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Field-theoretic simulations in the Gibbs ensemble
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View: Figures


Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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 .

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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).

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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 .

Image of FIG. 8.
FIG. 8.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Field-theoretic simulations in the Gibbs ensemble