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Langmuir 1995,11, 4225-4233
Articles Hydrodynamic Modes of Soluble Surfactant Films C.-Y. D. Lu” Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K.
M. E. Cates Department of Physics and Astronomy, King’s Buildings, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, U.K. Received November 9, 1994. I n Final Form: July 24, 1995@ Soluble surfactants in suspended soap films are distributed on the surface monolayers and inside the films. We study theoretically the effects of the dynamics redistribution of surfactants between these two places on the linearized soap film hydrodynamics. The interchange of surfactants can be incorporated into the existing hydrodynamic analysis as an effective complex surface modulus. The main effects are on Marangoni modes and the peristaltic modes. The presence of micelles at high surfactant concentration can be easily included for submegahertz frequencies provided the surfactant solution has a fast micelle fonninghreaking kinetics. The effects of any intrinsic surface viscosities in the monolayers are briefly discussed.
1. Introduction Suspended films have been studied for more than 30 years.l-1° They provided the first testing ground for the DLVO theory of interactions between monolayers, and hence, a n experimental Hamaker constant can be obtained. The interest in dynamics is related to the problem of film drainage1’ and rupture.12 Another problem which involves the film dynamics is the dynamics of dense emulsions or f ~ a m s , in ~ ~parJ~ ticular, the linear viscoelasticity which has been studied recently.15-17 The response of a deformed emulsion may involve the relaxation of the film thickness, which is driven by disjoining pressure; the relaxation of shape, which is driven by surface tension; and the relaxation of surfactant concentration, which may be achieved either by molecular @
Abstract published inAdvanceACSAbstracts, October 1,1995.
(1)Thin Liquid Films; Surface Science Series 29, Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988. (2) Vrij, A,; Joosten, J. G. H.; Fijnaut, H. M. In Adv. Chem. Phys.; Prigogine, I., Rice, S. A., Eds.; John Wiley & Son: New York, 1981; Vol. 48. (3) Vrij, A. Adv. Colloid Interface Sci. 1968,2,39. Lucassen, J.; van den Tempel, M.; Vrij, A,; Hesselink, F. T. Proc. K. Ned. Akad. Wet. B 1970,73,109. Vrij, A.; Hesselink, F. T.; Lucassen, J.;van den Tempel, M. Proc. K. Ned. Akad. Wet. B 1970, 73, 124. (4) Fijnaut, H. M.; Joosten, J. G. H. J . Chem. Phys. 1978,69,1022. ( 5 ) Joosten, J. G. H.; Fijnaut, H. M. Chem. Phys. Lett. 1979,60,483. (6)Young, C. Y.; Clark, N. A. J . Chem. Phys. 1981, 74,4171. (7) Joosten, J. G. H. J . Chem. Phys. 1984,80, 2363. (8) Joosten, J. G. H. J . Chem. Phys. 1984, 80, 2383. (9) Joosten, J. G. H. In ref 1. (10)Sens, P.; Marques, C.; Joanny, J. F. Langmuir 1993,9, 3212. (11)See, e.g., the review by I. B. Ivanov and D. S. Dimitrov in ref 1. (12) See, e.g., the review by C. Maldarelli and R. K. Jain in ref 1. (13) Kraynik, A. K. Annu. Reu. FZuid Mech. 1988,20, 325. (14) In addition to films, those systems also contain many fluid channel networks, the so-called Plateau borders which form the boundaries of the films. Whether the films or the Plateau borders dominate the viscoelasticity measured is still a n open question, but the analysis on the films dynamics will certainly be relevant. (15)Mason,T. G.; Bibette, J.; Weitz, D. A. Phys. Reu. Lett. 1996, 75, 2051. (16) Kahn, S. A.; Schnepper, C. A.; Armstrong, R. C. J.Rheol. 1988, 32, 69. (17) Buzza, D. M. A.; Lu, C.-Y. D.; Cates, M. E. J . Phys. II 1996,5, 37.
a. Symmetric solutions
LX
b. Asymmetric solutions
Figure 1. Symmetry types of the solutions.
