Coarsening of precipitation patterns in a moving reaction-diffusion front

Precipitation patterns emerging in a two-dimensional moving front are investigated on the example of NaOH diffusing into a gel containing ${\text{AlCl}}_3$. The time evolution of the precipitate $\text{Al}{\left(\text{OH}\right)}_3$ can be observed since the precipitate redissolves in the excess outer electrolyte NaOH and thus it exists only in a narrow optically accessible region of the reaction front. The patterns display self-similar coarsening with a characteristic length $\xi$ increasing with time as $\xi \left(t\right)\sim \sqrt{t}$. A theory based on the Cahn-Hilliard phase-separation dynamics, including redissolution, is shown to yield agreement with the experiments.

Figure 1

(Color online) Time evolution of the precipitation pattern in the reaction zone for the samples with $\left[\text{NaOH}\right]=2.5\text{M}$ (outer electrolyte) and $\left[{\text{AlCl}}_3\right]=0.52\text{M}$ (inner electrolyte). The front moves perpendicularly to the plane of the picture and the pictures were taken at $t_1=180\text{s}$ (a), $t_2=480\text{s}$ (b), and $t_3=960\text{s}$ (c) after the initiation of the reaction. The length of the scale bars is 1 cm.

Figure 2

(Color online) Visual demonstration of the scaling in the evolution seen in Fig. 1. Parts of panels (a) and (b) are magnified by factors ${\left(t_3/t_1\right)}^{1/2}$ and ${\left(t_3/t_2\right)}^{1/2}$, respectively. The scale bars are the same length of 1 cm.

Figure 3

(Color online) Square of the characteristic length ${\xi }^2$ of the precipitation patterns plotted as a function of time $t$ for three inner electrolyte concentrations and at fixed outer electrolyte concentration $\left(a_0=2.5\text{M}\right)$. We calculated ${\xi }^2$ from the second moment of the structure factor of the patterns.

Figure 4

(Color online) Simulation results for the spatial distribution of C integrated along the $z$ direction. The parameters used were $b_0/a_0=0.18$, $c_h/a_0=0.2$, $c_l/a_0=0$, $D={10}^{-9}{\text{m}}^2{\text{s}}^{-1}$, $k_1=8.6{\text{M}}^{-1}{\text{s}}^{-1}$, $k_2=0.13{\text{M}}^{-1}{\text{s}}^{-1}$, $\lambda =0.83\times {10}^{-9}{\text{m}}^2{\text{s}}^{-1}$, $\sigma ={10}^{-6}{\text{m}}^2$, the grid spacing, and the time step were 0.01 mm and 0.003 s, respectively. The snapshots were taken at 75 s, 600 s, and 1200 s.

Figure 5

(Color online) Simulation results demonstrating the diffusive growth ${\xi }^2\sim t$ of the characteristic length scale $\xi$.