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.     

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.     

Phase equilibrium in two-phase coherent solids

Phase equilibrium in a two-phase stressed coherent solid is analyzed using the thermodynamic equilibrium conditions directly. Subject to the constraints of constant temperature and external pressure, a bulk alloy composition is chosen and the corresponding volume fractions and phase compositions that satisfy the equilibrium conditions are obtained. We demonstrate unequivocally that, unlike fluids, a number of equilibrium states (combinations of volume fractions and phase composition) may exist that yield energy minima for a given temperature, pressure, and alloy composition and that these multiple metastable states may lead to a nonuniqueness in the observed physical state of the system. In addition, we show that discontinuous jumps in precipitate volume fraction both between 0 and 1 as well as jumps within the range 0 to 1 are possible with a smooth variation in alloy composition. These results are a consequence of the elastic energy, and thus the system energy, being a function of the volume fraction when two phases coexist as coherent solids. As it is difficult to display these effects on a coherent phase diagram, the concept of a phase stability diagram is introduced for both displaying and analyzing the equilibrium conditions in coherent solids. The influence of elastic inhomogeneity and the form of the free energy curves as a function of composition in the absence of stress on phase equilibrium are examined.     

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...     

Coherent phase equilibria: Effect of composition-dependent elastic strain

Recent investigations on coherent equilibria have shed much light on the nature of phase stability. For example, equilibrium phase concentrations are shown to depend on the alloy composition, and two-phase coexistence can terminate at a critical coherency strain, termed a Williams point. This work, however, presents that when lattice parameter, hence coherency strain, is taken to be a function of phase composition, Williams point can disappear, i.e. the two-phase termination occurs only at an infinite elastic strain. In particular, if lattice parameter depends linearly on phase composition following Vegard's law, the equilibrium phase compositions are found to be independent of alloy composition, similar to the incoherent equilibrium case, but the chemical potentials vary with alloy composition, underlying breakdown of the usual Gibbs phase rule. When the analysis is extended to a binary regular solution obeying Vegard's law, complete nature of both coherent solvus and coherent spinodal is revealed in accordance with Cahn's prediction made a generation ago.     

A graphical method for constructing coherent phase diagrams

The equilibrium conditions for a stressed two-phase coherent solid are employed to develop a graphical construction for coherent equilibrium. We show that in certain cases when the stress is generated by a misfit between the precipitate and matrix, the equilibrium coherent phase compositions can be found drawing parallel tangents to the molar free energy curves of the phases measured in the absence of stress. Total energy diagrams and phase stability diagrams can be constructed to determine the combinations of phase compositions and volume fractions which yield unstable, metastable, and stable states. Through these constructions it is possible to show that elastic stress can result in the appearance of cusps on a total energy plot, indicating the presence of an nonanalytic total free energy of the system as a function of alloy composition. Finally, we show how a stable and metastable coherent phase diagram can be determined qualitatively using these constructions.     

Intrinsic thermodynamic stability of stressed coherent systems

The criteria for intrinsic thermodynamic stability are obtained for an elastically stressed coherent system comprised of homogeneous phases at equilibrium using the thermodynamic approach of Tisza. The coherency constraint is an additional constraint on admissible thermodynamic perturbations. Thermodynamic densities cannot be perturbed independently and the extremized free energy at equilibrium is no longer convex in all cases. Multiple equilibrium states become possible for certain mechanical boundary conditions. The stability criteria are applied to a binary system possessing a consolute critical point and, in contrast to coherent equilibria among phases with different crystal structures, multiple equilibrium states are shown not to exist in a coherent binary syste, possessing a consolute critical point.     

