Classical Kinetic Equation Research Articles

Grain growth kinetics and mechanical properties of spark plasma sintered Fe-15Cr-2W powders

In recent years, spark plasma sintering (SPS) technique has emerged as an alternative full density consolidation process as rapid sintering / densification is achieved with restricted grain growth as well as better purification and activation of particle surfaces. In the present investigation, consolidation of mechanically alloyed powder of nominal composition Fe-15Cr-2W at two pressures (50 MPa and 60 MPa) and three sintering temperatures (950 °C, 1025 °C, and 1100 °C) was studied. The consolidated samples were characterized and tested for density, grain size / distribution, hardness, and compressive strength. Consolidated samples were further annealed at 1100 °C for different holding periods (0, 60, and 120 mins) and at different temperatures of 950, 1100, and 1250 °C at a fixed holding period (60 min) to study the grain growth kinetics. The different stages in the displacement profiles during SPS were discussed in depth in view of the densification of the alloys. Pressure was found to be dominant factor in terms of densification at lower sintering temperature (i.e. 950 °C) while at higher temperature (≥1000 °C), temperature activated diffusion was found to be dominant. The grain growth exponent (n) and activation energy (Q) determined for the alloy sintered at 1025 °C and 50 MPa via classical grain growth kinetic equation were 7.34 and 214.9 kJ/mol respectively. The deviation of grain growth exponent from its ideal value (2) has been discussed in light of various attributes of mechanical alloying.

A full probabilistic analysis of a randomized kinetic model for reaction–deactivation of hydrogen peroxide decomposition with applications to real data

The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and aspergillus niger catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.

Regarding Corrections to the Stokes Force in the Knudsen Number

With the help of the classical Boltzmann kinetic equation, a correction to the Navier–Stokes equation has been found in the form of a biharmonic term in the Laplace operator. It is shown that upon taking this term into account, corresponding corrections in the Knudsen number appear in the Stokes formula.

The basis of the semiclassical description of electron transport in solids

The theoretical account of electron transport in crystalline solids is often taught to graduate students through a semiclassical equation, which is a classical kinetic equation amended with elements of quantum physics in a non-systematic way. This blending of classical and quantal elements accounts for many experimental features of electron transport, save for a few ones observed over short distances. For students acquainted with solid-state quantum mechanics, it is preferable to derive the semiclassical equation from quantum dynamics so as to pinpoint the shortcomings and approximations underlying the semiclassical picture of transport. This paper explores the pathway leading from quantum dynamics to semiclassical kinetics within a simplified model not exceeding the proficiencies of a graduate student in physics or electronic engineering. In this model an electron moves in a crystal under the joint influences of the periodic crystal potential, an external electrostatic field and a set of lattice imperfections haphazardly distributed in the crystal volume. The first influence is treated exactly by means of standard energy-band theory and the other two are handled perturbatively. Following a procedure already tested in free-space issues, we replace Schrödinger’s wave function in position space with a Wigner function defined in the phase (position–pseudomomentum) space of semiclassical kinetics. The dynamical equation on the Wigner function is worked out and approximated so as to retrieve the semiclassical equation. The errors incurred in making use of the latter are assessed quantitatively. It is explained how a time-irreversible evolution equation—governing a phase-space occupancy—arises from reversible Schrödinger dynamics—governing a complex-valued probability amplitude. Besides clarifying the foundation of the semiclassical picture of electron transport and providing an explicit quantum-coherence correction to that picture, this paper may help students in general physics to better grasp, in a concrete issue, the interplay between classical and quantum concepts.

A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Levy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Levy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.

Photogalvanic effect in monolayer transition metal dichalcogenides under double illumination

We study the photogalvanic effect caused by a simultaneous action of two light sources: circular-polarized interband and linearly-polarized intraband. It is found that, in such conditions, the steady photocurrent appears. The effect originates from the valley-selective pumping by the circular-polarized light, the trigonal symmetry of the valleys and the linear polarization of intraband electromagnetic field, that produces a polar in-plane asymmetry of the electron and hole distribution functions, leading to the photocurrent. The approach is based on the solution of the classical kinetic equation for carriers with accounting for the quantum interband excitation.

Shear sound of two-dimensional Fermi liquids

We study the appearance of a sharp collective mode which features transverse current fluctuations within the bosonization approach to interacting two-dimensional Fermi liquids. This mode is analogous to the shear sound modes in elastic media, and, unlike the conventional zero sound mode, it is damped in weakly interacting Fermi liquids and only separates away from the particle-hole continuum when the quasiparticle mass becomes twice the transport mass $m^* \gtrsim 2 m$. The shear sound should be present in a large class of interacting charged and neutral Fermi liquids especially those proximate to critical points where the quasiparticle mass diverges. In metals this mode remains linearly dispersing in the presence of the long-ranged Coulomb force, unlike the conventional zero sound mode which becomes the plasma mode. We also detail a quick path between bosonization and classical Landau's Fermi liquid theory by constructing a mapping between the solutions of the classical kinetic equation and the quantized bosonic eigenmodes. By further mapping the kinetic equation into a 1D tight-binding model we solve for the entire spectrum of collective and incoherent particle-hole excitations of Fermi liquids with non-zero $F_0$ and $F_1$ Landau parameters.

