Free energy critical size

It is important to understand that critical behavior can only exist in the thermodynamic limit that is, only in the limit as the size of the system N —> = oo. Were we to examine the analytical behavior of any observables (internal energy, specific heat, etc) for a finite system, we would generally find no evidence of any phase transitions. Since, on physical grounds, we expect the free energy to be proportional to the size of the system, we can compute the free energy per site f H, T) (compare to equation 7.3)... 

The deposition of the first stem of thickness / > lmin is the critical stage of nucleation. However, further stems need to attach next to it before the patch reaches a stable size (see Fig. 3.10). This size can be determined by setting the change in free energy A v of formation of a surface strip possessing v stems to zero ... 

The relative magnitude of these two activation free energies determines the size and shape of the critical nucleus, and hence of the resulting crystal. If sliding diffusion is easy then extended chain crystals may form if it is hard then the thickness will be determined kinetically and will be close to lmin. The work so far has concentrated on obtaining a measure for this nucleus for different input parameters and on plotting the most likely path for its formation. The SI catastrophe does not occur because there is always a barrier against the formation of thick crystals which increases with /. 

The sample size in a real simulation is always finite, and usually relatively small. Thus, understanding the error behavior in the finite-size sampling region is critical for free energy calculations based on molecular simulation. Despite the importance of finite sampling bias, it has received little attention from the community of molecular simulators. Consequently, we would like to emphasize the importance of finite sampling bias (accuracy) in this chapter. 

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

It is often convenient to think of adsorption as occurring in three stages as the adsorbate concentration increases. Firstly, a single layer of molecules builds up over the surface of the solid. This monolayer may be chemisorbed and associated with a change in free energy which is characteristic of the forces which hold it. As the fluid concentration is further increased, layers form by physical adsorption and the number of layers which form may be limited by the size of the pores. Finally, for adsorption from the gas phase, capillary condensation may occur in which capillaries become filled with condensed adsorbate, and its partial pressure reaches a critical value relative to the size of the pore. 

Figure 7.1. Free energy of formation of a cluster as a function of size N (a cluster of N atoms) the size of the critical cluster (nucleus).

When a liquid is quenched to a temperature below its melting temperature, small nuclei of different sizes and shapes are initially bom as a result of fluctuations in the density of the liquid. These nuclei are stabilized by lowering of the free energy associated with the formation of the more stable crystalline phase, but destabilized by the increase in free energy associated with the creation of interfaces. As is well known, sub-critical-size nuclei dissolve back... 

Figure 4-1 Extra Gibbs free energy of clusters as a function of cluster radius. The critical cluster size is when the extra free energy reached the maximum. A5m c = 56 J/K/mol, Vc = 46 cm /mol, a = 0.3 J/m, Te = melting temperature = 1600 K, and system temperature = 1500 K. AGc mX ASm ( (r—Tg) = —5600 J/mol. The radius of the critical cluster is r = 2aVg/(AGm c) = (2) (0.3) (46 x 10 )/5600m = 4.93 nm. The Gibbs free energy of the critical cluster relative to the melt is AG = (16/3)7tCT /(AGm c/l g)2 = 3.05 x lO i J.

And the free energy of the critical cluster is still Equation 4-4. If the cluster is not spherical (e.g., the cluster could be a cube, or some specific crystalline shape), then the specific relations between i and cluster volume and surface area are necessary to derive the critical cluster size. 

Recall that this equation could be minimized with respect to particle radius to determine the critical particle size, r, as given by Eq. (3.35). This critical radius could then be used to determine the height of the free energy activation energy barrier, AG, as given by Eq. (3.36). A similar derivation can be performed for a cubic particle with edge length, a. 

In the nucleation-and-growth transitions, nuclei of the new phase possessing a critical size have to be first formed in the parent phase. The change in free energy, AG, due to the formation of spherical nuclei is given by... 

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