Anomalous tracer diffusion in hard-sphere suspensions

Abstract

Coupled equations describing diffusion and cross-diffusion of tracer particles in hard-sphere suspensions are derived and solved numerically. In concentrated systems with strong excluded volume and viscous interactions the tracer motion is subdiffusive. Cross diffusion generates transient perturbations to the host-particle matrix, which affect the motion of the tracer particles leading to nonlinear mean squared displacements. Above a critical host-matrix concentration the tracers experience clustering and uphill diffusion, moving in opposition to their own concentration gradient. A linear stability analysis indicates that cross diffusion can lead to unstable concentration fluctuations in the suspension. The instability is a potential mechanism for the appearance of dynamic and structural heterogeneity in suspensions near the glass transition.

Dynamic colloidal membranes

“…Quantitatively, the model results depend on the expressions for the cross-diffusion coefficients summarized in the Appendix. Expressions for D 11 and D 22 have been compared with experiment for the hard-sphere system [10,11], and similar equations for all four D ij are in good agreement with experiments on salt cross diffusion in polyethylene glycol solutions [12]. Perhaps the most uncertain aspect of the model is the expression for the colloidal reflection coefficient σ.…”

“…The main D 22 diffusion term acts to relax the tracer concentration gradient and move the particles through the membrane in the −x direction. The cross-diffusion term is driving them in the opposite direction, against their own concentration gradient (uphill diffusion) [11]. When the colloid volume fraction φ 1 approaches close-packing, hindered diffusion causes the tracer Fickian diffusion coefficient D 22 to reduce to zero (figure 7d ).…”

“…where D 11 and D 22 are the main Fickian diffusion coefficients of the colloidal particles and tracer particles, and D 12 and D 21 are cross-diffusion coefficients. Expressions for the D ij as functions of particle concentration have been obtained previously for hard-sphere suspensions [9][10][11] and for polymer solutions [12]. Here the hard-sphere expressions are used, as summarized in the Appendix.…”

“…For spherical particles approximate expressions for the friction and compressibility factors of the host-particle matrix are K = (1 − φ 1 ) 6.55 and (19a,b) where a = 4 − 1/φ p and φ p = 0.64 is the volume fraction at random close packing [11]. In the dilute limit φ 1 1 equations ( 19) reduce to the exact results K = 1 − 6.55φ 1 and Z = 1 + 4φ 1 [17,18].…”

“…In the dilute limit φ 1 1 equations ( 19) reduce to the exact results K = 1 − 6.55φ 1 and Z = 1 + 4φ 1 [17,18]. Given Z,Π 1 can be obtained aŝ where λ = R 2 /R 1 is the size ratio of the tracer and host particles, b 12 = (1+λ) 3 is a thermodynamic coupling coefficient, φ 2 = n 2 v 2 is the tracer volume fraction and v 2 = 4 3 πR 3 2 is the tracer particle volume [10,11].…”