{"id":"6f144682-e6ff-4703-988f-0906ce984a79","arxiv_id":"1908.01221","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The wetting pattern at a 90-degree outer corner is measurably changed by a nearby 270-degree inner corner, while the inner-corner rivulet shape follows a single universal curve described by a new model valid up to 90-degree contact angles.","lead":"This paper measures how a liquid film climbs over a step with one sharp outward corner and one sharp inward corner, and finds the two corners pull on each other's wetting pattern from surprisingly far away. It also proposes a unified model for the liquid rise in the inward corner that works across a wide range of contact angles, which matters for coating, printing, and textile wetting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 7 fails its stated α=90 limit by a factor of two; the claimed unified physical model is not correctly derived, leaving Eq. 8's quantitative predictions unsupported.","rationale":"In good faith, the paper's quantitative contribution hinges on Eq. 8 and its derivation from the slice model. The slice decomposition is acknowledged to be heuristic, but the paper makes a crisp, checkable claim: at α=90° Eq. 7 becomes Eq. 5. That claim is false by a factor of two, and the error is visible already at θ=0, where both limiting models coincide. This is an internal inconsistency, not merely a disagreement with another capillary model. The reader's weakest_assumption pointed to the same slice model and its lack of derivation; the present critique sharpens it into a concrete algebraic failure. The conditional verdict is retained, but the condition is now non-optional: the authors must correct Eq. 7 and re-validate Eq. 8 against the measured rivulet shape and against the high-contact-angle behavior. If the corrected model no longer matches the data, the strongest claim should be withdrawn. The experimental observations of cusp depth and corner interaction, along with the Surface Evolver checks, are useful and are not affected by this critique.","tokens_in":12198,"tokens_out":32720,"duration_ms":352876,"concrete_test":"Substitute α=π/2 and θ=0 into Eq. 7 and compare with Eq. 5: the result is σ/(2ρg x) versus σ/(ρg x), exposing the factor of two. Then re-derive the slice capillary-rise expression from the two-dimensional Young-Laplace equation for a meniscus between walls of opening angle α, requiring exact agreement with the α=0 and α=90 limits, and replace Eq. 7 with that corrected expression. Re-solve Eq. 8 with the corrected expression and compare the predicted rivulet profile with the measured ~10 lσ data in Fig. 7, especially for x/lσ ≲ 1. If the corrected profile differs from the published one by more than the measurement scatter, the experimental agreement is not evidence for Eq. 8 as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Directly substituting α=π/2 into Eq. 7 yields h = σ sin(π/2−θ−π/4) cos(π/4)/(ρg x) = σ(cosθ−sinθ)/(2ρg x), whereas Eq. 5 gives h = √2 σ sin(π/4−θ)/(ρg x) = σ(cosθ−sinθ)/(ρg x). The paper states that at α=90° Eq. 7 'becomes equal to Eq. 5', but it is exactly half. This is not a harmless convention: at θ=0 the two constituent models, Eqs. 3 and 5, are identical, so a consistent slice interpolation must stay at σ/(ρg x) in that limit, yet Eq. 7 drops to half. The mismatch indicates that the opening-angle geometry enters as cos(α/2) in the numerator instead of through the correct wedge denominator; any physically consistent slice derivation must reproduce both limits exactly. Since Eq. 8 inherits this factor, the quantitative rivulet profile near the corner and the high-contact-angle curves in Fig. 10 rest on an internally inconsistent equation. The low-θ agreement with Fig. 7 cannot validate the claimed physical unification until this limit is repaired.