J. Phys. Chem. 1986, 90, 3894-3895
3894
All metals seem to fall onto curves which are specific to their groups in the periodic table with the exception of the alkali earth elements. Volumetric data for this figure were once again taken at 1000 K, and configurational energy data were calculated at the melting points. Although a different basis could be justified, the relationships would not be affected greatly, since both quantities are not strongly temperature dependent. The heat of vaporization is calculated from the Clausius-Clapeyron equation by differentiating vapor pressure data. Some of the scatter in Figure 3 may be due to the temperatures used in the calculations, but scatter may also be due to inaccurate heat of vaporization data used in eq 6. The magnitude of the electron density and configurational
energy of gold deviate from the trends exhibited by the other elements in both Figure 2 and Figure 3 due to interactions of the inner d orbitals. In these respects gold seems to behave more like a transition metal. These figures demonstrate that the configurational energies, molar volumes, and valence are interrelated. Studies and correlations which neglect these relationships should be regarded with some skepticism. Further studies to understand the configurational energy behavior on a fundamental level should permit mathematical modeling. These figures indicate that an ideal gas reference state might be considered for fundamental study rather than the ionized ideal gas reference state used in the NFE approach.
Damping of Capillary Waves by Polymeric Monolayers. Comparison with Hydrodynamic Theory K. Dysthe,? G. Rovner, Center for Studies of Nonlinear Dynamics,$La Jolla Institute, La Jolla, California 92037
and Y. Rabin* Chemical Physics Department, The Weizmann Institute of Science, Rehovot, Israel 76100 (Received: May 19, 1986)
A combination of the hydrodynamic theory of damping of capillary waves by compressible monolayers and of the experimental equilibrium pressure vs. area curves was used to calculate the wave damping coefficient vs. area curves for poly(dimethylsiloxane)s. The results are in qualitative agreement with the complex wave damping behavior observed by Garrett and Zisman.
The hydrodynamic theory of damping of short surface waves by insoluble monolayers is based on the notion that the main effect of the thin film is to modify the tangential stress boundary condition at the liquid-vapor interface.’ In the simplest version of the theory which will be referred to as the elastic hydrodynamic theory (EHT), one neglects the effects of film viscosity and assumes that the only film parameter entering the expressions for the damping rate is its equilibrium compressibility (or the inverse compressibility, e.g. the elastic modulus)
where A is the area per film (surfactant) molecule and Il is the film pressure given by the difference between the surface tensions in the absence and in the presence of the film, respectively. It can be shown2 that an excellent approximation to the (temporal) damping coefficient y is given by 1
y = -kSw
2
1
+ 2@k8(@- 1) ( @ -1 ) 2 + 1
(2)
+
where k is the wavenumber and w = [gk (a/p)k3]’/* is the wave frequency (g is the acceleration of gravity, a is the surface tension in the presence of the film, and p is the density of the bulk liquid). In deriving the above expression we have neglected terms of order (kQ2(but kept terms of order @(kQ2which give the correct pure liquid limit) where S is the width of the viscous shear layer3 which is much smaller than the wavelength in both the capillary and gravity wave regimes. For a given wavenumber, the dependence on film properties comes through the relation between w Permanent address: Institute of Mathematical and Physical Sciences, De artment of Physics, University of Tromso, Tromso, Norway. PAffiliated with the University of California, San Diego.
