Effect of Buoyancy on Appearance and Characteristics of Surface

J. Phys. Chem. B 2005, 109, 15037-15047

15037

Effect of Buoyancy on Appearance and Characteristics of Surface Tension Repeated Auto-oscillations N. M. Kovalchuk† and D. Vollhardt*,‡ Institute of Biocolloid Chemistry, 03142 KieV, Ukraine, and Max Planck Institute of Colloids and Interfaces, 14424 Potsdam/Golm, Germany ReceiVed: April 15, 2005; In Final Form: June 10, 2005

The effect of buoyancy on spontaneous repeated nonlinear oscillations of surface tension, which appear at the free liquid interface by dissolution of a surfactant droplet under the interface, is considered on the basis of direct numerical simulation of the model system behavior. The oscillations are the result of periodically rising and fading Marangoni instability. The buoyancy force per se cannot lead to the oscillatory behavior in the considered system, but it influences strongly both the onset and decay of the instability and therefore, affects appearance and characteristics of the oscillations. If the surfactant solution density is smaller than the density of the pure liquid, then the buoyancy force leads to a considerable decrease of the induction period and the period of oscillations. The buoyancy force affects also the dependence of the oscillation characteristics on the system dimensions. The results of the simulations are compared with the available experimental data.

Introduction The phenomenon of spontaneous oscillations arising at liquid/ liquid interfaces by heat or solute transfer as well as the more general problem of dynamics of systems far from equilibrium has already attracted scientific attention for many decades. Apart from the fundamental aspect, the problem of instabilities at liquid interfaces is very important for many applications, such as, for example, foam and emulsion stability, liquid extraction, expanding alveolar layers, floating-zone process of crystal growth from the melt, and others. Despite numerous works published on this topic (for review, see, for example, refs 1 and 2), the mechanisms leading to development of nonlinear spontaneous oscillations at liquid/ liquid interfaces often remain unclear. This concerns the oscillations of the interfacial tension and the electrical potential by the transfer of ionic solutes through the oil/water interface discovered by Dupeyrat and Nakache3 and oscillations across the oil membrane, which were studied first by Yoshikawa and Matsubara.4 The comprehensive theoretical analysis of these systems is difficult because of their chemical complexity. Autooscillations of the surface tension produced by dissolution of the surfactant droplet under the water/air interface5 have much in common with those mentioned above. At the same time, they occur in systems that are more simple from the chemical point of view and, thus, permit detailed theoretical studies which can be the basis for the understanding of more complicated phenomena. Well-shaped and regular auto-oscillations can be observed, for example, by using the droplets of aliphatic alcohols6 or fatty acids7 with medium alkyl chain length. To obtain auto-oscillations of the surface tension, the surfactant droplet (diameter of 2-3 mm) is formed at the tip of a capillary immersed in the vessel (diameter of 40-70 mm, height 20-40 mm) filled with pure water. The precondition necessary for the appearance of oscillations is a limited area of the interface. A * Corresponding author. † Institute of Biocolloid Chemistry. ‡ Max Planck Institute of Colloids and Interfaces.

detailed description of the experimental procedure is given elsewhere.8 Similar oscillations of the interfacial tension and electrical potential difference through the interface have been observed at the water/nitrobenzene interface by injection of an ionic surfactant (sodium alkyl sulfate) with a syringe pump through a capillary in the water bulk.9-11 The auto-oscillations begin after a certain induction period, during which the surface tension remains nearly constant. They are essentially nonlinear with an asymmetric shape. Through the course of the oscillation, a sharp decrease in the surface tension is followed by its gradual increase. Fast convective motion at the interface, directed from the capillary to the vessel wall, accompanies the drop in surface tension. This confirms that the auto-oscillations appear as a result of the development of convective instability. It is well-known12,13 that in a horizontal liquid layer, where heat or solute transfer occurs in a direction normal to the surface, the conductive (no-flow) state becomes unstable, provided the temperature or concentration gradient exceed a certain threshold value. As a result of the instability development, the system passes to a regime with steady cellular or oscillatory convective motion. Linear stability analysis performed for surfactant transfer through the liquid/fluid interface of infinite horizontal extension shows that in the case of the air/liquid interface, oscillatory instability occurs only when the mass transfer is directed from the air to the liquid.14-16 There is a strong similarity to the heat transfer in a liquid layer with the upper surface open to the air (Benard-Marangoni problem), where the steady cellular convection occurs by heating from below, whereas the oscillatory regimes develop by heating from above. Nonlinear analysis predicts steady cellular convection by the surfactant or heat transfer from the liquid to the gaseous phase also far from the threshold. An increase of the concentration gradient over its threshold value leads only to the change of the pattern mode (for example, to the emergence of the roll cells instead of the hexagonal cells).17 In past decades, much attention has been paid to the experimental and theoretical study of the Benard-Marangoni