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports an experimental and theoretical study of spontaneous wetting on a flat vertical step bounded by a 90° outer corner and a 270° inner corner. For step sizes ranging from about 0.03 to 10 capillary lengths, the authors measure the full three-dimensional contact line contour, quantify the cusp at the outer corner and the rivulet at the inner corner, and track the early-time rivulet rise. The main theoretical contribution is a unified rivulet equation (Eq. 8) that is intended to combine a vertical-curvature model (Eqs. 2–4) with a horizontal-curvature model (Eq. 5), to reduce to each in the appropriate limits, and to be the first rivulet shape description that obeys the Concus–Finn criterion: infinite rise for contact angles below 45° and finite rise for angles above 45°. The paper also proposes an empirical cusp-depth relation and a simple model for the effect of small step size on rivulet rise speed.","tokens_in":12434,"tokens_out":5218,"duration_ms":52986,"significance":"If the unified equation were correctly derived and validated, the paper would provide a useful quantitative framework for wetting on surfaces containing interacting sharp corners, a setting relevant to coating, printing, and textured-surface wetting. The experimental work is careful: contact lines were measured manually with repeated averaging, error bars are shown, and step-size independence of the inner-corner rivulet shape is a clean and well-supported observation. The use of independent Surface Evolver calculations to bound the asymptotic cusp depth is a methodological strength. The claim about the Concus–Finn behavior is attractive but, as detailed below, is not currently supported because the central equation fails its own stated limit by a factor of two. The paper is therefore a promising experimental study whose main theoretical claim needs substantial repair.","major_comments":[{"comment":"The paper states that for α = 90°, Eq. (7) 'becomes equal to Eq. (5)'. Direct substitution gives h = σ sin(π/2 − θ − π/4) cos(π/4)/(ρg x) = σ(cos θ − sin θ)/(2ρg x), whereas Eq. (5) gives h = √2 σ sin(π/4 − θ)/(ρg x) = σ(cos θ − sin θ)/(ρg x). Thus Eq. (7) is exactly half of Eq. (5) at the stated limit. At θ = 0 both constituent models reduce to σ/(ρg x), but Eq. (7) gives σ/(2ρg x). Since Eq. (8) inherits this factor, the quantitative rivulet profile near the corner and the high-contact-angle curves in Fig. 10 rest on an internally inconsistent equation. The low-contact-angle agreement with the measured rivulet in Fig. 8 cannot validate the claimed unification until this limit is repaired.","section":"Rivulet behavior, Eq. (7)"},{"comment":"The slice model behind Eq. (7) is an ansatz rather than a derivation from the three-dimensional Young–Laplace equation. The manuscript asserts that each slice normal to the main curvature sees two walls whose opening angle is equal to the local slope α and that the second curvature along the contact line is negligible ('tests... showed that its effect is negligible'), but no quantitative support or error estimate is given. Because the α = 90° limit fails by a factor of two, the geometrical equivalence is not merely unproven; the interpolation does not correctly reproduce either limiting model, so the physical grounding of Eq. (7) is currently missing.","section":"Rivulet behavior, Figure 9 and derivation of Eq. (7)"},{"comment":"The Bullard correction term from Eq. (4) is applied in Eq. (8) to an inclined, nonparallel-wall slice geometry, although Eq. (4) was fitted to numerical simulations of menisci between parallel planar walls. The manuscript gives no justification for transferring this empirical correction to slices with opening angle α. The correction is not negligible at the intermediate distances where the model is compared with the measured rivulet in Fig. 8, so the apparent agreement shown there does not by itself validate Eq. (8).","section":"Rivulet behavior, Eq. (8)"},{"comment":"The conclusion that Eq. (8) is the first model able to 'quantitatively predict rivulet shapes for all contact angles between 0° and 90°' is overstated. The manuscript states that reproducible contact angles above 45° could not be achieved and that the high-angle comparison is only qualitative. Given the factor-of-two error in Eq. (7), the θ > 45° curves in Fig. 10 are unverified predictions rather than validated results, and the claim should be softened or supported by additional evidence.","section":"Conclusions, high-contact-angle predictions"}],"minor_comments":[{"comment":"The abstract contains a typo: 'in uenced' should read 'influenced'.","section":"Abstract"},{"comment":"In the text near Fig. 2 the phrase 'the contact line deviatesis no longer straight' appears corrupted; it should likely read 'the contact line deviates, i.e., it is no longer straight' or similar.","section":"Results and Discussion, General phenomena"},{"comment":"The caption of Fig. 9 uses 'slides' where 'slices' is meant; this should be corrected.","section":"Figure 9 caption"},{"comment":"The