The role of stress development and relaxation on crystal growth in glass

The role of internal stresses energy is usually neglected when crystallization kinetics is considered. The common argument is that stress is relaxing too fast to affect the process. In this article we develop a generalized formalism to describe steady-state growth kinetics in viscoelastic media. The residual stress energy results from interplay between the rate of stress development (due to the propagation of a crystal throughout the matrix) and rate of stress dissipation (due to relaxation of the viscous matrix). The degree to which the stress energy can relax depends on the ratio of τ_c/τ_r the characteristic time for crystal growth, τ_c, and the relaxation time, τ_r. The present results challenge the widespread fallacy that internal stresses relax too fast to affect crystal growth. Our model explains the often observed lack of agreement between the theoretical predictions (without taking into account the stresses energy development) and experimental data.     

Fundamental aspects of VLS growth

The kinetics and the mechanism of the vapor-liquid-solid (VLS) growth are discussed. Emphasis is placed on the dependence of the growth rate on the whisker diameter. It is found that the rate decreases abruptly for submicron diameters and vanishes at some critical diameter d_c ? 0,1 μm according to the Gibbs-Thomson effect. A new method for simultaneous determination of kinetic coefficients and of supersaturations has been developed. The method can be used to measure the coefficients of some materials as well as the temperature dependence of the coefficient for silicon and the activation energy of the process. From the dependence of supersaturation on the diameter we conclude that whiskers grow by a poly-nuclear mechanism. The periodic instability of the diameter is discussed and the rate-determining step is analysed. We conclude that phenomena on the liquid-solid interface are the decisive ones. In determining the role of liquid phase in vapor growth we measured the “liquid phase effectivity coefficient” as a function of crystallization condition; the coefficient typically was about 10²?10³. It is stressed that the liquid phase reduces the activation energy both on vapor-liquid interfaces (for chemical reactions) and on liquid-solid interfaces (for nucleation). The liquid phase ensures growth rates as high as 1 cm/sec, provided there are no barriers between the interfaces. The growth mechanism of the side faces was studied, and it was observed that the faces grow mainly by a chain mechanism rather than by two-dimensional nucleation. In work on surface diffusion in the VLS whiskers growth by CVD, we found that the whiskers grow mainly by direct deposition rather than by diffusion on the side faces. It is concluded that the VLS mechanism is important also for the vapor growth of platelets, films, and bulk crystals.     

Recent research on pseudobiological hydroxyapatite crystal growth and phase transition mechanisms

We have studied the growth and phase transition mechanisms of hydroxyapatite (HAP) and its interaction with a growth factor protein in a simulated physiological environment. Using atomic force microscopy (AFM) and real-time phase shift interferometry, we performed in situ observations of growth in simulated human body fluid solutions seeded with millimeter-sized HAP single crystals produced by hydrothermal synthesis, and we measured the normal growth rate. The step kinetic coefficient (derived from the velocity of growth steps) and the edge free energy (calculated from the variation in the normal growth rate with the degree of supersaturation) both deviated greatly from the standard values for typical inorganic salt crystals and were found to be close to those of protein crystals. This suggests that the growth units of HAP crystals are clusters rather than simple ions and that growth proceeds through the accumulation of these clusters. Observations using dynamic light scattering confirmed the presence of clusters with a diameter of about 1 nm in simulated body fluids. Ab initio analysis of the cluster energy stability indicated that calcium phosphate clusters based on Ca₃(PO₄)₂ units achieve an energy minimum for clusters of the form [Ca₃(PO₄)₂]₃. These clusters have S₆ symmetry, and, when they are used to build a HAP crystal structure, their structure is likely to become slightly modified, resulting in the formation of C₃ structures.Since these clusters would also be the building blocks of amorphous calcium phosphate (ACP), they provide vital clues to the phase transition from ACP to HAP. Using time-resolved static light scattering, we tracked the ACP–HAP phase transition process and found that the degree of coarseness inside a cluster aggregate changes abruptly within a specific time interval and HAP is formed and deposited in the final stages. This suggests that an ACP aggregate changes into HAP as its internal structure becomes regularized. Using this phase transition process, we produced a complex of HAP and the FGF-2 growth factor protein. This complex releases the FGF-2 into a physiological saline solution over a period of at least a week, meaning that it can be used as a pharmacologically active third-generation biomaterial.     

Crystallization of silica and titanium oxide-silica corning glasses (codes 7940 and 7971)

The crystallization kinetics of silica glass (Corning 7940) and titanium oxide-silica glass (Corning 7971) was investigated. Special attention was paid to prevent the contamination of the surface of the samples during their heat treatment. It was found that the thickness of the crystallized layer depends linearly on the time of heat treatment for both types of the investigated glasses. The beginning of the crystallized-layer growth is preceded by the quite considerable induction period. The temperature dependences of the induction period and of the rate of crystal growth for silica glass coincide practically and are characterized by the activation energies E_τ ≈ E_u ≈ 160 kcal/mol. For titanium oxide-silica glass E_τ >E_u. The measured value of the activation energy of viscous flow differs considerably from the activation energy of the crystal growth rate. This may be evidence of the fact that the crystallization and shear viscosity depend on different kinetic units of silica glass.     