10.1021/jp0519619 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/15/2005

15038 J. Phys. Chem. B, Vol. 109, No. 31, 2005 convection in containers with an aspect ratio (radius over height) of the order of unity.18-22 It was shown that the aspect ratio of the container strongly influences the shape and number of convective cells, but the presence of the lateral wall does not cause the appearance of oscillatory regimes by heat transfer from the liquid to the air. Oscillatory switching between two modes with different cells was observed for some values of the aspect ratio.23,24 Therefore, auto-oscillation of the surface tension cannot be understood on the basis of the available conceptions of stability of liquid layers with surfactant transfer and requires a detailed theoretical study. Numerical simulations of the processes occurring by dissolution of a surfactant droplet show that the auto-oscillations can be explained with a rather simple model, taking into account diffusion of the solute and its adsorption/desorption at the interface influenced by Marangoni convection.25-27 According to this model, the sharp decrease of the surface tension corresponds with the development of convective instability, after the normal to surface concentration gradient exceeds a threshold value. In this fast stage of system evolution, a large amount of the solute is supplied to the surface and spreads over it. The instability in the considered system fades after a rather short period of time, and the system passes to a slow stage of its evolution. During this stage, the solute desorbs from the interface and the surface tension decreases gradually. The main cause for instability fading is the accumulation of surfactant near the wall of the vessel. That leads to appearance of the Marangoni force that hinders the convective motion. Thus, it can be stated that auto-oscillations of the surface tension, similar to the phenomena reported in refs 23 and 24, are the result of switching between two different regimes, namely, between the basic slow convective regime and the fast convective regime, which appears due to the development of instability. In contradistinction to the cases,23,24 where switching between different modes reveals itself only by some fixed aspect ratios of the system, auto-oscillations of the surface tension can be observed in wide ranges of the aspect ratio values, namely, they occur when the system aspect ratio is less than a certain critical value. By large aspect ratios (by small capillary immersion depths) only a single oscillation appears which is followed by a monotonic decrease of the surface tension (quasistationary regime).8,26 The model considered in refs 25-27 assumes only the Marangoni effect as a driving force for instability. It is wellknown, however, that the density difference over the bulk of the liquid (buoyancy) can also cause the onset of instability by heat transfer in the normal to interface direction.28 In systems in which auto-oscillations of the surface tension were observed, the density of the solution depends on the concentration and, correspondingly, it is nonuniform in space and changes over time. Thus, the buoyancy force can affect the onset and fading of instability and, therefore, the appearance and characteristics of auto-oscillations. The comparison of the experimental results on surface tension auto-oscillations in systems with droplets consisting of aliphatic alcohols or fatty acids confirms the significance of the buoyancy force for the considered phenomenon.7 The objective of the present paper is to theoretically study the effect of buoyancy on surface tension auto-oscillations. Mathematical Formulation The geometry of the considered problem is shown in Figure 1. The model system represents a cylindrical container of radius R0 and height H0 filled with a solvent that is a viscous,

Kovalchuk and Vollhardt

Figure 1. System geometry and distribution of the streamlines in the stable quasi-stationary convective regime.

incompressible, Newtonian liquid. The upper liquid surface is in contact with a passive gas. A cylindrical capillary with a spherical surfactant droplet on the tip is immersed to the depth h in the liquid so that the capillary axis coincides with the container axis. The independence of all variables of the angular coordinate is assumed according to the system symmetry and the experimental observations of the surface flow visualized with talcum particles. The temperature is assumed to be constant; all thermal effects are neglected. System evolution is described by the Navier-Stokes, continuity, and convective diffusion equations which are rewritten in terms of vorticity and stream function in cylindrical coordinates. The dependence of the solution density on the surfactant concentration is taken into account only in the buoyancy term (Boussinesq approximation). Scaling of time, length, velocity, concentration, stream function, and vorticity, respectively, with L2/D, L, D/L, c0, LD, and D/L2 leads to the dimensionless form of the governing equations

∂ω ∂(Vrω) ∂(Vzω) + + ∂t ∂r ∂z ∂c ∂2ω ∂2ω 1 ∂ω ω + RaSc ) 0 (1) Sc 2 + 2 + r ∂r r2 ∂r ∂r ∂z

(

)

∂2Ψ ∂2Ψ 1 ∂Ψ + 2 - ωr ) 0 r ∂r ∂r2 ∂z

(

(2)

)

∂2c ∂2c 1 ∂c ∂c ∂(Vrc) ∂(Vzc) Vrc + + + + + )0 ∂t ∂r ∂z r ∂r2 ∂z2 r ∂r

(3)

where L is the characteristic length scale, D is the bulk diffusion coefficient of the surfactant, c0 is the surfactant solubility, t is time, r is the radial coordinate, z is the normal to the interface coordinate downward directed with z ) 0 at the interface, Vr and Vz are the velocity components radial and normal to the interface directions, respectively, Ψ is the stream function, defined in such a way that Vr ) (1/r)(∂Ψ/∂z), Vz ) -(1/r)(∂Ψ/ ∂r), ω ) (∂Vr/∂z) - (∂Vz/∂r) is the vorticity, Sc ) (ν/D) is the Schmidt number, Ra ) (gc0L3/F0νD)(∂F/∂c) is the Rayleigh number, ν is the kinematic viscosity of the liquid, F ) F0 (c/c0)(F0 - Fs) is the solution density, F0 is the solvent density, Fs is the density of the saturated surfactant solution, g is the acceleration due to gravity, and c is the surfactant concentration.

Effect of Repeated Auto-oscillations of Surface Tension

J. Phys. Chem. B, Vol. 109, No. 31, 2005 15039

TABLE 1: Dependence of the Auto-Oscillation Characteristics on Grid Resolution

a

grid resolution

induction period, mina

oscillation period, mina,b

oscillation amplitude, mN/ma,b

critical capillary immersion depth, mm

80 × 80 100 × 100 120 × 120

20.0 21.4 21.5

15.6 15.1 14.9

3.0 3.3 3.4

13.5 10 8

Capillary immersion depth h ) 14 mm. b The mean value for the third-sixth oscillations.