legend entries labeled 'ref.' are not defined in the captions; the definition given in the accompanying text should also appear in the figure captions for clarity.","section":"Figures 5 and 7"},{"comment":"Eq. (1) should be described more carefully as an empirical model with c_max as an input parameter taken from the Surface Evolver calculation, rather than as a prediction; the two outlier data points are attributed to calibration inaccuracies without supporting evidence, and this should be flagged as a limitation.","section":"Cusp behavior, Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":"The factor-of-two limit error in Eq. (7) is the central issue; it undermines the main theoretical claim and the high-angle predictions. The experimental dataset and the step-size independence observation are solid and worth publishing after the model is corrected or its scope is reduced. I do not see any other indication of problematic research practice."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the experiments are worth knowing about, but the 'unified' rivulet model has a genuine mathematical bug. The stress-test note is right: plug α=π/2 into Eq. 7 and you get half of Eq. 5. The authors say Eq. 7 'becomes equal to Eq. 5' at α=90°, but it doesn't. This is not a harmless typo; the slice construction puts the opening angle in the wrong place. Because Eq. 8 inherits the error, the quantitative rivulet profiles in Fig. 10 and the Concus-Finn claim rest on an inconsistent equation. At θ=0 the low-angle agreement with Fig. 7 is not enough to validate the unification.\n\nWhat's genuinely good: the contact-line measurements are careful. Manual detection is averaged, error bars are reasonable, the step-size range from 0.033 to 10 lσ is well chosen, and the collapse of the rivulet shape independent of step size (Fig. 7) is a solid empirical result. The cusp-depth vs step-size data and the Surface Evolver asymptotic limit are new and useful. The pinning transition at step sizes below lσ is also a clear finding.\n\nThe soft spots beyond the factor-of-two: the slice model is a heuristic ansatz, not derived from the 3D capillary equation; the 'second curvature is negligible' claim is asserted without showing the test; the Bullard correction is applied outside its parallel-wall domain; and the high-contact-angle predictions have no quantitative data behind them. The rivulet-dynamics speed correction uses a fitted tip width d, so it is descriptive rather than predictive.\n\nNet: this is a solid experimental paper with an overclaimed unified model. I would send it to peer review, but the referee should require the authors to fix or properly derive Eq. 7 and re-examine the α=90 limit before the model claims are accepted. I wouldn't cite Eq. 8 in its current form.","headline":"Solid experimental study of corner-wetting interaction, but the central 'unified' rivulet equation fails its stated α=90° limit by a factor of two, so the claimed unification is not yet established.","tokens_in":12975,"tokens_out":3451,"would_cite":false,"duration_ms":33200,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a unified differential equation for the static rivulet shape at a 270° inner corner, the first to reproduce the Concus-Finn criterion, and supports it with measurements showing the rivulet profile is independent of step…","keywords":["spontaneous wetting","capillary rise","rivulet","Concus-Finn criterion","contact line","inner corner","outer corner","cusp"],"falsifier":"Take the same step geometry with a liquid having a stable contact angle above 45°, which the authors could not obtain reproducibly, and compare the measured equilibrium rivulet profile to Eq. (8). A systematic departure in height or width, or a full numerical solution of the Young-Laplace surface that does not match the slice prediction, would show the slice decomposition fails for high contact angles.","tokens_in":1742,"feed_emoji":"💧","tokens_out":1863,"duration_ms":83338,"temperature":0.7,"pith_summary":"This paper studies what happens when a liquid spontaneously climbs a vertical surface with a 90° outer corner and a 270° inner corner nearby. It claims the inner-corner rivulet has a universal shape for steps from about a third of the capillary length up to ten capillary lengths, and it proposes a single differential equation that predicts that shape for contact angles from 0° to 90°. The equation is the first rivulet model that reproduces