![T-x -projection and k0(x) course of the near ideal system Ge-Si according Eq. (2.10) with h0 Si = 50 kJ mol -1 and TmSi = 1685 K. Full lines were calculated, dashed liquidus course was experimentally obtained [9].](https://www.researchgate.net/profile/P-Rudolph-2/publication/349087856/figure/fig5/AS:988286923325440@1612637372207/T-x-projection-and-k0x-course-of-the-near-ideal-system-Ge-Si-according-Eq-210_Q320.jpg)
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Crystal Growth Fundamentals. Thermodynamics, Kinetics and Transport
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Thermodynamics, kinetics and transport are the mainstays of the science of crystal growth belonging to the basic knowledge of each specialist or student dealing with practice of bulk crystallization or epitaxy. By using them in combination we are able to describe the complex processes of crystallization carefully and correctly.The exceptional importance of monocrystals and epitaxial layers in nearly all high-tech branches of economy is obvious. Therefore, it is the great hope of the author that the present chapter contributes to the improvement of the scientific level of students, young researchers, industrial co-workers and experienced scientists being active in this fascinating field.
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1. Introduction.- 2. Thermodynamics.- 3. Statistical Thermodynamics.- 4. Equilibrium Between Large Phases The Vapor Pressure of Solids.- 5. The Surface Tension of Crystals.- 6. Equilibrium Between Large Three- and Two-Dimensional Phases: Adsorption Phenomena.- 7. Thin Films, Surface Roughening, and Surface Alloys.- 8. Equilibrium Between a Small and a Large Phase.- 9. Equilibrium Shapes of Crystals.- 10. Homogeneous Nucleation the Phase Approach.- 11. Homogeneous Nucleation the Chemical Approach.- 12. Nucleation on a Foreign Substrate.- 13. Some Specific Cases of Nucleation.- 14. Time-Dependent Nucleation Kinetics.- 15. Elementary Processes on the Surface of a Crystal.- 16. Growth of a "Perfect" K Face.- 17. Growth of an F Face of a Perfect Crystal.- 18. Growth of an F Face of an Imperfect Crystal.- 19. Conclusion.- Appendices.- A. Legendre Transformations.- B. Method of Lagrange Multipliers.- C. Euler's Theorem.- D. Stirling's Approximation.- E. Maximum Term Approximation.- References.
Volume IA Handbook of Crystal Growth, 2nd Edition (Fundamentals: Thermodynamics and Kinetics) Volume IA addresses the present status of crystal growth science, and provides scientific tools for the following volumes: Volume II (Bulk Crystal Growth) and III (Thin Film Growth and Epitaxy). Volume IA highlights thermodynamics and kinetics. After historical introduction of the crystal growth, phase equilibria, defect thermodynamics, stoichiometry, and shape of crystal and structure of melt are described. Then, the most fundamental and basic aspects of crystal growth are presented, along with the theories of nucleation and growth kinetics. In addition, the simulations of crystal growth by Monte Carlo, ab initio-based approach and colloidal assembly are thoroughly investigated. Volume IB Handbook of Crystal Growth, 2nd Edition (Fundamentals: Transport and Stability) Volume IB discusses pattern formation, a typical problem in crystal growth. In addition, an introduction to morphological stability is given and the phase-field model is explained with comparison to experiments. The field of nanocrystal growth is rapidly expanding and here the growth from vapor is presented as an example. For the advancement of life science, the crystal growth of protein and other biological molecules is indispensable and biological crystallization in nature gives many hints for their crystal growth. Another subject discussed is pharmaceutical crystal growth. To understand the crystal growth, in situ observation is extremely powerful. The observation techniques are demonstrated. Volume IA: Explores phase equilibria, defect thermodynamics of Si, stoichiometry of oxides and atomistic structure of melt and alloys. Explains basic ideas to understand crystal growth, equilibrium shape of crystal, rough-smooth transition of step and surface, nucleation and growth mechanisms. Focuses on simulation of crystal growth by classical Monte Carlo, ab-initio based quantum mechanical approach, kinetic Monte Carlo and phase field model. Controlled colloidal assembly is presented as an experimental model for crystal growth. Volume IIB: Describes morphological stability theory and phase-field model and comparison to experiments of dendritic growth. Presents nanocrystal growth in vapor as well as protein crystal growth and biological crystallization. Interprets mass production of pharmaceutical crystals to be understood as ordinary crystal growth and explains crystallization of chiral molecules. Demonstrates in situ observation of crystal growth in vapor, solution and melt on the ground and in space.
