Pith. sign in

REVIEW 3 major objections 5 minor 29 references

Effective one-body interactions due to the presence of a liquid-vapor interface

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A liquid-vapor interface of a square-well solvent exerts a strong attractive one-body force on dilute nanoparticles, and a new weighted-density functional gives two agreeing routes to it.

desk verdict The paper's central equivalence between its new weighted-density functional and RPA fails for inhomogeneous densities, so the interface results rest on an unvalidated approximation, but the idea is worth a careful revision. read the letter →

arxiv 2608.09375 v1 pith:UEC27UF2 submitted 2026-08-10 cond-mat.soft cond-mat.stat-mech

Melih Gül , Roland Roth This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mech
keywords classicaldensityfunctionaltheorysquare-wellmixtureweighteddensitiesfundamentalmeasurerandomphaseapproximationeffectiveinteractionliquid-vaporinterfacesurfacetension
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

Classical density functional theory for square-well fluid mixtures usually treats the attractive part with the random-phase approximation, which is accurate but cumbersome to set up for many components. This paper rewrites the square-well perturbation functional in the weighted-density form of fundamental measure theory, Eq. (17), recovering the random-phase approximation exactly for additive mixtures with Lorentz-Berthelot mixing rules. Applying this functional to a binary square-well mixture at liquid-vapor coexistence, the paper finds that a dilute solute of nano particles is strongly attracted to the interface, with equilibrium density enhancements growing up to five orders of magnitude at the largest size ratio studied. It then establishes the paper's central quantitative claim: the effective one-body potential acting on the nano particles can be computed either from the solute density profile, Eq. (35), or from the Widom insertion theorem, Eq. (36), and the two routes agree closely. If correct, this gives a simple way to compute interface-induced forces on dilute solutes and the accompanying reduction of surface tension.

What carries the argument

The load-bearing object is the new weighted-density form of the square-well perturbation functional, Eq. (17), built from effective FMT-style weight functions $w_3^{\varepsilon,i}$, $w_2^{\varepsilon,i}$, $w_1^{\varepsilon,i}$, $w_0^{\varepsilon,i}$, and vector weights that carry factors of $\sqrt{\varepsilon_i}$ and are evaluated at the square-well range $\lambda_i R_i$. This form is an exact rewriting of the optimized random-phase approximation for additive square-well mixtures, so it keeps the full RPA thermodynamics while reducing the attraction term to the same convolution structure as the hard-sphere weighted densities. The second piece of machinery is the pair of formulas for the effective one-body potential, Eq. (35) from the dilute-solute density profile and Eq. (36) from Widom insertion; their agreement is the paper's main quantitative check on the mapping.

What would settle it

A Monte Carlo or molecular-dynamics simulation of the same additive binary square-well mixture ($\beta\varepsilon_1=1$, $\lambda_1=1.5$, $\sigma_2/\sigma_1$ from 1 to 3, with the symmetric and solvophilic solute energies used in Sec. 3) could measure the solute density profile across the planar liquid-vapor interface and the surface tension; if the interface accumulation at the largest size ratio does not approach five orders of magnitude, or the surface-tension reduction disagrees with the values in Table 1, the RPA solvent model underlying $V_{\mathrm{eff}}(z)$ would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the square-well attraction term of the excess free-energy functional, conventionally a double convolution in the optimized random-phase approximation, can be written exactly as a fundamental-measure-theory-style functional of weighted densities: $\beta F_{\mathrm{sw}}=\int d\mathbf{r}\,\beta(n_0^\varepsilon n_3^\varepsilon+n_1^\varepsilon n_2^\varepsilon-\mathbf{n}_1^\varepsilon\cdot\mathbf{n}_2^\varepsilon)$, with effective weights built from the square-well energy and range of each species. For an additive binary square-well mixture at liquid-vapor coexistence, this functional predicts that the interface exerts on a dilute solute of nano particles an effective one-body potential $V_{\mathrm{eff}}(z)$ that is strongly attractive at the interface. The paper shows that $V_{\mathrm{eff}}(z)$ can be obtained either from the solute density profile through $\beta V_{\mathrm{eff}}(z)=-\lim_{\rho_{2,0}\to 0}\log(\rho_2(z)/\rho_{2,0})$ or from the change in the solute's one-body direct correlation function through $\beta V_{\mathrm{eff}}(z)=\lim_{\rho_{2,0}\to 0}[c_2^{(1)}(\pm\infty)-c_2^{(1)}(z)]$, and it reports that both routes agree closely. The same calculations show that the interfacial accumulation lowers the surface tension, consistent with the Gibbs adsorption theorem.

