## 6 Effect of Surface Heterogeneity on the Transient Behaviour of IEMs
### Impact of Surface Heterogeneity on the Transition Time of IEM  
The Use of Electric Current Stream Function. Comparison with the Sand and Choi and Moon Theories  
Electrochemical responses of materials having heterogeneous surface mainly depend on the parameters of surface heterogeneity [15; 274; 291]: on the size and shape [292; 293], as well as the surface fraction [274; 292; 294; 295; 296] of conductive and nonconductive areas. Most of mathematical descriptions of ionic transport in electrochemical systems are restricted to 1D geometry [296], not relevant to a heterogeneous permselective surface. The passage from 1D to 2D and especially 3D geometry sharply increases mathematical difficulties. Nevertheless, there are a large number of systems with permselective surface for which multidimensional effects are important: mass transfer at partially blocked electrodes [292; 295], microelectrodes [297; 298], proton-exchange membrane fuel cells and batteries [299]; flow electrochemical devices, such as electrolysers and electrodialysiss [80; 300], electrochemically heterogeneous graphene [293] or carbon nanofilaments [274], heterogeneous IEMs [15; 274; 291] and so forth. Development of approaches for handling multidimensional modelling of interface phenomena represents an important issue. 2D/3D mathematical description of the PD response of membrane/electrode systems is complicated since an integral condition of current intensity is needed at the permselective surface. This constitutes a problem as the current density distribution over the conductive area is not known a priori. [PERSON] et al. [274], [PERSON] and [PERSON] [301; 302] used the condition of uniform current density distribution on the conductive regions of membrane surface. However, this condition was not verified using another approach. The use of the electric current stream function [303] seems promising for the solution of this kind of problems, since this approach does not require any assumption about the character of current density distribution. Stream functions are widely used in fluid mechanics [304]. The main feature of this approach is that it allows calculation of the fluid velocity distribution when  
Figure 14: Dependence of theoretical (lines) and experimental (points) transition time on the normalized current density for a Neosepta® CMX membrane in a 0.02 M NaCl, \(\tau_{Sand}\) refers to the transition time calculated using the Sand equation, Equation (76); \(\tau_{Model}\) refers to the finite-length time-dependent diffusion layer model [269]; the DBL thickness and the limiting current density calculated using the Lévéque Equations (60) and (61) are \(\delta_{Lev}=260\) μm, \(i_{\text{lim}}=2.0\) mA/cm\({}^{2}\), respectively. Reproduced with permission from [269], Elsevier, 2016.  
the volumetric flow rate is known in heterogeneous systems [305]. Another advantage is that any solution obtained with the use of stream function satisfies automatically the law of mass conservation. Apparently, [PERSON] et al. [306] were the first who has introduced the stream function for modelling electric current across heterogeneous surface. They used the mathematical analogy between the flow of fluid and that of electric current. Later on, this approach was applied for simulation of electric current density distribution in semiconductors [307], in superconductors [308], in thin circuits [309], on the earth's surface [310], in electromagnetic devices [311] and other systems. In membrane systems, this method was applied by [PERSON] et al. [312] and [PERSON] et al. [303]. 3D modelling in Ref. [303] was used for describing the chronopotinograms of a heterogeneous IEM; the results of computations were compared with experiment and with other known theories, namely, that by [PERSON] [284], [PERSON] [291] and [PERSON] et al. [274]. The geometry used in Ref. [303] is shown in Figure 15. A hexagonal distribution of conductive areas (the open circles in Figure 15) over the membrane surface is assumed.