Atomistic origin of the thermodynamic activation energy for self-diffusion and order-order relaxation in intermetallic compounds II: Monte Carlo simulation of B2-ordering binaries

Abstract

The validity of previously derived formulae expressing the activation energies for self-diffusion and ‘order–order’ relaxations in intermetallics in terms of the activation energies of more elementary processes involved in the phenomena is tested by simulation of particular binary systems. The simulation results were in good agreement with the tested formulae. It was shown that the relationship between the activation energies observed in triple-defect B2-ordering binaries, where the value of the activation energy for order–order relaxations is substantially lower than that for self-diffusion, does not hold in the case of non-triple-defect binaries. Using the tested formulae, the origin of the effect was elucidated and attributed to the atomistic origin of the tendency for triple-defect disordering.     

Relaxation kinetics of the long-range order parameter in a non-uniform system studied by the phase field method using the free energy obtained by the cluster variation method

The order-order relaxation process of the long-range order parameter in the non-uniform L1 0 ordered phase is investigated by the hybridized calculation using the phase field method (PFM) and the cluster variation method (CVM). The resultant kinetics are composed of three processes sufficiently differing in the relaxation rates. The first and second processes correspond to antisite ordering-disordering relaxation within the ordered domain and relaxation due to a wetting-antiwetting of the antiphase boundary respectively. The third process is due to a coarsening of the ordered domain and its relaxation rate is fairly slow compared with the relaxation rates of the first and second processes; this is consistent with experimental observation. It is noted that the present result of relaxation kinetics involving the three processes is obtained only by the hybridized calculation using the PFM and CVM.     

N.M.R. study of solid perfluorocyclobutane

spin-lattice relaxation times and linewidth measurements for fluorine nuclei in solid perfluorocyclobutane are presented. The results are discussed in terms of D _2d molecular species performing fast internal motion. The relaxation measurements corroborate the existance of four solid-solid phase transitions and give some insight into their nature. The activation energies for molecular reorientation and self-diffusion processes are found to be 28·0 and 32·2 kJ mol^-1, respectively.     

Defect-related relaxation processes in irradiated rare gas solids

Electronic and atomic relaxation processes in preirradiated solid Ar doped with N 2 were studied with a focus on the role of radiative electronic transitions in relaxation cascades. Combining methods of activation spectroscopy - thermally stimulated and photon-stimulated exoelectron emission, a new channel of relaxation induced by photon emission from metastable N atoms was detected. It was shown that in insulating materials with a wide conduction band photons of visible range can release electrons from both kinds of traps - shallow (lattice defects) and deep thermally disconnected ones. Correlation in the charge recombination reaction yield and the yield of low temperature desorption - important relaxation channel in a preirradiated solid - clearly demonstrates interconnection between atomic and electronic processes of relaxation.     

Autodiffusion dans les alliages concentres fer-palladium

Measurements of the diffusion coefficients of ³⁹Fe and ¹⁰³Pd in various iron-palladium alloys (between 0 and 100at.% iron) between 1100 and 1250°C, have shown that the activation energies for selfdiffusion for the two tracers are practically equal, independent of composition.The values obtained are very close to those corresponding to the activation energy of chemical diffusion. This result is in good agreement with the fact that the thermodynamic activities of iron and palladium have small temperature dependance and that the vacancy flow, as calculated with the model of Manning, is rather small.In spite of a strong ordering tendency in the solid solution of Fe-Pd, as indicated by a thermodynamic calculation, this model allows to calculate, with the aid of the measured self-diffusion coefficients, various parameters, such as the correlation factors for both iron and palladium, the vacancy correlation factors, and the atomic jump frequencies.     

