Weak and Strong Coupling Theories for Polarizable Colloids and Nanoparticles

Abstract

A theory is presented which allows us to accurately calculate the density profile of monovalent and multivalent counterions in suspensions of polarizable colloids or nano-particles. In the case of monovalent ions, we derive a weak-coupling theory that explicitly accounts for the ion-image interaction, leading to a modified Poisson-Boltzmann equation. For suspensions with multivalent counterions, a strong-coupling theory is used to calculate the density profile near the colloidal surface and a Poisson-Boltzmann equation with a renormalized boundary condition to account for the counterion distribution in the far-field. All the results are compared with the Monte Carlo simulations, showing an excellent agreement between the theory and the simulations.PACS numbers: 64.70.pv, 61.20.Qg, 82.45.Gj Colloidal suspensions are of great practical interest for biology, chemistry, and physics. The subject has a long history going back more than a hundred years. In spite of the intense effort, many interesting phenomena which are found in colloidal science have not been fully elucidated. For example, it is well known that the stability of a hydrophobic colloidal suspension depends specifically on the electrolyte present in suspension. Addition of multivalent counterions results in a rapid precipitation of colloidal particles. What is more surprising is that even for monovalent counterions, stability of colloidal suspensions depend strongly on the precise nature of the counterions. Thus, as one goes along the halogen series, the critical coagulation concentrations of positive colloidal particles can decrease by as much as an order of magnitude, when anion is changed from fluoride to iodide [1]. Another interesting phenomenon found in suspensions with multivalent ions is the reversal of electrophoretic mobility [2,3], or equivalently charge reversal [4][5][6][7]. Under some conditions, it is also possible to observe like-charge attraction between the colloidal particles of the same sign of charge [8][9][10][11][12]. Many of these interesting phenomena are the consequence of strong electrostatic correlations between the counterions. The role of electrostatic correlations has been studied using simple models of colloidal suspensions which neglect particle polarizability. The standard Poisson-Boltzmann equation (PB) -used extensively in colloidal science -fails to account for the induced charge at the particle-solvent interface, predicting that the counterion density should remain unaffected by the colloidal polarizability. In this Letter, we will show that the induced colloidal charge significantly modifies the ionic density distribution even for monovalent counterions. The theory developed in this Letter allows us to accurately predict the counterion density distribution both in the weak (monovalent counterions) and strong (multivalent counterions) coupling limits.We will use the primitive model of colloidal suspension in which colloidal particles are represented by hard spheres of radius a and dielectric constant ǫ c with the ...

On the theoretical description of weakly charged surfaces

“…It is also difficult to physically reconcile the absence of the image interaction from the Boltzmann factor in the weak-coupling limit with its "reemergence" in the strong coupling limit in the NKP approach. Furthermore, ion depletion near a weakly charged dielectric interface has been observed in Monte Carlo simulation 15,27 as well as predicted by the hypernetted chain approximation (HNC) integral equation theory that includes the image charge interactions. 28 In this work, we clarify the origin of these discrepancies by a re-examination of the role of the image charge interaction in the physical weak-coupling limit.…”

“…The self-energy due to image charge repulsion appears in the Boltzmann factor and is responsible for the depletion layer in the ion distribution near the surface, as shown in Figure 5. Note, however, in the original WOS theory as well as in subsequent treatments, 14,26,27,37,38 the image charge term was added to the Boltzmann factor ad hoc based on physical intuition, whereas in our theory, its appearance is the result of systematic derivation. Therefore, our theory not only recovers the WOS theory (upon making additional approximations, e.g., by using the constant bulk screening length for the image force potential) but also provides the means for systematically improving the WOS theory.…”

“…The presence of the depletion boundary layer inside the Gouy-Chapman length is obvious, and is consistent with results from Monte Carlo simulation. 15,27 Within the depletion boundary layer (z < d ∼ l B Ξ −1/2 ), image charge repulsion is dominant and ions are excluded from the plate surface. In the point-charge model, the self-energy diverges to infinity at the plate surface; thus, the ion concentration vanishes at z = 0.…”

“…To avoid the complexity of solving the equation for the Green function (2.12), previous work usually invoked approximate schemes, e.g., by replacing the spatially varying screening length by the bulk Debye length 14,26,27,37,38 or using a WKB-like approximation. [34][35][36] However, the screening on the image force at the dielectric interface is inhomogeneous, long-ranged, and accumulative, which cannot be captured fully by these approximate methods.…”

Strong‐Coupling Theory for a Polarizable Planar Colloid

“…[5], for a spherical macro-ion in a concentric spherical Wigner-Seitz cell. It was realized that while the image charge effects are generally small in the weak-coupling regime, they become relevant in the strong-coupling limit.…”