Strength of Crystals of Small Dimensions and their Triboluminescence

Stress-dependence of triboluminescence (TBL) in uranyl nitrate, tartaric acid and other nine crystals has been studied with the help of a new technique of crushing the crystals. As the appearance of TBL needs the creation of new surfaces in a crystal, the minimum stress at which TBL appears has been taken to be the fracture-strength of the crystal. The value of the fracture-strength and stress coefficient of binding energy determined from TBL measurements are found to be of correct order. It has been found that the rate of rise of TBL intensity with stress is higher for those crystals which have less value of the fracture-strength. The fracture strength (σ_f) is found to be higher for those crystals which have less value of stress coefficient of binding energy (β) and the product of σ_f and β is higher for those crystals which have higher melting point. It has been shown that the rate of rise of TBL intensity with stress decreases at higher values of stress due to strain hardening in the crystals.     

Analysis of ambient gas influence on silicon layers crystallized by means of LPE

Epitaxial growth of thin layers from the liquid phase can occur with the use of solutions saturated under different ambient gases. Most often this process takes place in a vacuum or gaseous atmosphere of hydrogen or argon. As the experimental data show, the morphology of crystallized layers is determined by the ambient type in which the process occurs.The cohesion energy responsible for epitaxial lateral deposition processes on the substrate surface depends on the surface free energy which is a measure of attraction of the solution atoms by substrate atoms. In the case of crystallization of an epitaxial lateral layer of Si on a substrate partially masked with dielectric, the chemical potentials of atoms in the neighboring phases (determining the interface evolution) are not without influence on the relaxation velocity of the saturated liquid phase, and on the horizontal and vertical growth rate.The aim of the investigation was to analyze experimentally the influence of the ambient gases used during the LPE growth on the cohesion of the Sn–Si solution with substrates applied for the lateral epitaxial growth of Si layers. This work presents comparative temperature analysis of the wetting angle of such surfaces as Si, SiO₂ and SiN_x by the Sn–Si solution.     

Capillary phenomena during solidification

Capillarity, broadly stated, is the collection of thermodynamic concepts which relate the state of a system to its size, per se. Usually, in crystal growth, one is most interested in relating state variables such as temperature, composition, chemical potential, and pressure to interfacial curvature. The ability to evaluate theories of crystal growth and morphological development has been stymied until recently by the unreliability of interfacial energy measurements. However, two systems — ice/water and succinonitrile — are now characterized with sufficient accuracy to permit critical evaluations to proceed. Measurements of the kinetics of dendritic growth in these systems confirm that a characteristic rate dependence arises. Specifically, In(V/V₀) = 2.65 In ΔT, where V is the axial growth rate, V₀ is a measured constant, and ΔT is the supercooling. The almost-cubic rate exponent indicated is independent of the material properties, and is predicted by diffusion and capillary theory, whereas the constant V₀ depends on the material system and is poorly estimated by current steady-state theory. Other examples to be reviewed include recent experimental assessments of linear morphological stability theory.     

Kinetics of hydrothermal crystallization of quartz in Na2CO3 solutions

The kinetics of crystallization of the {0001}c, {01 $\bar 1$ 1}r, and {10 $\bar 1$ 1}R faces of quartz in 0.5 M Na₂CO₃ (M is molarity) aqueous solutions has been studied in the temperature range 200–450°C. It is established that the dependence of the crystal growth rate on temperature in the logV-1/T, K coordinates is of a parabolic nature. It is most probable that the nonlinearity of this dependence is associated with a deficiency in the solution of silica monomers, taking part in the elementary event of quartz crystallization. The causes of a jumpwise decrease in the activation energy of the growth of the c, r, and R faces at t > 280–325°C are considered.     

Effects of structure instability of a V3Si single crystal from X-ray scattering data

Diffuse X-ray scattering from a V₃Si single crystal was studied at room temperature. It was demonstrated that the structure possesses instability regions associated with the formation of a new phase. The characteristic features of the q dependence of diffuse scattering are indicative of the presence of two-types of domains— those randomly distributed over the crystal and those forming a spatially periodic distribution.     

