Diffusioosmosis of electrolyte solutions in uniformly charged channels

TL;DR

This paper develops a semi-analytical theory for diffusio-osmotic flow in thick planar channels with constant surface charge and thin EDLs, deriving a compact, cross-section-averaged flow-rate relation $\mathcal{Q}$ that depends primarily on the Gouy-Chapman length $\ell_{GC}$ and the end-point salinity $c_1$. By solving the coupled Nernst-Planck, Poisson, and Stokes equations in the thick-channel limit, it links the bulk concentration $c_m(x)$ and midplane potential $\psi_m(x)$ to the flow, yielding nonlinear concentration profiles, nonuniform and sometimes alternating slip velocity $u_s(x)$, and a pressure gradient that ensures a uniform $\mathcal{Q}$ across the channel. The analysis reveals two key mechanisms for vanishing $\mathcal{Q}$ (zero surface potential or cancelation between electroosmotic and chemiosmotic contributions) and shows that two distinct $\sigma$ values can produce the same $\mathcal{Q}$, with the ion flux $\mathcal{J}$ coupling to $\mathcal{Q}$ and $c_m(x)$. The results provide compact analytical tools for interpreting diffusio-osmotic experiments, connect to diffusiophoresis and membrane electrochemistry, and offer a framework for testing numerical approaches and extending to related systems.

Abstract

When the concentration of electrolyte solution varies along the channel the forces arise that drag the fluid toward the higher or lower concentration region inducing a flow termed diffusio-osmotic. This article investigates a flow that emerges in channels with constant density of surface charge σ and thin compared to their thickness electrostatic diffuse layers. An equation for the fluid flow rate Q is derived and used to describe analytically the flux of ions, and local potentials and concentrations. This equation, which allows to treat the diffusio-osmotic problems without tedious and time consuming computations, clarifies that the global flow rate is controlled only by the surface charge and concentration drop between the channel ends, and indicates that there always exist two different values of σ that correspond to a particular Q. Our theory provides a simple explanation of the directions of the fluid flow rate and ionic flux depending on the surface charge and diffusivity of ions, predicts a non-linear concentration distribution along the channel caused by convection, and relates it to the local potential changes by a compact formula. We also present and interpret the variations of the diffusio-osmotic velocity profiles and the apparent slip velocity along the channel and show that the latter is highly non-uniform and could even becomes alternating. The relevance of our results for diffusio-osmotic experiments and for some electrochemistry and membrane science issues is discussed briefly.

Figures (12)

• Figure 1: Sketch of the microchannel of thickness $H$, length $L \gg H$ and constant surface charge density $\sigma$ that connects the "fresh" (left) and "salty" (right) bulk electrolyte reservoirs of concentrations $C_{0}$ and $C_{1}$. The extension of electrostatic diffuse layers, which is of the order of the Debye length $\lambda_D \ll H$ and takes its upper value $\lambda_D = \lambda_D^{\star}$ at $X=0$ by reducing along the channel. The electroneutral central area is of "bulk" concentration $C_m$ and potential $\Psi_m$, both depend on $X$.
• Figure 2: Surface potential $\phi_s$ as a function of $c_m$ computed using $\ell_{GC} = 1$ and 10 nm (solid curves from top to bottom) for $\mathcal{C}_{0} = 10^{-3}$ mol/l and $\mathcal{C}_{1} = 1$ mol/l. Open and filled circles show calculations from Eqs. \ref{['fs_DB']} and \ref{['fs_DB2']}.
• Figure 3: Electroosmotic mobility profiles computed for cross-sections of $\mathcal{C}_m = 4 \times 10^{-3}$ and $0.2$ mol/l (solid curves from top to bottom) for the upper curve in Fig. \ref{['fig:pot-charge']}. Circles show $m_e$ calculated from Eq. \ref{['ueo']} using \ref{['eq:pot-charge_hs']} and \ref{['eq:PBSWexact']}.
• Figure 4: Chemiosmotic mobility $m_{c}$ as a function of $z$ computed with the same parameters as in Fig. \ref{['fig:pot-profile']}. Filled and open circles are calculations from Eqs. \ref{['udo2']} and \ref{['mdo_sm']}.
• Figure 5: $\mathcal{Q}$ as a function of $\ell_{GC}$ computed using $c_1 = 10^3$ and 10 (solid and dashed curves) for KCH$_{3}$COO [$\beta = 0.286$] (a) and NaCl [$\beta = -0.208$] (b). Circles show calculations from Eq. \ref{['eq:q_lin']}.