Experimental validation of slope- and curvature-dependent disjoining pressure in microchannel thin-film evaporation

The steady thin-film evaporation model of Biswal, Som and Chakraborty (2013) extends the classical augmented Young–Laplace description of evaporating menisci in microchannels with a disjoining pressure that depends on the local film thickness, interfacial slope, and curvature, using formulations derived by Wu and Wong (2004) and Dai, Leal and Redondo (2008). The model predicts film profiles and Nusselt numbers that differ from traditional thickness-only ($\delta^{-3}$) disjoining-pressure models by about 20% in the average Nusselt number, but its only quantitative support is molecular-dynamics simulation of non-evaporating Lennard-Jones films, and it has never been tested against experiments or against molecular dynamics of evaporating films. The open problem is to determine, by well-controlled microchannel evaporation experiments and/or full molecular-dynamics simulation of evaporating films, whether the disjoining pressure of an evaporating thin film genuinely depends on interfacial slope and curvature, and whether the predicted profiles and the ~20% Nusselt-number difference are quantitatively correct.

Background

In microscale cooling devices (microchannel heat sinks, heat pipes, vapor chambers), a large share of phase-change heat transfer is carried by evaporation from the extended meniscus: the region where a wetting liquid meets a heated wall, comprising an ultrathin adsorbed film (a few nanometers, held by intermolecular forces and essentially non-evaporating), a thin-film transition region where the evaporative heat flux peaks, and the bulk intrinsic meniscus. In the transition region the film is thin enough that van der Waals attraction between liquid and solid produces a disjoining pressure — an additional pressure inside the film that suppresses evaporation — and this pressure, together with surface-tension-driven capillarity, sets the film shape and hence the local heat transfer. Classical models combine these ingredients in the augmented Young–Laplace formulation: the pressure jump across the liquid–vapor interface equals the capillary term plus the disjoining pressure, the flow is described by lubrication theory (the approximation that the film is thin and slowly varying so that the governing equations reduce to a one-dimensional balance along the wall), the evaporation flux at the interface follows the kinetic-theory Hertz–Knudsen–Schrage relation (a formula giving evaporative mass flux in terms of the interfacial temperature and the vapor-pressure driving force), and the local Nusselt number — the standard dimensionless heat-transfer coefficient — is taken as $Nu_x = x/\delta$ with $x$ the distance from the start of the film and $\delta$ the local film thickness. In these models the disjoining pressure is a function of local film thickness $\delta$ alone, typically $\Pi \propto \delta^{-3}$ (the Lifshitz/Hamaker form, where the Hamaker constant quantifies the strength of the van der Waals interaction between the phases). This thickness-only form has known inconsistencies: it diverges as $\delta \to 0$, so an artificial finite adsorbed-film thickness must be imposed as a boundary condition, and contact-line motion cannot be described without invoking unphysical interfacial slip. To remove these inconsistencies, Wu and Wong (2004) derived, by minimizing the intermolecular free energy of a non-uniform film, a disjoining pressure that also depends on the local interfacial slope $\delta'$ and curvature $\delta''$ (with the film profile $\delta(x)$), $P_d = \pm (B/\delta^3)[a^4 - \delta'^4 + 2\delta\delta'^2\delta'']$, where $B$ is a material constant and $a$ is the contact angle, both fixed in the theory by molecular interaction constants; Dai, Leal and Redondo (2008) derived the related form $\Pi = -A_{123}(4 - 3\delta'^2 + 3\delta\delta'')/(24\pi\delta^3)$, which reduces to the classical Lifshitz expression for parallel interfaces. Molecular-dynamics simulations of non-evaporating Lennard-Jones liquid films (a standard simple-liquid interaction model) by Yi and Wong (2007) support the slope-dependent theory. Biswal, Som and Chakraborty (2013) incorporated the Wu–Wong disjoining pressure into a steady, two-dimensional lubrication model of thin-film evaporation in a microchannel with kinetic-theory evaporation and conduction across the film, solving the resulting fourth-order nonlinear ordinary differential equation for the film profile, and reported that neglecting the slope and curvature dependence — i.e. traditional thickness-only models — overpredicts the average Nusselt number of the thin-film region by about 20%, with the largest deviations at low superheat (the wall-to-vapor temperature difference driving evaporation) and near the location of peak heat flux. However, the disjoining-pressure construct they used is an equilibrium, non-evaporating-film result, the only quantitative check was against molecular dynamics of non-evaporating films, and no experimental validation was performed. What remains unresolved is whether this higher-order disjoining-pressure physics is real and quantitatively important in actually evaporating films.

