^{1}, Péter L. Simon

^{1}, Mária Wittmann

^{1}and Zoltán Noszticzius

^{1,a)}

### Abstract

Until now acid-base diodes and transistors applied strong mineral acids and bases exclusively. In this work properties of electrolyte diodes with weak electrolytes are studied and compared with those of diodes with strong ones to show the advantages of weak acids and bases in these applications. The theoretical model is a one dimensional piece of gel containing fixed ionizable groups and connecting reservoirs of an acid and a base. The electric current flowing through the gel is measured as a function of the applied voltage. The steady-state current-voltage characteristic (CVC) of such a gel looks like that of a diode under these conditions. Results of our theoretical, numerical, and experimental investigations are reported in two parts. In this first, theoretical part governing equations necessary to calculate the steady-state CVC of a reverse-biased electrolyte diode are presented together with an approximate analytical solution of this reaction-diffusion-ionic migration problem. The applied approximations are quasielectroneutrality and quasiequilibrium. It is shown that the gel can be divided into an alkaline and an acidic zone separated by a middle weakly acidic region. As a further approximation it is assumed that the ionization of the fixed acidic groups is complete in the alkaline zone and that it is completely suppressed in the acidic one. The general solution given here describes the CVC and the potential and ionic concentration profiles of diodes applying either strong or weak electrolytes. It is proven that previous formulas valid for a strong acid-strong base diode can be regarded as a special case of the more general formulas presented here.

The authors would like to thank Jürgen Vollmer and Henrik Farkas for helpful discussions. This work was supported partially by OTKA Grant No. T-042708 and the ESF Program, “Reactor.”

I. INTRODUCTION

II. THEORY

A. Governing equations

B. Steady-state current-voltage characteristics

III. AN APPROXIMATE ANALYTICAL SOLUTION

A. Approximations, simplifications applied to find an analytical solution

B. Derivation of an analytical solution

C. Specific cases

IV. ILLUSTRATIVE NUMERICAL EXAMPLES CALCULATED WITH THE ANALYTICAL FORMULAS

V. CONCLUSION

## Figures

Scheme of an electrolyte diode experiment. The alkaline and the acidic (HA) reservoirs are connected by a capillary filled with a hydrogel to prevent advection. Constant concentrations are maintained in these reservoirs by feeding them continuously with fresh electrolyte solutions. The steady-state voltage-current characteristics measured in this setup depends on the polarity like in a semiconductor diode. In the case of a reverse-biased electrolyte diode (when the acidic reservoir is positive as in this figure) usually the gel can be divided into three different zones: an alkaline (length ), an acidic (length ), and a middle weakly acidic (length ) zone.

Scheme of an electrolyte diode experiment. The alkaline and the acidic (HA) reservoirs are connected by a capillary filled with a hydrogel to prevent advection. Constant concentrations are maintained in these reservoirs by feeding them continuously with fresh electrolyte solutions. The steady-state voltage-current characteristics measured in this setup depends on the polarity like in a semiconductor diode. In the case of a reverse-biased electrolyte diode (when the acidic reservoir is positive as in this figure) usually the gel can be divided into three different zones: an alkaline (length ), an acidic (length ), and a middle weakly acidic (length ) zone.

Analytical approximation of the current-voltage characteristic of a weak acid-weak base (a) and a strong acid-strong base diode (b), where acid and base concentrations were at both ends of a -long gel cylinder. The continuous line shows the validity range of the approximation, i.e., where the gel can be divided into three regions. The dashed line shows the potential range where there is no middle (weakly acidic) zone in the gel. (The three-zone approximation for a reverse-biased diode can be applied when the absolute value of the applied voltage is more than in the case of weak acids and bases and in the case of strong acids and bases.)

Analytical approximation of the current-voltage characteristic of a weak acid-weak base (a) and a strong acid-strong base diode (b), where acid and base concentrations were at both ends of a -long gel cylinder. The continuous line shows the validity range of the approximation, i.e., where the gel can be divided into three regions. The dashed line shows the potential range where there is no middle (weakly acidic) zone in the gel. (The three-zone approximation for a reverse-biased diode can be applied when the absolute value of the applied voltage is more than in the case of weak acids and bases and in the case of strong acids and bases.)

Analytical approximation of the concentration profiles and the voltage profile along the gel for voltage difference in a weak acid-weak base diode [(a), (b), and (c)] [ in the left and in the right reservoir] and in a strong acid-strong base diode [(A), (B), and (C)] ( in the left and HCl in the right reservoir). The characteristic length of the gel was , and the weakly acidic fixed group concentration was , whereas the dissociation constant of the fixed groups was .)

Analytical approximation of the concentration profiles and the voltage profile along the gel for voltage difference in a weak acid-weak base diode [(a), (b), and (c)] [ in the left and in the right reservoir] and in a strong acid-strong base diode [(A), (B), and (C)] ( in the left and HCl in the right reservoir). The characteristic length of the gel was , and the weakly acidic fixed group concentration was , whereas the dissociation constant of the fixed groups was .)

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