Abstract
Solid-state transformations can be classified into two main categories: (a) homogeneous reactions, which occur in the bulk crystalline (or amorphous) phases, and (b) heterogeneous ones, which occur at structural defects such as grain boundaries, dislocations, etc. This chapter covers homogeneous reactions only, and particularly those transformations that are caused by instabilities or meta-stabilities of solid solutions to local compositional changes. Such replacive reactions can be of two types: (a) clustering transformations, which favor like-atom bonds, and (b) ordering transformations, which favor unlike bonds. Only clustering reactions are treated here, although the formalism can easily be extended to cover order—disorder reactions as well.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. W. Gibbs, On the equilibrium of heterogeneous substances, in Scientific Papers, pp. 105 and 252, Dover, New York (1961).
M. Volmer and A. Weber, Nuclei formation in supersaturated states, Z. Physik. Chem. 119, 277–301 (1925).
R. Becker and W. Döring, Kinetic treatment of grain formation in supersaturated vapors, Ann. Physik. 24, 719–752 (1935).
D. Turnbull, Phase changes, in Solid State Physics ( F. Seitz and D. Turnbull, eds.), pp. 226–308, Academic, New York (1956).
J. W. Christian, The Theory of Phase Transformations in Metals and Alloys, Pergamon, London (1965).
K. C. Russell, Nucleation in solids, in Phase Transformations (H. I. Aaronson, ed.), pp. 219–268, American Society for Metals, Metals Park, Ohio (1970).
M. Hillert, A theory of nucleation of solid metallic solutions, D.Sc. Dissertation, Massachusetts Institute of Technology, Cambridge, Mass. (1956).
M. Hillert, A solid-solution model for inhomogeneous systems, Acta Met. 9, 525–535 (196i).
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys. 28, 258–267 (1958).
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, II. Nucleation in a two-compartment incompressible fluid, J. Chem. Phys. 31, 688–699 (1959).
J. W. Cahn, On spinodal decomposition, Acta Met. 9, 795–801 (1961).
J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Met. 10, 179–183 (1962).
J. W. Cahn, Coherent fluctuations and nucleation in isotropic solids, Acta Met. 10, 907–913 (1962).
H. E. Cook, D. de Fontaine, and J. E. Hilliard, A model for diffusion on cubic lattices and its application to the early stages of ordering, Acta Met. 17, 765–773 (1969).
D. de Fontaine and H. E. Cook, Early-stage clustering and ordering kinetics in binary solid solutions, in Critical Phenomena in Alloys, Magnets, and Superconductors (R. E. Mills, E. Ascher, and R. I. Jaffee, eds.), pp. 257–275, McGraw-Hill, New York (1971).
D. de Fontaine, A computer simulation of the evolution of coherent composition variations in solid solutions, Ph.D. Dissertation, Northwestern University, Evanston, Illinois (1967).
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions-I. Stability criteria, J. Phys. Chem. Solids 33, 297–310 (1972).
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions-II. Fluctuations and kinetics, J. Phys. Chem. Solids 34, 1285–1304 (1973).
J. E. Morral and J. W. Cahn, Spinodal decomposition in ternary systems, Acta Met. 19, 1037–1045 (1971).
D. de Fontaine, in Solid State Physics (H. Ehrenreich, F. Seitz and D. Turnbull, eds.), to be published.
J. W. Cahn, The later stages of spinodal decomposition and the beginnings of particle coarsening, Acta Met. 14, 1685–1692 (1966).
J. S. Langer, Theory of spinodal decomposition in alloys, Ann. Phys. (N.Y.) 65, 53–86 (1971).
L. H. Shendalman and J. T. O’Toole, Nucleation and coarsening in binary condensed phases, J. Colloid. Interface Sci. 27, 145–160 (1968).
L. A. Swanger, P. K. Gupta, and A. R. Cooper, Jr., Computer simulation of one-dimensional spinodal decomposition, Acta Met. 18, 9–14 (1970).
J. W. Cahn, Spinodal decomposition, Trans. AIME 242, 166–180 (1968).
J. W. Cahn, Unmixing in binary critical systems, in Critical Phenomena in Alloys, Magnets, and Superconductors (R. E. Mills, E. Ascher, and R. I. Jaffee, eds.), pp. 41–64, McGraw-Hill, New York (1971).
