Elastic moduli of multiblock copolymers in the lamellar phase

Stability and Elasticity of Ultrathin Sphere-Patterned Block Copolymer Films

Sphere-patterned ultrathin block copolymer films are potentially interesting for a variety of applications in nanotechnology. We use self-consistent field theory to investigate the elastic response of sphere monolayer films with respect to in-plane shear, in-plane extension, compression deformations, and bending. The relations between the in-plane elastic moduli are roughly compatible with the expectations for two-dimensional elastic systems with hexagonal symmetry, with one notable exception: The pure shear and the simple shear moduli differ from each other by roughly 20%. Even more importantly, the bending constants are found to be negative, indicating that free-standing block copolymer membranes made of only a sphere monolayer are inherently unstable above the glass transition. Our results are discussed in view of the experimental findings.

The stress deformation response influenced by the chain rigidity for mesostructures in diblock copolymers

A self-consistent field theory formalism based on the wormlike chain model is developed to investigate the stress-strain relation for mesostructures in diblock copolymers under the influence of chain rigidity, involving the adjustable simulation cell in the non-orthogonal coordinates by means of optimization of free energy. We elucidate the effect of the chain persistency broadly spanning from the Gaussian chain to the rigid rodlike chain on the elastic response of mesophases that deviate from the initial equilibrium structures. We analytically and numerically demonstrate that our current approach in the long chain limit recovers to the Gaussian-chain-based theory. Being ascribed to the distinct conformational behaviors for flexible chains and rigid rodlike chains, the tensile and compressive stresses applied to lamellae exhibit asymmetric deformation behaviors and the shear stress applied to the initial equilibrium hexagonal cylinders results in noticeable deviations in the shape and spatial arrangement of cylindroids for various chain rigidity values. For the zero stress, in addition, our approach can be straightforwardly utilized to explore the optimal size and shape of the simulation cell in order to achieve a stress free configuration of systems.

Directed assembly of non-equilibrium ABA triblock copolymer morphologies on nanopatterned substrates

The majority of past work on directed assembly of block copolymers on chemically nanopatterned surfaces (or chemical patterns) has focused on AB diblock copolymers, and the resulting morphologies have generally corresponded to equilibrium states. Here we report a study on directed assembly of ABA triblock copolymers. Directed assembly of thin films of symmetric poly(methyl methacrylate-b-styrene-b-methyl methacrylate) (PMMA-b-PS-b-PMMA) triblock copolymers is shown to be capable of achieving a high degree of perfection, registration, and accuracy on striped patterns having periods, L(s), commensurate with the bulk period of the copolymer, L(o). When L(s) is incommensurate with L(o), the triblock copolymer domains can reach dimensions up to 55% larger or 13% smaller than L(o). The range over which triblock copolymers tolerate departures from a commensurate L(s) is significantly larger than that accessible with the corresponding diblock copolymer material on analogous directed assembly systems. The assembly kinetics of the triblock copolymer is approximately 3 orders of magnitude slower than observed in the diblock system. Theoretically informed simulations are used to interpret our experimental observations; a thermodynamic analysis reveals that triblocks can form highly ordered, non-equilibrium metastable structures that do not arise in the diblock.

Mesoscopic dynamics of inhomogeneous polymers based on variable cell shape dynamic self-consistent field theory

In this paper, we combine variable cell shape method with dynamic self-consistent field theory and extend to study structure and dynamics under shear for triblock copolymer melts. Due to shear, the calculation cell shape is variable and no longer orthogonal. Pseudospectral method is employed to solve the diffusion equation for chain propagator on the nonorthogonal coordinate and the shear periodical condition can be easily designed in terms of the variable cell shape method. By using this strategy, the shear induced morphology evolution is investigated for topologically complex polymeric systems such as linear and star triblock copolymers; the morphology of linear ABC triblock copolymers is more shear sensitive than that of star triblocks. In particular, once the chain propagator is obtained, the microscopic elastic stress and spatial stress distribution can be derived and thus the dynamic mechanical property can be calculated under shear. By imitating the dynamic storage modulus G' corresponding to any given morphology in the oscillatory shear measurements, we explore the relationship between the morphology and the storage modulus G' and extend to study the mechanism of phase separation dynamics as well as order-disorder transition (ODT) for linear and star triblock copolymers. The results show that the chain architecture can be easily distinguished by investigating the ODT, though the systems such as AB symmetric diblock and ABA triblock copolymers by coupling AB precursors almost exhibit similar microstructures. In addition, the storage modulus G' and loss modulus G" can be simultaneously determined in frequency sweeps of oscillatory shear measurements and the dependence of the moduli on phase separated patterns and the chain topology is investigated. The simulation findings are in qualitatively agreement with the experimental results.

