Multivalent Ion-Mediated Attraction between Like-Charged Colloidal Particles: Nonmonotonic Dependence on the Particle Charge

Abstract

Ion-mediated effective interactions are important for the structure and stability of charged particles such as colloids and nucleic acids. It has been known that the intrinsic electrostatic repulsion between like-charged particles can be modulated into effective attraction by multivalent ions. In this work, we examined the dependence of multivalent ion-mediated attraction between like-charged colloidal particles on the particle charge in a wide range by extensive Monte Carlo simulations. Our calculations show that for both divalent and trivalent salts, the effective attraction between like-charged colloidal particles becomes stronger with the increase of the particle charge, whereas it gradually becomes weakened when the particle charge exceeds a “critical” value. Correspondingly, as the particle charge is increased, the driving force for such effective attraction transits from an attractive electrostatic force to an attractive depletion force, and the attraction weakening by high particle charges is attributed to the transition of electrostatic force from attraction to repulsion. Our analyses suggest that the attractive depletion force and the repulsive electrostatic force at high particle charges result from the Coulomb depletion which suppresses the counterion condensation in the limited region between two like-charged colloidal particles. Moreover, our extensive calculations indicate that the “critical” particle charge decreases apparently for larger ions and smaller colloidal particles due to stronger Coulomb depletion and decreases slightly at higher salt concentrations due to the slightly enhanced Coulomb depletion in the intervening space between colloidal particles. Encouragingly, we derived an analytical formula for the “critical” particle charge based on the Lindemann melting law.

Revisiting Contact Electrification at Polymer–Liquid Interfaces

“…Moreover, T t is found to be dependent on the cation charge to size ratio. Given the same amount of thermal energy at an elevated temperature, cations with larger charge-to-size ratios would be more likely to collide and be stuck onto the PET film . As a result, the decreasing CE charge due to increasing ion attraction takes over the trend at a lower temperature, resulting in a lower T t .…”

“…Given the same amount of thermal energy at an elevated temperature, cations with larger charge-to-size ratios would be more likely to collide and be stuck onto the PET film. 40 As a result, the decreasing CE charge due to increasing ion attraction takes over the trend at a lower temperature, resulting in a lower T t . From the discussion above, we conclude that the size of anions does not seem to affect T t significantly.…”

Effective Repulsion Between Oppositely Charged Particles in Symmetrical Multivalent Salt Solutions: Effect of Salt Valence

“…For the model system, the MC simulations with the Metropolis algorithm have been employed to calculate the PMFs between oppositely charged particles in symmetrical salt solutions. The simulation cell was generally taken as a rectangular box, and to diminish the boundary effect, the box size was kept larger than the center-to-center separation x between oppositely charged particles by at least six Debye-Hückel lengths [62,64]. Each MC simulation starts from an initial configuration with two particles constrained by a spring and randomly distributed ions, and the probability to accept a trial move of an ion or a particle is given by min[1, −ΔU/k B T], where ΔU is the interaction energy change associated with the trial move of the ion or particle in the simulation cell.…”

“…k B is the Boltzmann constant, and T is the absolute temperature in kelvin. The process is repeated until the system reached equilibrium, where the separation between two oppositely charged particles becomes stable [62,64].…”

“…Effective interactions between polyelectrolytes in salt solutions have attracted rather attention and have been studied extensively in recent years [42][43][44][45][46][47][48][49][50]. The intrinsic repulsions between like-charged polyelectrolytes can be modulated into effective attractions by multivalent counterions, which is generally attributed to the bridging effect or the Coulomb depletion effect of condensed counterions [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. For oppositely charged polyelectrolytes, the intrinsic attractive interaction can be mediated into apparent repulsions by nonsymmetrical 1:2 or 1:3 salt due to the respective underneutralization and over-neutralization of the oppositely charged polyelectrolytes by their respective counterions [66][67][68][69][70][71].…”