Defect Removal in the Course of Directed Self-Assembly is Facilitated in the Vicinity of the Order-Disorder Transition

The stability of prototypical defect morphologies in thin films of symmetric diblock copolymers on chemically patterned substrates is investigated by self-consistent field theory. The excess free energy of defects and barriers of defect-removal mechanisms are obtained by computing the minimum free-energy path. Distinct defect-removal mechanisms are illustrated demonstrating that (i) defects will become unstable at a characteristic value of incompatibility $\chi N_{*}$ above the order-disorder transition and (ii) the kinetics is accelerated at weak segregation. Numerical findings are placed in the context of physical mechanisms, and implications for directed self-assembly are discussed.

Figure 1

(a) Minimum free-energy path (2D-MFEP) between defect, $\alpha =0$, and lamellar structure, $\alpha =1$, without guiding pattern. The dimensionless free-energy difference $\mathrm{\Delta }f=R_{e0}\mathrm{\Delta }{F}_b/D{k}_{B}T\sqrt{\overline{\mathcal{N}}}$ per film thickness as a function of the contour parameter $\alpha$ along the 2D-MFEP path is shown. The excess free energy $\mathrm{\Delta }{F}_d$ and the highest barrier $\mathrm{\Delta }{F}_b$ are indicated for incompatibility $\chi N=28$. Additional, characteristic stages along 2D-MFEP are indicated by the numbered arrows, and the corresponding intermediate morphologies [isosurfaces of $A$ density of the central portion indicated by the red box in the morphology $\alpha =0$ of panel (a)] are shown in panels (b1) $\alpha \approx 0.250$, (b2) $\alpha \approx 0.428$, (b3) $\alpha \approx 0.577$, and (b4) $\alpha \approx 0.710$.

Figure 2

(a) MFEP2 paths for $\chi N=30$ and ${\mathrm{\Lambda }}_{0}N=0$ and 0.11. The inset enlarges the region near the barrier for ${\mathrm{\Lambda }}_{0}N=0.11$. The alternate ordinate shows the absolute free energy for $\sqrt{\overline{\mathcal{N}}}=128$. [(b1), (b2), and (b3)] Three intermediate morphologies for ${\mathrm{\Lambda }}_{0}N=0.11$ located at the barrier, $\alpha \approx 0.059$, and the two shoulders, $\alpha \approx 0.318$ and $\alpha \approx 0.420$, respectively. The morphology has been twice cleaved along an $xz$ plane [indicated by the blue lines in the morphology of panel (a)] to show the cross section of the $A$ connections.

Figure 3

Defect free energy $\mathrm{\Delta }{f}_d=R_{e0}\mathrm{\Delta }{F}_d/D{k}_{B}T\sqrt{\overline{\mathcal{N}}}$ (scaled by $1/8$) and free-energy barrier $\mathrm{\Delta }{f}_b=R_{e0}\mathrm{\Delta }{F}_b/D{k}_{B}T\sqrt{\overline{\mathcal{N}}}$, for different mechanisms as a function of $\chi N$. The inset presents the dependence of $\mathrm{\Delta }{f}_b$ of MFEP2 on ${\mathrm{\Lambda }}_{0}N$ at $\chi N=30$.

Figure 4

(a) MFEP for the removal of two apposing dislocations at $\chi N=30$. [(b1)–(b3)] Three intermediate morphologies located at $\alpha \approx 0.656$, $\alpha \approx 0.812$, and $\alpha \approx 0.844$, respectively.