Surfactant conservation equation

The balance of Marangoni and viscous stresses (8.153), reformulated in terms of T, is integrated to obtain the surfactant distribution and yields T as a function of ( ) and the dimensionless Marangoni number Ma. The surfactant distribution can be integrated over the cap region to obtain the total amount on the surface, M. The variable M is also computed independently from the surfactant conservation equations and equating the two expressions yields Once ( ) is specified, the drag coefficient and terminal velocity can be calculated. 

Surfactant conservation equation. This interfacial condition is a statement of solute balance on the interface. Several physicochemical effects can be modelled here including diffusion-controlled kinetics, adsorption-controlled kinetics, insoluble surfactants and surface solute production rate per unit area (for example by condensation or evaporation of solute across the interface). 

In general, the surfactant is distributed along the interface by a combination of convection and diffusion, as well as transport to and from the interface from the bulk solvents. However, in many cases, the solubility of a surfactant in the two solvents is very low, and a good approximation is that the transport from the solvents is negligible. In this case, it is said that the surfactant is an insoluble surfactant, and the total quantity of surfactant on the interface is conserved. We have notpreviously derived a bulk-phase conservation equation to describe the transport of a solute in a solvent. Hence in this section we adopt the insoluble surfactant case, and follow Stone50 in deriving a surfactant transport equation that relates only to convection and diffusion processes on the interface. 

It may be noted that the elTect of bubble convection was ignored in our earlier papers [35 38]. Although this does not affect the computed liquid profiles much, it cannot be ignored when the conservation equations for the surfactant are formulated because bubble convection produces a significant movement of the surface area and hence of the adsorbed surfactant. The balance equations for the surfactant are considered next. 

The basic conservation equations for the continuous-phase liquid and the surfactant have now been formulated. However, several additional details are required to specify the system completely. Expressions are needed for the movement of the boundaries Zi and zp. Furthermore, the mean values 5, Apxp, and R depend on the degree of coalescence. The movement of the... 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

Surface phenomena affect the particle sedimentation when a tangential surfactant concentration gradient exists near the particle surface. This situation is described by the stationary diffusion equation for the surfactant concentration outside the surface layer in the form = 0 [2,3], where the boundary condition, which reflects the substance conservation in a very thin surface layer, may be written as... 

The general boundary condition for conservation of mass of some species in the interfacial region is derived in Chapter 6. For the present case of an insoluble surfactant, and in the absence of surface diffusion, we anticipate that the first two terms of Equation 5.32 should suffice with T replaced by A ... 

On the surface of the liquid film equation for adsorption-desorption kinetics, the equations of mass conservation of surfactant and Pick s law are employed... 

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