Rigorous treatment of pairwise and many-body electrostatic interactions among dielectric spheres at the Debye–Hückel level

Abstract

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye–Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye–Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium. Graphic abstract

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson–Boltzmann Framework: An Analytical Treatment

“…Appendix I briefly summarizes the minimal necessary information on the modified Bessel functions used in the text. Let us also note, however, that many authors ,,, prefer to express potentials 7 in terms of complex-valued spherical harmonics Y nm (θ i , φ) = ( 2 n + 1 ) ( n − m ) ! 4 π ( n + m ) ! P n m ( cos ⁡ θ i ) normale ı m φ instead of using the real-valued ones (that is, cos( mφ ) P n m (μ i ), sin( mφ ) P n m (μ i )); this case and the corresponding re-expansion for the DH potential are discussed in Appendix G.…”

“…The main difficulty in determining expansion coefficients in and from the boundary conditions 4 is that the expansions for Φ out, i ( r̃ i , θ i , φ) and Φ out, j ( r̃ j , θ j , φ) refer to different spherical coordinate systems and corresponding spherical harmonics. For instance, in order to impose boundary conditions 4, the authors of recent refs − propose to re-expand the potential, say Φ out, j , in terms of coordinates (and corresponding orthogonal Legendre polynomials) of the other sphere i ; let us note that this is quite a conventional approach which is followed by many authors, see refs , , − , , , − , − , allowing one to handle the corresponding boundary conditions correctly from the mathematical point of view. Let us also note that, in contrast to the well-known works in refs , , and , the theory built in refs − does not make use of the additional reflection symmetry about the plane bisecting the line connecting the spheres’ centers and the corresponding equality of the expansion coefficients of Φ out, i and Φ out, j , which rely on the assumption that the radii of the spheres are equal.…”

“…Many authors ,,, prefer to express potentials 7 in terms of complex-valued spherical harmonics Y nm (θ i , φ) instead of using the real-valued ones; e.g., this makes sense if one wants to take advantage of the simplicity of the rotation of spherical harmonics using the Wigner functions , (which, in turn, allows one to re-expand the external potentials in multibody systems). It can be shown that re-expansion can be rewritten for this case in terms of Y nm as follows: .25ex2ex normalΦ out , j = prefix∑ n = 0 + ∞ ∑ m = − n n K n + 1 / 2 ( r̃ j ) r̃ j G n m , j Y n m ( θ j , φ ) infix= prefix∑ n = 0 + ∞ ∑ m = − n n ( ∑ l = false| m false| + ∞ b̆ n m l false( r̃ i , R̃ false) …”

“…This, however, does not diminish the importance of studying the behavior of the LPBE also in the case of highly charged objects: their electrostatics may still be correctly described at sufficiently long distances (as compared to the Debye length) by the usual DH approximation provided that the sources of the electric field are properly renormalized. − (See also the recent ref for additional comments concerning the ranges of applicability of the DH theory.) This once again emphasizes the importance of a thorough study of the DH approximations, both theoretically and numerically, and justifies the constant stream of works related to the LPBE (see recent refs , , and − , and references therein).…”

“…Recently, Filippov and Derbenev and co-workers have published several works ,, exploring the electrostatic force between two charged polarizable spheres immersed in an electrolytic solution or in equilibrium plasma. Other relevant studies important to mention here are refs , , , , and − . Let us note that the existing literature largely focuses on treating the case of a system of two spherical particles with special symmetries also in the charge (especially, azimuthal symmetry).…”