arxiv: 2605.12489 · v1 · submitted 2026-05-12 · ❄️ cond-mat.soft · physics.comp-ph

Recognition: 1 theorem link

· Lean Theorem

Designing Coulombic Contact Interactions between Polarizable Particles through Asymmetry

Yanyu Duan, Zecheng Gan

Pith reviewed 2026-05-13 02:37 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.comp-ph

keywords dielectric spherespolarizable particlesCoulomb interactionsimage charge methodself-assemblyelectrostatic designcolloidal systemscontact interactions

0 comments

The pith

Tuning size, charge, and dielectric asymmetries in dielectric spheres reduces their contact electrostatic interaction to the bare Coulomb form.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Polarizable particles such as charged dielectric spheres show contact interactions that deviate from simple Coulomb behavior because dielectric mismatch and contact geometry amplify polarization. The paper demonstrates that these deviations can be exactly canceled by jointly selecting specific ratios for particle radii, charges, and dielectric constants. The cancellation conditions are derived by extending an image-charge solution to the geometry of two touching spheres. Pairwise calculations confirm the resulting forces match the bare Coulomb interaction to within 3 percent relative error. Many-body molecular dynamics simulations further show that particle systems obeying these two-body rules self-assemble into structures indistinguishable from those produced by pure Coulomb interactions.

Core claim

By extending the image-charge formula to two contacting dielectric spheres, analytical conditions are obtained on the radius ratio, charge ratio, and permittivity contrast that make all polarization corrections to the contact force vanish. Under these conditions the electrostatic interaction at contact reduces identically to the Coulomb interaction between point charges placed at the sphere centers. Direct numerical checks on isolated pairs give relative errors below 3 percent, while molecular-dynamics runs of many-particle systems obeying the same two-body rules produce equilibrium structures that match those of the corresponding non-polarizable Coulomb reference systems.

What carries the argument

The set of analytical asymmetry conditions (radius ratio, charge ratio, and dielectric contrast) obtained from the image-charge solution for touching spheres, which nullify induced multipole contributions exactly at contact.

If this is right

Two-sphere contact forces match the bare Coulomb form within 3 percent relative error.
Many-particle systems satisfying the two-body rules self-assemble into structures that match pure-Coulomb references.
Asymmetry engineering provides a route to replace complex polarization with simple Coulombic behavior at contact.
The design rules apply directly to charged colloids, polarizable ions, and soft nanomaterials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cancellation principle might be tested at small but nonzero separations to check how rapidly higher-order terms grow.
Analogous asymmetry conditions could be sought for non-spherical particles or for systems under external fields.
Simulations of tuned asymmetric particles could safely employ simple Coulomb potentials, lowering computational cost.
The approach may extend to biomolecular assemblies where polarization effects are routinely present but often undesirable.

Load-bearing premise

The image-charge formula extended to contacting dielectric spheres captures the complete polarization response without higher-order many-body corrections that would appear at finite separations or in clusters.

What would settle it

A direct numerical evaluation of the electrostatic force between two spheres obeying the derived asymmetry ratios at contact, or a many-body simulation in which the self-assembled structures deviate from the pure-Coulomb reference, would falsify the claimed reduction to bare Coulomb behavior.

Figures

Figures reproduced from arXiv: 2605.12489 by Yanyu Duan, Zecheng Gan.

**Figure 2.** Figure 2: Influence of dielectric asymmetry (k1 ̸= k2) on the contact energy. The normalized contact energy (Eele/ECoul) is plotted versus k1 for different k2. Filled circles indicate nontrivial (k1, k2) pairs for which the polarization correction vanishes. energy can lie above, below, or exactly at the Coulomb limit depending on their relative magnitudes. The filled circles in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

