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REVIEW 4 major objections 5 minor 63 references

Sampling Free Energy Landscapes of Ionic Colloidal Crystal Systems using Machine-Learned Proxy Collective Variables

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Machine-learned proxies for environment-similarity order parameters let metadynamics compute relative free energies of competing colloidal crystal phases, showing that redistributing surface charges at fixed attraction switches the…

desk verdict A useful ML-proxy metadynamics recipe for colloidal crystals, with a qualitative story that holds up; the quantitative free energies need reweighting and convergence checks before I'd trust them. read the letter →

arxiv 2608.09714 v1 pith:VE6LRLYG submitted 2026-08-10 cond-mat.stat-mech cond-mat.softphysics.chem-ph

Michael S. Chen , Stefano Sacanna , Glen M. Hocky This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softphysics.chem-ph
keywords colloidalself-assemblyfreeenergylandscapemetadynamicscollectivevariablesmachine-learnedorderparametersSE(3)-equivarianttransformerioniccrystalsactivelearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that free-energy differences between competing ionic colloidal crystal structures can be computed even when standard order parameters cannot tell the structures apart. It does so by defining similarity order parameters that measure how closely each particle's local environment matches reference CsCl-like and Th3P4-like environments, then replacing these expensive-to-differentiate descriptors with cheap machine-learned proxies based on equivariant transformer networks. With the proxies as collective variables in metadynamics, the paper maps two-dimensional free-energy surfaces for a 256-particle polymer-attenuated Coulombic system. The payoff is a concrete physical prediction: for a fixed opposite-charge attraction, shifting the surface potentials from -40:+53 to -53:+40 to -70:+30 mV moves the favored phase from CsCl-like to Th3P4-like to disordered condensate, rationalized by like-charge repulsion between the larger negative particles.

What carries the argument

The central object is the rotationally aligned local environment similarity order parameter, which aligns each particle's neighborhood to reference environments for CsCl and Th3P4 and averages over positive and negative species to form two global collective variables. Computing this order parameter requires a rotational alignment per particle per frame, making it too expensive for biased molecular dynamics. The paper replaces it with an SE(3)-equivariant transformer network, a neural network whose predictions rotate correctly with the coordinate frame and that uses attention between particles, trained via active learning iterations that add mispredicted configurations from proxy-driven metadynamics. The mechanism carrying the argument is the proxy's ability to supply accurate values and gradients during the simulation, so that metadynamics on the two collective variables explores the relevant metastable states and yields relative free energies.

What would settle it

Reweight the biased metadynamics trajectories to the exact (non-proxy) environment-similarity order parameters and recompute the free-energy differences between the CsCl-like and Th3P4-like basins; if the reweighted differences no longer show CsCl favored at -40:+53 mV and Th3P4 favored at -53:+40 mV, the proxy bias is responsible.

Watch

Extended reading notes

Core claim

The central claim is that SE(3)-equivariant transformer networks trained by active learning reproduce the values and gradients of environment-similarity order parameters accurately enough to serve as collective variables in on-the-fly metadynamics, and that the resulting free-energy surfaces are physically meaningful. On the paper's own terms, the key result is Figure 5: at equal product of surface potentials (approximately equal pairwise attraction between unlike charges), the relative stability of CsCl-like and Th3P4-like crystals reverses as charge is transferred from the smaller positive particles to the larger negative ones, and at still higher negative charge both crystals are destabilized relative to a disordered condensate. The mechanism identified is that in the CsCl-like structure the larger negative particles sit closer together than in the Th3P4-like structure, so increasing their surface potential pushes them up the repulsive wall and disfavors CsCl before eventually disfavoring Th3P4 as well.

Load-bearing premise

The free-energy surfaces are taken to be the true landscape of the physical system, but they come from metadynamics driven by machine-learned proxies that are validated on sampled configurations; if the proxies are inaccurate in rarely sampled or high-barrier regions, the relative free energies and even the stability order could be biased.

