Universal Phenomenology of Symmetric Diblock Copolymers near the Order–Disorder Transition

Abstract

We present a simulation study of how properties of symmetric diblock copolymers depend on the invariant degree of polymerization N̅, focusing on the vicinity of the order–disorder transition (ODT). Results from several coarse-grained simulation models are combined to cover a range of N̅ ≃ 200–8000 that includes most of the experimentally relevant range. Results are presented for the free energy per chain, the value of χe N at the ODT, the latent heat of transition, the layer spacing, the composition profile, and compression modulus in the ordered phase. Universality (i.e., model independence) is demonstrated by showing that equivalent results for all properties are obtained from corresponding thermodynamic states of different simulation models. Corresponding states of symmetric copolymers are states with equal values of the parameters χe N and N̅, where χe is an effective Flory–Huggins interaction parameter and N is a degree of polymerization. The underlying universality becomes apparent, however, only if data are analyzed using an adequate estimate of χe, which we obtain by fitting the structure factor in the disordered state to recent theoretical predictions. The results show that behavior near the ODT exhibits a different character at moderate and high values of N̅, with a crossover near N̅ ≃ 104. Within the range N̅ ≲ 104 studied here, 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 χe N at the ODT. Results for the highest values of N̅ studied here agree reasonably well with FH predictions, suggesting that the theory may be accurate for N̅ ≳ 104. Self-consistent field theory (SCFT) grossly underestimates (χe N)ODT for modest N̅ 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.

Field-Theoretic Simulation of Block Copolymers at Experimentally Relevant Molecular Weights

“…The use of χ e2 also leads to good agreement with SCFT for the period and smectic compression modulus of well-ordered lamellae, consistent with recent particle-based simulations. 27 Furthermore, it results in good agreement with the empirical eq 2 for the ODT. 18 The difference between the two interaction parameters is that χ e2 in eq 13 involves an integral over the RPA structure function at χ = 0, S RPA,0 (k), while χ e1 in eq 8 uses its asymptotic k −2 behavior, eq 14.…”

“…30 The two definitions of χ e were tested for a range of N̅ by evaluating the disordered-state . 27 Dashed lines denote the FH predictions from eqs 20 and 21, and solid curves are simple fits to guide the eye. structure function, S(k), at different grid resolutions.…”

“…With these advances, we obtain a wealth of new results that match up nicely with those of recent particle-based simulations. 18,27 ■ THEORY Model and Simulation Method. Our study examines the standard Gaussian-chain model 4 for an incompressible melt of n AB diblock copolymers in a fixed volume, V = nN/ρ 0 = n/ρ c , where ρ c is the chain density.…”

“…Interestingly, similar consistency with SCFT was recently reported for particle-based simulations. 27 Order−Disorder Transition (ODT). Our scans of the lamellar period in the previous section extend to the point where the lamellar phase disorders, thereby providing us with estimates of χN and D at the ODT.…”

“…The only real exception was for h = 3, where {221} orientations were frequently observed, but this is because they possess the exact same domain spacing as the {300} ones. 27 Compression Modulus and Lamellar Profile. A convenient byproduct of the MC box move is the smectic (or layer) compression modulus B of the lamellae.…”