Equipos utilizados en el beneficio de minerales.

Cacciamani60 _{developed the flowchart displayed in Fig. 2.8 that outlines the steps}

involved in performing a CALPHAD assessment. Descriptions of each step will herein be provided.

Thermodynamic modeling of phases involves constructing a model of a phase that exists within a binary or higher order system. Thermodynamic solution models such as the

CEF and TSPIL are most often developed within software packages such as FactSage43 _or

Thermo-Calc.61 _{These models can be used to calculate thermodynamic properties of}

ordered and disordered solid solutions as well as liquids. 2.4.1. Evaluation of experimental data

It is necessary to collect and critically evaluate experimental data applicable to the system being assessed as inaccurate data will result in a poor optimization. In analyzing the quality of experimental data, details to be considered include the experimental technique used, phases present within the system, purity of the sample analyzed, experimental conditions, quantities measured, and accuracy of the measurements.62 _Per

Saunders & Miodownik,63 _{many experimental techniques can be utilized to obtain}

thermochemical data. Isothermal and isoperibol calorimeters can be used to measure heat contents of pure substances from which heat capacities may be derived whereas adiabatic and heat-flow calorimeters are more effective at directly measuring heat capacities and enthalpies of transformation. Calorimetric techniques such as the drop method and electromagnetic levitation calorimetry can also be utilized to measure enthalpies and heat capacities of pure substances or reactions. Differential scanning calorimetry (DSC) measures the heat absorbed or released during a transformation and thus is often used to quantify thermodynamic properties during phase transformations. Differential thermal analysis (DTA) is more sensitive to temperature changes in a sample than DSC and, consequently, is more often used to determine temperatures of material phase changes. Combustion bomb calorimetry has been successfully used to measure enthalpies of formation of carbides, borides, and nitrides. Gas phase equilibria techniques used to determine thermochemical properties from activities derived from measured vapor

pressures include static methods, dew-point and non-isothermal methods, as well as Knudsen effusion and Langmuir free-evaporation methods. Electromotive force (EMF) experiments can also be conducted to measure partial Gibbs energies.

Experimental techniques also exist to determine phase equilibria and can be categorized as non-isothermal and isothermal techniques. Non-isothermal methods include thermal analysis techniques such as generating cooling curves, DSC, and DTA as well as chemical potential techniques such as EMF, magnetic susceptibility measurements, resistivity methods, and dilatometric methods. Isothermal techniques include metallography that involves the use of optical or electron microscopy, x-ray diffraction, sampling and equilibration methods, and diffusion couples.

Thermodynamic data can also be estimated when experimental data is not available or sufficient for a system. Spencer64 _{details methods of thermodynamic data estimation of}

heat capacities, entropies, and enthalpies of formation for metallurgical applications. Also discussed are the thermophysical property data requirements for single phases as well as two phases that enable the correlation of this data to phase diagrams generated via the CALPHAD method.

Thermochemical data can also be generated from semi-empirical or ab-initio methods such as Density Functional Theory65 _{(DFT). Ab-initio calculations, however, are}

currently limited to relatively small unit cell analyses due to the computational requirements inherent to calculation methods such as DFT. Consequently, first principle methods to calculate thermochemical data remain relatively restricted within the CALPHAD community.

(2.1) 2.4.2. Sequential Bayesian estimation for optimization

The next step in the CALPHAD process is the optimization of the thermodynamic models to thermochemical data. Optimizations are most often conducted within software packages that have a module specifically for optimization procedures such as OptiSage within the FactSage software suite,43 _{which uses a sequential Bayesian parameter}

estimation technique as the main optimization routine.66

The goal of sequential Bayesian estimation within the context of the CALPHAD method is to determine values for unknown coefficients of a Gibbs energy function such that the function can output calculated thermodynamic values within the desired accuracy of a corresponding experimentally determined data point. The following mathematical description is based on texts authored by Walton,67 _{Konigsberger & Gamsjager,}68