Editorial extensions

If this is right

  • The method allows free-energy rankings of non-close-packed colloidal polymorphs that standard bond-orientational order parameters cannot resolve.
  • At fixed unlike-charge attraction, increasing the magnitude of the negative surface potential destabilizes CsCl first, then Th3P4, giving the phase sequence CsCl-like to Th3P4-like to disordered condensate.
  • Like-charge repulsion between the larger negative particles controls the balance, offering a design rule: changing size asymmetry or charge split can target one polymorph.
  • ML proxies trained for one set of reference environments must be retrained for new structures or particle sizes, but the active-learning procedure keeps data requirements modest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proxy approach could extend to other expensive structural descriptors, such as polyhedral template matching, whenever a learned surrogate can supply differentiable values and gradients.
  • The predicted phase sequence could be tested experimentally by directly measuring crystal prevalence under the three charge splits, since the paper notes qualitative consistency with a related study but does not show these exact conditions experimentally.
  • A natural validation is to run umbrella sampling or well-tempered metadynamics with the exact analytic order parameters on a small system to see whether the relative free-energy differences match the proxy results.
  • Because the proxy is trained against the analytic order parameter, the method inherits that descriptor's limitations: it can only resolve structures for which reference environments are supplied.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops rotationally invariant local environment similarity order parameters that distinguish CsCl-like and Th3P4-like structures in binary polymer-attenuated Coulombic self-assembly (PACS) colloidal systems, notes that these order parameters are too expensive for on-the-fly biasing, and trains SE(3)-equivariant transformer proxies for them using an active learning protocol. The proxies are then used as collective variables in untempered metadynamics to compute two-dimensional free energy surfaces for three surface-potential conditions at approximately fixed pairwise attraction. The main reported result is that the favored phase shifts from CsCl-like at (-40,+53) mV to Th3P4-like at (-53,+40) mV to a disordered condensate at (-70,+30) mV, consistent with a like-charge repulsion mechanism.

Significance. If the central claim holds, the paper offers a practically useful route to enhanced sampling with structurally sensitive collective variables that are otherwise prohibitively expensive to differentiate, and it connects computed free energy ordering to experimentally tunable parameters. The open release of training data, code, and simulation inputs is a clear strength, as is the explicit active-learning protocol with validation against analytic order parameters on held-out configurations and over biased trajectories. The qualitative consistency with the authors' separate experimental study is encouraging. However, the missing reweighting to the analytic order parameters, the absence of convergence analysis, and the out-of-distribution validation gap mean that the quantitative free energy ordering in Figure 5 is not yet established at the level the paper claims.

major comments (4)
  1. [Section II C-D, Section III, Figure 5] The free energy surfaces in Figure 5 are produced by metadynamics biased on the ML proxy collective variables, but the biased trajectories are never reweighted to the analytic environment-similarity order parameters defined by Eqs. 3-6. Because the proxy is not exact, the estimated free energies are those of the proxy projection rather than of the intended physical order parameter; a small systematic proxy error in a rarely sampled high-barrier region could shift the relative basin depths in Figure 5 and change the reported ordering. The authors should reweight the biased ensembles to the analytic order parameters, or otherwise quantify the proxy-induced error in the free energies, before claiming the stability ordering.
  2. [Section II C, Figure 5] No convergence analysis is provided for the metadynamics estimates. The simulations are untempered, and the paper reports no hill-height decay, no block averaging, and no replicate runs at the three surface-potential conditions; the F approximately equal to -V relation is only valid in the appropriate asymptotic limit, and the relative basin depths in the lower panels of Figure 5 carry no error bars. At minimum, the authors should show the time evolution of the basin depths or run multiple independent simulations to establish that the ordering is converged.
  3. [Section II D, Figures 3-5] The validation set is a random 10% split of active-learning configurations, and the paper does not report validation data from the -70,+30 mV condition used in Figure 5c; the initial training data come from the -40,+53 and -53,+40 mV unbiased runs and the active learning illustration in Figure 4 uses +46/-46 mV. The paper also reports only aggregate validation error, not error conditional on being inside the crystal basins, the disordered basin, or the high-barrier transition regions. Since the central claim concerns this specific state point, the authors should validate the proxy on trajectories and basins for -70,+30 mV and report error conditional on the relevant regions.
  4. [Section II D] Only the predicted values of Q_CsCl and Q_Th3P4 are compared against the analytic reference values; the gradients of the proxy collective variables with respect to particle positions, which enter the metadynamics forces, are not validated. If the proxy gradients are inaccurate in regions where the values are accurate, the biased dynamics can still be driven away from the basins defined by the analytic order parameters. The authors should compare proxy gradients to analytic gradients on the validation set or otherwise demonstrate that the biased trajectories are consistent with the analytic collective-variable dynamics.
minor comments (5)
  1. [Figure 3] The correlation plots would be more informative with numerical R-squared and RMSE values, and with the color density scale defined; as printed, the degree of scatter at the extremes is difficult to assess.
  2. [Equations 7-8] The neighbor averaging in Eqs. 7-8 combines positive and negative neighbor contributions with different normalization factors; the authors should clarify the derivation and state whether the resulting collective variable remains bounded on [0,1].
  3. [Section II C] The metadynamics parameters (hill height, width, deposition stride) are given, but the total simulation time, the number of hills, and the number of independent runs for each surface-potential condition are not stated; these details should be reported for reproducibility.
  4. [Sections I and III] The claim that standard crystalline order parameters cannot differentiate the relevant structures is only supported by a qualitative Q6 comparison in Figures 2b-c; specifying which order parameters were tested and with what numerical thresholds would strengthen the motivation.
  5. [Section II B and Figure 1] The cutoff used to define the local environment chi_i is not stated; the order parameter values depend on how many neighbors are included in the alignment and overlap sums, so this should be specified.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the free-energy ordering is an output of metadynamics biased on ML proxy CVs that are trained to analytic order parameters, not fitted to the free energies themselves.