diffisions or by the Marangoni effect,18J9in which surface concentration gradients of surfactant cause a differential surface tension which drives the fluid motion and, hence, provides a mechanism for surfactant transport. A good understanding ofthese dynamics is crucial for a successful theory of linear foam visc~elasticity.~’ Fortunately, the hydrodynamic mode analysis for the linearized dynamics of films is particularly well studied, and it seems to describe the equilibrium thermal fluctuations quite accurately as probed by light-scattering experiment^.^-^ Since most of the modes are mixtures of many hydrodynamic variables, it is useful to simplify the analysis by exploiting the symmetry of the system. As we will review in section 3.2, there is a dynamic symmetry which is the reflection with respect to the midplane of the film. One can then use the eigenvalues of this symmetry to classify the solutions into symmetric (sometimes referred to as “squeezing”) solutions and asymmetric (“bending”)solutions (see Figure 1). Because there are more than one mode in each class of solutions, we will avoid the use of squeezing and bending to refer to symmetry. (A restrictive usage of squeezing and bending modes in this paper will be defined later.) Also in this paper, we will refer to all independent relaxation processes (18)Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (19) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Son: New York, 1990.
0743-746319512411-4225$09.00/0 0 1995 American Chemical Society
4226 Langmuir, Vol. 11, No. 11, 1995
Lu and Cates
as “modes”,but reserve “hydrodynamicmodes” for modes instabilities. Here we have similar physical ingredients whose decay rates beomes zero as the wavelength inbut focus on their consequences for the decay rates of creases to infinity (as defined by Martin et a1.20in the thermal fluctuations in a stable film. This topic was context of generalized hydrodynamics). partially discussed by Vrij et a1.;2here we present a fuller treatment. Our analysis is valid close to the equilibrium As is known in the literature (see, e.g., the review by state (about which we linearize the dynamics), and hence, Joosteng and the references therein), depending on physithe results should be testable in light-scattering experical parameters, the soap film mode spectrum can be of ments. various types. Among those parameters, the surface In the following, we will calculate this effective surface dilation modulus, which drives the Marangoni effect, is modulus in section 2, study the film hydrodynamics in a n important one: when it is high, we have three section 3, look a t the mode structure mentioned above symmetric solutions consisting of a pair of peristaltic and consider the effect of the frequency-dependent surface modes (propagating Marangoni effect) and an overdamped modulus on the mode spectrum in section 4, discuss the thickness relaxation (squeezing)mode. When the surface effect of intrinsic viscosities in section 5, and finally give modulus is small, a pair of propagating thickness (squeeza short symmetry in section 6. Our new results, for the ing) modes and a single (symmetric) overdamped Macase of soluble bilayers, complement those of refs 3-9 for rangoni mode arise. On the asymmetric solutions side, insoluble bilayers and refs 23, 30, and 31 for soluble ”there are always a pair of bending modes and a (asymmonolayers. metric) Marangoni mode. In this paper, we study the linearized film hydrodynamic 2. Surface Modulus modes in the presence of diffusive equilibrium of surfactants between the two boundary monolayers and the fluid Since the surfactants have the effect oflowering surface in between. Due to this exchange of surfactant molecules, tension, a differential surface tension is caused by the boundary condition is altered, and additional convariation of surface concentration r as do = ( a d 8 In r) d centration variables are needed for the mode analysis. In r, where o is the surface tension. The negative However, we will find that for analyzing the hydrodynamic derivative is defined as a (positive) Gibbs’ elasticity modes, one can follow the analysis for insoluble surfactant if an effective complex surface modulus is introduced, E,=-ao which contains all the diffusive dynamics and hence a In r