phillips~Chemical Physics Letters 266 (1997) 161-168

for the escaping molecules is the Maxwell distribution corresponding to the temperature of the surface. However, this reasoning is over-simplified because it ignores the fact that evaporation is a dynamic process for which the probability of escape is not necessarily equal to the probability of possessing sufficient energy to surmount the activation barrier. In his book on the kinetic theory of gases, published in 1938, Kennard [7] asks the student to show as an exercise that the mean translational energy of molecules crossing a surface S with a component of velocity normal to the surface is 2 RT per mole, and not the ~RT per mole which corresponds to a Maxwell distribution of randomly directed velocities in the bulk gas. This result applies strictly to an effusive flow, the area of the surface S being assumed small in comparison with the mean free path. If we suppose that the surface S is located adjacent to a perfectly absorbing solid or liquid surface, it follows that 2 RT is the average translational energy per mole of molecules absorbed by the surface. At equilibrium, detailed balance requires that the average translational energy of molecules desorbing from the surface and passing through S must also be 2RT per mole. For a real gas-liquid interface the situation should be similar to that of the perfect absorber, provided the sticking coefficient of molecules striking the surface is close to unity and the imaginary surface S can be located at a distance from the liquid which is both much greater than the rms displacement of the liquid surface due to thermal motions and much less than the mean free path in the gas. Except under extreme conditions of high gas pressure, low surface tension, or the absence of a gravitational field, these conditions are easy to fulfil. If the velocity vectors of escaping molecules on the liquid side of surface S are randomly oriented, the situation from the point of view of an observer on the other side of S is not distinguishable from one in which the molecules passing through S originated from a gas with a Maxwellian distribution of velocities. If, however, the velocity vectors of molecules leaving the liquid have a tendency to be directed normal to the liquid surface, as we shall show to be the case, the average energy of 2 RT for molecules crossing S must correspond to an average kinetic energy of motion normal to the surface which is significantly higher than the value given by a Maxwellian distri-

bution for motion in a single degree of freedom. Experimentally, deviations from a Maxwell distribution of velocities for desorbed molecules are wellknown in the field of gas-solid interactions [8,9] but seem not to have been sought in the field of gasliquid interactions. In this Letter we first derive an explicit expression for the rms displacement of a liquid surface due to thermally excited capillary waves, and hence show that the rms displacement for a water surface at room temperature is much less than the mean free path in air at a pressure of one atmosphere. We then consider the evaporation process in the framework of statistical reaction-rate theory [10], as the thermally activated unimolecular dissociation of an element of surface over the potential barrier provided by the latent heat of vaporization, and so obtain an expression for k(e), the first-order rate constant for escape of a molecule with excess kinetic energy e. This leads to a mean value of 2RT for the translational energy of molecules departing at right angles to a perfectly flat liquid surface and hence to a velocity distribution which is both non-Maxwellian and strongly peaked around the normal to the surface. In the language of transition-state theory, the critical coordinate for an escaping molecule is at right angles to the liquid surface, and molecules which escape at other angles are required to possess higher total kinetic energy. In practice, the predicted angular dependence of the velocity distribution is likely to be somewhat modified by departures of the liquid surface from planarity in the presence of thermally excited capillary waves. Very large deviations from planarity might even result in a near-random distribution of velocity vectors over the hemisphere, and hence lead to a Maxwellian distribution of velocities for the evaporating molecules. The present discussion of capillary waves suggests that such deviations from planarity are likely to be unimportant for a water surface under ordinary conditions.

2. Capillary waves

For the purpose of discussion we regard the liquid surface as a membrane one molecule thick and clearly differentiated from the fluid phase beneath, a membrane which is resistant to deformation and therefore

L.F. Phillips / Chemical Physics Letters 266 (1997) 161-168

capable of supporting normal modes of vibration in three dimensions. In this section we consider only vibrational modes normal to the liquid surface. We suppose that the motions of individual molecules which comprise the membrane are correlated over a maximum distance L which is given by half the wavelength at which capillary waves (waves for which the restoring force is provided by surface tension) are superseded by gravity waves [11]. Thus

