Numerically investigating the gyroid structure of microphase separated block copolymer melts using self consistent mean field theory Donal O'Donoghue, Physics, National University of Ireland, Cork. Dr Nathaniel Lynd, Prof Glenn Fredrickson, Materials Research Laboratory. Project Background
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Block copolymers consist of two or more chemically distinct polymer species (i.e., blocks) covalently bound into a single molecule. Systems of these polymers in the melt phase are interesting as they 'microphase separate' into different monomer-rich domains depending on their composition and the relative immiscibility of their different components. We concentrate our investigation on the Gyroid phase for a mikto-arm star triblock copolymer.
Phase Diagram for an asymmetric ABA triblock
Methods II – Spectral Decomposition
To compute spatially varying functions such as monomer densities and interaction field strengths, we use a spectral decomposition method. We first pick a specific morphology we are interested in, then we expand all functions of interest in a basis relevant to the chosen morphology. This procedure is similar to expanding an arbitrary function as a Fourier Series of sines and cosines, the more terms of the series you include, the better your approximation to the true function. The accuracy of our model depends on the number of basis functions we use. The time of computation scales as n4 for every n basis functions that we include, therefore it is imperative to optimise the number of basis functions utilised. The convergence of the free energy of the gyroid phase of a particular block copolymer as the number of basis functions is increased, is shown below.
Modelling Block Copolymers
We investigate a block copolymer with a specific branched architecture called a mikto-arm star triblock copolymer. We can characterise such copolymers with four parameters. Firstly the number of branched arms, n, then the fraction of the total polymer which consists of monomer A, fA, the interaction strength between A and B species, cN, andt, a measure of how asymmetric the molecule is.
The phase behaviour of the mikto-arm star triblock copolymer is shown to be qualitatively similar to that of an asymmetric ABA triblock copolymer.
Methods I - From Particles to Fields
We employ self consistent mean field theory (SCFT) to investigate the equilibrium phase behaviour of block copolymer melts. This involves using a mathematical transformation to go from particle–particle interactions to particle–field based interactions. This changes functions like the individual monomer densities from highly discontinuous functions to continuos ones, and which are more easy to work with numerically.
Donal O'Donoghue would like to thank Dr Nathaniel Lynd for support & guidance on this project, Prof Glenn Fredrickson for facilitating research in his group, and all involved in the CISEI programme at UCSB. D O'D would like to thank Science Foundation Ireland for partial funding of this project under the SF1 UREKA International Exhchange program, and all in UCC and Tyndall National Institue who arranged the exchange. Partial support for this project was provided by the NSF Division of Materials Research through the MRSEC and IMI programs.