Interfacial Surfactant Concentrations on an Oscillating Droplet: Solution of a Singular Boundary-initial Value Problem

Using singular spectral theory, a boundary-initial value problem is solved for the interfacial concentration of a surfactant on a liquid droplet oscillating in a surrounding second immiscible liquid phase. Besides being of value to the specific application the methodology of this paper is useful for a variety of boundary-initial value problems of interest to chemical engineers in which material domains have infinite or semi-infinite extents. INTRODUCIION In this paper we solve a boundary-initial value problem in a semi-infinite domain featuring diffusion of a substance in two contiguous immiscible phases with an interface possessing a capacitance for the diffusing solute. There are several similar initial-boundary value problems of interest to chemical engineers and suitable methods for solving them are essential. We present here an effective approach for the solution of such boundary-initial value problems based on the spectral properties of singular second order differential operators. We do not present the development of the method in detail since it cannot be done concisely. However, somewhat similar boundary value problems have been solved by Parulekar and Ramkrishna (1984a, b, c) which feature similar details that are available in Parulekar’s doctoral dissertation (1983). Basically, the method relies on the development of continuous integral transforms from general singular second order differential operators, of which the wellknown infinite Fourier transform is a very special case. For detailed treatment of the application of the transform method the reader is referred to Parulekar and Ramkrishna (1984a). The boundary value problem of interest to this paper arises from the physical situation of a drop set into a spherically symmetric oscillation (such as by periodic injection and withdrawal of a liquid into the drop through a syringe) within a second immiscible external phase. A surfactant is allowed to diffuse between the bulk phases and the interface where it can accumulate by sorption or deplete by desorption. The oscillation in TPresent address: Department of Chemical Engineering, Illinois Institute of Technology. Chicano. IL 60616. U.S.A. ~To whom corresponden&hould be addressed.’ BPresent address: Department of Chemical Engineering, University of Tulsa, Tulsa, OK 74104, U.S.A. the droplet (and the surrounding phase) translates into an oscillation in the surfactant concentration in the bulk phases and at the interface. Of specific interest to the problem is the sensitivity of perturbations in the surfactant concentration at the interface to the frequency of drop oscillations. Thus the manner in which the amplitude in the concentration perturbation depends on the period of the oscillation of the droplet may be determined by the analysis. The relationship must depend as the sorption and desorption rate constants in the two phases. This physical system is important in the study of surface rheology and mass transfer at interfaces. For a detailed development of the mathematical formulation and simplifications of the system just described the reader is referred to Gottier (1980) and Gottier er al. (1986). Here we will merely present the boundaryinitial value problem arrived at by Gottier. Gottier’s solution to the problem was based on assuming the external phase to be finite in extent thus unable to exploit the advantage of an infinite medium which conveniently describes a region of indefinite vastness. Here we present a solution to the problem assuming the external phase to be infinite. We begin with the statement of the boundary-initial value problem of interest, present a reformulation essential for the development of the transform solution, obtain the solution and conclude with some numerical evaluation of the solution. BOUNDARY-INITIAL VALUE PROBLEM The diffusion of the surfactant in the drop phase and the external phase is described by the differential equations in suitably defined perturbation variables, 6, w and F. The differential equations are $=D-az: ’ O<z,<l (1)