{"id":"733481d3-b252-4d72-9101-4742c65704fe","arxiv_id":"2607.16670","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"An SPH droplet model whose cohesion and adhesion kernel amplitudes are fixed by the first moment of the kernel and tied to surface tension and the Young–Dupré equation.","lead":"The paper develops a smoothed particle hydrodynamics (SPH) model for droplet dynamics in which the pairwise forces are set directly from the liquid surface tension and a prescribed work of adhesion, rather than from fitted force constants. If it works as claimed, it would make droplet simulations on patterned surfaces easier to set up and more physically interpretable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The planar half-space first-moment mapping (Thm 3.1, Eq. 4.10) is applied to curved, finite-resolution interfaces with no Laplace-pressure or R_cho convergence check; the slope error in §5.1 (2.081 vs 2) suggests the effective surface tension is not exactly the prescribed γ_l.","rationale":"The reader's CONDITIONAL verdict is appropriate. I examined the central mathematical claim: Eqs. (4.10) and (4.14) are derived from Theorem 3.1's planar half-space first-moment identity, and the SPH discretization (4.16) uses these coefficients. The derivation itself is clean and internally consistent: integration by parts connects M1[W] to the virial integrand, so any compact kernel with the same first moment gives the same planar surface energy. The genuine soft spot is the transfer from the planar, constant-density continuum to a finite-resolution SPH droplet with curved and moving interfaces. This is precisely the assumption the authors flag in Remark 3.2 but do not test. Static contact-angle validation is insensitive to surface-tension magnitude because α_adh is the ratio W_sl/(2γ_l); the dynamic cases do depend on γ_l through the Weber number but are only compared visually. A Laplace-pressure benchmark on a spherical droplet at several R_cho/R values and resolutions would settle whether the effective γ equals the prescribed γ_l. The §5.1 slope 2.081 vs 2.0 already hints at a small systematic offset from the planar assumption. This does not overturn the paper; it strengthens the case for CONDITIONAL acceptance with a required convergence/Laplace test. The public code and DOI are a positive and make the proposed test directly feasible.","tokens_in":14753,"tokens_out":12852,"duration_ms":146581,"concrete_test":"Run a static spherical-droplet benchmark (no gravity, no solid) using the same cubic-spline kernel and Eq. (4.10), for droplet radii R = 2R_cho, 5R_cho, 10R_cho and two resolutions Δx = R_cho/4 and R_cho/8. Measure the steady pressure jump Δp across the interface and compare with Laplace's law Δp = 2γ_l/R. If Δp deviates from 2γ_l/R by more than ~5% at R = 10R_cho, or changes with Δx or R_cho/R, then the planar Theorem 3.1 coefficient does not transfer to curved interfaces at practical resolution, and the central 'physically prescribed' claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"At the core of the paper is Eq. (4.10), which fixes the liquid–liquid kernel potential by matching the first absolute moment of a planar interface between two half-spaces (Theorem 3.1). Remark 3.2 excuses the extension to curved interfaces only 'asymptotically' when the curvature radius is much larger than R_cho. In the simulations, R_cho = 0.20 mm while the droplet in §5.3 has radius ~2.5 mm (R/R_cho ≈ 12), and during impact the interface curvature is locally much sharper. For a short-range pair potential, the surface energy of a finite drop carries curvature corrections of order R_cho/R (Tolman-type), so the effective surface tension produced by the SPH force term can differ from the prescribed γ_l. Static contact angles in §5.1 do not test this: contact angle depends only on the ratio W_sl/γ_l, so a uniform shift in surface-tension magnitude is invisible. The dynamic validations (§5.3–5.4) are qualitative and have no error bars or convergence study. The observed least-squares slope 2.081 vs the theoretical 2.0 in §5.1 is a 4% systematic deviation consistent with a resolution-dependent effective interaction. Thus the central claim that γ_l is physically prescribed rather than effectively tuned is not yet established at the resolutions used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a single-phase SPH formulation for droplet dynamics on solid surfaces. It models liquid-liquid and liquid-solid interactions through effective kernel potentials whose prefactors are fixed by a first-moment relation derived for planar, constant-density interfaces (Theorem 3.1). In particular, Eq. (4.10) prescribes the liquid-liquid cohesion potential from the macroscopic surface tension coefficient γ_l and the kernel's first absolute moment, and Eq. (4.14) prescribes