J. Phys. Chem. B 2001, 105, 11283-11289
11283
Gaussian Local Modes of a Liquid Interface Leon F. Phillips* Chemistry Department, UniVersity of Canterbury, Christchurch, New Zealand ReceiVed: June 20, 2001; In Final Form: September 6, 2001
The time-dependence of an over-damped local mode of a liquid surface membrane is derived by a method that corrects a previous error. By expressing an arbitrary displacement of the surface in the form of a FourierBessel series, spectra of k values are obtained for Gaussian-dome and cylindrical displacements of the surface, where k is the quantity that corresponds to the wave vector of an under-damped oscillation. At high k values, the results are affected by the discrete small-scale structure of the liquid surface. The root-mean-square surface displacement and the rate of surface-bulk exchange are calculated for the Gaussian modes. Retardation effects, associated with the finite rate of spread of a disturbance from the origin of the displacement, are not significant with this model. An increase in surface roughness at high k values results from the form of the dependence of surface area on the height and variance of the Gaussian. At long times the more rapid decay of high k components causes an arbitrary displacement to tend to the form of a Gaussian dome.
1. Introduction Recent theoretical work on the small-scale motions of a liquid interface1,2 has shown that the majority of these motions are associated with over-damped local modes, as opposed to underdamped normal modes. We use a model in which the local modes are excited by random impacts on the surface membrane from molecules or groups of molecules in the subsurface liquid. The local-mode displacement of the membrane, which may be positive or negative for a given impact, can be represented as the product of a Bessel function J0(kr) with a time-dependent factor, which, for motion in the over-damped regime, results in a fast rise time and a slow fall time. Here k corresponds to the wave vector of an under-damped oscillation, and r is the radial distance from the point of impact. The derivation given in refs 1 and 2 contained an error, which led to an expression for the fall time that did not have the correct dimensions3 and was (incorrectly) independent of k. The first section of this paper obtains the time dependence of a local mode by a method that avoids this mistake. An arbitrary deformation of the surface with circular symmetry can be expressed as a Fourier-Bessel series on the basis of local modes. Examples of such series will be given for cylindrical and Gaussian deformations. If one assumes that impacts on the surface membrane represent the outcome of very many random motions of the molecules in the subsurface heat bath, then a Gaussian function probably provides the most realistic representation of the average response of the surface. In addition, we shall see that the time dependence of an arbitrary displacement is such that it must eventually evolve into a Gaussian dome. Hence it is of interest to calculate the rootmean-square (rms) displacement of the surface that results from the noncoherent overlap of thermally excited Gaussian modes. In this paper we also reexamine two assumptions of the previous work, namely: (1) that with excitation by a delta function impact on the surface membrane, all k values are equally likely; and (2) that a factor exp(-kr) can be used to account for retardation effects associated with the spreading out * Corresponding author e-mail:
[email protected].
of the surface displacement from the site of a sudden disturbance. Related experimental and theoretical work4-8 is discussed in refs 1 and 2, the most pertinent result for our present purpose being the observation by Fradin et al.6 that, for wave vectors between 107 and 108 cm-1, the surface is markedly rougher than the usual form of capillary-wave theory would predict, the difference increasing with increasing k. In ref 6 the effect was attributed to a large, progressive decrease in surface tension accompanying the transition to a short distance scale. In refs 1 and 2 this phenomenon was explained as a consequence of the form of the expression for the change in surface area associated with a transient local mode at large values of k, when retardation effects were included. In the over-damped regime there are no travelling waves, and the present model does not support retardation effects; however, similar effects might still result from differing transient behavior of the several Fourier-Bessel components of a Gaussian deformation, and that too is investigated in the present paper. 2. Basic Theory We begin with the velocity potential φ for irrotational motion near the surface of a liquid, which we write in the form
