Effect of ultrafast diffusion on adsorption, desorption, and reaction processes over heterogeneous surfaces

The effect of ultrafast diffusion is studied on reaction-diffusion processes in heterogeneous media, as encountered in catalysis and field emission microscopy. The reaction-diffusion equations describe adsorption, desorption, and reaction processes for two adspecies, for instance, on a paraboloidal heterogeneous surface in the presence of an external electric field. Using multiscale analysis, we show that the fast adspecies rapidly reaches a quasiequilibrium spatial distribution, characterized by a nonequilibrium chemical potential of the fast adspecies. An ordinary differential equation is derived for the time evolution of the nonequilibrium chemical potential. Numerical simulations are performed under different conditions, which confirm the convergence of the dynamics for finite diffusion toward the ultrafast diffusion limit predicted by our multiscale analysis. The numerical simulations also demonstrate that electric fields may induce important diffusive currents on heterogeneous surfaces under the conditions of field emission microscopy.

Figure 1

Schematic illustration of the discretization of a paraboloid in terms of $N$ rings. The middle of each ring $i$ is positioned at ${\eta }_i=i\Delta \eta$ (dashed lines), the apex being at $i=0$. Their boundaries are located at ${\eta }_{i+1/2}={\eta }_i+\Delta \eta /2$ (solid lines), they have the perimeters $P_{i+1/2}$, and they are crossed by the current or flux $I_{i,i+1}$ of adspecies defined by Eq. (A5). The distance between the middles of the rings $i$ and $i+1$ is denoted ${\Delta }_{i+1/2}$.

Figure 2

The partial coverage buildup curves for adspecies A on all the rings of a field emitter tip for the monomeric case ($m=1$) in the absence of both the $\mathrm{A}+$B reaction and an electric field, with an initial homogeneous distribution for adspecies B $({\theta }_{\mathrm{B}}=0.5$  ML). This plot includes curves obtained for $D_{\mathrm{A}}/R^2={10}^{\gamma }$ s${}^{-1}$, with $\gamma =1,2,3,4$, as well as the corresponding infinite-diffusion case. Notice that all the curves completely overlap. Parameters: $k_{\mathrm{a}}^{\mathrm{A}}=1$ s${}^{-1}$, $k_{\mathrm{d}}^{\mathrm{A}0}=1\times {10}^{10}$ s${}^{-1}$, $E_{\mathrm{d}}^{\mathrm{A}}=1$ eV, $T=350$ K and $P_{\mathrm{A}}(0)=100$ (in units setting $k_{\mathrm{a}}^{\mathrm{A}}=1$), ${\mu }_{\mathrm{A}}^{\mathrm{eq}}=-0.5$ eV, and ${\mu }_{\mathrm{A}}(t=0)=-1.5$ eV (see Model I in Table ).

Figure 3

The partial coverage buildup curves for adspecies A on rings $i=2,10,20,\dots ,100$ (ring $i=2$ curve at the bottom) for the monomeric case ($m=1$) in the absence of both the A $+$ B reaction and an electric field, with an inhomogeneous distribution for adspecies B given by Eq. (71). Similarly, as in Fig. 2, this plot exhibits overlapping curves for $D_{\mathrm{A}}/R^2={10}^{\gamma }$ s${}^{-1}$, with $\gamma =1,2,3,4$, as well as for $D_{\mathrm{A}}=\infty$. Same parameters as in Fig. 2 (Model I in Table ).

Figure 4

The partial coverage buildup curves for adspecies A at ring (a) $i=20$ and (c) $i=75$ for the dimeric case ($m=2$) in the absence of both the A $+$ B reaction and an electric field, with an inhomogeneous distribution for adspecies B as in Fig. 3. Curves obtained for $D_{\mathrm{A}}/R^2={10}^{\gamma }$ s${}^{-1}$, with $\gamma =1$ (dot), $\gamma =2$ (dash-dot), $\gamma =3$ (dash-dot-dot), $\gamma =4$ (dash), and for the quasiequilibrium distribution $(D_{\mathrm{A}}=\infty )$ (continuous line). Associated with plots (a) and (c), the evolution of the relative differences of partial coverage with respect to the quasiequilibrium distribution $[{\theta }_{\mathrm{A}}(D_{\mathrm{A}}=\infty )-{\theta }_{\mathrm{A}}(D_{\mathrm{A}})]/{\theta }_{\mathrm{A}}(D_{\mathrm{A}}=\infty )$: at (b) ring $i=20$ and at (d) ring $i=75$. Here we have considered ${\mu }_{\mathrm{A}}(t=0)=-1.4$ eV, and the other parameters as in Figs. 2 and 3 (see Model I in Table ).

Figure 5

For the same system as in Fig. 3 (monomeric case), in the presence of an electric field: at ring $i=20$ (a) the partial coverages and (b) the diffusion rates; at ring $i=75$ (c) the partial coverages and (d) the diffusion rates. Curves obtained for $D_{\mathrm{A}}/R^2={10}^{\gamma }$ s${}^{-1}$, with $\gamma =1$ (dot), $\gamma =2$ (dash-dot), $\gamma =3$ (dash-dot-dot), $\gamma =4$ (dash), and for the quasiequilibrium distribution $(D_{\mathrm{A}}=\infty )$ (continuous line). The electric field applied was $F=10$ V/nm, while $d_{\mathrm{d}}^{\mathrm{A}}=0.02$ eV nm ${\mathrm{V}}^{-1}$ and ${\alpha }_{\mathrm{A}}=1\times {10}^{-3}$ eV nm${}^2$ ${\mathrm{V}}^{-2}$ (Model II of Table ).

