LIQUID SURFACE AND INTERFACE WAVES

Transcription

1 LIQUID SURFACE AND INTERFACE WAVES R. Loudon To cite this version: R. Loudon. LIQUID SURFACE AND INTERFACE WAVES. Journal de Physique Colloques, 1984, 45 (C5), pp.c5-93-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1984 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE Colloque C5, suppl6rnent au n04, Tome 45, avril 1984 page C5-93 LIQUID SURFACE AND INTERFACE WAVES R. Loudon Physics Department, Essex University, Co Zchester C04 3SQ, U. K. Rgsumk - La fonction de Green et le spectre de fluctuations pour le d6placement d'interface entre deux diffgrents fluides, visqueux et compressibles, qui sont s6par6s par une couche mince Qlastique, ont QtB obtenus. Nous discutons les spectres calcules pour trois cas limites de la thiorie ggndrale, avec attention particuli3re sur la sgparation des contributions des ondes capillaires et des ondes acoustiques. Abstract - We derive the interface-displacement Green function and fluctuation spectrum for two different viscous and compressible fluids separated by a thin elastic film. The calculated spectra for three special cases of the general theory are discussed, with particular emphasis on the separation of the capillary and acoustic wave contributions. I - INTRODUCTION The well-known relation between the frequency w of a ripple and its surface wavevector Q for a free liquid surface is where g is the gravitational acceleration, a is the surface tension coefficient and p is the fluid density. The frequency is mainly determined by gravitation for small Q and by surface tension for large Q; the two forces have equal influence for water at the wavevector 360 mel. Longitudinal bulk acoustic waves propagated parallel to the surface with velocity vl and wavevector Q have the dispersion relation The main experimental tool in the study of thermally-excited ripples is lightscattering spectroscopy, where wavevectors in the range are most easily accessible. For such wavevectors, wr is typically two or three orders of magnitude smaller than wl. The interpretation of light-scattering experiments requires theoretical expressions for the power spectra of the thermally-excited fluctuations at the liquid surfaces or interfaces. The light-scattering cross section is in most cases proportional to the mean-square perpendicular displacement of the surface from its flat, static, equilibrium position. The plan of the paper is to derive in section Ilthe fluctuation spectrum for a rather general system of two fluids separated by a thin film. The subsequent sections are devoted to special cases of the general system. More complete details of the calculation and a review of experimental work are published elsewhere /I/. Article published online by EDP Sciences and available at

3 C5-94 JOURNAL DE PHYSIQUE I1 - INTERFACE FLUCTUATION SPECTRUM Fig.1 shows the arrangement of the liquids whose interface ripple spectrum is to be evaluated. The z-axis points vertically upwards, with the gravitational forces directed in the negative z direction. The lower half-space z < 0 is occupied by a liquid having density p and viscosity 11. The upper half-space z > 0 is occupied by a different liquid whose physical parameters are denoted throughout by the same symbols as for the lower liquid but with primes attached to them. Both liquids are assumed to have indefinite extent parallel to the z-axis; in practice, their z dimensions should be much larger than the normal decay lengths of the interface excitations. The z = 0 surface of separation of the liquids is also assumed to have indefinite extent, although it is convenient to assign to it a large area A for normalization purposes. The two liquids are separated by a thin film of some third material. The film thickness is taken to be very small compared to the wavelengths of the interface ripples and to the characteristic wavelengths of bulk excitations in the two liquids. Thus in applying macroscopic fluid mechanics to the system, the effects of the film are felt only in the boundary conditions at the fluid interface. A case of particular interest is that in which the film is formed from a monolayer of adsorbed molecules. An example of a system that has been studied by light scattering is a monolayer of fatty acid molecules on the surface of water with just air above the monolayer. Forces of repulsion between the molecules generate a surface pressure, which diminishes the surface tension a of the supporting liquid from its ordinary value. The surface pressure normally increases with the surface number density y of the adsorbed molecules, and the magnitude of the effect is described by a parameter /2,3/ The surface tension at a molecular density y is thus related to its value for a free surface by a = a free -: 1 (~/y)dy. (5) The parameter E plays the role of an elasticity since its product with the strain in a film deformed from equilibrium determines the restoring force. It is knownas the surface dilational modulus and it acts only for distortions that change the local molecular density. It is a complex quantity for dynamic deformations, and (5) shows that a is also complex in the presence of a viscoelastic film. Consider first the kinds of wave that can be propagated in a bulk liquid. The treatment below is restricted to motions that have a harmonic time dependence exp(-iwt) so that the liquid velocity v and displacement u are related by Fig. 1 - Geometrical arrangement of two liquids separated by a thin film.

