An analytical solution to the nonlinear evolutionary equations for nucleation and growth of particles

ABSTRACT

In this paper, we consider a mathematical model for the growing crystals in supersaturated or supercooled systems. An integro-differential equation of the Fokker-Planck type is analytically solved using the saddle point method for the Laplace integral. The solution describes an intermediate stage of the phase transition process with allowance for the power law of the growth rate of spherical particles and the nucleation mechanisms known as the Meirs and Weber-Volmer-Frenkel-Zeldovich kinetics. A dynamical dependencies for the supersaturation/supercooling and the distribution function of crystals by their size are obtained. The novelty of the theory is the use of a power law for the growth rate of crystals, which leads to new analytical solutions of the problem.     

Nucleation and growth of crystals at the intermediate stage of phase transformations in binary melts

An exact analytical solution of integro-differential equations, which describe the phase transitions in supercooled binary melts, is constructed with allowance for fluctuating rates of crystal growth. A complete solution of the corresponding Fokker–Planck and balance equations is found for arbitrary nucleation kinetics and growth rates of crystals. In limiting cases, the obtained solutions transform to earlier known solutions for a special form of diffusivity in the space of crystal sizes and purely kinetic and “diffusion-kinetic” growth rates of nuclei.     

Vacancy-decomposition-induced lattice instability and its correlation with the kinetic stability limit of crystals

Vacancy decomposition kinetics in crystals at elevated temperatures is analysed. It is found that lattice instability is induced by a significantly enhanced vacancy decomposition at a critical temperature (T?*). The critical temperature coincides with the kinetic instability limit (kinetic limit of superheating) of crystals in a variety of metals determined from the homogeneous nucleation catastrophe model.     

On the role of Ag in enhanced age hardening kinetics of Mg–Gd–Ag–Zr alloys

The addition of Ag to the age hardenable Mg–Gd–Zr alloy system dramatically enhances early stage age hardening kinetics. Using atom probe tomography (APT), Ag-rich clusters were detected in a Ag-containing Mg–Gd–Zr alloy immediately after solution treatment and water quenching. During subsequent isothermal ageing at 200 °C, a high density of basal precipitates was observed during the early stages of ageing. These basal precipitates were enriched with Ag and Gd, as confirmed by APT. It is posited that Ag-rich clusters in the context of quenched-in vacancies can attract Gd atoms, increasing diffusion kinetics to facilitate the formation of the Ag + Gd-rich basal precipitates. The rapid formation of Ag + Gd-rich precipitates was responsible for accelerated ageing.     

Precipitation of niobium carbonitrides in ferrite: chemical composition measurements and thermodynamic modelling

High-resolution transmission electron microscopy and electron-energy loss spectroscopy have been used to characterize the structure and chemical composition of niobium carbonitrides in the ferrite of a Fe–Nb–C–N model alloy at different precipitation stages. Experiments seem to indicate the coexistence of two types of precipitates: pure niobium nitrides and mixed sub-stoichiometric niobium carbonitrides. In order to understand the chemical composition of these precipitates, a thermodynamic formalism has been developed to evaluate the nucleation and growth rates (classical nucleation theory) and the chemical composition of nuclei and existing precipitates. A model based on the numerical solution of thermodynamic and kinetic equations is used to compute the evolution of the precipitate size distribution at a given temperature. The predicted compositions are in very good agreement with experimental results.     

A new counter-example to Kelvin's conjecture on minimal surfaces

A survey of the published literature on undercooled metallic and oxide melts suggests that phase selection during solidification can be categorized as nucleation controlled or growth controlled. Common characteristics governing the phase-selection pathway have been identified for various alloy systems. It is recognized that when competing stable and metastable phases share the same crystalline characteristics and have comparable interface kinetic coefficients, the principle of nucleation control applies for primary phase formation in a deeply undercooled melt. However, there can be a difference of two or three orders of magnitude in the interface kinetic coefficients for competing phases, either between an ordered intermetallic compound and a disordered solid solution, or between a crystalline phase with a high level of complexity and a simple crystal. In such cases, the principle of growth control will apply; more specifically, the phase with the faster growth kinetics should be favoured and the competing counterpart with sluggish interface kinetics should be suppressed at high undercoolings. Some simple predictions are suggested on the basis of this principle when considering stable and metastable phase diagrams. The specific conditions under which the present categorization is applicable are outlined. Future work is required to elucidate phase competition under conditions of very rapid solidification.     

