Simulations of five different coarse-grained models of symmetric diblock copolymer melts are compared to demonstrate a universal (i. e., model-independent) dependence of the free energy on the invariant degree of polymerization N , and to study universal properties of the order-disorder transition (ODT). The ODT appears to exhibit two regimes: Systems of very long chains (N > ∼ 10 4 ) are well described by the Fredrickson-Helfand theory, which assumes weak segregation near the ODT. Systems of smaller but experimentally relevant values, N < ∼ 10 4 , undergo a transition between strongly segregated disordered and lamellar phases that, though universal, is not adequately described by any existing theory.PACS numbers: 82.35. Jk,64.70.km,64.60.De Universality is a powerful feature of polymer statistical mechanics that allows the behavior of real systems to be predicted on the basis of simple generic models and scaling arguments. The paradigmatic example is the scaling theory of dilute and semidilute polymer solutions in good solvents [1][2][3], which predicts a universal dependence of all properties on two thermodynamic state parameters (an excluded volume parameter and an overlap parameter). Historically, this scaling hypothesis was verified by comparing experiments on diverse chemical systems with varied chain lengths and concentrations [3][4][5]. Here, we compare simulations of diverse models to verify an analogous scaling hypothesis about the equation of state and order-disorder transition (ODT) of symmetric diblock copolymers, and to characterize this transition.We consider a dense liquid of AB diblock copolymers, with N monomers per chain, and a fraction f A of A monomers. We focus on the symmetric case, f A = 1/2. Self-consistent field theory (SCFT) is the dominant theoretical approach for block copolymers [6][7][8]. SCFT describes polymers as random walks with a monomer statistical segment length b, which we take to be equal for A and B monomers. The free energy cost of contact between A and B monomers is characterized by an effective Flory-Huggins interaction parameter χ e . Let g denote a dimensionless excess free energy per chain, normalized by the thermal energy k B T . SCFT predicts a free energy g for each phase that depends only upon f A and the product χ e N , or upon χ e N alone for f A = 1/2. This yields a predicted phase diagram [6, 7] that likewise depends only on f A and χ e N . For f A = 1/2, SCFT predicts a transition between the disordered phase and lamellar phase at (χ e N ) ODT = 10.495.SCFT is believed to be exact in the limit of infinitely long, strongly interpenetrating polymers [9, 10]. The degree of interpenetration in a polymer liquid is characterized by a dimensionless concentration C ≡ cR 3 /N , in which c is monomer concentration, c/N is molecule concentration, and R = √ N b is coil size. Alternatively, interpenetration may be characterized by the invariant degree of polymerization. A series of post-SCF theories [10][11][12][13][14][15][16][17][18], starting with the Fredrickson...

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.

We present a detailed comparison of simulations of disordered melts of symmetric AB diblock copolymers to predictions of the renormalized one-loop (ROL) theory. The behaviors of the structure factor S(q) and of single-chain correlations are studied over a range of chain lengths (N = 16, ..., 128) for two models: one with harshly repulsive pair interactions and another with very soft interactions. The ROL theory is shown to provide an excellent description of the dependence of S(q) on chain length and thermodynamic conditions for both models, even for very short chains, if we allow for the existence of a nonlinear dependence of the effective interaction parameter χ e upon the strength of the AB repulsion. The decrease in peak wavenumber q* with increasing χ e is shown to be unrelated to changes in single-chain correlations. Results for all quantities are consistent with the hypothesis that the ROL theory gives an exact description of the dominant ̅corrections to RPA and random-walk predictions in the limit of infinite chain length N.

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 χN and N , where χ is an effective interaction parameter, N is a degree of polymerization, and N is a measure of overlap. We test whether simulation results for the structure factor S(q) obtained from several different simulation models are consistent with this two-parameter scaling hypothesis. We compare results from three models: (1) a lattice Monte Carlo model, the bond-fluctuation model, (2) a bead-spring model with harsh repulsive interactions, similar to that of Kremer and Grest, and(3) a bead-spring model with very soft repulsion between beads, and strongly overlapping beads.We compare results from pairs of simulations of different models that have been designed to have matched values of N , over a range of values of χN and N , and devise methods to test the scaling hypothesis without relying on any prediction for how the phenomenological interaction parameter χ depends on more microscopic parameters. The results strongly support the 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.

We present a detailed quantitative comparison of experimental results and theoretical predictions for the structure and thermodynamics of low molecular weight symmetric ( f L ≈ 1/2) poly(1,4-isoprene-b-DL-lactide) (IL) diblock copolymers near the order−disorder transition (ODT). Small-angle neutron and X-ray scattering (SANS and SAXS) measurements obtained in the disordered phase with IL degree of polymerization N = 39 were fit to the renormalized one-loop (ROL) theory in order to estimate the effective interaction parameter χ e (T). Calorimetric measurements of the latent heat of the ODT for the same copolymer compare well with that obtained from corresponding coarse-grained simulations, when the comparison is based on this estimate of χ e (T). The corresponding estimate of (χ e N) ODT at the experimental ODT of this polymer is much closer to the value obtained from simulations than to any theoretical prediction but differs from the simulation result by somewhat more than the bounds implied by experimental uncertainties. A larger discrepancy between simulation and experimental results for (χ e N) ODT is obtained for longer chains, with N ≥ 50, when also calculated using χ e (T, N = 39). We discuss possible reasons for this discrepancy, including the possibility of a significant end-group effect for these polymers. These results confirm the overwhelming importance of fluctuation effects in short diblock copolymers, and the usefulness of coarsegrained simulations as a starting point for quantitative modeling, but also indicate the need for attention to nonuniversal features of specific polymers that can also become more important with decreasing chain length.