A fluctuation theory of normal crystal growth for the very dilute binary metallic solid solution's crystallization in the case of macroscopic systems

A theory connected with spontaneous fluctuations of the limited spectrum solid state particle concentrations in the case of binary metallic macrosystems crystallizing from the binary melts is developed. Normal kinetics was found to be realized in the case of very dilute binary metallic solid solutions within the small ‘supercoolings’ region. A comparison between the theoretical and experimental data on the crystallization kinetics of very dilute binary metallic solid state solutions within the small ‘supercoolings’ region has been made. For some types of binary metallic macrosystems the quantitative restrictions of the supercooling regions within which a fluctuation mechanism and normal kinetics are assumed to act have been established. The restrictions are conditioned by some parameters of the binary macroscopic melt and two-phase transitional zone considered.     

A fluctuation theory of normal crystal growth in the case of one-component metallic macrosystems

A Model of spontaneous fluctuations of the limited spectrum solid state particle concentrations in the case of one-component metallic macrosystems is considered. Within the small supercoolings' region of the melt-crystal macrosystem a normal law of crystallization is established due to the action of fluctuation mechanism. For some metallic macrosystems a comparison on the crystallization kinetics between the theory and experiments has been realized. Some quantitative restrictions of the melt-crystal system supercoolings' region within which the mechanism of spontaneous fluctuations is expected to be realized are established for a collection of one component metallic macrosystems. Two-phase transitional zone length's influence on the values of admitted supercoolings in presented.     

A fluctuation mechanism of crystallization processes for the one-component metallic melts

In given work a mechanism of spontaneous fluctuations of the „unlimited”︁ spectrum solid state particle concentrations acting in some monolayers of two-phase transitional zone within the small supercoolings' region is considered. This mechanism is applied for some one-component metallic macrosystems crystallizing from the supercooled melts. It is established that given mechanism in the case of macrosystems transforms into action of only one spontaneous fluctuation occuring in each monolayer of two-phase transitional region as a result of which the solid state particle concentration becomes equal to nearly 1/2. It is shown that in this case within the small supercoolings' region given mechanism leads to so-called normal law of crystallization connected with one-component metallic macrosystems. In the particular scheme of mechanism presented an effective kinetic coefficient is estimated. In the vicinity of melting temperature for some metallic melts a comparison of theoretical and experimental data concerning the mean crystallization velocities has been fulfilled. In the case of spontaneous fluctuations of the „unlimited”︁ spectrum solid state particle concentrations two-phase transitional zone's sizes do not have influence on the one-component metallic melts' crystallization kinetics within the small supercoolings' region.     

Thermodynamic properties of strontium tungstate crystal growth from sodium tungstate melts

Energy (E), enthalpy (ΔH_a), entropy (ΔS_a) and free-energy (ΔG_a) of activation and the pre-exponential factor (k₀) of the ordinary Arrhenius equation k_Dl = k₀e–E/RT were estimated for diffusion-controlled crystal growth of SrWO₄ from Na₂WO₄ melts. E increased slightly with increased cooling rates (R_T). k₀ was parallel to k_Dl and increased with increasing R_T. ΔH_a, ΔS_a, and ΔG_a did not change with the changes in R_T and crystallization temperature (T₀). The distance (d₁₂), between a diffusing particle and its host necessary for a successful diffusion, was estimated. Such distances slightly increased with T₀ and R_T.     

Viscous behaviour of undercooled metallic melts

Viscosity data for non-crystalline Au_0.77Ge_0.136Si_0.094 alloys in the region of the glass transition temperature and above the melting point, are fitted to a single expression of the type proposed by Doolittle. This expression also leads to reasonable values for the activation energy for hole formation in the liquid and for the liquid free volume at the glass transition temperature. A similar procedure was applied to the viscosity data of a Pd_0.775Cu_0.06Si_0.165 non-crystalline alloy. By assuming values for the liquid free volume at both the glass transition temperature and the melting point, the analysis is also extended to Pd_0.82Si_0.18 alloys for which no viscosity data are available. Here, time-temperature-transformation (T-T-T) curves for crystallization are calculated for each of the three alloys, and used to determine the critical cooling rates for the formation of an amorphous solid.     

Induction periods and mechanism and kinetics of nucleation of barium tungstate crystal growth from sodium tungstate melts in platinum crucibles

Induction periods (t) and mechanism and kinetics of nucleation in barium tungstate crystallization from sodium tungstate melts in platinum crucibles were studied. A theoretical relation has been developed to express the dependence of t on the cooling rates (R_T) and the rate (R_c) of development of excess solute concentration. At any crystallization temperature the average rate-constant (k_n) for heterogeneous nucleation was related to R_c by (k_n/p)^1/(p+1) = 1/(tR_c^γ), where γ is a constant and p is the average number of particles in the critical nuclei. The critical temperature (T), critical supersaturation (S) and γ values were estimated.     