Load-bearing premise

The load-bearing premise is that treating the attractive forces among solvent molecules as a mean-field average, without accounting for correlations between those attractions, is accurate enough at liquid-vapor coexistence; the paper tests this against no simulation or experiment.

Editorial extensions

If this is right

  • The weighted-density functional recovers the random-phase approximation for any number of additive square-well components, so phase equilibria, density profiles, and surface tensions of multi-component square-well mixtures can be computed with the same convolution structure as hard-sphere fundamental measure theory.
  • At the liquid-vapor interface of the square-well solvent, dilute nano particles experience a strongly attractive one-body potential, with the equilibrium density enhancement growing from an order of magnitude at $\sigma_2/\sigma_1=1.5$ to five orders of magnitude at $\sigma_2/\sigma_1=3$.
  • The Widom-insertion route to the effective potential, Eq. (36), requires only the pure solvent interface profile, so the dilute-limit one-body force can be obtained without solving the full binary mixture.
  • The interfacial accumulation lowers the surface tension, and the reduction is consistent with the Gibbs adsorption theorem, growing with the size ratio and reaching about 0.6% of the bare solvent value in the dilute regime studied.

Reading between the lines

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

  • Going beyond the paper, the same weighted-density construction should extend directly to square-shoulder attractions (the paper notes the sign of the weights would flip) and to ternary or polydisperse square-well mixtures, since the convolution structure is no more costly than the hard-sphere terms.
  • The predicted interfacial accumulation is strong enough that the derived $V_{\mathrm{eff}}(z)$ could be used as an input to Brownian dynamics of drying droplets, making the coffee-stain transport mechanism quantitative in this model.
  • A direct test of the weakest point would be to compare the one-component square-well interface's density profile and surface tension against simulation; if the mean-field attraction model is inaccurate, the depth and sign of the predicted effective potential are the quantities most likely to change.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a weighted-density reformulation of the square-well perturbation contribution to classical density functional theory (Eq. 17), claims that this form is equivalent to the standard optimized random-phase approximation (RPA) functional (Eq. 9), and applies the new functional to an additive binary square-well mixture at liquid-vapor coexistence. The authors compute density profiles of a dilute nanoparticle component, report a reduction of the interfacial surface tension, and obtain an effective one-body interface potential V_eff(z) by two routes: from the solute density profile (Eq. 35) and from the Widom insertion expression (Eq. 36). The central physical claim is that the liquid-vapor interface attracts nanoparticles strongly, with density enhancements reaching five orders of magnitude for size ratio sigma_2/sigma_1 = 3.

Significance. If correct, the paper would deliver a computationally convenient FMT-style functional for multicomponent square-well fluids and a clean DFT route to interface-induced one-body effective potentials. The manuscript has genuine strengths: the bulk limit of the new functional reproduces the RPA free energy under Lorentz-Berthelot mixing; the Widom insertion route is elegant and allows the dilute limit to be taken analytically; and the numerical implementation is transparent. However, the claimed equivalence with RPA is not established for inhomogeneous densities, and the apparent validation in Fig. 4 is a formal identity rather than an independent check. As a result, the quantitative predictions, including the large density enhancement and the surface-tension reduction, are supported only by an unbenchmarked functional.