J. E. Hilliard, Spinodal decomposition, in Phase Transformations ( H. I. Aaronson, ed.), pp. 497–560, American Society for Metals, Metals Park, Ohio (1970).
D. de Fontaine, Development of fine coherent precipitate morphologies by the spinodal mechanisms, in Ultrafine Grain Metals (Burke and V. Weiss, eds.), pp. 93–131, Syracuse Univ. Press (1970).
A. Bonfiglioli, La decomposition espinodal, Comision Nacional de Energia Atomica PMM/I-91, Buenos Aires, Argentina (1972).
A. G. Khachaturyan and R. A. Suris, Theory of periodic distributions of concentrations in a supersaturated solid solution, Soviet Phys.-Crystallography 13, 63–67 (1968).
R. Kikuchi, Cooperative phenomena in the triangular lattice, J. Chem. Phys. 47, 1664–1668 (1967).
P. C. Clapp, A critical examination of the validity of the pair-wise interaction model for ordered alloys, in Ordered Alloys: Structural Applications and Physical Metallurgy (B. H. Kear, ed.), pp. 25–35, Clator’s Press, Baton Rouge, La. (1970).
D. W. Hoffman, Configurational entropy and solute correlation in disordered alloys, Trans. AIME, in press.
R. Cadoret, A statistical treatment of the free energy of binary nonhomogeneous solutions, Phys. Stat. Sol. (b) 46, 291–298 (1971).
J. F. Nye, Physical Properties of Crystals, Oxford Univ. Press (1957).
H. E. Cook and D. de Fontaine, On the elastic free energy of solid solutions-I. Microscopic theory, Acta Met. 17, 915–924 (1969).
A. G. Khachaturyan, Microscopic theory of diffusion in crystalline solid solutions and the time evolution of the diffuse scattering of X-rays and thermal neutrons, Soviet Phys.-Solid State 9, 2040–2046 (1968).
H. E. Cook and D. de Fontaine, On the elastic free energy of solid solutions-II. Influence of the effective modulus on precipitation from solution and the order-disorder reaction, Acta Met. 19, 607–616 (1971).
D. W. Hoffman, Concerning the elastic energy of dilute interstitial alloys, Acta Met. 18, 819–833 (1970).
W. H. Zachariasen, X-Ray Diffraction in Crystals, Wiley, New York (1945).
A. Guinier, A new type of X-ray diagram, Compt. Rend. 206, 1641–1643 (1938).
G. D. Preston, The diffraction of X-rays by age-hardening Al-Cu alloys, Proc. Roy. Soc. A167, 526–538 (1938).
M. Murakami, O. Kawano, and Y. Murakami, On the determination of the solvus temperature for G.P. zones in an AI-0.8 at. % Zn alloy, J. Inst. Metals (London) 99, 160 (1971).
J. Lasek, Über die Einfluss der durchschnittlichen Zusammensetzung auf die Lage der kohärenten Mischungslücke von Al-Zn Legierungen, Czech. J. Phys. 15, 848–857 (1965).
V. Gerold, Die Zonebildung in AI Zn Legierungen, Phys. Stat. Sol. 1, 37–49 (1961).
V. Gerold and W. Mertz, On the decomposition of an aluminum-zinc alloy, Scripta Met. 1, 33–35 (1967).
G. J. C. Carpenter and R. D. Garwood, Hardness reversion and the metastable phase boundary for G.P. zones in Al-Zn alloys, J. Inst. Metals (London), 94, 301–304 (1966).
T. Niklewski, P. Spiegelberg, and K. Sunbulli, The solvus curve for G.P. zones in Al-Zn alloys: A diffuse X-ray study, Metal Sci. J. 3, 23–25 (1969).
A. J. Ardell, K. Nuttall, and R. B. Nicholson, The decomposition of concentrated Al-Zn alloys, in The Mechanism of Phase Transformation in Crystalline Solids, pp. 22–26, The Institute of Metals, London, England (1968).
B. Golding and S. C. Moss, A recalculation of the gold nickel spinodal, Acta Met. 15, 1239–1241 (1967).
D. de Fontaine, An approximate criterion for the loss of coherency in modulated structures, Acta Met. 17, 477–482 (1969).