Fluctuation theory of the one-component and binary metallic melt's crystallization in the case of micro- and macro-systems

All types of fluctuation mechanisms resulting in a normal kinetics of the one-component and binary melt's crystallization within a small supercooling region for micro- and macro-systems are systematized. Three principal mechanisms of spontaneous fluctuations acting in the monolayers of two-phase transitional zone in the vicinity of melting temperature for one-component metallic crystals and of the liquidus temperature for binary ones are described. The formulas of kinetic coefficients figuring in the normal law of crystallization conformably to one-component and binary metallic micro- and macrosystems for all types of fluctuation mechanisms are presented. Some peculiarities of each fluctuation mechanism are presented. Some peculiarities of each fluctuation mechanism acting within a small supercooling region of the melt-crystal micro- and macrosystems are pointed out. A comparison of theoretical and experimental data concerning the one-component and binary melt's crystallization kinetics at some small supercoolings is presented.     

Dynamic stability analysis of the solidification of binary melts in the presence of a mushy region: changeover of instability

A dynamic linear instability analysis of steady-state binary melt solidification with a mushy region has been carried out. Such an instability differs from a conventional morphological one of a planar solid–liquid interface and is connected with the perturbations in the steady-state solidification velocity. Solidification with a narrow mushy region has been revealed to be absolutely unstable with a monotonous instability of a “hard” type. An increase of the mushy region width leads to an instability changeover from the “hard” type to the oscillatory “soft” one. Both the critical changeover width and the neutral stability curves have been determined as the functions of relevant physical and operating parameters. The steady-state solidification regime with a broad mushy region is absolutely stable. Thus, the mushy region width has been shown to represent a stabilizing factor.     

Periodic regrowth during crystal growth

Periodic remelting (or etching) and regrowth during crystal growth can significantly improve the quality of the resultant crystals. The periodic remelting and regrowth must be optimized both in amplitude and period, and approximately two-thirds to three-quarters of the growth during each cycle is remelted in our experiments. Significant improvement in crystal perfection can be achieved without changing the net growth rate. The mechanisms which produce the improvement are discussed.     

In-situ observation of crystal growth and the mechanism

The spatial and time resolution in the measurements of growth rates and the observation of surface morphologies and the associated transport phenomena reflecting their growth mechanism have been developed because advanced microscopes and interferometers have attained nano-scale resolution. The first part covers the historical background how in-situ observation of crystal growth at molecular-level by optical and other scanning methods had been developed for understanding of crystal growth by measuring crystal growth rates and by observing surface nano-topographies, such as growth steps and spiral hillocks, with the same vertical resolutions comparable to that of the scanning probe microscopic techniques. The potential of recently developed interferometric techniques, such as Phase-Shift Interferometry (PSI) is then reviewed with the principle of the optics. Capability of measuring growth rates of crystals as low as 10⁻⁵ nm/s (1 µm/year) is introduced. Second part of the article emphasizes basic interferometric technique for the understanding of crystal growth mechanism by measuring growth rate vs supersaturation. Utilization of these techniques not only in fundamental crystal growth fields but also in environmental sciences, space sciences and crystallization in microgravity would briefly be introduced. At the end, we select a few examples how growth mechanism was analyzed based on these kinetic measurements.     

Development of dendrite array growth during alternately changing solidification condition

To investigate competitive growth in a dendrite array a directional solidification study was carried out on a succinonitrile–acetone alloy. In the experiment the temperature gradient G applied is alternately altered over a wide range, increased from a lower to a higher limit, and then decreased back to the lower one. It was found that source dendrites, i.e. parent dendrites, are not superior to derived dendrites in terms of growth competition. The orientation deviation between neighbouring dendrites impacts both local dendrite creation and elimination. At lower gradients, the new dendrites grow out from the ternary arms, while at the higher gradients new dendrites originate directly from the secondary arms. During increasing and decreasing G, average primary dendrite spacing λ measured traces out different paths, revealing an unclosed hysteresis loop in the λ–G-diagram.     

