## 6 Effect of Surface Heterogeneity on the Transient Behaviour of IEMs
### Modelling of Chronopotetograms for Homogeneous Membranes. Effect of Finite-Length Diffusion Layer
In 1901 [PERSON] [284] deduced a simple equation for the transition time, \(\tau\), when considering diffusion in a semi-infinite 1D domain with a constant-flux boundary condition:  
\[\tau=\left(\frac{\pi D}{4}\right)\left(\frac{c_{0}z_{i}F}{T_{i}-t_{i}}\right)^ {2}\frac{1}{i^{2}} \tag{76}\]  
The adaptation of the [PERSON] equation to IEMs with non-ideal permselectivity expressed by \(T_{k}\) was first made apparently by [PERSON] and [PERSON] [285] and later by [PERSON] et al. [282]. In [PERSON]'s theory, the DBL thickness is assumed infinite; the forced or natural convection occurring in the membrane/electrode system is not taken into account. The transition time is defined as the moment where the interfacial concentration attains zero and the PD tends to infinity. The fact that the DBL thickness is assumed infinite leads to essential deviation between theoretical and experimental values of in certain conditions. The situation is similar to that in the theory of impedance. [PERSON] [286] proposed a simple theory for computing the impedance of solution/electrode system as a function of frequency assuming that the DBL thickness is infinitely large. However, taking into account the finite length of the DBL changes cardinally the shape of impedance spectrum [287; 288]. The finite length of the DBL at membrane surface was accounted by [PERSON] [289] and [PERSON] et al. [46], when basing on the Nernst-Planck equation and the LEN assumption. The value of the DBL thickness, \(\delta\), is considered as a fitting parameter. In Reference [46], the time dependence of \(\delta\) is implicitly taken into account in conditions of current-induced gravitational convection. Recently, [PERSON] et al. [290] presented a theory based on the Nernst-Planck-Poisson equations. The value of \(\delta\) was fixed as the distance between two parallel electrodes forming the cell; electron-transfer reactions across the Stern monolayer at the electrode surface were taken into account [290]. [PERSON] et al. [269] took into account the decrease of \(\delta\), occurring with time, due to the development of current-induced convection. The value of \(\delta\) is considered as a function of the Donnan PD (\(\Delta\phi_{Don}\)) at the membrane/depleted solution interface. When an overlimiting current density is applied and the time approaches the transition time, current-induced convection (namely, electroconvection) arises near the depleted interface and causes a decrease of \(\delta\). The \(\delta(\Delta\phi_{Don})\) function is assumed a linear one containing two adjustable parameters: a threshold value of \(\Delta\phi_{Don}\), which relates to the onset of electroconvection and a value of \(\Delta\phi_{Don}\) related to the steady state. As it is shown in Reference [269], the model describes well experimental chronopotentiograms of a CMX homogeneous CEM at all applied overlimiting current densities. The comparison of the calculation using the Sand theory and the model [269] with experimental transition times found for different current densities is shown in Figure 14. As it can be seen, the Sand transition time is less than the experimental value; this difference increases with decreasing (overlimiting) current density. The explanation is that the assumption of the semi-infinite DBL does not allow accounting for the contribution of forced (or natural) convection. The electrolyte delivery by convection slows down the growth of PD with time. However, when the current density is sufficiently high (>1.5 \(i_{\text{lim}}\) in Figure 14), the concentration profile develops so rapidly that the DBL thickness does not reach the values, where the contribution of forced convection is significant; therefore, at \(i>1.5\)\(i_{\text{lim}}\), the model of [PERSON] etal. [269] and the Sand theory give the same results and both agree with the experiment. The same conclusion concerning the agreement between the Sand theory and a model taking into account the finite length of \(\delta\) was obtained by [PERSON] et al. [290] in the case of electrode systems.