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REVIEW 3 major objections 5 minor 49 references

Sublimation and deposition at a solid sphere in the presence of a non-condensable gas

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Pith's one-line read This paper computes the three Onsager kinetic coefficients for sublimation and deposition at a solid argon sphere in a helium background gas, and shows that the cross-coupling term can change sign and deviate by more than 100% from its…

desk verdict Solid incremental study of sublimation at a sphere in a binary mixture; useful tables, but Eq. (47) has a sign error and the complete-condensation boundary condition deserves a caveat. read the letter →

arxiv 2507.10858 v1 pith:PQQ6MM6R submitted 2025-07-14 physics.flu-dyn

Denize Kalempa , Felix Sharipov , Irina Graur This is my paper · ORCID
classification physics.flu-dyn PACS 47.45.-n51.10.+y05.70.Ln
keywords sublimationdepositionmasstransferOnsagercoefficientslinearizedBoltzmannequationMcCormackmodelbinarygasmixtureabinitiopotential
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 computes the three Onsager kinetic coefficients $\Lambda_{PP}$, $\Lambda_{PT}$ and $\Lambda_{TT}$ that determine the mass and energy flow rates when a solid argon sphere sublimes or deposits in a mixture with non-condensable helium. The coefficients are obtained from the linearized Boltzmann equation with the McCormack collision model, for both hard-sphere and ab initio interatomic potentials, across free-molecular, transitional, and near-continuum regimes. The central finding is that the cross coefficient $\Lambda_{PT}$, which couples the temperature difference to the mass flow, is far more sensitive than the diagonal ones: it can deviate by more than 100% from its free-molecular value, depends strongly on the interaction potential and the helium fraction, and can even change sign as the rarefaction parameter, temperature, or species mass ratio varies. These tabulated coefficients let engineers predict the interface flow rates for arbitrary small driving forces without re-solving the kinetic equation.

What carries the argument

The load-bearing object is the symmetric Onsager matrix built from the fluxes $J_P=(1-C_0)u_2$ and $J_T=C_0q_1+(1-C_0)q_2$, evaluated with the linearized Boltzmann equation for the binary mixture, where $u_2$ is the mean bulk velocity of the sublimating species and $q_1$, $q_2$ are the heat fluxes of the two species. The collision integral is replaced by the McCormack model, which respects conservation laws and the H-theorem and gives correct mixture transport coefficients, while the interatomic forces enter through the $\Omega$-integrals for either hard spheres or ab initio potentials. To handle the discontinuity of the distribution function at the surface of a convex body, the solution is split into an analytically known free-streaming part and a numerical part solved with the discrete velocity method, and the reciprocity relation $\Lambda_{PT}=\Lambda_{TP}$ is used as a check of the numerical error. The free-molecular regime is solved analytically and serves as the reference for all deviations.

What would settle it

Measure the steady mass flow from a small argon sphere suspended in a helium-argon mixture at $T_0=50$ K with $C_0=0.5$ while sweeping the pressure so that the rarefaction parameter runs from about 0.1 to 10, and applying a small temperature difference of about 0.1 K. The predicted sign change of $\Lambda_{PT}$ would appear as a reversal of the net argon flow direction relative to the temperature gradient, and the particular value of $\delta$ at which the reversal occurs would distinguish the hard-sphere from the ab initio potential prediction.

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Extended reading notes

Core claim

For sublimation and deposition at a solid argon sphere surrounded by a helium-argon mixture, the interface mass and energy flow rates are linear functions of two thermodynamic forces, the vapor pressure difference and the temperature difference, with the proportionality encoded in a symmetric matrix of three kinetic coefficients: $\Lambda_{PP}$, $\Lambda_{PT}(=\Lambda_{TP})$ and $\Lambda_{TT}$. The paper establishes how these coefficients behave: $\Lambda_{PP}$ and $\Lambda_{TT}$ stay positive, while $\Lambda_{PT}$ can be positive or negative and its sign depends on the rarefaction parameter $\delta$, the helium molar fraction $C_0$, the temperature, the interatomic potential, and the mass ratio of the two gases. Unlike the planar case, where $\Lambda_{PP}$ decreases monotonically with rarefaction, for the sphere $\Lambda_{PP}$ can increase, peak, or decrease depending on the molar fraction. The cross coefficient is so sensitive that in some conditions the hard-sphere model predicts a sign opposite to the ab initio potential at the same state point, and the inverted temperature gradient appears near the sphere, with its presence controlled by the helium fraction.

