{"id":"2dabbd6b-b2f8-4548-b627-9cc8fc79df14","arxiv_id":"2411.15021","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a static-friction model, capillary force and plate separation in a two-dimensional liquid bridge are sufficient to deduce the Young's angle and the static friction coefficient without direct contact-angle measurement.","lead":"This paper proposes a way to find a liquid's Young's angle, the ideal contact angle a drop makes with a solid, by measuring capillary force and plate separation in a two-dimensional liquid bridge instead of measuring contact angles directly. The method also yields a formula for the static friction coefficient from the difference between advancing and receding contact angles.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The determination of θY hinges on Eq. (8)'s postulate of a single, direction-independent μs multiplying γ sinθ at both critical angles; without experimental or simulation evidence for this law, Eq. (10) is not established. A direct test of μs,adv = μs,rec would settle it.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing point: the entire extraction of μs and θY from critical angles in Section III rests on Eq. (8), which is imported without independent support. I checked the algebra from Eq. (8) to Eqs. (10) and (12); it is correct. I also checked the force-balance equations (A3) and (6) used for the inversion; they follow from the Young–Laplace equation and are internally consistent. The paper's analytical work on capillary-bridge profiles and the spring analogy is extensive, though I note that the energy expression in Eq. (3) appears to have the surface element inverted (it writes sqrt(1−f^2) where sqrt(1+f'^2) = 1/sqrt(1−f^2) would be expected from Eq. (B3)); this would affect the energy–force relation and spring constants, but not the central θY formula, which depends only on force balance and the friction law. Since the proposed method cannot be used until Eq. (8) is verified, the paper is best treated as a conditional theoretical proposal. No new evidence in the paper changes the reader's conditional verdict, so I recommend UNCHANGED.","tokens_in":14721,"tokens_out":12748,"duration_ms":120472,"concrete_test":"Conduct a capillary-bridge experiment (or MD simulation) on a well-characterized solid–liquid system with independently known θY (e.g., a smooth silanized surface with a pure alkane; θY from a Wilhelmy-plate or captive-bubble method). Measure the force–separation curve while slowly pushing down and pulling up the upper plate; at each depinning event, solve Eqs. (A3) and (6) for the instantaneous contact angles, identifying θa and θr. Then compute μs,a=(cosθY−cosθa)/sinθa and μs,r=(cosθr−cosθY)/sinθr. If |μs,a − μs,r| exceeds experimental uncertainty, Eq. (8)'s single-coefficient assumption is falsified and Eq. (10) cannot be used. If the two agree, repeat for 3–5 liquid–solid pairs spanning θY from 30° to 150° to confirm the sinθ scaling.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that θY is determined by Eq. (10) and μs by Eq. (12)—is an algebraic consequence of equating the two branches of Eq. (8), where μs is defined as (cosθY − cosθa)/sinθa for advancing and (cosθr − cosθY)/sinθr for receding. This equation imports a specific constitutive law from the authors' unpublished preprint [9]: the maximum static friction at a contact line is proportional to γ sinθ of the critical angle, with one coefficient μs valid for both advancing and receding motion. If the scaling with sinθ is wrong, or if advancing and receding pinning are characterized by different coefficients μs,a ≠ μs,r—a common situation on rough or chemically heterogeneous surfaces—then equating the two expressions is invalid and Eq. (10) does not yield the thermodynamic Young's angle. The paper presents no experimental force–gap data, no simulation, and no literature comparison testing Eq. (8); the only support is self-cited. The inversion from measured F and H to θa and θr (Eqs. (A3), (6)) also assumes the contact lines remain static until exactly one critical angle is reached, but the depinning signature in the F–H curve is never specified. The most load-bearing gap, however, is the untested single-coefficient friction law.