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Thermodynamic stability of droplets, bubbles and thick films in open and closed pores

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Pith's one-line read Open pores make thick films unstable except near perfect wetting

desk verdict A careful Hessian-stability study of confined droplets, bubbles, and films that delivers a genuinely new result on open-pore film instability, with an honest but unproved symmetry restriction. read the letter →

arxiv 1909.01088 v1 pith:NBSVO5Z7 submitted 2019-09-03 physics.comp-ph physics.chem-ph

Magnus Aa. Gjennestad , {O}ivind Wilhelmsen This is my paper · ORCID
classification physics.comp-phphysics.chem-ph
keywords thermodynamicstabilitycapillarymodelthickfilmsdropletsandbubblesopenclosedporescontactanglephasediagramsequationofstate
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

This paper asks which droplets, bubbles, and thick films of water can actually persist inside a pore, and why the answer depends on whether the pore is closed or connected to an outside reservoir. It argues that the deciding factor is not mechanical force balance alone but full thermodynamic stability: a configuration can satisfy the Young–Laplace equation and equal chemical potentials and still be a saddle point in the energy landscape. Its central results are that adsorbed droplets and bubbles can be stable in both open and closed pores, while films in open pores are chemically unstable except for very low film-phase contact angles and a limited range of external pressure; in large pores, the thermodynamic stability limit of adsorbed structures is governed by mechanical stability linked to pore shape. The paper also supplies a numerical method to test film stability by discretizing the energy functional and examining Hessian eigenvalues, and it maps the outcomes onto phase diagrams for pores of 10 µm and 0.01 µm.

What carries the argument

The load-bearing object is the Hessian matrix of the energy functional. For films, the first variation only produces stationary states, so the authors discretize the film energy with a midpoint quadrature rule, turning the variational problem into finite-dimensional minimization over the vector $y=(N_n,z_\ell,z_r,R_f^1,\dots,R_f^M)$; diagonalizing this Hessian gives eigenvalues whose signs decide local stability. Negative eigenvalues fall into two classes: translation modes, which shift the film or adsorbed phase along the pore axis, and condensation/evaporation modes, which grow or shrink the phase while exchanging particles with its surroundings. That separation is what lets the paper show that open-pore films fail through chemical exchange with the reservoir rather than purely through mechanical imbalance, which is why a purely mechanical stability analysis is insufficient.

What would settle it

Take a non-symmetric solution to the film Euler–Lagrange equation, such as the blue or red branch in the paper's Figure A.11, compute the Hessian of the discretized Helmholtz energy at that state, and test whether all eigenvalues are positive for some pore size, contact angle, and pressure; finding even one stable non-symmetric branch would falsify the symmetry restriction and make the open-pore film stability maps incomplete.

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Extended reading notes

Core claim

At fixed temperature, the paper studies water in a cylindrically symmetric pore whose radius varies sinusoidally, coupling a capillary description to the cubic-plus-association equation of state. Stationary states are found by requiring equal chemical potentials and Young–Laplace pressure differences; local stability is then decided by the Hessian matrix of the Helmholtz energy for closed pores and of the grand potential for open pores. The central discovery is that thick films are almost always unstable in open pores, because a condensation/evaporation eigenmode lets the film exchange particles with the reservoir; only for very low film-phase contact angles and a limited external-pressure window can a film survive, and then it is still metastable relative to a homogeneous phase. In closed pores, films and adsorbed droplets or bubbles show wide stable regions, with the stability boundary in large pores set by a translation, or mechanical, instability tied to the pore shape. The instability modes always separate into translation and condensation/evaporation classes, and the resulting phase diagrams show that in open pores only homogeneous phases and adsorbed droplets or bubbles are ever equilibrium structures.

Load-bearing premise

The stability maps for films rest on the unproved observation that film states not symmetric about the pore centre are always unstable, so only the symmetric film solution with the lowest pressure difference was analysed.