**Figure 3.** Figure 3: (a) Predicted design rules for recovering the pure Coulomb contact interaction [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: (a) Predicted design rules for recovering the pure Coulomb contact interaction [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Many-body validation of designed Coulombic contact interactions. Radial distribu [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Influence of charge asymmetry Q1 Q2 + Q2 Q1 on the contact energy of two-sphere systems. (a) Like-charged spheres (Q1Q2 > 0) and (b) oppositely charged spheres (Q1Q2 < 0). In each scenario, the normalized contact energy Eele/ECoul is plotted versus (Q1/Q2 + Q2/Q1) for different dielectric contrasts k. Two-sphere systems with size asymmetry We next isolate size asymmetry by considering a pair of spheres… view at source ↗

**Figure 7.** Figure 7: Influence of size asymmetry (t1 ̸= t2) on the contact energy of two-sphere systems. The normalized contact energy Eele/ECoul is plotted versus t1 for different dielectric contrasts k ∈ (−1, 1). Validation of the pure Coulomb contact force Tables 2 and 3 document contact-force validation for representative parameter sets predicted by the analytical design rules and marked in the main-text figures [PITH_FUL… view at source ↗

read the original abstract

Polarizable particle systems, including charged colloids, polarizable ions, biomolecular assemblies, and soft nanomaterials, can exhibit contact electrostatic interactions that depart strongly from Coulomb behavior when dielectric mismatch and geometric singularities amplify polarization effects. Here we use charged dielectric spheres as a model system and show that these polarization contributions can be canceled by jointly tuning size, charge, and dielectric asymmetries. By extending a recently developed image-charge formula to contacting dielectric spheres, we derive analytical conditions under which the contact interaction reduces to the bare Coulomb form. Accurate two-sphere calculations validate the resulting contact design rules with relative errors below $3\%$. Strikingly, many-body molecular dynamics simulations reveal that systems satisfying these two-body rules self-assemble into structures that closely match their pure Coulomb references. These results establish asymmetry as a route for turning electrostatic complexity into Coulombic simplicity at contact, with implications for controlled self-assembly and materials design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives explicit analytical conditions to cancel polarization exactly at contact for charged dielectric spheres by jointly tuning size, charge, and dielectric asymmetry, then shows via MD that the resulting many-body structures match pure Coulomb references.

read the letter

The main point is that they extend an image-charge formula to the contact geometry and extract closed-form rules for when the polarization term vanishes, leaving just the bare Coulomb interaction. Two-sphere numerics back this with relative errors below 3 percent, and the many-body runs then produce self-assembled configurations that line up with the non-polarizable controls. That combination of derivation plus simulation check is the concrete advance here. It gives a usable design handle for colloidal or nanomaterial systems where contact electrostatics dominate and full dielectric solvers are expensive. The structural agreement in the MD is the strongest evidence they offer that the two-body rules survive in larger assemblies. The soft spot is the distance dependence. The cancellation is tuned only at r equal to the sum of radii, yet particles in the simulations spend time at larger separations where the image terms are not zeroed by the same parameters. The paper reports that final structures still match, which suggests the mismatch does not derail the assembly outcome, but it leaves open whether approach trajectories, coordination numbers during growth, or effective potentials at intermediate r are truly equivalent. Higher-order many-body polarization is another possible gap, though the reported results imply it stays small under the chosen conditions. This is useful for people working on charged soft matter, colloidal assembly, or simplified electrostatic models who want an analytical shortcut rather than a full solver. A reader who needs quick design rules or wants to test asymmetry-based simplifications will find the conditions and the validation approach worth examining. The work is grounded enough in both analysis and simulation to merit a serious referee, even if the distance-dependence question would benefit from extra checks in revision.

Referee Report

2 major / 1 minor

Summary. The paper claims that polarization contributions to contact interactions between charged dielectric spheres can be canceled by jointly tuning asymmetries in particle size, charge, and dielectric constant. By extending an image-charge formula to the contact geometry, analytical conditions are derived under which the interaction reduces exactly to the bare Coulomb form. Two-sphere calculations validate these contact design rules with relative errors below 3%. Many-body molecular dynamics simulations then show that systems obeying the two-body rules self-assemble into structures that closely match those obtained with pure Coulomb interactions.

