J. Phys. Chem. B 2000, 104, 7987-7992
7987
Autooscillations of Surface Tension in Water-Alcohol Systems N. M. Kovalchuk† and D. Vollhardt*,‡ Max-Planck-Institute of Colloids and Surfaces, 14424 Potsdam/Golm, Germany, and Institute for Problems of Material Science, KieV, Ukraine ReceiVed: April 25, 2000
Autooscillations of surface tension at the water-air interface are observed when a droplet of an alcohol is formed at the tip of a capillary immersed under the water surface. The new phenomenon is studied for a number of aliphatic alcohols. It was revealed that the oscillations period increases in the homologous series from pentanol to nonanol. The system behavior depends on the immersion depth of the capillary. The observed phenomenon is discussed from the point of view of the transition processes in the system and the Marangoni instability.
Introduction In a fluid system with free interfaces, nonuniform distribution of temperature or concentration can lead to instability and formation of dissipative structures due to Marangoni effect.1,2 The known revelations of this instability are Benard cells,3 rolls4 and waves, including solitary waves.5,6 These phenomena are investigated rather comprehensively and have well developed theoretical backgrounds (see, for example, refs 7-12). Recently a new interesting type of instability has been found which is related to the surface tension driven instability. Autooscillations of surface tension have been produced by periodically arousing Marangoni instability.13 They were observed at the water-air interface when a drop of diethyl phthalate is carefully formed at the tip of a capillary immersed into pure water. After a certain induction period when surface tension remains constant, well defined and regular oscillations of the surface tension arise in the system. The oscillations can be observed over more then 8 h and have asymmetrical shape, in the course of which an abrupt decrease of the surface tension value is followed by a gradual increase. Some phenomena which have probably the same theoretical origin are mentioned in the literature. Roze and Gouesbet investigated the instability of an oscillatory type produced when a hot wire is placed under liquid-air interface.14,15 Free surface oscillations in form of propagating waves arise in the system if the temperature difference between the hot wire and the ambient liquid exceeds a critical value. The waves can appear above one end of the wire and disappear above the other end, or they can appear above the center of the wire and propagate to its ends. A model of such processes is given in ref 16 but it does not deal with the properties of the liquid and therefore cannot predict the conditions for the appearance of oscillations. The surface tension at the air-liquid interface was not measured. Another instability arising at the interface between an aqueous alkyltrimethylammonium halogenide solution and an organic phase (see section 2.2.1 in ref 1) was studied by Dupeyrat and Nakache. Solutions of potassium iodide or picric acid in nitrobenzene or nitroethane was used as the organic phase. The * Corresponding author. † Institute for Problems of Material Science. ‡ Max-Planck-Institute.
system is far from equilibrium because alkyltrimethylammonium halogenide is more soluble in the organic solvents than in water, and accordingly, potassium iodide and picric acid are more soluble in water. It was found that in this system the mass transfer between the phases is accompanied by interfacial tension oscillations. The existence of oscillations, their period and amplitude depend strictly on the solute concentration in both phases, the shape is similar to those described in ref 13. Analogous experimental results were obtained by Magome and Yoshikava17 by using aqueous solutions of trimethyloctadecylammonium and solutions of potassium iodide in nitrobenzene. A quantitative theoretical analysis that can explain the complicated chemical and electrochemical processes of these systems could not be given but also the qualitative analyze proposed is not quite complete as it does not take into account the role of the convective flow although the authors indicated the existence of sufficient convective streams. It seems to be possible that the mechanism of the oscillations described in the works discussed above is similar to the autooscillations mechanism of the surface tension.13 A deeper study of this mechanism could shed a light onto the conditions, where such a instability type can be realized. The understanding of