pith. sign in

arxiv: 2604.18189 · v1 · submitted 2026-04-20 · ❄️ cond-mat.stat-mech · cond-mat.soft

Density Profiles and Direct Correlation Functions from Density Functional Theory in Binary Hard-Sphere Crystals: Substitutional Solid and Interstitial Solid Solution

Alessandro Simon , Martin Oettel This is my paper

Pith reviewed 2026-05-10 03:34 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords density functional theoryhard-sphere crystalsdirect correlation functionsbinary mixturesvacancy concentrationsubstitutional solidinterstitial solid solutionfundamental measure theory
0
0 comments X

The pith

The large-large components of the inhomogeneous direct correlation functions in binary hard-sphere crystals scale as one over the vacancy concentration and admit a simple geometric picture.

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

This paper uses classical density functional theory and the White Bear II functional to compute fully resolved density profiles for two binary hard-sphere crystal structures. In the substitutional case the profiles consist of narrow peaks at lattice sites much like the single-component crystal. In the interstitial case the smaller spheres spread out through the unit cell rather than localizing at specific sites. From the resulting density profiles the authors extract the species-resolved two-body direct correlation functions and observe that the large-large part is controlled mainly by the vacancy concentration, reaching a magnitude roughly one over that concentration. They use the observation to sketch a geometric interpretation of the full six-dimensional function and note that small-sphere components behave differently in the two crystal types.

Core claim

Using the White Bear II fundamental measure theory functional, the equilibrium density profiles are computed for substitutional and interstitial binary hard-sphere crystals. For substitutional crystals the density profiles consist of narrow Gaussian peaks at fcc lattice sites similar to the one-component case, while for interstitial solid solutions the small species is delocalized in the unit cell. The inhomogeneous two-body direct correlation functions are calculated, revealing that the large-large components are mainly determined by the vacancy concentration and exhibit a characteristic magnitude of approximately 1/n_vac, leading to a proposed simple geometric picture of this function. The

What carries the argument

The species-resolved inhomogeneous two-body direct correlation functions, whose large-large components are controlled by vacancy concentration and admit a geometric interpretation.

If this is right

  • Density profiles in substitutional binary crystals remain narrow Gaussian peaks centered on fcc lattice sites.
  • Small spheres in interstitial solid solutions occupy a delocalized distribution throughout the unit cell.
  • Large-large components of the direct correlation function maintain a magnitude set by one over the vacancy concentration.
  • Components of the direct correlation function that involve the small spheres differ substantially between substitutional and interstitial crystals.
  • A geometric picture suffices to represent the six-dimensional large-large direct correlation function once vacancy concentration is known.

Where Pith is reading between the lines

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

  • The observed scaling may allow quick estimates of correlation functions in other crystal lattices without repeating the full density-functional calculation.
  • The geometric picture could simplify models of how vacancies affect ordering or defect motion in colloidal mixtures.
  • Similar vacancy-controlled correlations may appear in systems with continuous size polydispersity rather than two discrete species.
  • Direct comparison with molecular-dynamics trajectories of hard spheres could confirm or refute the predicted delocalization of the small species in interstitial solutions.

Load-bearing premise

The White Bear II functional supplies quantitatively accurate density profiles and direct correlation functions for these inhomogeneous binary hard-sphere crystals.

What would settle it

A molecular-dynamics simulation of the identical binary hard-sphere mixture that produces large-large direct correlation functions whose peak heights fail to scale inversely with the vacancy concentration would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.18189 by Alessandro Simon, Martin Oettel.

Figure 1
Figure 1. Figure 1: FIG. 1: Sketch of the [100] plane illustrating the substitutional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Free energy per particle ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Lagrange multipliers (canonical chemical potentials) [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The volumetric plot on the l.h.s. shows the total [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Equilibrium density profiles (large spheres in (a) and [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The DCF of the substitutional crystal along the [100] [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison of the PY binary DCF and the FMT [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Sketch of the ISS unit cell and the three [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Equilibrium DCF of the ISS phase with reference [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Volumetric plot of the DCF for the [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Value of [PITH_FULL_IMAGE:figures/full_fig_p007_13.png] view at source ↗
read the original abstract

