Rough capillary rise

The paper investigates equilibrium capillary rise in rough structures and how surface roughness alters classical wetting behavior. Rough capillary rise involves coupling two phenomena: capillary rise between confining walls and hemiwicking within surface textures. While Jurin’s law and classical treatments (Young-Laplace, Gibbs, de Gennes) describe smooth capillaries, and hemiwicking criteria on rough surfaces are established, their interplay has been oversimplified. Prior works often assumed hemiwicking and capillary rise act independently or used effective-angle models (Cassie-Baxter, Wenzel) without clear validity ranges. This leads to inaccurate predictions of rise heights, especially at narrow separations typical in many applications (porous media, geology, textiles, bioinspired systems). The authors aim to develop a rigorous model that captures the coupling between capillary rise and hemiwicking across general roughness, elucidate when hemiwicking is suppressed by confinement, quantify differences between smooth and rough systems (including perfectly wetting cases), and accurately predict raised liquid volumes. They introduce the Dual-Rise model derived by energy minimization, highlighting the importance of two length scales: roughness depth and plate separation.

Classical literature established equilibrium capillary rise in smooth systems (Jurin’s law; Young-Laplace; de Gennes) and hemiwicking criteria on textured surfaces. However, many studies in porosity, earth sciences, biophysics, textiles, and engineering treat roughness via effective contact angles (Cassie-Baxter, Wenzel) inserted into Jurin’s law, implicitly assuming independence of capillary rise and hemiwicking and lacking guidance on validity ranges. The Wenzel approach is frequently applied where its predicted angle saturates at 0°, which is common for modest roughness and wettability, yet the approximation underestimates rise in this regime. Literature on rough rise dynamics often uses θc (hemiwicking critical angle) as a sole criterion without accounting for hydrostatic pressure changes due to capillary rise, leading to errors at small separations. This work addresses these gaps with a unified, validated model.

The study combines theory, experiments, and phase-field simulations. Theoretically, the Dual-Rise model is obtained by minimizing the total free energy change ΔG between reservoir and equilibrium configuration, including changes in interfacial areas (solid-gas, solid-liquid, liquid-gas) and gravitational potential energy. The system is partitioned into capillary (capillary rise and meniscus) and hemiwicked regions. A circular arc approximation describes the meniscus (valid up to separations on the order of the capillary length). The model defines geometric parameters: capillary rise C, meniscus height M, hemiwicking height H (defined via twice the center of mass height to ensure finiteness), total height T, average roughness depth δ, average gap between hemiwicked plates d, and average total separation D=d+2δ. Roughness is characterized by r (surface area ratio), φ (dry projected area fraction), and f (wetted surface area fraction), with f=1/r for the non-overhanging pillars studied. Energy minimization yields conditions that naturally bifurcate solutions into H=0 (no hemiwicking) and H>0, thus predicting the presence/absence of hemiwicking rather than assuming it. The model uses Cassie-Baxter effective angle for pre-hemiwicked surfaces and Wenzel angle for rough wetting where appropriate to set the meniscus angle, and provides analytic expressions for C, H, T, and M in terms of system parameters, capillary length λ, and contact angles. Experiments: Two parallel, additively manufactured textured plates (5 cm × 5 cm), each with square or cylindrical micropillars, were aligned at controlled separations using micrometers. Liquids: distilled water (γ=72 mN/m, ρ=1000 kg/m³, λ=2.7 mm) and dodecane (γ=24 mN/m, ρ=745.2 kg/m³, λ=1.8 mm). Surface wettability was tuned by plasma treatment. Heights of capillary rise, meniscus, and hemiwicking were measured via imaging between plates; pinning was mitigated by tapping to approach the equilibrium (global minimum energy) state; uncertainties include ±1 post pitch in height and ±50 μm in separation. Simulations: A (bi/ternary) phase-field diffuse-interface model with gravity and fluid–solid interactions was minimized via L-BFGS to obtain equilibrium morphologies. For cylindrical pillars, a ternary formulation enabled smooth curved solids; for square pillars, a binary liquid–gas model with sharp solid boundaries sufficed. System scaling allowed exploration of very small separations. Parameters of simulated systems matched experiments within error (see system labels SQ1, SQ2, CY1).