full rationale

The central claim of the paper, that the relative free energies of CsCl-like and Th3P4-like structures shift as surface potentials are redistributed, is obtained from metadynamics simulations biased on ML proxy collective variables (Section II C-D, Figure 5). The proxy models are trained to reproduce analytic environment-similarity order parameters (Eqs. 3-8), with a loss function comparing predicted and reference local and global OP values (Eq. 9). The validation in Figure 3 is against those analytic reference values on a held-out set, not against the free-energy result. Therefore the simulation output is not a re-labeled fit; the free energies come from an independent dynamical sampling procedure. Using known CsCl and Th3P4 crystal structures as reference environments is standard, deliberate CV construction rather than circular reasoning. There are self-citations (e.g., Refs. 21, 48, 49) but they supply context, simulation software details, and an experimental consistency check; they are not used as the argument that the computed ordering is correct. The absence of reweighting to the analytic OPs and the lack of convergence analysis for the untempered metadynamics are legitimate accuracy and convergence concerns, but they are not circularity because the free-energy estimate is not defined as the training target or as the output of the cited prior work.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The listed free parameters are hand-chosen values (OP widths, loss weights, hill parameters) that affect the computed free energies but are not fitted to the target free energy result. The axioms are the physical model, the OP design, the ML surrogate fidelity, and the metadynamics estimator; each is load-bearing for the central claim.

free parameters (3)
  • OP overlap widths (sigma_N, sigma_P) = CsCl: 23, 23 nm; Th3P4 positive OP: 23, 45 nm; Th3P4 negative OP: 64, 32 nm
    Hand-chosen Gaussian breadths in the environment similarity OP (Eqs. 3-4) that determine how sharply local environments must match the reference; they affect the OP's discrimination and are not fitted to the target free energies.
  • ML loss weights lambda_Q, lambda_O = 0.1, 100
    Hand-set weights in Eq. 9 balancing global CV and per-particle OP errors during training; chosen to emphasize local OP accuracy.
  • Metadynamics hill parameters = height 1 kJ/mol, sigma 0.025, deposit every 100 ps
    Chosen by hand; no convergence analysis vs. hill size or deposition rate is reported.
assumptions (4)
  • domain assumption PACS coarse-grained pair potential (Eqs. 1-2) captures the physics of the polymer-attenuated Coulombic system.
    The entire simulation and all free energies rest on this potential taken from prior PACSim work; its accuracy is not revalidated here.
  • ad hoc to paper The environment-similarity OPs with the chosen reference structures and sigma values resolve the relevant metastable states.
    The OP functional form is from Ref. 34, but the reference environments and sigma values are selected specifically for CsCl and Th3P4 in this system; if these choices miss a relevant state, the FES is incomplete.
  • ad hoc to paper The SE(3)-equivariant transformer proxies predict the analytic OPs and their gradients accurately over the sampled configuration space.
    The FES is computed from simulations biased on the proxy CVs; proxy errors in values or gradients directly bias the free energies, and no gradient accuracy test or reweighting to the analytic OPs is provided.
  • domain assumption Untempered metadynamics with the reported hill parameters converges to the free energy surface in the proxy CV space.
    No convergence metrics (e.g., hill-size dependence, block analysis, multiple replicas) are reported, and untempered metadynamics can overfill basins.

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Cite this review

Pith. "Pith review of Sampling Free Energy Landscapes of Ionic Colloidal Crystal Systems using Machine-Learned Proxy Collective Variables." pith.science (2026). https://pith.science/paper/VE6LRLYG

@misc{pith2026260809714,
  author       = {Pith},
  title        = {Pith review of: Sampling Free Energy Landscapes of Ionic Colloidal Crystal Systems using Machine-Learned Proxy Collective Variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VE6LRLYG}},
  note         = {Machine review of arXiv:2608.09714}
}
read the original abstract