the Concus-Finn threshold: infinite rise below 45° and a finite rise above 45°, flattening at 90°. The authors also quantify how the cusp at the outer corner shrinks as the step becomes smaller and show the interaction is one-sided: the inner corner feels the outer one only when the step is much smaller than the capillary length.","feed_headline":"Unified equation predicts rivulet rise at every contact angle up to 90°","feed_subtitle":"A new differential equation unites vertical and horizontal capillary curvature and matches measured rivulet shapes.","key_machinery":"The central object is Eq. (8), a first-order nonlinear ordinary differential equation for the rivulet height $h(x)$ in terms of the distance $x$ from the corner. Its working parts are: the identification of the local slope angle $\\alpha$ as the effective opening angle between the walls seen by an inclined liquid slice; the geometric fact that the distance from a wall point to the 45° bisector equals the wall position $x$; the blend of parallel-wall capillary rise (vertical curvature) with horizontal-arc capillary rise (horizontal curvature); and an empirical correction for the difference between average meniscus height and deepest meniscus point. The equation's defining property is its limiting behavior: it becomes the parallel-wall law for $\\alpha=0$, the horizontal-arc law for $\\alpha=90^\\circ$, and it is the first such description that respects the Concus-Finn criterion over the full 0° to 90° contact-angle range.","core_discovery":"The central discovery is that the entire static contact line contour between a 90° outer corner and a 270° inner corner can be described by treating the rivulet as a stack of liquid slices normal to the main curvature. Each slice sits between two walls whose effective opening angle equals the local slope angle $\\alpha=\\arctan(-\\partial h/\\partial x)$. Combining the vertical-curvature rise of parallel-wall theory with the horizontal-curvature rise of arc theory gives a unified differential equation, and adding an empirical meniscus-height correction yields Eq. (8): $$h = \\frac{\\$\\sigma$ \\sin\\left(\\frac{\\pi}{2}-\\$\\theta$-\\frac{\\$\\alpha$}{2}\\right)}{\\cos\\left(\\frac{\\$\\alpha$}{2}\\right)\\rho g x} - 2x \\left(f(\\$\\theta$)g(\\sigma_B)$e^{{-4.48\\sigma_B^{1/8}}$}\\right).$$ This equation reduces to the parallel-wall law far from the corner ($\\alpha=0$) and to the horizontal-arc law at the corner ($\\alpha=90^\\circ$). Unlike either parent model, it satisfies the Concus-Finn criterion: it gives an infinite rivulet for $\\theta<45^\\circ$, finite heights for $\\theta>45^\\circ$, and a flat surface at $\\theta=90^\\circ$. Measurements with silicone oil show that the rivulet shape near the inner corner is independent of step size and matches this unified equation for the largest step at low contact angle.","pith_inferences":["Editorial extension: the same slice construction could be applied to inner corners with opening angles different from 90°, where the bisector-distance identity would take a different form; the predicted threshold would then follow the general Concus-Finn angle rather than 45°.","Editorial extension: Eq. (8) adds the empirical correction without blending near the corner, so a high-contact-angle experiment would reveal whether the correction is hiding a real geometric effect outside the parallel-wall regime where it was originally derived.","Editorial extension: the tip-only dynamics suggested by the velocity data could be tested by tracking a rivulet tip on a sample with a sudden step-width change; the curvature-ratio model predicts an immediate speed change tied to the local geometry rather than to the overall step size."],"forward_implications":["For contact angles below 45°, the inner-corner rivulet rises without bound; above 45° it reaches a finite equilibrium height that shrinks to a flat surface as the contact angle approaches 90°.","Near the inner corner, the rivulet shape is universal: it stays the same for step sizes from about one capillary length up to at least ten capillary lengths.","The outer corner's cusp becomes shallower as the two corners move closer; when the step is smaller than the capillary length, the contact line pins at the outer corner and follows the vertical edge.","The contact line contour on the face between the two corners can be computed from the unified equation together with the cusp-depth relation, giving a quantitative