## 6 Effect of Surface Heterogeneity on the Transient Behaviour of IEMs
### Impact of Surface Heterogeneity on the Transition Time of IEM
They had modified the Sand equation, Equation (76), replacing the average current density, \(i_{av}\), by the local current density, \(i_{local}=i_{av}/\gamma\), across the conductive areas:  
\[\tau_{Choi-Moon}=\left(\frac{\pi D}{4}\right)\left(\frac{c_{i0}z_{i}F}{T_{i}- t_{i}}\right)^{2}\frac{\gamma^{2}}{i_{av}^{2}} \tag{81}\]  
The relationship between the transition times expressed according to Sand and to Choi and Moon [291] is expressed as \(\tau_{Choi-Moon}=\tau_{Sand}\gamma^{2}\). The transition time calculated using the 3D model [303] is in a good agreement with the experimental data (Figure 16a) obtained for a specially designed membrane. The substrate is a polyethylene terephthalate track-etched membrane with cylindrical randomly distributed pores with a radius of 12.93 \(\pm\) 0.22 \(\upmu\)m. The substrate was coated on one side with a Nafion(r) perfluorinated resin solution of thickness \(d_{2}\) about 3 \(\upmu\)m, the Nafion(r) solution filled also the pores. Thus, a membrane with a smooth homogeneous permselective surface on one side and a heterogeneous surface on the other was obtained. The circular conductive areas filled with the Nafion(r) material occupy about \(\approx\)7.7% of the surface, 92.3% of the surface was non-conductive. No fitting parameters in the model of [PERSON] et al. [303] are used. The curve computed for the lower bound of the range of possible radii, \(R_{1}=12.71\)\(\upmu\)m, is below the experimental points;  
Figure 16: Transition time normalized by \(\tau_{Sand}\) as a function of (**a**) \(i/i_{av\text{ lim}}\): experimental results (circles) are compared with the simulation using the 3D model [303] with integral current condition (77) (curve 1) or condition [274] (80) (curve 2) and the Choi-Moon theory [291], Equation (81), (curve 3), \(R_{2}=46.53\)\(\upmu\)m; (**b**) \(i/i_{av\text{ lim}}\) at different fraction of conductive area \(\gamma=0.1;0.3;0.5;0.7;0.9;1.0\); \(R_{1}=12.71\)\(\upmu\)m; (**c**) \(R_{1}/\delta\) at different \(\gamma\) (the same as in b) at fixed \(i=2i_{av\text{ lim}}\); (**b**,**c**) are computed using the 3D model [303] with condition (77). The dashed straight lines show the transition time calculated by the Choi-Moon Equation (81). In all cases \(\delta=260\)\(\upmu\)m, \(i_{av\text{ lim}}=19.74\) A/m\({}^{2}\). Reproduced with permission from [303], American Chemical Society, 2016.  
that for the upper bound, \(R_{1}=13.15\)\(\upmu\)m, is above. The transition times calculated using the Sand equation, Equation (76), are essentially higher than the experimental ones and that calculated using the Choi-Moon equation are much lower (Figure 16a). The calculated values of \(\tau\) using the 3D model [303] but under condition (80) are lower too. In fact, Equations (76) and (81) give two extreme asymptotes of the model [303]. As we mention in the previous section, the Sand Equation (76) holds, when the surface is homogeneous and the current density \(i>(1.5\div 2)i_{\text{{\tiny{$i\infty$}}}}\)[269] (Figure 16b,c). The Sand equation holds also when \(\gamma<1\) and the ratio \(R_{1}/\delta\) is sufficiently small (Figure 16c). In this case, the concentration distribution far from the heterogeneous surface is the same as in the case of homogeneous surface. The Choi-Moon theory [291] assumes the current lines normal to the conductive surface, hence, the funnel effect [274] is not taken into account and tangential transport is considered negligible. This condition is satisfied, when \(R_{1}/\delta\) and \(i\) are both sufficiently great (Figure 16b,c).