φ ) φ0 exp(ky + nt)J0(kr)
(1)
where the vertical coordinate y is zero at the surface of the undisturbed liquid and takes negative values below the surface. Motions of the medium above the surface, which has its own velocity potential, are neglected. The potential φ has circular symmetry, as is appropriate for the response to a localized disturbance of the surface, and satisfies Laplace’s equation when the Laplacian operator is written as ∂2-/∂y2+ (1/r)∂-/∂r + ∂2-/ ∂r2, with no dependence of φ on the angular coordinate θ. The vertical component of the liquid velocity is V ) -∂φ/∂y, and the radial component u ) -∂φ/∂r. By integrating V at y ) 0 with respect to time t, we obtain the surface displacement ζ in the form
ζ ) -(φ0k/n) exp(nt)J0(kr) + C
10.1021/jp012367b CCC: $20.00 © 2001 American Chemical Society Published on Web 10/16/2001
(2)
11284 J. Phys. Chem. B, Vol. 105, No. 45, 2001
Phillips
where C is a constant of integration that we can set equal to zero by requiring the zeros of ζ to coincide with the zeros of the Bessel function. If this expression for the displacement is inserted into eq 3
0)-
∂φ ∂V + (g + γk2/F)ζ + (2η/F) ∂t ∂y
At time t ) 0 the right-hand side of eq 7 reduces to the product hnJ0(knr), where we have labeled the amplitude hn with the same index as the wave vector kn. Hence a general solution at t ) 0 is kmax
(3)
ζ(r,0) )
hn J0(knr) ∑ k
(10)
min
which is essentially Lamb’s equation for the surface pressure,9 and the gravitational acceleration g is neglected, the result is
0 ) φ0ent J0(kr)(n2F - 2ηk2n + γk3)
(4)
where kmin has still to be specified. An arbitrary function f(F) defined in the interval from F ) 0 to F ) a can be expressed as a Fourier-Bessel series, in the form
where F, η, and γ are density, viscosity, and surface tension, respectively. Equation 4 must hold for all r and t. Hence the quantity n is given by
n)-
k2η (1(x1 - Fγ/kη2) F
(5)
kc ) Fγ/η2
(6)
For water at 25 °C, this works out to be 7.3 × which is significantly higher than the value in ref 2 and corresponds to a wavelength of 8.6 µm. For k . kc, the two solutions for n become n1 ) -2k2η/r and n2 ) -kγ/2η, where |n1| . |n2|. Thus a general expression for ζ is 105
cm-1,
]
w - hn1 w - hn2 exp(n1t) exp(n2t) J0(kr) ζ) n1 - n2 n 1 - n2
(7)
where the factors before the exponentials have been chosen to give ζ ) h and V t ∂ζ/∂t ) w at (r ) 0, t ) 0). For a displacement from h ) 0 with w * 0, this becomes
ζ)
w [exp([n1 - n2]t) - 1] exp(n2t)J0(kr) n 1 - n2
(8)
which has a fast rise time and slow fall time, as in ref 2 (note that n1 and n2 are negative), but the fall time is now inversely proportional to k, in agreement with the experimental and theoretical results of Huang and Webb.8 The rate of surface to bulk exchange of molecules is therefore dependent on the form of the distribution over k, which will be considered in the next section. At low k values the relative magnitudes of n1 and n2 are reversed. For a water surface, the changeover occurs at k ) 1.8 × 105 cm-1, which is below the critical damping value, so this possibility can be ignored. 3. The Spectrum of k Values The first zero of J0(kr) occurs at kr ) 2.4048... When k () 2π/λ for a harmonic wave) takes its greatest value kmax, this should correspond to r ) σ/2, where σ is the size of the smallest discrete unit of surface area and is essentially a molecular diameter. Hence we have
kmax ) 4.8096/σ
(9)
and kmax is of the order of 108 cm-1. The quantity σ is best regarded as an adjustable parameter because, as has often been pointed out, a continuum treatment such as the present one must break down on a distance scale that is small enough for the discrete nature of the surface to become evident.