Molecular dynamics simulation of viscosity in supercooled liquid and glassy AgCu alloy

The viscosity of the model AgCu alloy is simulated by several methods using (i) correlation functions through the Green–Kubo formalism, (ii) a non-equilibrium molecular dynamics approach and (iii) creep tests under constant stress. Temperature dependences of the shear viscosity and the diffusion coefficient show the breakdown of the Stokes–Einstein relation well above the glass-transition temperature T_g. This observation is interpreted as a manifestation of the development of heterogeneities in the supercooled liquid approaching T_g. Based on a generalized Einstein formula for the viscosity of liquid, a temperature dependence of heterogeneity degree of supercooled liquid is estimated. Using the dependence of the deformation rate on external stress, the activation volume is evaluated to be four atomic volumes in liquid state. However, below the mode-coupling temperature T_c the activation volume increases by several times.     

Electrocrystallization of silver: Growth kinetics of screw-dislocation-free (100) crystal faces

At given conditions, especially at higher supersaturation, the growth rate of a close packed, perfect crystal face depends on the formation rate of two-dimensional nuclei and on the propagation rate of the monoatomic layers. This multinuclear multilayer growth as well as the advancement rate of growth steps have been studied experimentally on electrocrystallization of silver. The advancement rate of mono- and polyatomic growth steps has been measured on screw dislocation-free (100) crystal faces. For low overvoltages a linear dependence of the rate on overvoltage has been found. A strong influence of the surface condition of the crystal face — “fresh” or “aged” on the step advancement rate has been established. It was also found that on a “fresh” surface mono- and polyatomic steps advance with the same rate. The average monoatomic step spacing of the polyatomic step has been determined. The kinetic constants of the step growth rate are established and a conclusion regarding the mechanism of electrolytic deposition of silver is drawn. The initial current—time curves were recorded on applying potentiostatic pulses on a perfect crystal face. The shape of these curves coincides very well with those theoretically calculated for the cases of multinuclear growth. On the basis of the theoretical dependences, one can determine from these curves the formation rate J of two-dimensional nuclei at a given overvoltage η since the rate of step advancement is known. A linear dependence of log J on 1/η has been established. The values of the pre-exponential term in Volmer's equation and the specific edge energy of the two-dimensional nucleus have been determined. The surface condition of the crystal face influences strongly also the process of two-dimensional nucleation.     

Pressure and temperature dependence of the electrical resistivity of bulk Al23 Te77 glass

Electrical resistivity of bulk amorphous Al₂₃T₇₇ samples has been studied as a function of pressure (up to 80 kbar) and temperature (down to 77 K). At atmospheric pressure the temperature dependence of resistivity obeys the relation ? = π₀ exp(δE/RT) with two activation energies. In the temperature range 300 K ? T > 234 K the activation energy is 0.58 eV and for 234 >T ? 185 K the value is δE = 0.30 ev. The activation energy has been measured as a function of pressure. The electrical resistivity decreases exponentially with the increase of pressure and at 70 kbar pressure the electrical behaviour of the sample shows a metallic nature with a positive temperature coefficient. The high pressure phase of the sample is found to be a crystalline hexagonal phase.     

Thermal stress relaxation phenomenon through forming the interstitial region in CZ silicon pulled with rapid and slow cooling heat shields

This review article aims to clarify a mechanism of point defects formation in a CZ Si crystal through an experimental arrangement using the two kinds of heat shields with different slow-pulling periods. Point defects in a melt grown silicon crystal have been studied for a long time. The author and his co-researchers have reported about “Mechanism for generating interstitial atoms by thermal stress during silicon crystal growth” [in Progress in Crystal Growth and Characterization of Materials, 66 (2019) 36-46]. The experimental arrangement includes constant growing, changing pulling rate and finally detaching crystals from the melt. The two types of heat shields were used to change the cooling history of the grown crystals, for changing a temperature gradient at a bulk part in the grown crystal, G_b. In order to prove that the formation of an interstitial region or a boundary of vacancies (Vs)/interstitials (Is) in a silicon crystal is a phenomenon of relaxing thermal stress, the author explains that a G_b in a crystal forms thermal stress and causes some silicon atoms at lattice positions to move to the closest interstitial sites to relax the stress. The author defines a new term of metastable interstitial atom, I’, or I's as the plural of I’. The I’ coexists with the metastable vacancy V’ from where the I’ is displaced. The plural of V’ is defined to be V's. The author defines the above state to be a complex (I’+ V’), or (I ’+ V’)s as the plural of (I’+ V’), and explains that the (I’+ V’) s convert to Is and form the Is region. The (I’+ V’) is considered as the Frenkel pair-like complex.The crystals were firstly pulled with a high pulling rate, and the pulling rate was consequently decreased to a slow one. Then the crystals were pulled with the slow constant pulling rate for different periods making different cooling processes. Finally, the grown crystals were detached from the melt and cooled rapidly. Characterization of defects, such as Vs, Is, and defect-free (D-F) regions were identified in X-ray topographs (XAOP(s)). Wafer lifetime mapping (WLTM(s)) allows confirming dislocation loop (DL) regions. The results show that the Is are generated depending on the pulling period of the slow pulling and the shapes of the heat shields. The Is and DL regions are formed in a region at temperatures near the melting point. The Is form an Is region through a defect-free (D-F) region, forming the Vs/Is boundary. When the thermal stress weakens, the DL region changes to the Is region; the Is region changes to the D-F region; and the D-F region changes to the Vs region. Temperature gradient distribution is induced toward various directions at different parts of the growing crystal depending on the different slow-pulling periods. The temperature gradient, G_b, includes a temperature gradient from the cooled region shaded by the heat shield to the growth interface and a temperature gradient from the upper surface cooled during the long-time growth to the growth interface. The G_b exceeding a certain threshold at near the melting point forms thermal stress, generating Is to relax the stress.     

