On the inapplicability of gibbs phase rule to coherent solids

A relationship is derived between the number of degrees of freedom, independent chemical components, and number of phases for a specially constructed nonhydrostatically stressed coherent solid in complete thermodynamic equilibrium. The results obtained are based on the thermodynamics of crystalline solids, and indicate that the degrees of freedom are independent of the number of phases present. This model is used to demonstrate that coherent solids do not obey Gibbs phase rule. It is also shown that domains of what are the same phase in the absence of all stress effects may occupy different states of deformation and hence possess different equilibrium compositions when the system is constrained to behave coherently. The results derived here are in complete agreement with recent work on coherent phase diagrams.     

On the inapplicability of gibbs phase rule to coherent solids

Metallurgical and Materials Transactions A - A relationship is derived between the number of degrees of freedom, independent chemical components, and number of phases for a specially constructed...     

Characteristics of phase equilibria in coherent solids

Several new results concerning the characteristics of phase equilibria in nonhydrostatically stressed coherent systems are presented for systems in which the deformation of each phase is homogeneous. A critical point is identified, the applicability of Gibbs phase rule relating the number of phases and thermodynamic fields to the degrees of freedom in the system is established, and the existence of a common tangent construction in the appropriate thermodynamic space is proven. The intersection of the hyperplanes and the appropriate free energy of the individual phase in the common tangent construction are shown to form a set of density lines on which all tie lines must end but which need not correspond to the phase boundaries separating single- from two-phase fields. The quantitative results presented herein are in agreement with earlier predictions of phase equilibria in coherent solids.     

On the rule of additivity in phase transformation kinetics

It is commonly held that a sufficient condition for the rule of additivity to be valid is that the transformation rate depend only on temperature and volume fraction. This is not true in general.     

The coherent phase diagram of Cu-Ni-Zn

The phase diagram for the f.c.c. part of the CuNiZn system has been derived using the tetrahedron approximation of the Cluster Variation Method. The pair-wise interaction potentials have been obtained from a pseudopotential model for the stoichiometric Cu₂NiZn composition. The results obtained are combined with the cluster variation method in order to make predictions on the phase diagram of the CuNiZn system. In particular, this paper reports the influences which the various more-body forces exert on the shape of the regions in the phase diagram where ordering occurs. The predictions on the phase diagram of the ternary system CuNiZn are in reasonable agreement with experiments and are the least topologically correct.     

The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids

The elastic strain energy of coherent ellipsoidal precipitates (ellipsoids of revolution) in anisotropic crystalline solids has been calculated as a function of ellipsoid aspect ratio using the method of Eshelby. When the precipitate is eithermuch softer or harder, elastically, than the matrix, the results are similar to those previously obtained using isotropic elasticity. When this condition is not met, however, anisotropic elasticity can yield quite different results which vary markedly with the orientation relationship between precipitate and matrix. When the precipitate has a non-cubic crystal structure, the elastic strain energy often passes through a maximum or a minimum at shapes which are neither thin discs nor spheres. During this study, the isotropic elasticity result that the strain energy associated with a disc-shaped precipitate is independent of the matrix elastic constants was also shown to hold under the conditions of anisotropic elasticity, and in such circumstances it depends only on the elastic properties of the precipitate in the direction of the principal directions of the disc. Incorporation of the anisotropic elastic strain energy into the calculation of ΔG ^*, the free energy of activation for the formation of a critical nucleus for the basic case of homogeneous nucleation with boundary-orientation independent interfacial energy, showed that the ratio of the strain energy to the volume free energy change must usually be somewhat larger than 3/4 in order to cause the shape of the critical nucleus to differ from that of a sphere.     

On the gibbs energy interaction parameters of oxygen and nitrogen in liquid alloys

A number of theoretical models reported in the literature for predicting the values of the Gibbs energy interaction parameters of oxygen and nitrogen in liquid alloys is briefly reviewed. The validity of these models is tested using the large number of experimental data available in the literature. It is found that the Wagner model together with the correlations proposed by Chang and coworkers predict values in best agreement with the experimental values. Formerly Graduate Research Assistant, Materials Department, College of Engineering and Applied Science, University of Wisconsin-Milwaukee     

On the reaction of solids with mixed gases

A mathematical formulation is presented, describing the rate of reaction of porous solids with mixed gases in regimes when the overall rate is controlled by pore diffusion or by external mass transfer. The reduction of metal oxides with a hydro gen-carbon monoxide mixture is a typical example of such systems. In the formulation, proper account was taken of the coupled nature of the fluxes, by writing the Stefan-Maxwell equations to represent the diffusion process. The overall rates obtained from the numerical solution of these simultaneous differential equations were compared with those calculated using a much simpler procedure, involving the concept of the pseudobinary diffusion coefficient. The maximum discrepancy between the total reducing gas flux obtained by these two methods was only some 15 pct at intermediate gas compositions. However, much more substantial discrepancies were obtained regarding the transfer of the individual components. Formerly Graduate Student, Department of Chemical Engineering, State University of New York at Buffalo     

On the reaction of solids with mixed gases

A mathematical formulation is presented, describing the rate of reaction of porous solids with mixed gases in regimes when the overall rate is controlled by pore diffusion or by external mass transfer. The reduction of metal oxides with a hydrogen-carbon monoxide mixture is a typical example of such systems. In the formulation, proper account was taken of the coupled nature of the fluxes, by writing the Stefan-Maxwell equations to represent the diffusion process. The overall rates obtained from the numerical solution of these simultaneous differential equations were compared with those calculated using a much simpler procedure, involving the concept of the pseudobinary diffusion coefficient. The maximum discrepancy between the total reducing gas flux obtained by these two methods was only some 15 pct at intermediate gas compositions. However, much more substantial discrepancies were obtained regarding the transfer of the individual components. Y. EL-TAWIL, Formerly Graduate Student, Department of Chemical Engineering, State University of New York at Buffalo,     

On the coarsening of gas-filled pores in solids

An analytical study is presented of the asymptotic coarsening kinetics of both monatomic and diatomic-gas-filled pores in solids, the rate-controlling process being assumed to be the volume diffusion of gas atoms within the host solid. It is shown that the asymptotic size-distribution functions can be expressed in terms of appropriate similarity transformations, and exact expressions are derived for both the frequency and cumulative distributions for each of the two cases considered. The fact that the order in size between pores is maintained as coarsening proceeds is used, together with the cumulative distribution functions, to derive expressions which describe the temporal evolution of individual pores. The behavior of gross properties of the pore distributions, such as pore concentration, mean radius, and volume fraction is also evaluated.