This analyzer calculates the derivative of free energy with respect to parameter \(\chi_b N \) at fixed united cell, fixed polymer volume fraction, and fixed invariant degree of polymerization for AB diblock copolymer system.

See also: ChiDerivative (class API)

Mathematical Formula

\[ \frac{\partial F}{\partial \chi_b} = -\frac{\partial \ln Z}{\partial \chi_b} = \left\langle \frac{\partial H}{\partial \chi_b} \right\rangle + \frac{\partial \ln N}{\partial \chi_b} \]

\[ \frac{\partial H}{\partial \chi_b} = \frac{1}{v} \int \mathrm{d} \mathbf{r} \left\langle - \frac{W_{-}(\mathbf{r})^2}{\chi_b^2} + \frac{1}{4} \right\rangle \]

\[ \frac{\partial \ln N}{\partial \chi_b} = \frac{\text{MeshSize}}{2\chi_b} \]

where \( v \) denote the monomer reference volume and \(\text{MeshSize} \) denotes the total number of mesh size.

Parameter File

The full parameter file format, including all optional parameters, is shown below:

ChiDerivative{
  interval           int
  outputFileName     string
  hasAverage*        bool     (default true)
}

Meanings of the parameters are described briefly below:

Label	Description
interval	number of steps between data samples
outputFileName	name of output file
hasAverage	whether the average and error analysis are needed?

Output

During the simulation, \(\chi_b N \) and corresponding \( \frac{\partial F}{\partial \chi_b} \) are output to the file {outputFileName} every interval simulation steps.

At the end of the simulation, if hasAverage is true:

average info and error analysis are output to {outputFileName}.ave
error analysis info are output to {outputFileName}.aer