Dynamics of solidification in binary melts during rapid cooling I. The basic equations

In this paper we deal with the analysis of the general complex model which describes the solidification of the binary melt. Within the framework of this model we consider the mass and energy transport in the system and the kinetics of the phase transformation. The paper is divided into two parts. In the first part the set of equations is derived, which describe the evolution of the system within the framework of the theory of stochastic processes. The emphasis is laid on the rapid changes of temperature in the system. In the second part the analysis of the mentioned set of equations is presented. It results in the delimitation of the conditions under which the adiabatic approximation and the kinetic phase diagrams at the phase interface can be applied.     

On the Kinetic Description of Condensation in Binary Vapours

Based on a thermodynamic analysis and an earlier developed general growth equation for clusters of a new phase, the kinetics of growth of droplets in a binary gaseous mixture under isothermal and isobaric conditions is described. Differential equations for the time development of the mean radius of the droplets, the number of droplets, and the overall mass concentrated in the droplets are obtained. These equations describe the evolution of the system of droplets beginning after the nucleation period has finished. The equations can be easily solved numerically. For long times analytic solutions are derived. It is shown that the growth of droplets proceeds accordingly to the mechanism of Ostwald ripening.     

Stochastic theories and scaling relations for early-stage surface growth

Recent theories for the surface growth are reviewed, which describe the nucleation kinetics of binary alloys in the first layer and the nucleation on top of islands in the second layer. In the first part, describing the submonolayer regime, scaling relations for the number density of stable islands in dependence of the growth parameters are derived from an extension of the classical rate equation approach to binary systems. In the second part a general stochastic theory is presented that allows one to predict whether thin films become smooth or rough under non-equilibrium growth conditions.     

Adiabatic and isothermal elastic constants for highly anharmonic crystals

The mass operator of the phonon Green function for a highly anharmonic crystal in the ladder approximation is derived. In order to calculate the mass operator in the collision-dominated regime it is necessary to solve the Boltzmann-like equations. These solutions allow us to find the thermal conductivity coefficient and to prove that the thermodynamic relation between adiabatic and isothermal elastic constants is satisfied.     

Analysis of Adiabatic Approximation Using Stable Hamiltonian Method

In this paper, we deal with the adiabatic approximation of general Hamiltonians by splitting it into two parts, with one part a Hamiltonian that has at least one time-independent eigenstate up to a phase factor. We first develop the method of finding this kind of Hamiltonians. Then the relationship between adiabatic approximation and these Hamiltonians is discussed. Applying this to a general case, we give both a necessary condition and a sufficient condition for adiabatic approximation, followed by a spin-half example to illustrate.     

THERMODYNAMICS OF EQUILIBRIUM AND STABILITY

The dependence of the entropy of a homogeneous system on the composition is investigated with the help of a reversible adiabatic process which allows the change of composition by means of a semipermeable wall. The conditions of equilibrinm for phase transition and for homogeneous chemical reaction are derived in a new way. Next the criterion of minimum energy for constant entropy and volume is derived from the principle of increase of entropy. This criterion is then applied to obtain the conditions of equilibrium and stability with the help of Lagrange's multipliers. The conditions of stability are expressed in several alternative forms. Next the equilibrium properties of a binary system arc considered, and some types of phase diagram are explained by means of equations. The theory is extended to the general heterogeneous equilibrium of a system consisting of any number of independent components. A system of equations for the change of temperature, pressure, and composition are obtained and are solved by means of determinants. Next Planck's theory of a binary solution is extended to a solution consisting of several solnte components, with the same conclusion regarding the lowering of freezing point as for a binary solution. Finally Planck's theory on the number of coexisting phases for aone-component system is extended to a system consisting of k components with the result that a state with, σ coexisting phases is more stable than one with σ-1 phases: where σ is an integer not greater than k + 2.     

