Stability of frozen waves in the modified Cahn-Hilliard model

We examine the existence and stability of frozen waves in diblock copolymers with local conservation of the order parameter, which are described by the modified Cahn-Hilliard model. It is shown that a range of stable waves exists and each can emerge from a “general” initial condition (not only the one with the lowest density of free energy). We discuss the implications of these results for the use of block copolymers in templating nanostructures.

Figure 1

The existence region of frozen waves (those described by one-wave solutions) on the $\left(\alpha ,\lambda \right)$ plane [$\alpha$ is the parameter in the modified Cahn-Hilliard equation (1), $\lambda$ is the wavelength]. The boundaries of the region are given by (2).

Figure 2

A schematic illustrating linear two-wave solutions. The solid curve [determined by (11) and (12)] corresponds to linear frozen waves. The lengths ${\lambda }_{1,2}$ of the two “composite” waves are shown by dots and are connected by dashed lines. Two-wave solutions exist only for those values of $\alpha$ for which ${\lambda }_2/{\lambda }_1$ is a rational number (presented in the figure above the corresponding ${\lambda }_2$). The wavelength of a two-wave solution as a whole equals the lowest common multiple of ${\lambda }_1$ and ${\lambda }_2$.

Figure 3

Regions of existence (bounded by the dotted line) of two-wave solutions with ${\lambda }_2/{\lambda }_1=2$$:$1. Thin solid lines show the boundaries of the existence region calculated under the assumption of weak nonlinearity. The thick solid line shows the existence region for one-wave solutions (as in Fig. 1).

Figure 4

Examples of frozen waves with spatial period $\lambda =16$: (a) one-wave solutions with $\alpha =0.005,0.050,0.100$ (curves 1, 2, and 3, respectively); (b) two-wave solutions with $\alpha =0.07,0.09,0.11$ (curves 1, 2, and 3, respectively).

Figure 5

The stability region of one-wave solutions on the $\left(\alpha ,\lambda \right)$ plane (shown by the dashed line). The thick solid line shows the existence region of one-wave solutions (as in Fig. 1). The upper panel represents a blow-up of the shaded region of the lower panel [the thin solid line shows the stability region's asymptotic boundary, (46)].

Figure 6

Evolution of perturbed steady states within the stable region at $\alpha =0.1$. The traces show the limiting ($t\to \infty$) solution initialized by (47) for various $\beta$. The traces are incremented in steps of $0.5$, beginning with zero increment for $\beta =0$ (the lowest trace) and ending for $\beta =1$ (the highest trace). The curves are marked with the corresponding values of $\beta$.

Figure 7

The dispersion relation $s\left(\theta \right)$ for the eigenvalue problem (8) and (9) with $\alpha =0.24$, $\lambda =8.886$ (solid line). The dotted line shows the limiting dispersion curve (A2). Note the difference in the vertical axes’ scales of the upper and lower panels.

Figure 8

Examples of eigenfunctions of problem (8) and (9) with $\alpha =0.24$, $\lambda =8.886$. The eigenfunctions marked with $U$ and $L$ correspond to the upper and lower branches, respectively, of the dispersion relation illustrated in Fig. 6.

Figure 9

The dispersion curves (upper branches) for $\alpha =0.24$ and (1) $\lambda =8.886$ (stability for all $\theta$), (2) $\lambda =9.786$ (instability for sufficiently small $\theta$).