Problem Statement

Determine whether the disjoining pressure of an evaporating thin liquid film depends on the local interfacial slope and curvature, as the formulations of Wu and Wong (2004) and Dai, Leal and Redondo (2008) predict and as incorporated into the microchannel thin-film evaporation model of Biswal, Som and Chakraborty (2013) — and, in consequence, whether that model's quantitative predictions are correct: the thin-film profiles and the average Nusselt number of the thin-film region relative to traditional thickness-only ($\delta^{-3}$) disjoining-pressure models, for which Biswal et al. (2013) report a difference of about 20%. A resolution must settle both parts: whether the slope/curvature dependence is physically real, and whether the model's thin-film profiles and its ~20% average-Nusselt-number difference from thickness-only models are quantitatively correct; establishing one part alone does not resolve the other. Because the slope-/curvature-dependent disjoining-pressure closure is an equilibrium (non-evaporating-film) construct whose existing quantitative support comes from molecular dynamics of non-evaporating films, a resolution is sought through well-controlled microchannel thin-film evaporation experiments and/or full molecular-dynamics simulation of genuinely evaporating (non-equilibrium) films that tests the closure. The overarching objective is to establish whether the higher-order slope/curvature dependence of the disjoining pressure is physically real and quantitatively important for evaporating thin-film heat transfer in microchannels.

