Universality of Block Copolymer Melts

Abstract

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...

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

“…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. This approximation of S RPA,0 (k) adds a contribution to χ e1 from long-wavelength fluctuations that would appear to the molecules as a uniform shift in the field, which thus should not affect their statistics.…”

“…Shift in the ODT relative to mean-field theory evaluated using χ e1 from eq 8 (blue triangles) and χ e2 from eq 13 (red triangles). Also included are the ODTs from a Langevin FTS 24 (green star) and particle simulations 18 (black circles). The dashed and solid curves denote the FH prediction in eq 1 and the empirical fit in eq 2, respectively.…”

“…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.…”

Communication: One size fits all: Equilibrating chemically different polymer liquids through universal long-wavelength description

“…In contrast, the implementation to nonconcatenated ring-polymers is challenging since coarse-graining into blobs relaxes microscopic non-crossability of subchains. It is important to explore whether similar strategies are applicable to materials with complex long-wavelength structure, depending on more control parameters, 13,14 e.g., microphaseseparated block-copolymers.…”

“…10-12N (γ) plays a key role in more complex systems, e.g., it controls in a universal way thermodynamic properties in symmetric blockcopolymers. 13,14 Let us consider different melts (indexed by α), having the sameN (l) as the lth library melt (see Fig. 1).…”