Eur. Phys. J. E

**8**, 33-58 (2002)

DOI: 10.1140/epje/i2001-10091-9

## Simulations of counterions at charged plates

**A.G. Moreira and R.R. Netz**

Max-Planck-Institut für Kolloid- und Grenzflächenforschung, D-14424 Potsdam, Germany netz@mpikg-golm.mpg.de

(Received 6 November 2001)

** Abstract **

Using Monte Carlo simulations,
we study the counterion distribution close to planar charged walls
in two geometries: i) when only one charged wall is present and the
counterions are confined to one half-space, and ii) when the
counterions are confined between two equally charged walls.
In both cases the surface charge is smeared out and the
dielectric constant is the same everywhere.
We obtain the counterion density profile and compare it with
both the Poisson-Boltzmann theory (asymptotically exact in the limit of
weak coupling, *i.e.* low surface charge, high temperature
and low counterion valence) and the strong-coupling theory
(valid in the opposite limit of high surface charge, low
temperature and high counterion valence) and with previously calculated
correction terms to both theories for different values of the
coupling parameter, thereby establishing the domain of validity
of the asymptotic limits.
Gaussian corrections to the leading Poisson-Boltzmann behavior
(obtained *via* a systematic loop expansion) in general perform
quite poorly: At coupling strengths low enough so that
the Gaussian (or one-loop) correction does describe the
numerical deviations from the Poisson-Boltzmann result correctly,
the leading Poisson-Boltzmann term by itself matches the data
within high accuracy. This reflects the slow
convergence of the loop expansion. For a single
charged plane, the counterion pair correlation function indicates
a behavioral change from a three-dimensional, weakly correlated
counterion distribution (at low coupling) to a two-dimensional,
strongly correlated counterion distribution (at high coupling),
which is paralleled by the specific-heat capacity which displays a
rounded hump at intermediate coupling strengths.
For the case of counterions confined between two equally
charged walls, we analyze the inter-wall pressure
and establish the complete phase diagram,
featuring attraction between the walls for large enough coupling strength
and at intermediate wall separation.
Depending on the thermodynamic ensemble, the phase diagram
exhibits a discontinuous transition where the inter-wall distance
jumps to infinity (in the absence of a chemical potential coupling to the
inter-wall distance, as for charged lamellae in excess solvent) or a
critical point where two coexisting states with different
inter-wall distance become indistinguishable
(in the presence of a chemical potential,
as for charged lamellae with a finite fixed solvent fraction).
The attractive pressure decays with the inter-wall distance as an
inverse cube, similar to analytic predictions, although the amplitude
differs by an order of magnitude from previous theoretical results.
Finally, we discuss in detail our simulation methods and
compare the finite-size scaling behavior
of different boundary conditions (periodic, minimal image and open).

**PACS**

82.70.-y - Disperse systems; complex fluids.

61.20.Ja - Computer simulation of liquid structure.

61.20.Qg - Structure of associated liquids: electrolytes, molten salts, etc..

**©**

*EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2002*