THERMODYNAMIC SMEARING

It has been implied that the surface of the nanocrystal acts as a perturbation of the total nanocrystal free energy. The assumption has been that, from a thermodynamics viewpoint, the nanocrystal behaves merely as a fragment of the extended solid. However, evidence suggests that structure throughout a nanoparticle in very small particle size may be perturbed relative to the bulk (see chapters by Banfield and Zhang and Waychunas, this volume). Although the thermodynamic properties of the nanocrystal ensemble are well defined, structural phase transitions are rigorously defined only in an infinite medium. A lower limit must exist for the number of atoms required for the crystal in 

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Figure 4. Schematic showing a hysteresis loop for the CdSe nanociystals with the smearing of the thermodynamic transition pressure caused by the finite nature of the nanocrystal particle. The thermodynamic transition pressure is offset from the hysteresis center to emphasize that in first-order solid-solid transformations, this pressure is unlikely to be precisely centered. The lower plot shows the estimated smearing for CdSe nanocrystals as inversely proportional to the number of atoms in the crystal, at two temperatures, as discussed in the text. Note that nanocrystals are not ordinarily synAesized or studied in sizes smaller than 20 A in diameter. This figure shows that this thermal smearing is insignificant compared to the large hysteresis width in the CdSe nanociystals studied (25-130 A in diameter), such that the transition is bulk-like from this perspective. This means that observed transformations occur at pressures far from equilibrium, where there is little probability of back reaction to the metastable state once a nanociystal has transformed. In much smaller crystals or with larger temperatures, the smearing could become on the order of the hysteresis width, and the crystals would transform from one stmcture to the other at thermal equilibrium.

Figure 12. Proposed schematic low-coverage melting phase diagram of N2 on graphite according to the tricritical point model. Full lines correspond to the ideal thermodynamic system that is, AT = 0. Dashed curves were obtained with two different temperature smear-ings AT > 0. (Adapted from Fig. 3 of Ref. 276.)...

One should thus not regard LCEs as entirely below-critical or above-critical systems, but rather as systems in which both types of phase-transition behaviours may be found, the extent of which is determined by the values of (G) and Gg- We shall introduce the term smeared criticality for such behaviour. In Fig. 16a, the graphical representation illustrates the extent of each type of phase transition behaviour in two LCEs, one prevalently below critical and the other prevalently above critical. A straightforward manifestation of the smeared criticality is the presence of latent heat in many LCE systems that exhibit a supercritical, effective thermodynamic response (see Sect. 5). This latent heat is released by the below-critical component of the LCE whose extent is given by the surface of the shaded area in Fig. 16. 

We shall see in Sect. 5 that LCEs can be tailored to exhibit a thermodynamic response spanning from subcritical to supercritical. However, the classification in terms of subcritical, critical or supercritical is, in the presented picture of smeared criticality, applicable only for describing the average (effective) response of LCEs. 

Especially important is this question with respect to phase transformations. In a rigorous theoretical treatment, the phase transition (as a stepHke change of the first- or second-order derivatives of Gibbs potential) is defined only in the thermodynamic Emit No —> oo, V oo. No here is the number of atoms and V is the volume of the system. Nevertheless, everybody is applying the notion of phase transformations to rather small (even nano) systems. One should expect that the behavior of various parameters during such gradual transformations is smeared by fluctuations so that the transition point should become a transition interval, the solubility line should become a solubility band, and so on. 

We finally note that similar conclusions about the interplay of internal and external fluctuations are obtained when considering fluctuations in the source (o() or creation (<) parameters. Also in these cases a transition is smeared out by internal fluctuations. In these two cases the probability distribution in the thermodynamic limit is defined for values of x larger than a boundary value. A change of behavior at this boundary is found for a critical value of X. Fluctuations of o and X turn out to have qualitatively the same effects. They are qualitatively different from the... 

Thermodynamic properties of polyelectrolyte solutions are mostly determined by ionic distribution around the polyion skeleton. To explain thermodynamic properties, therefore, the rod-like model may be effective. It is assumed that the polyion is a rod of infinite length and has smeared charges distributed uniformly over the surfaces of the rod. That is, a real polyion chain consists of a series of discrete charges and each charge is surrounded by its own ionic atmosphere. If the radius of ionic atmosphere is... 

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