Time-dependent Ginzburg-Landau model for nonfrustrated linear $ABC$ triblock terpolymers

A time-dependent Ginzburg-Landau (TDGL) model is proposed to simulate the ordering of linear $ABC$ triblock terpolymers. The model, in its current form, is applicable to nonfrustrated triblock systems, with the specific condition that ${\chi }_{AC}\gg {\chi }_{AB}\approx {\chi }_{BC}$. Simulations are presented that demonstrate the model's ability to evolve a wide variety of morphologies throughout time, including tetragonal, core-shell hexagonal, three-phase lamellar, and beads-in-lamellar phases. The model also incorporates an interaction term to study templated substrates for directed self-assembly. The efficiency of the TDGL model enables large-scale simulations that allow investigation of self-assembly, and directed self-assembly, processes that may exhibit very small defect concentrations.

Figure 1

Triple-well free-energy surface as a function of ${\varphi }_A$ and ${\varphi }_C$. The three different energy barriers govern the interfacial energy of the three different binary interfaces. Note that the origin where ${\varphi }_A$ = ${\varphi }_C$ = 0 corresponds to the $B$ block on the $ABC$ triblock terpolymer.

Figure 2

Energy-density term $f_1({\varphi }_A,{\varphi }_C)$, expressed in Eq. (4), as a function of ${\varphi }_A$ for various values of $f_A$. The $x$ position of the saddle point is located at $f_A$, and the barrier height relative to the common tangent line is constant for different $f_A$ values. Also drawn is the Flory-Huggins energy assuming $\chi$ = 1.0 and $N$ = 60. The barrier height can be adjusted with ${\eta }_A$.

Figure 3

2D phase diagram for the proposed model considering an $AB$ diblock copolymer (i.e., $f_C$ = 0) for varying $f_A$ and ${\eta }_A$. Here, ${\eta }_A$ is a phenomenological parameter that is proportional to ${\chi }_{AB}N$. Small values of ${\eta }_A \le$ 1 result in homogeneous morphologies, and larger values of ${\eta }_A >$ 1 result in microphase separated patterns that depend on $f_A$.

Figure 4

Gibbs triangle for a ternary system. Each numbered circle corresponds to a composition that was simulated and displayed (in the same arrangement) in Fig. 5.

Figure 5

Simulated morphologies of $ABC$ triblock terpolymers obtained from the TDGL model developed herein. Each picture corresponds with a composition denoted in Fig. 4. The pictures along each of the three edges represent diblock copolymers (i.e., one of the blocks has zero monomer units). The green, blue, and yellow regions correspond with density distributions of $A, B$, and $C$ monomers, respectively.

Figure 6

Morphologies of a $f_A$ = 0.33, $f_B$ = 0.34, $f_C$ = 0.33 $ABC$ triblock terpolymer system that forms trilayer lamellae. The left-to-right images correspond to ${\eta }_A$ (and ${\eta }_C$) = 2.5, 3.2, and 4.0, respectively, each at simulation time $t$ = $1.5\times {10}^4$. Only the $A$ and $C$ domains are shown for clarity. The coarsening process for the ABC triblock terpolymer increases with decreasing ${\eta }_A$ and ${\eta }_C$ values, analogous to increasing temperature (or decreasing ${\chi }_{AB}$ and ${\chi }_{BC}$).

Figure 7

Morphologies of a $f_A$ = 0.4, $f_B$ = 0.4, $f_C$ = 0.2 $ABC$ triblock terpolymer system that forms a bead-in-lamellar morphology. All conditions, other than composition, are the same as Fig. 6.

Figure 8

Time evolution of a $f_A$ = 0.6, $f_B$ = 0.2, $f_C$ = 0.2 $ABC$ triblock terpolymer system that forms a core-shell hexagonal morphology at (a) $t$ = ${10}^3$ and (b) $t$ = $1.5\times {10}^4$ (using ${\eta }_A$ = ${\eta }_C$ = 4). This composition corresponds with number 38 in Fig. 4.

Figure 9

Time evolution of a $f_A$ = 0.2, $f_B$ = 0.6, $f_C$ = 0.2 ABC triblock terpolymer system that forms a tetragonal morphology at (a) $t={10}^3$ and (b) $t=1.5\times {10}^4$, (c) is a close-up of the inset in (b). This composition corresponds with number 12 in Fig. 4. The dashed lines indicate grain boundaries.

Figure 10

Directed self-assembly of a $f_A$ = 0.2, $f_B$ = 0.6, $f_C$ = 0.2 $ABC$ triblock terpolymer system with a square-array of posts indicated by the blue dots. The post spacing is $L_p=2L_o$, where $L_o$ is the lattice parameter of the tetragonal unit cell. The simulation time is $t$ = $1.5\times {10}^4$. Improved ordering is obtained, however, locally defected regions still exist in the domain.

Figure 11

Defect concentration versus time for a $f_A$ = 0.2, $f_B$ = 0.6, $f_C$ = 0.2 $ABC$ triblock terpolymer system, with an untemplated substrate and with the presence of posts ($L_p=2L_o$, as shown in Fig. 10). The untemplated system reduces the defect concentration in a two-stage manner, similar to that observed by Li et al. [33]. The presence of posts results in an initially higher defect concentration, followed by a more rapid decrease in defect concentration.