Intra-particle Pore Diffusion

Key Insight

Pore diffusion is a critical intra-particle mass transfer mechanism where solutes diffuse within liquid-filled pores sufficiently large to minimize interaction with pore walls. The transport is described by the effective pore diffusivity, D_e, which defines the protein mass transfer flux in the stationary phase via the protein concentration gradient. D_e is typically smaller than the free solution diffusivity, D naught, and is expressed as D_e = epsilon_p * D naught * psi_p / tau_p.

Key factors influencing D_e include intra-particle porosity (epsilon_p), which accounts for available space, and the tortuosity factor (tau_p), which reflects the longer, tortuous diffusion path due to random pore orientation; typical tau_p values for protein chromatography matrices range from 1.5 to 4. The diffusional hindrance coefficient (psi_p) specifically addresses reduced diffusion for biopolymers whose molecular size is comparable to pore size, resulting from steric exclusion near pore walls and viscous drag. To avoid excessive hindrance (e.g., psi_p < 0.5), the pore radius generally needs to be about 8 times the protein radius; for a 150 kilodalton protein with an approximate radius of 5 nanometers, a pore radius of around 40 nanometers is needed.

Estimation of D_e is complicated by real adsorbent properties such as non-uniform pore sizes, dead-end pores, reduction of available pore size by adsorbed protein, and electrostatic interactions between pore walls and proteins, especially at low ionic strengths where electrical potentials may overlap (with Debye lengths around 2.2 nanometers at 0.02 molar ionic strength). Despite these complexities, equations incorporating these factors provide useful predictions. Effective pore diffusivities (D_e) for proteins often show significant reduction compared to free solution diffusivities, with larger proteins like albumin and IgG exhibiting severely restricted diffusion in matrices with smaller pores, such as SP-Sepharose-FF.