Growth improvement and characterization of AgGaxIn1−xSe2 chalcopyrite crystals using the horizontal Bridgman technique

AgGa_xIn_1?xSe₂ single crystals with x=0.4 have been grown by the horizontal Bridgman technique for nonlinear optical application requires phase matching. High purity polycrystalline synthesis of AgGa_xIn_1?xSe₂ was carried out at 850 °C, which is a relatively lower temperature compared to those in earlier reports, thus reducing secondary phase formation. An average Ga:In ratio of 62:38 (±3%) was measured using energy dispersive spectroscopy (EDS). As grown, a single crystal shows very high IR transmission of ～65% in the spectral range of 4000–600 cm^?1. There was no significant change in its IR transmission after annealing it at 500 °C for 20 days in vacuum in the presence of AgGa_xIn_1?xSe₂ powder. This indicates a low concentration of defects in the crystal. The results demonstrate that the improved new synthesis method for crystal growth was promising and that the quality of the crystal was good.     

Phase transitions of p-type (Pb,Sn,Ge)Te-based alloys for thermoelectric applications

p-type (Pb,Sn,Ge)Te-based alloys for thermoelectric applications were prepared using Bridgman technique. Second-order, rhombohedral to cubic phase transitions are involved, as evaluated from anomalies in the temperature dependence values of Seebeck coefficient, electrical conductivity, heat flow and the elongation, in the vicinity of the phase transition temperature, T_c. The correlation between these anomalies in both the electronic and thermodynamic properties was interpreted by means of the relationship of Fermi energy to the chemical potential (or to the molar Gibbs free energy).     

Nanostructured crystals of fluorite phases Sr1 − x R x F2 + x and their ordering: 9. The defect crystal and real structure of quenched fluorite phases Sr1 − x Ce x F2 + x (x = 0–0.5)

A Sr_0.7Ce_0.3F_2.3 crystal (CaF₂ type, sp. gr. $Fm\bar 3m$ ), obtained by quenching from melt, has been studied for the first time by X-ray diffraction. Fluorine vacancies and interstitial anions are found in the 8c and 32f sites, respectively. The defect ratio in the Sr_0.7Ce_0.3F_2.3 structure corresponds to the tetrahedral cluster configuration of defects {Sr_{4 ? n} Ce _n F₂₆}. The defect structure of quenched (at a rate of ～25 K/min) crystal differs from that of a crystal grown from melt (cooling at a rate of ～3 K/min) by the displacement of some cations (presumably Ce³⁺) along the threefold axis to the 32f site and the anisotropy of thermal vibrations of ions in the cluster core (F _int(32f)3). The concentration dependence of the lattice parameters of quenched Sr_{1 ? x} Ce _x F_{2 + x} phases (x = 0–0.5) is described by a third-order polynomial: a = 5.80009 + 1.166518 × 10^?3 x ? 1.124969 × 10^?5 x ² + 8.258155 × 10^?8 x ³. The compositional dependence of microdistortions is also nonlinear; maximum microdistortions are observed in the SrF₂ crystal. They decrease with an increase in the cerium concentration x to ～ 0.35. The minimum in the range x = 0.30–0.35 correlates with a composition corresponding to the peak (at x ～ 0.29) in the melting curves of the fluorite phase estimated from the phase diagram of the SrF₂-CeF₃ system (the method of thermal analysis).     

A necessary condition for the formation of a bisystem molecular crystal

A hypothesis is formulated which states that the necessary condition for the existence of a bisystem molecular crystal is the approximate equality of the total energies of the interactions of two symmetrically independent molecules N and N′ with their environment in the crystal structure (U_Σ). The hypothesis was tested on four crystalline organic compounds of the structure class P2₁/c, Z = 8(1, 1). The computations in the atom-atom approximation demonstrated that the energies U_Σ(N) and U_Σ(N′) have very close values despite the different coordination numbers of the molecules and a difference in the structures of the molecular agglomerates that include the molecules N and N′.