J. Phys. Chem. B 2001, 105, 1041-1046
1041
Local Modes and the Surface-Bulk Exchange Rate at a Liquid Interface Leon F. Phillips† Chemistry Department, UniVersity of Canterbury, Christchurch, New Zealand ReceiVed: October 2, 2000; In Final Form: NoVember 13, 2000
The small-scale motions of a liquid surface due to thermally excited capillary waves are dominated by overdamped local modes of oscillation, which for a water surface at 300 K occupy the range of wave vectors between the critically damped kc ) 1709 cm-1 and kmax ≈ 10-8 cm-1. Values of k below kc correspond to under-damped, normal modes of oscillation. For values of k near kmax, the peak displacement produced by a given increment of surface energy, at constant surface tension, increases sharply with increasing k, an effect which probably contributes to the reported apparent decrease in surface tension at high k. The radial dependence of a local mode is of the form exp(-kr)J0(kr), where J0(kr) is a Bessel function and the exponential factor arises from the finite rate at which the displacement spreads out from the site of the initial disturbance. The time dependence of a local mode consists of a rapid initial displacement of the surface followed by a very slow recovery. Any element of the surface spends most of its time in the recovery phase, and the rate of recovery is independent of k when k . kc. The recovery process transfers molecules out of the surface layer; hence the first-order rate constant for transfer of a surface molecule into the bulk liquid is found to be kBT/ 2hσ2 s-1, where η is the viscosity and σ is the molecular size.
Introduction The theory of local-mode oscillations at a liquid surface has been presented elsewhere in a rather compact form.1 Here we give an expanded derivation of the theory, together with new calculations of the rate of exchange of molecules between the surface layer and the bulk liquid, and of correlation functions for the speed of vertical displacement. We adopt a macroscopic viewpoint, such that the surface of the liquid is regarded as a continuous medium and its small-scale motions are described by a superposition of capillary-waves, i.e., waves for which the restoring force is provided by surface tension. When the characteristic wavelength associated with the motion is comparable with molecular dimensions, the results of such a treatment are likely to be qualitatively correct, but quantitative conclusions must be regarded with the sort of caution that is necessary when, for example, a string of beads is represented by a catenary. Because of surface tension, a liquid surface behaves as though it were covered by a thin, elastic membrane. Capillary waves can be regarded as vibrational modes of this membrane, and thermally excited capillary waves correspond to Brownian motion of the membrane, which is subject to a continual barrage of random impacts from molecules below the surface. The present calculations are based on a model in which localized delta-function impacts from below give rise to sudden, almost Heaviside-function, changes in the position and velocity of the membrane, leading to excitation of localized oscillations. Depending on the magnitude of the wave vector, such localized oscillations may or may not give rise to oscillations of the entire surface. By analogy with the transition from normal modes to overtones in molecular spectroscopy, we refer to a lightly damped motion, which covers an extended region of the surface and represents “ringing” of the surface membrane, as a normal †
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mode, and refer to the over-damped response to a localized impact on the surface membrane as a local mode. Motions of the surface membrane of a liquid have almost invariably been discussed in terms of the normal modes, which for a planar surface are sinusoidal functions of distance and for the surface of a drop are spherical harmonics, multiplied by sinusoidal functions of time.2-5 However, such a basis set is not appropriate for describing very small-scale motions, because at moderately short wavelengths the damping by viscosity is so great that the majority of surface excitations do not persist for more than a fraction of a vibrational period, or extend in space for more than a fraction of a wavelength. Further, the random individual and collective motions of molecules in the liquid below the surface are poorly suited to exciting coherent motion of the surface membrane on anything approaching a macroscopic scale, so the actual small-scale motion