Light Scattering from the Free Surface of Water BY D. LANGEVIK Ecole Normale SupQieure Laboratoire de Physique, 24 Rue Lhomond, Pa.ris(Se), France Received 21st June, 1973 The spectrum of light scattered by thermally excited surface waves on water has been investigated and the value of the surface tension and, after careful analysis of instrumental broadening, the value of the viscosity determined. The results are in good agreement with those obtained by more con- ventional techniques. In contrast to a recent report by McQueen and Lundstrom (ref. (1)) who also performed a light scattering experiment, an anomalously high viscosity was not found. These authors invoked the concept of a highly structured surface region, in order to interpret their data. This region probably does not exist.In recent years, light scattering techniques have been used to study liquid inter- faces. Several theoretical and experimental papers have already been published on this ~ubject.l-~ The mechanism of scattering may be described as follows: thermal motion produces small asperities (about lOA high) on the liquid interface. The interface is submitted to a capillary restoring force and its motion is damped by viscosity (one can neglect gravity forces). Spatial and temporal evolution of the fluctuations can be described on a macroscopic scale by the laws of hydro- dynamics. Consider a surface vibration mode, or surface phonon, with a given wave vector q. The scattering process is simply : Scattered light is concentrated in a well defined direction, simply related to q by expressing momentum conservation in the surface plane, during the scattering process.Our experimental procedure, which involves heterodyne spectroscopy techniques, has already been de~cribed.~ We are presently interested in the study of water covered by monomolecular films and have computed the theoretical spectrum of the light scattered by such films.5 This spectrum depends on the surface tension, on the viscosity of pure water and on four viscoelastic parameters of the film : surface pressure, compression modulus and their two associated viscosities. We report here preliminary experiments on light scattering from the free surface of pure water. In this case the theoretical spectrum depends only on two parameters : the surface tension CT and the viscosity 71 of the liquid.However, the experimental spectra are broadened by instrumental effects resulting from the finite angular resolution of the optical system. The main contribution to the instrumental broaden- ing is the divergence of the laser beam wising from diffraction. The shape and the width of the experimental spectra differ slightly from the theoretical predictions, when using conventional values of viscosity.6 By studying another liquid, ethyl alcohol, and assuming that the viscosity is equal to its conventional value, we found that the instrumental function can be satisfactorily described by a Voigt function.’ The Voigt functions belong to a family of functions depending on one parameter a. They incident photon surface phonon 3 scattered photon.9596 LIGHT SCATTERING AT LIQUID SURFACES include Gaussian (a = 0) and Lorentzian (a = co) functions as special cases. For the instrumental function, we found a = 0.3 (fig. 1, curve 1). Using this instrumental function, we deduced from the line-shape and from the width of the experimental spectra for water, values of the viscosity in good agreement with the conventional value, and of the instrumental line-width, in good agreement with estimated values deduced from the angular divergence of the laser beam arising from diffraction. - 3 - 2 - 1 LJ - vsolALJ1 FIG. 1.-Instrumental functions: (1) Voigt function of parameter a1 = 0.3; (2) square pulse; (3) diffraction function by a circular aperture. Several workers have already studied light scattering by the free surface of water.Mann et aL8 studied mechanically excited surface waves because their signal to noise ratio was not sufficiently good to detect thermally excited waves. They observed two peaks in their experimental spectra and interpreted these strange results using an incorrect dispersion equation, the roots of which are not complex conjugate. More recently, McQueen and Lundstrom studied thermally excited surface waves on water with an experimental arrangement similar to ours. They found the same value of