Clusters equilibrium fraction

KAPRAL - The results for the equilibrium fractions of clusters in the electrolyte were obtained from long-time averages over the Brownian dynamics simulation. These results could be checked by direct Monte-Carlo simulations on the primitive model electrolyte. Has this been done ... 

A model has been developed to calculate the size distributions of the short lived decay products of radon in the indoor environment. In addition to the classical processes like attachment, plate out and ventilation, clustering of condensable species around the radioactive ions, and the neutralization of these ions by recombination and charge transfer are also taken into account. Some examples are presented showing that the latter processes may affect considerably the appearance and amount of the so called unattached fraction, as well as the equilibrium factor. 

Abstract The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range T < 30 MeV and the baryon density range ps < 1014 2 g/cm3. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle (1 < A < 4) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and (3 equilibrium is considered as a special case. 

Within the approach, given here, also the /5-equilibrium can be calculated and the influence of the cluster formation on the proton fraction can be considered. The formation of clusters will increase the proton fraction. 

In certain regions of the density-temperature plane, a significant fraction of nuclear matter is bound into clusters. The EOS and the region of phase instability are modified. In the case of /3 equilibrium, the proton fraction and the occurrence of inhomogeneous density distribution are influenced in an essential way. Important consequences are also expected for nonequilibrium processes. 

A puzzling problem was posed by Levinthal many years ago.329 We usually assume that the peptide chain folds into one of the most stable conformations possible. However, proteins fold very rapidly. Even today, no computer would be able, in our lifetime, to find by systematic examination the thermodynamically most stable conformation.328 It would likewise be impossible for a folding protein to "try out" more than a tiny fraction of all possible conformations. Yet folded and unfolded proteins often appear to be in a thermodynamic equilibrium Experimental results indicate that denatured proteins are frequently in equilibrium with a compact denatured state or "molten globule" in which hydrophobic groups have become clustered and some secondary structures exists.330-336 From this state the polypeptide may rearrange more slowly through other folding intermediates to the final "native conformation."3363 3361 ... 

When an emulsifier or soap is dissolved in water, the solute molecules associate to form small clusters called micelles. The hydrocarbon parts of the emulsifier molecules constitute the interior of the micelles, the surface of which is formed by the ionic groups of the emulsifier. A small fraction of the soap is molecularly dissolved in the water and there is a dynamic equilibrium between the micelles and these single molecules in the aqueous phase. Micelles are of colloidal size, consisting of a relatively small number of soap molecules of the order of 100 molecules. This corresponds to a diameter of about 50 A., if one assumes the cluster to be spherical. At the concentrations usually employed in emulsion polymerization, there are some 10 micelles per milliliter of water. 

The data in Table 1 show that clustering occurs in the water-rich region of solutions of propanols and tert-butyl alcohol, for alcohol molar fractions <0.3—0.4. Numerous models have been suggested to explain the properties of water-alcohol mixtures. They can be roughly subdivided in the following groups (a) Chemical models 29-33 qj) chemical equilibrium between clusters and the constituent components, which can explain some thermodynamic properties of these solutions, but involve... 

Calculate the mass concentration of water in the vaporphase present as equilibrium clusters (g > 2) at a relative humidity of 50% and a temperature of 20 C. Expre.ss your answer as nanograms/m-. Calculate the fraction of the total ma.ss of water vapor present in the form of equilibrium clusters under (he.se conditions. 

The shrinkage of the coexistence temperature interval is one factor that makes it difficult to detect dynamical coexistence in large clusters. The other is that the fraction of the unfavored phase decreases with the number of molecules in the cluster. This dependence appears in the exponent of the ratio of the amounts of the two phases—that is, in the exponent of the equilibrium ratio of concentrations [1]. One necessarily must compute an extremely long phase trajectory in order to distinguish between two different phases and especially to establish the equilibrium ratio of the amounts of the two phases. This explains why absence of coexistence was reported [42] in simulations of clusters containing more than 500 molecules. 

The liquid-cluster system appears to have equilibrium metastable or stable heterophase states, in which the volume fraction of solid clusters may vary from negligibly small values up to unity. To find these states, the thermodynamics of the liquid-cluster systems will be considered. 

It follows from (6.57, 58) that pmix — /iL can have one or two minima at 0 < x < 1, thus one or two (stable or stable + metastable) states of supercooled liquids can exist. At high temperatures, T > T, and at low temperatures, T < fm, only one equilibrium state exists. If two equilibrium states coexist, they differ by the degree of clusterization. If a clusterized fraction is large enough, the state must be treated as solid one. Indeed, in the system at x > xc 0.16 an infinite (percolated) solid cluster is formed and at (1 — x) > (1 — x)c 0.16 a percolated liquid cluster appears. So, at x > xg 0.84 the mixed state is really a solid with heterophase liquid fluctuations. The temperature at which the stable state with x > xg exists, is the thermodynamic glassing temperature, Tgh. 

---