L = rr~/(y/pg),

(1)

where y is the surface tension, p the density and g the gravitational acceleration. For wavelengths greater than 2 L the restoring force is provided by gravitation and the surface membrane plays no significant role. For a surface of dimensions greater than L x L we can imagine the surface to be divided into smaller areas with periodic boundary conditions, such that the maximum wavelength for pure capillary waves is governed by L. The value of L given by Eq. (1) is about 0.85 cm for water at room temperature. Widom [12] has termed the quantity L/~r the 'coherence length'; however, this is not to be confused with the coherence length ~ over which the motions of individual molecules are correlated and which amounts to a few molecular diameters in a liquid far from the critical point (see Ref. [13], part a). In the present work, the small size of ~ is regarded as a consequence of the incoherent superposition of the whole spectrum of capillary waves up to the maximum wavelength of 2L. The number of modes of vertical oscillation for a surface of area less than or equal to L2 is equal to the number of degrees of one-dimensional vibrational freedom for the discrete molecules in the surface, which is the same as the number of molecules N in the correlated area. When N has its maximum value this is equal to 7 r ( L / 2 o r ) 2, where ~ is the molecular diameter, and the correlated area is circular (although for convenience we shall use the solutions of the wave equation for a rectangular membrane as our basis set of vibrational modes). For a smaller surface the value of N is given by the actual area divided by o-2. Provided there is a non-zero gravitational field, this approach avoids the problem (see Ref. [13], part b) that the rms displacement and other properties of the surface diverge in the limit of infinite surface area.

163

The sum of kinetic and potential energies per unit area of surface for a capillary wave of amplitude a and wavenumber k ( = 27r/A) over deep water is given [ 11 ] by I

2_ 1.2

W = 7a y~ ,

(2)

where the elevation of the free surface ~ takes the form ~'= a c o s ( c o t - k x ) , and the mean-square displacement sr2 is ~I a .2 For thermal excitation, the energy WL2 per mode can be set equal to kBT, where k B is Boltzmann's constant, and the sum of ( 2 for all modes, which we label z 2, is then given by

z:=

(kBT/yL2)(Eg(k)/k2),

(3)

where g(k) is a degeneracy factor for the mode of wavenumber k and the sum is over all modes from k = kmin to k = kmax. We can write k = "trm/L, so the sum now runs from m = 1 to m = mma× and the degeneracy factor g(k) is replaced by g(m). Individual modes on an L × L surface in the xy plane are of the form

qg= qb,,,(t)sin(m~x/L)sin(n'rry/L),

(4)

where the range of n values is the same as the range of values for m and the energy is proportional to m 2 + n 2 = r 2, The number of states of a given energy is therefore equal to the length of the arc of a quarter-circle of radius r in the space of m and n. Hence the degeneracy factor g(m), which represents the spectrum of surface vibrations and should be accessible to experiment, is given by

g( m) = ~rm/2,

(5)

2 and the total number of modes N is ~rmmax/4, which fixes mmax. The sum in Eq. (3) can now be replaced by an integral over d m from 1 to rnmax, and the mean-square displacement becomes

z 2 = (kBT/4rr3,)

Ln(4U/v),

(6)

where N = "rr(L/2or) 2. This result differs from Eq. (4.243) of Ref. [13] mainly in the way that N is regarded as fixed by L. Substituting appropriate values for a water surface at 300 K and using o-= 3 × 10 -8 cm, we find N = 8 × 10 ~4, mmax = 3.2 × 1 0 7, which justifies the change from a sum to an integral, that the shortest wavelength A = 2L/mma x is 2tr (an algebraically exact result which is also physically

164

L.F. Phillips/Chemical Physics Letters 266 (1997) 161-168

reasonable) or 6 X 10 -8 cm, on which scale the macroscopic surface tension 3' is liable to be only an approximation to the true restoring force, and that the rms displacement of the surface is 4.0 × 10 -8 cm. The mean free path in air is about 10 -5 cm at atmospheric pressure, so there is no difficulty in locating Kennard's imaginary surface S at a height which is much greater than the rms displacement and much less than the mean free path. The small size of the rms displacement given by Eq. (6) indicates that the liquid surface is highly planar on the scale of the mean free path. This is unlikely to be true on the scale of the molecular diameter o-, so we must also consider the effect of capillary waves on the slope of the surface. For a wave of the form sr = a cos(o~t- kx) where the thermal amplitude a is {(2kBT/L2k2T), the mean square value of the slope over a whole number of wavelengths is easily shown to be kBT/L23" and is the same for every mode, which is quite an interesting result. If we consider the basis set of surface modes to comprise equal numbers of modes along the x and y axes, we can obtain an 'rms slope' of the surface along a fixed axis by multiplying the mean-square slope for a single mode by half the number of modes N and then taking the square root. The result is

( O(/Ox )r~s = ~/(rr kBT/ 83"o"2).