depends on the wavelength, frequency, and the symmetry where different isotherms will give different Gibbs’ eigenvalue. Also, the effect of surface diffusion can be elasticities as functions of surface concentrations. In this included similarly. Compared to the insoluble surfactant paper, we only consider linear dynamics, so only its value films studied previouslf-l0 for which the corresponding a t the equilibrium concentration is needed. modulus is simply a constant, the main consequences are To solve the dynamic equations ofthe film (next section), qualitative and quantitative changes in the modes driven connection has to be made between the surface concenby the surface modulus. Our treatment of surfactant tration and the movement of fluid close to the monolayers. diffusion is general and includes micellization effects for In general, when the fluid tangential velocity varies, the those surfactants whose micellar relaxation kinetics is surface area is forced to change, which then alters the fast. The nonaggregated limit (c < cmc) is included as a surface concentration. If the surfactants are insoluble special case. and surface diffusion is negligible, -8 In r/a1nA = 1, and Apart from the effective surface loss modulus caused one can simply write do = EG d In A. The local area by diffusion, some studies on monolayers suggest that an intrinsic surface dissipation could also be ~ i g n i f i c a n t . ~ l - ~ ~variation of dA can also be expressed by a n infinitesmal tangential displacement E as d 1nA = VLE, where VI is a We will later discuss briefly its effect on the damping of tangential derivative. Following the same line in the case hydrodynamic modes. of soluble surfactants, we define a surface modulus as The above effects are known to play nontrivial roles in some branches of film dynamics. The effects of solubility and the intrinsic surface shear viscosity have already been studied by Ivanov and Dimitrov (1974) in the quasistatic film drainage problem,11J2where hydrodynamics linearwhere a residual viscoelastic factor f is defined by f = -(a ized under a steady drainage perturbation was applied to In r/a In A). examine the drainage flow. The effects of surface diffusion This residual viscoelastic factor is a complex valued and the intrinsic surface shear viscosity were also function. (For insoluble surfactants without surface examined in the film stability problem by Maldarelli and diffusion,f = 1.)In general, it has to be found dynamically, Jain,12 where the attention is on the onset of various i.e., by applying a surface area variation and then solving for the surface concentration responding to it. The (20) Martin, P. C.; Parodi, 0.;Pershan, P. S. Phys. Rev. A 1972, 6 , surfactant response consists of an adsorptioddesorption 2401. (21) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport process on the monolayer, and the surfactants’ surface Processes a n d Rheology; Butterworth-Heinemann: Boston, 1991. and bulk diffusive dynamics. (These lead to phase shifts (22) Poskanzer, A. M.; Goodrich, F. C. J.Phys. Chem. 1976,79,2122. and, hence, complex values for f.) This factor in various (23) Ting, L.; Wasan, D. T.; Miyano, K. J.Colloid Interface Sci. 1986, monolayer geometries has been calculated by Levich and 107, 345. (24) Kao, R. L.; Edwards, D. A,; Wasan, D. T.; Chen, E. J. Colloid Lucassen et al.,l8Jowho used it to obtain a complex surface Interface Sci. 1992, 148, 257. modulus. Here we proceed along the same line but with (25) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1. a double-layer geometry of infinite extent as considered (26) Israelachvili, J. Intermolecular & Surface Forces, 2nd ed.; Academic: London, 1991. below. (27) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; The adsorptioddesorption process mentioned above is Kielmann, I.;Ulbricht, W.; Zana, R.; Lang, J.;Tondre, C. J.Phys. Chem. typically at the submicrosecond scale as probed by NMR,25 1976. ~. . . 80. 905. I
- - 7
(28) Turq, P.; Drifford, M.; Hayoun, M.; Perera, A,; Tabony, J. J. Phys. Lett. 1983, 44, L-471. (29) Evans, D. F.; Mukhejee, S.; Mitchell, D. J.;Ninham, B. W. J. Colloid Interface Sci. 1983, 93, 184.
(30)Lucassen, J.; Hansen, R. S. J. Colloid Interface Sci. 1967, 23, 319. (31) Bonfillon, A,; Langevin, D. Langmuir 1993, 9, 2172.