In the initial state, the liquid is supposedly motionless. The dimensionless surfactant concentration is equal to unity at the droplet/solvent interface and is equal to zero elsewhere. Noslip boundary conditions are used for the container wall and bottom, for the capillary, and for the droplet surface. To estimate the significance of the boundary condition at the droplet surface, some calculations were also performed employing the condition ω ) 0 at the droplet/solution interface. This study shows that the characteristics of the oscillations depend to a certain extent on this condition. Employing the vortex-free condition leads to a decrease of the induction and oscillation periods and to an increase of the oscillation amplitude of approximately 20-30%. The evaporation from the surface is neglected. The gas-liquid interface is supposed nondeformable. The intrinsic surface viscosity is neglected. Diffusion-controlled adsorption kinetics is assumed. Local equilibrium is assumed between the surface concentration and the sublayer concentration. It is described by the Langmuir isotherm, which is in dimensionless form

Γ)

c 1 + KLc0c

(4)

The surfactant mass balance on the free surface is described by the equation

(

) ()

∂c ∂Γ ∂(ΓVr) ΓVr DS ∂2Γ 1 ∂Γ + - NE )0 + + ∂t ∂r r D ∂r2 r ∂r ∂z

(5)

at z ) 0 where NE ) (L/KLΓm) is the exchange number that determines the effect of surfactant exchange between the surface and bulk (NE f 0 corresponds to the limiting case of insoluble monolayer behavior when exchange with the interface is absent), KL and Γm are the parameters of the Langmuir isotherm, DS is the surface diffusion coefficient, and Γ is the Gibbs adsorption scaled by c0KLΓm. The boundary condition for the vortex on the free surface is obtained from the tangential stress balance employing the Szyszkowsky-Langmuir equation of state for an adsorbed surfactant

ω ) Ma

∂Γ 1 (1 - KLc0Γ) ∂r

(6)

at z ) 0 Here, Ma ) (RTc0KLΓmL/µD) is the global Marangoni number27, R is the gas constant, and T is the temperature. Mathematical simulation was performed using the finite difference method on a regular grid. Equation 2 was solved by the Gauss-Seidel iterative method. For eqs 1 and 3 the two point forward difference approximation is used for the time derivatives, the three point centered differences are used for the diffusion terms, and the modified upwind differences are used for the convective terms.29 The following values of the geometrical parameters were used for the simulation: charac-

teristic length L ) R0 ) H0 ) 20 mm, capillary radius rc ) 1 mm, and droplet radius r0 ) 1.5 mm. The substance properties were chosen corresponding to the properties of the octanolwater system: solubility of the surfactant c0 ) 3.4 mol/m3, parameters of the Langmuir isotherm KL ) 3.23 m3/mol and Γm ) 6.6 × 10-6 mol/m2, volume diffusion coefficient D ) 6.7 × 10-10 m2/s, water density F ) 1 g/cm3, and solution viscosity µ ) 0.01 g/(cm/s). The density of the saturated surfactant solution is varied in this study, but here we consider only the case where the density of the surfactant solution is smaller than the density of water. By numerical simulation of the problem in the absence of buoyancy, the grid resolution 80 × 80 was found to be proper.25 To prove that the same grid can be used in this study, we calculated the main characteristics of the system evolution for different grids (Table 1) using the value for the density difference ∆Fs between water and a saturated solution of ∆Fs ) 10-5 g/cm3. It is seen from Table 1 that all the presented characteristics converge. The 80 × 80 grid gives quite reasonable values for the induction period, the oscillation amplitude (somewhat underestimated), and the oscillation period (slightly overestimated). The only exception is the critical capillary immersion depth, dividing oscillatory and quasi-stationary regime in the system evolution, which depends rather strongly on a grid resolution. This dependence decreases as the density difference between the saturated surfactant solution and water decreases. Thus, in the present work, we used the grid with 120 × 120 mesh points and restricted our study to the density difference between the solvent and saturated solution of ∆Fs ) 10-5 g/cm3 to obtain sufficiently correct results by the reasonable grid resolution. Results and Discussion The effect of the buoyancy force on the auto-oscillation characteristics is clearly seen from Figure 2 and Table 2. The increase of the buoyancy force leads to the decrease of the induction period and period of oscillations, and it practically does not affect the oscillation amplitude (see Table 2). It leads also to the increase in the critical immersion depth of the capillary, which divides the oscillatory and quasi-stationary regimes. The experiments with aliphatic alcohols6 display an increase of the critical immersion depth in the homologous series of alcohols from octanol to pentanol. Some contribution to this is due to the corresponding decrease of the surface activity within the homologous series of the alcohols.27 The present study shows that another cause is the increase of the density difference between water and the saturated alcohol solution with the decrease of the alcohol chain length. Curve 3 in Figure 2 represents the experimental results for the octanol/water system taken from ref 7 (oscillation period of ∼11 min). The experimental data available for the octanol/ water system6,7 show that the oscillation period can vary between 4 and 11 min. The model, which disregards buoyancy force, leads to a rather overestimated value for the oscillation period of ∼28 min (curve 1). The simulation employing the density difference ∆Fs ) 10-5 g/cm3 provides a value for the oscillation

15040 J. Phys. Chem. B, Vol. 109, No. 31, 2005


Figure 3. Temporal dependency of the velocity at the air/water interface (capillary immersion depth h ) 8 mm, distance to the cell axis r ) 10 mm): 1, KL ) 3.23 m3/mol, ∆Fs ) 0; 2, KL ) 0, ∆Fs ) 10-5 g/cm2; 3, KL ) 3.23 m3/mol, ∆Fs ) 10-5 g/cm2.

