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

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

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.

Project Background

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.

Microphase Separation of Block Copolymers

The competition between a polymer's entropic desire to curl up into a closely confined configuration, and the interaction between different monomer species repelling this tendency, leads to ordering on the nano scale. For example ;

● Equal amounts of strongly interacting A and B monomers will order into regularly spaced lamellar domains.● If one species makes up a small fraction of the total polymer, the minority components will arrange themselves into isolated domains, such as cylinders or spheres.● If the A and B species do not interact strongly enough there will be no long range order.

There are several classical phases which are stable, lamellar, hexagonally packed cylinders, and body centric cubic morphologies. A complex phase which has been found to be stable is the gyroid morphology. We are specifically interested in this phase.

Microphase Separation of Block Copolymers

The competition between a polymer's entropic desire to curl up into a closely confined configuration, and the interaction between different monomer species repelling this tendency, leads to ordering on the nano scale. For example ;

● Equal amounts of strongly interacting A and B monomers will order into regularly spaced lamellar domains.● If one species makes up a small fraction of the total polymer, the minority components will arrange themselves into isolated domains, such as cylinders or spheres.● If the A and B species do not interact strongly enough there will be no long range order.

There are several classical phases which are stable, lamellar, hexagonally packed cylinders, and body centric cubic morphologies. A complex phase which has been found to be stable is the gyroid morphology. We are specifically interested in this phase.

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, N, and , a measure of how asymmetric the molecule is.

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, N, and , a measure of how asymmetric the molecule is.

Phase Diagram for an asymmetric ABA triblockPhase Diagram for an

asymmetric ABA triblock

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.

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.

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.

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.

GyroidGyroid

AcknowledgementsDonal 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.

AcknowledgementsDonal 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.

Preliminary ResultsThe phase behaviour of the mikto-arm star triblock copolymer is shown to be qualitatively similar to that of an asymmetric ABA triblock copolymer.

Preliminary ResultsThe phase behaviour of the mikto-arm star triblock copolymer is shown to be qualitatively similar to that of an asymmetric ABA triblock copolymer.