∑1 Anν Jν(xnνF/a)
(11)
where the coefficients are given by
Anν )
and the wave vector for critical damping is
[
∞
f(F) )
2
∫aFf(F)Jν(xnνF/a) dF
2 a2Jν+1 (xnν) 0
(12)
the quantity xnν being the argument of the nth zero of the Bessel function Jν(x).10 Thus, the right-hand side of eq 10 amounts to a truncated Fourier-Bessel series for the case ν ) 0, in which the nonzero terms correspond to wave vectors given by
kn ) xn0/a
(13)
provided kn is between kmin and kmax, and kmin is fixed by the location of the first zero of the Bessel function. From eq 13, the minimum value of k that is able to be excited in an observation region of radius a is 2.4048/a. Successively higher values of k, up to k ) kmax, are separated by a number that rapidly approaches π/a, the first few separations being 3.1153/a, 3.1336/a, 3.1378/a. The restriction of k to the values given by eq 13 is a property of the Fourier-Bessel series that is the general solution of eq 3 at t ) 0 and is independent of the form chosen for the function f(F). However, it does not impose any physical limitations on k, because the value of a is not restricted in any way. Hence, within a limited range of k, it is correct to say that all k values are equally likely, and the distribution of k must be essentially continuous. However, the k distribution is not required to be flat; we return to this point shortly. At the upper end of the k spectrum, the truncation of the series (eq 10) at k ) kmax reflects the inability of the liquid surface to display the structure of f(F) on a scale finer than σ, and the actual value of kmax is fixed by σ. Thus the discrete structure of the surface imposes a low-pass filter on the spectrum of k values that can result from an impact, and the total surface area imposes a high-pass filter. Figures 1 and 2 show wave forms resulting from “squarewave” (i.e., cylindrical) and “Gaussian” (i.e., a Gaussian-shaped dome) displacements, respectively, of large and small dimensions, together with the corresponding spectra of k values. The quantity b in these functions is the radius of the cylinder, in the case of the square wave, or the rms displacement from the origin, in the case of the Gaussian, with b measured in units of σ/2. In Figure 1, b is varied and the radius a of the observation zone is always equal to 4b. The k spectra, which are the coefficients A of eq 12, run from kmin to kmax in these plots and have been normalized so that the largest value has A ) 1. Figure 2a-e shows the effect on the k spectrum of changing the size of the observation region a while keeping b constant at 10 in units of σ/2. Figure 2f shows the dense k spectra resulting
Gaussian Local Modes of a Liquid Interface
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11285
Figure 1. Effect of varying the size of the impact zone. Plots of Fourier-Bessel series for square-wave [range of peak value ) radius of impact zone] and Gaussian functions [exp(-r2/2b2)], and the corresponding spectra of k-values from kmin to kmax. Radius of impact zone b in units of σ/2, and ratio of observation zone radius a to impact zone radius b, as follows: (a) 1,4; (b) 2,4; (c) 4,4; (d) 8,4; (e) 16,4; (f) 32,4.
from square-wave and Gaussian functions that are highly concentrated near the origin, relative to the radius of the observation zone, and so approximate a delta function. These results lead to several conclusions, as follows. (1) For a Gaussian function, the distribution of k values is quite narrow (neglecting values whose contributions are too small to appear on the scale of these diagrams) and is clustered near the lower end of the spectrum, with a form that appears somewhat similar to the Maxwell distribution of speeds for a
gas. For a square wave, the distribution is broad and clearly shows the truncation of the series at kmax. As noted above, on simple physical grounds one might expect a Gaussian dome to be the most common form of displacement, and this expectation can be further rationalized as follows. Because the distribution of k values is narrow, the shape of a Gaussian displacement is not expected to vary greatly during the rise and fall processes. In contrast, the form of a nominally square wave mode is likely to vary considerably during both the rising and falling phases
11286 J. Phys. Chem. B, Vol. 105, No. 45, 2001
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Figure 2. Effect of varying the size of the observation zone. Plots as in Figure 1, with radius of impact zone b, in units of σ/2, and ratio of observation zone radius a to impact zone radius, b as follows: (a) 10,3; (b) 10,4; (c) 10,6; (d) 10,8; (e) 10,10; (f) 1,100.
because of the much shorter rise and fall times at high k. For an arbitrary displacement, the more rapid loss of high k modes during the falling phase will cause the k spectrum to evolve toward one in which the distribution is concentrated at the lowfrequency end, so that the displacement evolves toward a shape that approximates a Gaussian dome. (2) The density of the k spectrum is strongly dependent on the radius a of what we have termed the observation zone.