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a method to determine the static friction coefficient and the Young's angle for two-dimensional horizontal capillary bridges by measuring only the capillary force and the plate separation, avoiding direct contact-angle measurement. The authors derive exact solutions to the Young-Laplace equation for two-dimensional bridges with and without gravity, establish a spring-like relation between capillary force and energy, and derive formulas μs = tan((θa−θr)/2) and θY = cos^{−1}[sin(θa+θr)/(sinθa+sinθr)] from the assumption that a single static friction coefficient governs both advancing and receding contact-line motion. The appendices contain detailed energy-minimization and integration steps.","tokens_in":14980,"tokens_out":4356,"duration_ms":42246,"significance":"If the underlying constitutive assumption is valid, the proposed method would provide a practical experimental route to the Young's angle and a quantitative measure of contact-line friction, and the exact bridge profiles and spring relation are useful theoretical contributions. The appendices present detailed, checkable derivations, and the paper correctly identifies that direct contact-angle measurement is difficult. However, the central claim that the Young's angle can be determined rests entirely on an untested friction law imported from the authors' prior preprint, so the significance is conditional on that law.","major_comments":[{"comment":"The derivation of Eq. (10) and Eq. (12) follows by equating the two expressions for μs in Eq. (8). This equating presumes that the maximum static friction at the advancing and receding contact lines is given by γ μs sinθa and γ μs sinθr with a single coefficient μs. This constitutive law is imported from reference [9] and is not tested or independently justified in the present manuscript. No experimental force–gap data, no simulation, and no literature comparison are provided that would support the sinθ scaling or the equality of μs for advancing and receding motion. Since this assumption is the load-bearing element of the proposed method, the claim that the Young's angle 'can be determined' is not established. The authors should either provide direct evidence for Eq. (8) or explicitly frame the results as consequences of that model, with a discussion of how violations (e.g., μs,a ≠ μs,r) would affect the inferred θY.","section":"Section III, Eq. (8)"},{"comment":"The proposed experimental protocol requires deducing the critical angles θa and θr from measured capillary force and plate separation. The manuscript does not specify how the onset of depinning is identified in the F–H curve—whether it appears as a slope discontinuity, a force plateau, or a jump—nor does it discuss the stability of the pinned state before the critical angle is reached. This information is essential for the practical implementation of the method and for verifying that the measured angles indeed correspond to the advancing/receding critical angles used in Eq. (8).","section":"Section III and Eqs. (A3) and (6)"},{"comment":"The paper contrasts Eq. (10) with the traditional formulas in Eq. (11) and concludes that the traditional relations are valid only for small hysteresis. However, this comparison is made under the same unvalidated friction law; without an independent determination of θY, the numerical closeness to Adam and Jessop's value is not evidence for the correctness of Eq. (10). The authors should temper the conclusion or provide a separate argument that does not rely on the very assumption being tested.","section":"Section III, Eq. (10)"}],"minor_comments":[{"comment":"The function f(Y) is used before its definition is clearly stated; please define all symbols in a single notation table or at first use.","section":"Eq. (3)"},{"comment":"The captions for Figs. 4 and 5 do not identify which line corresponds to which branch of Eq. (8); please add explicit labels or a legend.","section":"Figures 4 and 5"},{"comment":"The statement that 'the maximum static friction takes different values' is confusing because Eq. (5) defines κu and κd as constraint forces, not maximum values; please clarify the connection between κu, κd and the maximum static friction used in Eq. (8).","section":"Section II, after Eq. (5)"},{"comment":"Reference [9] is cited as an arXiv preprint; if it has been published or accepted, please update the citation.","section":"Reference [9]"},{"comment":"The piecewise definition of Y0,max and the conditions for pinch-off would be clearer if the physical meaning of the two cases (no pinch-off vs. pinch-off) were explained in the main text.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central claim is conditional on an untested constitutive law from the authors' own preprint. The paper would be substantially strengthened by adding any available experimental or numerical evidence for Eq. (8), or by clearly re-scoping the claims as model-dependent predictions. The exact-solution and spring-relation material in the appendices is valuable and may be publishable even if the Young's-angle determination is presented as conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is real classical capillary theory: exact two-dimensional bridge profiles with gravity in elliptic functions, an energy–force relation, spring constants, and clear neck/bulge/pinch-off conditions. That part is careful, checkable, and genuinely useful. Second, the headline experimental proposal—determining the Young's angle from force–gap measurements via Eq. (10)—is not established. It depends on a one-coefficient static-friction law imported from the authors' own preprint [9], and no data or simulation in this paper tests that law.