Protonic Conductivity : a New Application of Soliton Theory

We propose a two sublattice model with a doubly periodic on-site potential to describe the proton transport in hydrogen-bonded quasi-one-dimensional networks. The discrete system is reduced to a continuum double Sine-Gordon equation for the protonic part plus a simple differential equation for the heavy ion part. Its two-component kink solitons correspond to the ionic and B j e m defects. The correct response of these solitonic defects to an externally applied dc electric field makes this system an excellent model for qualitative and quantitative description of the protonic conductivity in hydrogen-bonded networks. 1 INTRODUCTION Proton conductivity along quasi-one-dimensional hydrogen-bonded networks in some molecular systems can exceed the conductivity in the orthogonal direction by a factor of 1000. Ice-like structures of hydrogen-bonded systems show a proton mobility only an order of magnitude less than that in metals. For that reason, such a crystal is often considered as a "protonic semiconductor". Experimental evidence 11-41 show that high proton mobility in ice-like systems is due to their transfer along the hydrogen bonds. In spite of the profound importance that the hydrogen bond plays in Condensed Matter Physics and Biology, very little is known about the detailed mechanisms for proton transport along hydrogen-bonded networks. The hydrogen-bonded networks which we consider here are quasi one-dimensional clusters of molecular aggregates interacting with their first neighbors through hydrogen bonds. Considering the simplest case and focussing our attention on the main degrees of freedom, we obtain the following diatomic structure of hydrogen-bonded dimers : where the full line segments indicate a covalent or an, ionic bond, the dotted line segments indicate a hydrogen bond, and X denotes a negative ion. The atoms that are usually involved with hydrogen bonds representing the negative proton-accepting side of the heavy molecule or aggregate are 0, N, and F. In such a chain because of the symmetric one-dimensional environment of the hydrogen ion, protons can easily jump Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989301 C3-4 JOURNAL DE PHYSIQUE from their left equilibrium position to the energetically equivalent position at the right near the next X ion. This physical situation is usually modeled by the association with each proton of a double-well substrate potential representing the two degenerate proton equilibrium positions per unit cell [S]. Ice is a very common example of a hydrogen-bonded system. In the more stable Ih hexagonal ice structure 111, one can distinguish hydrogen-bonded networks either following a given crystallographic direction (for example, a zig-zag hydrogen-bonded linkage) or just following a Bemal-Fowler filarnental path [61. Before proceeding to better-defined but more complex quasi-one dimensional biological or polymeric hydrogen-bonded macromolecules, let us first study the zig-zag diatomic structure in Ih ice because of the extensive experimental data that are available for ice [I-31. One well-known property of ice, as well as of other hydrogen-bonded sytems, is the spontaneous formation of ionic and Bjenum-type defects. In the zig-zag hydrogen-bonded network (Fig.0, one can realize a configuration where a number of successive protons jumping in their second equilibrium position produce one hydroxyl ion (OH-) with negative effective charge and one hydronium ion (H30+) with a positive effective charge [Fig.l(a)l. In an 'infinite' chain we can generate a large number of negative and positive ionic defects without changing the number of protons in the system. On the other hand, following Bjerrum's theory 171, we can rotate a number of successive water dipoles in such a way that an 0-0 bond with two protons and positive effective charge and one without proton and with negative effective charge is generated [Fig.l(b)l. The number of positive (D) and negative (L) Bjerrum defects can increase or decrease by injection or removal of protons, respectively.