Modelling in relation to cation ordering

We review the methodology of using computer models to obtain quantitative information about cation ordering. Empirical interatomic potentials or ab initio electronic structure calculations are used to generate the energies for many configurations containing disordered arrangements of cations, and the parameters in model Hamiltonians can be determined from these energies. Monte Carlo simulations are then used to generate ensemble averages as functions of temperature or chemical composition. Analysis of the Monte Carlo ensembles directly yields the temperature dependence of long-range and short-range order, and thermodynamic quantities such as energy and heat capacity. Use of thermodynamic integration allows for the calculation of entropy and free energy. The methods are illustrated by examples showing long-range order/disorder phase transitions (feldspars), short-range order in solid solutions (pyrope-grossular), and non-convergent ordering (magnesium aluminate spinel); where comparisons with experimental data are possible, the model calculations are seen to give results that are reasonably accurate. The example in which ab initio electronic structure calculations are used show that it is now possible to extract accurate thermodynamic data for ordering processes using models that require no prior experimental data.     

Unified analysis of diffusion and relaxation processes in amorphous metallic alloys

Abstract

Self-diffusion data on amorphous metallic alloys determined in long-time radio-tracer experiments and activation-enthalpy spectra deduced from short-time structural relaxation studies on such materials are reviewed and analyzed in terms of current random transition rate models. It is shown that the seeming discrepancy between the self-diffusion enthalpies and the comparatively small enthalpies obtained from measurements of the magnetic after-effect arises from the existence of activation-enthalpy spectra. Whereas the short-time experiments reveal the small-activation-enthalpy parts of these spectra, the long-time self-diffusion experiments are controlled by the larger activation enthalpies. Assuming that the activation-enthalpy distributions are Gaussian, their characteristic parameters have been determined by comparing the two types of measurements. It is shown that the Arrhenius-type diffusion coefficients found by experiment are compatible with half-widths of the activation-enthalpy spectra of about 0.3 eV. Based on an analysis of the pre-exponential factors and other typical properties of the diffusion coefficients, potential mechanisms of the diffusion in amorphous alloys are proposed.     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.     

均匀合金自扩散的准化学模型

在准化学模型基础上,导出了具有面心立方及体心立方点阵的完全无序二元固溶体的自扩散激活能与组元浓度的关系。这种关系表明:(1)当两组元的原子尺寸接近相同时,两组元自扩散激活能之差与组元浓度的关系为一直线。这和Ag-Au系统自扩散实验数据符合很好。(2)在低浓度范围内,溶剂自扩散激活能与浓度亦应为一直线关系,其斜率的符号可由纯溶剂中自扩散和杂质扩散激活能的相对大小推知。结果与已知的银基固溶体中九种溶质原子对银自扩散影响的实验数据都符合。     

Dielectric and anelastic relaxation of crystals containing point defects

A theory is developed which describes the linear, reversible, time-dependent response of a crystal containing point defects to stress or electric fields, respectively known as anelastic and dielectric relaxation. Such relaxation occurs because of the redistribution of the defects among sites which are initially equivalent, but which becomes inequivalent in the presence of the external field. The macroscopic behaviour of such a crystal is found to be describable in terms of the symmetry which can be assigned to the defect. This defect symmetry determines whether or not the crystal will undergo dielectric or anelastic relaxation and, if relaxation can occur, which specific coefficients of elastic compliance or electric susceptibility show the relaxation effect. The latter information, called the ‘selection rules’ tells, in effect, which combination of stress or electric field components is capable of redistributing the defects. Tables are given for these selection rules for all possible defect symmetries in each of the 32 crystal classes. It is also shown that a hitherto unobserved phenomenon of piezoelectric relaxation may occur; the selection rules for this effect are also given.

Aside from its symmetry, the defect can be described as an electric dipole in terms of a suitable dipole moment vector μ, and as an ‘elastic dipole’ in terms of a tensor λ. It is shown that the defect symmetry determines the number of independent components of μ and λ. Finally, a thermodynamic theory is developed which permits calculation of the relaxation strengths for those compliance, susceptibility, and piezoelectric coefficients which undergo relaxation, in terms of the independent components of μ and λ. Applications of the theory to specific cases are then reviewed.     