model of the whole contour rather than a pointwise rise height.","The rivulet tip rise speed is independent of step size except for very small steps, and a model comparing the curvature of a pinned slice with an unhindered slice accounts for the slowdown at small steps."],"supporting_citations":[{"why":"Supplies the wedge criterion that infinite rivulet rise occurs only when contact angle plus half the opening angle is below 90°, the threshold Eq. (8) must satisfy.","marker":"[27]"},{"why":"Provides the capillary-tube rise model for corner rivulets that Eq. (8) extends by adding horizontal curvature.","marker":"[7]"},{"why":"Provides the horizontal-arc model of negative capillary pressure that gives the near-corner limit of Eq. (8).","marker":"[26]"},{"why":"Supplies the empirical correction between average meniscus height and deepest meniscus point that is added to the unified equation.","marker":"[31]"},{"why":"Provides the numerical energy-minimization results that set the asymptotic cusp-depth limit used in the cusp-depth model.","marker":"[30]"},{"why":"Numerically confirms the Concus-Finn threshold in a corner, supporting the claim that Eq. (8) is the first analytic shape description to reproduce it.","marker":"[28]"}],"fun_headline_variants":["Single equation unifies inner and outer corner wetting","Unified model predicts rivulet shapes at sharp corners","Corner wetting interaction distilled into one differential equation","One law links corner wetting, from close to far apart"],"cache_read_input_tokens":15104,"weakest_assumption_plain":"The prediction hangs on treating the rivulet as independent slices normal to the main curvature, with each slice seeing an opening angle equal to the local slope and with the second curvature along the contact line neglected; this geometric equivalence is asserted rather than derived from the full three-dimensional capillary equation.","fun_headline_variants_meta":{"raw":{"variants":["Single equation unifies inner and outer corner wetting","Unified model predicts rivulet shapes at sharp corners","Corner wetting interaction distilled into one differential equation","One law links corner wetting, from close to far apart"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000394,"raw_usage":{"total_tokens":2122,"prompt_tokens":1056,"completion_tokens":1066,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":1001}},"tokens_in":672,"tokens_out":1066,"duration_ms":10699,"temperature":1.0,"reasoning_tokens":1001,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:19:45.174835+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the same step geometry with a liquid having a stable contact angle above 45°, which the authors could not obtain reproducibly, and compare the measured equilibrium rivulet profile to Eq. (8). A systematic departure in height or width, or a full numerical solution of the Young-Laplace surface that does not match the slice prediction, would show the slice decomposition fails for high contact angles.","supporting_citations":[{"cited_title":"On the behavior of a capillary surface in a wedge","cited_arxiv_id":null,"evidence_quote":"Supplies the wedge criterion that infinite rivulet rise occurs only when contact angle plus half the opening angle is below 90°, the threshold Eq. (8) must satisfy."},{"cited_title":"A universal law for capillary rise in corners","cited_arxiv_id":null,"evidence_quote":"Provides the capillary-tube rise model for corner rivulets that Eq. (8) extends by adding horizontal curvature."},{"cited_title":"Capillary Rise in Tubes with Sharp Grooves","cited_arxiv_id":null,"evidence_quote":"Provides the horizontal-arc model of negative capillary pressure that gives the near-corner limit of Eq. (8)."},{"cited_title":"at Darmstadt, Darmstadt, Germany Maximilian Hartmann [NMF] Institute for Nano- and Microfluidics, Technische Universit\\","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical correction between average meniscus height and deepest meniscus point that is added to the unified equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the numerical energy-minimization results that set the asymptotic cusp-depth limit used in the cusp-depth model."},{"cited_title":"V.; Tropea, C.; Garoff, S","cited_arxiv_id":null,"evidence_quote":"Numerically confirms the Concus-Finn threshold in a corner, supporting the claim that Eq. (8) is the first analytic shape description to reproduce it."}],"review_version":1}