Parametric subharmonic instability in a narrow-band wave spectrum

Parametric subharmonic instability arising in a narrow-band wave spectrum is investigated. Using a statistical equation that describes weakly nonlinear interactions in a random wave field, we perform analytical and numerical stability analyses for a modulating wave train. The analytically obtained growth rate $\unicode[STIX]{x1D706}=(-\unicode[STIX]{x1D707}+\sqrt{\unicode[STIX]{x1D707}^{2}+4CE_{B}})/2$ agrees favourably with the results from direct numerical experiments, where $\unicode[STIX]{x1D707}$ is the half-value width of the background wave frequency spectrum, $E_{B}$ is the background wave energy density, and $C$ is a constant. This expression has two asymptotic limits: $\unicode[STIX]{x1D706}\sim \sqrt{CE_{B}}$ for $\unicode[STIX]{x1D707}\ll \sqrt{CE_{B}}$ and $\unicode[STIX]{x1D706}\sim CE_{B}/\unicode[STIX]{x1D707}$ for $\unicode[STIX]{x1D707}\gg \sqrt{CE_{B}}$. In the terms of weak turbulence, these two growth rates correspond to the ones occurring in the dynamic and kinetic time scales. In this way, our formulation successfully unifies the two conventional types of parametric subharmonic instability and offers a new criterion to determine the applicability of the classical kinetic equation in three-wave systems.

Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations

Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ($O(10^{2})$ wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description.

Electrical resistivity and thermal conductivity of liquid aluminum in the two-temperature state

The electrical resistivity and thermal conductivity of liquid aluminum in the two-temperature state is calculated by using the relaxation time approach and structural factor of ions obtained by molecular dynamics simulation. Resistivity witin the Ziman–Evans approach is also considered to be higher than in the approach with previously calculated conductivity via the relaxation time. Calculations based on the construction of the ion structural factor through the classical molecular dynamics and kinetic equation for electrons are more economical in terms of computing resources and give results close to the Kubo–Greenwood with the quantum molecular dynamics calculations.

Classical magnetoresistance of a two-component system induced by thermoelectric effects

Magnetotransport in a two-dimensional two-component system consisting of electrons and holes with the same concentrations is studied. Balance equations to describe charge carrier and heat transfer are derived from the classical kinetic equation. The charge-carrier density and temperature distributions and electric-current densities are calculated by solving the balance equations for a long strip sample. In a sufficiently high magnetic field, regions of increased and decreased charge-carrier density, temperature, and fluxes are formed near the sample edges. This leads to nontrivial positive magnetoresistance.

Modelling of grain refinement driven by negative grain boundary energy

Grain refinement can be described by the classical kinetic equation using a negative value of the specific grain boundary Gibbs energy. A respective overview is offered reporting according observations and simulations, particularly linked to grain boundary segregation. Classical grain growth model is used in the treatment of evolution of the distribution function during refinement. The adapted model requires defining nucleation rate of new grains, which significantly influences the kinetics of the system of grains. Moreover, a jump in the distribution function is allowed at a certain value of the grain radius RJ, which separates old grains from newly nucleated ones. Evolution equation for both the critical radius Rc and separation radius RJ (jump position) as well as for the dimension-free distribution (shape) function are derived. Illustrative examples for the evolution of the system parameters under various nucleation rates of newly generated grains are presented.

The [formula omitted]-theorem for the physico-chemical kinetic equations with discrete time and for their generalizations

In this paper the generalizations of the physico-chemical kinetics equations, including classical and quantum physico-chemical kinetics, coagulation–fragmentation equations, discrete velocity models of classical and quantum kinetic equation and, generally speaking, all discrete physico-chemical kinetics is considered. We make time discrete in these equations and prove the H-theorem.

Poly(n-octyl methacrylate) viscosity index improver: Kinetic study via on-line 1H-NMR technique

The free radical solution polymerization of n-octyl methacrylate has been studied in benzene-d 6 in the presence of 2,2′-azobisisobutyronitrile as thermal initiator. An on-line nuclear magnetic resonance spectroscopy was applied to record the reaction data and to determine the monomer conversion at different times during the polymerization reaction. Effect of monomer and initiator concentration as well as reaction temperature on polymerization rate was studied. The order of the reaction with respect to initiator (0.45) was consistent with the classical kinetic rate equation, while the order of reaction with respect to monomer (1.87) was much greater than unity. An overall activation energy (E a = 53.8 kJ/mol) was obtained over the temperature range 328−338 K. Also, the efficiency of the synthesized poly(n-octyl methacrylate) for improving the viscosity index of the lube oil was investigated.

