Transient anomalous diffusion regimes in reversible adsorbing systems

A solvable, minimal model of diffusion in the presence of a reversible adsorption site is investigated. We show that the diffusive particles are influenced by the adsorbing site on transient times when they have anomalous subdiffusive behavior. However, the particle dispersion law crosses over to the normal diffusive regime on asymptotically long times. The subdiffusive regime is characterized by a $t^{1/4}$ transient scaling with the same exponent as for the irreversible adsorption. On this transient time scale dominated by particle adsorption, there is a depletion of particles near the adsorbing site, and the typical width of the depletion zone grows in time as $t^{1/4}$ with the same exponent as the subdiffusive dispersion. We show that having a nonzero desorption probability for the adsorbed particles produces a crossover towards normal diffusion on time scales larger than a characteristic reactive time, which we show scales with diffusivity and the adsorption site reactivity.

Figure 1

Sketch of the minimal one-dimensional system of diffusion near an adsorption site, represented as a square symbol at $x=0$. Diffusing particles are depicted as points randomly scattered along the semi-infinite space $[0,\infty ]$. The concentration profile for an irreversible imperfect trap at a time $t=D/{\kappa }^2$ is superimposed on the drawing (solid red line) with an asymptotic concentration $c$ at $x\to \infty$. The average nearest distance to the adsorbing site $\langle L\rangle$ is drawn with a dashed line.

Figure 2

(a) Mean-square displacement $\langle \mathrm{\Delta }{x}^2\rangle$ and (b) time-dependent diffusivity $\tilde{D}\left(t\right)$ obtained with a reflective boundary condition (top long dashed bold line), a perfect trap (bottom long dashed bold line), an irreversible trap with $\kappa =1$ (bottom solid line), and a reversible trap with $\kappa =1$, $\mathrm{\Delta }=0.5\kappa$ (top solid line). The diffusive particle is initially located at $x_0=1$. The insets in (a) represent the velocity autocorrelation function (VACF) for every reactive boundary condition described above, in logarithmic scale for the temporal axis (top left inset), and in logarithmic scale for both axes (bottom right inset) for the absolute value of the VACF.

Figure 3

Probability density function $\varphi (L,t)$ obtained at different times $t=1$, 10, 100, and 1000 (see the indications next to each curve), for $\kappa =1$ and $\mathrm{\Delta }=0.9$. The inset gives the temporal evolution of the probability density function at $L=0$ in semilogarithmic scale for $t\in [{10}^{-3},{10}^5]$.

Figure 4

Mapping of the probability density function $\varphi (L,t)$ on the $(L,t)$ space, for a reversible adsorbing boundary with $\kappa =1$ and $\mathrm{\Delta }=0.9$. A logarithmic-scale axis is chosen for the temporal direction. Isoprobability contours are drawn with solid black lines. Orange disks point the extremal probability valley and ridge on each contour line.

Figure 5

Temporal evolution of the average nearest neighbor distance $\langle L\rangle$ to an adsorbing site in logarithmic scale for various desorption parameters, shown with symbols whose intensity is determined by $\mathrm{\Delta }$ (connected symbols), with a maximal value $\mathrm{\Delta }=\kappa$ corresponding to irreversible adsorption (disconnected symbols). The velocity value $\kappa =1$ is considered here, with $\mathrm{\Delta }$ values ranging in $[0.1,1]$, and a concentration $c=1$. The solid lines give the asymptotic temporal behavior of the average nearest neighbor distances to a reversible adsorbing site obtained in Eq. (21) characterized by $\kappa =1$, $\mathrm{\Delta }=0.9$ and $\kappa =1$, $\mathrm{\Delta }=0.99$. The inset gives the same curve obtained with the heuristic scaling laws presented in Fig. 6 with ${\langle L\rangle }^{*}=(\langle L\rangle \left(t^{*}\right)-1/c)/(max\left(\langle L\rangle \right)-1/c)$, and $t^{*}=t/t_{max}$, and for parameter values $\kappa =\{1,2\}$ and $\mathrm{\Delta }\in [0:\kappa ]$.

Figure 6

Dimensionless scaling functions $\sigma (\mathrm{\Delta }/\kappa )$ vs ${\sigma }^{\prime }(\mathrm{\Delta }/\kappa )$ obtained for $(\kappa ,\mathrm{\Delta })\in {[0.1,9]}^2$. The solid line shows the best fit of the form $\sigma (\mathrm{\Delta }/\kappa )=a{\left[{\sigma }^{\prime }(\mathrm{\Delta }/\kappa )\right]}^b$, with $a=1.14019$ and $b=0.188763$. Insets show the collapse of (a) $max(\langle L\rangle )$ and (b) $t_{max}$ obtained with the heuristic scalings in Eqs. (22) and (23).