Editorial extensions

If this is right

  • A film in an open pore can be mechanically balanced yet thermodynamically unstable, so stability maps for porous media, fuel-cell water management, and membrane science must include particle exchange with the reservoir.
  • In large pores, the stability limits of adsorbed droplets and bubbles coincide with mechanical limits set by pore shape, making pore geometry the controlling factor for where condensation or evaporation transitions occur.
  • The open-pore phase diagrams contain only homogeneous phases and adsorbed droplets or bubbles; free droplets, free bubbles, and films are not equilibrium structures there.
  • Smaller pores widen the pressure range over which adsorbed droplets and bubbles are stable, so confinement favors heterogeneous adsorption; they also enlarge the metastable regions around homogeneous pore filling in closed pores.

Reading between the lines

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

  • Beyond the paper: the same discrete-Hessian route should apply to thin films with a disjoining pressure, with the expectation that open pores add a chemical instability mode that height-only stability analyses miss unless the disjoining pressure couples strongly to reservoir exchange.
  • Beyond the paper: because only symmetric film solutions were checked, a search for stable non-symmetric film branches would settle whether the phase diagrams omit extra states near the pore walls; the unstable branches in the paper's Figure A.11 are the natural starting points.
  • Beyond the paper: the open-pore instability of films should be testable in nanofluidic or fuel-cell experiments by varying the external vapor pressure around the predicted window and observing whether films persist or collapse to adsorbed droplets.
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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

2 major / 4 minor

Summary. The paper develops a thermodynamic framework for heterogeneous fluid structures (homogeneous phases, free droplets/bubbles, adsorbed droplets/bubbles, and thick films) in cylindrically symmetric pores, treating both closed (canonical) and open (grand canonical) systems. The fluid is water at 358 K described by the CPA-SRK equation of state, and the pore geometry is a periodic neck defined in Eq. (1). For droplets and bubbles, stationary states are obtained from Young-Laplace and chemical-potential equality, with stability assessed from Hessians of F or Ω. For films, the authors introduce a discretized functional method (Section 3.3) in which the film shape is represented on a grid and the Hessian of the discretized energy is computed, enabling a local stability classification. Results are presented as stability maps (Figures 4-7) and phase diagrams (Figures 8-10). The main claims are: (i) for large pores the stability limits of adsorbed droplets and bubbles are governed by mechanical (translational) stability, while open pores also exhibit a condensation/evaporation instability; (ii) in closed pores, films can be stable, whereas in open pores films are chemically unstable except in strongly wetting cases where they are metastable; and (iii) metastable regions appear in small closed pores due to insufficient interfacial-energy compensation.

Significance. If the results are correct, the paper makes a useful contribution by providing a consistent capillary-plus-EOS framework for comparing the thermodynamic stability of multiple confined fluid configurations in both ensembles. The new discrete method for film stability is a practical tool that goes beyond the customary restriction to perturbations of the film height, and the distinction between mechanical (translational) and chemical (condensation/evaporation) instabilities is conceptually valuable. The explicit phase diagrams for open and closed pores are novel and illustrate how pore size, contact angle, and boundary conditions determine equilibrium structures. The paper also ships a reproducible numerical machinery (shooting method, discrete Hessian, Newton iteration) and validates the discrete film solver against the variational solution in Appendix B. However, the central claim about open-pore film instability depends on an unproved restriction to symmetric films, and the stability classification itself is not validated against an exact second-variation criterion. These gaps, if addressed, would make the framework considerably more rigorous.