major comments (3)
  1. [Sec. 2.2, Eq. (17)] The statement that Eq. (17) is 'the RPA of the SW fluid, now in the fashion of FMT' is not correct for non-uniform densities. For a one-component fluid with a = lambda R, the second functional derivative of Eq. (17) has the Fourier kernel K_new(k) = -4 pi epsilon a^3 [j0(ka)j1(ka)/(ka) + j0(ka)^2 + j1(ka)^2], while the RPA functional (9) with the potential (10) has K_RPA(k) = -4 pi epsilon (2a)^3 j1(2ka)/(2ka). The two kernels agree only at k = 0, which is why the bulk free energy (20) matches (12); at finite k they differ, with the magnitude ratio around 1.65 at ka = 1. Since the interface profile is intrinsically inhomogeneous, the calculations in Sec. 3 use a different perturbation functional from the RPA described in Sec. 2.1. The paper must either prove the functional identity for arbitrary densities or explicitly reframe Eq. (17) as an FMT-inspired approximation and benchmark it against Eq. (9) in the planar-interface geometry.
  2. [Sec. 3.2, Eqs. (35)-(36)] The 'excellent agreement' between the density-profile route and the Widom-insertion route is a formal consequence of the DFT Euler-Lagrange equation, not an independent validation. Substituting Eq. (7) into Eq. (35) gives beta V_eff(z) = c_2^(1)(infinity) - c_2^(1)(z), which is exactly Eq. (36). Thus Fig. 4 primarily checks the numerical consistency of the solver, and the paper should state this explicitly rather than presenting the agreement as evidence for the accuracy of V_eff.
  3. [Sec. 2.1 and Sec. 3] No external benchmark is provided for the RPA-based perturbation term, which controls the phase diagram, the interface structure, the surface tension, and hence V_eff. The manuscript cites Archer et al. for the general accuracy of RPA but does not test the square-well solvent at coexistence against simulation or experiment. Given that the quantitative claims, such as the five-orders-of-magnitude density enhancement in Fig. 3(d) and the surface-tension reductions in Table 1, are sensitive to this term, a comparison with simulation or with the full RPA functional (9) for at least one state point is needed before those magnitudes can be accepted.
minor comments (5)
  1. [Sec. 4] In the Summary and Outlook, 'Wigner's insertion theorem' should be 'Widom's insertion theorem'.
  2. [Eq. (16)] The same symbol w^{epsilon,i}_2 is used for both the scalar weight and the vector weight; please distinguish the vector weight with boldface or an arrow.
  3. [Table 1] The DFT and Gibbs columns differ by roughly a factor of four at sigma_2/sigma_1 = 3, so 'good agreement' overstates the match; the discrepancy should be discussed or the wording softened.
  4. [Sec. 3.1, around Eq. (25)] The sentence about including rho_2(z -> infinity) = rho_{2,0} as a 'fourth equation to Eq. (25)' is confusing because rho_{2,0} is already fixed when solving the coexistence conditions; please clarify the numerical procedure.
  5. [Sec. 3.1, Eq. (28)] The sigmoid parameter a is obtained from a fit, but the fit range and quality are not reported; since Eq. (29) is used in the Gibbs adsorption analysis, please provide this information.

Circularity Check

1 steps flagged · score 4.0 of 10

Only the Eq. (35)/Eq. (36) two-route agreement is genuinely circular (an identity from Eq. (7)); the central interface-attraction result is not circular, though its advertised RPA equivalence is a separate correctness overclaim.

  1. self definitional [Section 3.2, Eqs. (35)-(36), Fig. 4]
    "βVeff(z) = −lim_{ρ2,0→0} log(ρ2(z)/ρ2,0) ... βVeff(z) = lim_{ρ2,0→0}[ c_2^{(1)}(±∞) − c_2^{(1)}(z) ] ... We observe an excellent agreement between the two routes of calculating the effective potential which also holds for other size ratios that we have considered here."

    From the paper's own Euler-Lagrange equation, Eq. (7), applied to component 2 with V_ext=0: log(ρ_2(z)/ρ_2,0) = c_2^{(1)}(z) + βμ_2^ex. Because the bulk value c_2^{(1)}(∞) = −βμ_2^ex, Eq. (35) becomes c_2^{(1)}(∞) − c_2^{(1)}(z), which is exactly Eq. (36). The two routes are therefore the same quantity by construction in any self-consistent DFT solution; the 'excellent agreement' in Fig. 4 is a numerical consistency check of the solver, not an independent confirmation of the effective potential or of the new functional.