of the surface bears no resemblance to the normal modes. The sort of coherence that would allow the surface displacement to be expressed as a superposition of spatially extended normal modes is absent, so that superposition of a very large number of normal modes would be required in order to reproduce a single local mode. As noted in ref 1, in the case of a water surface, the high shear rates and resulting high dissipation rates encountered at very short wavelengths lead to behavior which resembles boiling porridge, or a geothermal mud pool, rather than the familiar mobile liquid. In this paper we first derive the form of an individual local mode, and then consider the effect of local-mode behavior on some properties of the surface as a whole. One significant difference between the results of local-mode and normal-mode treatments is that the local modes give rise to greater smallscale roughness of the surface. Jeng et al.6 derived expressions for the roughness of a planar liquid-liquid or liquid-vapor interface in the presence of damping by viscosity, by solving the linearized Navier-Stokes equation in a basis of sine and cosine functions. They distinguished strong-damping (short
10.1021/jp003590w CCC: $20.00 © 2001 American Chemical Society Published on Web 01/10/2001
1042 J. Phys. Chem. B, Vol. 105, No. 5, 2001
Phillips
wavelength) and weak-damping regimes, separated by a critical wave vector kc, and simply omitted the contribution of waves with k > kc to the mean-square displacement. In this way they obtained the relatively low value of 2.3 × 10-8 cm for the rootmean-square (rms) displacement of a water surface, compared with the usual value of about 3.5 × 10-8 cm that is obtained by including the whole spectrum of k. A somewhat similar approach was used by Simpson and Rowlen7 in their calculation of surface correlation functions, where they omitted waves of wavelength shorter than x when calculating the correlation function for vertical displacements of two points on the surface separated by a distance x. The calculations of Simpson and Rowlen, which included the effect of a predicted increase in surface tension at high k, also led to a smaller interfacial roughness than is given by standard capillary-wave theory. In contrast, recent experimental studies by Tikhonov et al.8 and Fradin et al.9 found greater small-scale roughness than is given by the normal mode treatment. For wave vectors between 107 and 108 cm-1, Fradin et al. found the surface to be markedly rougher than the usual form of capillary-wave theory would predict, and interpreted their results in terms of a dramatic decrease of the effective surface tension from its macroscopic value. The present work provides an alternative explanation for this phenomenon, as a necessary consequence of the form of the expression for the change in surface area associated with a transient local mode at large values of k. The behavior of the surface tension at high k values is still unclear and in this paper the surface tension is regarded as a constant. Theory of Local-Mode Capillary Waves. We begin with the Navier-Stokes equation in the form10
F
[
]
∂V 1 + grad V2/2 + (curl V)xV ) ∂t 2 -grad[p + Fgz] + γ
(
)
∂2z ∂2z + + η∇2V (1) ∂x2 ∂y2
where V is the velocity vector of the contents of an infinitesimal volume element of fluid whose density is F and viscosity η, and we have added a term which includes the coefficient of surface tension γ, since the volume elements of interest are all located in the surface layer.11 One approach to a theory of surface-bulk exchange is by way of the nonlinear “convective” terms on the left-hand side of eq 1, terms which describe motion of the sort that is envisaged in the surface-renewal model of gas-liquid exchange.12 The approach followed here considers surface-renewal on a much smaller scale, involving motion of individual molecules into or out of the surface, rather than the motion of macroscopic parcels of liquid, so we can omit the troublesome nonlinear terms. For small-scale motions, we can also assume that pressure is constant and gravitational effects may be neglected. This gives the linearized capillary-wave equation:
(
)
∂2z ∂2z ∂V F ) γ 2 + 2 + η∇2V ∂t ∂x ∂y
(2)
We now restrict our attention to the vertical component of the velocity V, rewrite the equation in terms of the vertical displacement ζ ) z-z0 of the surface from its equilibrium position z0, so that V becomes ∂ζ/∂t, and we assume that ζ and its time-derivatives are independent of z, which amounts to assuming that cavitation is negligible. This last assumption makes the term involving γ on the right-hand side of (2) equivalent to γσ2z ≡ γ∇2ζ. Next we change to polar coordinates