• The Dual-Rise model quantitatively matches experiments and simulations across textures (square and cylindrical pillars), liquids (water, dodecane), and separations, accurately predicting capillary rise C, hemiwicking H, meniscus M, total height T, and raised volume.
• Three regimes with varying plate separation d relative to roughness δ:
  1. Large d: Rough plates are non-interacting; capillary rise negligible; meniscus and hemiwicking replicate single-plate behavior. For SQ1 (water, θ=58°), single-plate meniscus M ≈ 2.6 mm predicted (experiment 2.1 ± 0.7 mm; simulation 2.8 ± 0.7 mm), total height T ≈ 12.1 mm predicted (experiment 11.5 ± 0.7 mm; simulation 11.6 ± 0.7 mm).
  2. Intermediate d: Capillary rise increases as plates approach, hemiwicking decreases, but total height T remains constant with separation; capillary rise can be treated independently using Cassie-Baxter effective angle for the pre-hemiwicked surfaces (model reduces to C_CB when meniscus is small).
  3. Small d: Capillary rise exceeds hemiwicking height; hemiwicking is suppressed (H=0); total height increases with decreasing separation; largest rise heights occur here. This regime was previously unreported.
• Hemiwicking criticality depends on separation: The classical criterion cos θc=(1−φ)/(fr) (gravity-free, zero hydrostatic head) is insufficient between plates. Including hydrostatic pressure yields a separation-dependent condition (derived by setting H=0), which shows hemiwicking can always be suppressed by decreasing d. In the small-meniscus limit, cos θ/(1−...) relation simplifies (Eq. 10), indicating confinement-controlled loss of hemiwicking even for highly wetting liquids.
• Perfectly wetting liquids (θ=0°): Rough and smooth systems can exhibit different rise heights at the same nominal contact angle; for CY1 (dodecane, θ=0°), total height T remains constant while H is finite and can be overtaken by capillary rise at small d.
• Model comparisons: • Jurin’s law (smooth) generally fails except when C=0 at θ=90°, or when hemiwicking fully shields roughness at θ=0° with sufficiently large d. • Cassie-Baxter insertion into Jurin’s law is accurate only when hemiwicking is present and meniscus is small (regime 2); it overestimates C when H=0 (regime 3) and cannot predict whether hemiwicking occurs. • Wenzel insertion requires an effective separation choice; using the average total separation D improves agreement in regime 3. However, when r cos θ ≥ 1 (θ_W→0°), Wenzel severely underestimates C; in contrast, Dual-Rise predicts continuing increase of capillary force with r cos θ, revealing widespread misuse of Wenzel in this common regime.
• Raised liquid volume: The total volume V_T is independent of separation and substantially larger on rough surfaces than smooth predictions. The model gives V_T = 2 λ r cos θ (per unit width), showing enhancement by roughness r; smooth approximations underestimate by factor r, and Wenzel underestimates when r cos θ > 1; Cassie-Baxter volume is unreliable since it neglects hemiwicked volume.
• Pinning yields many metastable states, but the equilibrium (global minimum) state occurs at intermediate heights with minimal pinning forces; experiments approximate this via tapping; simulations identify it by energy minimization.
• Transition separations observed: For SQ1 (water, θ=58°), hemiwicking lost near log10(d/δ)=0.5 (d≈0.75 mm). For CY1 (dodecane, θ=0°), transition near log10(d/δ)=−0.046 (d≈0.30 mm).

The Dual-Rise framework resolves how capillary rise and hemiwicking couple through two key length scales (roughness depth and plate separation), overturning the common assumption of independence at small separations. It explains separation-dependent loss of hemiwicking and the abrupt change in capillary rise slope across the transition, clarifying why the largest rise heights occur without hemiwicking at tight gaps. The model reconciles discrepancies among effective-angle approaches, defines when Cassie-Baxter-based estimates are valid (pre-hemiwicked regime with small meniscus), and shows Wenzel can severely underestimate rise when r cos θ ≥ 1. The separation-independence of total raised volume, and its scaling as V_T ∝ r cos θ, is crucial for applications that rely on fluid uptake/storage or interfacial area generation, such as solar desalination, water purification, and liquid-infused carbon capture. The findings underscore the need to design rough capillaries by balancing plate packing density with sustained hemiwicking where desired, and by correctly accounting for roughness-enhanced capillary forces and volumes.

The study introduces and validates the Dual-Rise model, a general, analytic energy-minimization framework that accurately predicts equilibrium rough capillary rise heights (C, M, H, T) and raised volumes across textures, liquids, and wettabilities. Key contributions include: (i) identification of three regimes governed by plate separation and roughness, with confinement-driven suppression of hemiwicking; (ii) demonstration that perfectly wetting liquids can exhibit different rise behaviors on rough versus smooth surfaces; (iii) proof that total raised volume is independent of separation and amplified by roughness (V_T = 2 λ r cos θ); and (iv) delineation of the limits of Cassie-Baxter and Wenzel adaptations to Jurin’s law, highlighting frequent misuse of Wenzel when r cos θ ≥ 1. Future research directions include quantifying how roughness geometry controls the curvature and maximum height of the hemiwicking interface, and predicting pinning forces and metastable configurations in both hemiwicking and capillary regions. The Dual-Rise model is poised to inform the design and control of rough capillary systems in diverse technologies, including sustainable platforms such as liquid-infused catalytic and carbon capture systems and the durability of cementitious materials.

• Meniscus modeled via circular arc approximation, accurate for separations up to the order of the capillary length; deviations at larger separations are not covered.
• Long-range surface interactions are neglected relative to interfacial and gravitational energy scales.
• Hemiwicking height H is represented via twice the center of mass height to regularize divergences; detailed tail morphology depends on specific geometries and is not explicitly resolved.
• Average roughness depth δ is treated as constant; spatial variations and overhanging or re-entrant structures are not included in the main analysis (f=1/r for the studied non-overhanging textures).
• Meniscus contact angle θ_M selection uses a simple piecewise choice (Cassie-Baxter when hemiwicking, Wenzel otherwise); more precise treatments are possible but reduce model closure.
• Experiments face pinning-induced metastability; equilibrium is approximated via tapping with an uncertainty of about ±1 pitch length in height and ±50 μm in separation at moderate gaps (larger relative errors at very small gaps).
• Simulations rely on diffuse-interface approximations and discretization; exact replication of experimental geometries is approximate though within reported errors.