Charged colloids coated with a polymer brush can be designed to preferentially self-assemble into different crystal structures by varying easy-to-tune experimental conditions. For a given set of conditions, we have observed in experiments and simulations a distribution of thermodynamically (meta)stable self-assembled crystal structures. Properly quantifying the free energy landscape of these colloidal systems is essential for rationally choosing conditions to preferentially target particular crystal structures. For some of the structures we have formed, standard crystalline order parameters are not able to differentiate between crystals or between crystals and amorphous aggregates. We show that local environment similarity descriptors are able to distinguish the relevant metastable states, but are too expensive for use in biased MD simulations. Here, we adopt an approach from machine-learned interaction potentials showing that SE(3)-equivariant transformer networks can serve as an efficient-to-evaluate machine-learned proxy. As a result, we can compute the relative free energies of accessible colloidal structures as a function of different experimentally-relevant physical knobs that can steer our system between two observed crystal types. As an example application, we then show how changing surface potentials of positive and negative colloids while maintaining the same attractive energy can shift which crystal structure is favored.

Figures

Figures reproduced from arXiv: 2608.09714 by the authors.

Figure 1
Figure 1. FIG. 1. Energy-minimized structures for CsCl-like (a) and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. a) Our CVs, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The accuracy of predictions obtained from our ML [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Time traces of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. 2D free energy surfaces obtained from our metadynamics simulations employing our ML models of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Reference graph

Works this paper leans on

63 extracted references · 44 canonical work pages

  1. [1]

    Jansen , author W

    author author M. Jansen , author W. A. \ Tisdale ,\ and\ author V. Wood ,\ title title Nanocrystal phononics , \ @noop journal journal Nat. Mater. \ volume 22 ,\ pages 161--169 ( year 2023 ) NoStop

    work page 2023

  2. [2]

    Michelson , author T

    author author A. Michelson , author T. J. \ Flanagan , author S.-W. \ Lee ,\ and\ author O. Gang ,\ title title High-strength, lightweight nano-architected silica , \ @noop journal journal Cell Rep. Phys. Sci. \ volume 4 ( year 2023 ) NoStop

    work page 2023

  3. [3]

    Li , author H

    author author Y. Li , author H. Jin , author W. Zhou , author Z. Wang , author Z. Lin , author C. A. \ Mirkin ,\ and\ author H. D. \ Espinosa ,\ title title Ultrastrong colloidal crystal metamaterials engineered with DNA , \ https://doi.org/10.1126/sciadv.adj8103 journal journal Sci. Adv. \ volume 9 ,\ pages eadj8103 ( year 2023 ) NoStop

    work page doi:10.1126/sciadv.adj8103

  4. [4]

    Qian , author E

    author author C. Qian , author E. Stanifer , author Z. Ma , author L. Yao , author B. Luo , author C. Liu , author J. Li , author P. Pan , author W. Pan , author X. Mao ,\ and\ author Q. Chen ,\ title title Nanoscale phonon dynamics in self-assembled nanoparticle lattices , \ @noop journal journal Nat. Mater. \ volume 24 ,\ pages 1616--1625 ( year 2025 ) NoStop

    work page 2025

  5. [5]

    Pattabhiraman , author G

    author author H. Pattabhiraman , author G. Avvisati ,\ and\ author M. Dijkstra ,\ title title Novel pyrochlorelike crystal with a photonic band gap self-assembled using colloids with a simple interaction potential , \ https://doi.org/10.1103/PhysRevLett.119.157401 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 157401 ( year 2017 ) NoStop

    work page doi:10.1103/physrevlett.119.157401

  6. [6]

    He , author J

    author author M. He , author J. P. \ Gales , author \'E . Ducrot , author Z. Gong , author G.-R. \ Yi , author S. Sacanna ,\ and\ author D. J. \ Pine ,\ title title Colloidal diamond , \ https://doi.org/10.1038/s41586-020-2718-6 journal journal Nature \ volume 585 ,\ pages 524--529 ( year 2020 ) NoStop

    doi:10.1038/s41586-020-2718-6

  7. [7]

    author author J. P. \ Gales , author M. J. \ Kim , author G. M. \ Hocky ,\ and\ author D. J. \ Pine ,\ title title Crystallization of non-convex colloids: the roles of particle shape and entropy , \ @noop journal journal Soft Matter \ volume 21 ,\ pages 7021--7033 ( year 2025 ) NoStop

    work page 2025

  8. [8]

    author author V. L. \ Colvin , author M. C. \ Schlamp ,\ and\ author A. P. \ Alivisatos ,\ title title Light-emitting diodes made from cadmium selenide nanocrystals and a semiconducting polymer , \ @noop journal journal Nature \ volume 370 ,\ pages 354--357 ( year 1994 ) NoStop

    work page 1994

Show all 63 references
  1. [9]

    author author B. O. \ Dabbousi , author M. G. \ Bawendi , author O. Onitsuka ,\ and\ author M. F. \ Rubner ,\ title title Electroluminescence from CdSe quantum‐dot/polymer composites , \ https://doi.org/10.1063/1.113227 journal journal Applied Physics Letters \ volume 66 ,\ pa...

