Electrostatic Interaction between Dissimilarly Charged Membranes in Salt-Free Solution

Abstract

A theoretical study on the electrostatic interaction between the dissimilarly charged membranes in a salt-free solution has been presented in this paper. The results show that the electric double-layer force is always repulsive for positively charged planar surfaces regardless of surface charge density (or potential) and separation; however, a long-range attraction is observed between surfaces with unequally opposite charge densities. Such attractive force also exists and is independent of the separation when both surfaces carry unequally opposite potential.

Colloidal Stability of PFSA-Ionomer Dispersions. Part I. Single-Ion Electrostatic Interaction Potential Energies

“…Figure 6 reports the case previously considered in the literature of single-ion electrostatic energies for repulsion between isolated spherical particles with a C L of 0, thereby neglecting the backside osmotic force. 7,[13][14][15][16]25 Repulsive energies are now considerably larger, as seen by the enlarged ordinate scale; all single-ion potential energies displayed in Figure 5A are negligible in comparison. Thus, neglecting the backside osmotic forces leads to single-ion dispersions that do not aggregate.…”

“…With no added electrolyte, the effective Debye length depends on the presence of nearby particles even for a dilute dispersion, making the interaction pair potential a function of the suspension volume fraction. Our work here in Part I thus deviates substantially from prior efforts ,− ,− by considering the effect of the diffuse double layers emanating from surrounding particles on the pair interaction energy. Nearby diffuse double layers provide a background osmotic force on the interacting particle pair.…”

“…To address the general single-counterion problem, we consider a colloidal dispersion of unequal-size spheres with cation-exchange sites immersed in a polar solvent with no added electrolyte (the case of anion-exchange sites follows by direct analogy). Following others, 7,[13][14][15][16]25 the Poisson− Boltzmann equilibrium relation provides an exact solution for zero-added-electrolyte diffuse double layers for flat plates. We then utilize the Derjaguin approximation 7,13,16,36 to extend the flat-plate pairwise electrostatic interaction to interacting spheres.…”

“…For finite L and from eq , this result implies that Φ′( L ) = 0 in addition to Φ( L ) = 0. Equations – are then solved to yield ,− ,,− .25ex2ex Φ = k B T z e ln true[ cos 2 ( x − L λ L 2 ) true] with a surface charge density of .25ex2ex q = − sign false( z false) 2 ε R T C L tan true( L λ L 2 true) and an effective Debye length of .25ex2ex λ normalL 2 = ε k B T z 2 e 2 C L …”

“…When the opposing particles separate significantly beyond the Debye length, the intervening solution is electrically neutral. Added positive and negative electrolyte ions allow electroneutrality to extend to distances infinitely far away where the electrostatic potential reaches a reference constant, normally taken to be zero. ,,,− …”