(7)

For a water surface at 300 K the rms slope works out to be 0.5, which is far from negligible. The high rms slope on a molecular scale will have the effect of blurring the angular distributions that we derive later in this Letter. Nevertheless, it is still insufficient to produce a random orientation of velocity vectors for molecules escaping from the liquid surface, and so not invalidate the conclusion that the escaping molecules have a non-Maxwellian velocity distribution.

3. Kinetics of evaporation

number of states accessible to the transition state of the system between the top of the barrier and energy E in degrees of freedom other than the critical coordinate for reaction, h is Planck's constant, and p(E) is the density of energy levels of the initial 'reactant' state. For our system, E is the total vibrational energy of an area of liquid surface over which the vibrational motions of N molecules are correlated in the sense that they participate in normal modes of vibration for the area as a whole. The actual value of N proves to be unimportant in this context, provided only that N is large enough for (1 + x/3N) 3N to be replaced by e "~ in the derivation of Eq. (12). The much more numerous molecules in the fluid beneath the surface serve both as a heat bath, with which the surface layer can exchange energy, and to provide replacements for molecules which are lost by evaporation. The value of k(c) = k(E*) is to be calculated for a potential surface of the form shown in Fig. 1, where E * is the total energy and e the kinetic energy in the critical coordinate for the process, i.e. the coordinate for translation at right angles to the liquid surface, and the barrier height q is the enthalpy of vaporization of one molecule. In the transition state, the critical coordinate is assumed to have the character of a vibrational mode of frequency v *. For this system we use a modified version of Eq. (8), namely

k(e) =

w + - ( e ) w ( E - e * , 3 N - 1) hp(E, 3N)

(9)

for one of the N equivalent dissociation channels, where 3N is the number of vibrations contributing to

E¢

Here we use the basic formula of RRK, RRKM, transition-state and other statistical reaction-rate theories, namely

k( e) = w( e ) / h p ( e ) , (8) where k(E) is the rate of crossing the potential barrier for a system of total energy E, w(E) is the

Fig. 1. Section through the gas-liquid potential energy surface, where E is the energy in the critical translational coordinate for an escaping molecule, q is the enthalpy of vaporization per molecule, and E ~ is the total energy in the reaction coordinate.

L.F. Phillips~Chemical Physics Letters 266 (1997) 161-168

the density of states p(E, 3N). The number of states is written as a product of we(e), the number of states associated with surface reformation when the escaping molecule has a kinetic energy e = E ~ - q , and w(E - E*, 3N - 1), the number of states for a correlated area of surface with energy E - E ~ and possessing only 3 N - 1 vibrations because it lacks the critical mode u ¢. When the emitted molecule departs at right angles to a perfectly flat surface (Fig. 2a) the quantity w~(s) is simply e / h v ~, because the escaping molecule is replaced by a molecule from the fluid region with energy e in the vibrational coordinate u S. This is one extreme case. The other extreme case occurs when the emitted molecule is lost from the tip of a sharp-pointed wavelet (Fig. 2b) or from a very small cluster, when there appear to be no extra states associated with restoration of the surface and one might be tempted to write w~(8) = 1. However, additional states must be created by the addition of new molecules to the surface during formation of a wavelet such as that in Fig. 2b, so it cannot be accurate to put w*(e) = 1 in this case. For emission from a small droplet or cluster a correct treatment would have to include the motion of molecules beneath the surface and such cases are not considered here.

Escaping molecule

Surface

l

Liquid

165

The number of states w ( E - E * , 3 N - 1) is the integral over d E from 0 to E - E * of the corresponding density of states p(E - E ¢, 3N - 1), where the number N is large enough for the system to be treated classically. The classical density of states p(E) for a system with 3N vibrational modes of frequency 1,,i is given [10] by

p ( E ) = i=,FI

(3N-

1),'.