Breakdown of classical nucleation theory near isostructural phase transitions

We report simulations of crystal nucleation in binary mixtures of hard spherical colloids with a size ratio of 1:10. The stable crystal phase of this system can be either dense or expanded. We find that, in the vicinity of the solid-solid critical point where the crystallites are highly compressible, small crystal nuclei are less dense than large nuclei. This phenomenon cannot be accounted for by either classical nucleation theory or by the Gibbsian droplet model. We argue that the observed behavior is due to the surface stress of the crystal nuclei. The observed effect highlights a general deficiency of the most frequently used thermodynamic theories for crystal nucleation. Surface stress should lead to an experimentally observable expansion of crystal nuclei of colloids with short-ranged attraction and of globular proteins.     

Photon statistics of the optical parametric oscillator including the threshold region

Starting from an effective Hamiltonian the derivation of a set of classical Langevin equations for the amplitudes of signal, idler, and pump is briefly reconsidered. From these equations all variables except those describing the signal mode are eliminated with the help of an adiabatic approximation and certain others, which are valid in the threshold region and somewhat above (i.e. photonumbers ? 10¹⁴). The signal mode amplitude then satisfies a van der Pol equation in the rotating wave approximation and is driven by a fluctuating force. With the exception of a slight difference due to the undamped phase diffusion of the pumping laser, the same Langevin equation has been derived earlier for the amplitude of a laser mode near threshold. We present the stochastically equivalent Fokker-Planck equation, whose solution is reduced to the known solution of the laser Fokker-Planck equation. Thus the complete photon statistics of the signal mode is revealed at once. In particular we obtain the stationary distribution and the amplitude and intensity correlation functions as well as the transient solution.     

光学双稳性的强迫振动模型

本文提出一个新的光学双稳性模型.基于平均场近似,慢变振幅近似和绝热近似的思想,把光学双稳系统看成一个“黑箱”,类比非线性振动理论,指出不同的光学双稳过程(包括不同的工作物质和不同的腔(F-P腔或环腔)),能用一个恰当的强迫振动方程统一地描述.方程中所包含的都是可测量的宏观参量,使得可能直接用实验拟合,与理论结果进行定量对照.用振动理论的方法,把方程演变成自治方程组,从而方便地得到光学双稳性稳态和动态解的一般形式.并用若干已报道的具体例子验证了这个模型的普适性.     

An evolution of adiabatic matter: a case for the quasistatic regime

We establish the connection between the standard ADM 3+1 treatment of matter with its characteristic equivalent, in the context of spherical symmetry. The flux-conservative rendition of the fluid equations are obtained. Considering adiabatic distributions of perfect fluid, we evolve the system using the so-called post-quasi-static approximation in radiation coordinates. We obtain an adiabatic matter evolution in the quasi-static regime or slow motion, which is not shear-free nor geodesic.     

Diffusion-stress coupling in liquid phase during rapid solidification of binary mixtures

An analytical model has been developed to describe the diffusion-viscous stress coupling in the liquid phase during rapid solidification of binary mixtures. The model starts with a set of evolution equations for diffusion flux and viscous pressure tensor, based on extended irreversible thermodynamics. It has been demonstrated that the diffusion-stress coupling leads to non-Fickian diffusion effects in the liquid phase. With only diffusive dynamics, the model results in the nonlocal diffusion equations of parabolic type, which imply the transition to complete solute trapping only asymptotically at an infinite interface velocity. With the wavelike dynamics, the model leads to the nonlocal diffusion equations of hyperbolic type and describes the transition to complete solute trapping and diffusionless solidification at a finite interface velocity in accordance with experimental data and molecular dynamic simulation.     

Synchronisationseffekte in einer linearen Anordnung aus N Josephson-Verbindungen

Synchronization Effects in a Linear Array of N Josephson Junctions Within the frame of the RSJ model we investigate the synchronization of the oscillations in a linear array of N identical Josephson junctions shunted by an electromagnetic resonator. Using an adiabatic approximation to the first order of the parameter I_cI the reduced equations of the slowly varying phases are derived. These equations allow the detailed investigation of all the stationary states of the system. Only the coherent state in the inductive regime and the radiationless state in the capacitive regime are found to be stable. Including noise effects we discuss the order parameter concept for the resonator current in the case N ? 1.     