Konigsberger,69 _{and Konigsberger & Eriksson.}66

According to Bayesian estimation theory, an error function can be established consisting of a term accounting for the difference between values calculated by the model and corresponding experimental data as well as a term accounting for the difference between original and final model parameters:

𝐸𝐸(𝑃𝑃) = � �𝑓𝑓j(𝑃𝑃)− 𝑦𝑦j� n j,k=1 𝐶𝐶yjk−1(𝑓𝑓k(𝑃𝑃)− 𝑦𝑦k) + � �𝑃𝑃j− 𝑃𝑃jo� m j,k=1 𝐶𝐶_P−1o_jk(𝑃𝑃_k− 𝑃𝑃_ko)

where yj and fj(P) are the jth experimental value and corresponding model calculated value

as a function o2f the model parameters P1,…,Pm, respectively. The variables 𝑃𝑃₁o,…, 𝑃𝑃_mo

represent a priori model parameters to use as initial values for the minimization process. Cyjk and 𝐶𝐶_po_jk are covariance matrices of experimental function values y and a priori

(2.2)

(2.3)

(2.4) enable different weightings for the two parts of the error function to represent the confidence of the assessor in the accuracy of the experimental data as compared to the original model properties.

Eq. (2.1) can be rewritten in matrix form as:

𝐸𝐸(𝑃𝑃) = (𝑓𝑓(𝑃𝑃)− 𝑦𝑦)T_𝐶𝐶

y−1(𝑓𝑓(𝑃𝑃)− 𝑦𝑦) + (𝑃𝑃 − 𝑃𝑃o)T𝐶𝐶P−1o(𝑃𝑃 − 𝑃𝑃o)

where f(p) and P are the model calculation vector function and model parameter vector, respectively.

Minimizing eq. (2.2) according to a Newton-Raphson method67 _{algorithm yields}

the recursive formula:

𝑃𝑃i+1_{− 𝑃𝑃}𝑖𝑖 _{=�(𝑆𝑆}T_)𝐶𝐶

y−1𝑆𝑆+𝐶𝐶P−1o� −1

∙ �𝐶𝐶_P−1o�𝑃𝑃o− 𝑃𝑃i��+𝑆𝑆T𝐶𝐶_y−1�𝑦𝑦 − 𝑓𝑓�𝑝𝑝i��

where S is the sensitivity matrix of derivatives of the model calculations, f(P), with respect to the model parameters, P. For an element jk of the sensitivity matrix:

𝑆𝑆jk =𝜕𝜕𝑓𝑓j(𝑃𝑃)⁄𝜕𝜕𝑃𝑃k

Pi+1 _{values are iteratively determined until the P}i+1 _{– P}i _{delta is less than a prescribed}

convergence limit at which point a final best set of model parameters is obtained. 2.4.3. Lagrange multiplier method for Gibbs energy minimization

As previously stated, the Lagrange multiplier method is used to minimize the Gibbs energy of thermodynamic models.

The following characterization of the Lagrange multiplier method is based on the texts of Hillert70 _{as well as Lukas et al.}26 _{The total Gibbs energy of a system is defined by}

summing the number of moles of the phase α, nα_{, multiplied by the integral molar Gibbs}

(2.6) (2.9) (2.8) (2.7) (2.5) 𝐺𝐺 = � 𝑛𝑛α_{∙ 𝐺𝐺} mα α

The equilibrium condition of the system may then be expressed as:

min[𝐺𝐺(𝑛𝑛α_{,𝑇𝑇,𝑝𝑝,𝑦𝑦}

kαs)] = min�� 𝑛𝑛α∙ 𝐺𝐺mα(𝑇𝑇,𝑝𝑝,𝑦𝑦kαs) α

�

where T and p are the temperature and pressure of the system, respectively, while 𝑦𝑦kαs represents the site fraction of the k species on sublattice s of the phase α.

