This analyzer calculates the spherical average of the structure factor for all wavevectors for a AB diblock copolymer system.

See also: BinaryStructureFactorGrid (class API)

Mathematical Formula

This class computes the structure factor \( S({\bf q}) \) for an incompressible system containing two monomer types of monomer type indices 0 and 1, defined as

\[ S(\mathbf{q}) = \frac{1}{4V}\langle |\hat{C}_{-}({\bf q})|^{2} \rangle \quad, \]

where \( v \) is monomer volume, \( V \) is total system volume, and

\[ \hat{C}_{-}({\bf q}) \equiv \int \! d{\bf r} \; e^{-i{\bf q}\cdot {\bf r}} [ c_{0}({\bf r}) - c_{1}({\bf r}) ] \quad. \]

Here, \( c_{i}({\bf r}) \) is the local concentration of monomers of type i, for i = 0 and i =1.

In a field-theoretic simulation, this is computed using the identity

\[ S(\mathbf{q}) = \frac{1}{v^2\chi_{b}^2} \frac{1}{V} \left\langle |\hat{W}_{-}(\mathbf {q})|^{2} \right\rangle - \frac{1}{2v \chi_b} \]

where \( \chi_{b} \) is the bare Flory-Huggins parameter, and

\[ \hat{W}_{-}({\bf q}) = \int \! d{\bf r} \; e^{-i{\bf q}\cdot {\bf r}} W_{-}({\bf r}) \]

is the transform of the exchange field \( W_{-}({\bf r}) = [w_{0}({\bf r}) - w_{1}({\bf r})]/2 \). The Fourier transform \( \hat{W}_{-}({\bf q}) \) defined above as an integral is approximated internally by \( V \) times the normalized discrete Fourier transform (DFT) of \( W_{-}({\bf r}) \) as computed by the function FFT::forwardTransform.

Parameter File

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

BinaryStructureFactorGrid{
  interval           int
  outputFileName     string
  nSamplePerBlock*   int      (default 1)
}

Meanings of the parameters are described briefly below:

Label	Description
interval	number of steps between data samples
outputFileName	name of output file
nSamplePerBlock	number of samples per block average

At the end of the simulation, time-averaged values of \( S({\bf q}) \) are averaged over groups of wavevectors of the same magnitude to obtain a spherically averaged structure factor \( S(q) \) at a discrete set of values of the wavenumber \( q \equiv |{\bf q}| \).

Output File

After completion fo the simulation, the output file {outputFileName} will contain two columns of numbers, in which the first column contains an ordered list of values of wavenumber \( q \), and the second column contains a corresponding list of values of the dimensionless, spherically averaged structure factor \( S(q) \).