the liquid-solid adhesion from the work of adhesion W_sl. The Young-Dupré relation is rewritten in terms of an adhesion coefficient α_adh = W_sl/(2γ_l), giving cos θ_C = 2α_adh - 1. The paper validates the method with static contact-angle simulations, a Tanner-spreading test, droplet impact and rolling rebound on a patterned wettability surface, and coalescence-induced jumping.","tokens_in":15202,"tokens_out":8098,"duration_ms":77766,"significance":"If the proposed mapping remains valid at finite SPH resolution and on curved interfaces, the model would be a useful step toward reducing empirical parameter calibration in pairwise-force SPH. The proof of Theorem 3.1 is correct and transparent, and the derivation from the pair-potential energy through Eq. (3.24) is clean. The public availability of the source code and validation videos is also a strength. However, the current evidence does not yet establish the central claim that γ_l is physically prescribed rather than effectively tuned: the static contact-angle test is a self-consistency check, and the dynamic tests are qualitative or use a prescribed fitting exponent without convergence studies.","major_comments":[{"comment":"The static contact-angle validation is a self-consistency test, not an independent check of the parameter mapping. α_adh is defined as W_sl/(2γ_l), and W_sl is prescribed from the target angle through Young-Dupré; Eq. (4.22) is therefore an identity. The measured slope 2.081 vs 2 and intercept -1.104 vs -1 in Table 1 show a systematic deviation, yet no error bars, repeated-realization statistics, or resolution study are reported. Because cos θ_C depends only on the ratio W_sl/γ_l, uniform errors in the effective surface tension are invisible to this test. An independent measure of γ_l (e.g., Laplace pressure, capillary wave frequency, droplet oscillation) and a resolution study are needed.","section":"§5.1, Eq. (4.22)"},{"comment":"The calibration fixes the first absolute moment for a planar, constant-density half-space interface. In the impact simulations the droplet radius is about 2.5 mm while R_cho = 0.20 mm, and during spreading/recoil the local interface curvature is much sharper; the film can become comparable to or thinner than R_cho. Remark 3.2 merely asserts the asymptotic regime. No R_cho- or h-convergence study is supplied. Since the first-moment matching does not control curvature corrections of order R_cho/R, the effective surface tension produced by Eq. (4.16) may differ from the prescribed γ_l at the resolutions used. Please quantify this with a convergence study or provide a curved-interface correction/error estimate.","section":"Theorem 3.1 / Remark 3.2, Eq. (4.10)"},{"comment":"The Tanner-law test prescribes the exponent 0.1 in the fitting function R(t) = 2(t - t_min)^0.1 rather than fitting it. The text states that 'the fitted exponent' is 0.1, but the only free parameter is t_min. With t_min = 15 ms and no sensitivity or uncertainty analysis, the data do not provide evidence for the 1/10 exponent. Report a free-exponent fit with confidence intervals, or compare models with different fixed exponents using an appropriate information criterion.","section":"§5.2, Eq. (5.2)"},{"comment":"The comparisons for rolling rebound and coalescence jumping are qualitative: visual shape agreement is claimed, but no quantitative metrics are reported (e.g., spreading factor, contact-line position time series, jumping velocity, rebound angle), and there are no error estimates or grid-convergence checks. Given the transient, strongly curved interfaces in these tests, they cannot substitute for the planar-interface validation and do not yet support the claim of 'good agreement' with the experimental references [2,31].","section":"§5.3–5.4"}],"minor_comments":[{"comment":"Specify how M1[W_cs_{Rcho}] is evaluated—analytically or by numerical quadrature—and report its value or the quadrature error.","section":"Eq. (4.10)"},{"comment":"Please state how contact angles are extracted from simulated droplet profiles (e.g., circular fit, tangent at the contact line) and report the measurement uncertainty.","section":"§5.1"},{"comment":"The momentum equation uses the cubic-spline kernel for interaction forces and the Wendland kernel for SPH interpolation. Clarify why these two kernels are used and whether the choice affects the moment-calibration consistency.","section":"§4.2, Eq. (4.16)"},{"comment":"The phrase 'physically prescribed parameters' is stronger than what is demonstrated, since R_cho, h, Δx, and the artificial viscosity coefficient remain numerical parameters. Consider softening to 'reduced empirical calibration of interaction prefactors.'","section":"Abstract / Conclusion"},{"comment":"Show error bars and define how the spreading radius is measured. Log-log plots with a