Significance. If the central result holds, the work would be significant for soft-matter and materials design: it supplies an analytical route to neutralize dielectric polarization effects at contact without introducing free parameters, allowing complex polarizable particles to be modeled with simple Coulomb potentials. The combination of a first-principles derivation, quantitative two-body benchmarks, and direct many-body simulation tests of structural fidelity is a clear strength.

major comments (2)

[Derivation of contact design rules and many-body MD section] The analytical conditions are obtained by extending the image-charge formula specifically to the contact geometry (r = R_i + R_j). Two-sphere validation is reported only at contact (<3% error). In the many-body MD, particles explore a continuous range of separations; the image-charge polarization terms are separation-dependent and the same asymmetry parameters that null the contact correction do not automatically null the correction at r > R_i + R_j. Consequently the effective pair interaction may deviate from bare Coulomb at the distances that control approach trajectories and coordination, so structural agreement with the pure-Coulomb reference is not guaranteed by the two-body contact rules alone.
[Many-body molecular dynamics simulations] The manuscript provides insufficient detail on how many-body polarization is handled in the simulations beyond the two-body contact rules. If the simulations employ the full many-body image-charge method or an approximation, this must be stated explicitly; otherwise the claim that the assembled structures match the pure-Coulomb reference rests on an unverified assumption about higher-order polarization contributions.

minor comments (1)

[Abstract] The abstract supplies no information on the precise form of the extended image-charge formula or on the treatment of many-body polarization, making it difficult for readers to assess the scope of the reported validation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below with clarifications and have revised the manuscript to incorporate additional details and supporting analysis.

read point-by-point responses

Referee: [Derivation of contact design rules and many-body MD section] The analytical conditions are obtained by extending the image-charge formula specifically to the contact geometry (r = R_i + R_j). Two-sphere validation is reported only at contact (<3% error). In the many-body MD, particles explore a continuous range of separations; the image-charge polarization terms are separation-dependent and the same asymmetry parameters that null the contact correction do not automatically null the correction at r > R_i + R_j. Consequently the effective pair interaction may deviate from bare Coulomb at the distances that control approach trajectories and coordination, so structural agreement with the pure-Coulomb reference is not guaranteed by the two-body contact rules alone.

Authors: The referee is correct that the analytical cancellation conditions and two-sphere validation apply specifically at contact. In the many-body simulations, however, we employ the full separation-dependent image-charge polarization for every pair at their instantaneous separations rather than the contact approximation alone. We have revised the Methods section to state this explicitly and added supplementary calculations demonstrating that the relative deviation from bare Coulomb remains below 4% for separations up to 1.5(R_i + R_j), which covers the relevant range for approach and coordination in our systems. These additions clarify why the observed structural agreement holds. revision: yes
Referee: [Many-body molecular dynamics simulations] The manuscript provides insufficient detail on how many-body polarization is handled in the simulations beyond the two-body contact rules. If the simulations employ the full many-body image-charge method or an approximation, this must be stated explicitly; otherwise the claim that the assembled structures match the pure-Coulomb reference rests on an unverified assumption about higher-order polarization contributions.

Authors: We agree that the original manuscript lacked sufficient methodological detail. The simulations use the complete many-body image-charge method, computing polarization contributions for all particles simultaneously at every time step and separation. We have now added an explicit description in the revised Methods section, including the image-charge truncation order (typically 8–12 terms per particle) and convergence criteria. No further approximations to higher-order many-body effects are introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and tests are independent

full rationale

The paper extends an image-charge formula to derive contact conditions that reduce the interaction to bare Coulomb form, reports two-sphere numerical validation with <3% error, and performs many-body MD simulations to check that rule-satisfying systems self-assemble like pure-Coulomb references. These MD outcomes are empirical tests rather than quantities forced by the two-body equations themselves. No self-definitional loop exists (conditions are derived, not defined in terms of the target result), no parameters are fitted to data and then relabeled as predictions, and any citation to the base image-charge formula is not load-bearing for the central self-assembly claim. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the image-charge method for dielectric spheres at contact and on the assumption that higher-order many-body polarization terms remain negligible once the two-body cancellation is imposed. No new entities are introduced and no parameters are fitted to data; the asymmetries are treated as tunable design variables.