We determine the fully resolved equilibrium density profiles for two binary hard-sphere crystal structures using classical density functional theory through the White Bear II functional from fundamental measure theory. While for the substitutional crystal, in which some hard spheres are replaced by spheres of slightly smaller diameter, the density profiles are rather similar to the single-component case (narrow Gaussian peaks centered at fcc lattice sites), we observe a more complex behavior for the case of interstitial solid solutions, where the small species is fairly delocalized in the unit cell. Further, we compute the species-resolved inhomogeneous two-body direct correlation functions for these two types of binary crystals. The large-large components are mainly determined by the vacancy concentration $n_\text{vac}$ and show a characteristic magnitude $~1/n_\text{vac}$. Based on this observation, we propose a simple geometric picture of this six-dimensional function. The components of the direct correlation function involving the small spheres substantially differ in interstitial solid solutions from those of the substitutional crystal.

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

2 major / 1 minor

Summary. The manuscript applies classical density functional theory with the White Bear II functional from fundamental measure theory to compute fully resolved equilibrium density profiles for substitutional and interstitial binary hard-sphere crystals. It then evaluates the species-resolved inhomogeneous two-body direct correlation functions, reporting that the large-large DCF components are primarily determined by the vacancy concentration n_vac with magnitude scaling as ~1/n_vac. A simple geometric picture is proposed for this six-dimensional function, while noting greater delocalization of the small species and differing DCF components in the interstitial case relative to the substitutional solid.

Significance. If the White Bear II results are accurate, the reported scaling of the large-large DCF with n_vac and the associated geometric picture would offer a useful simplification for understanding vacancy effects in binary crystals and could inform approximations in inhomogeneous liquid theory. The distinction between crystal types highlights the impact of species delocalization. Credit is given for the detailed numerical computation of species-resolved density profiles and fully inhomogeneous DCFs in these systems.

major comments (2)
  1. [Results (DCF analysis)] The central claim that large-large DCF components have magnitude ~1/n_vac and are mainly set by n_vac follows from the second functional derivative of the White Bear II excess free-energy functional after minimization for the equilibrium profiles. The manuscript provides no direct Monte Carlo or molecular-dynamics benchmarks for these inhomogeneous c_ij(r,r') in binary crystals, so any systematic error in predicted vacancy fractions or peak shapes propagates directly into the DCF scaling and geometric picture (see abstract and results on DCF computation).
  2. [Methods / Computational details] Numerical details on the minimization of the functional, convergence criteria for the density profiles, and specific choices of lattice parameters or n_vac values are not reported. These are load-bearing for assessing whether the ~1/n_vac magnitude is robust or sensitive to discretization and boundary conditions.
minor comments (1)
  1. [Abstract] The abstract would benefit from specifying the diameter ratios and range of n_vac values examined to give immediate context for the reported behaviors.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Results (DCF analysis)] The central claim that large-large DCF components have magnitude ~1/n_vac and are mainly set by n_vac follows from the second functional derivative of the White Bear II excess free-energy functional after minimization for the equilibrium profiles. The manuscript provides no direct Monte Carlo or molecular-dynamics benchmarks for these inhomogeneous c_ij(r,r') in binary crystals, so any systematic error in predicted vacancy fractions or peak shapes propagates directly into the DCF scaling and geometric picture (see abstract and results on DCF computation).

    Authors: We acknowledge that the manuscript does not include direct Monte Carlo or molecular-dynamics benchmarks for the inhomogeneous species-resolved direct correlation functions. The White Bear II functional has been validated extensively in the literature for hard-sphere systems, yet we agree that targeted benchmarks for these particular DCFs would provide additional support. Such simulations for the fully inhomogeneous six-dimensional functions are computationally demanding and lie beyond the scope of the present work. In the revised manuscript we have added an explicit discussion of this limitation in the conclusions section. revision: partial

  2. Referee: [Methods / Computational details] Numerical details on the minimization of the functional, convergence criteria for the density profiles, and specific choices of lattice parameters or n_vac values are not reported. These are load-bearing for assessing whether the ~1/n_vac magnitude is robust or sensitive to discretization and boundary conditions.