4 The liquid is assumed to be compressible with non-vanishing stiffness components related to the velocity of longitudinal acoustic waves by The thermally excited waves have small amplitudes and it is permissible to use the linearized equations of fluid mechanics. Then Euler's equation expressed in terms of the liquid displacement is /4,5/ where and q are respectively the coefficients of bulk and shear viscosity, assumed to be cons-tants for a given substance. Consider a plane wave excitation of the infinite liquid in which the displacement has a spatial dependence exp(iq.r_). The equation of motion (8) has two kinds of solution, corresponding to transverse and longitudinal polarization. The transverse solutions have The wavevector magnitude qt satisfies (8) if and the square root is chosen to have positive real and imaginary parts so that These transverse viscous waves are therefore critically damped in space. In applicationsto spectra measured by light-scattering spectroscopy, the waves of interest have their wavevectors in the zx-plane with an x-component Q that is required to be real by the conditions of the experiment. Let the z-component magnitude be denoted T, so that and from (10) The longitudinal solution of (8) has ~ L x ~ L = o, where the wavevector magnitude ql is given by 4 q2 = pu2/[pv2 - iw(5 + 7 n)j. L L Positive real and imaginary parts are again chosen in taking the square root. imaginary part is normally much smaller than the real part and the propagating longitudinal acoustic waves are only lightly damped. The For wavevectors in the zx-plane with the x-component Q required to be real, the z-component magnitude is denoted L, so that The longitudinal damping has a small effect on the interface fluctuation spectrum so that (15) gives

5 JOURNAL DE PHYSIQUE We now consider the boundary conditions at the interface between the two liquids. Since the fluctuations of the interface out of the z = 0 plane are small, the boundary conditions can be expressed in terms of the fluid displacements evaluated at z = 0. The conditions that the two fluids remain in contact are The remaining boundary conditions express the matching of forces at the interface. They are derived by Landau and Lifshitz /4/, by arc fa-~oliner /6/, and in greater detail by Lucassen-Reynders and Lucassen /3/. The conditions are expressed in terms of the stress tensors a and a'. A component of the stress tensor - for the lower fluid is The forces exerted on unit area of interface by the two fluids must be equal, except in so far as there are additional forces at the interface itself. There are three kinds of interface force. One of these is the ordinary surface tension directed parallel to the surface normal and with a magnitude proportional to the curvature of the interface. For ripples propagated parallel to the x-axis, whose amplitudes are much smaller than their wavelengths, the surface tension forces act parallel to the z-axis. The gravitational forces on the interface also act in the z-direction. The z-component boundary condition is therefore The third kind of interface force results from the viscoelastic properties of the thin film; a distortion of the interface leads in general to a spatially-dependent molecular density y, and hence to a spatially-dependent surface tension a. Thus for the ripple geometry assumed here there is a force aa/ax per unit area directed parallel to the x-axis, and the x-component boundary condition is With the use of (4) we have Suppose that the number of adsorbed molecules remains unchanged when the film suffers a displacement ux that varies with x. An initial surface density y is changed to and hence for small changes in density. use of (22) and (24) to read The right-hand side of (21) is transformed with the Consider the effect of subjecting the lower liquid to an applied force

