Geometric Theory of Diblock Copolymer Phases

Abstract

We analyze the energetics of sphere-like micellar phases in diblock copolymers in terms of wellstudied, geometric quantities for their lattices. We argue that the A15 lattice with P m3n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the A15 phase is stable.The ability to control the self-assembly of complex lattices by manipulating molecular architecture remains an essential aspect in the creation of new, functional materials. With only a few tunable parameters, diblock copolymer melts exhibit a wide variety of equilibrium phases which can be understood via the mean-field Gaussian chain model of "AB" diblock copolymers composed of immiscible A and B blocks [1,2]. Indeed, in a system where the A and B-blocks are otherwise identical, there are only two thermodynamic variables, φ, the volume fraction of A type monomers, and χN , where χ is the Flory-Huggins parameter characterizing the repulsive interactions between the A-and B-type monomers and N is the degree of polymerization [3]. In this letter, we present a model which predicts that the the A15 (shown in Fig. 1a) lattice of diblocks is stable relative to other sphere-like phases for sufficiently large φ or, in other words, sufficiently symmetric diblocks. We corroborate this prediction by recalculating the phase diagram for symmetric diblocks (Fig. 1b) via a self-consistent field theory (SCFT) for diblock copolymer melts [4].The "classical" diblock phases are well-understood: near the order-disorder transition (ODT), Leibler developed a Landau-like theory in the weak-segregation regime to establish the stability of a body-centered cubic (BCC) phase, a hexagonal phase of columns and a lamellar phase [5]. Moreover, Semenov's picture of spherical micelles interacting through a disordered copolymer background when φ ≪ 1 accounts for the appearance of the facecentered cubic (FCC) lattice near the ODT in the meanfield phase diagram [6]. The more exotic gyroid phase was discovered [7,8,9] and was explained successfully by Matsen and Schick via SCFT [4]. In our study of the A15 lattice, we find that the hexagonal and gyroid phases intervene and thus there should be no stable A15 lattice for simple diblocks. However, sphere-like topologies are favored by branched diblock copolymers [10,11,12] and dendritic polymers [13,14]; with this in mind we predict that sphere-like phases are stabilized and that the A15 phase is a ground state for this class of structures. By implementing, to our knowledge, the first full SCFT treatment of branched molecules (shown in Fig. 1c) we have verified our theory.In the dilute regime, Semenov's picture treats each micelle as an undistorted sphere so that the outer block extends to a spherical unit cell of radius R S . This unit-cell approximation pr...

Mesoscopic structure prediction of nanoparticle assembly and coassembly: Theoretical foundation

“…Up to now, we have limited the discussion to the case of chains with a linear configuration. The extension to more complex configurations is identical to that reported for other polymer systems such as branched polymers [47][48] and is not repeated here. Since, in our work, monomer species can be soft molecules and hard molecules, multi-component molecular structures can be readily described by defining the monomer-monomer connectivity and specifying monomers with a parameter, i or s .…”

Novel Phase Morphologies in a Microphase-Separated Dendritic Polymer Melt

“…26,[29][30][31] As with many of these model systems, we assume the melt of these copolymers is incompressible and the interactions are described as contactlike Flory-Huggins repulsions between dissimilar blocks. While much of our analysis is now standard in the theory of inhomogeneous polymers, 26 we focus on the aspects that are pertinent to the connectivity and branching architecture unique to this system.…”

“…In a recent investigation of a dendritic copolymer using this approach, the stability of a new cubic phase was demonstrated. 29,30 In the present paper, we apply this mean-field methodology to two dendritic "pitchfork-like" block copolymers, which are shown in Figure 1. This investigation was designed in order to compare these two molecular architectures and understand how their similar dendritic connecting blocks influence the regions of stable morphologies.…”

Crystalline free energies of micelles of diblock copolymer solutions

“…Indeed we examine systems of small number of micelles (32,54 and 64) and we can expect a finite size effect on the computed free energies. Such effects will not change the ordering of structures far from the melting transition but could be relevant for an accurate location of the melting line and, because of the small free energy differences observed near melting, to determine the stable structure upon crystallization.…”

“…In this approach micelles are mapped onto point particles interacting through density dependent effective pair potentials obtained by inverting the micelle-micelle radial distribution function. This work suggests that in the range 0.4 ≤ f ≤ 0.6 the OD transition density decreases upon increasing f , and that the A15 structure [28], already proposed as the stable structure for spherical micelles in linear block copolymer melts [31,32], in start block copolymer melts [33,34] and experimentally observed for dendrimic clusters [35], is the most stable structure among the few explored, namely BCC, FCC, diamond, SC, A15. This is somehow in contradiction with the expectation that the favored structure should depend on f = M A /(M A + M B ), with micelles with smaller coronae (crew-cut) preferring close packed structures such as FCC or HCP, while micelles with larger coronae (hairy) prefer less packed structures such as BCC and A15 [5,6,12].…”