Hence

w(E-E*,3N-

1) = i= ,

~/

×(1 - e * / e )

Escaping molecule

Surface

b

Fig. 2. Diagrammatic sections through a liquid surface for extreme cases of w~(~), the number of states associated with restoration of the surface when a molecule escapes with kinetic energy e. (a) Escape at right angles to a planar surface, w~(~)= ~ / h v ~. (b) Escape from the tip of a sharp wavelet, w~(~) = I (ignoring new states that result from formation of the wavelet).

(3N-

1)!

(ll)

and we note that for the limiting case of Fig. 2a the missing factor 1 / h v ~ in the extended product of Eq. (11) will reappear when we multiply by w~(e). If we now use E = 3NkBT for the correlated area of surface and put 3 N - 1 - - 3 N in the exponent, we obtain, for the rate of emission at right angles to a perfectly flat surface,

k(e) = (e/h)exp(-[e+q]/kBT),

(12)

where E ~ = e + q and the units of k ( e ) are s -~ Integration of Eq. (12) over d e from zero to infinity and multiplication by the number of equivalent dissociation channels N, followed by division by the integral over d e alone, which is NkBT for the single degree of freedom corresponding to the critical coordinate, gives the thermally averaged rate constant for this process in the usual transition-state theory form

k( T) = ( kBT/h)exp( - Q / R T ) , Replacement

(10)

(13)

where we now use Q, the enthalpy of vaporization per mole, and R, the ideal gas constant, instead of q and k B. In arriving at Eq. (13), the upper limit of integration over e for the right-hand side of Eq. (12) was taken to be infinity rather than NkBT. This can be justified by noting that N is a large number (--~ 10~4), whereas q is only of the order of 10kBT and the effective upper limit imposed on e by the Boltzmann factor in Eq. (12) is of similar magnitude. Therefore NkBT is so much larger than the range of e values for which k ( e ) is significantly greater than zero that it can be replaced by infinity in the integration limit without loss of accuracy.

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L.F. Phillips / Chemical Physics Letters 266 (1997) 161-168

For a polyatomic liquid the right-hand side of Eq. (12) should include an additional factor fr(e) for the ratio of the number of rotational states available to the escaping molecule to the corresponding density of torsional modes of the molecule when it is part of the liquid, but this is unnecessary for our present purpose. Multiplication by appropriate Boltzmann factors and integration over ~ must convert this ratio into the 'free angle ratio' of partition functions which appears in the transition-state theory treatment of the sticking coefficient [14]. Numbers and densities of states and partition functions for internal vibrations and rotations can be introduced in the same way. They approximately cancel from the expression for

k(T). The ratio of k(e) to k(T) is the fractional rate of emission of molecules having energy e in the critical coordinate, and so is a distribution function n(e) for the kinetic energy of evaporating molecules. Hence we obtain, for the limiting case of Fig. 2a,

n(e) = ( t:/kaT)exp( - e/kBT ) ,

(14)

which differs from a Maxwell distribution by the presence of the factor s before the exponential. As in the derivation of Eq. (13), the integral of Nn(~)de is normalized to unity by division by Nk~T. In view of the discussion following Eq. (7), it is considered that evaporation from a free liquid surface will closely approximate this case, but this is a point which needs to be tested experimentally. For the limiting case of Fig. 2b with the (doubtful) assumption that w*(e) can be set equal to 1, the equations corresponding to Eqs. (12), (13) and (14) are

k ( e ) : ~,*-exp( - [ e + q]/kBT),

(15)

k(T) = t , ~ e x p ( - Q / R T ) ,

(16)

and n ( e ) = exp( - 6/kBT),

(17)

and the escaping molecules do have a Maxwellian distribution of velocities.