Stress distributions in diblock copolymers

We demonstrate how a generalized self-consistent field theory for polymer melts that includes elastic stress and strain fields can be applied to the study of AB diblock copolymers melts. By obtaining the stress distributions for volume conserving strain loadings where lamellar and hexagonal morphologies are stable, we show that the local stress is reduced at the domain interface but slightly enhanced in the immediate vicinity of the interface. The overall stress profile is the result of the combined effects of chain connectivity across the interface, which yields a positive contribution, and the immiscible nature of the monomers, which leads to a stress reduction because of interfacial tension.

Single chain in mean field simulations: Quasi-instantaneous field approximation and quantitative comparison with Monte Carlo simulations

The description of fluctuations by single chain in mean field (SCMF) simulations is discussed and the results of this particle-based self-consistent field technique are quantitatively compared to Monte Carlo simulations of the same discretized Edwards-Hamiltonian providing exact reference data. In SCMF simulations one studies a large ensemble of noninteracting molecules subjected to real, external fields by Monte Carlo simulations. The external fields approximate nonbonded, instantaneous interactions between molecules. In the self-consistent mean field theory the external fields are static and fluctuation effects are ignored. In SCMF simulations, the external fields fluctuate since they are frequently recalculated from the instantaneous density distribution of the ensemble of molecules. In the limit of infinitely high density or instantaneous update of the external fields, the SCMF simulation method accurately describes long-wavelength fluctuations. At high but finite updating frequency the accuracy depends on the discretization of the model. The accuracy is illustrated by studying the single chain structure and intermolecular correlations in polymer melts, and fluctuation effects on the order-disorder transition of symmetric diblock copolymers.

Effect of Architecture on the Tensile Properties of Triblock Copolymers in a Lamellar Phase

We have used self-consistent field theory to calculate the tensile moduli of triblock copolymers in lamellar microstructures prepared from linear and star architectures. The extensional moduli K(33) are the main contributors to the tensile moduli, and the contribution of K(U)33 (the internal energy contribution to K(33)) is the main component of the value of K(33). We find that the tensile moduli of ABC three-miktoarm star terpolymers are smaller than those of ABC linear triblock copolymers having identical components, presumably for two main reasons. First, for the ABC three-miktoarm star terpolymers, the contributions of K(U)33 are larger than those of the linear triblock copolymers; we attribute this phenomenon to the star terpolymers having smaller lamellar domain sizes at equilibrium relative to those of the linear triblock copolymers. Second, conformational entropies play an important role in affecting the tensile moduli, mainly because of the different degrees of freedom of the various chains. In contrast, the shear moduli contribute negligibly to the tensile moduli.

Effect of polydispersity on the tensile modulus of diblock copolymers in a lamellar phase

We have calculated the tensile moduli of nanomaterials having lamellar microstructures prepared through the self-assembly of a polydisperse diblock copolymer. We observed that the extensional moduli K33 provided the major contribution to the tensile modulus and that the value of K33 depended mainly on the contribution of K33(U) (the internal energy contribution to K33). We found that a larger polydispersity index (PDI) weakens the material for our polydisperse model; we attribute this phenomenon to the larger lamellar domain size at equilibrium when the polydispersity of the block increases and to the competition between short and long chains. We found that longer chains in this system strengthen the material, but shorter chains weaken it as a result of the influence of the PDI. The shear modulus contributed negligibly to the extensional moduli.

Introducing Variable Cell Shape Methods in Field Theory Simulations of Polymers

We propose a new method for carrying out field-theoretic simulations of polymer systems under conditions of prescribed external stress, allowing for shape changes in the simulation box. A compact expression for the deviatoric stress tensor is derived in terms of the chain propagator, and it is used to monitor changes in the box shape according to a simple relaxation scheme. The method allows fully relaxed, stress free configurations to be obtained even in nontrivial morphologies, and it enables the study of morphology transitions induced by external stresses.