Published: January 26, 2011 r2011 American Chemical Society 1422 dx.doi.org/10.1021/jp106224j | J. Phys. Chem. B 2011, 115, 1422–1428 ARTICLE pubs.acs.org/JPCB Temperature at Small Scales: A Lower Limit for a Thermodynamic Description J.-M. Simon* ,† and J. M. Rubi ‡ † Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR-5209 CNRS-Universit  e de Bourgogne, 9 av. A. Savary, 21000 Dijon, France ‡ Department de Física Fonamental, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028, Barcelona, Spain ABSTRACT: We analyze the concept of equilibrium tempe- rature in a set of interacting argon atoms, confined in a nanostructure, a zeolite with an intricate distribution of channels through which the atoms may move. The tempera- ture is computed following two procedures: by averaging over the kinetic energy of the particles and over the forces acting on them. It is shown that for external surfaces and for regions which do not fall under the whole pattern of potential energy distribution, smaller than a quarter of a crystal unit cell, both temperatures, kinetic and configurational, show significant differences. The configurational temperature accounts for the different interactions on the particles in the different parts of the channels which makes them move in an energetically heterogeneous environment. The kinetic temperature is practically not affected by these inhomogeneities. The observed disparity between both temperatures disappears when averages are taken over larger regions of the zeolite. The size of these regions imposes a lower limit for a consistent thermodynamic description of a small-scale systems such as nanostructured materials, catalytic cells, and nano heat-exchangers. ’ INTRODUCTION The current interest in the study of small-scale systems has raised questions as to the validity of thermodynamic concepts and functionalities, originally set forth in the study of large-scale systems, 1,2 to treat processes taking place at small scales. 2,3 In some cases, despite its reduced dimension, the system can be treated thermodynamically. 4 This is the case of some proteins of nanometric size but still containing enough particles to emulate thermal processes occurring at large scales. 5 However, in the race toward the progressive miniaturization of the systems, a funda- mental question arises. Up to what sizes is a thermodynamic description valid? The case of small-scale systems removed from equilibrium due to the intervention of external forces 6 shares a similar problem- atic heightened by the fact that these external factors may increase the heterogeneity of the interactions even more. This article aims to answer this fundamental question. We analyze the concept of temperature, whose consistency is a key ingredient to validate a thermodynamic description, at the molecular scales and elucidate under what conditions this quantity can univocally be defined. The example treated is an adsorbed phase, an ensemble of argon atoms confined in a zeolite. 7 The peculiar form of the zeolite composed of a given distribution of channels oriented in the way indicated in Figure 1 introduces fluctuations of the interaction forces acting on each atom giving rise to an energetically heterogeneous environment in which the atoms move. We will show that under these conditions the kinetic temperature obtained from averages over the kinetic energy of the particles through the equipartition law and the configurational temperature, 8,9 inferred from averages of the forces, differ when averages are taken over very small regions, smaller than the crystal unit cell. The coincidence of both temperatures is regained at large sizes. This result thus establishes a lower limit for the consistency of the temperature concept and therefore for the validity of a thermodynamic description. The paper is structured as follows. In the next section we describe the system and the simulation method. In the following sections, we introduce the concept of kinetic and configurational temperatures and we present results on the distributions of the interaction potential inside the pores and of both temperatures. Finally, in the Conclusion we summar- ize our main results. ’ SYSTEM AND SIMULATION METHODS We consider a set of argon atoms confined in a siliceous (SiO 2 ) zeolite, silicalite, composed of pores with size of about 6 Å. The pores are built so as to shape two types of interconnected channels: straight channels and zigzag channels. Figure 1 shows the porous structure of a unit cell. The peculiar structure of the channel makes the strength of the interactions between the silicalite and the adsorbed molecules change dramatically over Received: July 6, 2010 Revised: December 9, 2010