r,θ in the X-Y plane and assume that z is independent of the polar angle θ. The result is
(
)(
)
∂2ζ ∂ζ 1 ∂ ∂2 ) γζ + η + 2 2 r ∂r ∂t ∂t ∂r
F
(3)
With the substitution ζ ) u(r)V(t), eq 3 separates into the two ordinary differential equations
d2u 1 du + k2u ) 0 + dr2 r dr
(4)
dV d2V + k2η + k2γV ) 0 dt dr2
(5)
and
F
where the separation constant is -k2. Equations 4 and 5 are standard forms, for which the required solutions are
u(r) ) J0(kr)
(6)
and
V(t) )
(h +RτwτS(Rt) + hC(Rt))exp(-t/τ)
(7)
where the Bessel J function is unity at the origin and the corresponding Bessel Y function, which is also a solution of eq 4, is not acceptable because it goes to minus infinity at the origin. The result (eq 7) for V(t) conforms to the boundary conditions V ) h and dV/dt ) w at t ) 0. In eq 7, τ stands for 2F/k2η, and R stands for x|R2|, where R2 is (1/τ2 - k2γ/F). If R2 is negative, the motion of the surface is under-damped and the functions S(Rt) and C(Rt) are sin(Rt) and cos(Rt). If R2 is positive, the motion is over-damped and the functions S(Rt) and C(Rt) become sinh(Rt) and cosh(Rt). The critical value kc ) (4γF/η2)1/2 corresponds to R ) 0. When k is much larger than kc, we can put τ ≈ 1/R, but the term -k2γ/r in R2 can never be entirely neglected in comparison with 1/τ2, because without it the value of ζ does not return to zero after a small displacement. Although the difference between R and 1/τ may become very small when k is large, R must always be smaller than 1/τ. The program Maple13 was used to obtain the solutions of eqs 4 and 5 and also to carry out most of the numerical calculations involved in this work. At large values of kr the Bessel function in eq 6 has the limiting form14
(2/πkr)1/2 cos(kr - π/4)
(8)
which shows that k corresponds to the usual wave vector when kr is large, and also provides a mechanism for excitation of normal modes when the effect of damping by viscosity is small. The limiting form15 of the spherical harmonic Ylm(θ,φ) at large l is
{exp(imφ)/(π sinθ)} cos(lθ+mπ/2) ≈ (R/πr) exp(imφ) cos(lr/R+mπ/2) (9) where R is the radius of the drop, and we have used r/R ≈ θ ≈ sinθ, so in the case of a spherical drop at large values of kr the quantity k corresponds to l/R, and again we have a mechanism for excitation of the normal modes. The maximum value of k ()2π/λ for a sinusoidal wave) is expected to correspond to a wavelength 2σ, where σ is the size
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J. Phys. Chem. B, Vol. 105, No. 5, 2001 1043
Figure 2. Radial dependence of a local mode, with allowance for decay by a factor exp(-t/τ) ) exp(-kr) during spreading of the displacement from the site of the initial sudden disturbance. The oscillations of the Bessel function are almost entirely damped out by the decay factor exp(-kr). Figure 1. Characteristic time dependence of a local mode.
of the smallest unit of surface area that can move independently. Thus, σ is expected to be of the order of a molecular diameter. Hence we obtain
kmax ) π/σ
(10)
and kmax ≈ 108 cm-1. The first zero of the Bessel function in eq 6 actually occurs at kr ) 2.4048, rather than at kr ) π/2 as it would for a cosine function. If the location of the first zero is supposed to correspond to the distance σ when k ) kmax, then kmax becomes 4.81/σ instead of π/σ. However, σ is best regarded as an adjustable parameter, and in practice it makes little difference whether we use 4.81/σ or π/σ for kmax. For a water surface at room temperature, the transition from underdamping to over-damping occurs at kc ) 1709 cm-1, corresponding to l ) 37 µm, whereas the maximum value of k is greater than 108 cm-1. Hence the vast majority of surface modes are over-damped and therefore localized, which implies that the small-scale properties of the surface are likely to be dominated by the local modes. Form and Amplitude of the Local Modes. First we consider a single local mode in isolation. The time dependence of ζ given by eq 7, for a positive initial velocity and with initial displacement h ) 0 is plotted in Figure 1 for k . kc, with time measured in units of τ, and R ) 0.95τ. There is seen to be a rapid initial displacement, followed by a slow return to the starting level. In practice the sign of the displacement ζ can be positive or negative, a sudden downward motion of the subsurface liquid being just as likely as a sudden upward motion. We suppose that any point on the surface membrane experiences a random series of such impacts, each impact exciting the motion associated with a single value of k, and all k values being equally likely. For a single excursion, the maximum displacement occurs at the time given by