Multistep Differential Transform Method for Numerical Solution of Classical Neutron Point Kinetic Equation

In this paper, the differential transform method is applied to compute the numerical solution for classical neutron point kinetic equations [5, 6, 7] in a nuclear reactor. The differential transform method (DTM) is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. The new algorithm multistep differential transform method (MDTM) is applied here for solving the classical neutron point kinetic equation. The multistep DTM is treated as an algorithm in a sequence of intervals for finding simple and accurate solutions. Moreover, numerical examples with variable step reactivities, ramp reactivity, and sinusoidal reactivity are used to illustrate the preciseness and effectiveness of the proposed method.

Entropic lattice Boltzmann method for turbulent flow simulations: Boundary conditions

We introduce a new concept of boundary conditions for realization of the lattice Boltzmann simulations of turbulent flows. The key innovation is the use of a universal distribution function for particles, analogous to the Tamm–Mott-Smith solution for the shock wave in the classical Boltzmann kinetic equation. Turbulent channel flow simulations demonstrate that the new boundary enables accurate results even with severely under-resolved grids. Generalization to complex boundary is illustrated with an example of turbulent flow past a circular cylinder.

A classical but new kinetic equation for hydride transfer reactions

A classical but new kinetic equation to estimate activation energies of various hydride transfer reactions was developed according to transition state theory using the Morse-type free energy curves of hydride donors to release a hydride anion and hydride acceptors to capture a hydride anion and by which the activation energies of 187 typical hydride self-exchange reactions and more than thirty thousand hydride cross transfer reactions in acetonitrile were safely estimated in this work. Since the development of the kinetic equation is only on the basis of the related chemical bond changes of the hydride transfer reactants, the kinetic equation should be also suitable for proton transfer reactions, hydrogen atom transfer reactions and all the other chemical reactions involved with breaking and formation of chemical bonds. One of the most important contributions of this work is to have achieved the perfect unity of the kinetic equation and thermodynamic equation for hydride transfer reactions.

Aqueous free-radical polymerization of PEGMEMA macromer: kinetic studies via an on-line 1H NMR technique

Free-radical polymerization of polyethylene glycol methyl ether methacrylate macromer (PEGMEMA) was studied in aqueous media and in the presence of potassium persulfate (KPS) as water soluble initiator. An on-line nuclear magnetic resonance (NMR) method was applied to record the reaction data and determine the monomer conversion at various times during the polymerization reaction progress. 1H NMR spectrum of reaction mixture containing monomer, initiator and resultant polymer was continuously recorded in NMR instrument with the increase of reaction time. By processing the obtained data from NMR spectrum, the rate equation can be derived and reaction order can be determined with regard to monomer and initiator concentration. In other words, to determine the order of polymerization with regard to the concentration of reactants in free-radical polymerization of PEGMEMA, macromer samples with different amounts of monomer and KPS were prepared and polymerized at 50 °C. Orders of reaction with respect to monomer and initiator molar concentrations were equal to 1.025 and 0.480, respectively. The obtained values for reaction orders in this study were consistent with the classical kinetic rate equation in which the dependency of polymerization rate (Rp) on monomer and initiator concentrations was equal to 1 and 0.5, respectively. To measure polymerization activation energy (Ea), the effect of reaction temperature on the polymerization rate was investigated and Ea = 37.08 kJ/mol was obtained at the temperature range of 40–50 °C.

An Explicit Finite Difference scheme for numerical solution of fractional neutron point kinetic equation

In the present article, a numerical procedure to efficiently calculate the solution for fractional point kinetics equation in nuclear reactor dynamics is investigated. The Explicit Finite Difference Method is applied to solve the fractional neutron point kinetic equation with the Grunwald–Letnikov (GL) definition ( Podlubny, 1999; Oldham and Spanier, 1974). Fractional Neutron Point Kinetic Model has been analyzed for the dynamic behavior of the neutron motion in which the relaxation time associated with a variation in the neutron flux involves a fractional order acting as exponent of the relaxation time, to obtain the best operation of a nuclear reactor dynamics. Results for neutron dynamic behavior for subcritical reactivity, supercritical reactivity and critical reactivity and also for different values of fractional order have been presented and compared with the classical neutron point kinetic (NPK) equation as well as the results obtained by the learned researchers Espinosa-Paredes et al. (2011).

On the application of an approximate kinetic equation of heat and mass transfer processes: the effect of body shape

An approximate kinetic equation of heat and mass transfer has been presented. It is an ordinary differential equation which is easy to integrate. The derivation of this approximation was based on an analysis of the available analytical solution of the problem. The proposed equation can be applied to bodies (pellets) in the shape of an infinite slab, infinite cylinder and sphere. A generalization of this equation to cases where the transfer resistance occurs both in the body and in the surrounding fluid has been proposed. The equation has been tested in various conditions both for thermal and diffusive processes. Radiative cooling of bodies has been considered as a thermal process and adsorption in a single pellet—as a diffusive process. All tests showed high accuracy of the approximate equation; in many cases the results were indistinguishable from the results of the exact model. A special feature of the proposed equation is its high accuracy for short times of the process, what significantly differentiates it from the classical approximate kinetic equation Linear Driving Force.