Effect of coherent strain energy on γ/γ′ phase equilibria in NiAlTi alloys

The characteristic influences of elastic strain energy on the phase equilibria of the γ + γ′ two phases in NiAlTi alloys have experimentally been investigated by means of analytical transmission electron microscopy. The results obtained are as follows; the tie-line ends of phase decomposition do not always coincide with the equilibrium phase boundaries and move inside the miscibility gap when the precipitates are coherent with the matrix, but move both to the right hand for the incoherent precipitates and are finally fixed for the large incoherent precipitates on the equilibrium compositions of the phase diagram. These experimental results are theoretically rationalized on the basis of the free energy of the microstructure proposed by us.     

Sequences of phase formation in multiphase stressed plates

A Cahn-Hilliard equation is derived and used to explore the effect of compositional strain on the sequence of phase formation in a binary alloy capable of sustaining five distinct, but isostructural, phases in its stress-free state. The system is configured as a thin plate capable of bending and initially has a layered composition profile. The system evolves to reduce the sum of its chemical and elastic energies when strains arise from the nonlinear dependence of the lattice parameter on composition. The compositional strain can stabilize the growth of phases that are nonequilibrium in the absence of stress, give different phase trajectories to various equilibrium states depending on the initial configuration of the system, and can induce a sequence of phase formation in which only one or two intermediate phases grow at any time. These sequences are especially pronounced when deviations from Vegard’s law are realized.     

还原气氛下渣-锍两相平衡反应

通过考察还原气氛下熔渣与锍相之间的平衡情况,研究了渣-锍两相反应的影响因素及机理,探讨了还原产物的分配规律。结果表明,熔渣碱度提高有利于锍相还原反应的进行,添加CaS抑制锍相的还原。由于锍相成分及密度不同出现分层现象,造成锍相在熔渣中的夹杂损失。CaS与金属铁具有互斥性,当熔渣中CaS质量分数超过0.9%之后,渣相中几乎没有金属铁的存在;在选取的实验条件下,CaS在渣、锍两相的平衡分配比维持在0.5左右。     

Multiphase binary diffusion in infinite and semi-infinite media: Part II. On the numerical calculation of the rate constants for formation of product phases

The equations describing multiphase binary diffusion derived in the preceding article are arranged in an order such that the unknowns in the equations are expressed as two series of sequential functions of the corresponding two independent variables. The conditions to be satisfied between one series of functions and the other are given, and a numerical method for solving this type of equations of two unknowns is developed. For the numerical calculation of the parabolic rate constants for formation of product phases, data of average interdiffusion coefficients, partial molal volumes, and phase boundary compositions are required for an infinite medium. For a semi-infinite medium, in addition to these data, information on the equilibrium surface composition and the ratio of mole transfers of the two components at the surface is required.     

Experimental Investigation of Gas/Slag/Matte/Tridymite Equilibria in the Cu-Fe-O-S-Si System in Controlled Atmospheres: Development of Technique

The majority of primary pyrometallurgical copper making processes involve the formation of two immiscible liquid phases, i.e., matte product and the slag phase. There are significant gaps and discrepancies in the phase equilibria data of the slag and the matte systems due to issues and difficulties in performing the experiments and phase analysis. The present study aims to develop an improved experimental methodology for accurate characterisation of gas/slag/matte/tridymite equilibria in the Cu-Fe-O-S-Si system under controlled atmospheres. The experiments involve high-temperature equilibration of synthetic mixtures on silica substrates in CO/CO₂/SO₂/Ar atmospheres, rapid quenching of samples into water, and direct composition measurement of the equilibrium phases using Electron Probe X-ray Microanalysis (EPMA). A four-point-test procedure was applied to ensure the achievement of equilibrium, which included the following: (i) investigation of equilibration as a function of time, (ii) assessment of phase homogeneity, (iii) confirmation of equilibrium by approaching from different starting conditions, and (iv) systematic analysis of the reactions specific to the system. An iterative improved experimental methodology was developed using this four-point-test approach to characterize the complex multi-component, multi-phase equilibria with high accuracy and precision. The present study is a part of a broader overall research program on the characterisation of the multi-component (Cu-Fe-O-S-Si-Al-Ca-Mg), multi-phase (gas/slag/matte/metal/solids) systems with minor elements (Pb, Zn, As, Bi, Sn, Sb, Ag, and Au).     