The minimum of the total Gibbs energy as described by eq. (2.5) can be obtained through the application of the Lagrange-multiplier method with the system subjected to the following three constraints:

𝑛𝑛i− � 𝑛𝑛α� 𝑎𝑎s� 𝑏𝑏ik∙ 𝑦𝑦kαs k s α = 0 � 𝑦𝑦_kαs k −1 = 0 � 𝑎𝑎αs s � 𝜈𝜈k k ∙ 𝑦𝑦_kαs_{= 0}

where ni is the total content of each element in the i system, aαs is the number of sites on

the sublattice s in one mole of phase α, 𝑏𝑏_ik _{is the number of i atoms per unit of k species,} and 𝜈𝜈k is the valency of k species. Eqs. (2.7) - (2.9) establish that the total amount of each i component in phase α remains constant, the site fractions in each sublattice of phase α sum to unity, and the charge of each ionic species in phase α sums to zero, respectively.

Each constraint can be multiplied by a Lagrange multiplier and then added to the total Gibbs energy of the system to form a target function to be minimized:

(2.11) (2.12) (2.10) min[𝐿𝐿(𝑇𝑇,𝑝𝑝,𝑛𝑛i,𝑦𝑦kαs,𝑛𝑛α,𝛽𝛽,𝛾𝛾,𝛿𝛿)] =� 𝑛𝑛α_{∙ 𝐺𝐺} mα(𝑇𝑇,𝑝𝑝,𝑦𝑦kαs) α +𝛽𝛽 �𝑛𝑛i− � 𝑛𝑛α� 𝑎𝑎s� 𝑏𝑏ik∙ 𝑦𝑦kαs k s α � +𝛾𝛾 �� 𝑦𝑦_kαs k −1�+𝛿𝛿 �� 𝑎𝑎αs s � 𝜈𝜈k k ∙ 𝑦𝑦_kαs_�

where β, γ, and δ are the Lagrange multipliers.

A set of nonlinear equations is then obtained by setting the first derivatives of L with respect to each of the unknowns to zero. The partial derivatives with respect to the Lagrange multipliers β, γ, and δ will yield Eqs. (2.7) - (2.9) while the partial derivatives with respect to nα _{and 𝑦𝑦}

kαs result in Eqs. (2.11) & (2.12), respectively:

∂𝐿𝐿 ∂𝑛𝑛α =𝐺𝐺mα − 𝛽𝛽 � 𝑎𝑎αs� 𝑏𝑏ik∙ 𝑦𝑦kαs k s = 0 ∂𝐿𝐿 ∂𝑦𝑦_kαs= 𝑛𝑛α ∂𝐺𝐺m α ∂𝑦𝑦_kαs− 𝛽𝛽 � 𝑛𝑛α� 𝑎𝑎αs� 𝑏𝑏ik k s α +𝛾𝛾+𝛿𝛿 � 𝑎𝑎αs s � 𝜈𝜈k k = 0

As the set of equations defined by Eqs. (2.7) - (2.9), (2.11), & (2.12) yield equilibrium conditions for the unknowns 𝑦𝑦_kαs_{, n}α_{, β, γ, and δ, the variables T, p, and ni must}

be given initial values and held constant during the minimization routine. Values for these unknown variables for the system at equilibrium can be determined through the use of a root-finding algorithm such as the Newton-Raphson method.

2.4.4. Model validation and database formation

Once the thermodynamic models are optimized, phase diagrams can be generated and compared with phase equilibria experimental and/or derived data. As indicated in Fig. 2.9, intensive and extensive thermochemical data can be deduced from the Gibbs energies of the optimized models and thus can also be compared to corresponding input data. The accuracy of the model to predict empirical data, assuming the data is of a quality

(2.14) nature, indicates the reliability of the model to predict phase equilibria or thermochemical data for which no experimental measurements exist.

Thermodynamic models that have been successfully optimized can then be added to databases from which other users can conduct thermochemical equilibrium analyses of systems that include the components of the optimized system. The caveat to this ability being that the subsystems of higher order system databases must be self-consistent. Consequently, higher-order systems are developed by first modeling and optimizing pseudo-binary and then -ternary systems that are then combined to form a complex database of quaternary or greater components.