Figure 2. Auto-oscillations of the surface tension in the octanol/water system (capillary immersion depth h ) 10 mm, cell radius R0 ) 23 mm, distance to the cell axis r ) 10 mm): 1, numerical results in the absence of buoyancy (∆Fs ) 0); 2, numerical results with ∆Fs ) 10-5 g/cm2; 3, experimental results.7

period of ∼15 min (curve 2), which is much closer to those observed experimentally. The density difference between water and the saturated aqueous octanol solution, calculated with the assumption that the solution is ideal and taking the water density Fw ) 1 g/cm3 and the octanol density Fo ) 0.827 g/cm3, is about ∆Fs ) 9 × 10-5 g/cm3. Simulations performed on the grid 300 × 300 mesh points for this density difference give a value for the oscillation period of ∼6 min. Thus, the mathematical model, accounting for the buoyancy force in the numerical simulations permits us to obtain values for the oscillations period close to those observed experimentally. Effect of Buoyancy on Instability Onset. In the case considered here, the surfactant solution has a smaller density than that of water. However, also the reverse case, where the density of the surfactant solution is larger than that of water, is possible, namely, in the case of diethylphthalate solutions.5 In this case, the appearance of auto-oscillations is not prevented, but their characteristics are affected. In such systems, the buoyancy convection results in a slower formation of a normal concentration gradient and in an increase of the induction period and oscillation period. The behavior of such systems is not discussed here.

If the solution density is independent of the concentration (buoyancy is absent), then the liquid initially remains at rest (diffusional regime of the solute transfer) during the dissolution of a surfactant droplet under an air/water interface. The surfactant gradually reaches the interface, and convection develops in the system due to the Marangoni force. The surface concentration of the solute has the highest value near the capillary. Therefore the liquid at the surface moves from the capillary to the wall. The distribution of the streamlines in the bulk by this convective regime is presented in Figure 1. The velocity of the convective motion increases, but its growth rate (d ln(V)/dt) decreases with time.27 It should be noted, that the convection during this stage is rather weak and the mass transfer in the system is predominantly diffusional. The concentration distribution in the system gradually changes with time. The concentration gradients near the droplet decrease whereas those near the surface increase. The onset of the Marangoni instability in the system is determined just by the normal concentration gradient near the surface. Instability develops due to feedback existing in the system. An increase of the normal concentration gradient at the surface near the capillary leads to the increase in the solute supply to the interface in this region, thus resulting in the increase of the surface concentration gradient as well as the surface and bulk velocities, which, in turn, leads to the increase of the solute supply to the interface (see Figure 1). Therefore, when the normal concentration gradient exceeds a threshold value, the slow convective regime becomes unstable and the system passes to the fast convective regime (Figure 3, curve 1). The convective flow keeps the previous form shown in Figure 1, but the velocity increases by some orders of magnitude during a very short time, and the convective mass transfer becomes predominant in the system. Fast convection brings a large amount of the solute from

TABLE 2: Effect of Buoyancy on the Auto-Oscillation Characteristics density difference, g/cm3

critical immersion depth, mm

induction period, mina

oscillation period, mina,b

oscillation amplitude, mN/ma,b

maximum surface velocity, mm/sa,c

minimum surface velocity, mm/sa,c

0 1 × 10-6 5 × 10-6 7.5 × 10-6 1 × 10-5

6 6.25 6.75 7.25 8

32.5 28.0 17.4 14.5 12.7

24.4 21.0 16.1 14.5 13.2

3.4 3.45 3.5 3.5 3.5

7.0/1.1 9.3/1.2 11.8/1.5 12.9/1.7 13.2/1.9

-0.024/-0.025 -0.024/-0.024 -0.017/-0.025 -0.014/-0.025 -0.011/-0.024

a For the capillary immersion depth h ) 8 mm. b Mean value for third-sixth oscillations. c Numerator, for the first oscillation; denominator, the mean value for third-sixth oscillations.



TABLE 3: Characteristic Surface Velocities for the Octanol/ Water Systema velocity of the quasi-steady Marangoni convection at onset of instability (∆Fs ) 0) maximum of velocity, reached by development of Marangoni instability maximum of velocity of buoyancy-driven convection (KL ) 0) by ∆Fs ) 10-6 g/cm3 ∆Fs ) 10-5 g/cm3 ∆Fs ) 10-4 g/cm3

3 × 10-5 mm/s 7-13 mm/s

2 × 10-3 mm/s 6 × 10-3 mm/s 2 × 10-2 mm/s

a Capillary immersion depth h ) 8 mm, distance to the capillary r ) 10 mm.

the droplet to the interface and spreads the solute over the interface. This corresponds to the abrupt decrease in surface tension. The evolution of a system, where a droplet of a surface inactive substance dissolves under the air/water interface but the solution density depends on the concentration (the action of the buoyancy force in the absence of the Marangoni force), occurs in a completely different way. There is no purely diffusional stage in system evolution. Convection begins already during droplet formation due to the radial density gradient. The velocity of the buoyancy-driven convection is determined by the concentration gradients in the bulk, which decrease with time due to the smoothing of the concentration distribution, and the growth rate of the surface velocity decreases all the time. At a certain time, the growth rate of the velocity becomes negative and the surface velocity begins to decrease (Figure 3, curve 2). Figure 3 allows a comparison of how the surface velocity changes with time in three distinctive systems. In the absence of the buoyancy force, the convection velocity due to the Marangoni effect is very small during the induction period and increases sharply after the onset of instability (curve 1). If only the buoyancy force acts in the system, then the convection is more intensive at the initial period (curve 2), but after some time it begins to decrease due to smoothing of the bulk concentration. In the system subjected to both buoyancy and Marangoni forces (curve 3), the surface velocity vs time dependence is the same as in system 2 up to the moment of the onset of instability. However, during the development of instability, the increase of the velocity becomes very similar to that obtained for system 1. Typical values of the surface velocity are given in Table 3. In the case of buoyancy-driven convection, the maximum of the surface velocity increases as the density difference between pure water and the saturated aqueous solution increases. This velocity is some orders of magnitude smaller than the velocity reached during the development of Marangoni instability, but it is much larger than the velocity of the quasi-steady Marangoni convection during the slow stage of the system evolution in the absence of buoyancy. The time dependencies in surface velocity over the long time interval, presented in Figure 4 (curves 1 and 2 correspond to the curves 1 and 2 in Figure 2), confirm that the auto-oscillations of the surface tension correlate with the rising and fading of convective instability. It is seen that the velocity peaks coincide with an abrupt decrease of the surface tension in time. Figure 4 also clearly shows the difference in the velocity scales during the fast and slow stages of system evolution. The velocity during the induction period and during the times corresponding to the gradual increase of the surface tension is negligible in comparison to the maximum velocity reached in the fast stages. This is justified both in the presence and in the absence of the