Load-bearing premise

The surface is assumed to be a perfect phase-change interface: every incident argon or krypton atom is absorbed and the surface re-emits vapor atoms with a Maxwellian distribution at the surface temperature and saturation pressure, while helium is fully accommodated; if the real condensation or sticking coefficient is less than one, the tabulated coefficients change.

Editorial extensions

If this is right

  • Any small driving force (pressure or temperature difference) at an argon sphere in helium now yields the mass and energy flow rates directly from the relations $\dot M=4\pi R_0^2 n_0 v_0 m_2 J_P$ and $\dot E=4\pi R_0^2 v_0 p_0\left(J_T+\tfrac52 J_P\right)$ using the tabulated coefficients.
  • In the transitional and continuum regimes the hard-sphere model is not reliable for the cross effect: at $C_0=0.5$ and $\delta=10$ the ab initio and hard-sphere potentials give $\Lambda_{PT}$ values that differ by more than 100% and even have opposite signs.
  • Adding helium as a background gas suppresses the mass flow coefficient $\Lambda_{PP}$ strongly, by about 60% when the helium fraction grows from 0.1 to 0.5 at $\delta=1$, so even a modest amount of non-condensable gas slows sublimation.
  • The inverted temperature gradient near the sphere is confirmed and shown to depend on the helium molar fraction and the rarefaction parameter, not on the interatomic interaction potential.
  • The helium-krypton comparison shows that the mass ratio of the sublimating species strongly shifts the heat-flow coefficient $\Lambda_{TT}$, with deviations up to about 43%, so results cannot be freely transferred from one sublimating species to another.

Reading between the lines

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

  • A sign change in $\Lambda_{PT}$ implies a regime where a temperature difference alone drives argon mass flow in the direction opposite to the saturated-pressure gradient; this could be searched for by levitating an argon particle and imposing a small thermal asymmetry while sweeping the background pressure.
  • If the real surface condensation or sticking coefficient is not exactly unity, the published tables would shift quantitatively, but the qualitative structure (sign change of $\Lambda_{PT}$ and non-monotonic $\Lambda_{PP}$) likely persists, so the free-molecular reference values could be rescaled by an effective sticking fraction as a first-order correction.
  • The same Onsager-matrix methodology extends naturally to dust grains, aerosol droplets, and sublimating particles in vacuum systems and planetary atmospheres, where the sphere is the natural geometry; the strong potential sensitivity warns that realistic collision cross-sections are needed in those applications too.
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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

3 major / 5 minor

Summary. The paper studies sublimation/deposition at a solid argon sphere in a helium-argon mixture, using the linearized Boltzmann equation with the McCormack collision model. The authors compute the Onsager kinetic coefficients ΛPP, ΛPT, ΛTT that determine mass and energy flow rates at the interface, over a range of rarefaction parameter δ, molar fractions C0, and two temperatures (50 K and 70 K), using both hard-sphere and ab-initio potentials. They also give analytic free-molecular results, solve the transitional regime numerically by a discrete velocity method, report flow fields and temperature/pressure jump behavior, and include examples of mass and energy flow rates. The main claims are that the cross coefficient ΛPT is highly sensitive to rarefaction, can deviate by more than 100% from its free-molecular value, and can change sign with δ.

Significance. If the results hold, the paper provides a useful tabulation of spherical-geometry kinetic coefficients for sublimation/deposition in a binary mixture, based on ab-initio interaction data rather than fitted transport parameters. The free-molecular formulas are explicit and check against the tables, and the numerical reciprocity check ΛPT=ΛTP is a valuable internal consistency test. The comparison of hard-sphere and ab-initio potentials, and of argon and krypton, gives a concrete picture of where the interaction model matters. However, the central quantitative claims are conditional on the assumed complete phase transition boundary condition, and no sensitivity analysis for that assumption is provided; this is the main scientific weakness.

