Mathematical Model of Current‐Polarized Ionophore‐Based Ion‐Selective Membranes: Large Current Chronopotentiometry

Abstract

The properties of ionophore-based ion-selective membranes were investigated theoretically and experimentally for the first time in large-current chronopotentiometric experiments. In these experiments, when either the free ionophore or the ion-ionophore complex concentration drops to zero at one of the membrane boundaries, a characteristic drop in voltage (breakpoint) appears in the chronopotentiometric transients at a transition time t.Based on a mathematical model of current polarized membranes, theoretical equations were derived and tested. These equations describe the correlation between the transition time t and the concentration of free ionophore, the concentration of ion-ionophore complex, and the diffusion coefficients of species in the membrane. The simulated concentration profiles correlated well with concentration profiles recorded experimentally prior to the transition time using spectroelectrochemical microscopy (SpECM). Diffusion coefficients calculated from (i) the transition times, (ii) curve fitting, and (iii) the initial ohmic resistance of the studied membranes were also compared. Similar to chronoamperometry, the chronopotentiometric transition times provided possibilities for assessing the loss of free ionophore and lipophilic anion from cation selective electrode membranes during continuous use.

Assessment of Ion‐Ionophore Complex Diffusion Coefficients in Solvent Polymeric Membranes

“…These concentration profiles change when a current is passed through the membrane. When the applied current is sufficiently large, the boundary concentrations of the free ionophore, the ion-ionophore complex, or both will reach zero at some finite transition time t, which we have previously described and called the breakpoint time [3,4,6]. Theoretically, an infinite voltage drop is expected at the transition time in CP experiments.…”

“…However, due to deviations from the ideal membrane behavior [3], the voltage change is finite. The equations used to calculate the chronopotentiometric breakpoints for the free ionophore t L,free and the ion-ionophore complex (t IL nþ k ) have been derived previously both for 1 : 1 [6] and for 1 : k ion:ionophore stoichiometries with primary ions of any charge n [3,4]. The equations for the free ionophore and ionionophore complex breakpoint times are given by Equations 1 and 2, respectively.…”

“…More recently, it was shown that CP can be used in the same way. However, CP has both theoretical and experimental advantages [3,6]: i) the constant current assumption is only strictly true in CP experiments; ii) it is usually easier to determine the transition times from the CP voltage-time transients than from the CA current -time curves.…”

“…More recently, it has been shown that breakpoints in the CA or CP transients of ion-selective membranes can emerge either when the free ionophore (L) or the ion-ionophore complex (IL nþ k Þconcentration reaches zero at one of the membrane boundaries [3,6]. In a specific experiment, the relative concentrations and the diffusion coefficients of the ionophore and the ion-ionophore complex determine which species concentration will drop first to zero at one of the membrane boundaries and cause the breakpoint in the CA or CP transients.…”

“…However, a variety of non-equilibrium techniques have been devised to probe membrane characteristics or alter membrane response. For example, chronoamperometric (CA) [1 -5] and chronopotentiometric (CP) [3,6] methods have been used to determine either the concentration or the diffusion coefficient of the free ionophore inside the membrane. Determination of the free ionophore concentration in membranes can be used to assess leaching of the ionophore or lipophilic anion from the membrane over time.…”