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arxiv: 2606.21184 · v1 · pith:3FNXL32Snew · submitted 2026-06-19 · ❄️ cond-mat.soft

Many-body attractions do not stabilize gas-liquid phase separation in aqueous dispersions of charged colloids within the Poisson-Boltzmann framework

Thijs ter Rele , Ren\'e van Roij , Marjolein Dijkstra This is my paper

Pith reviewed 2026-06-26 12:57 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords charged colloidsPoisson-Boltzmannmany-body interactionsphase separationmachine-learned potentialsgas-liquid transitioncharge regulation
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0 comments X

The pith

Higher-order many-body terms cancel triplet attractions and eliminate gas-liquid phase separation in Poisson-Boltzmann models of charged colloids.

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

The paper tests whether reported attractive three-body forces between like-charged colloids can produce gas-liquid coexistence when charge regulation and all higher-order terms are retained. Machine-learned potentials are fitted directly to finite-element Poisson-Boltzmann solutions computed on clusters of 13 and then 48 colloids. Three-body contributions remain attractive, yet four-body and higher contributions are net repulsive, progressively reducing overall cohesion. Molecular-dynamics runs with the 13-particle potentials still exhibit phase separation, while those trained on 48-particle clusters show no broad separation. Pair and triplet potentials of mean force from explicit primitive-model simulations match the Poisson-Boltzmann results, confirming the mean-field electrostatic treatment.

Core claim

Machine-learned many-body potentials trained on Poisson-Boltzmann calculations for colloid clusters show that the strongly attractive three-body term is over-compensated by repulsive four-body and higher-order contributions; when clusters of 48 colloids are included in training, the resulting potentials produce no gas-liquid phase separation in molecular-dynamics simulations of aqueous dispersions.

What carries the argument

Machine-learned many-body interaction potentials fitted to finite-element Poisson-Boltzmann solutions on finite clusters of charge-regulating colloids.

If this is right

  • Attractive triplet forces alone are insufficient to drive phase separation once four-body and higher terms are retained.
  • Potentials extracted from small clusters systematically overestimate cohesion relative to those from larger clusters.
  • Primitive-model simulations validate the Poisson-Boltzmann many-body potentials for pairs and triplets.
  • Macroscopic dispersions of like-charged colloids remain stable against gas-liquid separation within the Poisson-Boltzmann framework.

Where Pith is reading between the lines

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

  • Effective pair potentials for charged colloids may require systematic many-body corrections before being used for phase-diagram predictions.
  • The result questions earlier suggestions that three-body attractions alone can explain clustering observed in low-salt suspensions.
  • Extending the training clusters to several hundred particles would provide a direct numerical test of convergence.

Load-bearing premise

Finite clusters of at most 48 colloids already contain all relevant many-body contributions without appreciable truncation error from still larger groups.

What would settle it

Direct observation of stable gas-liquid coexistence in molecular-dynamics trajectories driven by potentials trained on clusters substantially larger than 48 particles, or experimental confirmation of macroscopic phase separation under the same low-salt, charge-regulated conditions.

Figures

Figures reproduced from arXiv: 2606.21184 by Marjolein Dijkstra, Ren\'e van Roij, Thijs ter Rele.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (b) is more substantial. The ML potential yields a significantly reduced attractive three-body contribution, βU(3) ML, compared to direct PB calculations of three col￾loids, βΩ (3). This indicates that higher-order interaction terms, which contain partially repulsive contributions, are now redistributed into the effective two- and three￾body components of the ML potential, leading to the observed discrepan… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

Attractive three-body interactions have been reported for like-charged colloids in low-salt suspensions, based on both finite-element Poisson-Boltzmann calculations and direct experimental measurements, and have been proposed as a mechanism to drive colloidal clustering. However, these Poisson-Boltzmann calculations typically neglect charge regulation and higher-order many-body effects. Here, we construct machine-learned (ML) many-body interaction potentials for charge-regulating colloids, trained on finite-element Poisson-Boltzmann calculations, to accurately capture three-body and higher-order contributions. We find that the three-body contribution to the many-body potential as obtained from Poisson-Boltzmann calculations on isolated colloid triplets is strongly attractive, consistent with previous work, whereas the four-body contribution for an equilateral pyramid configuration of four colloids is repulsive. We then construct ML many-body potentials for charged colloids using finite-element Poisson-Boltzmann calculations on clusters of 13 colloids, and find that the incorporation of higher-body interactions weakens the cohesive nature of the interactions. We identify a parameter regime exhibiting gas-liquid or gas-solid phase separation using the ML potentials in molecular dynamics simulations. However, when we include clusters of 48 colloids in the training data, the cohesion diminishes further, and molecular dynamics simulations using these potentials no longer include broad phase separation in aqueous dispersions of charged colloids. Finally, we compute the potential of mean force of pairs and triplets of colloids using primitive model simulations. We find that the resulting potentials are in good agreement with those obtained from the Poisson-Boltzmann calculations, thereby supporting the validity of the Poisson-Boltzmann approach for determining many-body interactions.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims that attractive three-body interactions in Poisson-Boltzmann (PB) calculations for like-charged colloids are counteracted by repulsive higher-order many-body terms. Machine-learned potentials trained on finite-element PB solutions for clusters of 13 colloids still permit gas-liquid or gas-solid phase separation in MD, but training on 48-colloid clusters further weakens cohesion such that broad phase separation is absent. Pair and triplet potentials of mean force from primitive-model simulations agree with the PB results, supporting the validity of the PB framework.