\n\nThe analytic contributions are solid. Appendices A and B are thorough, the derivation of F = −∂E/∂H with positive second derivative looks right, and the B=0 circular solution plus Eq. (B1) are standard. The exact elliptic solutions go beyond the numerical treatment of Teixeira and Teixeira, and the spring-constant expressions (different from the axisymmetric 3D results) are a nice addition. I would cite it for those results.\n\nThe soft spot is Section III, and it is load-bearing. Equation (8) defines μs as (cosθY − cosθa)/sinθa and (cosθr − cosθY)/sinθr, which assumes the maximum static friction scales as γ sinθ with a single μs valid for both advancing and receding motion. Equating the two expressions gives Eq. (10). But if advancing and receding pinning have different coefficients—common on rough or heterogeneous surfaces—or if the sinθ scaling fails, Eq. (10) will not yield the thermodynamic Young's angle. The paper offers no experimental or numerical test of this constitutive law. The abstract says the method \"dispens[es] with the need for direct measurement\" and that the determined Young's angle differs from Adam-Jessop and Drelich; as it stands that is a model prediction, not a measurement. The abstract's phrasing overclaims slightly, though the body does frame it as a proposal.\n\nA smaller but real issue: the inversion from measured F and H to θa and θr assumes the contact lines remain pinned until exactly one critical angle is reached, but the depinning signature in the force–gap curve is never specified. That is minor relative to the friction-law gap.\n\nWho is this for? People working on capillary bridges or contact angle hysteresis will find the exact solutions and the spring analogy worth their time. The Young-angle method needs validation before being accepted. It deserves a serious referee—the mathematics is strong enough to warrant expert scrutiny, and the friction-law assumption should be flagged as the key thing to test experimentally. I would not reject it outright; I would send it to review with a request that the empirical claim be rephrased as conditional on the friction law.","headline":"Solid analytic work on 2D capillary bridges, but the proposed Young's-angle determination rests on an untested friction law and is not yet a measurement.","tokens_in":15515,"tokens_out":1972,"would_cite":true,"duration_ms":21364,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"By treating contact-line pinning as static friction, this paper derives $\\mu_s = \\tan((\\theta_a-\\theta_r)/2)$ and a Young's-angle formula that can be applied using only capillary force and plate separation measurements.","keywords":["contact angle hysteresis","static friction coefficient","Young's angle","capillary bridge","capillary force measurement","contact line pinning","Young-Laplace equation","two-dimensional liquid bridge"],"falsifier":"Run repeated push-down and pull-up cycles on a single capillary bridge of known liquid and solid while recording capillary force and plate separation; extract the advancing and receding angles and compute $\\mu_s$ from Eq. (12) separately for the two directions. If the two values disagree by more than experimental error, or if the Young's angle from Eq. (10) contradicts an independent determination obtained by a different method (for example, the value approached as hysteresis width goes to zero on the same surface), the friction law and the proposed derivation fail.","tokens_in":14451,"feed_emoji":"💧","tokens_out":11659,"duration_ms":96788,"temperature":0.7,"pith_summary":"This paper proposes a way to determine the coefficient of static friction at a liquid-solid contact line and the Young's angle without ever measuring contact angles directly. The route is to use a two-dimensional horizontal capillary bridge between two plates: record only the capillary force and the plate separation, deduce the advancing and receding critical angles from those readings, and then compute the friction coefficient as $\\mu_s = \\tan((\\theta_a-\\theta_r)/2)$ and the Young's angle as $\\theta_Y = \\cos^{-1}[\\sin(\\theta_a+\\theta_r)/(\\sin\\theta_a+\\sin\\theta_r)]$. If correct, this sidesteps a measurement widely regarded as difficult and gives a quantitative, hysteresis-aware definition of the Young's angle. The paper also derives exact bridge profiles, locates necks, bulges, and pinch-offs, and shows the bridge responds to vertical