Hydrodynamic Modes of Soluble Surfactant Films
Langmuir, Vol. 11, No. 11, 1995 4227
whereas the diffusion across a thick film (with thickness lo3 A say) is of the order of a few microseconds. So we shall assume that the adsorptioddesorption processes happen instantaneously. This links the surface concentration to the nearby bulk concentration by the equilibration of the surfactant chemical potentials pz$olayerpbulk(Z = f h l 2 ) where the superscript Z denotes the upper monolayer, whose equilibrium position is taken as the plane z = hl2, and ZZ means the bottom layer, whieh is at z = -h/2. Using the Gibbs' adsorption equation aolap = -r, we have
ao -r 4bulk = - dpmonolayer = d o = -E, aP
d In
- T dc(n) = -rich)
T
dctot Cn2c(n)
(4)
n
where c(n) is the concentration of micelles containing n monomers. The total amount of surfactant is given by ctot = Cnnc(n). Notice the extra n in the denominator sum, which means that when the surfactants form micelles of typical size (n),the chemical potential behaves like that of a solution containing far more pure monomer, with concentration of the order (n)cbt. The fast-exchange assumptions also simplify greatly the surfactant bulk diffision into a conventional diffusion equation for cbt with a n effective diffusion constant
for nonionic surfactants as derived by Turq et al. and Evans et al.,28,29 where D(n)is the diffusion constant of micelle with n surfactants. For spherical micelles, CJ is never very different from D(1). For ionic surfactants with high salt, the effective diffusion coefficient is similar but contains a small correction from salt diffusion,29which, for simplicity, we will neglect in this paper. In the case of slow eq~ilibration,~' a much more complicated multispieces diffision-reaction theory would have to be applied. Now we solve for the residual viscoelastic factor f(12,o). Let the upper and the lower monolayers undergo a periodic tangential displacement 5' and 511,respectively, where
The amplitude 5~ represents the in-phase tangential movement which will correspond to the symmetric modes of the next section. The amplitude EB corresponds to the opposite (asymmetric) case. In response to this displacement, the surface concen= ro+ ( r A f rg) exp(ikx iwt), where trations are rlJ1
+
Ctot
= c0
+ (cAcos(qz) + cBsin(qz))e
i(kx+wt)
(7)
where q is given by
io = - n q 2
+ k2)
(8)
The integration constants CA,CB,r A , and r B are related by eqs 3 and 4 as
(3)
To relate +bulk to the bulk concentration, we consider a general situation where micelles are present (the lowconcentration regime is included as a special case). In addition to the fast-exchange assumption between the single surfactants and the monolayers, we further assume the relaxation of the micellar size distribution is fast. The second assumption seems to be valid for surfactants such as sodium octanoate, where the slowest equilibration time is less than microseconds;2showever, it is unreliable in some other cases like SDS,where the slowest equilibration time is milliseconds or more, as studied by Aniansson et ale2' Based on these assumptions, there is a local equilibrium and the differential chemical potential at a point in the bulk fluid isz6 dpbulk
To is the equilibrium surface concentration. The surfactant diffusion equation has the solution
where we define
r = (E&'To)~n2c(n)h/2
(9)
n
In the dilute surface concentration regime where the surface pressure obeys an ideal gas law,26we have EdZTo = 1. In general, it could be-of the order of 10 or less. If we take it as unity, then r represents the number of "effective monomers'' of surfactant per unit area in half the bulk fluid making up the bilayer. For a solution above cmc, s_ayctot M, with (n) = 50 and h 3 10-7m, we have r (70 A2)-l which, despite the low concentration, is of the order of To. The conservation of surfactants on the monolayers to linear order reads
- -
-ar _ - --row + D,V:r at at
F GJv,c,,,
h 2
for z = f-
(10) where D, is the surface diffision constant.lsJg Substituting eqs 6 and 7 gives
(io
+ D,k2)(rAf r B ) = -io(ikgA f ik6,)ro +
with the solutions rA/TO = -fA(k,o)(ik(d and TflO = -fB' (k,w)(i&). This equation defines the residual viscoelastic factors for symmetric and asymmetric modes. We discuss these now in turn. 2.1. Symmetric Case. For a symmetric displacement where 6~ = 0, the residual viscoelastic factor is
fA(k,U)=
io
io
+ D,k2 - @2(2/qh) tan(qh/2)l;/r0
(11)
A very similar result was discussed by Vrij et al. (see eqs 5.37 and 5.40 of ref 2). At long wavelengths and low frequencies, eq 11 becomes