4. Mean energy and velocity of escaping molecules We first consider motion in the direction of the critical coordinate, i.e. at right angles to the liquid

surface, for the limiting case of Fig. 2a. By integrating e n ( e ) d e from zero to infinity we obtain the mean kinetic energy of an evaporating molecule as er~ = 2 k B T ,

(18)

which is the same as the mean energy given by Kennard [7] for molecules crossing a surface S. By integrating u n ( e ) d ~ from zero to infinity we obtain

u m = g(9"rrkBT/8m )

(19)

for the mean velocity u m along the normal to the liquid surface. The mean energy E m refers to kinetic energy of translational motion in a single degree of freedom, so the value given by Eq. (18) is four times the value that would correspond to a Maxwell distribution at the same temperature. Similarly, the velocity u m should be compared with the mean velocity in a fixed direction, ~/(2kaT/'rrm), which is smaller by a factor of 3"rr/4. In practice these conclusions will be modified by the presence of horizontal modes of vibration of the surface, which will cause the emitted molecules to have a component of velocity parallel to the liquid surface. In view of Kennard's result, Eq. (18) must then give the total translational energy, and the manner in which this energy is distributed between vertical and horizontal components of velocity will depend on the detailed dynamics of the surface. The critical coordinate for escape is at right angles to the liquid surface and we are assuming that it is only the component of velocity along the normal which is effective in overcoming the activation barrier. Therefore the total kinetic energy of a molecule leaving the surface at an angle 0 to the normal is equal to the energy in the critical coordinate divided by cos20, and the intensity of emission into a small element of solid angle at an angle 0 to the normal, relative to that at 0 = 0, with a fixed amount of energy e in the critical coordinate, is given by

I( O, e)/l(O, e ) = e x p ( - e tanZO/kBT).

(20)

In Fig. 3a the right-hand side of Eq. (20) is plotted as a function of 0 for several values of e/kBT, and the emission is seen to be quite concentrated around the normal to the surface. For a fixed angle 0 to the normal, the mean velocity is urn/cos 0 and the mean energy is Em/COszO. TO obtain the relative probability of emission at an angle 0 to the normal for an

L.F. Phillips/Chemical Physics Letters 266 (1997) 161-168

F.JRT = 4

\, " \ ,

\\

0.2

~

=

,

0 L,_ . . . . . . . . . . . 0 10

-

~. . . . 30

20

degrees

i .,~.~'~-~- ."~:~--- ~ 40 50 60

,

:

' 70

.

have non-random directions of velocity and high kinetic energy in comparison with those in the gas phase at the same nominal temperature. The transfer of heat from the liquid to the gas phase, in the form of excess energy carried by the evaporating molecules, provides a plausible mechanism by which the heat of vaporization might become involved in the 'heat of transport' for the process throughout a coupling region which includes the liquid surface, as was previously concluded on the basis of macroscopic arguments [6] and expressed in the equation

away from normal

J = -DmCm{(Q*/RTm)7"/T on

" =

/,/

-

0.6 ~

/) /

J

,\\ \

I/ ~~ / 0

i

'/

o., t 0.~'

!

•

10

20

30 degrees

167

40

50

60

70

away from normal

Fig. 3. (a) Intensity of emission of vaporizing molecules at an angle 0 to the normal relative to intensity along the normal, with constant kinetic energy e in the critical coordinate. Curves calculated for E / R T = e/kBT values I, 2, 3 and 4. (b) Relative probability of emission at an angle 0 to the normal for the same E/RT values as in (a).

individual molecule, the right-hand side of Eq. (20) must be multiplied by sin 0. The resulting function, normalized to a peak value of unity, is plotted in Fig. 3b. It is apparent that the extreme case of Fig. 2a actually has a vanishingly small probability of occurrence. Nevertheless, the most probable direction of emission is not far from the normal to the surface. As noted in the introduction, deviations from planarity of the surface need to be very large in order to modify these conclusions to such an extent that the distribution of velocity vectors over the hemisphere becomes random.

5. Discussion

The main conclusions which can be drawn from this work are that the newly evaporated molecules

m + C'm//fm}.

(21)

Here J is the steady-state gas flux through the surface, a subscript m signifies a gas-phase value measured adjacent to the surface, D m is the diffusion coefficient, Cr, is concentration, a prime signifies a derivative with respect to distance from the surface, Q* = Q - C p Tm is the heat of transport, and Cp is the constant-pressure heat capacity of the gas. The region over which flux coupling can occur by this mechanism is the region where the flux of energetic molecules from the surface is significant, and its thickness is therefore limited to a small number of mean free paths in the gas. This is consistent with recent model calculations [15] for the experimental data of Liss et al. [16] in which the thickness of the coupling zone was allowed to vary over the range from many thousands of mean free paths down to a fraction of a mean free path. In order to reproduce the experimental data it was necessary to use Eq. (21) to describe the flux through the layer of gas immediately above the surface, but the thickness of the coupling zone was required to be only a few mean free paths in air at atmospheric pressure. (Earlier calculations with the same data [17-19] assumed that the coupling region comprised the whole of the stagnant air layer between the liquid surface and the zone of turbulent transport.) The phenomenon of thermal diffusion in the gas phase is inherent in the Boltzmann equation, and appears in all kinetic-theory treatments of transport in a homogeneous gas. However, the coupling of heat and matter fluxes via the heat of vaporization Q for a heterogeneous system derives from Onsager's reciprocal relations and is not deducible from the

