0 Copyright 1989 American Chemical Society
The ACS Journal of
Surfaces and Colloids MAYIJUNE, 1989 VOLUME 5, NUMBER 3
Reviews Observation of Unusual Effects in the Damping of Capillary Waves on Monolayer-Covered Surfaces R. C. McGivern and J. C. Earnshaw* Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 l N N , Northern Ireland Received September 27, 1988. In Final Form: January 6, 1988 Quasi-elastic light scattering has been used to study the temporal propagation of thermally excited capillary waves upon monolayers of glycerol monooleate at the air-water interface, over a wide range of wavenumbers. Cases were observed in which the damping of the capillary waves on the monolayer-covered surface was less than that for the free subphase. Further investigation of the theoretical capillary wave dispersion, based on the values of the physical properties of the monolayer determined from the light-scattering measurements, showed that these unexpected effects are in fact predicted by accepted theory.
Introduction Liquid surfaces are agitated by random thermal molecular motions. The consequent rough surface can be Fourier decomposed into a complete set of surface waves. The propagation of these capillary waves is governed by the physical properties of the liquid, including the surface tension (y),density ( p ) , and viscosity ( 7 ) . However, when the surface is covered by a monomolecular film, the capillary waves couple to longitudinal or compression modes (governed by a surface dilational modulus), changing their propagation characteristics. In particular, for the monolayer-covered surface, the wave damping is altered. It is generally accepted that the damping is increased, a maximum having been found in many previous studies (e.g., ref 1 and 2). This increase is well understood: the local surface tension varies as the waves alternately compress and dilate a particular surface element, producing an alternating tangential force on the subphase which resists the wave motion, increasing the rate of dissipation of energ^.^ In contrast to this increase, this paper describes the observation for monolayers of glycerol monooleate (GMO)of a decrease in the damping compared to that for ~~
~~
(1) Lucassen-Reynders, E. H.; Lucassen, J. Adu. Colloid Interface Sci. 1969, 2, 347. (2) Lucassen, J.; Hansen, R. S. J . Colloid Interface Sci. 1966,22, 32. (3) Lamb, H. Hydrodynamics; Dover: New York, 1945; p 631.
the clean subphase. An explanation of these observations based on established theory is presented. The thermally excited waves, whilst of microscopic amplitude, scatter light appreciably (e.g., ref 4). The spatial characteristics of the surface waves determine the angle of scatter, while their temporal evolution affects the spectrum of scattered light. In this work, waves of known, real wavenumber were experimentally selected, their temporal evolution being observed by photon correlation.
Theoretical Background The only fluctuations of a monomolecular film on a liquid surface causing significant scattering of light are the thermally excited capillary waves. The relevant theory is well established, and only a brief summary will be given here. A surface fluctuation of wavenumber q (= 2alA) can be described by the departure of the surface from its equilibrium plane: f((r,t) = cOei(qr+wt)
(1)
Experimentally, waves of real q propagating in the x direction were selected for observation, their temporal evolution being characterized by the complex frequency w (= (4) Langevin, D. J. Colloid Interface Sci. 1981, BO, 412.
0743-7463/89/2405-0545$01.50/0 0 1989 American Chemical Society
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546 Langmuir, Vol. 5, No. 3, 1989
+
wo ir). This frequency is related to q via the surface wave dispersion equation. For the case of a fluid surface supporting a monolayer, this is' D(W) = [cq2 + iw.rl(q + m ) ] [ y q 2+ iwdq + m ) - w2p/q1
I
I
1
I
2
4
6
8
=o
(2) where
m = (42
+
FYI2;
Re(m) > 0
(3)
In eq 2, e is the dilational elastic modulus, defined classically as dr (4) d In A A being the area per molecule in the monolayer. The moduli y and e may be viscoelastic,6 incorporating surface viscous effects as e=-
y = yo + iwy' e = to iwe'
(5)
+
(6)
The elastic responses of the system to shear stress normal to the surface and to dilational stress within the plane of the surface are expressed via the elastic moduli yo and eo while the correspondingdissipative processes are described by the surface viscosities y' and e'. The dilational modulus affects the capillary waves indirectly, via their coupling to longitudinal waves within the monolayer. This coupling and its consequences have been discussed el~ewhere,'*~*~ and a brief summary suffices. The frequency w and particularly the wave damping r depend upon e. Most previous analyses of the effects of t upon r have concentrated on low-frequency capillary waves, in contrast to the high-frequency waves involved in light scattering. At relatively low q, the accepted picture holds: as eo increases relative to a fixed yo,the damping passes through a maximum at eo/yo= 0.16, after which it varies very slowly with e* The maximum value of I7 is twice that at eo = ~ 0 . l As q rises the resonance between the two modes moves to higher e,, (for fixed yo)due to the different dispersion behaviors of the capillary and the longitudinal wavesS4The viscosity e' tends to reduce the changes of I' due to eo, its effects saturating at a given q. The spectra of the thermally excited capillary and longitudinal waves of given q are affected by the four surface properties. These spectra have been computed4 and are given by
b
4
for capillary waves and I
for longitudinal waves. The spectrum of light scattered by the surface fluctuations is just P,(w). It is approximately Lorentzian in shape, the exact form depending on the surface properties. The spectra of the two modes are shown in Figure 1for selected values of the surface properties, the coupling being evident
0
Figure 1. Longitudinal (a) and capillary wave (b) spectra evaluated for yo = 40 mN/m, eo = 10 mN/m, and y' = c' = 0.0
at q = 1000 cm-I. Note that spectra are arbitrarily normalized to ease comparison of shapes. around the resonant frequencies.
