Pith. sign in

REVIEW 3 major objections 4 minor 47 references

Elastic Properties of Symmetric Liquid-Liquid Interfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper claims that for symmetric liquid-liquid interfaces the mean bending rigidity is positive everywhere, the Gaussian rigidity turns negative at strong segregation, and these signs control interface fluctuations and droplet formation.

desk verdict A clean SCF calculation that for the first time finds positive mean bending rigidity from a molecular model, but the sign is tied to a specific interface-pinning convention that the authors do not test. read the letter →

arxiv 1908.02522 v1 pith:3KZJ5I2E submitted 2019-08-07 cond-mat.stat-mech physics.chem-ph

Ramanathan Varadharajan , Frans A M Leermakers This is my paper · ORCID
classification cond-mat.stat-mechphysics.chem-ph PACS 68.05.n68.35.Md05.70.Np31.15.Ne
keywords liquid-liquidinterfacebendingrigidityGaussianself-consistentfieldtheorygrandcanonicalensembleinterfacialtensionscalingpolymernon-localinteractions
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 claims to settle a long-disputed sign question: for a symmetric interface between two immiscible liquid-like phases of equal chain length, does bending the interface cost or release free energy? Using a lattice mean-field (self-consistent field) model in the grand canonical ensemble, it reports that the mean bending rigidity $\kappa$ is positive for all interaction strengths, so a flat interface is a local free-energy minimum. It further finds that the Gaussian bending rigidity $\bar{\kappa}$ is positive near the critical point and becomes negative at strong segregation, when the interfacial width is only about three or four segment sizes. If these results hold, they imply that short-wavelength height fluctuations are damped, that strongly segregated interfaces resist saddle-shaped deformations, and that the crossover length $\lambda = \sqrt{\kappa/\gamma}$ is a molecular-scale quantity accessible to experiments and simulations.

What carries the argument

The load-bearing device is the non-local average $\langle \phi_B(r)\rangle = \phi_B(r) + \frac{1}{6}\nabla^2 \phi_B(r)$ appearing in the polymer-solution interaction free energy, together with a Lagrange multiplier that pins the interface at the equimolar surface. The grand potentials of planar, cylindrical, and spherical interfaces are computed at fixed chemical potentials, and the standard curvature expansion of the tension yields $\kappa$ from the cylinder and $2\kappa + \bar{\kappa}$ from the sphere. The non-local term is decisive: when it is replaced by the local approximation $\langle\phi\rangle \to \phi$, the same numerical machinery reproduces earlier self-consistent-field results with negative $\kappa$ and positive $\bar{\kappa}$.

What would settle it

Compute the same grand potentials with a different discretization of the Laplacian, for instance by including next-nearest-neighbor lattice sites in the gradient stencil, and check whether $\kappa$ stays positive and $\bar{\kappa}$ still changes sign. Alternatively, simulate a symmetric polymer blend at fixed chemical potential and extract the capillary-wave spectrum: the predicted positive $\kappa$ and $\lambda$ of order a few segment sizes would be confirmed or refuted by the wavelength dependence of the height fluctuations.

Watch

Extended reading notes

Core claim

The central discovery is that in a mean-field treatment that keeps non-local corrections to the contact interaction, the elastic moduli of symmetric $A_N$-$B_N$ interfaces have definite signs: $\kappa > 0$ for all $\chi$ above the critical value, while $\bar{\kappa} > 0$ near the critical point and $\bar{\kappa} < 0$ at strong segregation. The ratio $\bar{\kappa}/\kappa$ takes plateau values of $1/2$ in weak segregation and $-3/2$ in strong segregation, and both rigidities scale with $\Delta\chi = \chi - \chi_c$ exactly as the interfacial tension does, as $N(\Delta\chi)^{3/2}$ near criticality and $(\Delta\chi)^{1/2}$ far from it. The sign switch of $\bar{\kappa}$ occurs when the interfacial width shrinks to a few segment lengths, and the paper maps this switch in the $(\sqrt{\Delta\chi}, 1/\sqrt{N})$ plane.

