Surfactant adsorption to spherical particles: The intrinsic length scale governing the shift from diffusion to kinetic-controlled mass transfer

When a drop or bubble of radius b is formed in surfactant solution, surfactant adsorbs, diffuses from solution, and desorbs to establish the equilibrium surface concentration. The transport coefficients for diffusion, adsorption, and desorption are fundamental parameters. However, the transport mechanisms that control the interface far from equilibrium are highly context dependent. Thus, no surfactant has universal "diffusion-controlled transport. Here we identify a new length scale, RD-K, that depends on surfactant physicochemistry, and which ranges from roughly 15 to 65 microns. For drops or bubbles with bRD-K, mass transfer is kinetically controlled. For bRD-K, mass transfer is diffusion controlled. Simulations of adsorption to quiescent spherical interfaces support the importance of RD-K in determining the controlling transport mechanism for surfactant solutions with concentrations below the critical micelle concentration (CMC). While the case of surfactant adsorbing to a bubble is discussed in detail, the arguments presented are quite general and should apply to adsorption of any solute to any spherical particle.