For a given k, this factor is a function of frequency. (Note that q is defined by (8)in terms of 12 and o.)The relevant properties of f are examined separately for overdamped and propagating modes. To discuss the effect of surfactant solubility cf* 1)for an overdamped mode, we need to know the behavior o f f at pure imaginary the values of fA frequency. On this decay axis (0 < -io), are of the order of 1or less, except when the frequency is
Lu and Cutes
4228 Langmuir, Vol. 11, No. 11, 1995 close to a series of diffusive poles. The first pole of f A is at
where we define a total tangential diffusion constant DT = (ras T@Y(To r). For 0 < -iw .c ADA, f~is always negative. This means that a mode driven by Gibbs’ elasticity can not exist in this range. The higher poles are from the diffusion processes across the film, with the slowest one having a decay rate of the order of Qn2/h2. In the range ADA LA. We will refer to the resulting spectrum as type 1 (Figure 2 ) . Equation 43 then describes a mode of frequency iw3 = (2II’h - uk2)h2k2/24rwhich was first derived by Vrij in 1966.12 We will refer to this mode as a n overdamped squeezing mode, as it is mainly a thickness relaxation mode drived by disjoining pressure (and surface tension a t higher k ) and damped by the Poiseuille flow. The surface concentration in this mode is close to the equilibrium value. The other two modes are the peristaltic modes iw1,Z fi J m k - 2vk2as derived by Vrij et al. in 1970.3 They are propagating Marangoni processes in which the fluid inertia are driven by the Gibbs’elasticity (which controls AM), so the surface concentration is strongly involved.33 The mode also involves modulation of the thickness variable whose intrinsic relaxation rate is slower than that of surface concentration. A different type of spectrum (type 2, see Figure 2 ) can arise if one can find a surfactant with a very low Gibbs’ elasticity or apply a large disjoining pressure so that AA > AM. A monolayer close to its liquid-gas critical point37 (if it has one) could be a good candidate. In such a system, the spectrum will consist instead of a slow Marangoni mode driven by Gibbs’ elasticity with little thickness variation and two propagating squeezing (thickness) modes (Vrijet a1.3)where thickness propagation modulates the slow surface concentration variable. We now discuss the effect offAwhich describes the effect of solubility and diffusion. The presence of fA is always associated with A M and, hence, has a large influence on the modes driven by Gibbs’elasticity, whereas other modes are affected less strongly. The type of spectrum present is now determined by comparing l l A with AMfA, the two contributions of the third term in eq 38. Since the the viscoelastic factor fA becomes smaller as the concentration increases, a n existing type 1spectrum can be switched to type 2 by increasing the surfactant concentration to exceed a critical amount a t which the above two contributions (33) In multilayer systems, i.e., lyotropic smectic liquid ~ r y s t a l s , ~ ~ - ~ ~ the second sound modes are similar to the propagating peristaltic modes found here for the simple film, whereas the baroclinic mode is similar to the overdamped squeezing mode mentioned above. (34)Brochard, F.; de Gennes, P.-G. Pramana Suppl. 1975, 1 , 1. (35)Nallet, F.; Roux, D.; Prost, J. J . Phys. Fr. 1989, 50, 3147. (36)Ramaswamy, S.; Prost, J.; Cai, W.; Lubensky, T. C. Europhys. Lett. 1993,23,271. (37) Knobler, C. M.; Desai, R. C. Annu. Rev.Phys. Chem. 1992,43, 207.
Langmuir, Vol. 11, No. 11, 1995 4231
Hydrodynamic Modes of Soluble Surfactant Films
Symmetric modes Overdamped
Tvw 2 Propagating
Symmetric surfactant
-
Asymmetric modes
Bending modes
Asymmetric surfactant transport mode
1H-I
Figure 2. Eigenmodes of the spectra. The thickness of the lines represents the surface concentration on the monolayers. There are two types of spectra in symmetric solutions. Type 1spectrum contains a pair of (propagating)peristaltic modes and an overdamped squeezingmode. Type 2 spectrumcontains a pair of propagating squeezing modes and a (overdamped) surfactant transport mode which becomes the symmetric Marangoni mode in the insoluble films. There is only one type of spectrum in asymmetric solution. The spectrum consists of a pair of (propagating) bending modes and an asymmetric surfactant transport mode which becomes the asymmetric Marangoni mode in the insoluble films.
Figure 3. Schematic plot of the concentration dependence of the symmetric modes. The real coefficientsplotted are defined by i q 2 = fiClk - _C2k2-and i o 3 = -C3k2. In the lowconcentration region (r