Figure 6

Convergence toward the infinite-diffusion limiting case for the same system as in Fig. 5 (monomeric case): (a) the partial coverages at ring $i=20$, (b) the relative differences of coverage evolution with respect to the quasiequilibrium distribution, (c) the difference of the generalized chemical potential (30) between the rings $i=20$ and $i=75$. Curves for different values of $D_{\mathrm{A}}/R^2={10}^{\gamma }$ s${}^{-1}$, as follows: $\gamma =1$ (gray dot), $\gamma =1.5$ (gray dot-dash), $\gamma =2$ (gray dash-dot-dot), $\gamma =2.5$ (gray dash), $\gamma =3$ (black dot), $\gamma =3.5$ (black dash-dot), $\gamma =4$ (black dash-dot-dot), $\gamma =4.5$ (black dash), $\gamma =5$ (black short dot), $\gamma =6$ (black short dash), and for the quasiequilibrium distribution $(D_{\mathrm{A}}=\infty )$ (continuous line) (Model II in Table ).

Figure 7

For the same system as in Figs. 5 and 6 (monomeric case), comparison for the quasiequilibrium solution using two different values of $w$ in Eq. (59) as indicated in the figure: (a) the partial coverages at ring $i=20$ and (b) the relative difference of partial coverages with respect to the quasiequilibrium distribution using the two different values of $w$. The continuous grey line in the upper panel is the quasiequilibrium solution with $w=1$, while the other line types have the same meaning as in Fig. 6 for the value of $D_{\mathrm{A}}/R^2$ that was used (Model II of Table ).

Figure 8

The partial coverage buildup curves of adspecies A at ring (a) $i=20$ and (c) $i=75$ for the same system as in Fig. 4 (dimeric case), but now in the presence of an electric field. The evolution of the relative differences of the partial coverages of adspecies A with respect to the infinite-diffusion case, at rings (b) $i=20$ and (d) $i=75$. Curves for different values of $D_{\mathrm{A}}/R^2$ as in Fig. 4. Here we have considered an external electric field of $F=10$ V/nm, while $d_{\mathrm{d}}^{\mathrm{A}}=0.02$ eV nm ${\mathrm{V}}^{-1}$ and ${\alpha }_{{\mathrm{A}}_2}=1\times {10}^{-3}$ eV nm${}^2$ ${\mathrm{V}}^{-2}$ (Model II of Table ).

Figure 9

Evolution of the diffusion rates corresponding to the processes illustrated in Figs. 4 and 8 (dimeric case): in the absence of an electric field, (a) at ring $i=20$ and (b) at ring $i=75$; in the presence of an electric field, (c) at ring $i=20$ and (d) at ring $i=75$. Curves for different values of $D_{\mathrm{A}}$ and $D_{\mathrm{A}}=\infty$ as in Figs. 4 and 8 (Model II of Table ).

Figure 10

Monomeric model ($m=1$) in the presence of an $\mathrm{A}+$B reaction, in absence of an electric field ($F=0$), and with the same inhomogeneous initial distribution (71) as in Fig. 3, for rings $i=10,20,\dots ,100$: (a) partial coverages (the curve at the bottom of the plot corresponds to the apex and the one at the top to the base of the tip), (b) adsorption rates [as in plot (a), from bottom to top, the adsorption rates of the rings between the apex and the lowest flank], and (c) reaction rates [contrary as in figures (a) and (b), from top to bottom, the rings between the apex and lowest flank]. Here we consider a reaction energy barrier $E_{\mathrm{r}}=0.7$ eV and corresponding prefactor $k_{\mathrm{r}}^0=1\times {10}^{10}$ s${}^{-1}$ (Model III in Table ).

Figure 11

Monomeric case ($m=1$) in presence of both an $\mathrm{A}+$B reaction and an electric field, with the same inhomogeneous initial distribution (71) as in Fig. 3: partial coverages at ring (a) $i=20$ and ring $i=75$; relative differences of the partial coverages with respect to the infinite diffusion limiting case at (b) ring $i=20$ and (d) ring $i=75$. Curves for different values $D_{\mathrm{A}}$ and $D_{\mathrm{A}}=\infty$ as in Figs. 4 and 8. The parameter values used are $F=10$ V/nm, $d_{\mathrm{d}}^{\mathrm{A}}=0.02$ eV nm ${\mathrm{V}}^{-1}$, ${\alpha }_{\mathrm{A}}=1\times {10}^{-3}$ eV nm${}^2$ ${\mathrm{V}}^{-2}$, and $d_{\mathrm{r}}=0.01$ eV nm ${\mathrm{V}}^{-1}$ (Model IV of Table ).

Figure 12

(a) Evolution of the adsorption rate corresponding to the coverage process depicted in Fig. 11a (ring $i=20$). (b) Relative differences of the adsorption rate relative to the infinite diffusion limiting case. Curves for different values of $D_{\mathrm{A}}$ as in Fig. 4.

Figure 13

For the same system as in Fig. 11, the partial coverage at ring $i=20$. Now with dipole values (a) $d_{\mathrm{r}}=0.012$ eV nm ${\mathrm{V}}^{-1}$ and (b) $d_{\mathrm{r}}=-0.002$ eV nm ${\mathrm{V}}^{-1}$. Curves for different values of $D_{\mathrm{A}}$ as in Fig. 4.