Diffusiophoretic and diffusioosmotic velocities for mixtures of valence-asymmetric electrolytes

Diffusiophoresis and diffusioosmosis are electrokinetic phenomena where relative motion is induced between a charged surface and a surrounding electrolyte due to a concentration gradient of ions. In the literature, a relative velocity between a surface and the electrolyte has been derived for a valence-symmetric $\left(z\text{:}z\right)$ electrolyte. In this article, we reformulate the governing equations in a convenient form based on a systematic generalization of the nonlinear Poisson-Boltzmann equations in the limit of a thin double layer, which allows us to derive results for diffusiophoretic and diffusioosmotic velocities for a mixture of electrolytes with a general combination of cation and anion valences. We find that the relative motion depends significantly on ion valences. We also provide analytical approximations for the diffusiophoretic and diffusioosmotic velocities and discuss their accuracy and applicability. Further, we tabulate diffusiphoretic velocities for some common cases, which highlights the importance of asymmetry in cation and anion valences. Finally, we discuss the validity of our assumptions and the importance of effects such as finite ion size, dielectric decrement, and surface conduction for typical experimental conditions.

Figure 1

Sketch and notation for the derivation of the diffusioosmotic velocity. The valence of the $i\mathrm{th}$ ion is denoted as $z_i$, where $z_i>0$ for a cation and $z_i<0$ for an anion, here illustrated in shorthand by $\oplus$ and $\ominus$, respectively. The concentration gradient of ions in the $x$ direction induces a relative motion between the electrolyte and the surface. The surface charge density is assumed to be constant and is denoted as $q$ (the schematic shows the scenario of $q>0)$.

Figure 2

Approximate solutions for the diffusioosmotic velocity as compared to predictions calculated by numerical integration. The solid lines are the results of numerically integrating Eq. (28), circles are results from Eq. (30) with accuracy up to $O\left({\mathrm{\Psi }}_D^2\right)$, squares are evaluated through Eq. (30) with accuracy up to $O\left({\mathrm{\Psi }}_D^3\right)$, and triangles are generated from Eq. (31) with ${\mathrm{\Psi }}_{\ell }$ values given by Eq. (32). The parameters $z_{{}_{+}}=1,z_{{}_{-}}=-2,\beta =0.72$ correspond to ${\mathrm{H}}_2{\mathrm{SO}}_4,z_{{}_{+}}=1,z_{{}_{-}}=-2,\beta =0.08$ correspond to ${\mathrm{Na}}_2{\mathrm{SO}}_4$, and $z_{{}_{+}}=2,z_{{}_{-}}=-1,\beta =-0.34$ correspond to ${\mathrm{CaCl}}_2$; see Table 2.

Figure 3

Diffusioosmotic velocity for some common $z\text{:}z$ and $z_{{}_{+}}\text{:}{z}_{{}_{-}}$ electrolytes as given by Eq. (28).

Figure 4

Comparison of $U_{\mathrm{D}0}$ obtained from Eq. (31) with numerical solution of Eq. (28) for different values of ${\mathrm{\Psi }}_{\ell }$. Parameter values of $\beta =0.72,z_{{}_{+}}=1$ and $z_{{}_{-}}=-2$ (corresponding to ${\mathrm{H}}_2{\mathrm{SO}}_4)$ were used.