Langmuir 1991, 7, 2419-2421
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High-Frequency Capillary Waves on the Clean Surface of Water J. C. Earnshawl and C. J. Hughes The Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland Received May 31, 1991. In Final Form: August 21, 1991 Thermally excited capillary waves on the free surface of water have been studied by laser light scattering. In the present work the range of surface wavenumbers observed has been extended by a factor of about 2 compared to previous studies. Over the entire experimental range the capillarywave propagation accorded with theoretical expectation. This permits a more stringent upper limit to be set on possible surface viscosities for water.
Introduction Some 10years ago the basis for a fundamental approach to understanding interfacial or surface viscosities of fluids was laid down.lI2 In this approach the surface viscosities were regarded as surface excess quantities. This rather naturally yielded the possibility that up to five independent viscoelastic moduli might exist for fluid surfaces or interfaces.2 There is no unique way to define these five moduli, but one physically plausible interpretation is that of Goodrich,3 who defined the five modes of motion of a fluid interface as shear and dilation (both normal to and within the interfacial plane) and slip of one fluid over the other. Some of the corresponding viscoelastic properties are known from studies of surfaces or interfaces supporting molecular films, but others are less familiar. The present point of interest concerns surfaces of pure fluids: if surface viscosities are really surface excess properties, then there is no reason a priori why free surfaces of pure liquids might not display nonzero values. In fact, apart from one or two reports for water$6 which have not been verified:.’ no experiment has demonstrated a nonzero surface viscosity for a pure fluid. This might, however, arise from purely experimental considerations, as shown by the following argument. Following the usual viscoelastic convention, a given surface modulus can be written as (for example) (1) y = yo+ iwy’ In this example y is the modulus governingshear transverse to the surface, yo being identified as the surface tension and y’ as the transverse shear surface visc0sity.l The other surface moduli can be similarly expanded as response functions; the only other one which might be relevant here is t (=to i d ) , governing in-plane dilation. Now a given value of a surface viscosity will have greater effect upon the corresponding mode of surface motion as the frequency w of the perturbation used to probe the surface response is raised, as wy’ rises relative to yo (for example). In attempting to set limits upon surface viscosities of pure fluids, it is, therefore, important to study the highest frequency perturbations possible. The modulus y directly governs the dispersive behavior of capillary waves on fluid surfaces. The dispersion
+
(1) Goodrich, F. C. h o c . R. SOC.London, A 1981,374,341. ( 2 ) Baus, M. J. Chem. Phys. 1982, 76, 2003. (3)Goodrich, F. C.J. Phys. Chem. 1962, 66, 1858. (4)Earnshaw, J. C.Nature 1981 292,138. ( 5 )Vila, M. A.; Kuz, V. A.; Rodriguez A. E. J. Colloid Interface Sci. 1985, 107, 314. (6) Earnshaw, J. C.; McGivern, R. C. J. Phys. D Appl. Phys. 1987, 20, 82. (7) Hdrd, S.; Neuman, R. D. J. Colloid Interface Sci. 1987, 115, 73.
relation between the frequency w and the wavenumber q of capillary waves on the free surface of a fluid is8
where m = (q2 + i w p / q ) 1 / 2 . Quasi-elastic light scattering can be used to observe the dispersion of thermally excited capillary waves of rather high frequency (>kHz): a real value of q is selected experimentally and the complex frequency (w = wo i r ) is measured.6 Comparison of the measured wo and I’values with those found by solution of eq 2 enables the adequacy of the theory to be tested. In particular if the observed wave damping (I’) exceeds that predicted for a free liquid surface, it may indicate that the transverse shear surface viscosity is nonzero. This can most simply be inferred6 by noting the standard analytical approximations for w in the regime where the capillary waves are under-damped8
+
(3) (4)
If now we substitute the response function form of 7 , as given in eq 1,into eq 3, the resulting expression contains an imaginary part which is evidently a contribution to the wave damping
AI’ = yfq3/2p (5) This reinforces the point made earlier that the effect of a given value of y‘ will be greater for waves of high frequency (therefore high, q, see eq 3). Experimental Section Previous experiments on capillary waves upon the surface of pure water have been limited in the range of q covered, the widest being 195 < q < 1530 cm-1.6 Recent advances involving the refinement of several experimental factorss have permitted a substantial extension to larger q: values -3000 cm-’ are now accessible. Consideration of eq 5 suggests that this should allow the upper limit upon y’ to be improved by about an order of magnitude.
Detailed discussions of experimentalconsiderationswould be out of place in this brief report. In summary light from an Ar+ laser is directed on to the fluid surface where it is scattered by thermally excited capillary waves. The spectrum of the scattered light is just the power spectrum of the waves of the selected wavevector and can be measured by heterodyne detection and (8) Lamb, H. Hydrodynamics; Dover: New York, 1945. (9) Hughes, C. J.; Earnshaw, J. C. To be submitted for publication.
0743-7463/91/2407-2419$02.50/0 0 1991 American Chemical Society
2420 Langmuir, Vol. 7, No. 11, 1991
Letters
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I
k 0
1000
2000
3000
Q (cm-')
Figure 2. Differences between the observed and predicted capillary wave damping values as a function of q. The data are
randomly scattered about zero except at the lowest three q values (see text).
Figure 1. Frequency (0) and damping (X) of thermally excited capillary waves on the surfaceof clean water. The lines represent the dispersion behavior predicted from eq 2 for the accepted properties of water at 20.0 "C. The data are the weighted means of the results of fitting 20 (40 at q = 2990 cm-1) independent correlation functions. The errors are smaller than the plotted points.
photon correlation spectroscopy.6 The observed correlation functions can be fitted with the form g(7) = B
+ A cos
+
(6) to yield unbiased estimates of the frequency and damping of (~0~171 I$)e-rre-82r*/4
waves of the chosen q.10 The phase term I$ accounts successfully for the departureof the spectrumfrom an exact Lorentzian form. The final Gaussianfactor represents instrumentalline-broadening due to the illumination by the incident laser beam of a finite number of wavelengths of the surface wave. For high frequency waves (wg 1 MHz) a rapid after-pulsing phenomenon in the photomultiplier becomes apparent. This exhibits a quasiexponential time dependence for correlator sample times O.5lI2, the relative change in I' will be largest at small WO. This is not observed, so this possibility cannot can be eliminated. The alternative of 007 I be rejected. This leads, using the largest capillary wave frequency, to a limit of