Figure 4. Temporal dependencies of the surface velocity in the octanol/ water system corresponding to Figure 2 (capillary immersion depth h ) 10 mm, cell radius R0 ) 23 mm, distance to the cell axis r ) 10 mm): 1, ∆Fs ) 0; 2, ∆Fs ) 10-5 g/cm2 (numerical results).

buoyancy force in the system. Maximum and minimum surface velocities calculated for some density differences are given in Table 2. It is seen that the maximum of the surface velocity increases with the increase of the density difference for the first as well as for the following oscillations. The minimum surface velocity (that is the maximum negative velocity) depends essentially on the density difference only for the first oscillation. From the numerical results discussed above, it can be concluded that, in systems with a dissolving surfactant droplet, the buoyancy itself is not the driving force of the instability of the convective flow that initially exists. Instability develops only due to the Marangoni effect, resulting in a system transition from the basic convective regime with a relatively low velocity and, therefore, a low mass transfer rate to a convective regime with a much higher mass transfer rate. At the same time, the intensity of the buoyancy convection at the basic state determines the rate of the solute distribution over the system and, correspondingly, it influences rather strongly the onset and development of instability. In particular, buoyancy reduces the induction period (Figure 3) and oscillation period (Figures 2 and 4). In the absence of buoyancy, instability begins to develop in the capillary region, where the maximum of the normal concentration gradient occurs near the surface.25 The velocity of the buoyancy-driven convection has the maximum in the bulk near the droplet at a distance of 1-2 mm from the capillary. Correspondingly, the maximum of the normal to surface concentration gradient and the maximum of the solute supply to the surface due to convective mass transfer occur also at a certain small distance from the capillary. Thus, convection driven by the buoyancy force does not prevent the formation of the solute distribution near the capillary, caused by diffusion



TABLE 4: Dependence of the Critical Dimensionless Normal Concentration Gradients at the Onset of Instability on the Density Difference between the Solvent and the Saturated Surfactant Solution ∆Fs, g/cm3 ∇n c, 10-7

0 1.6

10-6 1.5

5 × 10-6 2.2

7.5 × 10-6 3.3

10-5 4.9

and Marangoni convection, but acts in the same direction. The buoyancy-driven convection leads to the faster formation of a large normal concentration gradient near the surface desired for the development of Marangoni instability. Instability develops in the considered system when the local Marangoni number, proportional to the normal concentration gradient, exceeds a certain critical value.27 The onset of instability can be approximately attributed to the time when the growth rate of the surface velocity begins to increase. The growth rate of the surface concentration demonstrates the same behavior. For the case ∆Fs ) 0, the analysis of the time dependencies of (d ln(Vs)/dt) and (d ln(c)/dt) shows that first the growth rate of the surface concentration begins to increase. The time interval between the minima of the corresponding curves is about a few seconds. This result agrees with the mechanism of the feedback acting in the system. According to the analysis performed above, it can be assumed that to logically and correctly establish the time of instability onset for ∆Fs * 0, one should take into account only the additional contribution to the velocity growth due to the Marangoni effect ∆Vs (i.e., the difference between curves 3 and 2 in Figure 3). Indeed, when we consider, for example, the system with ∆Fs ) 10-5 g/cm3 and h ) 8 mm, the growth rate of the surface concentration has a minimum at 11 min 39 s, and the corresponding time for (d ln(∆Vs)/dt) is 11 min 41 s, whereas the minimum of the full growth rate of the velocity (d ln(Vs)/dt) is reached much earlier, at ∼10 min 24 s. Thus, the time corresponding to the minimum of the curve (d ln(∆Vs)/dt) can be accepted as the time of the instability onset. Of course, such an estimation is rather rough and gives slightly overestimated values of the critical concentration gradient, but it allows one to trace the tendencies. The values of the critical concentration gradients, corresponding to the time of the instability onset defined as described above, are presented in Table 4. For small density differences (∆Fs ) 10-6 g/cm3), the critical gradient coincides practically with that for ∆Fs ) 0. A further increase of ∆Fs leads to a certain increase of the critical gradient (and of the corresponding critical value of the local Marangoni number). Such a result agrees with the mechanism for the development of instability. Obviously, the development of instability becomes appreciable when the feedback between the growth of the velocity and the growth of the concentration gradient begins essentially to influence the system evolution. That means, the mass transfer due to Marangoni convection becomes comparable (approximately 10%, according to ref 27) with the mass transfer by other mechanisms (diffusion and buoyancy driven convection). The increase of the density difference leads to the increase of the mass transfer by buoyancy convection and, therefore, to the increase of the velocity and the corresponding concentration gradients desired for the onset of instability. However, the increase of the critical concentration gradient with the increase of the density difference is rather small, whereas the acceleration of the mass transfer rate due to buoyancy convection is quite large, resulting in an essential decrease of the induction period and the oscillation period (Table 2). Effect of Buoyancy on Development and Decay of Instability. Convection develops mainly in the capillary region during

Figure 5. Radial distribution of the surface velocities at the onset of instability: 1, ∆Fs ) 10-5 g/cm2, t ) 11 min 39 s; 2, ∆Fs ) 10-6 g/cm2, t ) 24 min 4 s; 3, ∆Fs ) 0, t ) 27 min 22 s.

