Predictive model for convective flows induced by surface reactivity contrast

Concentration gradients in a fluid adjacent to a reactive surface due to contrast in surface reactivity generate convective flows. These flows result from contributions by electro- and diffusio-osmotic phenomena. In this study, we have analyzed reactive patterns that release and consume protons, analogous to bimetallic catalytic conversion of peroxide. Similar systems have typically been studied using either scaling analysis to predict trends or costly numerical simulation. Here, we present a simple analytical model, bridging the gap in quantitative understanding between scaling relations and simulations, to predict the induced potentials and consequent velocities in such systems without the use of any fitting parameters. Our model is tested against direct numerical solutions to the coupled Poisson, Nernst-Planck, and Stokes equations. Predicted slip velocities from the model and simulations agree to within a factor of $\approx 2$ over a multiple order-of-magnitude change in the input parameters. Our analysis can be used to predict enhancement of mass transport and the resulting impact on overall catalytic conversion, and is also applicable to predicting the speed of catalytic nanomotors.

Figure 1

Schematic showing the model problem. The electrocatalytic surface reaction pair consists of gold (Au) and platinum (Pt) at a periodic width of $\lambda$ in contact with the electrolyte containing protons, hydroxide ions, hydrogen peroxide, and oxygen.

Figure 2

Field quantities for ${\text{Da}}_a=1.48, {\text{Da}}_c=2.68$. (a) $H^{+}$ concentration field. (b) ${\rho }_e$ field with electric field lines. (c) $\varphi$ field with electric field lines. (d) Velocity magnitude field with streamlines.

Figure 3

Electrostatic potential fields with electric field lines plotted for several values of the Damköhler numbers: (a) ${\text{Da}}_a=2.5, {\text{Da}}_c=.25,$ (b) ${\text{Da}}_a=.25, {\text{Da}}_c=.25,$ and (c) ${\text{Da}}_a=.025, {\text{Da}}_c=.25.$

Figure 4

Schematic showing conceptually our approximate model and how we evaluate it against the numerical simulations. The $\zeta$ potential just outside the double layer here shown as positively charged. $\zeta \left(x\right)$ is calculated as a sum of two terms, the far-field potential ${\varphi }_{\infty }$ and a component which varies along the surface ${\varphi }_{\tan }\left(x\right)$. Current in the bulk travels along semicircular lines which act as resistors. The model-predicted slip velocity $u_{\mathrm{slip}}$ and tangential electric field $E_{\tan }$ are compared against the maximum $x$ velocity and $x$ electric field along the line $x=0$.

Figure 5

Scaling of ${\varphi }_{\infty }$ and comparison of simulation with model (29). (a) Scaling with Damköhler ratio showing the predicted logarithmic scaling when ${\text{Da}}_a$ is small. (b) Scaling with ${\text{Da}}_a$ for two fixed values of ${\text{Da}}_a/{\text{Da}}_c$.

Figure 6

Scaling of tangential electric field with Damköhler numbers. Panel (a) shows that as predicted $E_{\tan }$ scales linearly with ${\text{Da}}_a$. Panel (b) shows the independence of $E_{\tan }$ from ${\text{Da}}_c$.

Figure 7

Comparison of maximum tangential velocity predicted by our model (dashed lines) with simulations (symbols). (a) Comparison for ${\text{Da}}_a/{\text{Da}}_c\ne 1$. (b) Comparison for ${\text{Da}}_a/{\text{Da}}_c=1$.