displacement like a spring.","feed_headline":"Friction formula turns capillary force readings into the Young angle","feed_subtitle":"No direct contact-angle readings needed: force and gap give the friction coefficient and Young's angle.","key_machinery":"The load-bearing machinery is the static-friction law for contact lines, taken from the authors' earlier droplet analysis: the maximum static friction at a critical angle is proportional to $\\sin\\theta_c$ with one coefficient $\\mu_s$. This law turns the modified Young's equations (the horizontal force balances at the pinned contact points) into two independent expressions for $\\mu_s$; since $\\mu_s$ is unique for a solid-liquid pair, equating them produces the Young-angle identity and then the hysteresis-width formula. The companion experimental machinery is the capillary force expression $F_{\\rm capillary} = 2\\sin\\theta_u + 2BY_0 + (\\cos\\theta_u+\\cos\\theta_d)/Y_0$, with the no-gravity reduction $F_{\\rm capillary} = 2(\\sin\\theta_C + \\cos\\theta_C/Y_0)$, together with the area constraint and the height-contact-angle relation (Eq. (B1)) that let contact angles be inferred from force and separation instead of optics. Finally, the energy-force relation $F = -\\partial E_{\\rm total}/\\partial H$ with $\\partial^2 E/\\partial Y_0^2 > 0$ supplies the spring-like stability and effective spring constants.","core_discovery":"The central claim is that contact-line pinning is static friction with a single coefficient per solid-liquid pair, and that this friction can be extracted from capillary-bridge force measurements. At each contact point the horizontal force balance gives modified Young's equations $\\kappa_u = \\gamma(\\cos\\theta_u-\\cos\\theta_Y)$ and $\\kappa_d = \\gamma(\\cos\\theta_Y-\\cos\\theta_d)$; when the contact line is about to move, the maximum friction is written in terms of the critical angles, giving $\\mu_s = (\\cos\\theta_Y-\\cos\\theta_a)/\\sin\\theta_a$ on the advancing side and $\\mu_s = (\\cos\\theta_r-\\cos\\theta_Y)/\\sin\\theta_r$ on the receding side. Equating the two expressions yields the Young-angle formula, and substituting it back gives $\\mu_s = \\tan((\\theta_a-\\theta_r)/2)$, so the hysteresis width alone fixes the friction coefficient. The experimental content is that $\\theta_a$ and $\\theta_r$ need not be measured optically: measuring capillary force (Eq. (6)) and plate separation, with the area constraint (Eq. (A3)), determines the critical angles, and in zero gravity the contact angle follows from the separation alone. The paper also derives exact solutions to the Young-Laplace equation for the bridge profiles, shows the capillary force is the negative derivative of the energy with respect to height, and uses the positive second derivative to establish spring-like stability.","pith_inferences":["A testable extension the paper does not spell out: repeated push-down and pull-up cycles on one bridge provide two independent estimates of $\\mu_s$ via Eq. (12); agreement would support the single-coefficient friction law, and disagreement would localize where it fails.","The identity $\\mu_s = \\tan((\\theta_a-\\theta_r)/2)$ implies that for a fixed hysteresis width the friction coefficient is independent of the absolute values of the contact angles; one could test this by comparing surfaces engineered to share $\\theta_a-\\theta_r$ while differing in $\\theta_a$ and $\\theta_r$.","Because the bridge is a stable spring around equilibrium, small oscillations of the upper plate could extract the spring constants and thereby infer contact angles and friction dynamically, a route the paper leaves for future work.","The force-separation protocol should transfer to axisymmetric and vertical bridges, where the competition between pinch-off and critical-angle motion would need to be characterized before routine use."],"forward_implications":["For a given solid-liquid pair, the Young's angle can be obtained from capillary force and plate-separation measurements alone, without optical contact-angle readings.","In the absence of gravity the contact angle is determined by the plate separation via Eq. (B1), so the experimental protocol simplifies to measuring one length and one force.","The widely used estimates $\\theta_Y = \\cos^{-1}[(\\cos\\theta_a+\\cos\\theta_r)/2]$ and $\\theta_Y \\approx (\\theta_a+\\theta_r)/2$ hold only when hysteresis is small; the paper's identity replaces them in general.","The bridge is a stable spring with linear and quadratic constants $k_1 = -2\\sin\\theta_{\\rm eq}\\tan\\theta_{\\rm eq}$ and $k_2 = -2\\sin\\theta_{\\rm eq}\\tan^2\\theta_{\\rm eq}$ in zero gravity, so effective stiffness is predictable from the equilibrium contact angle.","Because pinch-off can occur before a contact line reaches a critical angle for