Ryoichi Kikuchi

Kinetics of order-disorder transformations in alloys☆

Abstract Relaxation of order (both long-range and short-range) in b.c.c. binary alloys is theoretically worked out based on the vacancy mechanism of atomic migration, using the path-probability method of time-dependent cooperative phenomena and choosing the pair approximation. With this approximation, the system reduces to the equilibrium state specified by the Bethe approximation as the time dependence ceases. The relaxation times τ k for order parameters are calculated as functions of temperature. Necessity of more than one τ k is demonstrated. One of the τs shows the critical slowing down, and all of the τs increase sharply at low temperatures. Using τ k , temperature dependence of thermodynamic properties are calculated as the system is first cooled down to the freezing region, and is then heated with a constant rate of temperature change in both ways. Theoretical results show hysteresis. It is noteworthy that a second peak of specific heat appears in addition to the peak at the order-disorder transition point. The second peak is the consequence of “melting” of the “frozen in” state. The results are compared with experiments for FeCo alloys. The limitation of the “corresponding state” concept, often used in the discussion of phase transitions as well as in glass transitions, is shown by calculating the isothermal annealing behavior of the “frozen in” state at the corresponding (fictive) temperature. Since the glass transition is a “frozen in” phenomenon, the theory presented in the paper is also to be used in interpreting some experimental vitrification processes.

Kinetics of order–disorder transformations in alloys. III

As a continuation of previously published work, isothermal ordering and disordering processes (of B2 type structure) in bcc binary alloys from initial states both near and far from equilibrium are investigated theoretically. The vacancy mechanism for atomic migration is assumed, interactions among nearest neighbors are taken into account, and the pair approximation of the path probability method of irreversible, cooperative phenomena is utilized. This leads to a set of nonlinear rate equations derived in a systematic, statistical fashion which describes the dynamical behavior of the system. The rate equations can be linearized near equilibrium and are shown to obey in this form the formalism of irreversible thermodynamics of Onsager. The driving forces are identified as the partial derivatives of the free energy function of the system with respect to state variables. Several relaxation times are obtained which characterize near equilibrium relaxation, and their meanings are discussed. Nonlinear coupling o...

Correlation factor in tracer diffusion for high tracer concentrations

Abstract The meaning of cross terms in the Onsager equation for diffusion in multicomponent systems is clarified based on the Path Probability method of irreversible statistical mechanics. The problems involved in increasing the concentration of tracer atoms in tracer diffusion experiments are discussed. Some comments on the correlation function approach with respect to the present problem are also added.

Soret effect in solids

Abstract Thermal diffusion in solids is studied for a one-component system by the path probability method of irreversible statistical mechanics based on the vacancy mechanism from the atomistic point of view. The treatment allows a microscopic description of so far controversial concepts such as the heat of transport and leads to analytical expressions of the Soret effect. Concentration gradient or thermoelectric effect (in ionic conductors) created as a response to the temperature gradient was found to be proportional to the energy minus enthalpy of transport ( E ∗ − H ∗ ) rather than to the heat of transport Q ∗ (=E ∗ − μ , where μ is the chemical potential). The results are applied to thermotransport problems of interstitial solutes and thermoelectric effects in superionic conductors. The results depend sensitively on the detailed model of the assembly of particles. The application of the Monte Carlo simulation to the same model is discussed.

A theoretical model for mixed alkali effect in ionic conduction in β-aluminas and glasses

Abstract Ionic conductivity of a binary system consisting of two kinds of monovalent ions in the two dimensional honeycomb lattice has been calculated by the Path Probability Method of irreversible statistical mechanics in order to gain insight into the probable cause of the mixed alkali effect in glasses and β-aluminas. It is shown that, if the interaction between the two components is attractive, one can expect the existence of a sharp minimum in the conductivity at a certain composition. The minimum is mainly due to the percolation difficulty created by mutual arrangements of two kinds of ions. Relations between the percolation difficulty and the mixed alkali effect are discussed.

Statistical mechanics of thermal diffusion

Expressions of the flows of atoms A and B of a binary system in a crystal are derived as the response to the imposed gradients of temperature and chemical potentials. The formulation is done using the pair approximation of the Path Probability Method of irreversible statistical mechanics and atomic migration is assumed to be via the vacancy mechanism. The energy carried by photons (and electrons) under the temperature gradient is assumed to be independent of the atomic flux. For the case near equilibrium, linear relations are derived among the atomic fluxes, the energy flux (associated with atomic flux) and the gradients. The Onsager reciprocal relations are proved to hold among the coefficients, including those related to energy flows. The heat of transport (energy carried by a diffusing atom) and the heat conduction due to atomic flux are thus unambigously derived.

Convergence of the cluster-variation method for a system on a triangular lattice

The paper studies the convergence of the cluster-variation method to the rigorous result as the cluster size increases. The calculation is done on the phase boundary at T = 0 between the A2B-type ordered phase and the disordered phase on a two-dimensional triangular lattice with nearest-neighbor interaction. It is shown that the phase boundary at (T = 0) is obtained by maximizing the entropy under the constraint that only a limited number of atomic configurations are allowed. Formulations are developed for clusters of n = 3, 5, 7, 9,11, and 13 points. When thermodynamic quantities which are calculated using these clusters are plotted against 1/n, they approach the known rigorous (n = ∞) results more or less linearly but with a pseudo-period of δn = 6. An exception is the square of the long-range order, which bends down as 1/n tends to zero.

Percolation threshold for electronic conduction in β-alumina-type compounds

Abstract It is found that the addition of more than a certain amount of mixed valent ions would be necessary to render mixed conducting characteristics to β-alumina-type compounds. Based on the assumption that the electronic conduction is due to the hopping of small polarons to the nearest neighboring ions on the octahedral sites in the spinel block of β-alumina similar to that in magnetite, the pair approximation of the cluster variation method indicates that the threshold composition of such a mixed valent additive must exceed at least 0.24 of the number of octahedral ions. A new method of calculating the percolation limit is shown.

Percolation threshold for electronic conduction in β-alumina type compounds: II

Abstract The possibility of making β-alumina compounds mixed conducting by the addition of a certain amount of mixed valent ions has been examined from a theoretical point of view using the cluster variation method (CVM). Electronic conductivity in these substances is assumed to be due to the hopping of small polarons to the nearest neighboring ions on the octahedral sites in the spinel block of β-alumina in a manner similar to that in magnetite. The pair approximation of the cluster variation method indicates the composition of the mixed valent ions necessary to induce the electronic conductivity should be at least 0.24 of the number of octahedral sites. In this paper, we report the results of a higher order approximation of the CVM, namely the triangle+tetrahedron approximation. Our results indicate that a more accurate calculation increases this number to 0.396 thereby indicating that an extremely high level of doping would be necessary to render these compounds mixed conducting.