the mechanism that explains the autooscillation phenomenon observable in a system with the diethyl phthalate droplet in water is based on the mutual influence of diffusion and convection resulting in periodical switching between diffusion and convective mass transfer in the system. The surface tension changes according to the change in the mass transfer and play an active role in the generation of convective flow. According to ref 13, the main characteristics determining the behavior of such a system are the solubility of the surfactant in water and its surface activity. To corroborate the proposed mechanism and to investigate the phenomenon in more detail it was necessary to provide evidence that similar autooscillation can be induced also by other amphiphilic substances. We focused on the study of the system behavior with substances having surface activity and solubility in water of the same order as those for diethyl phthalate but being essentially different from diethyl phthalate in chemical structure, molecule size, density, and other physicochemical properties. The aliphatic alcohols are good candidates to fulfill these requirements (see Table 1). The results of the investigations of
10.1021/jp001582+ CCC: $19.00 © 2000 American Chemical Society Published on Web 07/29/2000
7988 J. Phys. Chem. B, Vol. 104, No. 33, 2000
Kovalchuk and Vollhardt
TABLE 1. Properties of the Used Surfactants13,18,19 substance
purity, %
density, g/cm3
Da × 109, m2/s
solubility in water, mol/m3
Γm × 10 mol/m2
Kl, m3/mol
R × 106, m
n-pentanol n-hexanol n-heptanol n-octanol n-nonanol diethyl phthalate
>99 >99 98 >99.5 98 98
0.814 0.819 0.822 0.827 0.828 1.118
0.861 0.781 0.718 0.667 0.625 0.572
200 58 14 3.4 0.8b 6.7
6.5 6.2 7.7 6.6
0.07 0.23 0.62 3.23
5.3
1.3
0.46 1.4 4.8 21 120b 6.9
a
The diffusion coefficients are determined according to the Wilke-Chang correlation, using Le Bas additive volumes for the calculation of molar volumes of the solute.20 b Data about the surface properties of nonanol were not found in the literature. Its solubility in water is not given in ref 19. In other sources nonanol is indicated as insoluble. Nevertheless it was decided to investigate the behavior of the nonanol, accepting its properties as extrapolation from the data of the foregoing alcohols.
Figure 1. Experimental setup for the investigation of the autooscillation of the surface tension. 1, Measuring cell with water; 2, covering glass; 3, capillary; 4, surfactant droplet; 5, Wilhelmy plate.
Figure 2. Autooscillations of the surface tension in the system waterdiethyl phthalate. Cell diameter d ) 45 mm, immersion depth of the capillary h ) 6.2 mm.
systems with an alcohol droplet under the free water surface are presented in this paper.
surface tension was measured by using a freshly annealed platinum Wilhelmy plate. After the surface was cleaned for removing possible contaminations, an alcohol droplet was formed at the tip of the capillary under the water surface and at once the surface tension measurement was started. During the experiment the container was covered with a glass plate having orifices for the capillary and the Wilhelmy plate. The distance between the capillary and the Wilhelmy plate was chosen to be 15 mm. The drop diameter was approximately 2 mm. The experiments were carried out at room temperature (21-22 °C) without using a thermostat.
Materials and Methods All alcohols used were obtained from Fluka and Aldrich. Their properties are given in Table 1, where D is the diffusion coefficient of the surfactants in water, Γm and Kl are the parameters of the Langmuir isotherm, and R ) Γm Kl is the Henry coefficient. For comparison the properties of diethyl phthalate are added.13 The parameter values of the Langmuir isotherm for alcohols adduced in the literature have some distinctions (in Table 1 are used the mean values presented in ref 18). Although the measuring temperature is not always exact reported, the common tendency is incontrovertible. The Γm values have rather small difference in the considered alcohol series, but Kl increases with increasing molecular weight. The solubility of the alcohols in water are taken from ref 19. At small surfactant concentrations the Henry adsorption isotherm can be used instead of the Langmuir isotherm. The Henry coefficients given in the last column of Table 1 reflect the surface activity of the substance. Ultrapure water with the specific resistance of 18.2 MΩ/cm was used as solvent. It was obtained by purification of the distillated water in a Millipore desktop unit. A schematic representation of the measuring cell is shown in Figure 1. In the experiments cylindrical glass containers with inner diameters of 45, 48, 57, and 78 mm and heights of 3545 mm were used. A conic container with a diameter in the top of 50 mm, in the bottom of 25 mm and a height of 40 mm were tested as well. The inner diameter of the glass capillaries was 1 mm. In the experiment the measuring cell was filled with pure water, placed in the measuring box of the tensiometer and the capillary was immersed into the water. The immersion depth of the capillary was controlled by a micrometrical screw. The