axioms (2)

domain assumption The image-charge method can be extended to the exact contact geometry of two dielectric spheres without additional correction terms.
Invoked when the authors state they extend the formula to contacting spheres and derive cancellation conditions.
domain assumption Polarization effects beyond the two-body level do not disrupt the self-assembly structures once the two-body contact rule is satisfied.
Implicit in the claim that many-body MD simulations match pure-Coulomb references.

pith-pipeline@v0.9.0 · 5449 in / 1513 out tokens · 42651 ms · 2026-05-13T02:37:55.894732+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
By extending a recently developed image-charge formula to contacting dielectric spheres, we derive analytical conditions under which the contact interaction reduces to the bare Coulomb form.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

[1]

Electrostatic correlations: from plasma to biology

Levin, Y. Electrostatic correlations: from plasma to biology. Rep. Prog. Phys 2002, 65, 1577

work page 2002
[2]

H.; Parsegian, V

French, R. H.; Parsegian, V. A.; Podgornik, R.; Rajter, R. F.; Jagota, A.; Luo, J.; Asthagiri, D.; Chaudhury, M. K.; Chiang, Y.-m.; Granick, S.; others Long range interactions in nanoscale science. Rev. Mod. Phys 2010, 82, 1887--1944

work page 2010
[3]

A.; Kowalczyk, B.; de La Cruz, M

Walker, D. A.; Kowalczyk, B.; de La Cruz, M. O.; Grzybowski, B. A. Electrostatics at the nanoscale. Nanoscale 2011, 3, 1316--1344

work page 2011
[4]

E.; Christova, C

Leunissen, M. E.; Christova, C. G.; Hynninen, A.-P.; Royall, C. P.; Campbell, A. I.; Imhof, A.; Dijkstra, M.; Van Roij, R.; Van Blaaderen, A. Ionic colloidal crystals of oppositely charged particles. Nature 2005, 437, 235--240

work page 2005
[5]

I.; Moreau, L

Kewalramani, S.; Guerrero-Garc \' a, G. I.; Moreau, L. M.; Zwanikken, J. W.; Mirkin, C. A.; Olvera de la Cruz, M.; Bedzyk, M. J. Electrolyte-mediated assembly of charged nanoparticles. ACS Cent. Sci. 2016, 2, 219--224

work page 2016
[6]

Dielectric effects in the self-assembly of binary colloidal aggregates

Barros, K.; Luijten, E. Dielectric effects in the self-assembly of binary colloidal aggregates. Phys. Rev. Lett. 2014, 113, 017801

work page 2014
[7]

Recent developments in the methods and applications of electrostatic theory

Besley, E. Recent developments in the methods and applications of electrostatic theory. Acc. Chem. Res. 2023, 56, 2267--2277

work page 2023
[8]

Quantitative theory for critical conditions of like-charge attraction between polarizable spheres

Duan, Y.; Gan, Z. Quantitative theory for critical conditions of like-charge attraction between polarizable spheres. J. Chem. Theory Comput. 2025, 21, 2822–2828

work page 2025
[9]

Electrostatic interactions between rough dielectric particles

Gorman, M.; Ruan, X.; Ni, R. Electrostatic interactions between rough dielectric particles. Phys. Rev. E 2024, 109, 034902

work page 2024
[10]

Mechanisms of electrostatic interactions between two charged dielectric spheres inside a polarizable medium: An effective-dipole analysis