6 ip'w directed parallel to the z-axis, where Q is a real and positive wavevector. The force couples to a liquid displacement u(z)exp(iqx) with an interaction energy H = -u (zo)*fexp(-iwt). (27) int The effect of the force on the motion of the liquids is obtained by solution of Euler's equation (8) with (26) inserted on the right-hand side of the z-component of the equation. The total displacement produced in the lower liquid by the applied force has components 2 ux (2) = (iq~/2apw) {-exp(i~ Iz-z0 1 ) + exp ( i~ Iz-z 1) )sgn (z-z ) where factors exp(-iwt + iqx) are understood throughout and 1 for z > z 0-1 for z < z. The y-component of the displacement is zero. In the above solutions for the x and z components, the first terms are the driven solutions and they satisfy the driven Euler's equation. The remaining two terms in each component are the free solutions; their coefficients are as yet unknown quantities to be determined from the interface boundary conditions. The displacement produced in the upper liquid contains only free solutions since the applied force does not appear in Euler's equation for z > 0. The components are thus with the signs of theecponents chosen to produce decay of the solutions at large positive z. The wavevector z components are given by (13) and (17) with the parameters for the upper liquid substituted on the right-hand sides. It is convenient to remove the x-components of the four free-wave amplitudes in the solutions for the displacements with the use of (9) and (14). The boundary conditions (181, (20) and (25) then provide four simultaneous equations for the four unknown z-components. The solutions determine the displacement-displacement Green function at the interface as I-. m 1 where F is taken at z = 0 and 2 O 2 ipw L + L' Q2 + TL I 2 ipw T ip'w T' p2+tl Q2+-r'~n ' (34) EQ2] [ctq2 + (0-pl)g nt (TI 2-Q 2-2TtL') Q2 + T'L'

7 C5-98 JOURNAL DE PHYSIQUE An equivalent denominator has been derived by ~arcia-~oliner /6/. The mean-square displacement of the interface at wavevector Q, frequency w, and temperature T is obtained from the fluctuation-dissipation theorem The expression for the interface fluctuation spectrum provided by (35) can be compared rather directly with measured light-scattering spectra. The generality of the expression however obscures the main features of the theoretical predictions. Subsequent sections consider three representative special cases of the general theory. I11 - SINGLE LIQUID WITH A FREE SURFACE It is a simple matter to remove the upper liquid and the film frorr, the theory of the previous section by setting their parameters equal to zero. The surface tension a is a real quantity in the absence of the film. The fluctuation spectrum (35) can be written with use of (10) and (13) in the form il (36) [(a~*/p) +g] +w +4 (nq/p) T (T-L) This form is particularly convenient for constructing graphs, and Fig. 2 shows the surface spectrum of liquid mercury calculated numerically. The quantity plotted is the spectrum (36) divided by the factor kb~/*~p. The spectrum has two main features, in the regions of the capillary wave and acoustic wave frequencies, and it is necessary to show them in separate parts of the graph because of the large disparities in their strengths and frequencies. The parts form of course a single continuous spectrum and the sharply rising feature on the left of Fig. 2(b) is the continuation of the high-frequency wing of the peak shown in Fig. 2(a) It is not difficult to extract from the theoretical spectrum (36) parts that correspond to the features shown in the two parts of Fig. 2. The component in the vicinity of the ripple frequency wr(2 x 108s-1) agrees with an expression of Bouchiat and Meunier /7/; this part of the spectrum has been measured by Bird and Hills /8/. The component in the vicinity of the acoustic frequency w (1.5 x 10l0s-l) L Fig. 2 - Calculated surface fluctuation spectrum of liquid mercury for a wavevector Q = 107m-'. (a) Region of capillary-wave frequency WR; (b) region of acoustic wave frequency wl (from /5/).

8 also agrees with earlier results /9,10/; measured by Dil and Brody /11/. this part of the spectrum has been IV - SINGLE LIQUID COVERED BY A FILM The fluctuation spectrum in this case is easily obtained from (34) and (35) by removal of the parts of the equations that involve the upper liquid. The spectrum now depends additionally on the film dilational modulus E. As discussed in section 11, both E and the surface pressure produced by the film are generally complex quantities. We assume for simplicity however that the imaginary parts of these quantities are negligibly small. An interesting effect /12,13/ occurs in the component of the fluctuation spectrum centred on the ripple frequency wr in the limit of small viscosity where Fig. 3 shows the predicted variation of the linewidth Au of the ripple spectrum with Fig. 3 - Dependence of the ripple spectrum linewidth on the ratio of film dilational modulus to liquid surface tension for water at Q = 5 104,-1. the film dilational modulus E. The maximum in the linewidth variation is associated with the propagation of longitudinal waves in the film. It can be shown /3,14/ that these waves have the frequency and Fig. 4 shows the variation of uf with E. The film waves are heavily damped, their frequency having an imaginary part equal to the real part w, which is much larger than the damping of the free-surface ripples shown by the gashed line in Fig. 4. The maximum in the linewidth in Fig. 3 results from coupling of the film and ripple modes via the fluid viscosity, the coupling effects being particularly strong at the value of E for which wf = There is the usual mode splitting at ur. the crossover point but this is not shown In Fig. 4. Application of the theory to the interpretation of measured spectra requires allowance for the imaginary parts of a and E /15-19/.

