Noise-induced kinetic model for autonomous motion of the contact line in oil-water systems with chemical reactions

A model for autonomous motion of the contact line of the oil-water interface along a solid surface is proposed. The present model is inspired by the spontaneous wave generation and the contact line motion of an oil-water interface composed of cationic surfactant and oil-soluble anions. The motion is created through wetting by an adsorption of surfactant followed by an autocatalytic process with a chemical reaction and also dewetting due to the desorption of the remaining monolayer. The wetting process is accelerated by the contact line motion itself through the convection-enhanced transport of reaction constituents, which is autocatalytic in nature. These processes are expressed by nonlinear time-evolution equations for the velocity and the amplitude. Following the model, the autonomous motion is essentially excitable, and hence the spatiotemporal pattern is of the noise-induced type. The present model explains well the diverse features of the experimental results with a few parameters.

Figure 1

Schematic representation of contact line motion. (a) Each process constituting a periodic change, (b) microscopic view of chemical desorption of outer leaflet.

Figure 2

(Color online) Calculated $u\left(t\right)$ [upward and downward spikes (dark curve) on the left-hand side], $v\left(t\right)$ [upward spikes (light curve) on the left-hand side], and $\text{cos}\theta \left(t\right)$ (right-hand column) are shown for four kinds of parameters: (a) $\mu =10$, $\chi =1$, $\kappa =20$; (b) $\mu =10$, $\chi =5$, $\kappa =20$; (c) $\mu =10$, $\chi =1$, $\kappa =15$; (d) $\mu =5$, $\chi =1$, $\kappa =20$. Other parameters are $k=1$ and $-{10}^{-5}<\xi <{10}^{-5}$ for all cases. Quantities of the abscissa and the ordinate are dimensionless.

Figure 3

(Color online) Dependency of calculated $u\left(t\right)$ [upward and downward spikes (dark curve) on the left-hand side], $v\left(t\right)$ [upward spikes (light curve) on the left-hand side], and $\text{cos}\theta \left(t\right)$ (right-hand column) on the reaction rate constant, $k=0.5$ (upper) and $k=2$ (lower). Other parameters are $\mu =10$, $\chi =1$, $\kappa =20$, $-{10}^{-5}<\xi <{10}^{-5}$. Quantities of the abscissa and the ordinate are dimensionless.

Figure 4

Effect of noise intensity on pattern of $v\left(t\right)$ and $\text{cos}\theta \left(t\right)$: $-0.1<\xi <0.1$ (a), $-1.0<\xi <1.0$ (b), and $-3.0<\xi <3.0$ (c). Other parameters are $\mu =10$, $\chi =1$, $\kappa =20$, and $k=1$. Quantities of the abscissa and the ordinate are dimensionless.

Figure 5

Experimentally observed $\gamma \text{cos}\theta$ (Ref. [12]). Concentrations of ${\text{C}}_{18}\text{TAC}$, bis(2ethylhexyl)phosphoric acid (DEHPA), and phenylboronic acid (PB) are $1.15\mathrm{m}M$, $50\mathrm{m}M$, and $0\mathrm{m}M$, respectively, for $\text{DEHPA}:\text{PB}=50:0$. They are $1.15\mathrm{m}M$, $25\mathrm{m}M$, and $25\mathrm{m}M$, respectively, for $\text{DEHPA}:\text{PB}=50:50$.

Figure 6

Time course of distance from a fixed point to the droplet periphery, $z\left(t\right)$ as shown in the inset. Concentrations of ${\text{C}}_{18}\text{TAC}$ and DEHPA are $1.15\mathrm{m}M$ and $100\mathrm{m}M$, respectively. Inset is a schematic illustration of change in the droplet shape. The droplet $\left(100\mu \text{l}\right)$ changes its shape without a significant translation. The details are reported elsewhere [12].

Figure 7

Pattern of traveling wave simply estimated by the $v\left(t\right)$ value. The abscissa is simply estimated by the time (the abscissa of Fig. 2 or Fig. 3) multiplied by the traveling velocity. Since we assume a constant traveling velocity, the time can be equal to the distance in an appropriate unit. Quantities of the abscissa and the ordinate are dimensionless. (a) Pattern of $v\left(t\right)$ for parameters shown in Fig. 2. (b) Parameters are chosen for reproducing the experimental result of Fig. 4 in Ref. [4].