    Authors: We thank the referee for highlighting this omission. The revised manuscript now contains a new subsection in the Methods section that specifies the numerical minimization procedure (including the Picard iteration scheme and mixing parameter), the convergence tolerance applied to the density profiles, the lattice constants chosen for the fcc unit cell, and the discrete set of vacancy concentrations n_vac that were examined. These additions confirm that the reported scaling remains stable under the employed discretization and periodic boundary conditions. revision: yes

standing simulated objections not resolved
  • Direct Monte Carlo or molecular-dynamics benchmarks for the inhomogeneous c_ij(r,r') in binary crystals

Circularity Check

0 steps flagged

No circularity in derivation chain; results are direct numerical outputs from established functional

full rationale

The paper minimizes the White Bear II FMT functional to obtain equilibrium density profiles for the two crystal structures, then computes the inhomogeneous DCFs explicitly as the second functional derivative c_ij(r,r') = -δ²F_ex/δρ_i(r)δρ_j(r'). The reported scaling of large-large DCF components with vacancy concentration n_vac (and the subsequent geometric picture) is stated as an observation extracted from these computed values, with n_vac entering as a parameter of the crystal model rather than a post-hoc fit. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked to force the result; the central claim remains an independent computational finding from the functional's output.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The calculations rest on the White Bear II FMT functional as the sole approximation for the excess free energy; vacancy concentration n_vac is an explicit model parameter that sets the scale of the reported DCF magnitude.

free parameters (1)
  • vacancy concentration n_vac
    Controls the magnitude of large-large DCF components; chosen to define the crystal structure.
axioms (1)
  • domain assumption White Bear II functional from fundamental measure theory accurately captures the excess free energy of inhomogeneous binary hard-sphere mixtures.
    Invoked as the basis for all density profiles and correlation functions.

pith-pipeline@v0.9.0 · 5481 in / 1287 out tokens · 29328 ms · 2026-05-10T03:34:21.108802+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Santos ,\ https://doi.org/10.1007/978-3-319-29668-5 title A Concise Course on the Theory of Classical Liquids \ ( publisher Springer ,\ year 2016 ) NoStop

    author author A. Santos ,\ https://doi.org/10.1007/978-3-319-29668-5 title A Concise Course on the Theory of Classical Liquids \ ( publisher Springer ,\ year 2016 ) NoStop

    work page doi:10.1007/978-3-319-29668-5 2016

  2. [2]

    \ Hansen \ and\ author I

    author author J.-P. \ Hansen \ and\ author I. R. \ McDonald ,\ https://doi.org/10.1016/C2010-0-66723-X title Theory of Simple Liquids: With Applications to Soft Matter ,\ edition 4th \ ed.\ ( publisher Academic Press ,\ address Oxford ,\ year 2013 ) NoStop

    work page doi:10.1016/c2010-0-66723-x 2013

  3. [3]

    o rig , author A. H \

    author author M. Oettel , author S. G \"o rig , author A. H \"a rtel , author H. L \"o wen , author M. Radu ,\ and\ author T. Schilling ,\ title title Free energies, vacancy concentrations, and density distribution anisotropies in hard-sphere crystals: A combined density functional and simulation study , \ https://doi.org/10.1103/PhysRevE.82.051404 journa...

    work page doi:10.1103/physreve.82.051404 2010

  4. [4]

    Density functional theory for hard-sphere mixtures: the White Bear version mark II.Journal of Physics: Condensed Matter, 18(37):8413, 2006

    author author H. Hansen-Goos \ and\ author R. Roth ,\ title title Density functional theory for hard-sphere mixtures: The White Bear version Mark II , \ https://doi.org/10.1088/0953-8984/18/37/002 journal journal J. Phys.: Condens. Matter \ volume 18 ,\ pages 8413--8425 ( year 2006 ) NoStop

    work page doi:10.1088/0953-8984/18/37/002 2006

  5. [5]

    \ Lin , author M

    author author S.-C. \ Lin , author M. Oettel , author J. M. \ H \"a ring , author R. Haussmann , author M. Fuchs ,\ and\ author G. Kahl ,\ title title Direct correlation function of a crystalline solid , \ https://doi.org/10.1103/PhysRevLett.127.085501 journal journal Phys. Rev. Lett. \ volume 127 ,\ pages 085501 ( year 2021 ) NoStop

    work page doi:10.1103/physrevlett.127.085501 2021

  6. [6]

    author author C. P. \ Royall , author P. Charbonneau , author M. Dijkstra , author J. Russo , author F. Smallenburg , author T. Speck ,\ and\ author C. Valeriani ,\ title title Colloidal hard spheres: Triumphs, challenges, and mysteries , \ https://doi.org/10.1103/RevModPhys.96.045003 journal journal Rev. Mod. Phys. \ volume 96 ,\ pages 045003 ( year 2024...