Inter- and intra-limb coordination in arm tremor

Inter- and intra-limb coordination in arm tremor was examined in adult subjects under vision and no vision conditions using accelerometery techniques. The accelerometer data were analyzed using standard time and frequency domain analyses and the regularity of the acceleration time series was determined using an approximate entropy (Ap En) measure. The data analysis was structured to examine the hypothesis that there is a functional compensatory relation between the motion (tremor) of the limb segments in the arm coordination postural pointing task. The results showed that the level of acceleration increased in a proximal to distal direction within a single arm and was symmetrical across homologous arm segments. The frequency analysis showed the established power spectral profiles for each limb segment in postural tremor tasks, but the finger motion included (beyond the normal 8-12 Hz and 20 Hz tremor) a third slower peak at around 2-3 Hz, due possibly to the reactive forces of the other arm links. There was no effect of vision on the level or frequency patterns of accleration in the limb segments. The coordination analysis showed that there was no linkage between the arms in either the time or frequency domain in the execution of this postural task. This result would tend to suggest that the neuronal commands underlying normal tremor are not derived from a common central oscillator within the central nervous system but are organized in a parallel fashion. The strength of the coupling of intra-limb coordination varied according to the particular adjacent limb links. There were significant correlations in the time domain and coherence in the frequency domain in the acceleration signals between upper arm and forearm, and between hand and finger. The phase lag of the arm units within each of these respective segment pairs was close to in phase or 0 deg. Significant coherence in the frequency domain was also evident between upper arm and hand motion, with the phase lag between these segments being close to 180 deg out of phase. The Ap En analysis of the acceleration signals revealed that there was more regularity to the upper arm and hand accelerometer signals than the forearm and finger signals. The findings show that the intra-limb coordination of the arm links in a two-limb postural pointing task is effected by a compensatory synergy organized about the action of the wrist and shoulder joints. This compensatory synergy reduces the coordination of the 4 within-limb degrees of freedom (arm links) to, in effect, a single degree of freedom arm control task that is not coupled in organization to the motion of the other limb or the torso. It is proposed that this coordination solution reduces the degrees of freedom independently regulated for realization of the task goal but preserves independent body segment control in critical degrees of freedom for potential adaptation to postural perturbations.     

Effect of elastic stress on two-phase binary diffusion couples

A front-tracking, finite difference approach has been used to examine the influence of misfit strain and applied stress on interdiffusion in binary, coherent, two-phase planar diffusion couples assuming local thermodynamic equilibrium at the interface. The phases are cubic; possess different lattice parameters, elastic constants, and diffusivities; and can be oriented in either the [001] or [111] direction. Interface compositions, which are time independent in the stress-free case, become time dependent when stresses are present and are affected by both the elastic state of the system and the relative diffusivities. Interfacial compositions can vary by up to a few atomic percent with time and can be either greater or less than the stress-free values for the same set of materials parameters, depending on the volume fraction of the phases. At sufficiently small times, the interfacial position can be approximated as proportional to the square root of time. Interfacial velocities in this regime can differ by up to a factor of 2 from an otherwise equivalent unstressed system. The nonlinear equations resulting from the coupling of stress and composition were linearized in the bulk phases and could be solved either implicitly or explicitly. Equations governing the interface motion and compositions were not linearized and were solved implicitly at each time-step.     