surface tension, though with a higher uncertainty (three times higher than ours), but a different viscosity value : rj = 4 cP. They used an instrumental function having the shape of a square pulse (fig. 1 , curve 2). We have used this second instrumental function also, and have deduced, from our experimental spectra on water, viscosity values lower than 4 CP but higher than 1 cP, which decrease when the scattering angle increases.This clearly indicates that the instrumental broadening is not well corrected for by their method. McQueen and Lundstrom invoked the concept of a highly ordered water surface region in order to interpret their data. It follows that the exis- tence of such a surface zone now appears unlikely. Accordingly this concept is re- j ected. In the following, we describe the experimental conditions, discuss the procedure used for the instrumental broadening correction, and present our experimental results for water at 21°C.D. LANGEVIN 97 EXPERIMENTAL We did not observe any difference between the spectra obtained using com- mercially available bidistilled water and extremely pure water tridistilled in a quartz apparatus.Light from a helium-neon laser (approximatively 60 mW output power) is incident from above the liquid surface at an angle of 3". We select with a diaphragm light scattered by fluctu- ations having a well defined wave vector q. This diaphragm has the shape of a circular annulus.* If 6 is the angle between scattered and reflected beams one has, k, being the wave vector of the incident light : The experimental equipment is described else~here.~ The light scattered by the fluctuations is collected on a photomultiplier tube which also receives light scattered elastically by the windows of the liquid container which thus fulfils the role of a local oscillator for optical heterodyning. In such conditions, it can be shown that the photocurrent power spectrum reflects exactly the power spectrum for the fluctuations of the vertical displacement of the free surface.The photocurrent is amplified, squared, frequency-analysed and recorded. The theoretical spectrum for the free surface of a simple liquid is given by :2 1 D(S) = 0 being the dispersion equation for surface waves : D(S) = y+(1 +s>'-JiTzs and introducing the following parameters : P , z o = - y = - OP 4q2q 2w2 v is the frequency, p the density, 0 the surface tension and q the viscosity; J is the square root determination having a positive real part. When the liquid viscosity is low, as in the case of water, the parameter y is much greater than unity and the roots of the dispersion equation, which are complex conjugates, can be approximated by S , = s;* = iyf [ 1+0 (31 a - [ 1 - $i +o($)] - Pg(v) takes significant values only when v z yh/2nzO. spectrum can be approximated by For these frequencies the 1 * It can be shown that for small incidence angles, the scattering is symmetric around the reflected beam and that a circular annulus selects light scattered with a wave vector q constant in modulus (its direction varies, but the spectrum does not depend on this direction).For higher incidence angles, this is no longer valid. For example, with an incidence angle of 45" and a circular annulus, one detects a finite range of wave vectors : 1-498 LIGHT SCATTERING AT LIQUID SURFACES The spectrum is Lorentzian, centred at the frequency v, = ISq1/2nzo and of half width Avq = -Re Sq/271z0.It follows The scattered intensity varies as q-2 : kT 271 P,(v) dv = -2. s aq One is thus restricted to small scattering angles, i.e., to s.mal1 wave vectors q. In the present experiment 153 cm-l < q < 722 cm-1 which corresponds to 5' < 8 < 25'. For larger scattering angles, the signal to noise ratio becomes very smdl and the interpretation of the spectra is difficult. In order to reduce mechanical vibrations, the optics and the sample were placed on a heavy (one ton) table, which rested on five rubber tyres. INSTRUMENTAL BROADENING CORRECTION When surface waves are not strongly damped (as is the case for water) experimental spectra are broadened by important instrumental effects.Several causes contribute to the instrumental width. (a) The finite aperture of the diaphragm, which therefore selects wave vectors inside a finite range qo & Aq, qo corresponding to the mean radius of the diaphragm. In our experiment, typically Aq/qo - 10 %. The experimental