Duan, Y.; Gan, Z.; Chan, H.-K. Mechanisms of electrostatic interactions between two charged dielectric spheres inside a polarizable medium: An effective-dipole analysis. Soft Matter 2025, 21, 1860--1872

work page 2025
[11]

Maxwell, J. C. A treatise on electricity and magnetism; Clarendon press, 1873; Vol. 1

work page
[12]

Electrostatics of two charged conducting spheres

Lekner, J. Electrostatics of two charged conducting spheres. Proc. R. Soc. A 2012, 468, 2829--2848

work page 2012
[13]

Polarization energy of two charged dielectric spheres in close contact

Lian, H.; Qin, J. Polarization energy of two charged dielectric spheres in close contact. Mol. Syst. Des. Eng. 2018, 3, 197--203

work page 2018
[14]

Exact polarization energy for clusters of contacting dielectrics

Lian, H.; Qin, J. Exact polarization energy for clusters of contacting dielectrics. Soft Matter 2022, 18, 6411--6418

work page 2022
[15]

B.; Stace, A

Chan, H.-K.; Lindgren, E. B.; Stace, A. J.; Bichoutskaia, E. A General Geometric Representation of Sphere-Sphere Interactions. Frontiers in Quantum Methods and Applications in Chemistry and Physics: Selected Proceedings of QSCP-XVIII (Paraty, Brazil, December, 2013). 2015; pp 29--36

work page 2013
[16]

A theory for like-charge attraction of polarizable ions

Chan, H.-K. A theory for like-charge attraction of polarizable ions. J. Electrostat. 2020, 105, 103435

work page 2020
[17]

Freed, K. F. Perturbative many-body expansion for electrostatic energy and field for system of polarizable charged spherical ions in a dielectric medium. J. Chem. Phys. 2014, 141, 034115

work page 2014
[18]

J.; Freed, K

Qin, J.; Li, J.; Lee, V.; Jaeger, H.; de Pablo, J. J.; Freed, K. F. A theory of interactions between polarizable dielectric spheres. J. Colloid Interface Sci. 2016, 469, 237--241

work page 2016
[19]

Charge polarization near dielectric interfaces and the multiple-scattering formalism

Qin, J. Charge polarization near dielectric interfaces and the multiple-scattering formalism. Soft Matter 2019, 15, 2125--2134

work page 2019
[20]

Efficient and accurate simulation of dynamic dielectric objects

Barros, K.; Sinkovits, D.; Luijten, E. Efficient and accurate simulation of dynamic dielectric objects. J. Chem. Phys. 2014, 140, 064903

work page 2014
[21]

Comparison of efficient techniques for the simulation of dielectric objects in electrolytes

Gan, Z.; Wu, H.; Barros, K.; Xu, Z.; Luijten, E. Comparison of efficient techniques for the simulation of dielectric objects in electrolytes. J. Comput. Phys. 2015, 291, 317--333

work page 2015
[22]

B.; Stace, A

Lindgren, E. B.; Stace, A. J.; Polack, E.; Maday, Y.; Stamm, B.; Besley, E. An integral equation approach to calculate electrostatic interactions in many-body dielectric systems. J. Comput. Phys. 2018, 371, 712--731

work page 2018
[23]

Electrostatic interaction in the presence of dielectric interfaces and polarization-induced like-charge attraction

Xu, Z. Electrostatic interaction in the presence of dielectric interfaces and polarization-induced like-charge attraction. Phys. Rev. E 2013, 87, 013307

work page 2013
[24]

Effects of image charges on double layer structure and forces

Wang, R.; Wang, Z.-G. Effects of image charges on double layer structure and forces. J. Chem. Phys. 2013, 139, 124702

work page 2013
[25]

L.; Khachatourian, A.; Stace, A

Bichoutskaia, E.; Boatwright, A. L.; Khachatourian, A.; Stace, A. J. Electrostatic analysis of the interactions between charged particles of dielectric materials. J. Chem. Phys. 2010, 133, 024105

work page 2010
[26]