full rationale

The paper's central physical claim, that a dilute square-well solute is strongly attracted to the solvent liquid-vapor interface, is a computed consequence of the proposed weighted-density functional Eq. (17) and is not itself circular. The genuine circular element is the advertised cross-validation of the two effective-potential routes: Eq. (35) is obtained from the density profile and Eq. (36) from the one-body direct correlation function, but Eq. (7) makes these expressions identical, so their agreement is guaranteed up to numerical tolerance. I do not score the Section 2.2 claim that Eq. (17) is 'a functional equivalent to Eq. (9)' as circularity: that is a mathematical-equivalence assertion, and for non-uniform densities the two functionals have different second-derivative kernels (matching only in the bulk k=0 limit), so this is a correctness/overclaim issue rather than a reduction of a prediction to its input. The self-citations to FMT and to the optimized RPA literature are to standard, externally validated prior work and are not load-bearing in a circular way. Overall, the derivation is largely self-contained; the circular step affects only the 'two routes agree' validation narrative, not the main interface-adsorption result.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim depends on standard FMT/RPA approximations and model restrictions (additive SW, Lorentz-Berthelot mixing). The only fitted parameter is the sigmoid slope used in an optional analytic illustration; no free parameters are fit to the effective potential or to experimental data. No new entities are introduced.

free parameters (1)
  • sigmoid slope parameter a = a ≈ 1.79/σ_1
    Eq. (28), fit to the solvent density profile from the DFT calculation; used to derive the analytic relation Eq. (29). Not load-bearing for the numerical density profiles.
assumptions (5)
  • standard math White Bear FMT provides an accurate hard-sphere reference system
    Used in Eq. (5) as the hard-sphere excess free energy; standard in the DFT community.
  • domain assumption Optimized RPA for the square-well attraction, with the potential extended inside the hard core, captures the attractive physics
    Eqs. (9)-(10); the paper relies on this approximation for coexistence, interface, and effective potential, with no simulation benchmark.
  • domain assumption Lorentz-Berthelot mixing rules are embedded in the weighted-density functional (epsilon_12 = sqrt(epsilon_1 epsilon_2), lambda_12 = (lambda_1 sigma_1 + lambda_2 sigma_2)/(sigma_1+sigma_2))
    Eqs. (21)-(22); the paper notes this is a restriction when one component prefers another over itself.
  • domain assumption The solvent density profile is unaffected by the dilute solute and can be approximated by a sigmoid
    Eq. (28) with fit parameter a; used for the analytic solubility argument in Sec. 3.1.
  • domain assumption Two- and higher-body effective interactions are negligible in the dilute limit
    Stated in Sec. 3.2; only the one-body effective potential is computed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effective one-body interactions due to the presence of a liquid-vapor interface." pith.science (2026). https://pith.science/paper/UEC27UF2

@misc{pith2026260809375,
  author       = {Pith},
  title        = {Pith review of: Effective one-body interactions due to the presence of a liquid-vapor interface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEC27UF2}},
  note         = {Machine review of arXiv:2608.09375}
}
read the original abstract

In this study we investigate the behavior of additive binary square-well mixtures within the framework of classical density functional theory. By leveraging on the geometrical structure of the square-well interaction, we propose a novel form of the perturbation theory contribution to the density functional in terms of weighted densities, inspired by fundamental measure theory. We apply this functional in order to study the effective one-body interaction due to the liquid-vapor interface of the solvent acting on a dilute component of dissolved nano particles. The effective one-body interaction attracts the nano particles strongly to the interface. We show that this effective interaction potential can be calculated either from the density profiles of the full mixture, at low but non-vanishing concentrations of the nano particles, or by employing the Widom insertion theorem in the dilute limit of vanishing density of nano particles. Both routes display excellent agreement.

Figures

Figures reproduced from arXiv: 2608.09375 by the authors.

