Instability and reorientation of block copolymer microstructure by imposed electric fields

The influence of electric fields on lamellar block copolymer microstructure is studied in the context of a density functional model and its sharp interface limit. A free boundary problem for domain interfaces of strongly segregated polymers is derived, which includes coupling of interface and electric field orientation. The linearized dynamics of lamellar configurations is computed in this context, leading to quantitative criteria for instability as a function of pattern wavelength, field magnitude, and orientation. Numerical simulations of the full model in two and three dimensions are used to study the nonlinear development of instabilities. In three dimensions, sufficiently large electric field magnitude always leads to instability. In two dimensions, the field has either stabilizing or destabilizing effects depending on the misorientation of the field and pattern. Even when linear instabilities are present, the dynamics can lead to stable corrugated domain interfaces which do not align with the electric field. Sufficiently high field strengths, on the other hand, produce topological rearrangement which may lead to alignment.

Figure 1

Neutral stability curves for wriggled and varicose perturbations in the absence of electric field ($\overline{\beta }=0$). The unstable region lies above the curves.

Figure 2

Neutral stability curve for wriggled modes. As the electric field parameter $\overline{\beta }$ increases, instability becomes more likely. Negative values $\overline{\beta }<0$ (possible only in two dimensions) have a stabilizing effect. The unstable region lies above the curves.

Figure 3

Neutral stability curve for varicose modes. As the electric field parameter $\overline{\beta }$ increases, instability becomes more likely. The unstable region lies above the curves for $\overline{\beta }\le 0.5$, whereas for large $\overline{\beta }$ is large enough, this region extends to small $L/L_{\mathrm{eq}}$ as well.

Figure 4

Nonlinear evolution of the electric-field induced instability of a two-dimensional lamellar film. Lateral increments at $t=0.15,0.20,0.25$, and 0.30 (from left to right) and vertical increments for the intensity of the electric field $\overline{\beta }=5,6.5,8$, and 9.5 (top to bottom).

Figure 5

Three-dimensional simulations of (5) subject to an electric field aligned along the $y$ axis. Left: Snapshot at time $t=400$ with $\overline{\beta }=0, \alpha =15$, and $\epsilon =0.1$. Right: Snapshot taken at $t=800$ and where field strength is $\overline{\beta }=10$, showing reorientation of the microstructure in the direction of the field.

Figure 6

Three-dimensional simulations of (5) subject to an electric field that alternates in the $x$ and $y$ direction. Top left: Initial lamellar state at time $t=0$ with $\overline{\beta }=15, \alpha =18$, and $\epsilon =0.1$. Top right: $t=25$, where the electric field is oriented along the $x$ axis. Bottom left: Time $t=50$, where the electric field is oriented along the $y$ axis. Bottom right: Time $t=75$ where the electric field is again oriented along the $x$ axis. Alternating the field direction produces a reoriented lamellar pattern.

Figure 7

Three-dimensional solution same as Fig. 6 but here the orientation of the electric field is at 20 degrees with respect to the initial lamellar orientation. The nonlinear evolution is captured for the different times $t=32,40,48,56$ from left to right and top to bottom.

Figure 8

Effects of finite $\epsilon$ on the instability of lamellar patterns. The top and bottom dashed lines correspond to the analytical sharp-interface prediction, and solid lines represent the numerically determined threshold values for various $\epsilon$.