Bulk and phase boundary effects

In the stationary state (t = oo) the voltage is determined by the total resistance R - U(t=c )/I if interfacial effects are neglected. Evaluation of the electrode area (a) and the distance between the electrodes (L) yields (Section 6.3) 

At the start of our d.c. experiment, i.e. immediately after switching on the cmrent, in addition to the pure conduction current Ir there is also a capacitive displacement current flow (Ic), which is responsible for the time dependence. For small stimuli it follows that (Ic = (5/ )Qc Qc=capacitor charge)  

As the electrical current can be carried by conduction or displacement ciurrent, the circuit elements, electrical resistance and capacitance are coimected in parallel in the equivalent circuit. Since now the voltages across the resistance (U = IrR) and capacitance (integral of Uc = Ic/C) are equal, and the contributing currents add to give a constant total current (I = (Ir +.Ic))) this yields the differential equation 

By substituting V = U — IR (i.e. V = U) and using the abbreviation r = RC we obtain a homogeneous differential equation (V + V/r = 0) with the solution V = V(t=0) exp —t/r. If we reverse the substitution and take account of the fact that immediately on switching on the current the capacitor does not provide any resistance to current flow, i.e. U(t=0) = V(t=0) 4-IR = 0, then we obtain a monotonic asymptotic voltage behaviour rising to the stationary value IR according to 

initially the whole current is a displacement current. As time passes the conduction current increases until it is representing the total current when.the star tionary state is reached. The capacitor is then completely impermeable and Ic = 0. The breakdown of the conduction current into electronic and ionic current (the resistances Reon and Rjon are in parallel, i.e. 0 = 0 eon+ 7 ion or R =R -t-R ) leads to (t transference number) 

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