A closed-form rescaling based on confinement geometry brought predictions from the Carnahan–Starling equation of state, a formula for hard-sphere thermodynamics, into close agreement with simulations of hard-sphere systems. The approach tracked the effective packing fraction, which accounts for the interaction space, across the full tested range, while the usual apparent packing fraction diverged substantially from it. The result offers a parameter-free route to free energies, pressures, chemical potentials and unmixing forces in the geometries examined.
The paper asks whether errors in bulk hard-sphere equations of state under nanoscale confinement are geometric, and whether a closed-form mapping can restore Carnahan–Starling thermodynamics. Its proposed closure, written as sλ(y), maps an apparent packing fraction y to an effective fraction η. It matches the exact dilute-limit relation and the dense boundary-layer estimate without adjustable parameters. In tests that varied particle number or confinement radius, sλ(y) closely followed simulated η, whereas y departed substantially from it.
The geometry behind the calculation
The main validation used two anchored hard-sphere subsystems whose centers were confined around their anchors. Representative simulations used particles with a radius of 2.5 nm and a confinement radius of 30 nm. As the anchors changed separation, the subsystems shared part of their accessible region while retaining exclusive regions. At each separation, the calculation determined local packing fractions by requiring equal chemical potentials in shared and exclusive regions and conserving the total particle number. It then obtained the force per particle by differentiating the equilibrium free energy with respect to separation.
In this setup, unmixing describes how particles redistribute between the shared and exclusive regions. The unmixing free-energy change was negative at every tested density, and its magnitude increased sharply as density rose, a pattern consistent with spontaneous unmixing. The rescaled free-energy expression closely matched Brownian-dynamics results across the density range. Naive bulk Carnahan–Starling theory, evaluated at the apparent fraction, increasingly overstated the magnitude of unmixing at higher packing fractions.
The same accounting held inside cavities
A separate test examined particles inside spherical cavities. The finite-N free energy included an N−1 excess factor that recovered the ideal N=1 limit. Predicted wall pressures agreed closely with the exact N=2 result and with simulations at larger particle numbers. The cavity tests covered confinement ratios from 1/12 to 1/3.
The finite-cavity free energy also generated the leading surface response without an independent surface coefficient. At a center-accessible cavity radius five times the particle radius, comparison with density-functional theory (DFT) gave normalized differences of 2.22% for the PY treatment and 4.80% for the CS treatment.
A force profile that follows the simulations
Force-profile tests used 50 to 600 particles per subsystem. Across those sizes, the theory captured both force magnitude and distance dependence. Its extended-boundary version closely matched simulations over the full range, including the short-distance onset. The correction used a density-dependent effective protrusion length, stretched the physical interaction range and used an odd cubic near zero to enforce symmetry while preserving the total free-energy integral; no force parameter was fitted.
The force-profile simulations increased separation in 0.1 nm steps, re-equilibrated for approximately 2 µs and averaged anchor forces for 100 ns during production.
The boundary of the result
The reported validation remained within a defined range. It included the exact N=2 result, spherical cavities with confinement ratios from 1/12 to 1/3, a DFT benchmark at a ratio of 1/5 and overlapping populations reaching an effective packing fraction of about 0.575, above bulk freezing. All theoretical curves were presented as parameter-free predictions.
The main caveat is the magic-number regime, from 1/3 to 1, which the spherical-cavity validation left out because discrete cluster packing governs the physics there and numerical treatment becomes unavoidable. The framework was developed for hard-sphere interactions and the tested geometries, so its accuracy for attractive systems, other equations of state or other geometries remains an open question. The evidence is computational and analytical, not direct experimental or biological force measurement. Formal uncertainty intervals were not reported for the main comparisons.
Paper data and sources
Original title: Closed-form hard-sphere thermodynamics under nanoscale confinement: from equations of state to unmixing forces
Authors: Jose M. G. Vilar, Leonor Saiz
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text