tmax ) (1/2R) ln{(1 + Rτ)/(1 - Rτ)} ≈ τ ln{kη/x(γF)}
where the mean decay time, τmean ) η/γ, is independent of k. We shall use this result later. Exponential decays were observed and attributed to overdamped surface waves by Huang and Webb16 during studies of light-scattering from a liquid-liquid interface near the critical solution temperature, where the surface tension is extremely small. In their experiments, they measured the rate of decay of a surface autocorrelation function. The decay time so obtained, which did not correspond to our τmean, varied as the reciprocal of the wave vector k, and could be as large as 20 s. Huang and Webb wrote the vertical displacement of the interface as a function of distance x, rather than radius r, and a solution equivalent to theirs can be obtained from eq 2 by expressing ζ as a product of the form u(x)V(y)w(t). The quantity τ ) 2F/k2η is the same in both treatments. The radial dependence of ζ given by eq 6 is that of an ordinary Bessel J0 function. However, if the initial disturbance which gives rise to the waveform of Figure 1 approaches a deltafunction in both time and space, it becomes necessary to allow for the finite time δt during which the disturbance spreads radially over a distance r, and hence for decay of the impulse by a factor exp(-δt/τ) during this time. If we write r ) cδt, and put the speed c equal to R/k, where R corresponds to the circular frequency of an under-damped wave, and use R ≈ 1/τ, the factor for decay during propagation of the initial disturbance becomes exp(-kr) and the radial function at the peak of the excursion takes the form exp(-kr)J0(kr). This function is plotted against the product kr in Figure 2. The slope of the Bessel J function is zero at the origin; the slope of the function exp(kr)J0(kr) is -1 at r ) 0. The peak potential energy associated with the disturbance is given by the product of the surface tension with the surface area of the solid of rotation obtained by rotating the function exp(-kr)J0(kr) around the axis of θ, less the product of surface tension and area for the undisturbed surface. The integral required to calculate the maximum change in surface area due to a sudden displacement h at the origin is therefore either
(11) To obtain the rate of decay after the maximum we use eq 7 with w ) 0 and h nonzero, and expand the hyperbolic functions in terms of their component exponentials. For k . kc, this leads to the remarkably simple result
ζ(t>tmax) ) h exp(-tγ/η) ) h exp(-t/τmean)
(12)
d 2πr e-kr(x1 + k2h2J1(kr)2 - 1)dr ∫ 0 df∞
∆A ) lim
(13)
or
∆A ) lim
df∞
∫0d2πr(x1 + k2h2e-2kr[J0(kr)+J1(kr)]2 - 1)dr
(14)
1044 J. Phys. Chem. B, Vol. 105, No. 5, 2001
Phillips
Figure 3. Square of the peak amplitude V0 versus wave-vector k for a local mode on a water surface at 300 K, with the surface energy set equal to kBT at the peak displacement and σ ) 2 × 10-8 cm.
where -J1(r) is the derivative of J0(r) with respect to r. Equation 13, which was used in ref 1, results from scaling the length of arc of the function J0(kr) at radius r by a factor exp(-t/τ) ) exp(-kr). This is incorrect. Equation 14, which is correct, results from using the length of arc of the function exp(-kr)J0(kr) to generate the area of the solid of rotation. The difference between the results obtained with these two equations is small but significant. The limiting values of integrals such as these were found by using the floating-point evaluation routine in Maple to calculate the integral with d set to successive zeros of J0(r) and observing the convergence to a fixed value. The ratio ∆A/h2 is a function of the product kh. For values of kh less than about 0.1, the quantity k2h2 exp(-2kr)[J0(kr) + J1(kr)]2 is much smaller than unity and the square root in eq 14 can then be replaced by the first two terms of its binomial expansion. The area ∆A then becomes
∆A ) h2(0.955049...)
(15)
which is independent of k (in ref 1, the corresponding formula had a numerical factor 0.57166...). Setting the peak potential energy equal to kBT for this case makes v02 equal to 1.047(kBT/ g). At the other extreme, for large values of the product kh, the quantity k2h2 exp(-2kr)[J0(kr) + J1(kr)]2 is much greater than unity over most of the important range of the integral. In the limit kh . 1, we obtain
∆A ) (h/k)(4.44288...)
