Abstract. -Realistic charged macromolecules are characterized by discrete (rather than homogeneous) charge distributions. We investigate the effects of surface charge discretization on the counterion distribution at the level of mean-field theory using a two-state model. Both planar and cylindrical geometries are considered; for the latter case, we compare our results to numerical solutions of the full Poisson-Boltzmann equation. We find that the discretization of the surface charge can cause enhanced localization of the counterions near the surface; for charged cylinders, counterion condensation can exceed Oosawa-Manning condensation.Introduction. -The interaction between charged macromolecules and counterions is a crucial component of the physics of charged systems [1]. Although realistic macromolecules are often composed of discrete charges, theoretical models typically assume a homogeneous surface charge distribution. In the case of cylindrical polyelectrolytes (PEs), this assumption leads to Oosawa-Manning (OM) condensation [2,3], in which counterions become closely associated with the macromolecule, effectively lowering its overall charge. A similar effect also occurs for spherical macromolecules [4]. Theoretically, the effects of surface charge inhomogeneities on the counterion distribution has been studied for planar [5,6], cylindrical [7][8][9][10], and spherical [11,12] geometries. In particular, both Moreira et al. [5] and Lukatsky et al. [6] have shown analytically that the heterogeneity of a planar charge distribution leads to an enhanced localization of the counterions near the macromolecular surface. Thus, theoretical models that assume homogeneous surface charge distributions -including OM condensation theory -must be modified for realistic charge distributions.In this letter, we explore the effects of surface charge discretization on the counterion distribution in the absence of added salt. For small charge modulation, it is appropriate to use a perturbative expansion of the Poisson-Boltzmann (PB) equation to describe the meanfield behavior of the counterion distribution. It has already been shown that such an expansion

A model of randomly grafted AB copolymer melts is constructed in which flexible B polymer grafts are statistically attached at three possible sites along flexible A polymer backbones. An incompressible melt of such molecules is examined theoretically at equilibrium for two situations: (1) the grafting is irreversible so that the chemical disorder associated with the statistical placement of the grafts is quenched, and (2) the grafting is reversible so that the disorder is annealed. Because of the simplicity of the model, we are able to exactly carry out the two types of disorder averages, yielding effective field theories for the quenched and annealed cases. These field theories are investigated in the mean-field approximation, but without further invoking the usual weak-amplitude random phase approximation. Our results clarify the conditions for which quenched and annealed averages can be interchanged.