Self-Consistent Field Theory for Melts of Low-Molecular-Weight Diblock Copolymer

This paper applies self-consistent field theory (SCFT) to discrete polymer chains consisting of a finite number of beads, N, joined together by freely jointed bonds of arbitrary potential, b(R). The numerics of this SCFT can be performed efficiently using spectral or pseudospectral algorithms, permitting its application to complex morphologies. To demonstrate its effectiveness, we examine diblock copolymer melts where the polymer bonds have a fixed length, a, and the nonbonded interactions have a finite range, σ, with a strength controlled by the standard Flory−Huggins χ parameter. Although the results reduce to those of the usual SCFT for Gaussian chains in the limit of large N and small χ, there are some notable differences for short chains with strong interactions. The most significant involves the internal interfaces, which in turn affects the size of the domains. Furthermore, the finite range of the nonbonded interactions, necessary to properly treat the internal interfaces, causes a noticeable shift of the ODT toward larger χN. As χ becomes very large, particularly at small N, the finite extensibility of the freely jointed chains restricts the size of the domains, which leads to a preference for the lamellar phase. ■ INTRODUCTION Self-consistent field theory (SCFT) has proven to be a remarkable theory for predicting the equilibrium behavior of structured polymers. This has been well demonstrated by its success regarding diblock copolymer melts. Vavasour and Whitmore produced the first SCFT phase diagram, but it was limited to the classical lamellar (L), cylindrical (C) and bcc spherical (S) phases. Matsen and Schick then extended it to include complex phases, predicting the gyroid (G) phase to be more stable than the perforated-lamellar (PL) phase as confirmed later by experiment. In a subsequent calculation by Matsen and Bates, a narrow closed-packed spherical (Scp) phase was predicted along the order−disorder transition (ODT), which has since been associated with a region of densely packed spherical micelles. Most recently, the Fddd (O) phase was predicted by Tyler and Morse and later observed in experiment. The standard SCFT applies mean-field theory to coarsegrained Gaussian chains, for which the statistical mechanics of a single chain in external fields is evaluated by solving a simple modified diffusion equation in three-dimensional space. This simplicity results because the polymers are treated as ideal elastic threads, where the elasticity accounts for the entropy associated with the microscopic degrees of freedom integrated out of the system by the coarse graining. However, approximating polymers by elastic threads results in a number of unphysical properties such as an unbounded end-to-end length of the molecules and a diverging entropic penalty for narrow interfaces. As such, the Gaussian chain model must be limited to situations to where the average end-to-end length of a polymer is much shorter than its contour length and where the local environment changes slowly on the monomer scale. These conditions are satisfied by polymers of high molecular weight, but not necessarily by ones of low molecular weight. These limitations can, in principle, be overcome by applying SCFT to the worm-like chain model, where the polymers are treated as semiflexible threads of fixed contour length. While the worm-like chain reduces to the Gaussian chain as its flexibility increases, the unphysical behavior described above is avoided so long as the rigidity of the worm-like chain remains finite. Unfortunately, the statistical mechanics of a worm-like chain in external fields involves a much more complicated diffusion equation that couples the orientation of the chain to spatial variations in the field. For simple problems where the field only varies in one direction such as the lamellar phase, the diffusion equation involves just two coordinates (one spatial and one orientational) and thus it can be readily solved. In the general case, however, there are three spatial and two orientational coordinates, resulting in a five-dimensional diffusion equation that has so far remained numerically intractable. Another alternative is to apply SCFT to discrete chains involving a finite number of beads joined together by freely jointed bonds. In this case, the diffusion equation is replaced by an iterative equation involving an integration over the three Received: August 26, 2012 Revised: October 4, 2012 Published: October 11, 2012 Article pubs.acs.org/Macromolecules © 2012 American Chemical Society 8502 dx.doi.org/10.1021/ma301788q | Macromolecules 2012, 45, 8502−8509 spatial coordinates. Although the dimensionality of the problem does not increase because the orientations of the bonds are not coupled, the triple integral is difficult to evaluate accurately in real space. Consequently, this model has so far been restricted to systems in which the fields vary in only one direction. However, we will demonstrate that if the numerics are solved using spectral or pseudospectral methods, then the SCFT of freely joined chains becomes equally efficient to that of Gaussian chains. This then allows us to examine the effects of finite molecular weight on the full range of periodic morphologies exhibited by diblock copolymer melts. ■ THEORY Here we formulate the freely jointed chain version of SCFT for a melt of n AB diblock copolymers, each with an A-block of NA monomers joined to a B-block of NB monomers giving a total polymerization of N ≡ NA + NB and an A-monomer composition of f = NA/N. We assume a uniform monomer density, ρ0, such that the total volume of the melt is V = nN/ρ0. The monomers are treated as featureless beads connected by a bonded potential, b(R). The natural length of a bond, a, is given by ∫ ≡ a R g R R ( ) d 2 2