Experimental Section The heterodynespedrometer used in the present work has been described in detail elsewhere.' Briefly, the monolayer-supporting liquid surface was illuminated with light from an Ar+ laser. Light scattered at small angles ( 750 cm-l fall below the predictions for the clean surface. The statistical significance of this has been examined. On the basis of the Wilcoxon matched pairs signed rank test? the hypothesis that r was not less than the clean surface variation is rejected at the 97.5% level. The present data clearly establish that under the appropriate conditions the damping of capillary waves on a monolayer-covered surface can be less than that for the clean surface. The term "underdamping" is used here to describe this situation. To the best of our knowledge, this is the first time that such unusual effects of monomolecular films upon the propagation of capillary waves have been remarked, although similar effects are apparent in the results of one previous experiment.'O
Discussion The present work derives from the temporal propagation of high-frequency capillary waves. The spatial propagation of mechanically generated capillary waves has been widely used to study low-frequency waves; here we merely cite representative studies on the surfaces of surfactant solu(9) Siegel, S. Nonparametric Statistics; McGraw-Hill: New York, 1956; pp 75-83. (10) Birecki, H.; Amer, N.M. J. Phys. (Les Ulis, I+.) 1979,40, C3-433. (11) Sauer, B. B.;Chen, Y.-L.;Zografi, G.; Yu, H. Langmuir 1986,2, 683.
(12) Eamshaw, J. C.;McGivern, R. C.; Winch, P. J. J. Phys. (Les Ulis, Fr.) 1988,49,1271.
5
105
I
105
2
104
0.01
0
1
(molPrulr/Az)
x
X
0 . 0
0 0 0
I
X
0 .
0
I
I
..*....
0.04
0.03
0.02
r,
e o
X
x x
X
.*.
x x
x x x x
x x
0 0
I
I
I
+
*+
+ + + + + +
+ + + *
103 0
0.01
0.02
0.03
0.04
r, (molrridr/A2) Figure 2. Capillary wave frequencies (a) and damping constants (b) as a function of monolayer surface concentration for five q = 745; ( 0 ) different q values: (+) q = 320.6; ( 0 )q = 533.8; (0) q = 968.5; (X) q = 1160 cm-'.
tions2 and monolayer-covered surfaces.' In these studies, the capillary wave propagation was in accord with accepted behavior, the spatial damping passing through a maximum value, thereafter decreasing to a value in the fully condensed phase which exceeded that for the clean subphase. To understand the present, unusual observations, we turn to the known dispersion of capillary waves. For the present, it is useful to consider the dispersion equation in
548 Langmuir, Vol. 5, No. 3, 1989
Reviews
Y Figure 4. Reduced complex frequency S as a function of the Y parameter for a clean liquid surface (see text).
102
103 '1
(d')
Figure 3. Capillary wave frequencies (X) and damping constants (+) for a fully condensed monolayer-covered surface as a function of q. See text for explanation of solid lines.
