**Size and shape effects on the thermodynamic properties of nanoscale
volumes of water**

BA Strom and JM Simon and SK Schnell and S Kjelstrup and JY He and D Bedeaux, PHYSICAL CHEMISTRY CHEMICAL PHYSICS, 19, 9016-9027 (2017).

DOI: 10.1039/c7cp00874k

Small systems are known to deviate from the classical thermodynamic
description, among other things due to their large surface area to
volume ratio compared to corresponding big systems. As a consequence,
extensive thermodynamic properties are no longer proportional to the
volume, but are instead higher order functions of size and shape. We
investigate such functions for second moments of probability
distributions of fluctuating properties in the grand-canonical ensemble,
focusing specifically on the volume and surface terms of Hadwiger's
theorem, explained in Klain, Mathematika, 1995, 42, 329-339. We resolve
the shape dependence of the surface term and show, using Hill's
nanothermodynamics **Hill, J. Chem. Phys., 1962, 36, 3182**, that the
surface satisfies the thermodynamics of a flat surface as described by
Gibbs **Gibbs, The Scientific Papers of J. Willard Gibbs, Volume 1,
Thermodynamics, Ox Bow Press, Woodbridge, Connecticut, 1993**. The Small
System Method (SSM), first derived by Schnell et al. **Schnell et al., J.
Phys. Chem. B, 2011, 115, 10911**, is extended and used to analyze
simulation data on small systems of water. We simulate water as an
example to illustrate the method, using TIP4P/2005 and other models, and
compute the isothermal compressibility and thermodynamic factor. We are
able to retrieve the experimental value of the bulk phase
compressibility within 2%, and show that the compressibility of
nanosized volumes increases by up to a factor of two as the number of
molecules in the volume decreases. The value for a tetrahedron, cube,
sphere, polygon, etc. can be predicted from the same scaling law, as
long as second order effects (nook and corner effects) are negligible.
Lastly, we propose a general formula for finite reservoir correction to
fluctuations in subvolumes.

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