Allowance For Fluctuations Research Articles

How the intermediate stage of a phase transition process transforms to the concluding stage of Ostwald ripening

Abstract A theoretical description of the intermediate stage of the phase transformation process, passing into the Ostwald ripening stage, is presented. First, we consider the intermediate stage of nucleation and growth of crystals in the supercooled melts or supersaturated solutions with allowance for fluctuations in their growth rates. The evolutionary behavior of the particle-size distribution function and the metastability degree is described analytically for the Meirs kinetic mechanism. The asymptotic solution for the “tail” of the particle-size distribution function is deduced in the form of the power-dependent function. This distribution function is used as the initial condition for the concluding stage where Ostwald ripening occurs. On this basis, the concluding stage of Ostwald ripening is analytically described. Namely, the maximum of the distribution function is lower and shifted to the left in comparison with the Lifshitz-Slyozov (LS) asymptotic solution. In addition, the distribution function obtained has a “tail” on the right of the LS blocking point. The theory is in good agreement with experimental data.

The Rate of Spontaneous Formation of Microscopic Nuclei in Supersaturated Vapor

The equilibrium nucleus-size distribution determined by the method of statistical physics has been analyzed. The analysis has shown that nuclei composed of 1000 or fewer molecules are microscopic objects. They are described by partition functions and cannot be described by thermodynamic methods. An approach has been proposed that makes it possible to determine a partition function over internal degrees of freedom of a nucleus and express the aforementioned distribution via commonly accepted thermodynamic parameters. The solution of the problem is reduced to the determination of the evaporation rate of clusters by extrapolating the evaporation rate, which has been calculated for a macroscopic droplet of an incompressible liquid in terms of thermodynamic concepts with allowance for fluctuations, to the sizes of nuclei. As a result, a theory has been formulated for homogeneous stationary nucleation. The comparison of the proposed theory with experimental data has shown that the calculated sizes of critical nuclei coincide with the measured ones and that the theoretical nucleation rates either coincide with the measured rates or agree with them within one or two decimal orders of magnitude.

Phase transitions and fluctuations in classical statistical mechanics

It is shown that allowance for fluctuations leads to the appearance of additional terms in the Hamiltonian of the system and in the thermodynamic parameters of the substance. As a result, it becomes possible to describe the critical phenomena and phase transitions of liquid–vapor within classical statistical mechanics (without taking into account fluctuations, this is impossible).

Modeling of phase decomposition of supersaturated solid solutions using the free-energy density functional method

The nucleation and growth of particles of the second phase in a one-dimensional binary alloy is considered based on the Cahn–Hilliard equation with allowance for fluctuations. Using the results of modeling, it is shown that the second phase is occupied by the mechanism of solid solution decomposition, which involves simultaneous processes of the fluctuational nucleation of the second phase, diffusion-type growth of precipitations, and absorption of small clusters by coarse ones at the coalescence stage. Composition fluctuations are among the main factors influencing the distribution of solid solution phases.

A synergistic approach to the polydispersity of polymers

A scheme for calculating polydispersity coefficients of polymers during chain propagation in living polymerization is proposed on the basis of the synergistic approach (the master equation) with allowance for fluctuations in the number of free radicals at the stage of initiation. The molecular-mass-distribution function of living chains is calculated. The form of this function is shown to be dependent on the ratio between the constants of the elementary stages of chain initiation and propagation.

Criterion for the onset of ostwald ripening stage with allowance for the particle number fluctuations in a nucleus

A theory of condensation is proposed for an open system in which the growth rate varies with the number of particles and the total particle concentration (supersaturation flux) varies with the time as described by power laws with arbitrary exponents (indices). It is established that the distribution of nuclei with respect to their sizes z, which is formed at the nucleation stage, is subject to fluctuational blurring. The dependence of the distribution width on the average size z is related in a nontrivial manner to the values of the growth and flux indices. Conditions for the onset of the Ostwald ripening (OR) stage in the system are analyzed with allowance for fluctuations in the number of particles. It is shown that the Kukushkin-Osipov criterion remains valid, since the transition curve occurs in a region where the distribution width grows slower than z. At the same time, the form of the initial condition and the character of evolution of the distribution at the OR stage significantly vary under the action of fluctuations.

Phase diagrams of polyelectrolyte solutions in poor solvents and of polyelectrolyte globules with allowance for microphase separation and fluctuation effects

Polyelectrolyte solutions in poor solvents have been studied theoretically in the weak crystallization ability approach with allowance made for fluctuation corrections calculated in terms of Brazovskii’s approximation. Phase diagrams have been constructed, and the regions of stability of phases possessing various crystal symmetry and the regions, where macrophase separation takes place, have been determined. In contrast to the results obtained previously within the framework of the mean-field weak crystallization theory, it has been shown that the allowance for fluctuations leads to the refinement of the shape and configuration of phases in the phase diagram. In the region where the fraction of charged chain units is small, globules characterized by the body-centered cubic structure may exist.