Load-bearing premise

The entire sign result rests on the non-local gradient term $\frac{1}{6}\nabla^2\phi$ in the interaction energy; if the physically correct contact interaction in a density gradient is instead the purely local average, the predicted positive $\kappa$ and the $\bar{\kappa}$ sign switch disappear.

Editorial extensions

If this is right

  • Short-wavelength height fluctuations of a symmetric liquid-liquid interface are damped: every deviation from the planar state raises the grand potential, so the interface is an elastic sheet down to the crossover length $\lambda = \sqrt{\kappa/\gamma}$.
  • The Gaussian rigidity sign switch means strongly segregated interfaces resist saddle deformations, making pinch-off and droplet formation more difficult, while near-critical interfaces promote such deformations.
  • Simulations that extract $\kappa$ from capillary-wave spectra should find their best signal near the maximum of $\lambda$, which grows linearly with chain length as $\lambda_{\max} \simeq 0.02N + 6.54$ in segment units.
  • Near the critical point the elastic constants inherit the tension's scaling, so the universal ratios $\bar{\kappa}/\kappa = 1/2$ and $-3/2$ in the weak- and strong-segregation regimes are testable predictions.

Reading between the lines

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

  • A testable extension of the paper's logic: measuring $\lambda$ from the capillary-wave spectrum of a symmetric polymer blend should show a non-monotonic dependence on $\chi$, peaking at the weak-to-strong segregation crossover.
  • The identification of interfacial width as the control parameter suggests the sign switch is not unique to symmetric blends; any interface whose width can be tuned through a few segment sizes may show the same $\bar{\kappa}$ sign switch.
  • If the non-local gradient term is as decisive as claimed, polymer self-consistent-field codes that drop it may still give accurate interfacial tensions but should not be trusted for elastic coefficients, at least not without including such a term.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This manuscript presents Scheutjens–Fleer self-consistent field (SF-SCF) calculations of the mean and Gaussian bending rigidities of symmetric AN–BN liquid–liquid interfaces in the grand canonical ensemble. The interface is pinned at the equimolar surface through the Lagrange term Δ_{r;r0} in Eq. (2), and the non-local Cahn–Hilliard contact interaction is retained via Eq. (4). Bending moduli are extracted from the grand-potential differences between cylindrical/spherical and planar interfaces using the Helfrich expansion, Eq. (1). The central claims are: κ>0 for all interactions studied; κ̄>0 near the critical point and negative at strong segregation, with a sign-switch boundary described by (Δχ)^{1/2}=0.2+0.5/√N; the scaling of γ, κ, and κ̄ is ∝N(Δχ)^{3/2} in weak segregation and ∝(Δχ)^{1/2} in strong segregation; and the cross-over length λ=√(κ/γ) is of order the segment size, with a maximum that grows linearly with N. The SI shows that setting ⟨φ_B⟩=φ_B in Eq. (4) reproduces the earlier negative-κ results of Matsen, and hence that the sign difference is caused by the non-local interaction term.

Significance. If the results are robust, they would resolve a long-standing controversy about the sign of κ for symmetric polymer interfaces, provide a quantitative prediction for the sign change of κ̄, and yield falsifiable scaling laws and a sign-switch diagram. The paper has notable strengths: the SCF solutions are internally consistent; the careful benchmark in the SI against Matsen’s classical SCF demonstrates that the numerical machinery is accurate and isolates the physical ingredient (non-local interactions) responsible for the difference; and κ and κ̄ are computed directly from the grand potential rather than fitted to a target. The main caveat is that the central sign result is tied to a specific dividing-surface pinning and to the treatment of non-local interactions; the manuscript does not yet show that the sign is independent of those choices.