Results Diethyl Phthalate. Recent results reported in ref 13 were obtained in a thermostated cell at a temperature of 30 °C. To compare more correctly the data obtained for the alcohols with those for diethyl phthalate the experiments with diethyl phthalate were repeated at the conditions described above. An example is shown in Figure 2. The study was carried out for capillary immersion depths between 6 and 7 mm. The amplitude of the auto-oscillation was usually approximately 2 mN/m; its period was between 11 and 14 min. But sometimes the period changed suddenly. For example, it can be doubled as shown in Figure 3. The auto-oscillations presented in Figure 3 are obtained at a distance of 30 mm between the capillary and the Wilhelmy plate. It is seen that the amplitude of these auto-oscillations is smaller in comparison to those presented in Figure 2, where the distance is 15 mm. The induction time before the beginning of the oscillations was approximately 12 min on average. That is smaller than reported in ref 13 because of the smaller immersion depth of the capillary. The average value of the surface tension gradually decreases with time according to the accumulation of diethyl phthalate on the water surface. Nonanol. The system water-nonanol reveals autooscillations of the surface tension (Figure 4) but their shape is not so regular
Surface Tension in Water-Alcohol Systems
J. Phys. Chem. B, Vol. 104, No. 33, 2000 7989
(a)
Figure 3. Autooscillations of the surface tension in the system waterdiethyl phthalate. Cell diameter d ) 57 mm, immersion depth of the capillary h ) 6.0 mm, distance between capillary and Wilhelmy plate r ) 30 mm.
(b)
Figure 5. Autooscillations of the surface tension in the system wateroctanol. Cell diameter d ) 45 mm, immersion depth of the capillary h ) 6.1 mm. Figure 4. Autooscillations of the surface tension in the system waternonanol. Cell diameter d ) 45 mm, immersion depth of the capillary h ) 6.25 mm.
and their duration is shorter than those obtained for the system diethyl phthalate-water. Autooscillations were observed 3 h maximum. Afterward the surface tension decreased gradually with time. The induction time is nearly 10 min, the oscillation period averages 15 min; the amplitude is 1-2 mN/m. Octanol. Long time regular autooscillations of the surface tension were observed when using octanol as the surface active substance. The induction time was 4-10 min in all processes investigated in the system water-octanol. For example, in Figure 5a the part of the autooscillation process is presented which took place in the 40th hour after onset of the process. Approximately after 60 h, the oscillations became less regular, sometimes they revealed changes in period and amplitude (Figure 5b). The experiment was stopped after 72 h, although the autooscillations still existed. During this time interval, the diameter of the octanol drop diminished roughly in two times. The oscillation period increased with time. It was equal to 7 min at the beginning of the process and 22 min after 72 h. The amplitude decreased from 5.5 mN/m to 2.5 mN/m. Further experiments with octanol were performed to obtain information on the effect of the immersion depth on the autooscillation process. At immersion depths of the capillary from 8.5 to 5.5 mm, autooscillations with periods of approximately 4-7 min and amplitudes of 5-6 mN/m were observed when using measuring cells with diameters of 45 and 48 mm. Interestingly only one oscillation arose in the system if the immersion depth of the capillary was reduced below 5.5 mm, and afterward the surface tension decreased gradually according to the octanol adsorption at the surface (Figure 6). For a measuring cell with a diameter of 57 mm, the critical
Figure 6. Single oscillation of the surface tension in the system wateroctanol at small immersion depth of the capillary. Cell diameter d ) 48 mm, immersion depth of the capillary h ) 5.3 mm.
