Making a Simple $A+B\to C$ Reaction Oscillate by Coupling to Hydrodynamic Effect

We present a new mechanism through which chemical oscillations and waves can be induced in batch conditions with a simple $A+B\to C$ reaction in the absence of any nonlinear chemical feedback or external trigger. Two reactants $A$ and $B$, initially separated in space, react upon diffusive contact and the product actively fuels in situ convective Marangoni flows by locally increasing the surface tension at the mixing interface. These flows combine in turn with the reaction-diffusion dynamics, inducing damped spatiotemporal oscillations of the chemical concentrations and the velocity field. By means of numerical simulations, we single out the detailed mechanism and minimal conditions for the onset of this periodic behavior. We show how the antagonistic coupling with buoyancy convection, due to concurrent chemically induced density changes, can control the oscillation properties, sustaining or suppressing this phenomenon depending on the relative strength of buoyancy- and surface-tension-driven forces. The oscillatory instability is characterized in the relevant parametric space spanned by the reactor height, the Marangoni (${\mathrm{Ma}}_i$) and the Rayleigh (${\mathrm{Ra}}_i$) numbers of the $i$th chemical species, the latter ruling the surface tension and buoyancy contributions to convection, respectively.

Figure 1

(a) Sketch of the $A+B\to C$ configuration used to generate chemohydrodynamic oscillations. The two reactants $A$ and $B$ have the same density ${\rho }_R$ and surface tension ${\gamma }_R$ while the formation of $C$ in the reactive zone can decrease locally the density ${\rho }_P\le {\rho }_R$ and increases the surface tension ${\gamma }_P>{\gamma }_R$. Typical initial topology of the chemically induced velocity field (b) in the presence of pure Marangoni flows and (c) when both buoyancy and surface tension forces are antagonistically at play. The black square in panel (c) indicates the stagnation point, $Sp$.

Figure 2

Chemo-Marangoni-driven oscillations in an $A+B\to C$ system ($\mathrm{\Delta }M=199.5$, $L_z=20$). (a),(c) Snapshots of the spatiotemporal evolution of the chemical ($c$) and hydrodynamic ($\psi$) fields, respectively, during two typical oscillatory cycles. The arrows at times 50–100–150 indicate in (a) the flow-driven compression relaxation compression of $c$ and in (c) the corresponding hydrodynamic morphology. (b),(d) Space-time plots ($L_x=256\times t=350$) of the dynamics, built along the white line traced in the first snapshot of panels (a) and (c). $c$ ranges between 0 (blue areas) and 0.5 (red areas), while $\psi$ varies between $-15$ (blue areas) and 15 (red areas). $T$ and $\lambda$ indicate the characteristic oscillation period and spatial length, respectively. (e) Time series of $c$ (black, left axis) and $\psi$ (gray, right axis) taken at ($x_0-20$, $L_z/2$). f) Power–law dependence of the oscillation period, $T$, and the spatial length, $\lambda$, on $L_z$ for different $\mathrm{\Delta }M$.

Figure 3

Chemo-Marangoni-buoyancy-driven oscillations ($\mathrm{\Delta }M=199.5$, $\mathrm{\Delta }R=1.75$, $L_z=20$). (a) Snapshots of the spatiotemporal evolution of $c$ during two typical oscillatory cycles. (b) Space-time plot ($L_x=256\times t=600$) of the dynamics, built along the white line traced in the first snapshot of panel (a). $c$ ranges between 0 (blue areas) and 0.5 (red areas). (c) Time series of $\psi$, taken at ($x_0-20$, $L_z/2$), for an oscillatory case (gray curve, $\mathrm{\Delta }M=199.5$, $\mathrm{\Delta }R=1.75$, $L_z=20$). The inset of panel (c) shows the evolution of the stagnation point, $Sp$, for a typical oscillatory (gray) and a nonoscillatory (black) dynamics ($L_c$ indicates the critical reactor height for the onset of oscillations in the corresponding Marangoni-driven case). (d) Power–law dependence of the oscillation period on $\mathrm{\Delta }R$ for various $\mathrm{\Delta }M\in (149.5,249.5)$.