H. Goldstein, Classical Mechanics, p. 53, Addison-Wesley, Reading, Mass. (1950).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, pp. 830–841, McGraw-Hill, New York (1968).
R. Kikuchi, A theory of cooperative phenomena, Phys. Rev. 81, 988–1003 (1951).
R. Kikuchi, Boundary free energy in the lattice model-III Solution of the paradox, J. Chem. Phys. 57, 787–798 (1972).
L. D. Landau and E. M. Lifshitz, Statistical Physics, pp. 366–369, Addison-Wesley, Reading, Mass. (1958).
P. C. Clapp and S. C. Moss, Correlation functions in disordered binary alloys I, II, Phys. Rev. 171, 418–427; 754–763 (1968).
M. A. Krivoglaz, Theory of X-Ray and Thermal-Neutron Scattering by Real Crystals, Plenum, New York (1967).
S. Wilkins, Determination of long-range interaction energies from scattering of X-rays by disordered alloys, Phys. Rev. B 2, 3935–42 (1970).
D. de Fontaine, Bose-Einstein condensation of concentration fluctuations in binary solid solutions, in Critical Phenomena in Alloys, Magnets, and Superconductors (R. E. Mills, E. Ascher, and R. I. Jaffee, eds.), pp. 277–287 McGraw-Hill, New York (1971).
P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, New York (1971).
E. L. Huston, J. W. Cahn, and J. E. Hilliard, Spinodal decomposition during continuous cooling, Acta Met. 14, 1053–1062 (1966).
K. B. Rundman and J. E. Hilliard, Early stages of spinodal decomposition in an aluminum-zinc alloy, Acta Met. 15, 1025–1033 (1967).
A. Guinier, Nouvelle interpretation des diagrammes à “side-bands,” Acta Met. 3, 510–512 (1955).
A. Bonfiglioli and A. Guinier, La structure des zones G.P. dans les alliages aluminium-zinc au premier stade de leur formation, Acta Met. 14, 1213–1224 (1966).
H. E. Cook, The kinetics of clustering and short-range order in stable solid solutions, J. Phys. Chem. Solids 30, 2427–2437 (1969).
H. E. Cook, Brownian motion in spinodal decomposition, Acta Met. 18, 297–306 (1970).
Y. Yamauchi, Doctoral Dissertation, Northwestern Univ., Evanston, Ill. (1973).
R. Acuna Laje, Transformaciones de fases coherentes en aleaciones de Al-Zn, Doctoral Dissertation, IMAF, Universidad Nacional de Córdoba (1971).
D. Turnbull and J. C. Fisher, Rate of nucleation in condensed systems, J. Chem. Phys. 17, 71–73 (1949).
M. Avrami, Kinetics of phase change-I. General theory, J. Chem. Phys. 7, 1103–1112 (1939).
W. A. Johnson and R. F. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. AIME 135, 416–458 (1939).
C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen, Z. Electrochem. 581–591 (1961).
I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19, 35–50 (1961).
E. P. Butler and G. Thomas, Structure and properties of spinodally decomposed Cu—Ni—Fe alloys, Acta Met. 18, 347–365 (1970).
R. Cadoret and P. Delavignette, Etude de la décomposition spinoidale au microscope electronique dans les alliages CuNiFe, Phys. Stat. Sol. 32, 853–865 (1969).
J. W. Cahn, A correction to spinodal decomposition in cubic crystals, Acta Met. 12, 1457 (1964).
R. B. Nicholson, unpublished work.
J. W. Cahn, Phase separation by spinodal decomposition in isotropic systems, J. Chem. Phys. 42, 93–99 (1965).
J. W. Cahn and R. J. Charles, The initial stages of phase separation in glasses, Phys. Chem. Glasses 6, 181–191 (1965).
J. Zarzycki and F. Naudin, Spinodal decomposition in the B2O3—PbO—Al2O3 system, J. Non-Cryst. Solids 1, 215–234 (1969).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1975 Springer Science+Business Media New York
About this chapter
Cite this chapter
de Fontaine, D. (1975). Clustering Effects in Solid Solutions. In: Hannay, N.B. (eds) Changes of State. Treatise on Solid State Chemistry, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1120-2_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-1120-2_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-1122-6
Online ISBN: 978-1-4757-1120-2
eBook Packages: Springer Book Archive