Interfacial instability gradients

Interfacial turbulence [60] Due to a nonuniform distribution of surfactant molecules at the interface or to local convection currents close to the interface, interfacial tension gradients lead to a mechanical instability of the interface and therefore to production of small drops. 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

FIGURE 1.8. (a) Schematic representation of the device used to study capillary surface instabilities. A polymer-air bilayer of thicknesses /ip and /ia, respectively, is formed by two planar silicon wafer held at a separation d by spacers. A capillary instability with wavelength k = 27t/q is observed upon applying a voltage U or a temperature difference AT. (b) Dispersion relation (prediction of Eq. (1.6)). While all modes are damped (r < 0) in the absence of an interfacial pressure pei, the application of an interfacial force gradient leads to the amplification of a range of k-values, with /.m the maximally amplified mode. 

Another role of the surfactant is to initiate interfacial instability, e.g., by creating turbulence and Raykleigh and Kelvin-Helmholtz instabilities. Turbulence eddies tend to disrupt the interface since they create local pressures. Interfacial instabilities may also occur for cylindrical threads of disperse phase during emulsification. Such cylinders undergo deformation and become unstable under certain conditions. The presence of surfactants will accelerate these instabilities as a result of the interfacial tension gradient. 

Numerous studies have shown that mass transfer of solute from one phase to the other can alter the behavior of a liquid-liquid dispersion—because of interfacial tension gradients that form along the surface of a dispersed drop. For example, see Sawistowski and Goltz, Trans. Inst. Chem. Engrs., 41, p. 174 (1963) BaWcer, van Buytenen, and Beek, Chem Eng. Sci., 21(11), pp. 1039-1046 (1966) Rucken-stein and Berbente, Chem. Eng. Sci., 25(3), pp. 475—482 (1970) Lode and Heideger, Chem. Eng. Sci., 25(6), pp. 1081—1090 (1970) and Takeuchi and Numata, Int. Chem. Eng., 17(3), p. 468 (1977). These interfacial tension gradients can induce interfaci turbulence and circulation within drops. These effects, known as Marangoni instabilities, have been shown to enhance mass-transfer rates in certain cases. 

Hartland and Jeelani performed a theoretical study on the effect of interfacial-tension gradients on emulsion stability (71). Dispersion stability and instability were explained in terms of a siuface mobility which is proportional to the siuface velocity. When the interfacial tension gradient is negative, the siuface mobility is negative under some conditions, which greatly reduces the drainage so that the dispersion is stable. This is a normal situation as surfactant is present at the interface. Demulsifier molecules penetrate the interface within the film, thereby lowering the interfa-... 

The surface renewal phenomena occur as a result of Marangoni instability or forced convection. Marangoni instability results if an interfacial tension gradient produced by a perturbation is amplified by the simultaneously occurring mass transfer process. 

Interfacial phenomraia involving heat and mass transfer are described and analyzed in Chapter 6. Much of the chapter again deals with stability, in this case the Marangoni instability produced by interfacial trasion gradients associated with temperature and eoneentration gradients along the interface. Time-dependent variation of interfadal tension resulting from diffusion, adsorption, and desorption of various species is also diseussed. 

When there is no free surface, this range is substantially reduced. Nevertheless, we still observe the formation of structures (37). Since we can no longer invoke an interfacial instability in this case, since a B nard instability (due to a vertical gradient of temperature or composition) appears less probable, it is thus wise to envisage (38) the intervention of a double diffusion process (39). In any case, we thus enter the domain of conjecture. 

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

Finally, we consider the problem of Marangoni instability, namely convection in a thin-fluid layer driven by gradients of interfacial tension at the upper free surface. This is another problem that was discussed qualitatively in Chap. 2, and is a good example of a flow driven by Marangoni stresses. 

C. V. Stemling and L. E. Scriven, Interfacial turbulence hydrodynamic instability and the Marangoni effect, AIChE J. 5, 514 (1959) L. E. Scriven and C. V Stemling, On cellular convection driven by surface tension gradients effects of mean surface tension and surface viscosity, J. Fluid Mech. 19, 321 (1964). 

A schematic of an instability caused by temperature gradients between the two phases resulting in surface tension gradient along the interface is shown in Fig. 3C.2. A small initial disturbance like interfacial convection from R to S will be amplified by additional mass transport at higher temperature, because this leads to y < y j on condition dy / dT < 0. 

---