References

1. Laxmidhar Biswal, S. K. Som, Suman Chakraborty, 'Thin film evaporation in microchannels with slope- and curvature-dependent disjoining pressure', International Journal of Heat and Mass Transfer 57 (2013) 402-410. DOI: 10.1016/j.ijheatmasstransfer.2012.10.022, https://doi.org/10.1016/j.ijheatmasstransfer.2012.10.022.
2. Qingfang Wu, Harris Wong, 'A slope-dependent disjoining pressure for non-zero contact angles', Journal of Fluid Mechanics 506 (2004) 157-185. DOI: 10.1017/S0022112004008420, https://doi.org/10.1017/S0022112004008420.
3. Bing Dai, L. Gary Leal, Antonio Redondo, 'Disjoining pressure for nonuniform thin films', Physical Review E 78 (2008) 061602. DOI: 10.1103/PhysRevE.78.061602, https://doi.org/10.1103/PhysRevE.78.061602.
4. Taeil Yi, Harris Wong, 'Theory of slope-dependent disjoining pressure with application to Lennard-Jones liquid films', Journal of Colloid and Interface Science 313 (2007) 579-591. DOI: 10.1016/j.jcis.2007.05.015, https://doi.org/10.1016/j.jcis.2007.05.015.
5. Michael S. Hanchak, Marlin D. Vangsness, Larry W. Byrd, Jamie S. Ervin, 'Thin film evaporation of n-octane on silicon: Experiments and theory', International Journal of Heat and Mass Transfer 75 (2014) 196-206. DOI: 10.1016/j.ijheatmasstransfer.2014.03.063, https://doi.org/10.1016/j.ijheatmasstransfer.2014.03.063.
6. Saumya Singh, Soubhik Kumar Bhaumik, Pawan Kumar Singh, 'Experimental investigation of thickness profile in an extended meniscus of thin film evaporation using reflectometry', Experimental Thermal and Fluid Science 151 (2024) 111069. DOI: 10.1016/j.expthermflusci.2023.111069, https://doi.org/10.1016/j.expthermflusci.2023.111069.
7. Saumya Singh, Ankit Kumar, Soubhik Kumar Bhaumik, Pawan Kumar Singh, Subramanian Narayanan, 'Thickness Profile Measurements of Acoustic Induced Maneuvering of Thin Film Evaporation', Heat Transfer Engineering 47 (2025) 382-397. DOI: 10.1080/01457632.2025.2452749, https://doi.org/10.1080/01457632.2025.2452749.
8. Ali Mostafazade Abolmaali, Mohamad Bayat, Jesper Hattel, 'A multiscale analytical-numerical method for the coupled heat and mass transfer in the extended meniscus region considering thin-film evaporation in microchannels', International Journal of Heat and Mass Transfer 222 (2024) 125145. DOI: 10.1016/j.ijheatmasstransfer.2023.125145, https://doi.org/10.1016/j.ijheatmasstransfer.2023.125145.
9. Ali Mostafazade Abolmaali, Mohamad Bayat, Jesper Hattel, 'Thin-film evaporation in microchannels: A combined analytical and computational fluid dynamics approach to assessing meniscus curvature impact', International Journal of Thermal Sciences 208 (2025) 109455. DOI: 10.1016/j.ijthermalsci.2024.109455, https://doi.org/10.1016/j.ijthermalsci.2024.109455.
10. Xiaoman Wang, Yang Li, Jonathan A. Malen, Alan J. H. McGaughey, 'Assessing the impact of disjoining pressure on thin-film evaporation with atomistic simulation and kinetic theory', Applied Physics Letters 116 (2020) 213701. DOI: 10.1063/5.0010467, https://doi.org/10.1063/5.0010467.
11. Kishan Bellur, Ezequiel F. Médici, Chang Kyoung Choi, James C. Hermanson, Jeffrey S. Allen, 'Multiscale approach to model steady meniscus evaporation in a wetting fluid', Physical Review Fluids 5 (2020) 024001. DOI: 10.1103/PhysRevFluids.5.024001, https://doi.org/10.1103/PhysRevFluids.5.024001.
12. Mojtaba Norouzisadeh, Philippe Leroy, Cyprien Soulaine, 'A lubrication model with slope-dependent disjoining pressure for modeling wettability alteration', Computer Physics Communications 298 (2024) 109114. DOI: 10.1016/j.cpc.2024.109114, https://doi.org/10.1016/j.cpc.2024.109114.
13. Florentina Samoila, Lucel Sirghi, 'Disjoining Pressure in Partial Wetting on the Nanoscale', Langmuir 33 (2017) 5188-5196. DOI: 10.1021/acs.langmuir.7b01156, https://doi.org/10.1021/acs.langmuir.7b01156.
14. Haiyu Fu, Leping Zhou, Xiaoze Du, 'Multiscale interfacial dynamics in evaporation and boiling: A critical review of governing factors and heat transfer mechanisms', Advances in Colloid and Interface Science 352 (2026) 103828. DOI: 10.1016/j.cis.2026.103828, https://doi.org/10.1016/j.cis.2026.103828.
15. Hyung Ju Lee, Jinghao Jin, Won Yeong Hwang, Chang Kyoung Choi, Seong Hyuk Lee, 'Time-invariant characteristics of evaporating thin film during droplet evaporation', Physics of Fluids 37 (2025) 062114. DOI: 10.1063/5.0271214, https://doi.org/10.1063/5.0271214.
16. Shahnawaz Ahmed, Manmohan Pandey, 'New insights on modeling of evaporation phenomena in thin films', Physics of Fluids 31 (2019) 092001. DOI: 10.1063/1.5112139, https://doi.org/10.1063/1.5112139.
17. Sukumar Pati, S. K. Som, Suman Chakraborty, 'Combined influences of electrostatic component of disjoining pressure and interfacial slip on thin film evaporation in nanopores', International Journal of Heat and Mass Transfer 64 (2013) 304-312. DOI: 10.1016/j.ijheatmasstransfer.2013.04.049, https://doi.org/10.1016/j.ijheatmasstransfer.2013.04.049.