    work page doi:10.1103/revmodphys.96.045003 2024

  7. [7]

    Filion , author M

    author author L. Filion , author M. Hermes , author R. Ni , author E. C. M. \ Vermolen , author A. Kuijk , author C. G. \ Christova , author J. C. P. \ Stiefelhagen , author T. Vissers , author A. van Blaaderen ,\ and\ author M. Dijkstra ,\ title title Self-assembly of a colloidal interstitial solid with tunable sublattice doping , \ https://doi.org/10.11...

    work page doi:10.1103/physrevlett.107.168302 2011

  8. [8]

    author author R. Evans ,\ title title The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids , \ https://doi.org/10.1080/00018737900101365 journal journal Adv. Phys. \ volume 28 ,\ pages 143--200 ( year 1979 ) NoStop

    work page doi:10.1080/00018737900101365 1979

  9. [9]

    Filion ,\ title Self-assembly in colloidal hard-sphere systems ,\ http://hdl.handle.net/1874/192603 type Ph.\,D

    author author L. Filion ,\ title Self-assembly in colloidal hard-sphere systems ,\ http://hdl.handle.net/1874/192603 type Ph.\,D. thesis ,\ school Utrecht University ( year 2011 ) NoStop

    work page 2011

  10. [10]

    a rtel , author M. Oettel , author R. E. \ Rozas , author S. U. \ Egelhaaf , author J. Horbach ,\ and\ author H. L \

    author author A. H \"a rtel , author M. Oettel , author R. E. \ Rozas , author S. U. \ Egelhaaf , author J. Horbach ,\ and\ author H. L \"o wen ,\ title title Tension and stiffness of the hard sphere crystal-fluid interface , \ https://doi.org/10.1103/PhysRevLett.108.226101 journal journal Phys. Rev. Lett. \ volume 108 ,\ pages 226101 ( year 2012 ) NoStop

    work page doi:10.1103/physrevlett.108.226101 2012

  11. [11]

    Oettel , author S

    author author M. Oettel , author S. Dorosz , author M. Berghoff , author B. Nestler ,\ and\ author T. Schilling ,\ title title Description of hard-sphere crystals and crystal-fluid interfaces: A comparison between density functional approaches and a phase-field crystal model , \ https://doi.org/10.1103/PhysRevE.86.021404 journal journal Phys. Rev. E \ vol...

    work page doi:10.1103/physreve.86.021404 2012

  12. [12]

    author author M. Oettel ,\ title title Mode expansion for the density profiles of crystal--fluid interfaces: hard spheres as a test case , \ https://doi.org/10.1088/0953-8984/24/46/464124 journal journal J. Phys.: Condens. Matter \ volume 24 ,\ pages 464124 ( year 2012 ) NoStop

    work page doi:10.1088/0953-8984/24/46/464124 2012

  13. [13]

    author author S. K. \ Kwak , author Y. Cahyana ,\ and\ author J. K. \ Singh ,\ title title Characterization of mono- and divacancy in fcc and hcp hard-sphere crystals , \ https://doi.org/10.1063/1.2889924 journal journal J. Chem. Phys. \ volume 128 ,\ pages 134505 ( year 2008 ) NoStop

    work page doi:10.1063/1.2889924 2008

  14. [14]

    Hansen-Goos , author M

    author author H. Hansen-Goos , author M. Mortazavifar , author M. Oettel ,\ and\ author R. Roth ,\ title title Fundamental measure theory for the inhomogeneous hard-sphere system based on santos' consistent free energy , \ https://doi.org/10.1103/PhysRevE.91.052121 journal journal Phys. Rev. E \ volume 91 ,\ pages 052121 ( year 2015 ) NoStop

    work page doi:10.1103/physreve.91.052121 2015

  15. [15]

    author author J. F. \ Lutsko ,\ title title Explicitly stable fundamental-measure-theory models for classical density functional theory , \ https://doi.org/10.1103/PhysRevE.102.062137 journal journal Phys. Rev. E \ volume 102 ,\ pages 062137 ( year 2020 ) NoStop

    work page doi:10.1103/physreve.102.062137 2020

  16. [16]

