Hillerts Model

In our case, the more convenient form of dependence is the following  

Interrelations between the growth rate k (or k) and mobility M are all (for all mentioned kinds of models) linear. 

1) whether our simplified model of flux-driven lateral grain growth due to the chosen procedure (without thermally activated motion of GBs) predicts a reasonable kinetic law for the average lateral sizes 

2) and, if the answer is yes, then what kind of model (after replacement of GB mobihty by Equation 6.66) provides the lateral size distribution close to experiment 

In Hillert s model [17], Equation 6.65 for individual GB motion translates into an equation for growth/shrinkage of individual grains with some effective size R of 

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According to the Hillert model (Hillert, 1980), the excess Gibbs free energy of mixing of a ternary mixture is given by... 

The two coefficients KL and Ks are derived empirically. They are related through the entropy of transition and constrained to reproduce the total enthalpy and entropy increments accompanying the phase transition. Since, the Inden model demands a series expansion in order to calculate the entropy, a simpler related equation by Hillert and Jarl [21] is used in many computer programs. 

Non-stoichiometry in solid solutions may also be handled by the compound energy model see for example a recent review by Hillert [16]. In this approach the end-member corresponding to vacancies is an empty sub-lattice and it may be argued that the model loses its physical significance. Nevertheless, this model represents a mathematically efficient description that is often incorporated in thermodynamic representations of phase diagrams. 

Evidence relevant to the phase stability problem has been given by Massalski (1989). The so-called compound energy formalism was constructed by Hillert and Staffansson (1970) in order to describe models of the thermodynamic properties of phases with two or more sublattices showing a variation in composition, which therefore belong to the class of solution phases. A review of this formalism and a summary of its applications have been recently published by Hillert (2001) and Frisk and Selleby (2001). 

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

To overcome this problem an extension of the sublattice model was proposed by Hillert et al. (1985) which is now known as the ionic two-sublattice model for liquids. As in the previous case it uses constituent fractions as composition variables, but it also considers that vacancies, with a charge corresponding to the charge of the cations, can be introduced on the anion sublattice so that the composition can move away from the ideal stoichiometry and approach an element with an electropositive character. The necessary neutral species of an electronegative element are added to the anion sublattice in order to allow the composition to approach a pure element. The sublattice formula for the model can then be written as... 

To complete this section it is interesting to show the equivalence between the ionic two-sublattice model and the associate model as demonstrated by Hillert et al. (1985). Equation (5.62) can be simplified for a system (A - )p(B+ , J3°)q... 

They can also be made identical in the general case if the conditions t = —vB/vA and j — 1. The equivalences break down in ternary and higher-order systems as there is the introduction of more compositional variables in the associate model than for the two-sublattice case. This was considered (Hillert et al. 1985) to demonstrate the advantages of the sub-lattice model, but as mentioned previously it turns out that the number of excess terms to describe Fe-Mn-S is very similar. 

Subsequent reassessments now took divergent routes. Agren (1979) used thermodynamic data largely drawn from Orr and Chipman (1967) and re-characterised the magnetic component of the a-phase with the Hillert and Jarl model (see Chapter 8). The concept of two competing states in the 7-phase was abandoned as Orr and... 

Integration of experimental Cp data with temperature can be used to bypass the plethora of models described in the previous sections, as it provides both the critical ordering temperature and the ratio of sro to Iro. This route has been extensively used to describe magnetic transformations (Inden 1977a, Hillert and Jarl 1978, Nishizawa et al. 1979) and generalised through the develq)ment of a series of approximate... 

S.4.3.2 Model of Hillert and Jarl. In his original treatment, Inden (1976) used a complicated but closed expression for the enthalpy, but had to use a series expansion in order to calculate the entropy. Hillert and Jarl (1978) therefore decided to convert the Cp expression directly through a series expansion which substantially simplifies the overall calculation and leads to a maximum error of only 1-2 J/mol at the Curie temperature of Fe. The equivalent equations to those used by Inden (1976) are given by... 

This contrasts with the assumption made in virtually all other magnetic models that the value of is independent of temperature. Several variants of the Shottky model have been developed by Miodownik (1977, 1978a) to take into account the situation where one of the two states subsequently undergoes magnetic ordering (Fig. 8.S). In such cases it may also be necessary to consider a temperature-dependent AE (Miodownik and Hillert 1980). 