168

L.F. Phillips / Chemical Physics Letters 266 (1997) 161-168

Boltzmann equation. Further work is needed to find whether inclusion of this coupling in a kinetic-theory model leads to a result which is equivalent to Eq.

(21). With regard to the apparent paradoxes encountered in the two-surface problem, one may note that the heating effect produced by energetic molecules leaving the surface might, under some conditions, cause the temperature of the gas adjacent to the surface to be higher than the surface temperature, even though the overall temperature gradient were such that the gas would be expected to have a lower temperature than the surface. Also, under conditions where the main driving force for gas transport in Eq. (21) was the temperature gradient rather than the concentration gradient, it would be possible for the concentration adjacent to the surface from which evaporation was occurring to be greater than that corresponding to the saturated vapour pressure. Conditions such as these are approached by some of the wind-tunnel data of Liss et al. [16] and probably are present in the field data of Smith and coworkers [17,20-22]. Thus the paradoxes do appear to have some basis in reality.

Acknowledgements I am grateful to J.S. Rowlinson and I.W.M. Smith for their helpful criticisms of this Letter. This work was supported by the Marsden Fund.

References [1] D. Bedeaux, J.A.M. Smit, L.J.F. Hermans and T. Ytrehus, Physica A 182 (1992) 388. [2] L.D. Koffman, M.S. Plesset and L. Lees, Phys. Fluids 27 (1984) 876. [3] Y.-P. Pao, Phys. Fluids 14 (1971) 306. [4] T. Matsushita, Phys. Fluids 19 (1976) 1712. [5] J.W. Cipolla, Jr., H. Lang and S.K. Loyalka, J. Chem. Phys. 61 (1974) 69. [6] L.F. Phillips, J. Chem. Soc. Faraday Trans. 87 (1991) 2187. [7] E.H. Kennard, Kinetic theory of gases (McGraw-Hill, New York, 1938) p. 64. [8] J.C. Tully, in: Kinetics of interface reactions, eds. M. Grunze and H.J. Kreuzer (Springer-Verlag, Berlin, 1987) p. 37. [9] C. Ning and J. Pfab, Chem. Phys. Lett. 223 (1994) 486. [10] R.G. Gilbert and S.C. Smith, Theory of unimolecular and recombination reactions (Blackwell, Oxford, 1990) p. 150. [11] J. Lighthill, Waves in fluids (Cambridge University Press, Cambridge, 1978). [12] B. Widom, Faraday Symp. Chem. Soc. 16 (1981) 7. [13] (a) J.S. Rowlinson and B. Widom, Molecular theory of capillarity (Clarendon Press, Oxford, 1984) p. 65; (b) p. 117. [14] H. Eyring, D. Henderson, B.J. Stover and E.M. Eyring, Statistical mechanics and dynamics (Wiley, New York, 1964) p. 477. [15] L.F. Phillips, unpublished work. [16] P.S. Liss, P.W. Balls, F.N. Martinelli and M. Coantic, Oceanol. Acta 4 (1981) 129. [17] L.F. Phillips, J. Geophys. Res. 99 (1994) 18577. [18] S.C. Doney, J. Geophys. Res. 100 (1995) 14347. [19] L.F. Phillips, J. Geophys. Res. 100 (1995) 14351. [20] S.D. Smith and E.P. Jones, J. Geophys. Res. 90 (1985) 869. [21] S.D. Smith and E.P. Jones, J. Geophys. Res. 91 (1986) 1O529. [22] S.D. Smith, R.J. Anderson, E.P. Jones, R.L. Desjardins and R.M. Moore, J. Geophys. Res. 96 (1991) 8881.