Temperature dependence of the field induced magnetization reorientation in dzialoshinsky-moriya type weak ferromagnets

Using a Hamiltonian consisting of isotropic, anisotropic, and antisymmetric exchange terms plus a Zeeman term, the temperature dependence of the weak ferromagnetic-spin-flop phase transition is calculated in the molecuar field approximation. For all temperatures less than the tricritical one, the phase transition is of first order. The stability limits of the two phases and the thermodynamic phase boundary are calculated for this temperature region. For temperatures greater than the tricritical one, the second order transition boundary between the phases is calculated. A set of equations precisely defining the tricritical point in the field-temperature plane is derived and results are presented for the location of this point as a function of the parameters of the system. The methods developed in this work are general and can be applied directly to similar problems.     

Equilibrium state of charged particles in a wave propagating along the magnetic field

This investigation examines the adiabatic motion of a charged particle near the equilibrium state in a field of a plane, circularly polarized, electromagnetic wave which is propagating with a changing velocity phase along the magnetic field. Approximate equations are found which describe the behavior of the equilibrium state parameters when the wave leaves the medium and enters a vacuum. It is shown that compared to the equilibrium value in this situation under the adiabatic approximation there is a decrease in amplitude of the particle energy fluctuation; this establishes the possibility of a prolonged acceleration of the particle to high energies. It is further demonstrated that a particle moving close to equilibrium state can appear to be in the autoresonance regime when the wave enters vacuum.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 117–122, October, 1976.     

Differential cross sections for chemically reactive systems

A quantum mechanical theory is developed for the differential and total elastic and reactive scattering cross sections for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line, but with the whole system free to rotate in three dimensions. The introduction of a set of natural collision coordinates, together with a vibrationally adiabatic approximation, is used to reduce the scattering problem to the solution of one-dimensional Schrödinger equations. Semi-classical expressions for the elastic and reactive phase shifts are derived. The partial wave summations that occur in the theory are evaluated by semi-classical techniques and elastic and reactive differential cross sections are calculated for three different kinds of potential surface. A feature of the calculations is the appearance of a new kind of rainbow, which is named a ‘cubic’ rainbow since it arises when the deflection function varies cubically with impact parameter. The classical and semi-classical theory of cubic rainbows is developed.     

Dynamical fluctuations in classical adiabatic processes: General description and their implications

Dynamical fluctuations in classical adiabatic processes are not considered by the conventional classical adiabatic theorem. In this work a general result is derived to describe the intrinsic dynamical fluctuations in classical adiabatic processes. Interesting implications of our general result are discussed via two subtopics, namely, an intriguing adiabatic geometric phase in a dynamical model with an adiabatically moving fixed-point solution, and the possible “pollution” to Hannay’s angle or to other adiabatic phase objects for adiabatic processes involving non-fixed-point solutions.     

A stochastic model for solidification

A 3-dimensional (2-space, 1-time) model relating the diffusion of heat and mass to the kinetic processes at the solid-liquid interface, using a stochastic approach is presented in this paper. This paper is divided in two parts. In the first part the basic set of equations describing solidification alongwith their analysis and solution are given. The process of solidification has a stochastic character and depends on the net probability of transfer of atoms from liquid to the solid phase. This has been modeled by a Markov process in which knowledge of the parameters at the initial time only is needed to evaluate the time evolution of the system. Solidification process is expressed in terms of four coupled equations, namely, the diffusion equations for heat and mass, the equations for concentration of the solid phase and for rate of growth of the solid-liquid interface. The position of the solid-liquid interface is represented with the help of a delta function and it is defined as the surface at which latent heat is evolved. A numerical method is used to solve the equations appearing in the model. In the second part the results i.e. the time evolution of the solid-liquid interface shape and its concentration, rate of growth and temperature are given.     

The Stochastic Adiabatic Approximation of Multidimensional Non-Markovian Systems Driven by the Ornstein-Uhlenbeck Noises

A set of nonlinear stochastic differential equations (NSDE'S) that describes a large class of nonlinear multidimensional non-Markovian dynamical systems driven by the Ornstein- Uhlenbeck(O-U) noises is studied. By virtue of the stochastic generalization of usual adiabatic approximation, we obtain the equation for the order parameter. The statistical properties of the new stochastic variables occurred are studied. The effective Fokker-Planck equation (EFPE) corresponding to the equation for the order parameter is derived and the stationary solution of EFPE is calculated.