immersion depth was changed to 6.5-7 mm. It was possible to get regular auto-oscillations at a capillary immersion depth of 6.65 mm, although they were continued only 1.5 h. The oscillation period was nearly 5 min, their amplitude decreased from 2.5 to 1.3 mN/m. Afterward the oscillations were discontinued. If the immersion depths of the capillary is larger than the critical depth longtime autooscillations were always observed, but their period was not so stable as for cell diameters of 45 and 48 mm. The period changed between 7 and 24 min, the amplitude was 2-4 mN/m. For a cell diameter of 78 mm only single oscillations were observed with amplitudes nearly 2 mN/m (at the capillary immersion depths between 4 and 8 mm). Using conic measuring cells the autooscillations were also observed (Figure 7). They have periods of nearly 14.5 min but the amplitude changes essentially with continuation of the autooscillation process. Heptanol. The water-heptanol system also revealed longtime and regular autooscillations of the surface tension if the immersion depth of the capillary was larger than 9 mm (Figure
7990 J. Phys. Chem. B, Vol. 104, No. 33, 2000
Kovalchuk and Vollhardt
Figure 7. Autooscillations of the surface tension in the system wateroctanol for a conic cell. Immersion depth of the capillary h ) 6.75 mm.
Figure 10. Autooscillations of the surface tension in the system waterhexanol. Cell diameter d ) 57 mm, immersion depth of the capillary h ) 13.25 mm.
Figure 8. Autooscillations of the surface tension in the system waterheptanol. Cell diameter d ) 57 mm, immersion depth of the capillary h ) 10.9 mm.
Figure 11. Autooscillations of the surface tension in the system waterpentanol. Cell diameter d ) 57 mm, immersion depth of the capillary h ) 13.7 mm.
was 10 mm and larger (Figure 10). The induction time was approximately 1 min. Pentanol. The autooscillations in the water-pentanol system began already during the drop formation, i.e., the induction time is practically absent. The autooscillations were observed over more than 4 h (Figure 11) and were accompanied by considerable wave motion on the surface. The initially formed pentanol droplet was fully dissolved within 2 h, but even thereafter, the oscillations continued owing to consumption of the pentanol rest in the capillary. The oscillation period for this system was near 40 s, the amplitude was 1.5-2.5 mN/m. Discussion Figure 9. Monotonic decrease of the surface tension in the system water-heptanol at small immersion depth of the capillary. Cell diameter d ) 48 mm, immersion depth of the capillary h ) 6.1 mm.
8). The oscillation period was 2.3-3.8 min, the amplitude 2-4 mN/m, and the induction time 2-3 min. For this system the maximum time of observation was chosen to be 24 h although regular oscillations still existed when the measurements were stopped. When the depth of the capillary immersion was between 7 and 9 mm the system revealed only one oscillation similar to those shown in Figure 6 or sometimes two or three oscillations. If the capillary depth was less than 7 mm the surface tension in the system decreased rather quickly and monotonically without any oscillations (Figure 9). Hexanol. Autooscillations with oscillation periods of 1.31.7 min and amplitudes of 1.5-3 mN/m were obtained for the water-hexanol system if the immersion depth of the capillary
Experimental study showed that it is possible to obtain the autooscillations of the surface tension similar to those observed in the system with diethyl phthalate by putting of a alcohol droplet from the series from pentanol to nonanol under the free water surface. Over a certain range the solubility and surface activity of these alcohols are similar to those of diethyl phthalate. Thus, the experiments confirm that the development of autooscillations is not a specific feature of diethyl phthalate, but rather according to the theoretical analysis performed in ref 13 it is mainly the consequence of a particular interplay of solubility and surface activity. At the same time the experiments demonstrated that autooscillations can occur in a wide range of the surfactant properties (see Table 1). The phenomenon of the autooscillations can be understood in the frame of the model proposed recently.13 After the drop formation, a slow diffusion transfer of the dissolved mediumchain alcohol begins in the system. Gradually the surfactant reaches the surface and causes convective flow directed from