spectrum is then If Aq/qo is small enough, q/vq which is proportional to q-+ varies little with q within the range A4. If we neglect also the variation of Ava with q, the observed spectrum is simply the convolution of Pq(v) with an instrumental function having the shape of a square pulse (fig. 1, curve 2), of half width : vqo. Av, = - - 3 A4 2 40 We can easily compute P(v) with this approximation : when The half width of P(v) is then : AV = J(Av," +AvT). (b) The divergence of the laser beam due to diffraction is also a cause of imperfect definition of q.In our experimental set-up, the reflected beam is focused at the diaphragm centre, and has at this point a diameter roughly equal to the diaphragm aperture. Taking into account the diffraction effect only, the instrumental function would be close to the diffraction function for a spherical aperture (fig. 1, curve 3).D. LANGEVIN 99 (c) Mechanical vibrations cause a small random displacement of the reflected beam, and therefore add a small contribution to Aq. This effect modifies the instrumental width AvI, and probably also the shape of the instruniental function. As the results will show, this effect is relatively small. (d) Finally, the electronic system used for detection might contribute to the instrumental broadening.But in the present experiment this contribution is negligible. These different points were tested using experimental spectra obtained by light scattering from the free surface of ethyl alcohol, We choose this liquid since the damping conditions of surface waves were similar to the water case. Moreover, intermolecular forces and therefore surface zone organisation (if this zone exists) are expected to be different. We first fitted experimental spectra for ethyl alcohol with eqn (6). The viscosities we then deduced from eqn (4) and (7) were higher than the values obtained by conventional methods (y = 1.18 CP at 21°C). Moreover they decreased when q increased. We used Voigt f~nctions,~ which have interesting properties for the present problem : they form a one parameter dependent family of functions, including Gaussian and Lorentzian functions, and the convolution of a Lorentzian function with any Voigt function is another Voigt function.The experimental spectrum. is the convolution of the Lorentzian theoretical spectrum (eqn (2)) with the instrumental Voigt function : it is therefore a Voigt function having a shape characterized by the parameter a. We fitted every experimental spectrum with a Voigt function characterized by a parameter a. Assuming that Ava has the value obtained from eqn (4) using the conventional value of viscosity, we deduced the parameter ai for the instrumental Voigt function and the half width Av,, from the following set of equations :7 We then tried to find a different but still simple instrumental function.Tables for functions c(aI) and P(Av/AvI, al) are given in ref. (7). The instrumental function is in each case close to a Voigt function having a, = 0.30. This function is represented in fig. 1 (curve 1). It is closer to the diffraction function than to the square pulse. This indicates that the instrumental broadening arises mainly from diffraction. In the following, when dealing with the interpretation of data corresponding to the free surface of water, we assume that aI is still equal to 0.30 and allow Avl to vary in order to take into account the effect of mechanical vibrations. The parameter a, which characterizes the shape of an experimental spectrum, depends on q since instru- mental broadening is more important at small q.As q increases, the experimental spectra become closer to Lorentzian spectra, so a increases. We see also that there is a certain dispersion of a values deduced from different spectra for a given q : this comes from the random broadening effect of vibrations. Finally, from the width of the experimental spectrum Av and its shape characterized by the parameter a, we deduce Avq and A+ using eqn (8) and (9). RESULTS WATER AT21'C Measurements of the position of the peak of the spectra v,, and of their half widths Av are presented in table 1, and in fig. 2 using a logarithmic scale.c 0 0 q/cm- 1 153 240 304 366 469 615 772 vq/Hz 2540 2560 5030 5000 7200 7230 7300 9610 9600 9400 9460 9480 13600 13700 13400 13500 20200 28400 Av/Hz 319 362 500 500 880 770 660 727 710 796 853 810 1210 1210 1170 