Universality of Block Copolymer Melts

Simulations of five different coarse-grained models of symmetric diblock copolymers are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy and order-disorder transition (ODT) on the invariant degree of polymerization $\overline{N}$. The actual values of $\chi N$ at the ODT approach predictions of the Fredrickson-Helfand (FH) theory for $\overline{N}\gtrsim {10}^4$ but significantly exceed FH predictions at lower values characteristic of most experiments. The FH theory fails for modest $\overline{N}$ because the competing phases become strongly segregated near the ODT, violating an underlying assumption of weak segregation.

Figure 1

Plot of $g^{\prime }\equiv \partial g/\partial ({\chi }_{e}N)$ vs ${\chi }_{e}N$ for models $S1$-64 and $S2$-16 with matched $\overline{N}\simeq 960$. The dashed curve shows the SCFT prediction for $g^{\prime }({\chi }_{e}N)$. Vertical arrows mark the positions of the ODTs. Inset: Analogous plot using the simple linear approximation ${\chi }_e(\alpha )\simeq z\alpha /k_{B}T$.

Figure 2

Free energy per chain $g$ vs ${\chi }_{e}N$ at four different values of $\overline{N}$, plotted using a nonlinear approximation for ${\chi }_e(\alpha )$. Solid lines are SCFT predictions for $g({\chi }_{e}N)$. The straight solid line is the SCFT prediction $g({\chi }_{e}N)={\chi }_{e}N/4$ for a homogeneous phase. Vertical dotted lines show the SCFT ODT at ${\chi }_{e}N=10.495$. Vertical dashed lines show actual ODTs. Where data are shown for two systems, the ODT is shown for the system with larger $N$.

Figure 3

Values of ${\chi }_{e}N$ at the ODT vs $\overline{N}$, for all simulations. Bead-spring model results are shown as open symbols, with labels for specific systems. Lattice model results for $N=20$, 30, 40, 60, 90, 120, and 180 are filled gray circles. The short-dashed curve is the empirical fit in Eq. (3), the solid curve is the FH prediction, and the horizontal long-dashed line is the SCFT prediction. Inset: Analogous plot using the linear approximation ${\chi }_e^{(1)}(\alpha )\equiv z\alpha /k_{B}T$ to plot the ODT vs $\overline{N}$.

Figure 4

Maximum of the average volume fraction ${\varphi }_A(z)$ of $A$ monomers in the ordered phase at the ODT plotted vs $\overline{N}$ for all bead-spring simulations. The solid curve is the FH prediction. The horizontal dashed line is the bound ${\varphi }_A(z)=1$, above which the prediction becomes unphysical. Inset: ${\varphi }_A(z)$ for model $S1$-64, where the coordinate $z$ is normal to the layers and $L_z$ is the size of the simulation cell.