Significance. If the finite-cluster results hold, the work shows that many-body effects beyond triplets do not stabilize gas-liquid separation in this system, providing a concrete counter-example to proposals based on isolated triplet attractions. Strengths include direct finite-element calculations, ML training on sampled clusters, and independent primitive-model validation of pair/triplet PMFs.

major comments (1)
  1. [ML training with 13- and 48-particle clusters] ML training section (13- and 48-particle clusters): the central claim that many-body attractions do not stabilize phase separation rests on the assumption that clusters of 48 colloids capture all relevant higher-order contributions. No explicit convergence test with cluster size, error bound on omitted >48-body terms, or assessment of truncation effects at macroscopic densities is provided; the reported loss of phase separation could therefore be an artifact of the finite cutoff.
minor comments (2)
  1. [Abstract] Abstract: no error bars, convergence diagnostics, or details on ML model construction and training data sampling are reported, making it difficult to assess the robustness of the cohesion-diminution result.
  2. The manuscript should clarify how the equilateral-pyramid four-body configuration was chosen and whether other four-body geometries were examined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the importance of assessing convergence with respect to cluster size. We address the major comment below.

read point-by-point responses

  1. Referee: [ML training with 13- and 48-particle clusters] ML training section (13- and 48-particle clusters): the central claim that many-body attractions do not stabilize phase separation rests on the assumption that clusters of 48 colloids capture all relevant higher-order contributions. No explicit convergence test with cluster size, error bound on omitted >48-body terms, or assessment of truncation effects at macroscopic densities is provided; the reported loss of phase separation could therefore be an artifact of the finite cutoff.

    Authors: We acknowledge that an explicit convergence test with clusters larger than 48 colloids would provide stronger evidence against truncation artifacts. Finite-element PB calculations on clusters significantly exceeding 48 particles become computationally prohibitive due to mesh complexity and memory requirements. However, the trend we observe—from the 13-colloid training set (which still permits phase separation) to the 48-colloid set (which eliminates broad phase separation)—indicates that additional many-body contributions are systematically repulsive and further weaken cohesion. We will revise the manuscript to include a dedicated paragraph discussing this trend, estimating the expected magnitude of omitted terms by extrapolating the observed reduction in effective attraction, and noting the computational limitations. We also emphasize that the independent primitive-model PMF calculations for pairs and triplets remain in quantitative agreement with the PB results, providing external validation that the many-body framework up to the included orders is physically consistent. While we cannot perform the requested larger-cluster test at present, these elements support that the loss of phase separation is not an artifact of the 48-particle cutoff. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim follows from a direct computational pipeline: finite-element Poisson-Boltzmann solutions on explicit clusters (13- and 48-colloid), ML fitting of many-body potentials to those solutions, MD simulations driven by the resulting potentials, and separate primitive-model PMF calculations for validation. No equation or result is shown to equal its own input by construction, no fitted parameter is relabeled as an independent prediction, and no load-bearing premise rests on a self-citation chain. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Poisson-Boltzmann mean-field description for the electrostatics and the assumption that finite-cluster calculations capture the dominant many-body contributions.

free parameters (1)
  • ML model parameters
    Hyperparameters and weights of the machine learning model are fitted to the finite-element PB data on clusters.
axioms (1)
  • domain assumption The Poisson-Boltzmann equation accurately describes the electrostatic potential and charge regulation around the colloids.
    Invoked for all finite-element calculations used to generate training data.

pith-pipeline@v0.9.1-grok · 5845 in / 1377 out tokens · 33219 ms · 2026-06-26T12:57:36.063547+00:00 · methodology

discussion (0)

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Reference graph

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