Figure 6. Development of instability in the system with ∆Fs ) 10-5 g/cm3, h ) 8 mm: 1, t ) 12 min 38 s; 2, t ) 12 min 39 s; 3, t ) 12 min 40 s; 4, t ) 12 min 40.7 s; 5, t ) 12 min 41 s.

the stable period in system evolution. The intensity of convection in the distant parts of the system is rather weak, but the larger the density difference, the larger intensity difference (Figure 5). Instability develops initially within a certain small region, where the normal concentration gradient has a maximum. The development of instability results in a fast increase of the surface velocity and an extension of the region of intensive convection. This process follows clearly from Figure 6. It is seen that the velocity decreases very steeply at the leading edge of the instability wave. Particularly, a large velocity gradient occurs when this wave reaches the container wall (curve 4 in Figure 6). The velocity gradient causes the local surface contraction in the wall region and, therefore, an increase of the surface concentration here. As a result, a reverse concentration gradient appears at the surface near the wall (Figure 7). This gradient leads to the decrease of the surface velocity (curve 5 in Figure 6), which, because of the feedback between the surfactant supply and the velocity, results in a decrease of the surfactant supply to the surface and a decay of instability. With time, the reverse concentration gradient decreases and the region of its existence extends (compare curves a and b in Figure 7). When the reverse gradient acts during a sufficiently long period of time, it causes the surface motion in the wall region directed from the wall to the capillary (corresponding negative peaks of the surface velocity are clearly seen in Figure 4 just after the large positive peaks), thus forming the reverse convective roll. The reverse roll, extending toward the capillary, breaks the surfactant supply to the surface, which is the precondition for the development


Figure 7. Surface concentration distributions (with reverse gradients in the wall region): 1, ∆Fs ) 10-5 g/cm3, h ) 8 mm, a, t ) 12 min 41 s, b, t ) 12 min 41.7 s; 2, ∆Fs ) 0, h ) 4 mm, a, t ) 7 min 54.4 s, b, t ) 7 min 55.1 s.

Figure 8. Streamline distribution in the bulk by the decay of instability: 1, direct convective roll; 2, reverse convective roll (cf. with Figure 1).

of the following oscillation.26 For example, the reverse convective roll, reaching the capillary region is shown for the case, ∆Fs ) 0, h ) 8 mm, in Figure 8. When the surface velocity, which is reached during the rise of instability, is large enough, the reverse concentration gradient cannot overcome the motion directed from the capillary to the wall, and the reverse convective roll does not extend to the capillary. In this case, the surfactant is supplied continuously to the interface supporting this motion and the system passes into a quasi-stationary regime with a monotonic decrease in the surface tension after the first oscillation.25 The maximum of the surface velocity increases with the decrease of the immersion depth of the capillary. Hence, a quasi-stationary regime occurs when the immersion depth of the capillary is smaller than a certain critical value.25,26 It should be stressed that buoyancy affects not only the onset of instability but also the processes during its development and decay. The velocity of convection at the onset of instability increases as the density difference increases (Figure 5). With the development of instability, the surfactant is supplied to the surface mainly in the capillary region and then it spreads over the whole surface. This process is illustrated by Figure 9. The rapid concentration increase in the bulk near the capillary causes a fast growth of the radial concentration gradients in the space