Effect of correlated temperature fluctuations on the phase dynamics in an ultrathin lubricant film

The melting of an ultrathin lubricant film at friction between atomically smooth surfaces is studied with allowance for fluctuations of its temperature, which are described by the Ornstein-Uhlenbeck process. The behavior of the most probable types of shear stresses arising in the lubricant is considered, and phase diagrams for second- and first-order phase transformations (the melting of an amorphous lubricant and that of a crystal- line lubricant, respectively) are constructed. It is shown that, in the former case, lubricant temperature fluctua- tions lead to the formation of a stick-slip friction domain separating the domains of dry and sliding friction, which is typical of first-order transitions. In the latter case, three domains of stick-slip friction arise, which mark the transitions between dry friction and metastable and stable sliding friction. As the time of correlation of lubricant temperature fluctuations gets longer, the temperature of rubbing surfaces rises to the point where sliding friction sets in.

Analysis of packing factors of crystalline polymers in terms of the glass transition to melting temperature radio

An expression for the ratio of the glass transition temperature to the melting temperature, Tg/Tm, was derived with allowance for fluctuations of the packing factor in the amorphous state. This relationship made it possible to describe the actual range of variation in Tg/Tm depending on a change in the packing factor in the crystalline state ko, cr. The proposed approach forms the basis for the method of determination of increments in the packing ratio ko, cr and formulation of the principles of selection of experimental data for calculations. The glass transition temperatures in polyethylene depending on the type of unit cell were calculated.

Sea-level muon spectrum in the energy range 1–10 TeV from the data of underground experiments

Within an adjoint approach, the properties of the muon component in various types of rock and in water are calculated with allowance for fluctuations of energy losses in all muon-interaction processes. The behavior of the muon spectrum at sea level is established on the basis of a comparison of the muonabsorption curves obtained in this way and experimental data from underground facilities. It is shown that the deficit of the calculated muon flux obtained with the aid of data from direct measurements of the spectra of primary nuclei and Reggeon models of nuclear interactions is not less than 50% in the energy range 1–10 TeV.

Temperature and mass dependence of anharmonic defect dynamics

AbstractWithin the framework of an approach with allowance for fluctuations of the mean energy of a particle interacting with a thermostat, the partial density of vibrational states (PDVS) of a crystal lattice defect located in a two‐well potential has been investigated. The results obtained are in good agreement with those of a numerical study of the corresponding Langevin equation. It has been shown that the mass dependence of the anharmonic defect PDVS is quite complicated and is determined by temperature.

The Ion Charge Fluctuation Effect on Impurity Depth Distributions for High-Energy Ion Implantation. The Backward Transport Equation-Based Simulation

A model of high-energy ion implantation is formulated with allowance for fluctuations of the charge state of ions and the corresponding backward transport equation for post-implantation spatial distributions of an impurity. Simulation results of experiments concerned with boron ion implantation into silicon by numerical solving the backward transport equation are reported. It is shown that the account of ion charge fluctuations in the high-energy range leads to an increase of the range straggling of the implanted impurity and provides a better agreement with the experiment than the traditional methods of modelling based on the use of the Monte-Carlo method or the backward transport equations. [Russian Text Ignored].

Two-dimensional nonlocal hydrodynamics with allowance for fluctuations

Two-dimensional nonlocal hydrodynamics with allowance for fluctuations

Quantum-statistical calculation of the properties of a mixture of chemical elements allowing for fluctuations in the occupation numbers of electron states

A method is proposed for solving a system of equations which makes it possible to calculate the characteristics of a substance made up of a mixture of different chemical elements at a specified temperature and density. Allowance for fluctuations in the occupation numbers of the electron states enables one to determine the selfconsistent Hartree-Fock-Slater potentials and the relative concentrations of the different ions of the chemical elements in the mixture. A comparison is made with calculations carried out on SiO 2 using the Saha model.

Fluctuation phenomena in the tricritical behavior of two models with tensor order parameter

This paper uses the fluctuation theory of phase transitions to study tricritical behavior in the isotropic phase of two models in which the order parameter is p-dimensional symmetric traceless tensor. For arbitrary values of p, differential equations are obtained for the vertex functions from the second to the sixth orders as functions of the reciprocal susceptibility. The cases p = 1, 2, 3 are studied. For p = 1 and 2, the models exhibit tricritical behavior. For p = 3, allowance for fluctuations leads in the general case to the appearance of instability in the critical behavior of the models and to the replacement of tricritical behavior by a phase transition of the first kind.

Mass crystallization with allowance for fluctuations of crystal growth rate

Mass crystallization with allowance for fluctuations of crystal growth rate

Allowance for fluctuations of radiation flux in activation analysis

Allowance for fluctuations of radiation flux in activation analysis