Assessment methodology: modeling intermediate phases, crystal-structure information, compatibility of models, thermodynamic information, determining adjustable parameters, decisions to be made during assessment, checking results of an optimization and publishing it

CT – 14: Assessment methodology

Modeling intermediate phases

The term „intermediate“ phases means all phases that do not extend to the pure components of a system.

Identification of the structure type is the first step in the modeling.Intermediate phases often have more complicated crystal structures:e.g. , , Laves etc.- see Fe-Mo system

Example:For phase the Gibbs energy of end member oGA:B:C could be estimated with

respect to the similarity of structure with the same coordination number as: 10oGA

FCC + 4oGBBCC + 16oGC

BCC (CT-12)

(Ab initio calculated energies of formation are advantageous here.)

Solubility described by this model, e.g. for phase (10:4:16), (when one sublattice is occupied with the one kind of atoms only), does not cover the whole concentration range and model (A,B)10 (A)4 (A,B)16 should be extended to (A,B)10 (A,B)4 (A,B)16 in spite of that it increases the number of end members (e.g. in Ni-V system on the V-rich site).

Modeling intermediate phases - example

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Modeling intermediate phases with ordering

Some intermediate ordered phases can transform to a disordered state:

L12, L10, and D022 to A1 (FCC)

B2, D03, and L21 to A2 (BCC)

D019, B19 to A3 (HCP)

Those should be modeled by the same model (ordered and disordered state-partitioning)

Gibbs energy expression describing disordered part depends only on mole fractions, expression describing ordered part depends only on site fractions (partitioning).

For those phases which never transform directly into disordered phase: models for ordered and disordered phases should be different (e.g. D022 in some cases).

Crystal-structure information

Phase having four or five sublattices – it may be necessary to reduce this to two or three – if it is not enough experimental information available to determine all necessary model parameters. There are seldom enough data to fit more than two end members for binary phase.

Preferrential occupation of crystallographically different positions may give an enthalpy curve with a sharp minimum („V“-shape) at ideal (stoichiometric) composition (revealing „associate“).

In other cases – model of substitutional solid solution should be used.

Crystallographic data give further the information whether the range of homogeneity is created by anti-site atoms, vacancies, or interstitials.

The relative occupation of sublattices by different atoms may be also found from X-ray data (Fourier synthesis from the intensities of various reflections of the X-ray pattern – Rietveld refinement. Pycnometric study of density may help.)

Occupation of sublattices should be in agreement with the crystallographic data.

Crystal-structure information - exampleSite occupation in the phase Re-W calculated at 1500 K by CEF model with ab initio calculation results. Experimental points, CVM calculations dotted lines(In sublatices: 2a, 4f, 8i, 8i’, 8j, sublattice 4f is preferrentially occupied by W.)

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Compatibility with models used in databasesExample – model for phase:Old model (8:4:16) was used till about 1996, when new model (10:4:18)

was introduced. Old model is frequently used in databases.

In new assessments one should use the recommended model (new) but it may be necessary to fit also the old model for the sake of backward compatibility. When sufficient number of revised assessments is available, old model of phase may be removed from the database. (Similarly also for other phases – e.g.Laves.)

Regard to extrapolation of binary assessment to ternary one:

Example: Laves phase C15 – HfCr2 , Cr is in the second sublattice

- CrTa2 , Cr is in the first sublatticeIn ternary Cr-Hf-Ta system – Cr has to be modeled on both sublattices even in the binary assessments

Thermodynamic information

Heat capacity = source of data for the molar Gibbs energy expression: Gm

- biHiSER = ao + a1T + a2Tln(T) + a3T2 + a4T-1 + a5T3 +…T1TT2

bi is the stoichiometric factor of element i in phase and biHiSER represents the

sum of the enthalpies of the elements in their reference state (usually the stable state at 298.15 and 1 bar – SER).

The coefficients ai describe the heat capacity. They can be adjusted experimentally:

a0, a1 by heat content [H(T) – H(298)] data

a2, a3, etc. by heat capacity Cp(T) data

(Cp(T) - semiempirical estimates, Einstein function, Debye function.)

Transition from Debye or Einstein function to Dulong-Petit function: term a4T-1

Low temperature heat-capacity measurements: for integration to standard entropy: ST = 0

T(Cp(T)/T) dT.