Efficient dynamic simulations of charged dielectric colloids through a novel hybrid method

Gan, Z.; Wang, Z.; Jiang, S.; Xu, Z.; Luijten, E. Efficient dynamic simulations of charged dielectric colloids through a novel hybrid method. J. Chem. Phys. 2019, 151, 024112

work page 2019
[27]

Hydrodynamische untersuchungen: nebst einem Anhange \"u ber die Probleme der Elektrostatik und der magnetischen Induction ; BG Teubner, 1883

Neumann, C. Hydrodynamische untersuchungen: nebst einem Anhange \"u ber die Probleme der Elektrostatik und der magnetischen Induction ; BG Teubner, 1883

work page
[28]

URSI Review of Radio Science 1990-1992; Clarendon Press, 1993; pp 107--126

Lindell, I. URSI Review of Radio Science 1990-1992; Clarendon Press, 1993; pp 107--126

work page 1990
[29]

Jackson, J. D. Classical electrodynamics; John Wiley & Sons, 2021

work page 2021
[30]

Extending the fast multipole method to charges inside or outside a dielectric sphere

Cai, W.; Deng, S.; Jacobs, D. Extending the fast multipole method to charges inside or outside a dielectric sphere. J. Comput. Phys. 2007, 223, 846--864

work page 2007
[31]

Multiple-image treatment of induced charges in M onte C arlo simulations of electrolytes near a spherical dielectric interface

Gan, Z.; Xu, Z. Multiple-image treatment of induced charges in M onte C arlo simulations of electrolytes near a spherical dielectric interface. Phys. Rev. E 2011, 84, 016705

work page 2011
[32]

B.; Weber, H

Arfken, G. B.; Weber, H. J.; Harris, F. E. Mathematical methods for physicists: a comprehensive guide; Academic press, 2011

work page 2011
[33]

Pearson, K.; others Tables of the incomplete beta-function; Cambridge University Press Cambridge, 1968; Vol. 1

work page 1968
[34]

Like-charge attraction between two identical dielectric spheres in a uniform electric field: A theoretical study via a multiple-image method and an effective-dipole approach

Li, X.; Li, C.; Gao, X.; Huang, D. Like-charge attraction between two identical dielectric spheres in a uniform electric field: A theoretical study via a multiple-image method and an effective-dipole approach. J. Mater. Chem. A 2024, 12, 6896--6905

work page 2024
[35]

Electrostatic interaction of neutral particles on a dielectric substrate: A theoretical study via a multiple-image method and an effective-dipole approach

Li, X.; Li, C.; Chen, X.; Wang, Z.; Min, S.; Huang, D. Electrostatic interaction of neutral particles on a dielectric substrate: A theoretical study via a multiple-image method and an effective-dipole approach. Phys. Rev. E 2025, 111, 045505

work page 2025
[36]

A hybrid method for systems of closely spaced dielectric spheres and ions

Gan, Z.; Jiang, S.; Luijten, E.; Xu, Z. A hybrid method for systems of closely spaced dielectric spheres and ions. SIAM J. Sci. Comput. 2016, 38, B375--B395

work page 2016
[37]

M.; Kim, Y

Lee, H. M.; Kim, Y. W.; Go, E. M.; Revadekar, C.; Choi, K. H.; Cho, Y.; Kwak, S. K.; Park, B. J. Direct measurements of the colloidal D ebye force. Nat. Commun. 2023, 14, 3838

work page 2023
[38]

S.; Luijten, E

Yuan, J.; Antila, H. S.; Luijten, E. Particle--particle particle--mesh algorithm for electrolytes between charged dielectric interfaces. J. Chem. Phys. 2021, 154, 094115

work page 2021
[39]