major comments (3)
  1. [Eq. (2) and the interface-pinning procedure] The extraction of κ and κ̄ uses the Helfrich expansion for an interface pinned by the Lagrange term Δ_{r;r0} to the coordinate where φ_A=φ_B. This pinning is a gauge choice: the Helfrich moduli are not invariant under a change of dividing surface. The authors themselves note (paragraph around Refs. [34]–[36]) that Blokhuis found κ to depend on the interface-position convention and to be negative at the surface of tension. The SI benchmark shows that the same pinning without Eq. (4) yields Matsen’s negative κ, but no calculation is provided for another legitimate pinning (surface of tension, Gibbs equimolar surface, or a local external field) within the full SF-SCF model. Unless such a test is supplied, the statement “κ is strictly positive for L/L interfaces” remains contingent on a specific, unvalidated gauge.
  2. [Eq. (4) and the SI final section] The entire sign of κ and the sign switch of κ̄ rest on the gradient term (1/6)∇²φ_B in Eq. (4). The SI demonstrates that the local approximation ⟨φ_B⟩→φ_B exactly reproduces Matsen’s results, including negative κ and positive κ̄. That is an informative contrast, but it also means the positive-κ conclusion is not robust across two closely related SCF treatments. The manuscript gives no independent check that Eq. (4) is the correct non-local interaction for bending moduli—for example, a comparison with a different lattice discretization, an off-lattice calculation, or a continuum-limit check. Because this term is the load-bearing difference from all previous negative-κ results, this missing validation is a central concern.
  3. [Figs. 1 and 2 and Fig. 3(b)] The scaling exponents in Figs. 1(a) and 1(b), the sign-switch boundary in Fig. 3(b), and the linear fit λ_max=0.02N+6.54 in Fig. 2(b) are presented without error bars or convergence checks (e.g., with respect to the lattice discretization, the system size, or the radius r0). The paper states that SCF solutions are accurate to nine significant digits, but that does not quantify the error in the fitted slopes or in the location of the sign switch. Since the scaling laws and the sign switch are central quantitative claims, the authors should provide standard errors for the extracted exponents and fit parameters, at least for representative cases.
minor comments (4)
  1. [SI title] The supplementary information title contains a typo: “Elastic Poperties” should read “Elastic Properties.”
  2. [SI, section on finite chain length effects] The sentence beginning “small maximum and the height of this maximum is a weakly linear function of the chain length N” appears to be an incomplete fragment; it should be completed or merged with the previous sentence.
  3. [References] Reference [27] is incompletely specified (“which includes Refs. [4, 13, 20 & 28]”); the full citation information should be provided.
  4. [Conclusion] The wording “We have proved that the fluctuations from L/L interface away from the planar interface indeed cost free energy” overstates the evidential status of a numerical mean-field calculation; “shown” or “demonstrated” would be more appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: κ and κ̄ are computed outputs of the SF-SCF grand potential, not fitted inputs or consequences of self-citation.

full rationale

The derivation chain starts from the SF-SCF free energy (Eqs. 2–4), solves the SCF equations numerically in planar, cylindrical, and spherical geometries, and obtains κ and κ̄ from the curvature dependence of the computed grand potential via the Helfrich expansion (Eq. 1). No parameter entering the reported sign results is fitted to those results: κ and κ̄ are outputs, and the only linear fits in the paper (Figs. 2b and 3b; SI Figs. 3, 4, 6) are later summaries of already-computed quantities. The self-citation to the authors’ earlier microemulsion work (Ref. [32]) is a motivational comparison, not a load-bearing premise: the liquid/liquid sign switch is established by the SF-SCF calculation presented here. The SI comparison with Matsen’s negative-κ results is a numerical consistency test in the local limit ⟨φ⟩→φ, not an input; it demonstrates that the sign results depend on the non-local Cahn–Hilliard term, which is a model assumption, not a circular reduction. Sensitivity of the sign to the chosen dividing-surface pinning or to Eq. (4) would be a correctness/robustness concern, but no step of the derivation defines its conclusion into its premises.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The core calculation introduces no adjustable parameters: chi and N are control variables, and the elastic constants are computed rather than fitted. The main load-bearing choices are the Cahn-Hilliard gradient term, the Helfrich truncation, and the equimolar pinning convention. The fitted linear coefficients in the figures are auxiliary summaries, not inputs to the derivation.