Thermodynamic information-examplesPeculiar shape of heat-capacity data

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Thermodynamic information-examplesPeculiar shape of heat-capacity data

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Thermodynamic information-cont. Einstein model: CV,m = 3R [x2.ex/(ex-1)2], where x = E / T

E is Einstein temperature = hE/k. ([x2.ex/(ex-1)2] is Einstein function)

Debye model: CV,m = 3R [4D() - 3 /(e - 1)], where = D / T.

D is Debye temperature = hD/k. ([3/30 (x3/(ex-1))dx] is Debye function , where x = h/kT).

Dulong-Petit function: Cp,m = 3 R

Phonon spectra as a source of Cp data

Further important thermodynamic data: -Enthalpy of formation at 298.15 K data (H298) based on calorimetric data, -Standard entropies S298,

calculated from low-temperature heat-capacity data. They both (Cp and S298 ) must be positive!

(Example – system Al-B (S298 of AlB2 = -1.79 J.mol-1K-1) have to be re-assessed for this reason – Mirkovic et.al. 2004)

Further important thermodynamic data: -Chemical potential measurement (vapour pressure data, EMF data), value of this type of data is similar as phase-diagram data (no change in two phase region)

Thermodynamic information-example Einstein model

Atkins P.W.:Physical Chemistry

Stoichiometric phases For stoichiometric phases the function Gm

- biHiSER = ao + a1T + a2Tln(T) + a3T2 + a4T-1 + a5T3 +…T1TT2

can be adjusted to experimental data, similarly as for solutions. (Additional parameters are needed for magnetic phase.) If Gibbs energy is determined for narrow temperature range only –

only a single coefficient (or linear combination of coefficients) can be adjusted for ∆G = ∆H - T∆S.

One of two coefficients ∆H and ∆S (ao and a1) may be estimated – Tanaka‘s rule: ∆H / ∆S Tmelt.

For stoichiometric phase AmBn, partial Gibbs energy measurements (in special case) are equivalent to a direct measurerment of its integral molar Gibbs energy: if the phase is in equilibrium with the element A in nearly pure state, partial Gibbs energy A = 0 and, from condition Gm

AmBn = m A + n B, it follows Gm

AmBn = n. B.

Wagner-Schottky modelIt was first model using the crystallographic positions of different atoms in

sublattices (Wagner and Schottky 1930, Wagner 1952), developed for binary intermediate phases with small homogeneity ranges.

The „ideal phase“ is defined to have on each sublattice only one occupant. In „real phase“ - there are also defects on the sublattice.

Simplifications containing in Wagner-Schottky model:

- The defects are dilute, that interactions between them can be neglected

Gibbs energy of defects formation can be treated as independent of

composition

- Defects with lowest Gibbs energy of formation are only considered on both sides of stoichiometry (anti-structure atoms, vacancies, and interstitials – no clusters).

- Random mixing of the constituents, separately on each sublattice, is asssumed

CEF may be interpreted as a generalization of the Wagner-Schottky model

(crystallographic information is taken into account).

Carbides and nitridesInterstitial carbides (nitrides) are of types „MC“ and „M2C“

(interstitial solution of carbon or nitrogen in FCC and HCP lattice, respectively.)

Special type of carbide:

Carbide with more inequivalent sites for metallic elements: „M23C6“ carbide

Model: (Cr, Fe, …)21 (Cr, Fe, Mo, W, ...)2 C6.

Maximum solubility of W and Mo is 2 / 23 = 8.7 at.% in M23C6 determines the value of parameter oGCr:W:C

M23 , for the „end member“ Cr21W2C6.

From the phase diagram, it follows that M23C6 dissolve considerable amount of Fe – it is important for determination the parameter oGFe:Fe:C

M23 for the „end member“ Fe23C6.

Additional „end members“: Cr21Fe2C6 and Fe21Cr2C6. The oGCr:Fe:CM23 and oGFe:Cr:C

M23 behave as interaction parameters – (curve fitting result). It is not enough experimental information available, therefore their enthalpies of formation are set equal.

