## 6 Effect of Surface Heterogeneity on the Transient Behaviour of IEMs
### Impact of Surface Heterogeneity on the Transition Time of IEM
The three-layer system'membrane + depleted DBL + enriched DBL' adjacent to the membrane from its both sides is considered as a system of cylindrical cells (Figure 15). The membrane upper surface layer consists of a conductive disk of radius \(R_{1}\) within a nonconductive ring of external radius \(R_{2}\). The DBL thickness, \(\delta\), is assumed identical from both membrane sides. Using cylindrical symmetry, the system under study may be presented in two coordinates \(r\) and \(z\). The governing equations are the same as in the case of ED channel with heterogeneous membrane described in Section 5.3: the Nernst-Planck transport equation (without convective term), the continuity equation for ion transfer and the LEN condition. However, the time-dependent problem is considered here, that is, \(\partial c_{i}/\partial t=-div\overset{\rightarrow}{j}_{i}\
eq 0\). As in Section 5.3, the flux continuity condition at the conductive surface/solution boundaries is applied. At the non-conductive boundaries, the zero flux condition like Equation (65) is used. In galvanostatic regime, the average current density across the membrane surface, \(i_{av}\), is a known parameter. It is linked with the unknown local current density as [303]:  
\[i_{av}=\frac{1}{R_{2}}\underset{0}{\overset{R_{2}}{\underset{0}{\int}}}i_{z} (r,\delta,t)dr=\frac{1}{R_{2}}\underset{0}{\overset{R_{2}}{\underset{0}{ \int}}}i_{z}(r,-d_{1}-d_{2}-\delta,t)dr=const \tag{77}\]  
Figure 15: Distribution of round conductive areas over the membrane surface in hexagonal geometry (**a**). The DBL of thickness \(\delta\) is assumed independent of lateral position (**b**). Conversion of the 3D system (**b**) into a 2D simulation problem (**c**). Reproduced with permission from [303], American Chemical Society, 2016. No condition on the distribution of normal current density over the conductive regions is imposed. Instead, the electric current stream function [306; 312]\(\eta\) is introduced as follows:  
\[\frac{1}{r}\frac{\partial\eta}{\partial z}=-j_{r},\frac{1}{r}\frac{\partial \eta}{\partial r}=j_{z} \tag{78}\]  
The charge conservation (continuity) equation in the cylindrical geometry is written in the form:  
\[\frac{\partial\rho_{e}}{\partial t}=-div\overset{\rightarrow}{j}=-\left( \frac{1}{r}\frac{\partial}{\partial r}(rj_{r})+\frac{\partial}{\partial z}j_{ z}\right)=0 \tag{79}\]  
Note that Equation (79) holds, if the rate of the space charge density (\(\rho_{e}\)) variation with time is negligible. Equation (78) assures that Equation (79) is satisfied at any function \(\eta\), if only the condition for equality of mixed derivatives of \(\eta\) is met. The condition of uniform current density distribution over the conductive areas according to Refs. [274; 313] is expressed as:  
\[i_{z}(r,0+,t)\big{|}_{r=0\dots R_{1}}=i_{av}/\gamma \tag{80}\]  
However, as it is shown in Ref. [303], when integral condition, Equation (77), is applied, the current lines are concentrated within conductive areas at the boundary with the neighbour non-conductive region. The application of the uniform current density distribution, Equation (80), results in too fast decrease with time of the interfacial electrolyte concentration in the centre of conductive areas, which leads to a shorter transition time in comparison with the model [303]. The model [303] formulated above allows computation of chronopotenticograms and the transition times. As the LEN assumption and a fixed DBL thickness are used in this model, the transition time corresponds to the state where the PD tends to infinity, as in the model [269] described in Section 6.1, where the current-induced transport is implicitly taken into account. It is known, that the transition time for heterogeneous membranes is lower than that for homogeneous ones [291; 15]. However, the only mathematical description of the transition time for an electrically non-homogeneous permselective interface was made by [PERSON] and [PERSON] [291].