fitted t_min can be misleading without these details.","section":"Figure 6"}],"recommendation":"major_revision","confidential_remarks":"The core derivation is sound and the method is potentially useful, but the manuscript's main selling point—that γ_l is physically prescribed—is not established by the presented tests. The contact-angle validation is circular by construction, and the dynamic validations are qualitative. I would support acceptance after the authors add an independent measurement of the effective surface tension, a resolution/R_cho convergence study, and a properly fitted Tanner exponent. The paper is within the journal's scope; the issue is the gap between the claim and the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a solid incremental SPH paper with one clean mathematical kernel, an honest but overreaching claim, and some self-consistency validation presented as prediction. Worth refereeing, but it needs more work before I'd trust the dynamic results quantitatively.\n\nThe genuinely new piece is Theorem 3.1: for a planar interface, the interfacial energy from a pair potential is exactly one quarter of its first absolute moment. That identity is correct, and pushing it through to a closed-form calibration of the SPH cohesion amplitude (Eq. 4.10) removes the usual empirical knob-fitting for pairwise-force SPH. That is a real contribution for practitioners. Bonus: they ship the code and data on Zenodo, so the numerics are reproducible. The Tanner-law fit and the two-droplet adhesion tests are nice sanity checks.\n\nThe soft spots are exactly the ones flagged. Section 5.1 is not a prediction of the contact angle: W_sl is set to hit a chosen θ_C, so recovering Young–Dupré is a consistency test. The slope 2.081 vs 2 and intercept −1.104 vs −1 are fine as numerical diagnostics, but they do not validate the physics. More seriously, Theorem 3.1 is only proven for a planar interface, and Remark 3.2 waves at curvature without testing it. With R_cho = 0.20 mm and droplets of radius ~2.5 mm, curvature corrections at the interface are not negligible, and the observed slope error is exactly the kind of effect you'd expect. There is no R_cho-sensitivity or resolution-convergence study. The dynamic validations (rolling rebound, jumping) look plausible qualitatively, but they have no error bars and no quantitative comparison. That's the gap between a useful method and a validated one.\n\nI also note the artificial viscosity α and the t_min offset in the Tanner fit are free parameters; not a flaw per se, but they modulate the claims.\n\nWho should read this? SPH/CFD people who want to set cohesion parameters from macroscopic surface tension without empirical fitting. I'd use the kernel-moment formula in my own work, but I wouldn't cite the paper as a validated predictive tool for dynamic contact angles until the curvature/resolution question is answered.\n\nMy call: send it to peer review. A competent referee could ask for the missing convergence study and quantitative dynamic metrics; the core idea is sound and publishable after major revision.","headline":"A clean first-moment theorem gives SPH cohesion forces from macroscopic surface tension; the contact-angle validation is self-consistency, and dynamic claims need convergence support before they are quantitative.","tokens_in":15675,"tokens_out":2658,"would_cite":true,"duration_ms":28866,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76M28","76D45","65M99","76T10"],"pacs":[],"model":"deepseek-v4-flash","headline":"SPH droplet forces now follow from surface tension and adhesion energy alone.","keywords":["smoothed particle hydrodynamics","surface tension","contact angle","wetting","droplet impact","Young–Dupré equation","pairwise interaction force","interfacial energy"],"falsifier":"Run the static-droplet test of Section 5.1 at several kernel support radii Rcho (e.g., 0.1, 0.2, 0.4 mm at fixed h/Δx) and measure cos θC at fixed αadh. If the measured angle shifts away from 2αadh - 1 as curvature-to-Rcho ratio increases, the planar moment calibration does not generalize to curved interfaces.","tokens_in":14622,"feed_emoji":"💧","tokens_out":2756,"duration_ms":26852,"temperature":0.7,"pith_summary":"The paper proposes a single-phase SPH model for droplet dynamics in which the strength of every pairwise interfacial force is computed directly from two macroscopic quantities: the liquid surface tension γl and the liquid–solid work of adhesion Wsl. The key step is a theorem stating that the interfacial energy density across a planar interface equals one quarter of the first absolute moment of the pair potential. Inverting this relation gives the kernel amplitude Φll = -8γl / M1[W], so no empirical fitting of interaction strength is needed; the equilibrium contact angle follows from the Young–Dupré