The Concept of Randomized Particles as the Basis of Cluster and Associate Theory Viscosity and Flow

Using thermal barriers at the melting point?Hm and RTm,it is shown that the latter directly reflects the chaotic process,since it is equal to the kinetic energy reserve of chaotic(thermal)particle motions,and the first additionally takes into account the energy expenditure for overcoming the potential energy of the interconnection of particles,which is typical for inorganic compounds.Therefore,to determine the share of crystal-mobile particles responsible for the viscosity of the melt,the chaotization barrier of RTm should be used,since in the virtual clusters the potential binding energy is conserved,thereby compensating for the heat expense of breaking these bonds upon melting.Therefore,to analyze the share of crystal-mobile particles,it is necessary to use the formula of their share in the form:Pcrm=1-exp(-tm/t).On the basis of the distribution of clusters previously found by the authors in terms of the number of crystal-mobile particles included in them,it was shown that all non-single crystal-mobile particles are responsible for the viscosity,and for flowability all single particles,including crystal-mobile,liquid-mobile and vapor-mobile.This ensures the superiority of the share of single particles over the share of crystal-mobile particles arranged in non-single clusters at the melting point,and thereby the fluidity of the melt.Based on the share distribution of clusters in terms of the number of particles entering into them,the share of non-single clusters responsible for the viscosity of the melt is expressed as:Pct=p2crm=[1-exp(-Tm/T)2].The probabilistic meaning of the formation of clusters from non-single crystal-mobile particles is extended to the formation of associates,which made it possible to disclose the meaning of the second level of the exponential dependence of viscosity in the cluster and associate model:η=η1(T1/T)a2(T2/T)b,where the first level is responsible for the formation of clusters,and the second—for associates.This form corresponds to the physical hierarchy when combining crystal-mobile particles.The previously proposed method for processing viscosity data for the cluster and associate model assumed the use of three reference points from the available experimental array of values of viscosity at different temperatures.This method is supplemented by using the entire set of data on the viscosity with the preservation of two reference points and processing the rest to determine the exponent b,which has the meaning of aggregation degree of associates,from the linearized dependence:ln in(η/η1)in(T1/T2)/IN(T1/T)in(η/η1)=bin(T1/T).The new method was tested on reference data and showed its high statistical adequacy.

Quantization of Particle Energy in the Analysis of the Boltzmann Distribution and Entropy

The Boltzmann equilibrium distribution is an important rigorous tool for determining entropy, since this function cannot be measured, but only calculated in accordance with Boltzmann＇s law. On the basis of the commensuration coefficient of discrete and continuous similarly-named distributions developed by the authors, the article analyses the statistical sum in the Boltzmann distribution to the commensuration with the improper integral of the similarly-named function in the full range of the term of series of the statistical sum at the different combination of the temperature and the step of variation （quantum） of the particle energy. The convergence of series based on the Cauchy, Maclaurin criteria and the equal commensuration of series and improper integral of the similarly-named function in each unit interval of variation of series and similarly-named function were estab- lished. The obtained formulas for the commensuration coefficient and statistical sum were analyzed, and a general expres- sion for the total and residual statistical sums, which can be calculated with any given accuracy, is found. Given a direct calculation formula for the Boltzmann distribution, taking into account the values of the improper integral and commensuration coefficient. To determine the entropy from the new expression for the Boltzmann distribution in the form of a series, the conver- gence of the similarly-named improper integral is established. However, the commensuration coefficient of integral and series in each unit interval turns out to be dependent on the number of the term of series and therefore cannot be used to determine the sum of series through the improper integral. In this case, the entropy can be calculated with a given accuracy with a corresponding quantity of the term of series n at a fixed value of the statistical sum. The given accuracy of the statistical sum turns out to be mathematically identical to the fraction of particles with an energy exceeding a given level of the energy barrier equal to the activation energy in the Arrhenius equation. The prospect of development of the proposed method for expressing the Boltzmann distribution and entropy is to establish the relationship between the magnitude of the energy quantum Ae and the properties of the system-forming particles.

Cluster-Associate Model of Viscosity of Silver Chloridein Comparison with Frenkel Model

The purpose of the research is to develop the temperature dependence of the dynamic viscosity for silver chloride. The data were calculated on the basis of a new cluster and associate equation, which was derived using the concept of randomized particles. The calculated data are given in the temperature range from the melting point to the boiling point. The cluster and associate model is compared with the Frenkel’s equation in logarithmic coordinates, showing the mutual correspondence and complementarity of these models.