Outside the observation zone, interference between the longrange parts of the constituent Bessel functions can give rise to quite large excursions of the surface. Such excursions do not appear in practice, because the incoherent overlap of excursions from many observation zones must produce cancellation. Thus it is not necessary for the effective value of a to correspond to the whole surface area of the liquid. Fluctuations in the effective value of a will cause the spectrum of k values
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J. Phys. Chem. B, Vol. 105, No. 45, 2001 11287
to be essentially continuous, even when k approaches kmax. However, the results in Figure 2f show that the distribution of k values is not flat for an individual Gaussian displacement. For a fixed area of surface, the number of separate observation zones that can be fitted into the area is inversely proportional to the area of a zone, and so is proportional to k2min. For a Gaussian displacement, the main Fourier-Bessel components are clustered near kmin. Hence, for Gaussian modes, we can write the density of the k spectrum as gk2, where the proportionality constant g is fixed by the requirement that the integral of gk2 dk from kc to kmax, which gives the total number of local modes per unit area, has to be equal to 1/σ2 less the number of normal modes per unit area. Because kc , kmax and the number of normal modes is much smaller than the number of local modes, we can take the integral of gk2 from 0 to kmax as equal to 1/σ2, which gives g ) 3σ/(4.8096)3 ) 5.39 × 10-10 cm for σ ) 2 × 10-8 cm. With this distribution of k values we can calculate mean rise and fall times for the local modes given by eq 8. The results for σ ) 2 × 10-8 cm are 1.44 × 10-15 s and 1.54 × 10-12 s; for σ ) 1 × 10-8 cm the mean rise time and fall time are 3.86 × 10-16 s and 7.70 × 10-13 s. The fall times are smaller than those found before2 by a factor of the order of kmean, but the previous conclusion, that any given area of the surface spends most of its time in the recovery mode after a sudden excursion, is unaffected. (3) Truncation of the Fourier-Bessel series when the radius of the impact zone is very small prevents the surface from faithfully assuming the shape of f(F), to such an extent that nominally Gaussian and square-wave excitations produce the same response; for b ) 0.335 in units σ/2, the k spectrum reduces to a single line at k ) kmax for both wave forms. The discrete nature of the surface at high k forces the first node of J0(kmaxr) to occur at r ) σ/2, but does not result in a discrete distribution of k values because the centers of molecules other than the one at r ) 0 are free to follow the contours of the Bessel function, regardless of the locations of the zeros of J0(kr). (4) Retardation effects, associated with the finite rate of spreading out of a disturbance from the site of the initial impact, cannot exist in the flat observation zone surrounding a localized displacement, because there is no displacement to spread out over that region. Also, once the initial wave form has been established, there is little or no scope for retardation in the impact zone. However, transient effects akin to retardation might be expected during the short time interval while the initial wave form is being established, because of the markedly different rise times of different Fourier-Bessel components of the initial wave form, and also, to a lesser extent, during the decay of the wave form, as a consequence of the different fall times of different components. Effects of this sort are considered in section 5. 4. RMS Displacement and the Surface-Bulk Exchange Rate A Gaussian mode of the form h exp(-r2/2b2) has an excess surface area, over the undisturbed surface, that is given by
∆A )
∫0∞2πr(x{1 + [d(h exp(-r2/2b2))/dr]2} - 1) dr (14)
The rms displacement h due to the combined effect of all the thermally excited surface modes is known to be of the same order of magnitude as the quantity σ, whereas b can be a
macroscopic quantity. For displacements h that are very small in comparison with b, i.e., for values of b that are much greater than σ, eq 14 reduces to
∆A) πh2/2
(15)
which is independent of b and σ. In practice, this result is good to within about 5% for b ) h and to within a factor of 2 when b ) h/5. Hence for an isolated Gaussian mode with b . σ, taking the peak value of the mean excess area due to thermal excitation to be kBT/γ, as usual, and the average value to be half of this, we find the rms displacement h ) x(kBT/πγ) to be 1.34 × 10-8 cm for a water surface at 300 K. If the mode is not isolated, this has to be supplemented by the effect of all the other local modes, which introduces the multiplier (eq 16) for h2
1+
1 σ
∫0∞2πr exp(-r2/b2) dr
(16)
where the division by σ reflects the minimum separation of modes around the circumference 2πr. Equation 16 cannot be evaluated without making some assumptions about the value of b2. We first assume that the expression will be sufficiently accurate if b2 takes its average value, and then we calculate the average by assuming that the probability of observing a given value of b2 is proportional to the number of modes with the given b that can be fitted into unit area. Hence the average of b2 must be calculated with a weighting factor g/b2, where the value of g is fixed by setting the maximum value of b at 1 for unit area of surface and the minimum value at σ/2 and requiring the integral of gdb/b2 to be 1/σ2 per unit area. The result is g ) 1/2σ, so b2mean ) σ/2. With b2 set equal to σ/2, the factor (eq 16) becomes independent of σ and takes the value 1 + π/2, so the rms displacement due to the Gaussian local modes becomes 2.15 × 10-8 cm. Combining this with the rms displacement of 2.83 × 10-8 cm due to the normal modes,4 we obtain 3.55 × 10-8 cm as the total rms displacement due to thermally excited capillary waves. In view of the uncertainty associated with the averaging process for b2, this is not significantly different from the usually accepted value of about 3.8 × 10-8 cm for the rms displacement at a water surface. It is also similar to the result given in ref 2, because the rise and fall times do not enter into the calculation of the displacement. From eq 15, the number of molecules N in the excess surface area created by one Gaussian dome with displacement h is πh2/ 2σ2. We have dh/dt ) -h/τfall, and there are 1/σ2 modes per unit area, so the average rate of loss of molecules from unit area is given by
-
2 dN π〈h 〉 ) 3 dt σ τ fall
(17)
With σ ) 2 × 10-8 cm and τfall ) 1.54 × 10-12 s, and using the rms h value of 2.15 × 10-8 cm, this gives the loss rate due to the local modes as 1.18 × 1020 molecule cm-2 s-1. For an individual surface molecule the rate constant is 4.7 × 104 s-1. This loss rate has to be balanced by the rate of creation of new surface during the rising phases of the local modes. It was found previously2 that the exchange rate due to the under-damped normal modes was negligible in comparison with that due to the local modes. That point now needs to be checked with the new, larger value of kc. In ref 2 it is shown that, when damping is neglected, the rate constant for surface-bulk
11288 J. Phys. Chem. B, Vol. 105, No. 45, 2001
Figure 3. Excess surface area of a Gaussian dome of fixed height h, as a function of the impact zone radius b, from eq 14.