Figure 1
Figure 1. Liquid-vapor phase separation of the SW binary mixture with corresponding interface [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Normalized density profile ρ2(z)/ρ2,0 of the second component at the interface z = 0 for three energies ε2 with range λ2 = 1.5 and size ratio σ2/σ1 = 1. For reasons of clarity we do not show the density profile of the solvent. The lowest value of ε2 corresponds to the case of (slightly) solvophobic nano particles, that prefer the vapor phase, while the highest value of ε2 corresponds to case of (slightly) solvophili… view at source ↗
Figure 3
Figure 3. Normalized density profiles ρ2(z)/ρ2,0 for several size ratios σ2/σ1 = 1.5, 2.0, 2.5, 3.0 and energies ε2 slightly below (dotted blue) and above (dashed red) the sym￾metric solutions (solid black). By increasing the size ratio the adsorption at the interface be￾comes stronger and enhances the density by several orders of magnitude. Note the logarithmic scaling of the ordinates. below and above the energy β ε2 = 0.42… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Effective potential Veff(z) near the interface at z = 0 once obtained via Eq.(35) (red dots) and Eq.(36) (solid black) of the asymmetric mixture σ2/σ1 = 2 evaluated for the symmet￾ric case β ε2 = 1.3918 (a) and the case β ε2 = 1.50 (b) where the nano particles prefer t…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 14 canonical work pages

  1. [10]

    URLhttps://link.aps.org/doi/10.1103/PhysRevLett

    1103/PhysRevLett.63.980. URLhttps://link.aps.org/doi/10.1103/PhysRevLett. 63.980. R. Roth, R. Evans, and S. Dietrich. Depletion potential in hard-sphere mixtures: Theory and applications.Phys. Rev. E, 62:5360–5377, Oct

    doi:10.1103/physrevlett

  2. [19]

    URLhttps://dx.doi.org/10.1088/ 0953-8984/28/24/244012

    doi: 10.1088/0953-8984/28/24/244012. URLhttps://dx.doi.org/10.1088/ 0953-8984/28/24/244012. Howard Reiss, H. L. Frisch, E. Helfand, and J. L. Lebowitz. Aspects of the Statistical Thermo- dynamics of Real Fluids.The Journal of Chemical Physics, 32(1):119–124, 07

    work page doi:10.1088/0953-8984/28/24/244012

  3. [20]

    doi: 10.1063/1.1700883

    ISSN 0021-9606. doi: 10.1063/1.1700883. URLhttps://doi.org/10.1063/1.1700883. Yaakov Rosenfeld. Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing.Phys. Rev. Lett., 63:980–983, Aug

    work page doi:10.1063/1.1700883

  4. [23]

    URL https://link.aps.org/doi/10.1103/PhysRevE.64.051202

    doi: 10.1103/PhysRevE.64.051202. URL https://link.aps.org/doi/10.1103/PhysRevE.64.051202. R Roth, R Evans, A Lang, and G Kahl. Fundamental measure theory for hard-sphere mix- tures revisited: the white bear version.Journal of Physics: Condensed Matter, 14(46): 12063, nov

    work page doi:10.1103/physreve.64.051202

  5. [24]

    URLhttps://dx.doi.org/10

    doi: 10.1088/0953-8984/14/46/313. URLhttps://dx.doi.org/10. 1088/0953-8984/14/46/313. 17 Roland Roth. Fundamental measure theory for hard-sphere mixtures: a review.Journal of Physics: Condensed Matter, 22(6):063102, jan

    doi:10.1088/0953-8984/14/46/313

  6. [26]

    doi: 10.1021/acs.iecr.6b04551. V . Talanquer, Crystal Cunningham, and D. W. Oxtoby. Bubble nucleation in binary mixtures: A semiempirical approach.The Journal of Chemical Physics, 114(15):6759–6762, 04

    doi:10.1021/acs.iecr.6b04551

  7. [27]

    doi: 10.1063/1.1357798

    ISSN 0021-9606. doi: 10.1063/1.1357798. URLhttps://doi.org/10.1063/1.1357798. B. Widom. Some topics in the theory of fluids.The Journal of Chemical Physics, 39(11): 2808–2812, 12

    work page doi:10.1063/1.1357798

  8. [29]

