微环电极理论

THEORY OF MICRORING ELECTRODES

近几年来,微电极的应用得到迅速发展,但由于混合边界条件的存在,微电极稳态和暂态问题的数学处理极为复杂.过去,微电极上不可逆和准可逆过程主要采用近似方法研究.自文献利用贝塞尔级数处理微盘电极稳态行为以后。

In this work, expressions for concentration and current in steady-state and tran-sient-state experiments are given for microring electrodes. For the electrode reactionO+ne→R, assuming bulk concentration of R c_R^b=0, under steady-state conditions,we have G_O=G_R=sum from μ=1 to ∞ α_μ integral from n=0 to ∞ f_μ (α)J_0 (αr/r_2) exp (-αz/r_2) dαwhere f_μ(α)=J_μ(α)/α~μ-sum from r=1 to μ (r_1/r_2)~p[1-(r_1/r_2)~2]^(μ-r)/2^(μ-ν)(μ-ν)! J,(αr_1/r_2)/α~ν G_O=1 - (C_O/C_O^b), G_R=D_RC_R/(D_OC_O^b)where c_O^b is the bulk concentration of species O. r_2, r_l are outer and inner radiirespectively. The α_μ coefficients depend on the boundary condition on the surfaceof the electrode, and can be calculated by using either method previously reportedin [3, 4]. For the transient-state problem, we have G_O=sum from μ=1 to ∞ α_μ (s) integral from n=0 to ∞ f_(μ,O) (α, s) J_0 (αr/r_2) exp (-(α~2+sr_2~2/D_Oz/r_2)^(1/2)) dα G_R=sum from μ=1 to O α_μ (s) integral from n=0 to ∞ f_(μ,R) (α, s) J_0 (αr/r_2) exp (-(α~2+sr_2~2/D_Rz/r_2)^(1/2)) dαwhere f_(μ,i)(α,S)=1/(α~2+sr_2~2/D_i)^(1/2)[J_μ(α)/α^(μ-1)-sum from v-1 to μ(r_1/r_2)~v[1-(r_1/r_2)~2]^(μ-ν)/2^(μ-ν)(μ-ν)! J_ν(αr_1/r_2)/α^(ν-1)]G_i is the Laplace trasformation of G_i(r,z,t), i=O or R. s is the variable ofLaplace tansformation, the α_μ(s) coefficients can be determined by the boundarycondition on the microring electrodes [3, 4].