free parameters (2)
  • Linear fit coefficients for lambda_max(N) = a=0.02, b=6.54
    Least-squares fit of numerical lambda_max values in Fig. 2b; used to state that lambda_max grows linearly with N for N>20, but not used to derive kappa or kappa_bar.
  • Sign-switch boundary fit coefficients = 0.2 and 0.5 in (Delta chi)^(1/2) = 0.2 + 0.5/N^(1/2)
    Fit to the numerically determined kappa_bar=0 boundary in Fig. 3b; used to speculate that the interfacial width controls the sign switch, but not needed to establish the sign switch itself.
assumptions (4)
  • domain assumption The mean-field free energy functional in Eq. 2, with Flory-Huggins interactions, incompressibility, and freely jointed chains, accurately captures interface thermodynamics.
    Invoked throughout; the theory ignores fluctuations beyond mean field, which may affect bending rigidities and critical exponents.
  • domain assumption The non-local interaction is exactly <phi_B(r)> = phi_B(r) + (1/6) nabla^2 phi_B(r) (Eq. 4), with the gradient coefficient fixed by the lattice.
    The sign of kappa flips when this term is replaced by the local approximation, as shown in the supplementary comparison with Matsen; the entire positive-kappa result rests on this functional form.
  • domain assumption Helfrich's expansion (Eq. 1) to quadratic order in curvature with constant coefficients is valid at molecular length scales and for the curvatures used.
    Used to convert computed grand potentials of curved interfaces into kappa and kappa_bar; no check of higher-order curvature terms is reported.
  • domain assumption Pinning the interface at the equimolar surface via the Lagrange parameter Delta_{r;r0} (Eq. 2) is a physically legitimate and unique choice for extracting fixed-mu bending rigidities.
    Blokhuis's discussion, cited in the text, shows that kappa depends on how the interface position is defined; the paper does not prove that the equimolar pinning convention is the correct one.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Elastic Properties of Symmetric Liquid-Liquid Interfaces." pith.science (2026). https://pith.science/paper/3KZJ5I2E

@misc{pith2026190802522,
  author       = {Pith},
  title        = {Pith review of: Elastic Properties of Symmetric Liquid-Liquid Interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KZJ5I2E}},
  note         = {Machine review of arXiv:1908.02522}
}
abstract

The mean ($\kappa$) and Gaussian ($\bar{\kappa}$) bending rigidities of liquid-liquid interfaces, of importance for shape fluctuations and topology of interfaces, respectively, are not yet established: even their signs are debated. Using the Scheutjens Fleer variant of the self-consistent field theory, we implemented a model for a symmetric L/L interface and obtained high precision (mean field) results in the grand canonical $(\mu, V, T)$-ensemble. We report positive values for both moduli when the system is close to critical where the rigidities show the same scaling behavior as the interfacial tension $\gamma$. At strong segregation, when the interfacial width becomes of the order of the segment size, $\bar{\kappa}$ turns negative. The length scale $\lambda \equiv \sqrt{\kappa/\gamma}$ is of order the segment size for all strengths of interaction; yet the $1/\sqrt{N}$ chain length correction reduces $\lambda$ significantly when the chain length $N$ is small.

Figures

Figures reproduced from arXiv: 1908.02522 by the authors.

Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Ratio of Mean and Gaussian bending rigidities [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 1
Figure 1. FIG. 1. (a) Volume fraction profile ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figures from the paper (6 more)
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Interfacial tension [PITH_FULL_IMAGE:figures/full_fig_p009_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Cross-over length [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Interfacial width and (b) Density difference as a fu [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: b) W = (0.82 + 0.84/ √ N)∆χ −1/2 . Note that only in the strong segregation case we have the 1/ √ N-type finite chain length correction for W. This dependence is consistent with the N-dependence for the sign switch of ¯κ (analysed for sufficiently long chains). This le…
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of SCF results for polymeric interfaces w [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 45 canonical work pages

  1. [34]

    He identified the so-called equi- librium bending mode where µ controls the curvature

    must be expected. He identified the so-called equi- librium bending mode where µ controls the curvature. In this case, the bending of the interface is accompa- nied by the development of a Laplace pressure inside the ‘droplet’ phase. γ is then typically computed at the so- called surface of tension (SOT). The position of the inter- face is taken to be at t...

