^{1}and Glenn H. Fredrickson

^{2,a)}

### Abstract

Calculating phase diagrams and measuring the properties of multiple phases in equilibrium is one of the most common applications of field-theoretic simulations. Such a simulation often attempts to simulate two phases in equilibrium with each other in the same simulation box. This is a computationally demanding approach because it is necessary to perform a large enough simulation so that the interface between the two phases does not affect the estimate of the bulk properties of the phases of interest. In this paper, we describe an efficient method for performing field-theoretic simulations in the Gibbs ensemble, a familiar construct in particle-based simulations where two phases in equilibrium with each other are simulated in separate simulation boxes. Chemical and mechanical equilibrium is maintained by allowing the simulation boxes to swap both chemical species and volume. By fixing the total number of each chemical species and the total volume, the Gibbs ensemble allows for the efficient simulation of two bulk phases at equilibrium in the canonical ensemble. After providing the theoretical framework for field-theoretic simulations in the Gibbs ensemble, we demonstrate the method on two two-dimensional modelpolymer test systems in both the mean-field limit (self-consistent field theory) and in the fluctuating field theory.

This work was supported by NSF Grant No. DMR09-04499 and made use of MRL Central Facilities supported by the MRSEC Program of the NSF under Grant No. DMR05-20415.

I. INTRODUCTION

II. THEORY AND NUMERICAL METHODS

A. Gibbs ensemble

B. Mean-field approximation

C. CL sampling

D. Compressible binary homopolymer blend

1. Field-theoretic model

2. Binary blend in the Gibbs ensemble

E. Block copolymer/homopolymer blend

1. Field-theoretic model

2. Copolymer/homopolymer blend in the Gibbs ensemble

F. Implementation details

III. RESULTS

A. Binary homopolymer blend

B. Homopolymer/copolymer blend

IV. DISCUSSION

V. SUMMARY

### Key Topics

- Mean field theory
- 34.0
- Field theory models
- 24.0
- Polymers
- 23.0
- Block copolymers
- 22.0
- Phase diagrams
- 11.0

## Figures

Phase diagram for the binary homopolymer blend. The solid line represents the result from the analytic Flory–Huggins theory, the circles are mean-field Gibbs ensemble results, and the diamonds are the results from CL simulations. The inset also shows a point for CL results with and (△). The error bars in all cases are smaller than the size of the symbols.

Phase diagram for the binary homopolymer blend. The solid line represents the result from the analytic Flory–Huggins theory, the circles are mean-field Gibbs ensemble results, and the diamonds are the results from CL simulations. The inset also shows a point for CL results with and (△). The error bars in all cases are smaller than the size of the symbols.

Continuously varying box length (heavy black line) and the mapping of this length onto the nearest grid point (thin red line/squares) from the CL simulations of the binary homopolymer blend at .

Continuously varying box length (heavy black line) and the mapping of this length onto the nearest grid point (thin red line/squares) from the CL simulations of the binary homopolymer blend at .

Mole fraction of species in the -rich phase for the binary homopolymer blend at plotted against the inverse grid resolution, . The dashed horizontal line corresponds to the mean-field result, and the solid line is a guide to the eye.

Mole fraction of species in the -rich phase for the binary homopolymer blend at plotted against the inverse grid resolution, . The dashed horizontal line corresponds to the mean-field result, and the solid line is a guide to the eye.

The real and imaginary parts of (solid black line/circles) and (dashed red line/squares) during a CL simulation of the binary homopolymer blend with and a grid. The imaginary parts of both and exhibit small fluctuations about zero.

The real and imaginary parts of (solid black line/circles) and (dashed red line/squares) during a CL simulation of the binary homopolymer blend with and a grid. The imaginary parts of both and exhibit small fluctuations about zero.

The volume of each box in the Gibbs ensemble (a) and the mole fraction of species in box I (b) during a CL simulation of the binary homopolymer blend with . The calculations were performed in a simulation box with on a grid. Note that for (b), only results from early in the simulation are shown to more clearly show the drift of the mole fraction; these calculations continued to the CL time of 125 00, as shown in (a).

The volume of each box in the Gibbs ensemble (a) and the mole fraction of species in box I (b) during a CL simulation of the binary homopolymer blend with . The calculations were performed in a simulation box with on a grid. Note that for (b), only results from early in the simulation are shown to more clearly show the drift of the mole fraction; these calculations continued to the CL time of 125 00, as shown in (a).

Color maps of the local mole fraction of the section of the block copolymer (top), species (middle), and the homopolymer (bottom) for , , , , , and . The left column of figures corresponds to the mean-field results, while the right column shows a single configuration from a CL trajectory. The real part of the density fields were used to generate the maps derived from the CL simulations. Note that the two simulation boxes are not in contact during the simulations and are plotted in this manner to show the two phases in equilibrium.

Color maps of the local mole fraction of the section of the block copolymer (top), species (middle), and the homopolymer (bottom) for , , , , , and . The left column of figures corresponds to the mean-field results, while the right column shows a single configuration from a CL trajectory. The real part of the density fields were used to generate the maps derived from the CL simulations. Note that the two simulation boxes are not in contact during the simulations and are plotted in this manner to show the two phases in equilibrium.

The real and imaginary parts of (black solid line/circles) and (red dashed line/squares) during a CL simulation for the homopolymer/block copolymer mixture with , , , , , and . The lines correspond to the real parts of , while the points that fluctuate around 0 correspond to the imaginary parts of .

The real and imaginary parts of (black solid line/circles) and (red dashed line/squares) during a CL simulation for the homopolymer/block copolymer mixture with , , , , , and . The lines correspond to the real parts of , while the points that fluctuate around 0 correspond to the imaginary parts of .

Schematic of the interpolation scheme used when we increase the box dimensions during a CL simulation in the Gibbs ensemble in one dimension. In this simple example, the number of grid points is growing from four to five grid points. First, the fields are Fourier transformed [denoted by ]; the four Fourier components are denoted in the top line of this cartoon, and the arrays have the same wavevector ordering that is employed by the FFTW Fourier transform package (Ref. 32). Next, we increase the number of elements in the array by one, and finally insert a zero intensity mode at the newly created wavevector position. Finally, we inverse Fourier transform into the larger simulation box.

Schematic of the interpolation scheme used when we increase the box dimensions during a CL simulation in the Gibbs ensemble in one dimension. In this simple example, the number of grid points is growing from four to five grid points. First, the fields are Fourier transformed [denoted by ]; the four Fourier components are denoted in the top line of this cartoon, and the arrays have the same wavevector ordering that is employed by the FFTW Fourier transform package (Ref. 32). Next, we increase the number of elements in the array by one, and finally insert a zero intensity mode at the newly created wavevector position. Finally, we inverse Fourier transform into the larger simulation box.

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