some areas and Bond numbers, the proposed protocol must choose parameter ranges where contact-line motion precedes pinch-off."],"supporting_citations":[{"why":"Supplies the constitutive static-friction law for contact lines and the definition of the friction coefficient that the whole derivation rests on.","marker":"[9]"},{"why":"The century-old hysteresis-as-friction reference whose averaged-angle estimate the paper's Young-angle formula is compared against.","marker":"[1]"},{"why":"The recent proposal about how the Young's angle should be measured; the paper's result is numerically closer to the older estimate than to this one.","marker":"[36]"},{"why":"Shows capillary-bridge force measurements can determine contact angles, the experimental route the proposed method relies on.","marker":"[47]"},{"why":"Extends the capillary-bridge force-based contact-angle determination to hydrophobic surfaces, supporting the general measurement approach.","marker":"[48]"},{"why":"Provides the two-dimensional liquid-bridge shape analysis whose energy-capillary force relation the paper derives quantitatively.","marker":"[76]"},{"why":"Gives spring constants of capillary bridges in the axisymmetric case, the comparison point for the paper's different two-dimensional spring constants.","marker":"[80]"},{"why":"Provides analytical spring-constant expressions for capillary bridges that the paper contrasts with its two-dimensional result.","marker":"[81]"}],"fun_headline_variants":["Capillary bridge friction yields Young's angle without angle readings","Force and separation determine friction and Young's angle directly","Static friction in bridges reveals Young's angle from force data","Measure bridge force and gap, get friction and Young's angle","No contact angles needed: friction from force gives Young's angle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole method rests on the assumption, carried over from the authors' earlier droplet paper, that the maximum static friction at a contact line grows with the sine of the critical contact angle and is described by one coefficient shared by both moving directions.","fun_headline_variants_meta":{"raw":{"variants":["Capillary bridge friction yields Young's angle without angle readings","Force and separation determine friction and Young's angle directly","Static friction in bridges reveals Young's angle from force data","Measure bridge force and gap, get friction and Young's angle","No contact angles needed: friction from force gives Young's angle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1461,"prompt_tokens":1009,"completion_tokens":452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":370}},"tokens_in":625,"tokens_out":452,"duration_ms":4796,"temperature":1.0,"reasoning_tokens":370,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:35:56.831540+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run repeated push-down and pull-up cycles on a single capillary bridge of known liquid and solid while recording capillary force and plate separation; extract the advancing and receding angles and compute $\\mu_s$ from Eq. (12) separately for the two directions. If the two values disagree by more than experimental error, or if the Young's angle from Eq. (10) contradicts an independent determination obtained by a different method (for example, the value approached as hysteresis width goes to zero on the same surface), the friction law and the proposed derivation fail.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constitutive static-friction law for contact lines and the definition of the friction coefficient that the whole derivation rests on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The recent proposal about how the Young's angle should be measured; the paper's result is numerically closer to the older estimate than to this one."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows capillary-bridge force measurements can determine contact angles, the experimental route the proposed method relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the capillary-bridge force-based contact-angle determination to hydrophobic surfaces, supporting the general measurement approach."},{"cited_title":"Cooray, H","cited_arxiv_id":null,"evidence_quote":"Provides the two-dimensional liquid-bridge shape analysis whose energy-capillary force relation the paper derives quantitatively."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives spring constants of capillary bridges in the axisymmetric case, the comparison point for the paper's different two-dimensional spring constants."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides analytical spring-constant expressions for capillary bridges that the paper contrasts with its two-dimensional result."}],"review_version":1}