Alignment of Copolymer Morphology by Planar Step Elongation during Spinodal Self-Assembly

Using simulation and numerical self-consistent field theory of an unentangled diblock copolymer melt, we study the interplay between relaxation of molecular conformations from a highly stretched, nonequilibrium state and structure formation of the local, conserved density during self-assembly from a disordered state. We observe that the planar elongation of molecular conformations in the initial, disordered state results in an alignment of lamella normals perpendicular to the stretch direction during the subsequent self-assembly. Although thermodynamically the parallel orientation is favored, the alignment of the lamella normal perpendicular to the stretch direction is characterized by the larger growth rate of composition fluctuations during the spinodal ordering process.

Figure 1

Top: Time evolution of composition fluctuations in a slice perpendicular to the $x$ direction as obtained by MC simulation. The bottom image presents, for comparison, the fingerprint pattern (at $t/{\tau }_R=997$) obtained by quenching the system from $\chi N=0$ to 20 without initial deformation. Bottom: Time dependence of the collective structure factor along the stretch direction (lines) and perpendicular (symbols) to it, averaged over 32 independent simulation runs. The inset presents $S_{\text{coll}}(q_z)/N$ on a logarithmic scale to indicate the exponential growth for $t\le 2{\tau }_R$.

Figure 2

Relaxation of variance of Rouse modes $X_{py}^2$ along the stretch direction during structure formation. Similar to the disordered system, the individual modes relax exponentially with characteristic time ${\tau }_p={\tau }_R/p^2$. The right inset quantifies the time-dependent non-Gaussianity of the molecular conformations.

Figure 3

Free energy $\mathrm{\Delta }f$ vs lamellar spacing $L$ for molecular conformations with constraint variance of the first Rouse mode along the lamella normal $y$: Gaussian chains ${⟨X_{1y}^2⟩}_o=0.01715$ and non-Gaussian conformations ${⟨X_{1y}^2⟩}_o=1$ at $\chi N=20$. The equilibrium spacings are $L_0/R_{eo}\approx 1.67$ and 7.60, and the (base) chain extensions along the lamella normal are $R_{oy}/R_{eo}=1/\sqrt{3}$ and 4.0, respectively. The left inset depicts the negative, inverse structure factor $-N/S_{\text{coll}}$ for various values of ${⟨X_{1y}^2⟩}_o$ as indicated in the key.

Figure 4

Left: Illustration of parallel and perpendicular ordering. The arrows indicate the typical displacement necessary to form the lamellar structure. Right: Contour plot of the growth rate $R(\mathbf{q})\sim -\{{\mathbf{q}}^2{R_{eo}}^2/[S_{\text{coll}}(\mathbf{q})/N]\}$ of composition fluctuations in the stretch direction $q_y$ and the perpendicular direction $q_z$ at $\chi N=20$ and ${⟨X_{1y}^2⟩}_o=1$. Only wave vectors, for which composition fluctuations spontaneously grow, are shown.