Flow and streaming potential of an electrolyte in a channel with an axial temperature gradient

Abstract

The effect of an axial temperature gradient on the flow profile and the induced streaming potential of a pressure-driven symmetric electrolyte in a slit channel is investigated. Based on the non-isothermal Nernst–Planck equations, as well as the Poisson equation in the lubrication approximation, expressions for the ion distribution in the electric double layer (EDL) are derived. It is found that thermophoretic ion motion and a temperature-dependent electrophoretic ion mobility increase the local EDL thickness with temperature, whereas a temperature-dependent permittivity shrinks the EDL. Within the Debye–Hückel approximation, the Navier–Stokes equation with the corresponding electric body force terms is solved. Analytical expressions for the flow profile and the induced (streaming) field under non-isothermal conditions are derived. It is shown that for such a situation the induced electric field is the linear superposition of at least seven individual contributions. For very wide channels, only the thermoelectric field typically present in bulk electrolytes when subjected to a temperature gradient (Soret equilibrium) as well as the conventional pressure-induced streaming field are of importance. Counterintuitively, for the latter, while still being affected by the temperature dependence of the dielectric permittivity and local salt concentration, the temperature dependencies of the viscosity, Fickian diffusion coefficients and ion electromobilities exactly cancel each other. For narrow channels, five additional contributions become relevant, which – similar to the Soret voltage – do not vanish in the case that the externally applied pressure gradient is removed. The first is caused by selective thermo-electromigration driven by the interplay between the temperature-dependent electrophoretic ion mobility and the interaction of the ions with the surface wall charge. This non-advective effect is at its maximum under extreme confinement. For channels whose widths are of the same order as the EDL thickness, four thermoosmotic effects become significant. Besides the well-known thermoosmosis due to the temperature dependence of the dielectric permittivity in the (extended) Korteweg–Helmholtz force, it is demonstrated that – by contrast to isothermal conditions – a thermal gradient renders the ion cloud in the EDL out of mechanical equilibrium. In this context it is shown that a thermophoretic ion motion (i.e. the intrinsic Soret effect of the ions) and a temperature-dependent ion electromobility as well as a temperature-dependent permittivity not only cause an axial gradient of the EDL potential, but simultaneously lead to a pressure of thermal origin, which sets the fluid into an advective motion. Corresponding phenomena were not previously discussed in the literature and may be interpreted as an apparent, thermally induced slip velocity within the EDL. Subsequently, the ion advection affiliated with such thermoosmotic flow may induce a thermoelectric field of a similar order of magnitude to that caused by more conventional thermal effects.