Fluctuation correction for the order–disorder transition of diblock copolymer melts

Abstract

The order–disorder transition (ODT) of diblock copolymer melts is evaluated for an invariant polymerization index of N¯=104, using field-theoretic simulations (FTS) supplemented by a partial saddle-point approximation for incompressibility. For computational efficiency, the FTS are performed using the discrete Gaussian-chain model, and results are then mapped onto the continuous model using a linear approximation for the Flory–Huggins χ parameter. Particular attention is paid to the complex phase window. Results are found to be consistent with the well-established understanding that the gyroid phase extends down to the ODT. Furthermore, our simulations are the first to predict that the Fddd phase survives fluctuation effects, consistent with experiments.

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation

“…However, this requires numerical integration along the chain contours, which then leads to numerical inaccuracies. To avoid this, we will instead model the n polymers of the system by discrete bead-spring chains each with N monomers [ 42 ]. The position of monomer i of molecule will be denoted by the vector .…”

“…It will become convenient to reexpress the A and B concentrations in terms of the composition, , and total concentration, , defined by In terms of these new concentrations, the energy of the nonbonded interactions becomes As a result of incompressibility (i.e., ), the first term reduces to a constant, , and therefore is typically dropped. However, we retain it because of its dependence on , which will have implications regarding an ultraviolet divergence discussed later [ 42 ].…”

“…This factor is conveniently expressed as where is the total number of contacts experienced by a given monomer and is the probability that two monomers of an infinite chain, separated by i bonds along the contour, occupy the same cell of the grid. For orthorhombic simulation boxes [ 42 ], Note that . In the case of cubic grids with a resolution less than the bond length (i.e., ), Equation ( 43 ) reduces to which is the proportionality factor first used by Stasiak and Matsen [ 40 ].…”

Study of Segregation in Non-Dilute Solutions of Linear Diblock Copolymers and Symmetric Miktoarm (or Janus Star) Polymers Using Monte Carlo Simulations with the Bond Fluctuation Model

“…In the simplest RPA description, Equations (1) and (3) predict a linear variation of I AB ( q max , ε = 0)/ I AB ( q max , ε ) reaching the value 0 at the MST. The ROL theory provides a more detailed description [ 14 , 16 , 17 ], reflecting an upward deviation of the data with respect to the RPA prediction as the systems approach MST. This feature has previously been confirmed in previous Monte Carlo simulations for diblock copolymers in the melt state.…”

“…More recently, a renormalized one-loop (ROL) theory [ 14 , 16 , 17 ] was able to introduce corrections to the peak intensity and location of the structure-function of diblock copolymers, describing how the variation of 1/ I AB ( q ) with χ N shows a deviation upwards from the initial linear behavior when the systems approach the MST. This theory was shown to give a good description of simulation data showing the approach to the MST transition of symmetric linear diblock copolymers in the disordered melt state.…”