major comments (2)
  1. [Appendix A] The paper's film stability maps, and specifically the abstract's claim that open-pore films are chemically unstable except for very low film-phase contact angles, rest on the statement in Appendix A: "Since we observe that solutions that are not symmetric around the pore center are always unstable, we only need to consider the symmetric film solution with the lowest Δp." This is an empirical observation, not a proof, and the configuration space of films is infinite-dimensional with possibly non-symmetric stationary states. If a non-symmetric film were a local minimum of F or Ω for some parameters, the stability maps in Figure 6 and the phase diagrams in Figures 8-10 would be incomplete. The authors should justify this restriction more rigorously, for example by performing a symmetry-breaking perturbation analysis of the discrete Hessian at non-symmetric stationary states (which their shooting method can locate) across the full parameter range, or by proving that any non-symmetric stationary state is unstable via the structure of the Euler-Lagrange equations. Without such support, the completeness of the film stability analysis is not established.
  2. [Section 3.3 and Appendix B] The discrete film method is validated in Appendix B by showing second-order convergence of the film profile Rf to the variational solution. However, the central use of the method is to classify stability by the signs of the eigenvalues of the discrete Hessian (Section 3.4), and the paper does not demonstrate convergence of these eigenvalues with grid size M. Since stability boundaries in Figure 6 are determined by the smallest eigenvalues, spurious or poorly converged eigenvalues could shift the boundaries. The authors should report, at selected state points, the convergence of the lowest few eigenvalues with M, and ideally compare the discrete second variation against an exact or high-accuracy variational second-variation calculation for a simple film geometry. This would confirm that the discretization does not introduce artificial instabilities or mask real ones.
minor comments (4)
  1. [Section 4.2 (text near Figure 5)] "Absorbed droplets and bubbles" should be "adsorbed droplets and bubbles".
  2. [Section 4.3 (end of section)] "cannot be predicted form a purely mechanical analysis" should read "cannot be predicted from a purely mechanical analysis".
  3. [Section 3.3 (after Eq. (49))] "predefined gird" should be "predefined grid".
  4. [Abstract and Section 4.3] The phrasing "the film is chemically unstable except for very low film-phase contact angles and for a limited range in external pressure" could be slightly misleading because Figure 6a/6b show that in the strongly wetting regions the film is actually metastable (orange), not stable. Consider rewording to "unstable except for a narrow metastable region at very low contact angles" to match the stability maps.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability maps and phase diagrams are outputs of Hessian analysis of explicitly stated free-energy functionals, with the EOS and contact angle as independent inputs.

full rationale

The derivation is self-contained. The paper constructs the Helmholtz and grand canonical energies F and Omega for each listed configuration from explicit thermodynamic expressions (Sections 2.1-2.4), obtains stationary states from the Young-Laplace and Euler-Lagrange equations, and determines stability from the eigenvalues of the analytically and numerically differentiated Hessian (Sections 3.3-3.4). The external inputs are the CPA-SRK equation of state, the literature value of the gas-liquid surface tension, and a prescribed contact angle; none of these are fitted to the stability maps or phase diagrams that follow. The phase diagrams are generated by comparing the computed energies of the locally stable states, so the abstract's claims are outputs rather than inputs. Self-citations to Wilhelmsen et al. [13, 18, 19] and to the in-house EOS library [33] provide independent derivations, benchmarking, or standard EOS implementation, and the central film-stability result is established in this paper via Hessian eigenanalysis rather than by citing those works. The Appendix A restriction to symmetric films is an acknowledged limitation, stated as 'we only need to consider the symmetric film solution with the lowest Delta p', but excluding untested non-symmetric stationary states is an incompleteness or correctness risk, not a circularity: it does not make the derived stability map equal to an input. No fitted parameter is renamed as a prediction, and no self-definitional chain is present. Therefore no significant circularity was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central claims rest on classical capillary thermodynamics, a specific equation of state, and a numerical stability analysis. The most fragile items are the extension of the capillary model to adsorbed structures and films without independent validation, and the unproved restriction to symmetric film solutions.

