# Test PolyFTS with Polyorder

We will verify the PolyFTS calculation by comparing to the Polyorder results. The SCFT model we will test is miktoarm star block copolymer and homopolymer blends (AB3 + A).

# Polyorder Configuration File

The configuration file for Polyorder can be either in INI or in YAML format. Sample configuration files can be found in the example folder in the Polyorder project.

# Simpson's Rule and Other Fourth-Order Quadrature Formulas

## 1. Closed formulas

By dividing the closed interval $[a, b]$ into $N-1$ uniform subintervals with length of $h=(b-a)/(N-1)$, the extended Simpson’s rule is given by (1)

where $f_i = f(x_i)$ for $x_i = ih$, and $\xi$ is in $(a, b)$. The number of points, $N$, must be odd, meaning that there are even number of subintervals. This requirement is a consequence of the fact that the coefficients alternate in a specific pattern, namely there are $n$ pairs of $(4, 2)$ coefficient terms plus an extra $4$-coefficient term and two end-point terms. Therefore, there are $2n+3$ terms in total, while $2n+3$ is always odd.

# mpltex: A Tool for Creating Publication Quality Plots

Creating a publication-quality plot is not an easy job. One needs to consider a dozen of factors:

1. The figure size should be set explicitly to match journal specific value. For exmaple, journals published by American Chemical Society (ACS) allows a maximum 3.25-inch width figure for single-column and a maximum 7-inch width figure for double-column.
2. The font family should be customized. Most of the time, “Times New Roman” is a safe choice. You should consult the journal author guide for more information.
3. The font size also needs to be set properly.
4. The linewdith of axis, axis ticks, line arts, the format of legend, the colors are all important factors affects the final looking of a plot.
5. The file format of a figure should be chosen carefully. For most publishers, EPS is a good choice for line arts and other simple 2D arts, such as histograms, power spectra, bar charts, errorcharts, scatterplots.

# A Quick Guide to the Self-Consistent Field Theory in Polymer Physics

## 1. Introduction

The self-consistent field theory (SCFT) for many-chain systems is obtained by imposing a mean-field approximation to simplify the statistical field theories. The statistical field theories can be constructed from the particle-based model by carrying out a particle-to-field transformation.

The general approach for a particle-to-field transformation is to invoke formal techniques related to Hubbard-Stratonovich transformations, which have the effect of decoupling interactions among particles (or polymer segments) and replacing them with interactions between the particles and one or more auxiliary fields.