Effect of interfacial kinetic barriers on interface motion in binary diffusion couples

A finite difference scheme is used to examine the effect of misfit strains and interfacial kinetic barriers to the establishment of local thermodynamic equilibrium on the evolution of a coherent interface in a binary diffusion couple. Impediments to the transformation of one phase into the other and to attaining chemical equilibrium at the interface were considered using a matrix of interfacial kinetic coefficients which coupled thermodynamic driving forces with interfacial velocity and fluxes. Interfacial kinetic barriers result in a decrease in the interfacial velocity, a change in temporal power laws, and large shifts in the time-dependent interfacial compositions, sometimes up to 20 at. pct from the time-independent equilibrium values. The interfacial compositions can shift into either the two-phase field or single-phase field depending upon a number of materials parameters, including the initial compositions of the phases comprising the diffusion couple. This article is based on a presentation made in the symposium “Kinetically Determined Particle Shapes and the Dynamics of Solid:Solid Interfaces,” presented at the October 1996 Fall meeting of TMS/ASM in Cincinnati, Ohio, under the auspices of the ASM Phase Transformations Committee.     

Elemental Partitioning Characteristics of Equilibrium Phases in Inconel 718 Alloy at 600-1100 ℃

 The optimization of heat treatment and chemical composition for Inconel 718 alloy has been investigated uninterruptedly because of its excellent mechanical properties and metallurgical workability. The species, chemical compositions and content of equilibrium phases of Inconel 718 alloy in the temperature range of 600-1100 ℃ were calculated by using thermodynamic software “Thermo-Calc” and the latest relevant datebase of Ni-base superalloys. A concept of elemental partitioning fraction was used to study the partitioning characteristics of alloying elements in each equilibrium phase at different temperatures, such as Ni, Cr, Fe, Nb, Mo, Al, Ti and C, and some calculation results were confirmed under a scanning transmission electron microscope (STEM). The results showed that the elemental partitioning characteristics with the change of temperature revealed the selective partitioning characteristic of alloying elements in equilibrium phases at different temperatures, such as Nb was mainly distributed in δ and γ′ phase, C in carbides, Al and Ti in γ′ phase and Cr, Mo in Laves phase. At the same time, the effect of the change of component and quantity for each precipitated phase on matrix phase can be helpfully understood, which provided a theoretic foundation to optimize the chemical composition and heat treatment in different environments for Inconel 718 alloy.     

Extraction mechanism of copper by versatic 10

The extraction mechanism of copper from an aqueous ammonia-ammonium nitrate mixture by an n-hexane solution of Versatic 10 was studied from the standpoints of the equilibrium distribution of copper between aqueous and organic phases, the aqueous solubility and interfacial adsorption equilibrium of Versatic 10 and the extraction rate of copper.The equilibrium relations of the distribution of copper and the aqueous solubility and the interfacial adsorption of Versatic 10 were interpreted quantitatively and these equilibrium constants were evaluated.The extraction rate of copper was found to be proportional to the copper concentration in the aqueous phase, inversely proportional to the hydrogen ion concentration and independent of the Versatic 10 concentration in the organic phase. From the experimental results of the extraction rate and the interfacial adsorption equilibrium, it was inferred that the extraction rate was controlled by the elementary reaction between the copper ion in the aqueous phase and the anionic species of Versatic 10 adsorbed at the interface.     

A general model for the reaction of several minerals and several reagents in heap and dump leaching

The rate of recovery of metal values from heap and dump leach operations has been stimulated by mathematically modelling reactions among several minerals and reagents. The principal steps are: (i) representation of all possible reactions in a matrix, (ii) division of the system into coherent groups of reactions that proceed at the same rate, (iii) calculation of the rate of reaction of the coherent groups.The procedure models sequential and complementary reactions, including those which involve substances generated within the heap. (It is assumed that the rates of all reactions are controlled by molecular diffusion within the particles.) A general equation for multi-mineral, multi-reagent systems is derived and used in the algorithm.The procedure has been used with a number of systems of different complexity. It has shown that not all of the possible reactions occur under a particular set of conditions, times and location within the heap. The algorithm has been incorporated into a generalised, comprehensive simulation of heap and dump leaching processes, which is independent of any process testwork data. Comparison of predicted results and test results reported by others is included.