Molecular clusters, thermodynamic

Nucleation is the growth of clusters of molecules that become a thermodynamically stable nucleus. This process is dependent on the vapor pressure of the condensable species. The molecular clusters undergo growth when the saturation ratio, S, is greater than 1, where saturation ratio is defined as the actual pressure of the gas divided by its equilibrium vapor pressure. S > 1 is referred to as a supersaturated condition (14). 

Our interest is in the connection between the intermolecular forces that cause condensation and/or gas phase molecular clustering and thermodynamics. To set the stage consider the following simple model ... 

Nevertheless, the general principles of statistical thermodynamics are expected to extend to this weaker tier of clustering interactions. Molecular clustering is widely recognized to be... 

R. Ludwig, T. C. Farrar, and F. Weinhold. Quantum cluster equilibrium theory of liquids molecular clusters and thermodynamics of liquid ammonia. Ber. Bunsenges. Phys. Chem. 102, 197-204 (1998). 

When the supersaturation ratio S becomes greater than unit, the small liquid droplets (i.e. molecular clusters) commence to appear. Almost all the droplets are immediately destroyed due to evaporation and only small fraction of the droplets (critical clusters) with radii greater than a critical radius r have a chance to survive and grow by accretion of vapor molecules (monomers) onto their surface. It is assumed that macroscopic thermodynamics is applied to the critical clusters that are considered as liquid droplets containing the large number of monomers, that is nx>>i. The number of the critical clusters formed per unit time per unit volume is the nucleation rate J so that the number density of dust grains is Nd = JJdt. Expressions for calculation of the nucleation rate and other quantities can be found in the review paper by Draine (1981). 

Constrained nonlinear programming problems abound in a very large number of science and engineering areas such as chemical process design, synthesis and control facility location network design electronic circuit design and thermodynamics of atomic/molecular clusters. 

The production and growth of particles in the presence of condensable vapors is a major dynamic process. A considerable body of literature has accumulated on the subject, beginning with the thermodynamics of phase transition and continuing with the kinetic theory of molecular cluster behavior. 

Such calculations should eventually be able to provide a better understanding of the thermodynamic fundamentals of biochemical separation methods, particularly for such processes as membrane and chromatographic techniques a clearer understanding of the causes of molecular clustering will be important in this area. In the next five to ten years, these methods should be much more sophisticated and will play a major role in the design of drugs, affinity agents, and proteins that fold into desired patterns. 

The most widely used method for calculating the formation free energy is the classical nucleation theory (CNT) [12-16] based on thermodynamics. The molecular clusters are treated as spherical droplets of bulk liquid having sharp boundaries... 

The study of small, homonuclear clusters of atoms Is Important In understanding nucleatlon because such clusters are Intermediates In the formation of bulk condensed phases. The dynamic process of condensation from a gas must Initially Involve the formation of tiny aggregates of the new phase. This can be Illustrated by the reaction sequence A(g)—A2(g)— A3(g)— . . . — A(1). One of the major weak points In the present day understanding of such nucleatlon phenomena Is the unknown thermodynamic properties of clusters. Certainly, the common practice of treating a 2-200 atom cluster as a tiny piece of the bulk with a large surface Is Inexact. There Is a need for precise thermodynamic data on atomic and molecular clusters to better define nucleatlon kinetics. 

One of the more active and growing areas of research into the study of condensed matter is the investigation of the properties of atomic and molecular clusters. A detailed understanding of clusters is vital to the study of such diverse phenomena as condensation, the dispersion of supported catalysts, cloud formation, molecular generation on interstellar grains, - and the thermodynamic properties of powders. In addition, the study of clusters is of fundamental importance to the understanding of the transition from finite to bulk behavior. 

Size-dependent structure and properties of Earth materials impact the geological processes they participate in. This topic has not been fully explored to date. Chapters in this volume contain descriptions of the inorganic and biological processes by which nanoparticles form, information about the distribution of nanoparticles in the atmosphere, aqueous environments, and soils, discussion of the impact of size on nanoparticle structure, thermodynamics, and reaction kinetics, consideration of the nature of the smallest nanoparticles and molecular clusters, pathways for crystal growth and colloid formation, analysis of the size-dependence of phase stability and magnetic properties, and descriptions of methods for the study of nanoparticles. These questions are explored through both theoretical and experimental approaches. 

The discussions in the previous sections of this chapter have focused on the thermodynamics of single particles. However, there is an important class of problems involving the. statistical properties of interacting clouds of particles in the molecular cluster size range. The size distribution of these particles can be calculated using a simple spherical particle model as... 

Molecular clusters are formed due to weakly attractive forces between molecules, the Van der Waals forces. Except under conditions of low temperature, it is difficult to observe and study in the laboratory clusters containing more than a few molecules, so that details about their properties are sparse. Our understanding of the nucleation process consequently relies mainly on theoretical concepts based largely on the principles of statistical mechanics. The theory of homogeneous nucleation developed by Volmer and Weber (1926), Flood (1934), Becker and Doering (1935), and Reiss (1950) assumes that certain thermodynamic properties, such as the molar volume or the surface tension, that can be determined for bulk material... 

Giant clusters can serve as useful models for imderstanding the structure and chemical behavior of dispersed metals. Magnetic and thermodynamic measurements in the vicinity of absolute zero showed the Pdsei species to be the smallest particles which still have the properties of molecular clusters that distinguish them from bulk metal. 

The energies of the facetted molecular clusters of different sizes were plotted as a function of cluster size for both polymorphic forms of l-GA (Fig. 8.6c). The results were fitted with a power law function which enabled calculation of cluster energy for any molecular cluster size. The results revealed that the metastable a-form is a more thermodynamically stable form than the stable p-form at smaller cluster sizes with a crossover point of 240 molecules. 

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