Electrostatic correlation force of discretely charged membranes O. Gonza ´ lez-Amezcua, 1 M. Herna ´ ndez-Contreras, 1 and P. Pincus 2 1 Departamento de Fı ´sica, Centro de Investigacio ´n y Estudios Avanzados del Instituto Polite ´cnico Nacional, Apartado Postal 14-740, Me ´xico Distrito Federal, Mexico 2 Materials Research Laboratory, University of California, Santa Barbara, California 93106 ~Received 19 April 2001; revised manuscript received 18 June 2001; published 21 September 2001! The total force between two like charged surfaces is investigated as a function of counterion concentration in aqueous solution and surfaces distance of separation. A smooth and a discrete density of surface charge s s lead to differences in the force distance curve at high s s , which are negligible for low surface charge. The total force per unit area with divalent counterions is an oscillating function of s s . At fixed surfaces separation and region of attraction ~increasing s s !, there is a variation in its strength that results from a competition between the ideal kinetic and ion-ion correlation force components as predicted from the anisotropic hypernetted chain approximation. DOI: 10.1103/PhysRevE.64.041603 PACS number~s!: 68.15.1e, 82.65.1r, 68.43.2h I. INTRODUCTION Attractive forces between like charged objects due to the correlated fluctuations of their counterion concentration plays an important role in the adhesion of biological cell membranes @1–4#. They have been also recognized as an important mechanism for DNA to get tightly packed in a bacterial capsid, and in eukaryotic cells @5#. Experimentally these forces have been studied systematically with osmotic stress techniques in bulk aqueous solution of DNA @5#, and with the use of high resolution x-ray scattering techniques in an overall neutral stack of synthetic charged membranes made of a mixture of the surfactant sodium dodecyl sulfate and the pentanol cosurfactant, in equilibrium with their dis- sociated counterions @6,7#. Recently, it was demonstrated that these forces arise from the correlated fluctuations of multivalent counterions that form almost two-dimensional clouds closely bound to their compensating planar charged surfaces @8#. Also condensed monovalent counterions on highly charged surfaces with a smooth distribution of charge lead to this attraction, as shown through numerical calcula- tions of integral equation theory @9–12# and computer simu- lations @13–15#. Using a Gaussian fluctuation theory the asymptotic analytic expressions for this force at all separa- tion regimes that correspond to a few angstroms were found in a model of a pair of neutral membranes formed by an equimolar mixture of a simple salt, which corresponds to the case of strongly adsorbed counterions on discretely charged surfaces @16,17#. It has been also used for counterions delo- calized from the surfaces through an approximate mean field distribution @18#. In many real situations counterions are de- localized from the surfaces and distributed quite inhomoge- neously @19#; therefore, the correlated pressure is very sensi- tive to the actual ionic profile distribution and, it is expected the pressure curve to show a more complex behavior as a function of separation, presenting, however, the asymptotic force laws cited above. We performed numerical calculations of the hypernetted chain theory ~HNC! on the restricted primitive model ~RPM! for continuous and discretely charged membranes with their delocalized monovalent and divalent counterions in solution, in order to determine the total pressure between pairs of membranes under different thermodynamic conditions of charge and separation. Our cal- culations show that independent of the way the surface charges are distributed, the pressure curves with a low to moderate surface charge s s coincide quantitatively, but there appears quantitative differences at very high charge density. The total pressure of monovalent counterions is repulsive in the range of charge considered. For divalent ions it is found to be repulsive for low s s , and attractive for high values of surface charge with its strength varying nonmonotically as a function of the surface charge. These oscillations in the pres- sure curve derive from the competition between the ideal kinetic and ion-ion correlations pressure contributions. Counterion profiles at low s s are found to be different for discrete, and smooth s s . However, these differences dissa- pear at moderate and high charge in which case the ions deplete from the middle of the slit forming a cloud closer to the surfaces. It is also noticeable that radial distribution func- tions of counterions that are located in the same layer, par- allel, and just next to the surfaces display an accumulation of counterion pairs driven by the positional correlations with the oppositely charged neighbors. II. MODEL SYSTEM Two models of charged flat membranes are studied. The aqueous solvent is treated as a continuous dielectric, e 578.5. Smooth and discrete distribution ~of charged spheres on surfaces! are considered, neutralized by monovalent and divalent counterions that have identical size diameter s 54.25 Å. We shall not treat the case with added salt. Two such surfaces are separated by a mean distance h, Fig. 1. In this RPM model, N particles interact with the pairwise Cou- lomb potentials v i ~ r 3 D ! 5 H q i e r 3 D , r 3 D .s ‘ , r 3 D ,s , ~1! where q i 5eZ i is the total charge on particle i of valence Z i , and e is the elementary electric charge. r 3 D is the center to center distance of separation of two particles. Each of these model membrane systems satisfy the electroneutrality condi- PHYSICAL REVIEW E, VOLUME 64, 041603 1063-651X/2001/64~4!/041603~5!/$20.00 ©2001 The American Physical Society 64 041603-1