1060 1820 2100 Y 1220 786 614 509 397 308 242 TABLE WATER AT 21°C a/p/(c.g.s.units) (q/p)aa/(c.g.s. units) a 71.5 72.8 72.4 72 73.2 73.5 74.9 74.6 72.8 71.3 72.4 72.6 71.2 72.3 69.4 70.6 69.5 69.7 4.68 5.32 3 3.1 3.33 2.91 2.5 1.9 1.86 2.09 2.23 2.12 1.95 1.95 1.88 1.71 1.72 1.27 0.5 0.5 0.8 0.8 0.75 0.8 0.85 1.3 1.2 1.25 1.1 1.05 1.1 1.3 282 320 373 373 676 575 482 422 434 475 545 534 750 618 2Aq/cm- 1 22.8 25.6 23.8 23.7 38 32.2 26.8 21.3 22.2 24.8 28 27.4 35 28.6 Avq/HZ 56.4 54 186 186 304 288 275 422 3 89 452 43 6 400 600 618 E: q/p/(c.g.s. units) 0 3: 0.83 cl 0.94 1.11 > w 0 cl 1.11 H 1.15 m z 1.09 E 1.04 0 1.10 > 1.02 cl 1.18 z 1.14 rQ 5 1.04 UD. LANGEVIN 101 Taking the values obtained by conventional techniques at 21°C : (r = 72.6 dyn/ cm, q = 0.981 cP, p = 0.998 g/cni3, we computed the values of the parameter Y = 0p/4y2q.Using these y values,* we deduced from v, values and eqn (3), the quantity alp, and from Av values and eqn (4), a quantity ( r , ~ I p ) ~ ~ which would be equal to q / p if there were no instrumental broadening : Av = Av,. We obtain : o/p = (72$.2) c.g.S. units which agrees with conventional values. On the other hand ( ~ / p ) ~ ~ differs from the conventional value; it decreases rapidly as q increases and approaches 1 cgs. unit for high q. q/cnl-' FIG. 2.-Results of measurements of peak frequencies vq and half widths Av for spectra recorded on water at 21"C( +). Values of corrected half widths Av,. (0) Straight lines are the theoretical curves for vq and Avq obtained from conventional values of o, 11 and p.We fitted all the spectra with Voigt functions. Values of a axe shown in table 1 together with values of AvQ and AvI obtained from eqn (8) and (9); Aq values were deduced from AvI using eqn (5) and q / p values from Av4 values using eqn (4). For a given q, there is a lasge scatter in Av, a, and AvI values (precision of Av values is 10 Hz and of a values k0.05) this arises from the random effect of vibrations. We obtain u/p = (l.OS+O.lO) c.g.s. units which agrees with conventional values. Notice that the precision for q/p (10 %) is lower than for alp (3 %) because instrumental broadening is very important and * This procedure is justified so long as these values of y are used to compute relatively small corrections.102 LIGHT SCATTERING AT LIQUID SURFACES q/p depends in it crucial way on the spectral shape : frequency peak measurements are more precise than spectral shape measurements. Moreover the instrumental function is probably affected by vibrations and aI may not be rigorously constant.On the other hand we see from table 1 that 2Aq = 30+8 cm-1 and deduce from eqn (1) that the corresponding angular divergence of the laser beam is A0 = 2Aq/ko N 3 x rad, approximately equal to the divergence of the beam caused by diffraction.* Experimental broadening could not be corrected for several large wave vectors (see table 1) because the signal to noise ratio was low, and differences between a Voigt function and a Lorentzian become small when a increases.Fig. 3 and 4 represent typical spectra corresponding to wave vectors q = 240 cm-l and q = 469 cm-l. v/kHz FIG. 3.-Experimental spectrum of light scattered by the free surface of water, recorded at 21°C for a wave vector q = 240 cm-'. The solid line is the result of a best fit performed with a Voigt function having parameter a = 0.8. The dashed curve is a Lorentz curve. The dotted curve is the result of a fit with the function of eqn (6) where Avq has been computed using T i p = 0.983 c.g.s. units. The corresponding Voigt functions which fit these best have values of a = 0.8 and a = 1.3 respectively. The dashed line curve represents the Lorentzian (a = 03) of identical half width Av. The second spectrum (fig. 4) is closer to the Lorentzian curve : experimental broadening is 1ower.t * The laser beam is diffracted by the lenses and the mirrors of the optical set-up.The main angular divergence is produced by the first lens where the beam has its smallest diameter d, which is about 2.5 mm. The corresponding angular divergence is : Aediff = 1.22 Ald - 3 x rad. t Experimental spectra are not symmetrical with respect to the frequency v = vq because the theoretical spectrum is slightly different from a Lorentzian spectrum (eqn (2)), especially near zero frequency : it has values about four times higher at zero frequency. The differences become negligible only for v > v4- 2Av. To avoid difficult computations we restricted ourselves to this frequency range.D. LANGEVIN 103 Lorentzian function 0 13 15 v/kHz FIG.4.