J. Phys. Chem. B, Vol. 109, No. 31, 2005 15043 between the droplet and surface (Figure 10a, curves 1-3). This leads to an additional acceleration of the convective motion due to buoyancy. Therefore, the maximum of the surface velocity, which is reached during the development of instability, increases with an increase in the density difference. This should cause an increase in the critical immersion depth of the capillary (Table 2). Numerical simulations show that, for ∆Fs ) 10-5 g/cm3, the critical immersion depth of the capillary, by which the quasistationary regime is replaced by the oscillatory regime, is h ) 8 mm. The maximum surface velocity reached in this system is ∼13 mm/s. In the system, with ∆Fs ) 0, such a velocity is attained by the immersion depth of the capillary of h ) 4 mm and corresponds to the quasi-stationary regime. Repeated oscillations develop in the system with ∆Fs ) 0 and only with h ) 6 mm, when the maximum of the surface velocity decreases to 11 mm/s. In Figure 7, the reverse concentration gradients for the systems ∆Fs ) 10-5 g/cm3, h ) 8 mm (1) and ∆Fs ) 0, h ) 4 mm (2) are shown for time moments of 0.3 s (a) and 1 s (b) after reaching the maximum surface velocity gradient near the wall. It is seen that the reverse concentration gradients are practically equal for the two considered systems. Therefore it can be assumed that the buoyancy force promotes the fading of instability, which leads to the appearance of the regime with repeated oscillations by larger values of the surface velocity. Indeed, the surface contraction near the wall causes essential desorption of the surfactant in this region. The desorbed surfactant is distributed in the bulk by diffusion as well as by convection. Therefore, the surfactant concentration in the bulk near the wall increases with respect to the neighboring bulk regions (parts b and c of Figure 9). The reverse concentration gradient in the bulk (Figure 10b) acts in the same direction as the reverse surface concentration gradient, that means, it counteracts the flow in the direct convective roll and supports the decay of instability. Thus, the buoyancy force, acting in the system with the dissolving surfactant droplet, accelerates the growth of instability at the initial stage and accelerates the decay of instability after the surfactant wave reaches the container wall. It should be noted, however, that the accelerating effect of the direct concentration gradient in the bulk is still stronger than the retarding effect of the reverse concentration gradient in the bulk. As a result, the critical immersion depth of the capillary, dividing the oscillatory and quasi-stationary regime, increases as the buoyancy increases (Table 2). The dynamic regimes become more complicated in the presence of the buoyancy force. The redistribution of the streamlines for the system with ∆Fs ) 10-5 g/cm3 within an oscillation period is presented in Figures 11 and 12. Figure 11 corresponds to a large immersion depth of the capillary (h ) 16 mm). In this case, the growth of the reverse convective roll causes splitting of the direct roll into two separate rolls. The upper roll is driven by the surface tension gradient and the lower roll by buoyancy (t ) 29 min and t ) 30 min). The upper direct roll is then suppressed by the reverse roll, which leads to the breaking of the surfactant supply to the interface (t ) 32 min) and to the accumulation of the surfactant in the droplet region, creating in this way the precondition for the next oscillation. The motion in the direct convective roll is supported by dissolution of the surfactant droplet, whereas the concentration gradients in the reverse roll decrease quickly due to mixing in the solution. Therefore the direct roll extends and suppresses the reverse roll (t ) 33 min and t ) 40 min). After the disappearance of the reverse convective roll, the velocity in the direct roll begins to increase very quickly (instability develop-



Figure 10. Radial distribution of the surfactant concentration in the bulk (3 mm below the surface, ∆Fs ) 10-5 g/cm3, h ) 8 mm) a, capillary region, b, wall region: 1, t ) 12 min 30 s; 2, t ) 12 min 43 s; 3, t ) 12 min 58 s; 4, t ) 13 min 36 s; 5, t ) 15 min (maximum of the surface velocity corresponds to t ) 12 min 41 s).

Figure 9. Surfactant distribution in the bulk for ∆Fs ) 10-5 g/cm3, h ) 8 mm: a, t ) 12 min 30 s; b, t ) 12 min 41 s; c, t ) 13 min (1, c ) 2 × 10-1; 2, c ) 1.25 × 10-2; 3, c ) 10-3; 4, c ) 10-6; 5, c ) 10-10; 6, c ) 10-20).

ment); the system passes to the fast stage of its evolution marking the beginning of the following oscillation (t ) 43.5 min). Close to the critical immersion depth of the capillary (Figure 12), the reverse convective roll becomes less powerful and, after some time, splits into two rolls (t ) 17 min and t ) 20 min). The same takes place with the direct roll (t ) 18 min and t ) 20 min). The upper direct roll, driven by the surface tension gradient, does not disappear completely in this case, but the surfactant mixing in both direct rolls (upper and lower) occurs practically independent of each other. Therefore, also in this case, the surfactant supply from the droplet to the interface is broken. Upon a further decrease in the capillary immersion depth, the region, where the surfactant is supplied continuously to the interface, appears near the capillary and the system passes to the quasi-stationary regime after the first oscillation. Effect of System Dimensions. With a fixed density difference between the pure solvent and the saturated solution, the intensity of the buoyancy driven convection, as well as the bulk concentration gradients formed during growth and decay of Marangoni instability, depends on the system geometry. Therefore the effect of the system geometry on the development and characteristics of the surface tension auto-oscillation can depend on the density difference. The effect of the system geometry on the auto-oscillations of the surface tension in the absence of the buoyancy force was studied theoretically (numerically) in ref 26. Experimental studies of the problem were performed in



Figure 11. Streamline redistribution during the oscillation period for the system ∆Fs ) 10-5 g/cm3, h ) 16 mm.

Figure 12. Streamline redistribution during the oscillation period for the system ∆Fs ) 10-5 g/cm3, h ) 8 mm.

ref 8 for the system heptanol/water. The results of the simulations in the presence of buoyancy for different geometrical characteristics of the system are presented in Table 5. It was shown in ref 26 that, in the absence of buoyancy, the critical immersion depth of the capillary, dividing the oscillatory and quasi-stationary regime, increases as the cell radius increases. The same remains correct when buoyancy acts in the system. For example, in the system where ∆Fs ) 10-5 g/cm3 at a cell radius of R ) 20 mm, repeated oscillations appear at the immersion depth of the capillary of h ) 8 mm. However, at R ) 22 mm, the quasi-stationary regime with a single oscillation appears at the same depth. At R ) 23 mm, repeated oscillations still occur at h ) 10 mm, but at R ) 25 mm, they are also replaced by the quasi-stationary regime. The experiments

confirm the increase of the critical capillary immersion depth with the increase in cell radius.8 In the absence of buoyancy, an increase of the immersion depth of the capillary leads to an increase of the induction period and oscillation period. A similar dependency is obtained also for ∆Fs * 0, but the increase of both the induction period and the period of oscillations is smaller at higher density differences. For example, for the cell radius of 20 mm and when ∆Fs ) 0, the increase of the immersion depth from 8 to 16 mm causes the increase of the induction period by 4.1 times and the increase of the oscillation period by 2.6 times,26 whereas for ∆Fs ) 10-5 g/cm3, the corresponding values are 2 and 1.2 times. These results allow the supposition that by a further increase of the density difference the oscillation period will become almost