    work page

  2. [36]

    Varadharajan and F

    R. Varadharajan and F. A. M. Leermakers, Physical Re- view Letters 120, 028003 (2018)

    work page 2018

  3. [1]

    Laradji and O

    M. Laradji and O. G. Mouritsen, The Journal of Chemi- cal Physics 112, 8621 (2000)

    work page 2000

  4. [2]

    This proves the importance of the non-local interaction contributions to determine the rigidities

    ⟨ϕ (r)⟩ → ϕ (r), we do reproduce all results of Matsen to a high accuracy. This proves the importance of the non-local interaction contributions to determine the rigidities. In the absence of these non-local interactions, neither the sign nor the scaling dependen- cies are apparently properly predicted. (ii) A number of years ago Blokhuis has shown that a...

    work page

  5. [3]

    M. W. Matsen, The Journal of Chemical Physics 110, 4658 (1999)

    work page 1999

  6. [4]

    The seg- ment densities are found by the composition law, which for homopolymers read: ϕ i(r) = CiGi(r, s |1)Gi(r, N − s − 1|1)/G i(r). As the position of the interface is already controlled by a Lagrange parameter ∆ r;r0, we no longer need to specify a fixed amount of one of the components (as is needed in a canonical ensemble), but we can nor- malize the...

    work page

  7. [5]

    E. M. Blokhuis and D. Bedeaux, Physica A: Statistical Mechanics and its Applications 184, 42 (1992)

    work page 1992

  8. [6]

    E. M. Blokhuis, J. Groenewold, and D. Bedeaux, Molec- ular Physics 96, 397 (1999)

    work page 1999

Show all 47 references
  1. [7]

    F. A. M. Leermakers, The Journal of Chemical Physics 138, 124103 (2013)

    2013

  2. [8]

    S. M. Oversteegen and E. M. Blokhuis, The Journal of Chemical Physics 112, 2980 (2000)

    2000

  3. [9]

    M¨ uller and F

    M. M¨ uller and F. Schmid, in Advanced Computer Simu- lation Approaches for Soft Matter Sciences II (Springer,

  4. [10]

    A. E. Van Giessen and E. M. Blokhuis, The Journal of chemical physics 116, 302 (2002)

    2002

  5. [11]

    M¨ uller and G

    M. M¨ uller and G. Gompper, Physical Review E 66, 041805 (2002)

    2002

  6. [12]

    J. D. Van der Waals, Journal of Statistical Physics 20, 200 (1979)

    1979

  7. [13]

    Laradji and R

    M. Laradji and R. C. Desai, The Journal of chemical physics 108, 4662 (1998)

    1998

  8. [14]

    J. D. Van der Waals, Over de continuiteit van den gas- en vloeistof toestand, Ph.D. thesis, Leiden, the Netherlands (1873)

  9. [15]

    Helfrich, Z

    W. Helfrich, Z. Naturforsch. C 28, 693 (1973)

    1973

  10. [16]

    S. A. Safran, Statistical thermodynamics of surfaces, interfaces, and m embranes, Vol. 90 (Perseus Books, New York, 1994)

    1994

  11. [17]

    A. N. Semenov, Macromolecules 26, 6617 (1993)

    1993

  12. [18]

    X. Z. Wu, E. B. Sirota, S. K. Sinha, B. M. Ocko, and M. Deutsch, Physical review letters 70, 958 (1993)

    1993

  13. [19]

    J. S. Rowlinson and B. Widom, Molecular theory of cap- illarity (Courier Corporation, 2013)

    2013

  14. [20]

    D. M. Mitrinovi´ c, A. M. Tikhonov, M. Li, Z. Huang, and M. L. Schlossman, Physical review letters 85, 582 (2000)

    2000

  15. [21]

    J. M. H. M. Scheutjens and G. J. Fleer, J. phys. Chem 83, 1619 (1979)

    1979

  16. [22]

    B. M. Ocko, X. Z. Wu, E. B. Sirota, S. K. Sinha, and M. Deutsch, Physical review letters 72, 242 (1994)

    1994

  17. [23]

