Statistical mechanics of charged polymers in electrolyte solutions: A lattice field theory approach

Abstract

The Lattice Field Theory approach to the statistical mechanics of a classical Coulomb gas [R.D. Coalson and A. Duncan, J. Chem. Phys. 97, 5653 (1992)

On the reliability of mean-field methods in polymer statistical mechanics

“…In such a model (absent long-range electrostatic interactions, which are not treated in this paper) an essentially exact (up to statistical errors) solution of the statistical mechanics is available via standard Monte Carlo simulation methods, while, as we shall see below, the mean-field Schrödinger approach of [8,4] can be generalized to such a model, allowing a direct comparison and evaluation of the error incurred by the saddle-point approximation implicit in the mean-field equations.…”

“…To establish contact with the models studied in our previous work [8,4], we linearize the dependence on j by introducing an auxiliary field ω via a Hubbard-Stratonovich transformation. The path integral (3) then becomes…”

“…With the range √ γ set to zero, we recover the simpler model studied in our earlier mean-field papers [8,4] for the special case of uncharged polymer chains. Here we have specialized to the case of a single particle potential V excl in (3) which is either zero or +∞, and the sole effect of which is to restrict the fields in the path integral (5) to have support in the region where V excl = 0.…”

“…In order to study systematically the level of accuracy of the mean-field approach to polymer statistical mechanics introduced in some recent work of the present authors [8,4], it will be convenient to introduce a slightly generalized Hamiltonian which allows for a soft Yukawa type repulsive potential between the monomers of the polymer chain [9]. In such a model (absent long-range electrostatic interactions, which are not treated in this paper) an essentially exact (up to statistical errors) solution of the statistical mechanics is available via standard Monte Carlo simulation methods, while, as we shall see below, the mean-field Schrödinger approach of [8,4] can be generalized to such a model, allowing a direct comparison and evaluation of the error incurred by the saddle-point approximation implicit in the mean-field equations.…”

“…In some recent work [8,4] we developed a lattice field theory approach to the statistical mechanics of polymers in solution. This approach relied on several approximations: the treatment was at the mean-field level, which amounts to a saddle-point approximation to the system's partition function; also the problem was treated on a lattice, converting a continuous three dimensional (3D) problem to a discrete, finite matrix problem, and, finally, the polymer chain was assumed to be long enough so that it was approximated as a continuous object, ignoring the discrete character of the building monomers.…”