Intramolecular Correlations

Functions to compute intra-molecular correlation functions for Gaussian block polymer chains in a homogeneous system. More...

Classes
class	Pscf::Correlation::Mixture< WT >
	Correlations in a homogeneous ideal gas mixture. More...
class	Pscf::Correlation::Polymer< WT >
	Intramolecular correlation analysis for one polymer Species. More...

Functions
double	Pscf::Correlation::dt (double ksq, double length, double kuhn)
	Compute and return intrablock correlation function (thread model)
double	Pscf::Correlation::db (double ksq, double nBead, double kuhn)
	Compute and return an intrablock correlation function (bead model)
double	Pscf::Correlation::et (double ksq, double length, double kuhn)
	Compute and return one-sided factor for one block (thread model).
double	Pscf::Correlation::eb (double ksq, double nBead, double kuhn)
	Compute and return one-sided factor for one block (bead model).

Detailed Description

Functions to compute intra-molecular correlation functions for Gaussian block polymer chains in a homogeneous system.

Function Documentation

dt()

double Pscf::Correlation::dt	(	double	ksq,
		double	length,
		double	kuhn )

Compute and return intrablock correlation function (thread model)

This function returns the intramolecular correlation function for a homopolymer of specified length and statistical segment length. The result for the thread model can be expressed as a function

\[ d(k) = L^2 g(x) \]

where \( x = k^2 L b^2 / 6 \), for L = length, b = kuhn, and \( k^{2} \) = ksq, and

\[ g(x) \equiv 2[ e^{-x} - 1 + x ]/x^2 \]

is the Debye function. This function also gives the intra-block correlation function for a Gaussian block of contour given by the parameter "length" and statistical segment length given by "kuhn".

Parameters

ksq	square of wavenumber
length	contour length of polymer or block
kuhn	statistical segement length

Definition at line 17 of file Debye.cpp.

Referenced by Pscf::Correlation::Polymer< WT >::computeOmega(), and Pscf::Correlation::Polymer< WT >::computeOmegaTotal().

db()

double Pscf::Correlation::db	(	double	ksq,
		double	nBead,
		double	kuhn )

Compute and return an intrablock correlation function (bead model)

This function returns the intramolecular correlation function for a homopolymer of specified length and statistical segment length. The result for the bead model can be expressed as a function

\[ d(k) \equiv 2[ e^{-yN} - 1 + N(1-e^{-y}) ]/(1-e^{-y})^2 \]

where \( y = k^2 b^2 / 6 \), for N = nBead, b = kuhn, and \( k^{2} = \) ksq. This function also gives the intra-block correlation function for block of N beads.

Parameters

ksq	square of wavenumber
nBead	number of beads in the block (converted to double)
kuhn	statistical segement length

Definition at line 33 of file Debye.cpp.

Referenced by Pscf::Correlation::Polymer< WT >::computeOmega(), and Pscf::Correlation::Polymer< WT >::computeOmegaTotal().

et()

double Pscf::Correlation::et	(	double	ksq,
		double	length,
		double	kuhn )

Compute and return one-sided factor for one block (thread model).

This function returns the function

\[ e(k) = L [ 1 - e^{-x} ] / x \]

where \( x = k^2 L b^2 / 6 \), for L = length, b = kuhn, and \( k^{2} \) = ksq.

The intra-block correlation function \( \omega_{ij}(k) \) for two distinct blocks with block indices i and j can be expressed in the thread model as a product

\[ \omega_{ij}(k) = e^{-k^{2}R_{ij}^{2}/6} e_{i}(k) e_{j}(k) \]

where \( R_{ij}^{2} \) is the mean-squared end-to-end length of a sequence of other blocks that form a path connecting the two blocks of interest (if any), and where \( e_{i}(k) \) is the one sided factor returned by this function for block \( i \). In the thread model \( R_{ij}^{2} = 0 \) for blocks that both terminate at a shared vertex.

Parameters

ksq	square of wavenumber
length	contour length of block
kuhn	statistical segement length

Definition at line 53 of file Debye.cpp.

Referenced by Pscf::Correlation::Polymer< WT >::computeOmega(), and Pscf::Correlation::Polymer< WT >::computeOmegaTotal().

eb()

double Pscf::Correlation::eb	(	double	ksq,
		double	nBead,
		double	kuhn )

Compute and return one-sided factor for one block (bead model).

This function returns the function

\[ e(x) \equiv ( 1 - e^{-Ny} )/(1-e^{-y}) \]

where \( y = k^2 b^2 / 6 \), for N = nBead, b = kuhn, and \( k^{2} \) = ksq.

The intramolecular correlation function \( \omega_{ij}(k) \) for two distinct blocks with block indices i and j can be expressed in the bead model as a product

\[ \omega_{ij}(k) = e^{-k^{2}R_{ij}^{2}/6} e_{i}(k) e_{j}(k) \]

where \( R_{ij}^{2} \) is the mean-squared end-to-end length of the sequence of bonds that lie along a path connecting the two blocks, and where \( e_{i}(k) \) is the one sided factor returned by this function for block \( i \). In the bead model, the value of \( R_{ij}^{2} \) is a sum of squares of statistical segment lengths for all bonds along the path that connects the end monomers of the two blocks. In the bead model, if blocks \( i \) and \( j \) terminate at shared vertex, and have statistical segment lengths \( b_{i} \) and \( b_{j} \), then they are taken to be connected by a connecting bond with a effective squared statistical segment length \( R^{2}_{ij} = ( b_{i}^{2} + b_{j}^{2} )/2 \).

Parameters

ksq	square of wavenumber
nBead	number of beads in the block (converted to double)
kuhn	statistical segement length within the block

Definition at line 69 of file Debye.cpp.

Referenced by Pscf::Correlation::Polymer< WT >::computeOmega(), and Pscf::Correlation::Polymer< WT >::computeOmegaTotal().