This analyzer computes the partial derivative of the total Helmholtz free energy with respect to the chi parameter of a AB system, with two monomers types, for a system of fixed composition and fixed unit cell.

See also: BinaryChiDerivative (class API)

Theory

Consider a system with two monomer types with a Flory-Huggins parameter \( \chi \) and a total Helmholtz free energy \( F = -\ln Z \). This analyzer computes the derivative \( \partial F / \partial \chi \), which is given by an ensemble average

\[ \frac{\partial F}{\partial \chi} = \langle \Psi \rangle \quad\quad\quad\quad \Psi \equiv \frac{\partial H}{\partial \chi} + \frac{\partial \ln N}{\partial \chi} \quad, \]

in which

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

and

\[ \frac{\partial \ln N}{\partial \chi} = \frac{G}{2\chi} \quad. \]

Here, \( v \) is the monomer reference volume and \( G \) is the total number of nodes in the computational mesh.

Parameter file

Parameter file format:

BinaryChiDerivative{
  interval*          int  (1 by default)
  outputFileName     string
  nSamplePerOutput*  int  (1 by default)
}

Meanings of the parameters:

Label	Description
interval*	number of steps between data samples
outputFileName	base name of all output files
nSamplePerOutput *	Number of sampled values per block average output to file

Output

In the following, the symbol {outputFileName} is used to represent the string value of the outputFileName parameter within descriptions of output file names.

Output during a simulation or trajectory file analysis :

If nSamplePerOutput > 1, then block average values of \( \Psi \), averaged over blocks of nSamplePerOutput sequential samples are output to the file {outputFileName}.dat every interval * nSamplerPerBlock simulation steps or trajectory file frames.

If nSamplePerOutput == 1 (or if this optional parameter is omitted), then instantaneous values of \( \Psi \) are output every interval time steps or trajectory file frames, with no block averaging.

If nSamplerPerBlock = 0, then {outputFileName}.dat data file is not created.

Final output :

At the end of a simulation or trajectory file analysis:

The sequence average of \( \Psi \) and estimated error on this average are output to the file {outputFileName}.ave
Details of the error analysis are output to the file {outputFileName}.aer

See the discussion of error analysis for details.