    doi

  2. [10]

    author author C. B. \ Murray , author C. R. \ Kagan ,\ and\ author M. G. \ Bawendi ,\ title title Synthesis and characterization of monodisperse nanocrystals and close-packed nanocrystal assemblies , \ https://doi.org/https://doi.org/10.1146/annurev.matsci.30.1.545 journal jou...

    doi

  3. [11]

    author author S. C. \ Glotzer \ and\ author M. J. \ Solomon ,\ title title Anisotropy of building blocks and their assembly into complex structures , \ @noop journal journal Nat. Mater. \ volume 6 ,\ pages 557--562 ( year 2007 ) NoStop

    2007

  4. [12]

    Li , author D

    author author F. Li , author D. P. \ Josephson ,\ and\ author A. Stein ,\ title title Colloidal assembly: The road from particles to colloidal molecules and crystals , \ https://doi.org/https://doi.org/10.1002/anie.201001451 journal journal Angew. Chem. Int. Ed. \ volume 50 ,\...

    doi

  5. [13]

    Sacanna \ and\ author D

    author author S. Sacanna \ and\ author D. J. \ Pine ,\ title title Shape-anisotropic colloids: Building blocks for complex assemblies , \ https://doi.org/https://doi.org/10.1016/j.cocis.2011.01.003 journal journal Current Opinion in Colloid & Interface Science \ volume 16 ,\ p...

    2011 doi

  6. [14]

    Du \ and\ author R

    author author J. Du \ and\ author R. K. \ O ' Reilly ,\ title title Anisotropic particles with patchy , multicompartment and janus architectures: preparation and application , \ https://doi.org/10.1039/C0CS00216J journal journal Chem. Soc. Rev. \ volume 40 ,\ pages 2402--2416 ...

    doi

  7. [15]

    author author M. A. \ Boles , author M. Engel ,\ and\ author D. V. \ Talapin ,\ title title Self-assembly of colloidal nanocrystals: From intricate structures to functional materials , \ https://doi.org/10.1021/acs.chemrev.6b00196 journal journal Chemical Reviews \ volume 116 ...

    doi

  8. [16]

    Zhou , author Y

    author author W. Zhou , author Y. Li , author B. E. \ Partridge ,\ and\ author C. A. \ Mirkin ,\ title title Engineering anisotropy into organized nanoscale matter , \ https://doi.org/10.1021/acs.chemrev.4c00299 journal journal Chemical Reviews \ volume 124 ,\ pages 11063--111...

    doi

  9. [17]

    Hueckel , author G

    author author T. Hueckel , author G. M. \ Hocky ,\ and\ author S. Sacanna ,\ title title Total synthesis of colloidal matter , \ @noop journal journal Nature Reviews Materials \ volume 6 ,\ pages 1053--1069 ( year 2021 ) NoStop

    2021

  10. [18]

    author author M. E. \ Leunissen , author C. G. \ Christova , author A.-P. \ Hynninen , author C. P. \ Royall , author A. I. \ Campbell , author A. Imhof , author M. Dijkstra , author R. van Roij ,\ and\ author A. van Blaaderen ,\ title title Ionic colloidal crystals of opposit...

    2005

  11. [19]

    Hueckel , author G

    author author T. Hueckel , author G. M. \ Hocky , author J. Palacci ,\ and\ author S. Sacanna ,\ title title Ionic solids from common colloids , \ https://doi.org/10.1038/s41586-020-2205-0 journal journal Nature \ volume 580 ,\ pages 487--490 ( year 2020 ) NoStop

    doi

  12. [20]

    Zang , author A

    author author S. Zang , author A. W. \ Hauser , author S. Paul , author G. M. \ Hocky ,\ and\ author S. Sacanna ,\ title title Enabling three-dimensional real-space analysis of ionic colloidal crystallization , \ https://doi.org/10.1038/s41563-024-01917-w journal journal Nat. ...

    doi

  13. [21]

    Zang , author S

    author author S. Zang , author S. Paul , author C. W. \ Leung , author M. S. \ Chen , author T. Hueckel , author G. M. \ Hocky ,\ and\ author S. Sacanna ,\ title title Direct observation and control of non-classical crystallization pathways in binary colloidal systems , \ http...

    doi

  14. [22]

    van Kesteren , author N

    author author S. van Kesteren , author N. Smina , author S. Zang , author C. W. \ Leung , author G. M. \ Hocky ,\ and\ author S. Sacanna ,\ title title Light-controlled colloidal crystallization , \ https://doi.org/10.1016/j.chempr.2025.102917 journal journal Chem \ ( year 202...

