Langmuir 1991, 7, 403-406
403
Diffusiophoresis Caused by Gradients of Strongly Adsorbing Solutes John L. Anderson and Dennis C. Prieve* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received January 29, 1990. I n Final Form: June 12, 1990 A theory is developed for diffusiophoresis of rigid insulating particles in a gradient of an adsorbing solute for the case of strong adsorption. The range of the interaction between the solute and the particle’s surface is assumed small relative to the particle’s radius ( a ) ,but the adsorption length ( K ) ,defined as the thickness of a film of solution having the bulk concentration which would contain the same amount of solute per unit area as adsorbs, is allowed to be arbitrarily large. An expression relating the particle’s velocity to the solute concentration gradient is obtained in analytical form. If K >> a, then the particle’s velocity is determined by a rather than by K.
Introduction
same amount of solute per unit area as adsorbs. Adsorption lengths for surfactants can easily be tens or hundreds of When placed in a solution that is not uniform in the micrometers: which greatly exceed the upper bounds of concentration of some molecular solute, a colloidal particle particle size normally associated with colloids. We say will spontaneously migrate toward regions of higher or that solutes having K >> a are “strongly adsorbing”, where lower concentration of the solute as a result of physical a is the radius of the colloidal particle. The current theory3 interactions between the solute molecules and the particle. for diffusiophoresis caused by nonelectrolytes assumes Derjaguin et a1.l called this migration “diffusiophoresis”. K 1. dextran or of polymer-coated silica particles. In both cases, approximately the molecular dimension of the solute, say the interaction between the “solute” and the latex particle 1 nm, and therefore very small relative to the radius ( a ) is expected to be repulsive and the resulting diffusioof the particle. phoretic speeds are much slower than with electrolytes. In our previous a n a l y s i ~the ~ * ~relevant equations are On the basis of the previous theory, much larger speeds solved in an expansion in powers of L / a . The coefficient should be obtained with strongly attractive interactions, of the O ( L / a )term is proportional to K / L ; thus, the firstsuch as those occurring with surfactants. The strength of order correction for curvature of the particle’s surface is the adsorption can be characterized by the “adsorption actually O ( K / a ) ,and the following result for the particle’s length” ( K ) ,defined as the thickness of a film of solution velocity is valid in the limit L / a 0 and K / a