Surface Tension in Water-Alcohol Systems the capillary to the walls of the measuring cell owing to the nonuniform surface tension (Marangoni flow). Initially the convective flow is very small but when the concentration gradient reaches a critical value the system loses stability and convection abruptly increases. In this moment, a fast decrease in the surface tension is observed due to the fast transfer of the surfactant. However, this stage continues only during a rather short time. The convective stream spreads the surfactant uniformly over the surface. It also mixes the liquid in the bulk and creates a more uniform concentration distribution in the regions adjacent to the surface. The system returns to the stable state and the convection gradually disappears due to viscous dissipation. The surface tension in the remote parts of the interface gradually increases during this time because of both getting of more dilute solution to the surface and the partial desorption of the alcohol from the surface. Then in the region close to the capillary new diffusion concentration field begins to form until the concentration gradient reaches the critical value. Then all repeats again. The characteristics of the autooscillations change regularly in the studied series of the alcohols. The period of the oscillations has a well pronounced tendency to decrease with the decreasing chain length of the alcohol. It decreases from 15 min for nonanol to less than 1 min for pentanol. The induction time before the auto-oscillations start, has the same tendency. It is about 10 min for nonanol and is practically absent in the case of pentanol. The amplitude of the oscillations also decreases slightly in the series from octanol to pentanol but it is probably related to the increase of the immersion depth of the capillary in the experiments with lower chain alcohols. The observed autooscillations of the surface tension depend on the geometrical factors such as, the immersion depth of the capillary, the geometry of the measuring cell, and the distance between the capillary and the Wilhelmy plate. The induction time decreases with the capillary immersion depth. It was also found that at small capillary immersion depths below a critical value only a single oscillation was observed in the system. The autooscillations are absent at all by further decrease of the capillary immersion depth. The critical value of the immersion depth depends on the geometry of the measuring cell and possibly, on the drop diameter. It increases in the series from octanol to hexanol. The amplitude of the autooscillations decreases by increasing of the distance between the capillary and Wilhelmy plate. The autooscillations amplitudes obtained experimentally allow the estimation of the surfactant amount supplied to the surface during one oscillation. The calculated average values of (0.62.7) × 10-7 g are of the same order for all investigated substances. During the following diffusion stage, the greatest part of this surfactant amount (90-99%) dissolves in the bulk what leads to an increase of the surface tension. The part of the surfactant that remains on the surface after every oscillation decreases in the series from nonanol to pentanol in accordance with the decrease of the autooscillations period. The instability accompanied by an abrupt increase of the convection velocity, arises when the normal concentration gradient near the surface reaches its critical value that is dependent on the properties of both liquids (surfactant and solvent). The threshold of the instability in a system is usually characterized by the dimensionless Marangoni number. It should be noted that in the literature the Marangoni number is defined in two different ways7,11 depending on the conditions of the experiment. It can be defined as Mn ) h2/µD|dσ/dC|dC/ dz, where h is a characteristic length, µ the bulk viscosity of
J. Phys. Chem. B, Vol. 104, No. 33, 2000 7991 the liquid, D the diffusion coefficient, σ the surface tension, C the surfactant concentration, and z the normal to the surface coordinate. So the defined Marangoni number determines the threshold of the instability arousing due to the existence of the normal concentration gradient near the surface. On the other hand, the Marangoni number can be defined as Mt ) h2/µD|dσ/ dΓ|dΓ/dr, where dΓ/dr is the adsorption gradient along the surface. It determines the intensity of the convective flow produced by the surface tension gradient in the case of the tangential concentration gradient. The system considered here is characterized by a complicated time dependent concentration distribution in the bulk and on the surface. Both the tangential and the normal concentration gradients are presented here and each of them creates specific effects on the processes in the system. Therefore, both Marangoni numbers mentioned above should be used for the analysis of the system behavior.13 The intensity of the convection on the relatively stable diffusion stage is characterized by the number Mt, whereas the number Mn determines the transition of the system to the unstable state. At the beginning of the process when the convective fluxes in the system are still negligible the numbers Mt and Mn can be roughly estimated as13