0022-3654/86/2090-3894$01 .50/0
and k (e.g., a) and, more importantly, through the dimensionless parameter @ @=(2~)’/~pC,u~/~k-~
(3) where v is the kinematic viscosity of the bulk liquid. Notice that /3 varies from @ m in the absence of the film to @ 0 for an incompressible monolayer. The striking feature of eq 2 is the existence of a maximum at 0 = 1 which goes against the intuitive expectation that the damping is largest for “solid”, incompressible films. For frequencies in the capillary range (lo2-lo3 SI), maximal damping occurs when the compressibility is approximately 0.15 cm/dyn which, in many cases, corresponds to the transition region between the “gasesous” and “liquid-expanded” states of the monolayer: This maximum has been observed in monolayers of simple surfactants.l In other cases involving polymeric monolayers, highly complex variation of the damping rate as a function of area per film molecule has been ob~erved.~This complex behavior was attributed to conformational transitions in the film and, although no explicit analysis has been made, the experimentalists felt that the hydrodynamic theory could not predict the observed phenomena. In this Letter we report the results of the EHT analysis of damping of capillary waves on water by monomolecular films of linear poly(organosi1oxane)s(PDMQ5 In order to determine the film parameter p (eq 3), we have computed the compressibility as a‘function of the area per film molecule ( A ) from the exper-
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(1) Lucassen-Reynders, E. H.; Lucassen, J. Adu. Colloid Interface Sci. 1969, 2, 347, and references therein. (2) Dorrestein R. Proc. K.Ned. Akad. Wet., Ser. B: Palaeontol., Geol., Phys., Chem., Anthropol. 1951, B54, 260, 350. ( 3 ) Landau, L. M.; Liftshitz, E. M. Fluid Mechanics; Pergamon: London, 1959. (4) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience: New York, 1966. (5) Garrett, W. D.; Zisman, W. A. J . Phys. Chem. 1970, 74, 1796.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 3895
Letters
20
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0 0 100 200 300 400 500 MOLECULAR AREA (A2/molecule) Figure 1. Plot of the surface pressure II (broken line) and the wave damping coefficient K (solid line) as a function of the area per film molecule A, for PDMS heptadecamer.
2 0 1 , , , , 1 , , , , 1 I ,, ,, ,, , ,(
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200 300 400 MOLECULAR AREA (A 2/m o1e cul e) 0
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Figure 2. Same as Figure 1, for PDMS dodecamer.
imental ll vs. A curves, for PDMS heptadecamer and dodecamer (Figures 4 and 5, respectively, in ref 5). Then we have substituted B(A) into eq 2 and used the relation3 K
= y/(do/dk)
(4)
to obtain the spatial damping rate K . The results are given in Figures 1 and 2. The qualitative agreement between the experimental and theoretical K vs. A curves indicates that the simple ETH in which the only film parameters are the equilibrium Compressibility and surface tension is sufficient to describe the wave damping behavior of linear poly(organ0si1oxane)s. The agreement is quite remarkable in view of the difficulties associated with the determination of the compressibility from the ll vs. A curves in Figures 4 and 5 in ref 5 (the original
0 100 200 300 400 MOLECULAR AREA (A2/molecule) Figure 3. The dimensionless parameter j3 is plotted as a function of A for PDMS heptadecamer (solid line) and dodecamer (broken line).
data were not available) and the extreme sensitivity of the wave damping coefficient to variations in the compressibility. In order to understand the complex structure of the K vs. A curves we notice that the compressibility (and hence, (3) diverges when the monolayer undergoes a first-order phase transition. This singular variation is superimposed on the systematic decrease of the compressibility with increasing film density (decreasing A). Thus, as we decrease A, (3 decreases below the value corresponding to maximal damping ((3 = 1) but, when a phase transition is approached, it rapidly increases through unity (maximal damping) 01 (minimal damping). Upon further compression (3 to j3 decreases to (3 < 1 and another maximal damping point is reached as it passes through unity. This behavior is illustrated in Figure 3 where j3 is plotted vs. A for the PDMS heptadecamer and dodecamer cases. The success of the simple hydrodynamic theory in explaining the features of the damping vs. area coverage curves for PDMS monolayers suggests that wave damping experiments can be used to detect phase transitions occurring in such films. In view of the large variation of the damping coefficient in the neighborhood of a phase transition, this method is expected to be much more sensitive than equilibrium ll vs. A measurements. Finally, we comment on the limits of validity of the EHT approach to wave damping. This approach assumes that the energy dissipation in the viscous shear layer is much larger than in the monolayer and thus that film compressibility rather than film viscosity is the important parameter for wave damping. The above approximation is expected to be accurate if the time scale of the external perturbation (e.g., inverse wave frequency) is different from that of the dynamic processes in the film6 (diffusional exchange with the bulk, rearrangement in the film, etc.). However, when the two time scales coinside, viscous effects in the film can no longer be neglected and the more general viscoelastic theory has to be used.
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(6) van den Tempel, M.; Lucassen-Reynders,E. H. Adv. Colloid Interface Sci. 1983, 18, 281.