    Merkl, T

    C. Merkl, T. Pfohl, and H. Riegler, Physical review let- ters 79, 4625 (1997)

    1997

  18. [24]

    R. A. Kik, F. A. M. Leermakers, and J. M. Kleijn, Phys. Rev. E 81, 021915 (2010)

    2010

  19. [25]

    J. M. H. M. Scheutjens and G. J. Fleer, J. Phys. Chem. 84, 178 (1980)

    1980

  20. [26]

    F. A. M. Leermakers, J. Chem. Phys. 138, 04B610 (2013)

    2013

  21. [27]

    [4, 13, 20 & 28]

    See Supplemental Material http://link.aps.org/ which in- cludes Refs. [4, 13, 20 & 28]

  22. [28]

    Fleer, M

    G. Fleer, M. A. C. Stuart, J. M. H. M. Scheutjens, T. Cosgrove, and B. Vincent, Polymers at interfaces (Springer Science & Business Media, 1993)

    1993

  23. [29]

    O. A. Evers, J. M. H. M. Scheutjens, and G. J. Fleer, Macromolecules 23, 5221 (1990)

    1990

  24. [30]

    A. N. Semenov, Macromolecules 27, 2732 (1994)

    1994

  25. [31]

    Helfand and Y

    E. Helfand and Y. Tagami, Journal of Polymer Science Part B: Polymer Letters 9, 741 (1971)

    1971

  26. [32]

    Helfand, The Journal of Chemical Physics 62, 999 (1975)

    E. Helfand, The Journal of Chemical Physics 62, 999 (1975)

    1975

  27. [33]

    Mocan, M

    M. Mocan, M. Kamperman, and F. A. M. Leermakers, Polymers 10, 78 (2018)

    2018

  28. [35]

    M¨ uller, K

    M. M¨ uller, K. Binder, and W. Oed, Journal of the Chem- ical Society, Faraday Transactions 91, 2369 (1995)

    1995

  29. [37]

    E. M. Blokhuis, The Journal of Chemical Physics 130, 014706 (2009)

    2009

  30. [38]

    F. A. M. Leermakers, A. M. Skvortsov, and L. I. Klushin, Journal of Statistical Mechanics: Theory and Experi- ment 2004, P10001 (2004)

    2004

  31. [39]

    E. M. Blokhuis, J. Kuipers, and R. L. C. Vink, Physical review letters 101, 086101 (2008)

    2008

  32. [40]

    L. Xu, W. W. Zhang, and S. R. Nagel, Physical review letters 94, 184505 (2005)

    2005

  33. [41]

    A. L. Yarin, Annu. Rev. Fluid Mech. 38, 159 (2006)

    2006

  34. [42]

    Tekin, P

    E. Tekin, P. J. Smith, and U. S. Schubert, Soft Matter 4, 703 (2008)

    2008

  35. [43]

    C. Duez, C. Ybert, C. Clanet, and L. Bocquet, Nature physics 3, 180 (2007). arXiv:1908.02522v1 [cond-mat.stat-mech] 7 Aug 2019 Supplementary information for Elastic Poperties of Symmet ric Liquid-Liquid Interface Ramanathan Varadharajan ∗ and Frans A. M. Leermakers † Physical ...

    2007 arXiv

  36. [44]

    Matsen, The Journal of chemical physics 110, 4658 (1999)

    M. Matsen, The Journal of chemical physics 110, 4658 (1999)

    1999

  37. [45]

    Fleer, M

    G. Fleer, M. A. C. Stuart, J. M. H. M. Scheutjens, T. Cosgro ve, and B. Vincent, Polymers at interfaces (Springer Science & Business Media, 1993)

    1993

  38. [46]

    S. A. Safran, Statistical thermodynamics of surfaces, interfaces, and m embranes, Vol. 90 (Perseus Books, 1994)

    1994

  39. [47]

    Mocan, M

    M. Mocan, M. Kamperman, and F. A. M. Leermakers, Polymers 10, 78 (2018). 12

    2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.