The sub-lattice model is now the predominant model used in most CALPHAD calculations, whether it be to model an interstitial solid solution, an intermetallic compound such as 7-TiAl or an ionic solution. Numerous early papers, often centred around Fe-X-C systems, showed how the Hillert-Staffansson sub-lattice formalism (Hillert and Staffansson 1970) could be applied (see for example Lundberg et al. (1977) on Fe-Cr-C (Fig. 10.8) and Chatfteld and Hillert (1977) on Fe-Mo-C (Fig. 10.9)). Later work on systems such as Cr-Fe (Andersson and Sundman 1987) (Fig. 10.10) showed how a more generalised sub-lattice treatment developed by Sundman and Agren (1981) could be applied to multi-sub-lattice phases such as a. 

I.I The prediction of transformation diagrams after Kirkaldy et al. (1978). A model for the calculation of ferrite and pearlite was first presented by Kirkaldy et al. (1978) based on Zener-Hillert type expressions (Zener 1946, Hillert 1957). In this first effort, no attempt was made to differentiate between the diffusive and displacive transformations and a overall C curve was produced of the type shown schematically in Fig. 11.14. Kirkaldy ettd. (1978) used the formalism below where the general formula for the time (r) to transform x fraction of austenite at a temperature T is given by... 

Through their parallel and independent efforts on both sides of the Atlantic, which began in the 1950s with mathematically modeling known phase diagrams for unary and binary systems, Kaufman and Hillert are considered founding fathers of the CALPHAD method, the field of computational thermodynamics concerned with the extrapolation of phase diagrams for multicomponent systems. (Source L. P. Kaufman, personal communication, February 08, 2004.)... 

Hil] Hillert, M., Jarl, M., A Model for Alloying Effects in Ferromagnetic Metals , Calphad, 2, 227-238 (1978) (Calculation, Theory)... 

After the observation of DIGM and LFM in the 1970s," several models and mechanisms were proposed for the driving force of the phenomena. Among them, the coherency strain model of Hillert is now widely supported by some critical experiments of Yoon and others." " ... 

Hil] Hillert, M., Staffansson, L.-L, The Regular Solution Model for Stoichiometric Phases and Ionic Melts , Acta Chem. Scand., 24, 3618-3626 (1970) (Thermodyn., Calculation) as quoted by [1978Uhr]... 

Figure 9.13 The grain size distribution function for three theoretical distributions and that obtained from a computer simulation employing the Monte Carlo procedure lognormal distribution (solid curve), Hillert s model (dotted curve), Louat s model (dashed curve), and computer simulation (histogram). (From Ref. 22.)...

Models for grain boundary migration controlled by solute drag have been developed by Cahn (61), Stuwe (62), Hillert and Sundman (63), and others. We shall outline the model of Cahn which is more quantitative and concise than the others and has the advantage that the boundary mobility can be more directly related to the physical parameters of the process. The model analyzes the problem in one dimension and makes the following assumptions ... 

Hillert, M. (1961), A solid-solution model for inhomogeneous systems, Acta Metallurgica 9, 525-535. 

Hil61] Hillert, M., a Solid-Solution Model for Inhomogeneous Systems, Acta Metall.,Wo 9,1961, p. 525-535... 

A quantitative model was developed by Strawstrom, Hillert and Tedmon et al. [35,36]. This further strengthened the chromium depletion theory. Figure 4.31 shows a comparison of Cr-C-Cr23C6 chromium carbide equilibrium data for 304 stainless steel with experimentally obtained corrosion data for several temperatures. The bulk carbon concentration is shown on the abscissa. The chromium content in equilibrium with Cr23C6 in the steel is shown as the ordinate. 

The use of the sublattice model, developed by Hillert and Staffansson [70Hil] based on Temkin s model for ionic solutions [45Tem] and extended by Sundman and Agren [81Sun], allows a variety of solution phases to be treated, for example interstitial solutions, intermediate phases, carbides etc. All of these represent an ordering of the constituents on different sublattices. 

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