(16)
and at this limit, the ratio ∆A/h2 is proportional to 1/kh. Thus, for larger k values, larger values of h are required to produce the same value of ∆A. This has interesting consequences as follows. If the energy of a mode is set at the average value of kBT, the corresponding maximum value of ∆A, at the stage of the excursion when all of the mode’s energy is in the form of potential energy, is 5.63 × 10-16 cm2 for a water surface at T ) 300K, with γ ) 72 mN m-1. Fixing ∆A at this value makes h the rms value of the peak displacement V0 at r ) 0. Hence we obtain the plot of the square of the rms peak displacement versus k which is shown in Figure 3, where eq 15 holds for k values below about 2 × 106 cm-1. The quantity V02 is seen to increase quite sharply with increasing k for k values near 108 cm-1. In relation to the small-scale roughness of a water surface, this indicates that the same amount of energy per mode produces a larger initial displacement when k is very large, an effect which mimics the effect of a decrease in surface tension at high k, as proposed by Fradin et al.8 This effect, which is a basic property
of the integrals involved, is the same whether eq 13 or 14 is used for the excess surface area, but the effect is smaller with eq 14. In Figure 1 of ref 1, which corresponds to our Figure 3 and was based on the incorrect eq 13, the quantity plotted against k was (V0)rms rather than V02. Next we calculate the rms value of the displacement ζ. For large values of t/τ, the value of exp(-2t/τ) sinh2(t/τ) is 0.25, and the same is true if the sinh function is replaced by cosh, so the mean-square amplitude 〈ζ2〉 of a local mode, averaged over time and neglecting the rapid initial rise, is 0.25V02. With deltafunction excitation, all values of k between 1 and kmax are equally likely, so the average value of V02 can be obtained by numerical integration of V02dk up to k ) kmax, followed by division by kmax. Figure 3 shows that the mean value of V02 is dependent on the upper limit of integration, which in turn depends on the molecular size σ, so it is not possible to express the mean-square amplitude as a simple multiple of kBT/γσ. For σ ) 2 × 10-8 cm we obtain a value of 9.92 × 10-16 cm2 for the mean square value of V02, corresponding to 〈ζ2〉 ) 2.48 × 10-16 cm2, and a value of 1.57 × 10-8 cm for the rms displacement at the origin of an isolated local mode. If the mode is not isolated, the mean-square displacement has to be supplemented by the effects of impacts with the surface membrane at points away from the origin. The sum of all such effects is given by
V02
∞ 2πr J02(kr) exp(-2kr) dr ∫ 0 σ 2
(17)
for the peak displacement squared, where the factor 1/σ2 reflects the minimum spacing of local modes and again it is necessary to allow for decay of the Bessel function with distance from the point of excitation. Hence the required factor for multiplying 〈ζ2〉 is given by
1+
∫0∞2πR J02(R) exp(-2R) dR
1 k2σ2
(18)
where we have made the substitution R ) kr, and k2 will take its average value of (1/3)kmax2. The result of evaluating expression 18 with σ ) 2 × 10-8 cm is 1.302, so the rms displacement at the origin due to the local modes becomes 1.80 × 10-8 cm. To this must be added the rms displacement of 2.45 × 10-8 cm due to the normal modes up to k ) 1709.5,9 Taking the square root of the sum of the squares gives a total rms displacement of 3.04 × 10-8 cm. Any point on the liquid surface can serve as the origin, as defined by the location of a random impact from below, so these results apply to the surface as a whole. This calculated rms displacement of a water surface is significantly less than the usual capillary-wave value, but the difference is not very important because of the sensitivity of the result to the value assumed for s. For example, putting σ ) 1.4 × 10-8 cm increases the rms displacement to 3.51 × 10-8 cm. Changes in σ alter the upper limit kmax for integration over V02dk, without affecting the size of the factor (eq 18). However, changing from kmax ) π/σ to kmax ) 4.81/σ has almost no effect on the calculated rms displacement, because the increase in the integration limit kmax is almost exactly compensated by a simultaneous reduction in the factor given by expression 18. The Surface-Bulk Exchange Rate. The rapid initial displacement shown in Figure 1 creates new surface area, which is gradually lost during the slow return to the original level. The loss of surface area amounts to a transfer of molecules from the surface layer into the bulk liquid. From eqs 11 and 12, the ratio of the mean decay time τmean to the total rise time tmax for