major comments (3)
  1. [Sec. 6, Eq. (47)] Equation (47) for ν(P)_2 has the wrong sign. With h(P)_2=1 on the cone 0≤θ≤θ0 and zero outside (Eq. (43)), the angular average is (1−cos θ0)/2 = (1−sqrt(1−(r0/r)^2))/2, which is positive and tends to 0 as r→∞. The printed expression −1/2 − sqrt(1−(r0/r)^2)/2 is negative everywhere and tends to −1 at infinity; it also contradicts the positive ν(P)_2 profiles in Figure 1. Please correct Eq. (47) and check the companion moments (48)–(51) for consistency.
  2. [Sec. 4, Eqs. (26)–(27); Tables 2–4] The complete phase transition boundary condition (26)–(27) is an assumption that enters every entry of Tables 2–4. The headline conclusions—that ΛPT can deviate by more than 100% from its free-molecular value and can change sign with δ—are statements about the solution of this particular boundary-value problem. A partial sticking coefficient or non-Maxwellian emission could shift or remove the zero of ΛPT. The paper offers no sensitivity analysis and no experimental benchmark for the sticking coefficient. Please add a parametric study with a partial-condensation boundary condition (e.g., a Maxwell-type condition) or, at minimum, a quantitative discussion of how the tabulated coefficients would change; otherwise the conclusions should be presented as conditional on the complete-transition assumption.
  3. [Sec. 7; Tables 2–4] The claimed numerical error of 0.1% for the kinetic coefficients is stated but not demonstrated. The sign change of ΛPT occurs at values as small as 0.0040 (Table 2, HS, C0=0.5, δ=5), and the reported deviations from free-molecular values exceed 100%. The accuracy statement needs to be supported by a convergence study in Nr, Nθ, Nc and rmax, or by a supplementary table reporting the grid-variation results. Please include this information so that the sign changes in the tables can be assessed as genuine physical effects rather than numerical artifacts.
minor comments (5)
  1. [Sec. 4] There is a typo: “emmited” should be “emitted”.
  2. [Sec. 7] The sentence about rmax (“while the distance rmax was set as δ varied so that the increment in the radial distance ∆r ∼ 10−3”) is unclear and should be rewritten; please state the actual rmax values used for each δ.
  3. [Sec. 8.7] The text after Eq. (85) swaps the symbols for pressure and mole fraction deviations: ξ is defined as the pressure deviation in Eq. (84) and ζ as the mole fraction deviation in Eq. (85), but the paragraph says “The local mole fraction deviation, ξ, is always negative” and “the pressure deviation, ζ, is always positive.” This should be corrected in the text and checked against the panels in Figures 7–9.
  4. [Tables 2–4] Please state in the captions that the rows with δ=0 are obtained from the analytic free-molecular formulas, not from the discrete velocity method.
  5. [Appendix A, Eq. (A.9)] The choice γα=p0α/µ0α is introduced by analogy with the single-gas Shakhov model; a brief justification or a citation to a previous mixture validation of this specific choice would help the reader assess its effect on the transitional-regime results.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the kinetic coefficients are direct numerical outputs of a stated linearized kinetic boundary-value problem, with no parameter fitted to the reported results.

full rationale

The derivation chain is self-contained. The kinetic coefficients are moments of perturbation functions obtained by solving the linearized Boltzmann equation with the McCormack collision model (Eqs. 10-14) subject to explicitly stated boundary conditions (Eqs. 23-27), and the free-molecular limits (Eqs. 62-64) follow analytically from Eq. (41). No parameter is fitted to the reported values of the coefficients. The model inputs are independent: the ab initio omega-integrals come from Refs. [43,48], the hard-sphere diameter ratio is obtained from a standard viscosity relation (Eq. A.13), and the saturation pressures used in the illustrative examples come from Ref. [28]. The collision-frequency parameters gamma_alpha are chosen by analogy (Eq. A.9), not calibrated to the target flow rates. The thermodynamic-flux definitions (Eqs. 30-32) invoke the general reciprocal-relation framework of Refs. [37,38], which is an independent theoretical result; moreover, the paper verifies Lambda_PT = Lambda_TP numerically rather than imposing it on the solution. The complete-phase-transition boundary condition is a physical modeling assumption that can affect the numerical values, but it is not defined in terms of, or fitted to, the output coefficients, so it is a correctness risk rather than a circular step. The occasional self-citations are to benchmark calculations and prior independent results, not to a claimed uniqueness theorem or to the present target results. Therefore no significant circularity is present.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central results rest on standard kinetic theory plus several modeling choices: the McCormack closure, the ansatz for γα, complete phase transition, and dilute-gas transport data from prior papers. None of these are circular, but readers should treat them as assumptions that bound the validity of the tabulated coefficients.