On the theory of normal growth of crystals in binary systems

Crystallization processes of binary systems are considered, in which solid particles consisting of components A and B grow from the melt in monolayers of a two-phase boundary zone by a mechanism of spontaneous thermal equilibrium type concentration fluctuations. The calculation is based on probability distribution functions. The region of small supercoolings of the binary system melt-crystal proves most valid for this investigation. The change of the mean crystallization rate of the two-phase boundary zone with the supercooling of the melt shows a linear behaviour. The dependence of the kinetic coefficient on the degree of the atomic roughness of the phase boundary with different numbers of monolayers in the phase boundary, the total number of particles in each monolayer, and a certain roughness are connected with the energy of positional disorder in binary systems.     

Features of bicrystal growth during the directional crystallization of metal melts

The factors responsible for the formation of different configurations of boundaries between adjacent crystallites during their growth from melt by Bridgman and Czochralski methods have been considered by an of example Fe–20 wt % Ga alloy and Ni bicrystals. It is found that the configuration of intercrystallite boundary is related to the features of crystallite growth, caused by the strained state of intercrystallite and interphase (crystal–melt) boundaries, the difference in the linear thermal expansion coefficients of the crystallite boundaries and bulk, and the shape (geometry) of the bicrystal cross section. It is suggested that the strained state of boundaries and the formation of substructure in crystallites during directional crystallization from metal melt are significantly affected by their deformation under the melt weight.

     

The order-disorder transformation at supercooled melt/crystal transition region of binary melts (I) the master equation

Binary melts crystallizing with almost perfectly ordered structures if exposed to conditions close to thermodynamic equilibrium exhibit an increasing tendency to form metastable crystals with increasing disorder when these alloys crystallize on the growth conditions of supercooling. Based on the model of a two-phase transition zone separating a growing crystal from its non-solid surrounding a kinetic master equation is formulated describing the kinetics of competitive atomic exchange processes within the transition region. The theory considers the case of simple cubic structure with equal particle numbers of two components that form a NaCl-type lattice at the state of perfect order. In a subsequent paper solutions of the master equation obtained for steady-state conditions are discussed.     

The nature of phase separation in binary oxide melts and glasses. II. Selective solution mechanism

A comprehensive review of structural data in binary silicate systems indicates that the tetrahedral critical radius (87.2 pm) of binary silicate melts (or glasses) is associated with the silicon tetrahedral network that defines the structure of the melt. In a binary system, most of the cages present in the melt are made of six and five-membered rings of silicon tetrahedra. Cages bounded by six or more-membered rings can host cations of all sizes. However, cations that enter in cages made of five-membered rings are discriminated by their ionic radius. Cations with ionic radii larger than about 87.2 pm (network modifiers) cannot enter in pentagonal apertures; cations with radii smaller than 87.2 pm (amphoteric cations) can. Cages bounded by pentagonal rings play a key role in phase separation by selecting which cations can fit in them, adopt a four-fold coordination, and reduce the size of miscibility gaps, i.e. the cages permit explaining why some cations are amphoteric. This result is important because it shows that a structural control is exerted by the solvent (here SiO₂) upon immiscibility which creates a selective solution mechanism that affects small (<87.2 pm) cations in binary silica-rich melts.     

The mechanism of crystal growth in glass forming systems

A critical survey on experimental results on the mode of growth in simple glass forming melts is given, attention being mainly concentrated to data obtained at small undercoolings. Dissolution rates, change of interfacial conditions at constant undercooling as well as detailed structural determinations are considered as experimental evidences, complementary to a thorough analysis of growth-temperature dependences. For network glass formers (SiO₂, GeO₂, P₂O₅, Na₂B₄O₇) with melt structures, similar to those of the corresponding crystals, the normal mode of growth is typical. For a number of simple glass forming substances in which the crystallization is connected with a process of molecular reconstruction (NaPO₃, LiPO₃), spiral growth could be proved. Dislocation-free crystals of high entropy of melting glass forming substances (Na₂S₂O₃ · 5 H₂O, thymol) are obtained after prolonged annealing and growth in thin bored capillaries. Two-dimensional growth is verified for the resulting perfect crystals.     

The order-disorder transformation at supercooled melt/crystal transition regions of binary melts (II). The steady-state solution

In a preceding paper a kinetic master equation has been derived describing the kinetics of atomic exchange processes that occur within the transition region separating a growing binary crystal and its nonsolid surrounding. For simple cubic structures, with the lattice sites being occupied by equal particle numbers of two components (in the case of perfect order corresponding to an NaCl-type lattice) solutions for the steady-state conditions have been derived. According to the solutions the long-range order parameter is related to the atomic interaction energies, the arrival rate of particles at the crystal surface from the melt, and to temperature. A critical supercooling is predicted for the transition from a partly crdered structure to a disordered structure of the crystals growing from the melt. At the critical supercooling the temperature dependence of the crystallization rate reveals a characteristic change.