    URLhttps://doi.org/10.1021/acs.jpcb.7b10488

    1021/acs.jpcb.7b10488. URLhttps://doi.org/10.1021/acs.jpcb.7b10488. PMID: 29244951. Yang-Xin Yu and Jianzhong Wu. Structures of hard-sphere fluids from a modified fundamental- measure theory.The Journal of Chemical Physics, 117(22):10156–10164, 11

    work page doi:10.1021/acs.jpcb.7b10488

Show all 29 references
  1. [30]

    doi: 10.1063/1.1520530

    ISSN 0021-9606. doi: 10.1063/1.1520530. URLhttps://doi.org/10.1063/1.1520530. 18

    doi

  2. [1945]

    doi: 10.1063/1.1724036

    ISSN 0021-9606. doi: 10.1063/1.1724036. URLhttps://doi.org/10.1063/1.1724036. Ryuichi Okamoto and Akira Onuki. Density functional theory of gas–liquid phase separa- tion in dilute binary mixtures.Journal of Physics: Condensed Matter, 28(24):244012, apr

    doi

  3. [1954]

    doi: 10.1063/1.1740347

    ISSN 0021-9606. doi: 10.1063/1.1740347. URLhttps://doi.org/10.1063/1.1740347. Sho Asakura and Fumio Oosawa. Interaction between particles suspended in solutions of macromolecules.Journal of Polymer Science, 33(126):183–192,

    doi

  4. [1958]

    URLhttps://onlinelibrary.wiley.com/doi/ abs/10.1002/pol.1958.1203312618

    doi: https:// doi.org/10.1002/pol.1958.1203312618. URLhttps://onlinelibrary.wiley.com/doi/ abs/10.1002/pol.1958.1203312618. 15 Douglas J. Ashton, Nigel B. Wilding, Roland Roth, and Robert Evans. Depletion potentials in highly size-asymmetric binary hard-sphere mixtures: Compar...

    1958

  5. [1963]

    doi: 10.1063/1.1734110

    ISSN 0021-9606. doi: 10.1063/1.1734110. URLhttps://doi.org/ 10.1063/1.1734110. Patrick K. Wise and Dor Ben-Amotz. Interfacial adsorption of neutral and ionic solutes in a water droplet.The Journal of Physical Chemistry B, 122(13):3447–3453,

    doi

  6. [1969]

    doi: 10.1063/ 1.1672048

    ISSN 0021-9606. doi: 10.1063/ 1.1672048. URLhttps://doi.org/10.1063/1.1672048. Robert D Deegan, Olgica Bakajin, Todd F. Dupont, Greb Huber, Sidney R Nagel, and Thomas A. Witten. Capillary flow as the cause of ring stains from dried liquid drops.Nature, 389(6653):827–829, oct

    doi

  7. [1979]

    URLhttps://doi.org/10.1080/00018737900101365

    doi: 10.1080/ 00018737900101365. URLhttps://doi.org/10.1080/00018737900101365. R Evans.Density Functionals in the Theory of Non-Uniform Fluids, pages 85 –

    doi

  8. [1983]

    URLhttps://doi

    doi: 10.1080/00268978300102661. URLhttps://doi. org/10.1080/00268978300102661. J. L. Lebowitz, E. Helfand, and E. Praestgaard. Scaled Particle Theory of Fluid Mixtures.The Journal of Chemical Physics, 43(3):774–779, 05

    doi

  9. [1997]

    URLhttps://doi.org/10.1038/ 39827

    doi: 10.1038/39827. URLhttps://doi.org/10.1038/ 39827. Marjolein Dijkstra, René van Roij, and Robert Evans. Phase behavior and structure of binary hard-sphere mixtures.Phys. Rev. Lett., 81:2268–2271, Sep

    doi

  10. [2000]

    URL https://link.aps.org/doi/10.1103/PhysRevE.62.5360

    doi: 10.1103/PhysRevE.62.5360. URL https://link.aps.org/doi/10.1103/PhysRevE.62.5360. R. Roth, R. Evans, and A. A. Louis. Theory of asymmetric nonadditive binary hard-sphere mixtures.Phys. Rev. E, 64:051202, Oct

    doi

  11. [2001]

    doi: https://doi.org/10.1016/S0370-1573(00)00141-1

    ISSN 0370-1573. doi: https://doi.org/10.1016/S0370-1573(00)00141-1. URLhttps://www.sciencedirect.com/science/article/pii/S0370157300001411. G. A. Mansoori, N. F. Carnahan, K. E. Starling, and Jr. Leland, T. W. Equilibrium Thermo- dynamic Properties of the Mixture of Hard Spher...