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J. Phys. Chem. B, Vol. 105, No. 5, 2001 1045
water at T ) 300 K is 28 for k ) 104, 1 × 105 for k ) 106, and 6 × 108 for k ) 108, which shows that the rise time can usually be neglected in comparison with the decay time. In the steady state, the rates of creation and loss of surface area must be equal, so it follows that any given element of surface must spend most of its time in the decay mode described by eq 12. This prediction, that the mean decay time τmean ) η/γ is the same for all modes when k . kc, and that at any instant most of the surface is in the same slow decay mode, needs to be tested by experiment. Assuming that it is indeed correct, for molecules in the surface layer the first-order rate constant for transfer into the bulk liquid below the surface is 1/τmean ) η/γ, multiplied by the ratio of the excess area ∆A to the total projected area of surface. If we set the number of local modes per unit area equal to 1/σ2, and assume that half of the energy kBT per mode is present in the form of kinetic energy and half as the potential energy γ∆Amean when ∆A ) ∆Amean, we obtain
∆Amean ) kBT/2γσ
2
(19)
per unit area of surface, so this is also the ratio of excess area to total area. Hence the first-order rate constant for transfer of an individual surface molecule into the bulk liquid is kBT/2ησ2 s-1, which is independent of surface tension. For a solute molecule of similar size to the solvent molecules, this quantity will represent a frequency factor for the exchange process, the actual value of the rate constant being dependent on the size of the free energy barrier to solvation. If we put σ ) 2 × 10-8 cm for water at 300 K, the calculated rate constant is 5.2 × 103 s-1 and the rate of transfer, in molecules per second per square cm of macroscopic surface, is 1.3 × 1019. The foregoing calculation omits the contribution of the underdamped normal modes to the exchange rate. In view of the small size of kc relative to kmax, the omission appears reasonable, but we can check its validity as follows. For a drop of radius R, we write the time-dependent change in surface area, due to thermal excitation of a single normal mode (l,m), as (kBT/γ) cos(ωt), where ω2 ≈ l3γ/FR3.5 We now differentiate the area expression with respect to time, replace the resulting sine factor by its average value of 2/π, and then divide the result by 2σ2, to obtain the average rate of loss of molecules from the surface as ωkBT/ πγσ2. The factor of 2 is necessary because for normal modes the surface area spends only half of its time decreasing. The total number of molecules in the surface is 4πR2/σ2. Dividing by this quantity gives the first-order rate constant for transferring a molecule out of the surface, due to a single mode, as ωkBT/ 4π2R2γ. Next we substitute for ω, using the approximate formula given above, multiply the result by (2l + 1) ≈ 2l, and integrate over l from 2 to lmax to obtain the total rate constant for all modes up to lmax, where the contribution from the lower limit of integration is negligible. Replacing lmax/R by kmax then gives the first-order rate constant as (kBTkmax7/2)/(2π2F1/2γ1/2). With kmax ) 1709 cm-1 the value of the rate constant is 5 × 10-5 s-1, which is smaller by a factor of 108 than the value calculated for local modes. Correlation Functions for Local Modes. The derivation of a basic time-correlation function for the displacement due to local modes at some point on the surface follows the same lines as the derivation for normal modes in Appendix 1 of ref 5, with the difference that the expressions now involve sinh(Rit + i) and cosh(Rit + i) instead of the corresponding sine and cosine functions. If we ignore the effects of impacts at other locations that are expressed in eqs 17 and 18, the basic time correlation function at the origin of a local mode k is simply
Figure 4. Square root of the velocity correlation function 〈{w(0) w(r)}2〉 versus distance r for local modes on a water surface at 300 K, with σ ) 2 × 10-8 cm.