Mn ≈
| |
( )
2 2C0r0 dσ dΓ h2 -(xh2+rj2-r0)2/4Dt h e µD dΓ dC xπ(h2 + jr2) 4Dt
1/2
(1) Mt ≈
| |
( )
C0r0h dσ h3jr 4Dt x2 2 2 e-( h +rj -r0) /4Dt 2 2 2 2 µD dΓ xπ(h + jr ) h
1/2
(2)
where C0 is the solubility of alcohol in water, r0 is the droplet radius, h is the immersion depth of the capillary, and jr is the radial coordinate on the surface. At the beginning of the process the solution concentration near the surface is small and dσ/dΓ can be replaced by RT (R, universal gas constant; T, temperature) and dΓ/dC by the Henry constant R. It is seen from the data given in Table 1 that the product C0R changes insignificantly from pentanol to nonanol. In this case, the change of the normal Marangoni number (1) is mainly connected with the exponential factor, and therefore, the induction time (when at the first time the Marangoni number reaches its critical value) has to decrease in the series from nonanol to pentanol according to the increase of the diffusion coefficient. This conclusion is in line with the experimental data. For the same reason the induction times observed for diethyl phthalate are larger than those for octanol. In accordance with the experimental data, it is clear from the mechanism expounded above that the oscillation period have to be changed similar to the induction time; namely, it should also decrease in the alcohol series from nonanol to pentanol. It is also clear that the autooscillation periods have to increase gradually with time as far as the alcohol concentration near the surface increases after every oscillation, and more time is necessary to reach the threshold value of the normal concentration gradient. However, it should be emphasized that the real decrease of the induction period observed in the experiments for the medium-chain alcohols is much larger than it can be expected proceeding from eq 1. It can be supposed that also a significant buoyancy driven convective bulk flux takes place in the system water-alcohol which reduces the induction time. Indeed in these systems, the contribution of the buoyancy driven convection should increase as the density of the alcohol decreases and its
7992 J. Phys. Chem. B, Vol. 104, No. 33, 2000 solubility in water increases from nonanol to pentanol. For example, the density of the saturated octanol solution in water calculated according to ref 20 is only 0.009% less than that for pure water. It corresponds to an increase of the water temperature of approximately 0.4 °C and cannot be the reason for the noticeable buoyancy driven convection during the induction period. At the same time, the density of the saturated pentanol solution in water is 0.4% less than the density of pure water what corresponds to the difference of the water temperatures of nearly 15 °C, and this can, of course, produce the convective motion in the system. Thus, the buoyancy driven convection, negligibly small for less soluble substances as octanol or nonanol, should be taken into account for more soluble substances as pentanol, the solution density of which is sufficiently different from the density of pure water. It is seen from eq 2 that the tangential Marangoni number is independent of the surface activity. It increases in the homologous series from nonanol to pentanol, according to an increase of the solubility and the diffusion coefficient. Therefore, it is possible that the bulk convection, caused by surface Marangoni flow at the beginning of the process, leads to a more rapid increase of the normal concentration gradient. This can also be a reason for the rapid decrease of the induction time in the case of the lower chain alcohols. However, the influence of the convection caused by the tangential surface concentration gradient is not completely clear. It can accelerate the increase of the normal concentration gradient near the surface if the convective flux transfers slowly the solution from the regions adjacent to the droplet that are more rich on surfactant. However, more intensive convection can reduce the normal concentration gradient due to transfer of dilute solution from the distant regions and redistribution of the dissolved material over the large volume. In the last