Guiding fields for phase separation: Controlling Liesegang patterns

Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicrometer length scales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable pattern control based on the notion of driven precipitation, meaning that the phase separation is governed by a guiding field such as, for example, a temperature or pH field. The phase separation is modeled through a nonautonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front that separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a one-dimensional system with a diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity–local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields.

Figure 1

(Color online) Upper panel: Time evolution of the front position for $c_0=-0.05$ (solid line). The front moves diffusively for small $t$, $x_f\left(t\right)\sim \sqrt{t}$ (dotted line), while the large-time asymptote is given by Eq. (6) (dashed line). Lower panel: The velocity of the instability front $v_f$ as a function of the front position $x_f$ (solid line). The short-time asymptote $v_f\left(t\right)\sim 1∕\sqrt{t}$ (dotted line) and large-time behavior given by Eq. (7) (dashed line) are also displayed, with $P$ denoting the crossover point. The scaling of the spatial, temporal, and velocity variables is described in the text.

Figure 2

(Color online) Early stage of pattern formation: snapshots of the concentration field $c\left(x\right)$ (continuous line), guiding field $\epsilon \left(x\right)$ (dashed line), and spinodal lines $\pm \sqrt{\epsilon \left(x\right)∕3}$ (dash-dotted line). Dotted lines indicate the local equilibria $\pm \sqrt{\epsilon \left(x\right)}$. The big dot represents the position of the instability front. The parameters are $c_0=-0.05$, $L=1000$, $D=4$, and $t=4000$. The inset shows various possible profiles of the guiding field (see Sec. 6), namely, the usual diffusive configuration (dash-dotted line), a steplike profile (dashed line), and a rigid parabola (continuous line).

Figure 3

Numerical solution $c\left(x\right)$ of the nonautonomous Cahn-Hilliard equation (1) in the long-time limit $t\gg t_{\max }$, and for system size $L=4000$. Top panel: $c_0=-0.05$ and $D=1$; middle panel: $c_0=-0.05$ and $D=8$; and bottom panel: $c_0=-0.2$ and $D=1$.

Figure 4

(Color online) Position $x_n$ of the $n$th band with respect to its label $n$ for $c_0=-0.05$, $L=4000$, and different values of $D$.

Figure 5

(Color online) Distance ${\lambda }_n=x_{n+1}-x_n$ between two successive bands as a function of $n$, for $c_0=-0.05$, $L=4000$, and different values of $D$. One can notice outlier points at the beginning and ends of the curves. They correspond to the initial and final plug regions.

Figure 6

(Color online) Width $w_n$ of the $n$th band as a function of the band position $x_n$, for $c_0=-0.05$, $L=4000$, and different values of $D$. The vertical dashed line indicates the position of the crossover point $P$ (see Sec. 3). The jumps at the end points of the curves are due to the plugs.

Figure 7

(Color online) Comparison of the local wavelengths of the pattern as determined from numerical simulations (symbols) with the theoretical results based on Eq. (12) (continuous line). The wavelengths ${\lambda }_n$ and the band positions $x_n$ are rescaled as discussed in the text. Upper panel: $c_0=-0.05$ and $L∕D=1000$. Lower panel: $c_0=-0.05$ and $L∕D=125$. The vertical dashed lines indicate the position of the crossover point $P$ where the instability front has a minimal velocity (see Sec. 3). The outlier end points are due to the plugs.

Figure 8

(Color online) Local wavelengths of the pattern ${\lambda }_n$ versus band positions $x_n$, in appropriately rescaled variables. Symbols: numerical simulations for three different profiles of the guiding field (see the text). Continuous line: theoretical calculations based on Eq. (12). The parameters are $c_0=-0.05$, $D=16$, and $L=2000$. The vertical dashed line indicates the position, along the cylinder’s axis, of the crossover point $P$ where the instability front has a minimal velocity (see Sec. 3). The outliers at the beginning and end of the lines correspond to the plugs.