Kinetic study of hydrogen-induced direct phase transformation in Y2Fe17 magnetic alloy on the basis of Kolmogorov’s model

The kinetics of the hydrogen induced direct phase transformation in Y₂Fe₁₇ magnetic alloy has been analysed within the framework of Kolmogorov’s kinetic model. It is established that the transformation can be classified as a diffusion-controlled transformation, which occurs by a mechanism of nucleation and growth with a decreasing nucleation rate of new phases, namely α-Fe and YH₂. A kinetic equation has been obtained that well describes the isothermal kinetic curves of the phase transformation in Y₂Fe₁₇ as a function of the transformation temperature.     

Variation of Isometric Response Force in the Rat

Hungry, unrestrained rats (N = 7) were rewarded for pressing a response beam in excess of 11 different force requirements. Changes in peak force production as a function of peak force requirement were examined by analyses of the first four moments of distributions of peak response forces: constant error, the within-subject standard deviation, skewness, and kurtosis. Results were similar to those previously obtained with human subjects: Constant error was positive at low and negative at high force requirements, the within-subject standard deviation increased as a negatively accelerating function of force requirement, and skewness and kurtosis were positive at low force requirements and decreased to negative values at the highest increments. Additional analyses of response kinetics indicated that rats, like humans, meet increasing force requirements by altering the rate of rise of force. The performance similarities suggest that common processes are engaged by the human and rat motor control systems to solve the problem of generating forces that are appropriate to the prevailing environmental constraints.     

Linear-parabolic shift of initially sharp interface in AB diffusion couple above the ordering temperature

It is shown that initially sharp interface in AB diffusion couple, above the ordering temperature, can shift with anomalous kinetics, i.e. deviations from the parabolic growth kinetics are possible. The initial slopes of the shift versus time functions can be even unity for large diffusion asymmetry (the ratio of diffusion coefficients of the parent materials in orders of magnitude) and gradually decrease to 0.5 at larger distances (longer times). This is in accordance with earlier results obtained in phase separating systems. It is shown that from the results of simulations the crossover thickness, X*, between the linear and parabolic regimes can also be calculated, and the applicability of the linear–parabolic law is confirmed. Furthermore, it is illustrated that the crossover thickness depends exponentially on the diffusion asymmetry parameter, with the exponent close to the value obtained from analytical estimations.     

Temperature dependence of nucleation rate in a binary solid solution

The influence of regression (partial dissolution) effects on the temperature dependence of nucleation rate in a binary solid solution has been studied theoretically. The results of the analysis are compared with the predictions of the simplest Volmer–Weber theory. Regression effects are shown to have a strong influence on the shape of the curve of nucleation rate versus temperature. The temperature T_M at which the maximum rate of nucleation occurs is found to be lowered, particularly for low interfacial energy (coherent precipitation) and high-mobility species (e.g. interstitial atoms).     

Kinetics of Ostwald ripening in the presence of monomer diffusion along grain boundaries

The kinetics of an Ostwald ripening is studied where the prevailing mechanism of cluster growth is monomer diffusion along grain boundaries. The model describes self-consistently the situation when the delivery of matter to a growing cluster is carried out by monomer diffusion along grain boundaries. An analytical and numerical study of Ostwald ripening kinetics in a homogeneous supersaturated solution is performed. The possible transition modes arising during the Ostwald ripening process are discussed with reference to changes in the dominant monomer delivery mechanism.     

Wentzel–Kramers–Brillouin solution of cut-off frequency for horizontal shear (SH) waves in various inhomogeneous thin films

The Wentzel–Kramers–Brillouin method is employed to study the cut-off frequencies of the horizontal shear waves in a freestanding functionally graded piezoelectric–piezomagnetic material film with the electrically and magnetically open boundary conditions. An analytical solution, which could be used in analyzing the problems of various functionally graded materials, is proven to have high precision by analytical analysis and a numerical example. The results reveal that the set of cut-off frequencies is a series of approximate arithmetic progressions. A theoretical foundation based on the relationship between the cut-off frequencies and the materials’ gradient property is established for nondestructive evaluation.     

