Investigation of Universal Behavior in Symmetric Diblock Copolymer Melts

Abstract

Coarse-grained theories of dense polymer liquids such as block copolymer melts predict a universal dependence of equilibrium properties on a few dimensionless parameters. For symmetric diblock copolymer melts, such theories predict a universal dependence on only $\chi_e N$ and $\Nbar$, where $\chi_e$ is an effective interaction parameter, $N$ is the degree of polymerization, and $\Nbar$ is a measure of overlap. This thesis focuses on testing the universal behavior hypothesis by comparing results for various properties obtained from different coarse-grained simulation models to each other. Specifically, results from pairs of simulations of different models that have been designed to have matched values of $\Nbar$ are compared over a range of values of $\chi N$. The use of vastly different simulation models allows us to cover a vast range of $\Nbar \simeq$ 200 - 8000 that includes most of the experimentally relevant range. Properties studied here include collective and single-chain correlations in the disordered phase, block and chain radii of gyration in the disordered phase, the value of $\chi_e N$ at the order-disorder transition (ODT), the free energy per chain, the latent heat of transition, the layer spacing, the composition profile, and compression modulus in the ordered phase. All results strongly support the universal scaling hypothesis, even for rather short chains, confirming that it is indeed possible to give an accurate universal description of simulation models that differ in many details. The underlying universality becomes apparent, however, only if data are analyzed using an adequate estimate of $\chi_e$, which we obtained by fitting the structure factor $S(q)$ in the disordered state to predictions of the recently developed renormalized one-loop (ROL) theory. The ROL theory is shown to provide an excellent description of the dependence of $S(q)$ on chain length and thermodynamic conditions for all models, even for very short chains, if we allow for the existence of a nonlinear dependence of the effective interaction parameter $\chi_e$ upon the strength of the $AB$ repulsion. The results show that behavior near the ODT exhibits a different character at moderate and high values of $\Nbar$, with a crossover near $\Nbar \simeq {10}^4$. Within the range $\Nbar \lesssim {10}^4$ studied in this work, the ordered and disordered phases near the ODT both contain strongly segregated domains of nearly pure $A$ and $B$, in contrast to the assumption of weak segregation underlying the Fredrickson–Helfand (FH) theory. In this regime, the FH theory is inaccurate and substantially underestimates the value of $\chi_e N$ at the ODT. Results for the highest values of $\Nbar$ studied here agree reasonably well with FH predictions, suggesting that the theory may be accurate for $\Nbar \gtrsim {10}^4$. Self-consistent field theory (SCFT) grossly underestimates ${\left(\chi_e N\right)}_{\mathrm{ODT}}$ for modest $\Nbar$ because it cannot describe strong correlations in the disordered phase. SCFT is found, however, to yield accurate predictions for several properties of the ordered lamellar phase. A detailed quantitative comparison of experimental results to theoretical predictions and obtained simulations results is also presented. Experimental results for structure factor obtained from small-angle neutron and X-ray scattering (SANS and SAXS) measurements are analyzed using methods closely analogous to those used to analyze simulation results. Peak scattering intensity results of different chain lengths of a $AB$ pair are fitted to the ROL theory predictions in order to estimate the effective interaction parameter $\chi_e(T)$ of the chemical system. The resulting $\chi_e(T)$ estimates are used to obtain ODT values $(\chi_e N)_{\mathrm{ODT}}$ of different experimental systems, which we compare to the scaling law obtained from simulation results and to theoretical predictions. The results are largely consistent with the expected systematic decrease with increasing $\Nbar$ and lie closer to the simulations scaling law than to any theoretical prediction. These results confirm the overwhelming importance of fluctuation effects in systems with modest values of $\Nbar = 10^{2} - 10^{3}$, and the usefulness of coarse-grained simulations as a starting point for quantitative modeling.