equation. Static wetting tests confirm cos θC = 2αadh - 1, and dynamic tests reproduce Tanner spreading, rolling rebound on patterned surfaces, and coalescence-induced jumping.","feed_headline":"Surface tension now sets SPH wetting forces directly","feed_subtitle":"Pairwise force strengths come from γl and Wsl via a first-moment identity; static and dynamic wetting tests match.","key_machinery":"Theorem 3.1, the moment representation: for two half-spaces separated by a planar interface, the interfacial potential energy density JQ[φ] equals π∫r³φ(r)dr = ¼ M1[φ], the first absolute moment of the potential over R³. This identity carries the argument: it converts the continuum surface energy into a kernel amplitude, giving Eq. (4.10) and (4.14). The rest of the model is standard SPH with a single-phase treatment: long-range attraction via the kernel, short-range repulsion absorbed into pressure.","core_discovery":"The central claim is that macroscopic surface tension and work of adhesion can be prescribed exactly at the level of SPH pair potentials through the first absolute moment of the smoothing kernel. For a liquid–liquid interaction, the pair potential is set as -8γl W(r)/M1[W], and for liquid–solid adhesion as -4Wsl W(r)/M1[W]. These choices make the interfacial energy per unit area match γl and Wsl and make the equilibrium contact angle satisfy the Young–Dupré relation cos θC = Wsl/γl - 1. The paper validates this parameter-free calibration on static droplets and on three dynamic problems.","pith_inferences":["The moment theorem is derived for a planar interface; a convergence study in kernel radius Rcho near highly curved interfaces would determine how small Rcho must be relative to local curvature for the calibration to remain exact.","The same first-moment argument could be extended to liquid–liquid adhesion between two droplet phases, offering a parameter-free route for multi-component SPH simulations.","Because the calibration is energetic rather than force-curvature based, it may be more robust than CSF-type surface tension models for topological changes such as coalescence and pinch-off, but this advantage remains to be demonstrated at higher resolution.","The model's reliance on the first absolute moment suggests a testable extension where the kernel is chosen to match higher moments as well, which would control curvature-dependent corrections beyond the planar limit."],"forward_implications":["Static contact angles can be set by choosing one scalar, the adhesion ratio αadh = Wsl/2γl, without tuning force parameters.","The same calibrated kernel preserves the liquid surface tension γl and the work of adhesion Wsl, so dynamic wetting problems inherit the macroscopic energetics.","Single-phase treatment lowers cost because only the liquid is discretized; solid boundary effects enter through the calibrated adhesive force and a kernel-weighted hydrostatic pressure correction.","Reproduces the t^1/10 Tanner spreading law, rolling rebound on a patterned-wettability surface, and coalescence-induced jumping, suggesting the calibration transfers to transient, curved interfaces."],"fun_headline_variants":["SPH wetting forces now set by surface tension directly","Kernel moment ties SPH to surface tension and adhesion","Physically prescribed SPH for droplet wetting and impact","SPH model calibrates itself from surface tension and adhesion"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The contact-angle calibration assumes the interface is locally planar with a constant number density; if the kernel support radius is not much smaller than the local curvature radius, the moment theorem no longer predicts the surface energy correctly, and the paper provides no convergence study in Rcho or resolution for the curved, dynamic interfaces it simulates.","fun_headline_variants_meta":{"raw":{"variants":["SPH wetting forces now set by surface tension directly","Kernel moment ties SPH to surface tension and adhesion","Physically prescribed SPH for droplet wetting and impact","SPH model calibrates itself from surface tension and adhesion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3257,"prompt_tokens":689,"completion_tokens":2568,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":2501}},"tokens_in":433,"tokens_out":2568,"duration_ms":16995,"temperature":1.0,"reasoning_tokens":2501,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:16:29.491589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the static-droplet test of Section 5.1 at several kernel support radii Rcho (e.g., 0.1, 0.2, 0.4 mm at fixed h/Δx) and measure cos θC at fixed αadh. If the measured angle shifts away from 2αadh - 1 as curvature-to-Rcho ratio increases, the planar moment calibration does not generalize to curved interfaces.","supporting_citations":[],"review_version":1}