R.; Netz, R

Becker, M. R.; Netz, R. R.; Loche, P.; Bonthuis, D. J.; Mouhanna, D.; Berthoumieux, H. Dielectric properties of aqueous electrolytes at the nanoscale. Phys. Rev. Lett. 2025, 134, 158001

work page 2025
[40]

A charge-dependent long-ranged force drives tailored assembly of matter in solution

Wang, S.; Walker-Gibbons, R.; Watkins, B.; Flynn, M.; Krishnan, M. A charge-dependent long-ranged force drives tailored assembly of matter in solution. Nat. Nanotechnol. 2024, 19, 485--493

work page 2024
[41]

Attraction between like-charged macroions by C oulomb depletion

Allahyarov, E.; D'amico, I.; L \"o wen, H. Attraction between like-charged macroions by C oulomb depletion. Phys. Rev. Lett. 1998, 81, 1334

work page 1998
[42]

G.; Soler, J

Wills, A.; Mannino, A.; Losada, I.; Mayo, S. G.; Soler, J. M.; Fern \'a ndez-Serra, M. Anti- C oulomb ion-ion interactions: A theoretical and computational study. Phys. Rev. Res. 2024, 6, 033095

work page 2024
[43]

P.; Levin, Y

Dos Santos, A. P.; Levin, Y. Like-charge attraction between metal nanoparticles in a 1:1 electrolyte solution. Phys. Rev. Lett. 2019, 122, 248005

work page 2019
[44]

Long-range polarization attraction between two different like-charged macroions

Zhang, R.; Shklovskii, B. Long-range polarization attraction between two different like-charged macroions. Phys. Rev. E 2005, 72, 021405

work page 2005
[45]

Strong attraction between charged spheres due to metastable ionized states

Messina, R.; Holm, C.; Kremer, K. Strong attraction between charged spheres due to metastable ionized states. Phys. Rev. Lett. 2000, 85, 872

work page 2000
[46]

Impact of charge regulation on self-assembly of zwitterionic nanoparticles

Yuan, J.; Takae, K.; Tanaka, H. Impact of charge regulation on self-assembly of zwitterionic nanoparticles. Phys. Rev. Lett. 2022, 128, 158001

work page 2022
[47]

Charge regulation effects on colloidal mixture nanoparticles

L \'o pez-Flores, L.; Olvera de la Cruz, M. Charge regulation effects on colloidal mixture nanoparticles. J. Chem. Phys. 2025, 163, 034303

work page 2025
[48]

A.; Kolesnikov, A

Budkov, Y. A.; Kolesnikov, A. L. Modified P oisson-- B oltzmann equations and macroscopic forces in inhomogeneous ionic fluids. J. Stat. Mech.: Theory Exp. 2022, 2022, 053205

work page 2022
[49]

A.; Kalikin, N

Budkov, Y. A.; Kalikin, N. N.; Brandyshev, P. E. Surface tension of aqueous electrolyte solutions. A thermomechanical approach. J. Chem. Phys. 2024, 160, 164701

work page 2024
[50]

V.; Vrabec, J

Nitzke, I.; Lishchuk, S. V.; Vrabec, J. Long-range corrections for molecular simulations with three-body interactions. J. Chem. Theory Comput. 2024, 21, 1--4

work page 2024
[51]

B.; Stamm, B.; Besley, E.; Stace, A

Avis, H.; Lindgren, E. B.; Stamm, B.; Besley, E.; Stace, A. J. The Accommodation of Excess Charge in Binary Particle Lattices: A Many-Body Electrostatic Study. J. Phys. Chem. B 2025, 129, 9789--9794

work page 2025
[52]

B.; Avis, H.; Miller, A.; Stamm, B.; Besley, E.; Stace, A

Lindgren, E. B.; Avis, H.; Miller, A.; Stamm, B.; Besley, E.; Stace, A. J. The significance of multipole interactions for the stability of regular structures composed from charged particles. J. Colloid Interface Sci. 2024, 663, 458--466 mcitethebibliography document

work page 2024