(119b) Thermodynamics of a Small System In a μT Reservoir

To date, small systems have not received much attention in
thermodynamics. The reason seems obvious: small systems have few
particles, thus being far from the thermodynamic limit. However, the
thermodynamics of small systems will become increasingly important in
nanoscience. By using the formalism of Hill [1], we have derived the
finite-size dependence of the enthalpy and thermodynamic factor in a
μT reservoir [2]. We find this scaling to be dependent on 1/L,
where L is the linear size of the small system. The results are
verified using molecular simulations in 2 and 3 dimensions. It turns
out that one can compute the thermodynamic factor, adsorption enthalpy
and activity coefficients from fluctuations at a very small scale,
provided that the correct finite-size scaling is used. The method
provides an efficient way to compute activity coefficients in liquid
mixtures from MD/MC simulations in the NVE, NVT, or NPT ensemble. The
methodology also applies to non-homogeneous systems like argon
adsorbed in MFI-type zeolite. In this case the shape and size of the
small system compared to the unit cell of the zeolite framework needs
to be taken into account, it turns out that this should be in
multiples of a quarter unit cell of MFI-type zeolite.

We consider the situation of a large system of size L_t in 3
dimensions, so that the total volume equals (L_t)³. This large system can be in the
microcanonical, canonical or grand-canonical ensemble. Inside this
system, we construct an ensemble of smaller systems of size L and
volume L³. If L is much smaller than L_t, the small system can be considered to be in the grand-canonical
ensemble. By investigating density and energy fluctuations of the
small system of size L, we can obtain the molar enthalpy and
thermodynamic factor of the small system. The formalism of Hill [1]
can be used to show that the quantities obtained from the small system
show a very large finite-size effect, and that there is a scaling with
1/L in sharp contrast to the more usual finite-size scaling of
1/L³ [3].

We confirm this finite size scaling numerically for the WCA and
Lennard-Jones systems, as well as for argon adsorbed in MFI-type
zeolite. Clearly, by extrapolation of the linear regime, we obtain
the correct quantities corresponding to the macroscopic system. At
very small values of L, nook- and corner effects become dominant and
deviations from the 1/L scaling are observed. If L is close to
L_t, than the small subsystem will not be in the
grand-canonical ensemble and deviations from the 1/L scaling will
appear.

References

[1] T.L. Hill, Thermodynamics of Small Systems, Part 1, Benjamin,
    New York, 1963.

[2] S.K. Schnell, T.J.H. Vlugt, J.-M. Simon, D. Bedeaux, S. Kjelstrup,
    Chem. Phys. Lett., 504:199-201, 2011.

[3] D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics,
    Cambridge University Press, 2000.