〈ζk(0)ζk(t)〉 ) 〈ζk(0)2〉 exp(-t/τ)
(20)
Here τ is 2F/k2η and, in view of Figure 3, the average over k has to be obtained numerically. The corresponding distance correlation function for fixed k is
〈ζk(0)ζk(r)〉 ) 〈ζk(0)2〉 exp(-kr)
(21)
Plots of the functions 〈ζ(0)ζ(r)〉 and x〈{ζ(0) - ζ(r)}2〉 were given in ref 1. They confirm the conclusion, from Figure 3, that the roughness of the surface on a short distance scale is significantly greater than predicted by normal-mode calculations. Correlation functions for other quantities such as vertical velocity and slope of the surface are obtainable by the same methods as were used in ref 5, but simplifications are possible because of the change to hyperbolic functions, especially in the limit of large k. So, for example, we begin by writing
ζk(t) ) [a sinh(Rt) + b cosh(Rt)] exp(-t/τ)
(22)
where a and b are random numbers that can be either positive or negative. Squaring, expanding the hyperbolic functions as exponentials, and taking the long-time average gives
〈ζk2〉 ) 〈(a2 + b2)〉
(23)
The time-derivative of eq 22 is
wk(t) ) R(a- b)[cosh(Rt) - sinh(Rt)] exp(-Rt) (24) where we have used R ≈ 1/τ when k . kc. Hence, after expanding sinh(Rt) and cosh(Rt),
〈wk(t)2〉 ) R2〈(a - b)2〉 exp(-4Rt) ) R2〈ζk2〉 exp(-4Rt) (25) from eq 23, because the average of (a - b)2 is the same as the average of (a2+b2) when a and b are random quantities. For the time correlation function of vertical velocity we obtain
〈wk(0)wk(t)〉 ) R2〈ζk2〉 exp(-2Rt)
(26)
and for the distance correlation function we have
〈wk(0)wk(r)〉 ) R2〈ζk2〉 exp(-2kr)
(27)
These quantities can be averaged over k, with allowance for the dependence of 〈ζk2〉 on k that is shown in Figure 3, and the
1046 J. Phys. Chem. B, Vol. 105, No. 5, 2001 function 〈{wk(0) - wk(r)}2〉 can be derived from eq 27 in the standard way. The result of doing this for a water surface at 300 K is given in Figure 4, which shows that for local modes there is essentially no correlation of vertical velocity over distances greater than the molecular size σ. Acknowledgment. This work was supported by the Marsden fund. I am grateful to a referee for drawing my attention to ref 16. References and Notes (1) Phillips, L. F. Chem. Phys. Lett. 2000, 330, 15. (2) Lamb, H. Hydrodynamics, 6th ed. (reprinted); Cambridge University Press: New York, 1993. (3) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Sykes, J. B., Reid, W. H., Translators; Pergamon Press: Oxford, 1987. (4) Henderson, J. R.; Lekner, J. Mol. Phys. 1978, 36, 781. (5) Phillips, L. F. J. Phys. Chem. B 2000, 104, 2534.
Phillips (6) Jeng, U.-S.; Esibov, L.; Crow, L.; Steyerl, A. J. Phys.: Condens. Matter 1998, 10, 4955. (7) Simpson, G. J.; Rowlen, K. L. Chem. Phys. Lett. 1999, 309, 117. (8) Tikhonov, A. M.; Mitrinovic, D. M.; Li, M.; Huang, Z.; Schlossman, M. L. J. Phys. Chem. B 2000, 104, 6336. (9) Fradin, C.; Braslau, A.; Luzet, D.; Smilgies, D.; Alba, M.; Boudet, M.; Mecke, K.; Daillant, J. Nature 2000, 403, 871. (10) Le Mehaute, B. Introduction to hydrodynamics and Water WaVes; Springer-Verlag: New York, 1976. (11) Lighthill, J. WaVes in Fluids; Cambridge University Press: New York, 1978. (12) Higbie, R. Am. Inst. Chem. Eng. 1935, 35, 365. Danckwerts, P. V. Ind. Eng. Chem. Res. 1951, 43, 1460. Hasse, L. Tellus 1990, 42B, 250. (13) Maple V, Release 5.1; Waterloo Maple Software: Waterloo, Ontario, Canada. (14) Abramovitz, M.; Segun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965. (15) Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. Quantum Theory of Angular Momentum; World Scientific: Singapore, 1988. (16) Huang, J. S.; Webb, W. W. Phys. ReV. Lett. 1969, 23, 160.