Saddle-point energies and Monte Carlo simulation of the long-range order relaxation in CoPt

We present atomic-scale computer simulations in equiatomic L1₀-CoPt where Molecular Dynamics and Monte Carlo techniques have both been applied to study the vacancy-atom exchange and kinetics relaxation. The atomic potential is determined using a Tight-Binding formalism within the Second-Moment Approximation. It is used to evaluate the different saddle-point energies involved in a vacancy-atom exchange between nearest-neighbour sites. The potential and the saddle-point energies have been used to simulate the relaxation of the long-range order in CoPt using a Monte Carlo technique. A vacancy migration energy of 0.73±0.15 eV and an order-disorder transition temperature of 935 K have been found.     

Mechanisms of bulk diffusion at “high” and “low” temperatures

A phenomenological model has been proposed for bulk self-diffusion and diffusion of interstitial atoms in the ranges of high (T > T _D) and low (T < T _D) temperatures (where T _D is Debye temperature). It has been shown that the mechanisms of diffusion at high and low temperatures differ significantly. In the high-temperature range, the diffusion is provided by fluctuations, which can be described in terms of local melting, i.e., the formation of a “liquid diffusion channel.” In the low-temperature range, when melting for some reasons is hindered, the diffusion is due to the fluctuation formation of a “hollow diffusion channel.” The calculation of the activation energies of these processes in the case of self-diffusion agrees well with the experiment in the temperature range T > T _D and has demonstrated that the activation energy increases significantly at T < T _D. The calculation of the activation energy for diffusion of interstitial atoms in bcc metals agrees well with the experiment in the entire temperature range and provides an explanation of the decrease in the activation energy of diffusion at low temperatures.     

Reaction rate theory for supramolecular kinetics: application to protein aggregation

ABSTRACT

Probing reaction mechanisms of supramolecular processes in soft and biological matter, such as protein aggregation, is inherently challenging. This is because these processes involve multiple molecular mechanisms that are associated with the rearrangement of large numbers of weak bonds, resulting in complex free energy landscapes with many kinetic barriers. Reaction rate measurements at different temperatures can offer unprecedented insights into the underlying molecular mechanisms. However, to be able to interpret such measurements, a key challenge is to establish which properties of the complex free energy landscapes are probed by the reaction rate. Here, we present a reaction rate theory for supramolecular kinetics based on Kramers theory of diffusive reactions over multiple kinetic barriers. We find that reaction rates for protein aggregation are of the Arrhenius–Eyring type and that the associated activation energies probe only one relevant barrier along the respective free energy landscapes. We apply this advancement to interpret, in experiments and in coarse-grained computer simulations, reaction rates of amyloid aggregation in terms of molecular mechanisms and associated thermodynamic signatures. These results suggest a practical extension of the concept of rate-determining steps for complex supramolecular processes and establish a general platform for probing the underlying energy landscape using kinetic measurements.     

Thermal defect formation in undoped and Cr-and Mn-doped Bi12SiO20 crystals

The thermally stimulated depolarization currents and the temperature dependences of optical absorption are investigated. Interaction processes between defects are observed. A model consisting of association-dissociation of quasidipoles in the form of donor-acceptor pairs with a distribution of activation energies and relaxation times is proposed. Fiz. Tverd. Tela (St. Petersburg) 41, 1593–1596 (September 1999)     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Self-diffusion of aluminium in the intermetallic compound Fe-48 at.% Al

The self-diffusion coefficient of Al in the B2-type intermetallic compound Fe-48 at.% Al has been determined using the intrinsic diffusion coefficients of Fe and Al and the self-diffusion coefficient of Fe with the help of the Darken-Manning relation. The self-diffusion coefficient of Al in Fe-48 at.% Al is estimated to be a factor of about 0.6 smaller than that of Fe, and the activation energy for the self-diffusion of Al is obtained to be 280 kJ mol^?1 which is a little larger than the value of 262 kJ mol^?1 for the self-diffusion of Fe, indicating that the diffusion mechanisms for both components are nearly equal.     