  15. [23]

    Bussi \ and\ author A

    author author G. Bussi \ and\ author A. Laio ,\ title title Using metadynamics to explore complex free-energy landscapes , \ https://doi.org/10.1038/s42254-020-0153-0 journal journal Nature Reviews Physics \ volume 2 ,\ pages 200--212 ( year 2020 ) NoStop

    doi

  16. [24]

    Hénin , author T

    author author J. Hénin , author T. Lelièvre , author M. R. \ Shirts , author O. Valsson ,\ and\ author L. Delemotte ,\ title title Enhanced sampling methods for molecular dynamics simulations [article v1.0] , \ https://doi.org/10.33011/livecoms.4.1.1583 journal journal Living ...

    doi

  17. [25]

    author author M. E. \ Tuckerman ,\ @noop title Statistical Mechanics: Theory and Molecular Simulation, Second Edition \ ( publisher Oxford University Press ,\ address Oxford ,\ year 2023 ) NoStop

    2023

  18. [26]

    Giberti , author M

    author author F. Giberti , author M. Salvalaglio ,\ and\ author M. Parrinello ,\ title title Metadynamics studies of crystal nucleation , \ https://doi.org/10.1107/S2052252514027626 journal journal IUCrJ \ volume 2 ,\ pages 256--266 ( year 2015 ) NoStop

    doi

  19. [27]

    author author P. J. \ Steinhardt , author D. R. \ Nelson ,\ and\ author M. Ronchetti ,\ title title Bond-orientational order in liquids and glasses , \ https://doi.org/10.1103/PhysRevB.28.784 journal journal Phys. Rev. B \ volume 28 ,\ pages 784--805 ( year 1983 ) NoStop

    doi

  20. [28]

    Auer \ and\ author D

    author author S. Auer \ and\ author D. Frenkel ,\ title title Prediction of absolute crystal-nucleation rate in hard-sphere colloids , \ @noop journal journal Nature \ volume 409 ,\ pages 1020--1023 ( year 2001 ) NoStop

    2001

  21. [29]

    Trudu , author D

    author author F. Trudu , author D. Donadio ,\ and\ author M. Parrinello ,\ title title Freezing of a Lennard-Jones Fluid: From Nucleation to Spinodal Regime , \ https://doi.org/10.1103/PhysRevLett.97.105701 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 105701 ( year 20...

    doi

  22. [30]

    Filion , author M

    author author L. Filion , author M. Hermes , author R. Ni ,\ and\ author M. Dijkstra ,\ title title Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques , \ https://doi.org/10.1063/1.35...

    doi

  23. [31]

    Badin \ and\ author R

    author author M. Badin \ and\ author R. Marto n n \'ak ,\ title title Nucleating a different coordination in a crystal under pressure: A study of the b1 - b2 transition in NaCl by metadynamics , \ https://doi.org/10.1103/PhysRevLett.127.105701 journal journal Phys. Rev. Lett. ...

    doi

  24. [32]

    Lechner \ and\ author C

    author author W. Lechner \ and\ author C. Dellago ,\ title title Accurate determination of crystal structures based on averaged local bond order parameters , \ https://doi.org/10.1063/1.2977970 journal journal J. Chem. Phys. \ volume 129 ,\ pages 114707 ( year 2008 ) NoStop

    doi

  25. [33]

    Mickel , author S

    author author W. Mickel , author S. C. \ Kapfer , author G. E. \ Schröder-Turk ,\ and\ author K. Mecke ,\ title title Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter , \ https://doi.org/10.1063/1.4774084 journal journal...

    doi

  26. [34]

    author author P. M. \ Piaggi \ and\ author M. Parrinello ,\ title title Calculation of phase diagrams in the multithermal-multibaric ensemble , \ https://doi.org/10.1063/1.5102104 journal journal J. Chem. Phys. \ volume 150 ,\ pages 244119 ( year 2019 ) NoStop

    doi

  27. [35]

    Fijan , author M

    author author D. Fijan , author M. R. \ Ward Rashidi , author J. Bradley ,\ and\ author S. C. \ Glotzer ,\ title title Quantifying local point-group-symmetry order in complex particle systems , \ https://doi.org/10.1063/5.0312579 journal journal J. Chem. Phys. \ volume 164 ,\ ...

    doi

  28. [36]