exchange due to the normal modes is proportional to kmax7/2, so the increase in kc from 1709 to 7.3 × 105 causes the rate constant to increase from 5 × 10-5 s-1 to 8.1 × 104 s-1, which is larger by almost a factor of 2 than the contribution from the local modes. The total rate constant now becomes 1.3 × 105 s-1, and the total loss rate per square centimeter of surface is 3.3 × 1020 molecule cm-2 s-1. These numbers might usefully be compared with results of molecular dynamics calculations. 5. Surface Roughness at High k Values For small values of b, corresponding to high values of k, the increase in surface area given by eq 14, for a fixed displacement h at the origin, falls off quite sharply with decreasing b, as is shown in Figure 3. For b , h the limiting value of the excess area is hbx(2π3). This effect is in the right direction to account for the observations by Fradin et al.6 of increased surface roughness at high k values, since a larger vertical excursion is required to store the same amount of surface energy. However, Fradin et al. observed an apparent increase in surface energy, which amounted to a factor of 4 over the range of k values from 107 to 108 cm-1. The falloff in Figure 3 begins at a value of b that corresponds to a Gaussian function whose main Fourier-Bessel component has k ≈ 5 × 107 cm-1, and the falloff amounts to only about a factor of 2 between k ≈ 5 × 107 and k ≈ 2.4 × 108 cm-1. Increasing σ from the value 2 × 10-8 cm assumed here would shift the falloff to a lower range of k values, but the value of s required to match the observations would be unreasonably large and the effect would still be smaller than
Phillips that observed. Therefore we need to investigate whether the transient behavior of a displacement might help account for the extra roughness at high k. The transient behavior of square-wave and Gaussian displacements is shown in Figure 4, with rising wave forms on the left and falling wave forms on the right. Figure 4a shows results for the radius of the impact zone b equal to σ/2 (1 × 10-8 cm). Figure 4b shows results for b equal to 5σ (1 × 10-7 cm). The results for the square wave in Figure 4b clearly show the faster rise and fall of the high-k components, but the Gaussian waves show no particular effect, and there appears to be nothing akin to the retardation effects that were previously assumed to operate on the constituent Bessel functions. In the framework of the present continuum model, one possible explanation remains for the increased roughness at high k. Lamb’s derivation9 of his equation 12, which is equivalent to our eq 3, involves the assumption that the inclination of the surface to the horizontal is infinitely small. This is evidently not true if h is greater than b, which is quite likely to be the case when b approaches σ. At present it is not clear whether removal of this assumption would improve the agreement between theory and experiment at very high k, especially since the basic continuum assumption breaks down in the same range of k values. Nevertheless, it does seem obvious that the macroscopic surface tension can have very little to do with the restoring force which acts upon a single molecule, or even a small, isolated cluster of molecules, that has been lifted above the surface. On this scale, molecular dynamics calculations, such as those of Somasundaram et al.,11 are much more appropriate than a continuum treatment. 6. Correlation Functions for Gaussian Modes A quantity of particular interest to experimenters is the smallscale height-height correlation function, for which we write the mean-square value as 〈(ζ(0) - ζ(x))2〉. To calculate this quantity we need to know 〈ζ(0)ζ(x)〉, which we can estimate for the Gaussian modes, as follows. We first assume that the k spectrum of a Gaussian mode can be approximated by a single value of k ≈ kmin ) 2.4048/a, and we note that, if the effective radius a of the observation zone is assumed to be about 2.5b, where b corresponds to one standard deviation for the Gaussian, we have kmin ≈ 1/b. Fixing k fixes the rise and fall times of the component Bessel functions and ensures that a mode rises and
Figure 4. Transient behavior of nominally square-wave and Gaussian displacements, with rising waveforms on the left and falling waveforms on the right in each diagram. The top curve in the first frame of each pair of diagrams has the time-dependent factor omitted, so is the same as in Figures 1 and 2. Radius of impact zone b in units of σ/2, and ratio of observation zone radius a to impact zone radius b: (a) 1,3; (b) 10,3.