    Schoonen \ and\ author J

    author author C. Schoonen \ and\ author J. F. \ Lutsko ,\ title title Crystal polymorphism induced by surface tension , \ https://doi.org/10.1103/PhysRevLett.129.246101 journal journal Phys. Rev. Lett. \ volume 129 ,\ pages 246101 ( year 2022 ) NoStop

    work page doi:10.1103/physrevlett.129.246101 2022

  17. [17]

    author author J. F. \ Lutsko \ and\ author C. Schoonen ,\ title title Classical density-functional theory applied to the solid state , \ https://doi.org/10.1103/PhysRevE.102.062136 journal journal Phys. Rev. E \ volume 102 ,\ pages 062136 ( year 2020 ) NoStop

    work page doi:10.1103/physreve.102.062136 2020

  18. [18]

    author author J. L. \ Lebowitz ,\ title title Exact solution of generalized Percus--Yevick equation for a mixture of hard spheres , \ https://doi.org/10.1103/PhysRev.133.A895 journal journal Phys. Rev. \ volume 133 ,\ pages A895--A899 ( year 1964 ) NoStop

    work page doi:10.1103/physrev.133.a895 1964

  19. [19]

    van der Meer , author E

    author author B. van der Meer , author E. Lathouwers , author F. Smallenburg ,\ and\ author L. Filion ,\ title title Diffusion and interactions of interstitials in hard-sphere interstitial solid solutions , \ https://doi.org/10.1063/1.5003905 journal journal J. Chem. Phys. \ volume 147 ,\ pages 234903 ( year 2017 ) NoStop

    work page doi:10.1063/1.5003905 2017

  20. [20]

    Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing.Physical Review Letters, 63(9):980–983, 1989

    author author Y. Rosenfeld ,\ title title Free-energy model for the inhomogeneous hard-sphere fluid mixture , \ https://doi.org/10.1103/PhysRevLett.63.980 journal journal Phys. Rev. Lett. \ volume 63 ,\ pages 980--983 ( year 1989 ) NoStop

    work page doi:10.1103/physrevlett.63.980 1989

  21. [21]

    Roth ,\ title title Fundamental measure theory for hard-sphere mixtures: A review , \ https://doi.org/10.1088/0953-8984/22/6/063102 journal journal J

    author author R. Roth ,\ title title Fundamental measure theory for hard-sphere mixtures: A review , \ https://doi.org/10.1088/0953-8984/22/6/063102 journal journal J. Phys.: Condens. Matter \ volume 22 ,\ pages 063102 ( year 2010 ) NoStop

    work page doi:10.1088/0953-8984/22/6/063102 2010

  22. [22]

    author author W. A. \ Curtin \ and\ author N. W. \ Ashcroft ,\ title title Density-functional theory and freezing of simple liquids , \ https://doi.org/10.1103/PhysRevLett.56.2775 journal journal Phys. Rev. Lett. \ volume 56 ,\ pages 2775--2778 ( year 1986 ) NoStop

    work page doi:10.1103/physrevlett.56.2775 1986

  23. [23]

    author author W. G. T. \ Kranendonk \ and\ author D. Frenkel ,\ title title Computer simulation of solid-liquid coexistence in binary hard sphere mixtures , \ https://doi.org/10.1080/00268979100100501 journal journal Mol. Phys. \ volume 72 ,\ pages 679--697 ( year 1991 ) NoStop

    work page doi:10.1080/00268979100100501 1991

  24. [24]

    author author M. D. \ Eldridge , author P. A. \ Madden ,\ and\ author D. Frenkel ,\ title title Entropy-driven formation of a superlattice in a hard-sphere binary mixture , \ https://doi.org/10.1038/365035a0 journal journal Nature \ volume 365 ,\ pages 35--37 ( year 1993 ) NoStop

    work page doi:10.1038/365035a0 1993

  25. [25]

    Filion \ and\ author M

    author author L. Filion \ and\ author M. Dijkstra ,\ title title Prediction of binary hard-sphere crystal structures , \ https://doi.org/10.1103/PhysRevE.79.046714 journal journal Phys. Rev. E \ volume 79 ,\ pages 046714 ( year 2009 ) NoStop

    work page doi:10.1103/physreve.79.046714 2009

  26. [26]

    \ Hynninen , author L

    author author A.-P. \ Hynninen , author L. Filion ,\ and\ author M. Dijkstra ,\ title title Stability of LS and LS_2 crystal structures in binary mixtures of hard and charged spheres , \ https://doi.org/10.1063/1.3182724 journal journal J. Chem. Phys. \ volume 131 ,\ pages 064902 ( year 2009 ) NoStop

    work page doi:10.1063/1.3182724 2009