    Russo \ and\ author H

    author author J. Russo \ and\ author H. Tanaka ,\ title title Crystal nucleation as the ordering of multiple order parameters , \ https://doi.org/10.1063/1.4962166 journal journal J. Chem. Phys. \ volume 145 ,\ pages 211801 ( year 2016 ) NoStop

    doi

  29. [37]

    author author P. M. \ Larsen , author S. Schmidt ,\ and\ author J. Schi tz ,\ title title Robust structural identification via polyhedral template matching , \ https://doi.org/10.1088/0965-0393/24/5/055007 journal journal Modelling and Simulation in Materials Science and Engin...

    doi

  30. [38]

    author author E. G. \ Teich , author G. van Anders ,\ and\ author S. C. \ Glotzer ,\ title title Identity crisis in alchemical space drives the entropic colloidal glass transition , \ @noop journal journal Nat. Commun. \ volume 10 ,\ pages 64 ( year 2019 ) NoStop

    2019

  31. [39]

    author author F. M. \ Dietrich , author X. R. \ Advincula , author G. Gobbo , author M. A. \ Bellucci ,\ and\ author M. Salvalaglio ,\ title title Machine learning nucleation collective variables with graph neural networks , \ https://doi.org/10.1021/acs.jctc.3c00722 journal j...

    doi

  32. [40]

    author author V. J. \ Meraz , author Z. Zou ,\ and\ author P. Tiwary ,\ title title Simulating crystallization in a colloidal system using state predictive information bottleneck based enhanced sampling , \ https://doi.org/10.1021/acs.jpcb.4c02740 journal journal J. Phys. Chem...

    doi

  33. [41]

    Rogal , author E

    author author J. Rogal , author E. Schneider ,\ and\ author M. E. \ Tuckerman ,\ title title Neural-network-based path collective variables for enhanced sampling of phase transformations , \ https://doi.org/10.1103/PhysRevLett.123.245701 journal journal Phys. Rev. Lett. \ volu...

    doi

  34. [42]

    Karmakar , author M

    author author T. Karmakar , author M. Invernizzi , author V. Rizzi ,\ and\ author M. Parrinello ,\ title title Collective variables for the study of crystallisation , \ https://doi.org/10.1080/00268976.2021.1893848 journal journal Molecular Physics \ volume 119 ,\ pages e18938...

    2021

  35. [43]

    author author C. S. \ Zhao , author S.-T. \ Tsai ,\ and\ author S. C. \ Glotzer ,\ title title Hybrid Monte Carlo metadynamics (hybridMC-MetaD) , \ https://doi.org/10.1063/5.0296724 journal journal J. Chem. Phys. \ volume 164 ,\ pages 024107 ( year 2026 ) NoStop

    doi

  36. [44]

    author author F. M. \ Dietrich , author M. A. \ Bellucci ,\ and\ author M. Salvalaglio ,\ title title Committor-regularized learning of differentiable collective variables from non-differentiable structural descriptors , \ https://doi.org/10.1063/5.0311722 journal journal J. C...

    doi

  37. [45]

    Laio \ and\ author M

    author author A. Laio \ and\ author M. Parrinello ,\ title title Escaping free-energy minima , \ https://doi.org/10.1073/pnas.202427399 journal journal Proceedings of the National Academy of Sciences \ volume 99 ,\ pages 12562--12566 ( year 2002 ) ,\ https://arxiv.org/abs/http...

    doi

  38. [46]

    title title Materials Data on CsCl by Materials Project , \ https://doi.org/10.17188/1199031 \ ( year 2020 ),\ 10.17188/1199031 NoStop

    2020 doi

  39. [47]

    title title Materials Data on Th3P4 by Materials Project , \ https://doi.org/10.17188/1189608 \ ( year 2020 ),\ 10.17188/1189608 NoStop

    2020 doi

  40. [48]

    van Kesteren , author S

    author author S. van Kesteren , author S. Zang , author S. Sacanna ,\ and\ author G. M. \ Hocky ,\ @noop title Structure selection in ionic colloidal crystals via independent charge tuning , \ ( year In revision ) NoStop

  41. [49]

    H\"ollmer , author N

    author author P. H\"ollmer , author N. Smina , author J. P. \ Marquardt , author M. S. \ Chen , author S. van Kesteren , author S. Sacanna ,\ and\ author G. M. \ Hocky ,\ title title PACSim: A Flexible Simulation Framework for Polymer-Attenuated Coulombic Self-Assembly , \ @no...