9 JOURNAL DE PHYSIQUE Fig. 4 - Variation of the longitudinal frequency wf for water at Q = 5 x lo4,-l. The horizontal lines show the frequency w and damping 4nQ2/p of the ordinary capillary wave on tl?e free water surface. V - TWO LIQUIDS IN DIRECT CONTACT The fluctuation spectrum at the interface of two liquids in direct contact, with no separating film, is obtained by setting & equal to zero in (34) and (351, and by taking ct to be real. This clearly produces little simplification in the general results. However, further simplifications can be made as a result of the separation of the spectrum into two contributions in quite distinct regions of frequency. The situation is similar to that at the surface of a single fluid discussed in section I11 and illustrated in Fig. 2, with the first contribution in the region of the capillary wave frequency and a second contribution in the region of the acoustic wave frequency of the bulk liquid. We consider here only the second part of the spectrum, in the region of the acoustic frequencies wl and wi. These frequencies are much larger than the other characteristic frequencies of the two liquids and their interface. Suitable approximations in (34) and (35) lead to the interface fluctuation spectrum We suppose that both the density and the sound velocity for the upper liquid are smaller than for the lower liquid and the form of the predicted spectrum is shown in Fig. 5. With the help of (2) and (17), the spectrum (39) can be written p'v' (W -w ) (W -w,2)4 L L L = ,2 2 2 w L ' < W < w L naw p v (W -wi ) + P' VL (WL-W ) L 3 The ~0ntin~0~s curve in Fig. 5 shows this spectrum in units of kbtq/aapwl; the dashed curves show the free surface fluctuation spectra of the individual liquids in the same units, similar to the spectrum in Fig. 2(b).

10 Fig. 5 - Interface fluctuation spectrum in the acoustic frequency region for two liquids with v;/vl = 0.7 and p'v'/pv = 0.5. The acoustic contribution to the interface spectrum is much weaker than the capillary wave contribution, similar to the single-liquid surface spectrum discussed in section 111. There seem to be no measurements to date of the acoustic parts of interface fluctuation spectra with which the above theory can be compared. References LOUDON, R., in Surface Excitations (North Holland) LEVICH, V. G.,Physicochemical Hydrodynamics (Prentice Hall) LUCASSEN-REYNDERS, E.H. and LUCASSEN, J., J. Colloid Int. Sci. 2 (1969) 347. LANDAU, L.D. and LIFSHITZ, E.M., Fluid Mechanics (Pergamon Press) LOUDON, R., Proc. R. Soc. Lond. A372 (1980) 275. GARCIA-MOLINER, F., Annls. Phys. 2 (1977) 179. BOUCHIAT, M.A. and MEUNIER, J., J. Physique 32 (1971) 561. BIRD, M. and HILLS, G., in Physicochemical Hydrodynamics 609 (Advance Publications) DERVISCH, A. and LOUDON, R., J. Phys. C11 (1978) L291. ROWELL, N.L. and STEGEMAN, G.I., Phys. Rev. B18 (1978) DIL, J.G. and BRODY, E.M., Phys. Rev. B14 (1976) DORRESTEIN, R., Proc. Acad. Sci. Amst. B54 (1951) 260 and 350. HANSEN, R.S. and MANN, J.A., J. Appl. Phys. 35 (1964) 152. LUCASSEN, J., Trans. Faraday Soc. 64 ( GRIESMAR, C. and LANGEVIN, D., Coll. Nat. C.N.R.S. 938 ( BYRNE, D. and EARNSHAW, J.C., J. Phys. Dl2 (1979) LANGEVIN, D. and GRIESMAR, C., J. Phys. Dl3 (1980) LANGEVIN, D., J. Colloid Int. Sci. 80 (1981) 412. EARNSHAW, J.C., Thin Solid Films 99 (1983) 189.