-Spectrum of light scattered by the free surface of water at 21°C for q = 469 cm-l. solid line is the result of a best fit performed with a Voigt function having parameter a = 1.3. The The dashed curve is a Lorentz curve. We have also used the instrumental broadening correction explained in the preceding paragraph and fitted the curve of fig. 3 (q = 240 cm-l) with eqn (6). The best fit corresponds to AvJAv, = 1.2. Using eqn (7) and (4) one can deduce y/p = 1.92 c.g.s. units. Note that this y/p value is higher than ( ~ / p ) ~ ~ for larger wave vectors, and a fortiori higher than y / p values obtained after experimental broadening correction for these wave vectors. On the other hand, if we assume that q/p is equal to the conventional value, we obtain for the spectrum of fig.2: AvI = 3 Av4 = 492 Hz almost equal to the true half-width Av. The corresponding spectrum is shown in fig. 2 (dotted line); as can be seen, the corresponding fit is not satisfactory. McQueen and Lundstrom found, for wave vectors in the range : 353 < q < 707 cm-l that a constant viscosity q = 4 CP can account for their measurements. They invoked, therefore, the existence of a highly structured surface region. If this were true, we should similarly have obtained a value = 4cP had we adopted their method of correction. But we found different values, which decrease with q toward the conventional value q = 1 cP. This trend was observed for all our samples ; instrumental broadening is therefore not well corrected for by this method. Since completion of this work, McQueen and Lundstrom have indicated in a per- sonal communication that they have become aware of the probability that their estimates of the instrumental corrections are incorrect.At the present time they do not believe a highly structured surface region is necessary to explain their data. Av4 = 320 Hz CONCLUSION The spectrum of light scattered by thermally excited surface waves on water has been examined. We found the value of surface tension and, after a careful instrumen- tal correction, the value of the viscosity. These results agree well with those obtained by more conventional techniques. We used a broadening correction method, involving Voigt functions which are a generalisation of Gaussian and Lorentzian104 LIGHT SCATTERING AT LIQUID SURFACES functions.This method has already given satisfactory results with ethyl alcohol. We believe that the reason why it is appropriate is that the main contribution to instrumental broadening is the diffraction of the laser beam. The correction procedure of McQueen and Lundstrom leads to estimates of viscosity which, in our case, vary with scattering angle and which are greater than conventional values. These are however lower than estimates of McQueen and Lundstrom, which apparently do not depend on scattering angle. Such discrepancies clearly show that this type of broadening correction is not adequate. We conclude that it is not necessary to invoke the concept of a high viscosity surface zone to interpret the experimental results. D. McQueen and I. Lundstrom, J.C.S. Faraday I, 1973,69,694. M. A. Bouchiat and J. Meunier, J. Physique, 1971, 32, 561. M. A. Bouchiat and J. Meunier in Polarisation Mufigre et Rayonnernent, p. 121 Volume Jubilaire en l’honneur #Alfred Kastler, ed. by Soci6tk FranGaise de Physique, P.U.F. Paris (1969). For a review of some recent experimental work on light scattering by liquid interfaces, see for example, Supplement au J. Physique, 33. Fasc 2-3 C1(1972), especially the following papers : J. Zollweg, G. Hawkins, I. W. Smith, M. Giglio and G. Benedek, p. C1-135 ; M. A. Bouchiat and J. Meunier, (21-141 ; E. S. Wu and W. W. Webb, C1-149 ; D. Langevin and M. A. Bouchiat, C1-77. For a description of the experimental equipment see, for instance, the first reference in (3). D. Langevin and M. A. Bouchiat, Compt. rend., 1971,272, 1422. Handbook of Chemistry and Physics (Chemical Rubber publishing company, Cleveland, Ohio, 45th edn., 1964.) ’ H. W. Leidecker, Jr. and J. T. Lamacchia, J. Acoust. Soc. Amer., 1967, 43, 143. * J. A. Mann, J. F. Baret, F. J. Dechow and R. S. Hansen, J. Colloid Interface Sci., 1971,37,14. J. Meunier, Thesis (Paris, 1971).