a parametrized form by using the reduced complex frequency4
and a dimensionless parameter, expressing the balance of the driving force to the dissipative forces
y = -YOP
(12)
4v2q
As Y is reduced, both real and imaginary parts of S fall until eventually the imaginary part vanishes, the capillary waves then decaying through two modes. The present experiments only involve the propagating regime. Figure 4 shows the real and imaginary parts of S as a function of Y, the dashed lines indicating the established first-order approximations3 wo2
yOq3
,
llz -*
P
r = -2 w 2
in the propagating regime and
P
in the overdamped regime. These approximations only agree with the exact solutions far from critical damping. In the absence of any surface excess effects, the behavior of Figure 4 is universal, applying to all liquids. A monolayer upon a liquid surface introduces several surface elastic and viscous properties of which four (see eq 5 and 6) affect capillary wave propagation. The main effects are due to the surface pressure (r)and to e,,. These are first treated separately, and then the overall picture is considered. The primary effect of spreading a monolayer upon a liquid surface is to reduce the surface tension. This leads to a fall in Y , causing a reduction in both parts of S (and hence of the frequency w). The wave frequencies wo observed for the monolayer-covered surface (Figure 3) were, as expected, less than those for the clean surface. The lower yo also causes a reduction in r, so that capillary waves upon a monolayer-covered surface will apparently be less damped than those on the clean subphase surface at all q. This is observed for a liquid-liquid interface, where the effects of to are much reduced." However, it has normally not been observed for monolayer-covered liquid-air surfaces.lV2 The dilational modulus introduced by the monolayer also affects the wave damping. The coupling of the consequent dilational modes to the capillary waves leads to exchange of energy between the two modes. The longitudinal waves are generally more heavily damped than the capillary waves, so the net effect of the coupling is an increase in the rate at which the latter lose energy. The consequent increase in r usually exceeds the decrease due to T , explaining the increase in wave damping commonly observed in the presence of a surfactant film. The effects of a given to upon the capillary waves are not constant, but depend upon q. This is evident in Figure 5, the relative amplitude of the resonant peak decreasing as q increases. For a given value of to, I' is increased at low q. As q increases, the fractional change in r rises until the resonance is reached. At larger q , the damping falls again, ultimately dropping below the variation expected for the case of to = 0 (represented by the real part of S in Figure 4). These arguments are supported by Figure 5: at low q values ( 6 0 0 0 cm-') the capillary wave damping at to = a is larger than for eo = 0, whereas at higher q the reverse is true. The value of e,-, at which J? drops below that for to = 0 is q dependent, falling as q increases. Note that the solution of eq 2 when to # 0 depends on the relative magnitudes of to and yo,and thus no universal plot such
Langmuir, Vol. 5, No. 3, 1989 549
Reviews
20000
10000 106
-
c
Y
cn I
1
L
104;
4
’
j
*
to
i
50
8
*
*
I
100
(mN/m)
Figure 5. Capillary wave damping constants evaluated from eq 2 as a function of eo with yo = 72.75 mN/m and t’ = y‘ = 0.
as Figure 4 can be drawn for the monolayer case. Forming and compressing a monolayer at the liquid surface entails increases in both 7~ and eo. Thus the variations of r with 6 shown in Figure 5 will be combined with a progressive decrease in damping due to the falling yo. This causes the crossover from increased to underdamping to move to lower q, reaching values of the order of that evident in the experimental data at yo < 30 mN/m. The surface viscosity E‘ tends to reduce the variations of Figure 5: when it is large, I’ is nearly independent of to. With this background, we can return to the present observations. For fully compressed GMO monolayers, underdamping is observed for q 5 750 cm-l (Figure 3). This would be expected if A, eo, and e’ were all large. The present correlation functions have been analyzed to yield the monolayer viscoelastic properties yo,y’, eo, and
e’ directly.12 Unfortunately, when either to or t’ is large, these quantities cannot be estimated precisely. However, the present data sufficed to demonstrate that for fully compressed GMO monolayers both co and e’ were large, 5100 mN/m and mN.s/m, respectively, while yo = 28 mN/m and y’ was rather small. This combination of surface properties is just such as to lead to underdamping of capillary waves of relatively high q. Given the uncertainties in determination of the surface properties from the light-scattering data, it is probably too much to hope for quantitative agreement. However, calculations have shown that underdamping would occur for q I 800 cm-’ for yo = 28 mN/m, eo R= 150 mN/m, and E’ = mN.s/m. This q is slightly greater than the value of 750 cm-‘ at which the observed underdamping occurs (see Figure 3). However, the accord seems adequate, in view of other processes (e.g., intramembrane modes of molecular motion13)which may affect the capillary wave propagation.
Conclusions In conclusion, we have observed cases in which the damping of capillary waves on monolayer-covered liquid surfaces is less than that measured for the clean subphase surface. Subsequent investigation has shown that the accepted dispersion equation predicts that waves of sufficiently high q on a monolayer-covered surface must be underdamped. Previous experiments on mechanically generated waves (e.g., ref 1 and 2) involved waves of such low q that underdamping could never have been observed. Underdamping sets in at q values accessible to light scattering when eo is large relative to yo. The only other report of such phenomena of which we are awareloinvolved lecithin monolayers for which both A and c,, were large, as for the fully compressed GMO monolayers discussed here. Acknowledgment. R.C.McG. wishes to thank British Gas for financial support. Registry No. GMO, 25496-72-4. (13)Fan,C. J . Colloid Interface Sci. 1973, 44,369.