free parameters (1)
  • Hard-sphere atomic diameter ratio d2/d1 = 2.109 (He-Ar at 50 K), 2.031 (He-Ar at 70 K)
    Obtained from single-gas viscosities via Eq. (A.13); it sets the HS collision integrals and thus all HS-potential results in Tables 2 and 3.
assumptions (6)
  • domain assumption The McCormack collision model (Eq. A.1) accurately represents the linearized Boltzmann collision operator for the binary mixture.
    The paper relies on this to compute all kinetic coefficients; it cites prior benchmarks (Refs. 29-32) but does not re-derive or validate it here.
  • domain assumption Complete phase transition at the sphere surface for argon/krypton: all incident vapor atoms are absorbed and re-emitted as a Maxwellian at Ts and saturation pressure p2s (Eqs. 26-27).
    This boundary condition sets the mass and energy flow rates; a non-unit condensation coefficient would change all reported coefficients.
  • domain assumption Helium is diffusely scattered with full accommodation at the sphere surface (Eqs. 23-25).
    This sets the helium perturbation at the boundary and affects the heat flux JT.
  • ad hoc to paper The collision frequency in the McCormack model is chosen as γα = p0α/µ0α (Eq. A.9) by analogy with the single-gas Shakhov model.
    This choice pins the relaxation rate for each species; it is not derived from the collision integral and different choices would modify the transitional-regime results.
  • domain assumption The driving forces XP and XT are small (Eq. 5), justifying linearization around a single equilibrium.
    All results are linear-response coefficients; they do not cover finite temperature or pressure differences.
  • domain assumption Far from the sphere the perturbation vanishes (Eq. 28), modeling an unbounded equilibrium reservoir.
    Used to close the problem at rmax in the numerical domain; in a confined system the background composition and pressure would evolve.

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Pith. "Pith review of Sublimation and deposition at a solid sphere in the presence of a non-condensable gas." pith.science (2026). https://pith.science/paper/PQQ6MM6R

@misc{pith2026250710858,
  author       = {Pith},
  title        = {Pith review of: Sublimation and deposition at a solid sphere in the presence of a non-condensable gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQQ6MM6R}},
  note         = {Machine review of arXiv:2507.10858}
}
read the original abstract

The sublimation/deposition process at a solid sphere of argon into its vapor in the presence of helium as a background gas is modeled applying the linearized Boltzmann equation, in which the McCormack model is employed for the collisional term. The Onsager coefficients determining the mass and energy flow rates at the interface are calculated over a wide range of rarefaction parameter and for some values of molar fraction of the background gas in the mixture. Moreover, two values for the temperature of the mixture are considered on the basis of the sublimation curves of argon and krypton. To assess the influence of the interatomic interaction potential on the numerical results, the calculations are carried out for both the hard-spheres and \textit{ab-initio} potentials. The kinetic coefficients are presented as well as the flow fields around the sphere. The effect of a small temperature difference between the spherical surface and the gas mixture is analyzed.

Figures

Figures reproduced from arXiv: 2507.10858 by the authors.

Figure 1
Figure 1. Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]
Figure 5
Figure 5. Figure 5: Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]
Figure 6
Figure 6. Figure 6: Density and temperature deviations of species due [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 7
Figure 7. Figure 7: Temperature, pressure and molar fraction deviati [PITH_FULL_IMAGE:figures/full_fig_p036_7.png]
Figure 8
Figure 8. Figure 8: Temperature, pressure and molar fraction deviati [PITH_FULL_IMAGE:figures/full_fig_p037_8.png]
Figure 9
Figure 9. Figure 9: Temperature, pressure and molar fraction deviati [PITH_FULL_IMAGE:figures/full_fig_p038_9.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.