We study dynamics of filling of an initially empty finite medium by diffusing particles $A$ and $B$, which arise on the surface upon dissociation of $AB$ molecules, impinging on it with a fixed flux density $I$, and desorb from it by the reaction $A + B\to AB\to 0$. We show that once the bulk diffusivities differ ($p=D_{ A}/D_{ B}<1$), there exists a critical flux density $I_{ c}(p)$, above which the relaxation dynamics to the steady state is qualitatively changed: on time dependencies of $c_{ As}/c_{ e}$ ($c_{ e}$ being the steady state concentration at $t\to \infty$) a maximum appears, the amplitude of which grows both with $I$ and with $D_{ B}/D_{ A}$ ratio. In the diffusion-controlled limit $I \gg I_{ c}$ at $p \ll 1$ the reaction "selects" the { \it universal laws} for the particles number growth ${ \cal N}_{ A}={ \cal N}_{ B}\propto t^{ 1/4}$ and the evolution of the surface concentrations $c_{ As}\propto t^{ -1/4},c_{ Bs}\propto t^{ 1/4}$, which are approached by one of the { \it two characteristic regimes} with the corresponding hierarchy of the intermediate power-law asymptotics. In the first of these $c_{ As}$ goes through a comparatively { \it sharp} max$(c_{ As}/c_{ e})\propto I^{ 1/6}$, the amplitude of which is $p$-independent, in the second one $c_{ As}$ goes through a { \itplateau-like} max$(c_{ As}/c_{ e})\propto p^{ -1/4}$, the amplitude of which is $I$-independent. We demonstrate that on the main filling stage the evolution of the ${ \cal N}(t)/{ \cal N}_{ e}, c_{ As}(t)/c_{ e},$ and $c_{ Bs}(t)/c_{ e}$ trajectories with changing $p$ or $J$ between the limiting regimes is unambiguously defined by the value of the scaling parameter ${ \cal K}=p^{ 3/2}J$ ($J$ being the reduced flux density) and is described by the set of { \it scaling laws}, which we study in detail analytically and numerically.
Categories: cond-mat.stat-mech cond-mat.soft