Gaussian Local Modes of a Liquid Interface
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11289 (-r2/2b2) with the weighting factor 1/2σb2 that was used in section 4. The result is
〈ζ(0)ζ(r)〉 ) 〈ζ(0)2〉 x(2π) erf(x2 r/σ)/(4r/σ)
(21)
where, from section 4, 〈ζ(0)2〉 ) 4.62 × 10-16 cm2 is independent of σ. Here we have made the approximation of including the factor (eq 16) that was used in section 4 to allow for the effect of nearby modes. Hence, using12
〈{ζ(0) - ζ(r)}2〉 ) 2{〈ζ(0)2〉 - 〈ζ(0)ζ(r)〉} ) 2〈ζ(0)2〉{1 - x(2π) erf(x2 r/σ)/(4r/σ)} (22)
Figure 5. Root-mean-square height-height correlation function for the Gaussian modes, plotted against r/σ, from eq 22.
falls as a coherent unit. Next we assume that the rise time is negligible in comparison with the fall time, so that the persistence of a correlation is determined entirely by the fall time, whose value is τfall ) 2η/kγ ≈ 2bη/γ. For a fixed value of b, we then have, at time t,
ζb(0)ζb(r) )
∫-∞t ∫-∞t hi(b)hj(b) exp(-r2/2b2) exp([2t - ti tj]/τfall) dti dtj (18)
and
ζb(0)ζb(r) )
∫-∞t ∫-∞t hi(b)hj(b) exp([2t - ti - tj]/τfall) dti dtj (19)
where hi(b) is the displacement at r ) 0 due to an impact occurring at time ti with an impact-zone radius b. Hence
ζb(0)ζb(r) ) ζb(0)2 exp(-r2/2b2)
(20)
and the error associated with the assumption of a single k value largely cancels from both sides of eq 20. To eliminate the dependence on b, we now calculate the average value of exp-
we obtain the plot of 〈{ζ(0) - ζ(r)}2〉1/2 versus r/σ that is given in Figure 5. This curve is qualitatively very similar to correlations obtained previously, the main advantage over previous work being that the factor on the right-hand side of eq 22 is given in closed form, so it is not necessary to resort to numerical computations. Acknowledgment. I am grateful to Lioudmila Ametistova for drawing my attention to the incorrect dimensions of τfall as given in reference 2. References and Notes (1) Phillips, L. F. Chem. Phys. Lett. 2000, 330, 15. (2) Phillips, L. F. J. Phys. Chem. B 2001, 104, 1041. (3) Ametistova, L., personal communication. (4) Jeng, U.-S.; Esibov, L.; Crow, L.; Steyerl, A. J. Phys.: Condens. Matter 1998, 10, 4955. (5) Tikhonov, A. M.; Mitrinovic, D. M.; Li, M.; Huang, Z.; Schlossman, M. L. J. Phys. Chem. B 2000, 104, 6336. (6) Fradin, C.; Braslau, A.; Luzet, D.; Smilgies, D.; Alba, M.; Boudet, M.; Mecke, K.; Daillant, J. Nature 2000, 403, 871. (7) Mecke, K.; Dietrich, S. Phys. ReV. E 1999, 59, 6766. Napiorkowski, M.; Dietrich, S. Phys. ReV. E 1993, 47, 1836. (8) Huang, J. S.; Webb, W. W. Phys. ReV. Lett. 1969, 23, 160 (9) Lamb, H. Hydrodynamics, 6th edition reprinted; Cambridge University Press: Cambridge, U.K., 1953; p 626. (10) Jackson, J. D. Classical Electrodynamics; Wiley: New York, 1962; p 73. (11) Somasundaram, T.; Lyndon-Bell, R. M.; Patterson, C. H. Phys. Chem. Chem. Phys. 1999, 1, 143. (12) Simpson, G. J.; Rowlen, K. L. Chem. Phys. Lett. 1999, 309, 117.