    2026 arXiv

  42. [50]

    author author R. J. \ Hunter ,\ @noop title Foundations of Colloid Science ,\ edition 2nd \ ed.\ ( publisher Oxford Univ. Press ,\ year 2001 ) NoStop

    2001

  43. [51]

    ,\ title title Polymer adsorption on small spheres

    author author Alexander, S. ,\ title title Polymer adsorption on small spheres. a scaling approach , \ https://doi.org/10.1051/jphys:01977003808097700 journal journal J. Phys. France \ volume 38 ,\ pages 977--981 ( year 1977 ) NoStop

    1977 doi

  44. [52]

    \ Degennes ,\ title title Stabilit\'e de films polym\`ere/solvant , \ @noop journal journal Comptes rendus de l'Acad\'emie des sciences

    author author P.-G. \ Degennes ,\ title title Stabilit\'e de films polym\`ere/solvant , \ @noop journal journal Comptes rendus de l'Acad\'emie des sciences. S\'erie 2, M\'ecanique, Physique, Chimie, Sciences de l'univers, Sciences de la Terre \ ( year 1985 ) NoStop

    1985

  45. [53]

    Likos , author K

    author author C. Likos , author K. Vaynberg , author H. L \"o wen ,\ and\ author N. Wagner ,\ title title Colloidal stabilization by adsorbed gelatin , \ @noop journal journal Langmuir \ volume 16 ,\ pages 4100--4108 ( year 2000 ) NoStop

    2000

  46. [54]

    Bonomi , author G

    author author M. Bonomi , author G. Bussi , author C. Camilloni , author G. A. \ Tribello , author P. Banas , author A. Barducci , author M. Bernetti , author P. G. \ Bolhuis , author S. Bottaro , author D. Branduardi , author R. Capelli , author P. Carloni , author M. Ceriott...

  47. [55]

    author author G. A. \ Tribello , author M. Bonomi , author G. Bussi , author C. Camilloni , author B. I. \ Armstrong , author A. Arsiccio , author S. Aureli , author F. Ballabio , author M. Bernetti , author L. Bonati , et al. ,\ title title PLUMED Tutorials: A collaborative, ...

    2025

  48. [56]

    author author W. Kabsch ,\ title title A solution for the best rotation to relate two sets of vectors , \ https://doi.org/10.1107/s0567739476001873 journal journal Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography \ volume ...

    doi

  49. [57]

    Th \"o lke \ and\ author G

    author author P. Th \"o lke \ and\ author G. D. \ Fabritiis ,\ title title Equivariant transformers for neural network based molecular potentials , \ in\ @noop booktitle International Conference on Learning Representations \ ( year 2022 ) NoStop

    2022

  50. [58]

    author author M. S. \ Chen , author A. Robledo , author C. Sch\"afer , author K. Y. \ Han , author C. Clementi ,\ and\ author M. E. \ Tuckerman ,\ title title Machine learning-accelerated path integral molecular dynamics simulations of reactive organic electrolytes , \ https:/...

    doi

  51. [59]

    Munoz , author M

    author author M. Munoz , author M. S. \ Chen , author G. de Araujo Lima e Souza , author T. Simunovic , author V. Khokhar , author P. Qian , author J. Wainright , author R. Savinell , author A. Parnell , author S. Parnell , author R. C. \ Kilbride , author T. A. \ Zawodzinski ...

  52. [60]

    author author R. P. \ Pelaez , author G. Simeon , author R. Galvelis , author A. Mirarchi , author P. Eastman , author S. Doerr , author P. Th\"olke , author T. E. \ Markland ,\ and\ author G. De Fabritiis ,\ title title Torchmd-net 2.0: Fast neural network potentials for mole...

    doi

  53. [61]

    Vaswani , author N

    author author A. Vaswani , author N. Shazeer , author N. Parmar , author J. Uszkoreit , author L. Jones , author A. N. \ Gomez , author L. Kaiser ,\ and\ author I. Polosukhin ,\ title title Attention is all you need , \ in\ https://proceedings.neurips.cc/paper_files/paper/2017...

    2017

  54. [62]

    author author D. P. \ Kingma \ and\ author J. Ba ,\ title title Adam: A Method for Stochastic Optimization , \ in\ https://arxiv.org/abs/1412.6980 booktitle International Conference on Learning Representations (ICLR) \ ( year 2015 ) NoStop

    2015 arXiv

  55. [63]

    Krogh \ and\ author J

    author author A. Krogh \ and\ author J. Vedelsby ,\ title title Neural Network Ensembles, Cross Validation, and Active Learning , \ in\ https://proceedings.neurips.cc/paper/1994/file/b8c37e33defde51cf91e1e03e51657da-Paper.pdf booktitle Advances in Neural Information Processing...

    1994

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.