Variation of Isometric Response Force in the Rat

Hungry, unrestrained rats (N = 7) were rewarded for pressing a response beam in excess of 11 different force requirements. Changes in peak force production as a function of peak force requirement were examined by analyses of the first four moments of distributions of peak response forces: constant error, the within-subject standard deviation, skewness, and kurtosis. Results were similar to those previously obtained with human subjects: Constant error was positive at low and negative at high force requirements, the within-subject standard deviation increased as a negatively accelerating function of force requirement, and skewness and kurtosis were positive at low force requirements and decreased to negative values at the highest increments. Additional analyses of response kinetics indicated that rats, like humans, meet increasing force requirements by altering the rate of rise of force. The performance similarities suggest that common processes are engaged by the human and rat motor control systems to solve the problem of generating forces that are appropriate to the prevailing environmental constraints.     

Magnetic-field dependence of nucleation undercoolings in non-magnetic metallic melts

The nucleation undercoolings of non-magnetic metals like paramagnetic aluminium in high magnetic fields were measured by the differential thermal analysis technique. It was shown that the nucleation undercooling of pure aluminium increased with increasing the magnetic field, while its melting temperature was hardly changed. Based on the model of magnetic dipoles at the interface, it is proposed that the magnetic-field-induced interfacial energy mainly contributes to the increase in undercooling. The change in undercooling in the magnetic field is calculated theoretically, which is in comparison with experimental data. Additionally, the inhibition of atom diffusion in the magnetic field plays a role in the change of undercooling.     

Sensory Deprivation: A Symposium Edited by Philip Solomon

In psychoanalytical psychotherapy interpretation is the main activity of the therapist. It is related to revealing the latent structure of meanings of phenomena. The phenomena of analytical therapy are presented as generated by the interaction of two structures: the therapeutic setting and the theoretical models of the therapist. The analytical group setting is contrasted with the individual setting through the elaboration of its central characteristic: the lack of intimacy. This is seen as generating specific regressive phenomena in groups which the therapist will attempt to interpret following an object relations theory. A discussion of interpretations in analytical group therapy focuses on three levels: the group itself, the interpersonal, and the intrapsychic. These are presented as complementary rather than mutually exclusive. Finally, the therapists' function in groups is discussed through the notion of the oscillatory nature of their participation, from a passive–receptive stance to an active–synthetic function which leads to the delivery of an interpretation.     

Shear vibration of a crystal plate carrying an array of microbeams

We study shear vibration of a rotated Y-cut quartz crystal plate carrying an array of microbeams with their bottoms fixed to the top surface of the plate. The beams undergo flexural vibrations when the plate is in shear motion. The plate is modeled by the theory of anisotropic elasticity. The beams are modeled by the Euler–Bernoulli theory for beam bending. A frequency equation that determines the resonant frequencies of the structure is derived. An analytical solution on beam-induced frequency shift is obtained using a perturbation procedure. It is shown that the frequency shift may be used to measure geometric/physical properties of the beam array. A vibrating crystal plate carrying a beam array may also be considered for application as an ultrasonic brush.     

Reputation blackboard systems

Blackboard systems are motivated by the popular view of task forces as brainstorming groups in which specialists write promising ideas to solve a problem in a central blackboard. Here we study a minimal model of blackboard system designed to solve cryptarithmetic puzzles, where hints are posted anonymously on a public display (standard blackboard) or are posted together with information about the reputations of the agents that posted them (reputation blackboard). We find that the reputation blackboard always outperforms the standard blackboard, which, in turn, always outperforms the arrangement where the agents work independently. The asymptotic distribution of the computational cost of the search, which is proportional to the total number of agent updates required to find the solution of the puzzle, is an exponential distribution for those three search heuristics. Only for the reputation blackboard we find a nontrivial dependence of the mean computational cost on the system size and, in that case, the optimal performance is achieved by a single agent working alone, indicating that, though the blackboard organization can produce impressive performance gains when compared with the independent search, it is not very supportive of cooperative work.     

On the ultra-high-strain rate shock deformation in copper single crystals: multiscale dislocation dynamics simulations

Multiscale dislocation dynamics plasticity (MDDP) calculations are carried out to simulate the mechanical response of copper single crystals that have undergone shock loading at high strain rates ranging from 1?×?10⁶ to 1?×?10¹⁰?s^?1. Plasticity mechanisms associated with both the activation of pre-existing dislocation sources and homogeneous nucleation of glide loops are considered. Our results show that there is a threshold strain rate of 10⁸?s^?1 at which the deformation mechanism changes from source activation to homogeneous nucleation. It is also illustrated that the pressure dependence on strain rate follows a one-fourth power law up to 10⁸?s^?1 beyond which the relationship assumes a one-half power law. The MDDP computations are in good agreement with recent experimental findings and compare well with the predictions of several dislocation-based continuum models.