A simple model for scalar relativistic corrections to molecular total atomisation energies

ABSTRACT

Scalar relativistic corrections to atomisation energies of first- and second-row molecules can be rationalised in terms of a simple additive model, linear in changes in atomic s populations. In a sample of 200 first-and second-row molecules, such a model can account for over 98% of the variance (99% for the first-row subset). The remaining error can be halved again by adding a term involving the change in atomic p populations: those coefficients need not be fitted but can be fixed from atomic electron affinity calculations. This model allows a fairly accurate a priori estimate for the importance of scalar relativistic corrections on a reaction energy, at essentially zero computational cost. While this is not a substitute for explicit calculation of Douglas–Kroll–Hess (DKH) or exact two-component (X2C) relativistic corrections, the model offers an interpretative tool for the chemical analysis of scalar relativistic contributions to reaction energies.     

Self-diffusion dynamic behavior of atomic clusters on Re(0 0 0 1) surface

Using molecular dynamics simulations and a modified analytic embedded atom potential, the self-diffusion dynamics of rhenium atomic clusters up to seven atoms on Re(0 0 0 1) surface have been studied in the temperature ranges from 600 K to 1900 K. The simulation time varies from 20 ns to 200 ns according to the cluster sizes and the temperature. The heptamer and trimer are more stable comparing to other neighboring non-compact clusters. The diffusion coefficients of clusters are derived from the mean square displacement of cluster's mass-center, and diffusion prefactors D₀ and activation energies E_a are derived from the Arrhenius relation. It is found that the Arrhenius relation of the adatom can be divided into two parts at different temperature range. The activation energy of clusters increases with the increasing of the atom number in clusters. The prefactor of the heptamer is 2-3 orders of magnitude higher than a usual prefactor because of a large number of nonequivalent diffusion processes. The trimer and heptamer are the nuclei at different temperature range according to the nucleation theory.     

Application of random walk concept to the cyclic diffusion mechanisms for self-diffusion in intermetallic compounds

Huntington–Elcock–McCombie (HEM) mechanism involving six consecutive and correlated jumps, a triple-defect mechanism (TDM) involving three correlated jumps and an anti-structure bridge (ASB) mechanism invoking the migration of an anti-structure atom are the three mechanisms currently in vogue to explain the self- and solute diffusion in intermetallic compounds. Among them, HEM and TDM are cyclic in nature. The HEM and TDM constitute the theme of the present article. The concept of random walk is applied to them and appropriate expressions for the diffusion coefficient are derived. These equations are then employed to estimate activation energies for self-diffusion via HEM and TDM processes and compared with the available experimental data on activation energy for self-diffusion in intermetallic compounds. The resulting activation energies do not favour HEM and TDM for the self-diffusion in intermetallic compounds. A comparison of the sum of experimentally determined activation energies for vacancy formation and migration with the activation energies for self-diffusion determined from radioactive tracer method favours the conventional monovacancy-mediated process for self-diffusion in intermetallic compounds.     

First-principles investigations of ordering in binary α-Ti solid solutions

The ordering tendency in binary f-Ti solid solution containing 3sp or 4sp simple-metal (SM) or 3d transition-metal (TM) solute is investigated systematically by the linear muffin-tin orbital (LMTO) method within the atomic sphere approximation (ASA). We demonstrated that the effective pairwise interaction (EPI) energy in a solid solution is equal to half the solute-solute interaction energies and can be evaluated by a supercell total energy approach. The calculations of EPI energy both with and without volume relaxation of the supercells and local density of states (LDOS) show that the EPI energies of Ti-SM and Ti-TM solutions are dominated by different factors. For Ti-SM solutions, the EPI energies are of large absolute values with a negative sign, indicating strong ordering tendency in these solutions. The volume relaxation does not alter the EPI energy substantially. The calculated LDOS shows that the ordering tendency in Ti-SM solutions may be related to the hybridization between the electrons of the SM atoms when they are close to each other. For most Ti-TM solutions, if calculated without relaxation, the absolute EPI energies are very small; however, if calculated with relaxation, they are of relatively large positive values, indicating a clustering tendency in these solutions. By combining the calculated EPI energy and Flinn's model for short-range order (SRO) strengthening, the increase in critical shear stress sro due to SRO is estimated for Ti-SM alloys, and the results qualitatively agree with experiment.