    doi

  12. [2002]

    URLhttps://dx.doi.org/10.1209/epl/i2002-00137-2

    doi: 10.1209/epl/ i2002-00137-2. URLhttps://dx.doi.org/10.1209/epl/i2002-00137-2. A. J. Archer, R. Evans, R. Roth, and M. Oettel. Solvent mediated interactions close to fluid- fluid phase separation: Microscopic treatment of bridging in a soft-core fluid.The Journal of Chemica...

    doi

  13. [2003]

    doi: 10.1063/1.1675048

    ISSN 0021-9606. doi: 10.1063/1.1675048. URLhttps: //doi.org/10.1063/1.1675048. Jr. McMillan, William G. and Joseph E. Mayer. The statistical thermodynamics of multicompo- nent systems.The Journal of Chemical Physics, 13(7):276–305, 07

    doi

  14. [2004]

    doi: 10.1063/1

    ISSN 0021-9606. doi: 10.1063/1. 1696842. URLhttps://doi.org/10.1063/1.1696842. H.N.W. Lekkerkerker and R. Tuinier.Colloids and the Depletion Interaction. Lecture Notes in Physics. Springer Netherlands,

    doi

  15. [2005]

    doi: 10.1063/1.1855878

    ISSN 0021-9606. doi: 10.1063/1.1855878. URLhttps://doi.org/10.1063/1.1855878. Andrew J. Archer, Blesson Chacko, and Robert Evans. The standard mean-field treatment of inter-particle attraction in classical dft is better than one might expect.The Journal of Chemical Physics, 14...

    doi

  16. [2010]

    URLhttps://dx.doi.org/10.1088/0953-8984/22/6/063102

    doi: 10.1088/0953-8984/22/6/063102. URLhttps://dx.doi.org/10.1088/0953-8984/22/6/063102. Elmar Sauer and Joachim Gross. Classical density functional theory for liquid-fluid interfaces and confined systems: A new functional for the perturbed-chain polar statistical associating ...

    doi

  17. [2011]

    URL https://link.aps.org/doi/10.1103/PhysRevE.84.061136

    doi: 10.1103/PhysRevE.84.061136. URL https://link.aps.org/doi/10.1103/PhysRevE.84.061136. Edgar L. Camacho Vergara, Georgios M. Kontogeorgis, and Xiaodong Liang. Gas adsorption and interfacial tension with classical density functional theory.Industrial and Engineering Chemistr...

    doi

  18. [2014]

    Liquid-Liquid Phase Separation and Related Topics on Liquids

    ISSN 0167-7322. doi: https://doi.org/10.1016/j.molliq.2014.02.014. URLhttps://www. sciencedirect.com/science/article/pii/S0167732214000725. Proceedings of the 7th Mini-Symposium on Liquids “Liquid-Liquid Phase Separation and Related Topics on Liquids”. A. J. Archer, R. Evans, ...

    2014 doi

  19. [2016]

    URLhttps://dx.doi.org/10.1088/ 0953-8984/28/24/240401

    doi: 10.1088/0953-8984/28/24/240401. URLhttps://dx.doi.org/10.1088/ 0953-8984/28/24/240401. J.P. Hansen and I.R. McDonald.Theory of Simple Liquids. Academic Press,

    doi

  20. [2017]

    doi: 10.1063/1.4993175

    ISSN 0021-9606. doi: 10.1063/1.4993175. URLhttps://doi.org/10.1063/1.4993175. Sho Asakura and Fumio Oosawa. On interaction between two bodies immersed in a solution of macromolecules.The Journal of Chemical Physics, 22(7):1255–1256, 07

    doi

  21. [2019]

    URLhttps: //doi.org/10.1021/acs.iecr.9b00137

    doi: 10.1021/acs.iecr.9b00137. URLhttps: //doi.org/10.1021/acs.iecr.9b00137. Norman F. Carnahan and Kenneth E. Starling. Equation of state for nonattracting rigid spheres. The Journal of Chemical Physics, 51(2):635–636, 07

    doi

Pith tools

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