Yc. Chung et al., Calculation of the contribution to grain boundary diffusion in ionic systems that arises from enhanced defect concentrations adjacent to the boundary, J APPL PHYS, 87(6), 2000, pp. 2747-2752
The enhancement of the concentration of a defect in the space-charge region
near a grain boundary in an intrinsic ionic system was used to specify the
change in the corresponding diffusion coefficient as a function of distanc
e, x, normal to the boundary. This zone of enhanced and continuously varyin
g diffusivity serves as a vehicle for enhanced transport along the boundary
in addition to the expected contribution arising from the different struct
ure at the boundary core. A two-dimensional diffusion equation was establis
hed for concentration c(x,y,t) in the space-charged region, solved numerica
lly using the Crank-Nicolson finite-difference method, and integrated norma
l to the boundary to provide the average concentration gradient (c) over ba
r(y,t) along the boundary-the solute distribution usually measured by exper
iment. The present gradients were analyzed in the same fashion as experimen
tal data using a recent solution to the conventional model for the grain-bo
undary diffusion problem in which the interface is treated as a thin slab o
f half-width a and enhanced diffusivity D-' imbedded in a material with bul
k diffusivity D. The analyses provided effective values for the conventiona
l diffusion parameter beta = [(D'/D) - 1]left perpendiculara/Dtright perpen
dicular that describe the influence of space charge on enhanced diffusion a
long the boundary. The calculations were repeated for a number of values of
the parameter Z(i)e phi(infinity)/kT, where Z(i)e is the effective charge
of the defect and phi(infinity) is bulk electric potential far from the bou
ndary. In the finite-difference calculations the value of the "annealing ti
me," t, that appears in beta was set equal to the product of the square of
the Debye length, delta(2), divided by the bulk diffusion coefficient, D. G
iven a value of phi(infinity), the present results thus permit an estimatio
n of the effect of space charge on preferential diffusion along a grain bou
ndary in a material in terms of beta. If the Debye length is known for the
material or can be measured, the value of beta can readily be converted to
an effective value of the conventional grain boundary diffusion product aD'
/D. (C) 2000 American Institute of Physics. [S0021-8979(00)07806-3].