Counterion-Mediated Weak and Strong Coupling Electrostatic Interaction between Like-Charged Cylindrical Dielectrics

arXiv:1003.1726v1 [cond-mat.soft] 8 Mar 2010 Counterion-mediated weak and strong coupling electrostatic interaction between like-charged cylindrical dielectrics Matej Kanduˇc,1 Ali Naji,2 and Rudolf Podgornik1,3 1Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia 2Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom 3Institute of Biophysics, Medical Faculty and Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia We examine the effective counterion-mediated electrostatic interaction between two like-charged dielectric cylinders immersed in a continuous dielectric medium containing neutralizing mobile coun- terions. We focus on the effects of image charges induced as a result of the dielectric mismatch between the cylindrical cores and the surrounding dielectric medium and investigate the counterion- mediated electrostatic interaction between the cylinders in both limits of weak and strong elec- trostatic couplings (corresponding, e.g., to systems with monovalent and multivalent counterions, respectively). The results are compared with extensive Monte-Carlo simulations exhibiting good agreement with the limiting weak and strong coupling results in their respective regime of validity. I. INTRODUCTION Electrostatic interactions of charged macromolecules and colloids are often governed by small oppositely charged ions (counterions) that maintain global elec- troneutrality. These counterion-mediated interactions play a fundamental role in classical charged (Coulomb) fluids that are abundant in biological and soft matter context [1, 2] and include many charged macromolecules (such as nucleic acids DNA and RNA, actin filaments, microtubules and globular proteins), affecting their func- tional, structural and dynamical behavior. In spite of the importance of electrostatic interactions, there is no general method that would allow for an accurate pre- diction of electrostatic effects in all regions of the pa- rameter space, defined by the surface charge density of macroions, charge valency of counterions, dielectric mismatches between the often hydrophobic core of the macromolecule and the surrounding aqueous medium, etc. Often the electrostatic interactions are treated on the Poisson-Boltzmann (PB) level leading to effective in- teractions which turn out to be always repulsive between like-charged macromolecules. Conceptually, the PB ap- proach corresponds to a mean-field treatment of electro- static interactions and is asymptotically valid only for sufficiently large separations between macromolecules, low enough surface charge densities and low counterion valency [1]. It characterizes a situation where electro- static fluctuations and correlations due to the counteri- ons are negligible. There are other regions in the pa- rameter space of charged macromolecules where one ex- pects the mean-field framework to break down leading to the emergence of a completely different non-PB-type physics. A notorious example is the phenomenon of like- charge attraction, which emerges between highly charged macroions or in the presence of high valency counterions and has been at the focus of both experimental [3–10] and theoretical investigations in recent years (see Refs. [1, 11–20] for an extensive reference list). It appears to us that among the most important re- cent advances in this field has been the systematization of non-PB effects based on the notions of weak coupling (WC) and strong coupling (SC) approximations. These terms refer to the strength of electrostatic coupling in the system and may be understood conceptually in terms of the electrostatic interactions of mobile counterions with fixed external charges (macroions) in the system when compared with direct electrostatic interactions between the counterions themselves. This latter contribution may be characterized in terms of the Bjerrum length, ℓB = e2 0/(4πεε0kBT ), (1) which corresponds to the separation at which two unit charges, e0, interact with thermal energy kBT in a medium of dielectric constant ε (in water and at room temperature, the value is ℓB ≈0.7 nm). If the charge of the counterions is +qe0 then the Bjerrum length scales as q2ℓB. The interaction of counterions with macroion charges (of surface charge density −σs) can be charac- terized in terms of the so-called Gouy-Chapman length, µ = e0/(2πqℓBσs), (2) which gives the separation at which the counterion- surface interaction energy equals kBT . The ratio of these two fundamental length scales introduces a dimensionless parameter Ξ = q2ℓB/µ, (3) which is known as the (Netz-Moreira) electrostatic cou- pling parameter Ξ [14] and quantifies the strength of electrostatic coupling in the system. This parameter is closely related to the plasma parameter of ionic systems [21] and may be written also in terms of the typical lat- eral spacing, a⊥, between counterions in the proximity of a charged su

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