Bo Jönsson, Carsten Peterson and Bo Söderberg A Variational Approach to the Structure and
Thermodynamics of Linear Polyelectrolytes with Coulomb and
Screened Coulomb Interactions Journal of
Physical Chemistry 99, 1251-1266 (1995)Abstract:A variational approach, based on a discrete representation of the chain, is used to calculate free energy and conformational properties in polyelectrolytes. The true bond and Coulomb potentials are approximated by a trial isotropic harmonic energy containing force constants between all
monomer-pairs as variational parameters. By a judicious
choice of representation and the use of incremental matrix
inversion, an efficient and fast-convergent iterative
algorithm is constructed, that optimizes the free energy. The
computational demand scales as N^{3} rather
than N^{4} as expected in a more naive
approach. The method has the additional advantage that in
contrast to Monte Carlo calculations the entropy is easily
computed. An analysis of the high and low temperature limits
is given. Also, the variational formulation is shown to
respect the appropriate virial identities.The accuracy of the
approximations introduced are tested against Monte Carlo
simulations for problem sizes ranging from N=20 to
1024. Very good accuracy is obtained for chains with
unscreened Coulomb interactions. The addition of salt is
described through a screened Coulomb interaction, for which
the accuracy in a certain parameter range turns out to be
inferior to the unscreened case. The reason is that the
harmonic variational Ansatz becomes less efficient with
shorter range interactions. As a by-product a very efficient
Monte Carlo algorithm was developed for comparisons,
providing high statistics data for very large sizes -- 2048
monomers. The Monte Carlo results are also used to examine
scaling properties, based on low-T approximations to
end-end and monomer-monomer separations. It is argued that
the former increases faster than linearly with the number of
bonds.LU TP 93-15 |