free parameters (4)
  • Contact angle α = swept over [0, π]
    Sets the solid-fluid interaction via Young's equation with σ_ls = 0; not fitted, but is an input parameter controlling the phase diagrams.
  • Pore shape coefficients (0.2, 0.075) in Eq. (1) = 0.2, 0.075
    Chosen by hand to define the necked cylindrical pore; the claim that stability limits are closely linked to pore shape rests on this single family of shapes.
  • Pore length Lp = 10 µm and 0.01 µm
    Representative large and small sizes; results are shown for these two values only.
  • Solid-liquid interfacial tension σ_ls = 0
    Set to zero because only differences between solid-fluid tensions matter; with α it determines σ_gs via Young's equation.
assumptions (5)
  • standard math Equilibrium is a minimum of the Helmholtz energy (closed) or grand potential (open).
    Standard thermodynamics (Callen); used throughout Section 2.
  • domain assumption The capillary description with sharp interfaces, constant interfacial tensions, and no line tension is valid for the structures and pore sizes considered.
    Invoked in Section 2; justified by citing Wilhelmsen et al. [13] for free droplets and bubbles, extended here to adsorbed structures and films without new validation.
  • domain assumption Thick films have negligible disjoining pressure.
    Section 2.4; this limits the film results to thick films and may fail for the 10 nm pore where films could be only a few molecular layers.
  • ad hoc to paper Non-symmetric film stationary states are always unstable, so only the symmetric solution is analyzed.
    Appendix A; stated as an observation without proof, load-bearing for the film stability maps.
  • domain assumption The CPA-SRK equation of state accurately describes water at 358 K, including metastable and negative-pressure regions.
    Section 4; the EOS is tested against saturation data in Figure 2 but not against stability boundary data.

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Pith. "Pith review of Thermodynamic stability of droplets, bubbles and thick films in open and closed pores." pith.science (2026). https://pith.science/paper/NBSVO5Z7

@misc{pith2026190901088,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic stability of droplets, bubbles and thick films in open and closed pores},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBSVO5Z7}},
  note         = {Machine review of arXiv:1909.01088}
}
read the original abstract

A fluid in a pore can form diverse heterogeneous structures. We combine a capillary description with the cubic-plus-association equation of state to study the thermodynamic stability of droplets, bubbles and films of water at 358 K in a cylindrically symmetric pore. The equilibrium structure depends strongly on the size of the pore and whether the pore is closed or connected to a particle reservoir. A new methodology is presented to analyze the thermodynamic stability of films, where the integral that describes the total energy of the system is approximated by a quadrature rule. We show that, for large pores, the thermodynamic stability limit of adsorbed droplets and bubbles in both open and closed pores is governed by their mechanical stability, which is closely linked to the pore shape. This is also the case for a film in a closed pore. In open pores, the film is chemically unstable except for very low film-phase contact angles and for a limited range in external pressure. This result emphasizes the need to invoke a complete thermodynamic stability analysis, and not restrict the discussion to mechanical stability. A common feature for most of the heterogeneous structures examined is the appearance of regions where the structure is metastable with respect to a pore filled with a homogeneous fluid. In the closed pores, these regions grow considerably in size when the pores become smaller. [...] Complete phase diagrams are presented that compare all the investigated structures. In open pores at equilibrium, the most stable structure is either the homogeneous phase or adsorbed droplets and bubbles, depending on the type of phase in the external reservoir. Smaller pores allow for droplets and bubbles to adsorb for a larger span in pressure. In closed pores, most of the investigated configurations can occur depending on the total density, the contact angle and the pore shape. [...]

Figures

Figures reproduced from arXiv: 1909.01088 by the authors.

Figure 1
Figure 1. Illustration of the heterogeneous fluid structures under consideration: (a) a homogeneous fluid phase, (b) a free droplet or bubble that is not in contact with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Saturation properties of water, as predicted by SRK (green) and CPA-SRK (blue). Reference data from [35] are shown for comparison (black). Compared [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. CPA-SRK isotherm (solid blue) for water at 358 K. The densities at [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Stability maps for (a), (b) free droplets and (c), (d) free bubbles in closed pores of lengths (a), (c) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Stability maps of (left column) adsorbed droplets and (right column) adsorbed bubbles in open and closed pores with lengths 10 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Stability maps of (left column) liquid films and (right column) gas films in open and closed pores of lengths 10 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Maps of instability types for liquid films in (a) the 10 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Phase diagram showing equilibrium configurations in a closed pores of size (a) 10 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Phase diagram showing equilibrium configurations in an open pore of size (a) 10 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Phase diagram showing equilibrium configurations in an open pore of size (a) 10 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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