Surface diffusion of tungsten adatoms on several smooth, low‐index planes of the tungsten lattice has for the first time been followed by direct observation of individual atoms in the field‐ion microscope. Contrary to expectation, the mobility at room temperature is found to increase in the order (211) > (321) ∼ (110) > (310) ∼ (111). Migrating atoms are reflected at the boundaries of the (110), (211), and (321) planes; on the latter two, motion along atomic rows is favored over diffusion across lattice steps. From quantitative determinations of the rate of change of the mean‐square displacement, diffusion coefficients are obtained as follows: (110), D = 3×10−2exp(−22000/RT)cm2/sec; (321), 1×10−3exp(−20000/RT); (211), 2×10−7exp(−13000/RT). Differences in diffusion on the (211) and (321), planes of very similar structure, suggest a weakening of interatomic forces at lattice edges.
This chapter reviews a survey and organizes the extremely diverse phenomena coming under the heading of crystallization, and presents a physics framework for understanding their mechanisms. This framework consists of the application of statistical mechanics, thermodynamics, kinetic theory, and especially transport theory to crystallization problems. The chapter emphasizes the quantitative approach on well-defined systems, both for experiment and theory. The subject of the growth of crystals is an interdisciplinary one, in the sense that contributions to this field have been and continue to be made by scientists and engineers from many professional fields: solid-state physicists, mineralogists, crystallographers, physical chemists, mathematicians, chemical engineers, metallurgists, and probably many others. A similar diversity exists in the wide range of media of publication of the research, and in the departments of universities and research organizations in which the work is performed. The chapter discusses crystallization problems by analyzing them from the point of view of a particular discipline of physics. It begins with the more general disciplines—that is, thermodynamics and statistical mechanics and proceeds to more specialized ones—that is, mass transport and fluid flow.
The stability of the shape of a moving planar liquid‐solid interface during the unidirectional freezing of a dilute binary alloy is theoretically investigated by calculating the time dependence of the amplitude of a sinusoidal perturbation of infinitesimal amplitude introduced into the planar shape. The calculation is accomplished by using gradients of the steady‐state thermal and diffusion fields satisfying the perturbed boundary conditions (capillarity included) to determine the velocity of each element of interface, a procedure justified in some detail. Instability occurs if any Fourier component of an arbitrary perturbation grows; stability occurs if all components decay. A stability criterion expressed in terms of growth parameters and system characteristics is thereby deduced and is compared with the currently used stability criterion of constitutional supercooling; some very marked differences are discussed.
The known facts about nucleation phenomena in liquid metals are interpreted satisfactorily on the basis of the critical size and interfacial energy concepts. In large continuous masses nucleation is almost always catalyzed by extraneous interfaces. However, in very small droplets the probability that a catalytic inclusion is present is so much less that their minimum nucleation frequencies are reproducible and form a consistent set of values.
Interfacial energies, σ, between crystal nuclei and the corresponding liquids have been calculated from nucleation frequencies of small droplets on the basis of the theory of homogeneous nucleation. Energies of interfaces, σg, one atom thick and containing N atoms were calculated from the σ's. The ratio of σg to the gram atomic heat of fusion, ΔHf, was approximately 0.45 for most metals but ∼0.32 for H2O, Bi, Sb, and Ge.
The effect of relative complexity of crystal structure upon the supercooling behavior of pure metals apparently is a reflection of its effect upon ΔHf.
The results of developments of new crystalline media on the basis of rare-earth scandium borates doped with Nd, Er and Yb ions and designed for use in compact diode-pumped laser systems are presented. The problems of growing these crystals by the Czochralski technique as well as their X-ray structural and optical characteristics are discussed. The data of generation tests of diode-pumped lasers with active elements from lanthanum scandium borates doped with Nd, Er and Yb and cerium gadolinium scandium borates doped with Nd are presented.