Adsorbed Phase (Solid) Diffusion

Key Insight

Adsorbed phase diffusion, also referred to as solid diffusion, describes transport processes where diffusion occurs in the adsorbed state within a phase distinct from the bulk pore liquid. This mechanism encompasses surface diffusion (movement along pore surfaces without detachment), micropore diffusion (transport in cavities comparable to molecular size), and homogeneous diffusion (e.g., diffusion in an immiscible liquid-filled pore or a charged molecule in an oppositely charged gel with overlapping electrical double layers). The driving force for this type of diffusion is expressed in terms of the adsorbed-phase concentration, q, and quantified by an effective adsorbed-phase diffusivity, D_s.

Quantitatively, D_s values are generally smaller than effective pore diffusivities (D_e) because diffusion in the adsorbed phase is more restricted, and D_s values often exhibit concentration dependence, unlike pore diffusivities, due to much higher adsorbed-phase concentrations. However, if the adsorption capacity (q) is high, the overall mass transfer flux in the adsorbed phase can be substantially larger than that resulting from pore diffusion, even with a smaller D_s.

Understanding adsorbed-phase diffusion for proteins remains challenging, largely because mass transfer models based on vastly different transport mechanisms can provide nearly identical descriptions of macroscopic adsorption rate data, making it difficult to experimentally distinguish mechanisms. Despite this, some studies have shown apparent D_e values higher than free solution diffusivity, suggesting contributions from mechanisms beyond pore diffusion, such as protein interaction with charged dextran polymers in ion exchangers. Typically, D_s values are on the order of 10^-8 cm^2/s, and empirical measurements using the actual stationary phase are often necessary to establish governing mechanisms and the magnitude of relevant diffusional time constants, as well as their functional dependence on process variables.