Chemical potentials offer a new way to explain capillary pressure in tubes

Capillary pressure, the force that drives water into soil, oil through rock and sap up a tree, has been studied since the seventeenth century, yet researchers have long lacked a single consistent framework to describe it. A new theoretical study published in Discover Fluid Mechanics proposes one, built on a familiar concept from thermodynamics: the chemical potential. Dag Chun Standnes and Anders Kristoffersen argue that the capillary pressure governing how two immiscible fluids share space in a thin horizontal tube can be understood as the difference between the chemical potentials of the non-wetting and wetting phases, both inside and outside the system.

The framework rests on a simple principle. When a system is in what the authors call diffusive equilibrium, the change in total chemical potential for each fluid phase is zero, meaning the internal chemical potential within the tube must exactly balance the external chemical potential supplied by the surroundings, for example by a pump. The capillary pressure is then defined as the difference between the non-wetting and wetting phase internal chemical potentials, which equals the negative of the same difference externally. This definition generalizes the conventional expression, in which capillary pressure is simply the pressure difference between the two phases, because chemical potentials also capture the contribution of phase entropies, which the classical formulation ignores.

Because the tube geometry is simple, the authors could derive analytical expressions for every internal contribution to the capillary drainage and imbibition pressure curves. Two interfacial area terms appear: one associated with creating or removing the interface between the non-wetting and wetting fluids, and one associated with the interaction between the non-wetting phase and the solid tube wall, which depends on the wettability of the material. In their numerical example, the first term produces a threshold pressure of 1265 pascals at the highest wetting phase saturation, while the second requires 1138 pascals, so the total pressure needed for the non-wetting phase to enter and completely fill the tube exceeds 2403 pascals.

The most novel element is the inclusion of phase entropy terms. Under the assumption of non-interacting fluid phases, the entropy contribution takes a logarithmic form that depends on the fluid saturations and on dimensionless fit parameters. These fit parameters are necessary because the reference state cannot be calculated from first principles at the macroscopic level, where the quantum concentration used in microscopic statistical physics must be discarded. The authors stress that the expressions must therefore be regarded as phenomenological until quantitative predictions can be derived, and that the fit parameters must be found by comparison with experimental data, since bulk phase properties may change when fluids reside in narrow pores.

The entropy terms turn out to explain a striking feature of capillary pressure curves: the characteristic gradual increase in steepness at both high and low wetting phase saturations. The authors show that this shape appears even for a single tube, meaning it does not require a distribution of pore throat radii as often assumed for porous media. The entropy contribution also introduces a second source of hysteresis, which the authors call saturation hysteresis, because drainage is represented by one entropy term while imbibition, which always follows a drainage process, is represented by two. This adds to the better-known hysteresis caused by the difference between receding and advancing contact angles, which in their example produces a pressure jump of 1012 pascals when the process switches from drainage to imbibition.

The framework also explains why spontaneous imbibition occurs. When the external pressure is relaxed, a sharp gradient in the capillary imbibition pressure arises at the lowest wetting phase saturation, driven mainly by the disjoining pressure term associated with the thin wetting film trapped along the tube wall. This gradient creates a strong force acting along the tube that sucks water back in until the internal chemical potential again matches the reduced external condition. The authors note that the theory of disjoining pressure is incomplete for making reliable quantitative predictions, especially for strongly hydrogen-bonded fluids like water, so the remaining saturation values are used only qualitatively.

To test the model against real data, the authors compared their drainage expression with measurements on simple, homogeneous porous media, since capillary pressure curves are not commonly measured for single tubes. For a packing of unconsolidated dolomite particles treated with octanoic acid, reported by Morrow and colleagues, the calculated curve followed the measured data closely using an effective tube radius of 0.80 millimetres and two entropy fit parameters. A second comparison, to drainage tests on water-wet Ottawa sand reported by O’Carroll and colleagues, also produced a close visual match with an effective radius of 0.75 millimetres. The authors emphasize that the model matched the curves by adjusting three parameters, that no error analysis was possible because experimental uncertainties were not available, and that only drainage data were compared, leaving imbibition for future work.

Several additional observations support the framework. The authors show that Jurin’s law, the classical expression for capillary rise, can be derived from it, and that the law is strictly valid only when the wetting phase has a zero contact angle. They also demonstrate that the framework correctly accounts for the viscous pressure loss during stationary single-phase flow, recovering Hagen-Poiseuille’s equation for tubes and Darcy’s law for porous media. Furthermore, the inclusion of the phase entropy term explains the known temperature dependence of capillary pressures in air-groundwater systems: previous calculations found fractional temperature variations of -0.0082 per kelvin for drainage and +0.004 per kelvin for imbibition, the latter explaining why water tables rise when soil is heated.

The authors are candid about the limitations. The analytical expressions assume homogeneous wettability, so media with spatially distributed wettabilities cannot be described, and the framework is constrained to diffusive or stationary equilibrium processes, excluding irreversible, time-dependent thermodynamics. The assumption of complete reversibility in the interfacial area terms holds for the tube geometry but will generally fail in porous media, where fluid distribution is sensitive to how and how quickly fluids are supplied, potentially adding a third, kinetic-induced hysteresis mechanism. The main challenge, shared with much of interfacial science, is that surface energies and interfacial areas are difficult to measure, so the internal chemical potential cannot yet be calculated apriori and compared with imposed external values.

Even so, the authors argue that the framework could have important consequences for engineering and science disciplines where tube-like geometry serves as an idealized model of more complex systems, including the distribution of air and water in soils, storage of carbon dioxide in subsurface reservoirs, oil recovery, water rise in trees and water influx into narrow rock fractures. Because capillary pressure in porous media is frequently modelled as a bundle of tubes with different radii, fundamental understanding of the single tube case remains crucial. The work demonstrates that a thermodynamic treatment based on total chemical potential, capturing both interfacial energy changes and phase entropies, can unify a wide range of capillary phenomena under one consistent description.

Subject of Research: A thermodynamic framework describing capillary drainage and imbibition pressures in thin tubes using chemical potentials

Article Title: The chemical potentials governing fluid saturations in open constant volume systems

Article References: Standnes, D. C., & Kristoffersen, A. (2026). The chemical potentials governing fluid saturations in open constant volume systems. Discover Fluid Mechanics, 2(1), Article 8. https://doi.org/10.1007/s44369-026-00014-6

Image Credits: AI Generated

DOI: 10.1007/s44369-026-00014-6

Keywords: capillary pressure, chemical potential, thermodynamics, capillary hysteresis, spontaneous imbibition, phase entropy, interfacial tension, wettability, porous media, Jurin's law, disjoining pressure, drainage