TABLE 5: Effect of the System Geometry on the Auto-Oscillation Characteristics

a

oscillation perioda, min

oscillation amplitudea, mN/m

13.2 13.7 14.9 16.3

3.5 3.45 3.35 3.2

14.8

3.1

16.1 16.9

2.7 2.6

∆Fs ) 5 × 10-6 g/cm3 22.4 32.0 38.4 22.2 31.8 38.1

17.1 19.4 21.9 18.8 20.0 20.9

3.3 3.25 3.2 2.7 2.5 2.35

∆Fs ) 10-6 g/cm3 28.0 40.6 54.0 28.0 40.4 53.6

21.0 24.1 28.6 21.6 24.2 27.3

3.45 3.1 2.7 2.6 2.5 2.3

cell radius, mm

capillary immersion depth, mm

induction period, min

20 20 20 20 22 23 25 25 25

8 10 14 16 8 10 10 14 16

∆Fs ) 10-5 g/cm3 12.7 15.6 21.5 25.7 12.5 15.4 15.6 21.5 25.5

20 20 20 25 25 25

10 14 16 10 14 16

20 20 20 25 25 25

8 10 12 8 10 12

Mean value for third-sixth oscillations.

independent of the immersion depth of the capillary, as observed in the experiments with heptanol.8 The experiments with heptanol8 have also shown that the oscillation period increases as the cell radius increases, whereas simulations performed for the case ∆Fs ) 0 seem to give an opposite dependency.26 The analysis of the results listed in Table 5 shows that the increase of buoyancy leads finally (with increasing ∆Fs) to the dependency observed experimentally. It is interesting to note, however, that at small density differences, the oscillation period decreases with the increase of the cell radius in the case of large immersion depths of the capillary and increases at small immersion depths.

buoyancy or by its small contribution, the oscillation period decreases as the cell radius increases. At large density differences, the oscillation period increases with the increase of the cell radius. A comparison with the experimental data allows the conclusion that the agreement between the theory and the experiment is essentially improved by accounting for buoyancy. Taking into account the buoyancy force in the numerical simulations permitsthe values of the oscillations periods, and their dependence on the system geometry, to be much closer to those observed experimentally. As well, the increase of the critical immersion depth in the homologous series of alcohols from octanol to pentanol can be explained to a great extent by the effect of buoyancy convection.

Conclusions Numerical simulations show that in systems with a surfactant droplet dissolving under the liquid interface repeated oscillation of the interfacial tension can be observed due to periodically rising and fading Marangoni instability. Buoyancy force, acting in this system when the density of the solution depends on its concentration, does not cause the oscillatory behavior per se. At the same time, it influences strongly the onset and decay of Marangoni instability and, therefore, the oscillation characteristics. If the density of the solution is smaller than the density of the solvent, then an increase in the density difference leads to a decrease of the induction period, after which oscillations begin, as well as to a decrease of the oscillation period. The buoyancy force affects also the dependence of the oscillation characteristics on the system dimensions. An increase of the buoyancy force results in a decrease of the critical aspect ratio (ratio of the cell radius to the immersion depth of the capillary), which divides the two regimessinto single and repeated oscillations. The oscillation period increases with the increase of the immersion depth of the capillary in the absence of the buoyancy force according to numerical calculations. An increase of the density difference between the solvent and the saturated surfactant solution causes the weakening of this dependence and can probably lead to the independence of the oscillation period of the immersion depth. In the absence of

Acknowledgment. Financial assistance from the National Ukrainian Academy of Sciences (Project 9-003) is gratefully acknowledged. N.M.K. thanks the MPI of Colloids and Interfaces for the financial support. References and Notes (1) Colinet, P.; Legros, J. C.; Velarde, M. G. Nonlinear Dynamics of Surface Tension-DriVen Instabilities; Wiley-VCH: New York, 2001. (2) Srivastava, R. C.; Rastogi, R. P. Transport mediated by electrified interfaces. Studies in the linear, nonlinear and far from equilibrium regimes. In Studies in Interface Science; Mo¨bius D., Miller R., Eds.; Elsevier: Amsterdam, The Netherlands, 2003; Vol. 18. (3) Dupeyrat, M.; Nakache, E. Bioelectrochem. Bioenerg. 1978, 5, 134. (4) Yoshikawa, K.; Matsubara, Y. J. Biophys. Chem. 1983, 17, 183. (5) Kovalchuk, V. I.; Kamusewitz, H.; Vollhardt, D.; Kovalchuk, N. M. Phys. ReV. E 1999 60, 2029. (6) Kovalchuk, N. M.; Vollhardt, D. J. Phys. Chem. B 2000, 104, 7987. (7) Kovalchuk, N. M.; Vollhatdt, D. Mater. Sci. Eng., C 2002, 22, 147. (8) Grigorieva, O. V.; Kovalchuk, N. M.; Grigoriev, D. O.; Vollhatdt, D. J. Colloid Interface Sci. 2003, 261, 490. (9) Takahashi, T.; Yui, H.; Sawada, T. J. Phys. Chem. B 2002, 106, 2314. (10) Yui, H.; Ikezoe, Y.; Takahashi, T.; Sawada, T. J. Phys. Chem. B 2003, 107, 8433. (11) Ikezoe, Y.; Ishizaki, S.; Yui, H.; Fujinami, M.; Sawada, T. Anal. Sci. 2004, 20, 435. (12) Koschmieder, E. L. Benard cells and Taylor Vortices; Cambridge University Press: Cambridge, U.K., 1993.

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Effect of Buoyancy on Appearance and Characteristics of Surface

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