case, the normal concentration gradient cannot reach the critical value, so that only relatively stable continuous convection without oscillations is possible. This case is exactly observed if the immersion depths of the capillary are lower than the critical value. Here the solubility of the surfactant is important.13 The characteristic Marangoni number Mt increases with the increase in the solubility so that for the lower-chain alcohols continuous convection takes place in the broader interval of the capillary immersion depths. Nevertheless, this problem still requires more comprehensive theoretical and experimental investigations. Conclusions The studies of the instability in the system where a droplet of the homologous n-alcohol series from nonanol to pentanol is formed at the tip of a capillary under the free water surface show that all of them reveal autooscillations of the surface tension, similar to those developed in the water-diethyl phthalate system that is described in Ref. [13]. This provides clear evidence that auto-oscillations can evolve by surfactants having various chemical structure and physicochemical properties. The parameters of the auto-oscillations demonstrate the strong dependence on the surfactant properties. This dependence is most pronounced for the period of the oscillation and for the induction time but only slightly expressed for the amplitude of
Kovalchuk and Vollhardt the autooscillations. The parameters of the auto-oscillations are also strongly affected by the geometrical characteristics of the system. The very strong decrease of the induction time for the lower chain alcohols indicates probably the existence of a considerable convective bulk flux in the system before the instability arises. This convection can be driven by the buoyancy due to the rather large density difference between water and the saturated solution of the light alcohols especially of pentanol, and should be taken into account at the further analysis of such systems. Two qualitatively different types of the system behavior induced by the dissolution of the surfactant can occur, namely, autooscillations and continuous convection. The system behavior depends on the capillary immersion depth. Auto-oscillations are developed at the immersion depths larger than a critical value, whereas at the smaller immersion depths only continuous convection takes place. The critical immersion depth increases in the homologous alcohol series from octanol to pentanol in good correlation with the increase of the solubility. However, the conditions of the transition from one type of the system evolution to the other type depend on various factors and are, obviously more complex. The obtained experimental data should be a basis for further comprehensive studies of the new phenomenon of the autooscillations of the surface tension and the peculiarities of the instability in the described systems. Acknowledgment. We thank Dr. S. Siegel for his help in optimizing the experimental setup. N.K. thanks gratefully the Max-Planck Gesellschaft for the financial support of this work. References and Notes (1) ConVectiVe transport and instability phenomena; Zieper, J., Oertel, H., Eds.; Karlsruhe: Braun, 1982. (2) Velarde, M. G.; Rednikov, A. Ye. In Time-Dependent Nonlinear ConVection; Tyvand.-Suthampton, P. A., Ed.; Comput. Mech. Public. LTD: 1998. (3) Koshmieder, E. L. AdV. Chem. Phys. 1974, 26, 177. (4) Schwabe, D.; Mo¨ller, U.; Schneider, J. Scharmann, A. Phys. Fluids A 1992, 4, 2368. (5) Wierschem, A.; Velarde, M. G.; Linde, H.; Waldhelm, W. J. Colloid Interface Sci. 1999, 212, 365. (6) Linde, H.; Chu, X.-L.; Velarde, M. G.; Waldhelm, W. Phys. Fluids A 1993, 5, 3162. (7) Pearson, J. R. A. J. Fluid. Mech. 1958, 4, 489. (8) Stearling, C. V.; Scriven, L. E. A. I. Ch. E. J. 1959, 5, 514. (9) Smith, K. A. J. Fluid Mech. 1966, 24, 401. (10) Sorensen, T. S.; Hennenberg, M.; Sanfeld, A. J. Colloid Interface Sci. 1977, 61, 62. (11) Smith, M. K.; Davis, S. H. J. Fluid Mech. 1983, 132, 119; 1983, 132, 145. (12) Chu, X.-L.; Velarde, M. G. Phys. ReV. A 1990, 43, 1094. (13) Kovalchuk, V. I.; Kamusewitz, H.; Vollhardt; D.; Kovalchuk, N. M. Phys. ReV. E 1999, 60, 2029. (14) Roze, C.; Gouesbet, G.; Darrigo, R. J. Fluid Mech. 1993, 250, 253276. (15) Roze, C.; Gouesbet, G. Physics Lett. A 1997, 227, 79. (16) Gouesbet, G. Phys. ReV. A 1990, 42, 5928. (17) Magone, N.; Yoshikawa, K. J. Phys. Chem. 1996, 100, 19102. (18) Chang, C.-H.; Franses, E. I. Colloids Surf. A 1995, 100, 1. (19) Demond, A. H.; Lindner, A. S. EnViron. Sci. Technol. 1993, 27, 2318. (20) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977.