Carbides and nitrides - example

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Ionic crystalline phases

Phases with strongly ionic behavior (oxides, sulfides, chlorides) have often no compositional variation (they are compounds).

Elements have fixed values of their valencies and phase must be electrically neutral.

A variation in composition is usually due to vacancy formation and to the fact, that elements may have multiple valencies.

Examples:

Fe-O (wustite): oxygen ions form FCC lattice with Fe ions in the octahedral interstitial sites. Model is (Fe2+, Fe3+, Va)1 (O2-)1

MgO – Al2O3 (spinel): Al2MgO4. Oxygen ions form FCC lattice. The Al3+ ions occupy half of the octahedral interstitial sites and the Mg2+ ions occupy one eight of available tetrahedral interstitial sites. However, both Al and Mg can be at the „wrong“ sites and electroneutrality condition requires following model (Hallstedt 1992):

(Mg2+, Al3+)1 (Al3+, Mg2+, Va)2 (Va, Mg2+)2 (O2-)4

Ionic crystalline phases - example

H11 spinel structure

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Semiconductor compoundsMost commonly used semiconductor materials –

structures A4 (diamond) and B3 (zincblende).

A4 (diamond): Fd6m, all sites belong to Wyckoff position (8a)

B3 (zincblende):F4‘3m, these sites form two different Wyckof positions (4a), (4c)

Compounds are nearly perfect ordered compounds III. and V.

groups of the periodic table or from the II. and VI. groups.

Deviations from ordering are 10-7 to 10-5 – they can be

described as defects in Wagner-Schottky model, creating

electron holes or free electrons (both obey Fermi-Dirac

statistics, do not contribute to mixing enthalpy and must not

be treated with extra sublattices).

Best routes for production III/V semiconductors: for strongly directed bonding and extremely slow diffusion rate - homogeneization by annealing of crystallized material is nearly impossible. Therefore, growing single crystals or epitaxial films from liquid is adviceable.

Phases with miscibility gaps

Miscibility gap: the same phase appears on both sides of a two-phase region

in a binary system. At high enough temperature – it normaly closes

Term “composition set” with a number: identify the two different instances of the

same phase (the same phase with different compositions in the

same equilibrium –e.g.two terminal phases of the same structure

Miscibility gap can be described with the parameters of single phase.

It can be expected in systems with positive interaction between components, but also in an ideal reciprocal systems.

During optimization it may happen that miscibility gap appear where there should not be any (“inverted“ miscibility gap at high temperatures).

At the end of assessment, it need to be tested at high temperature (4000 K) to ensure that no solid phase reappear.

Inverted miscibility gap at high temperatures-example

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Phases with miscibility gaps - example

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Hume - Rothery phases

For Hume-Rothery (1969) phases, the Gibbs energy has a significant contribution from the energy of the electron gas (metallic bonding).

It predicts the stability of phases at some values of electron concentration (e.g. 7/4, 21/13 ). It should be explained in terms of density of states (DOS) and valence-

electron concentrations (VEC).

RK polynomial has limited accuracy for this class of phases for the electron-gas contribution to the enthalpy H. (Electron-gas contribution to (-TS) is very small. Contributions from other sources may be described well by RKM formalism).

In extrapolation to ternary systems, VEC should be a (single) composition variable instead of Muggianu formalism.

Selecting parameters – practical hints

Adjustable coefficients depend on many of the diverse measurements Each measured value contributes to many of the coefficients.

Least-square method should select the best possible agreement of all the coefficients and all experimental values.

To many coefficients may not improve the fit between measurements and descriptions significantly (they follow more or less scatter than physics).

It must be discussed for each coefficient, if it really improves the fit between calculation results and the experimental dataset.

Quick guide for the model selection for a phase

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Reducing the number of parameters

It is recommended to start the assessment with as few coefficients as possible and then include additional coefficients.

A systematic misfit between some series of experimental points and the corresponding calculated curve may give hints which coefficients should be added or excluded.

Some coefficients may describe well measured values and calculated curves, but may have bad influence on the behavior of extrapolated calculations into areas not covered by experiments. It need to be tested!

Constraining parameters For phases with several sublattices on each, there are many reciprocal relations. If

there are not enough information – possible method is to assume that the reciprocal energy is zero.

If three end members of reciprocal relation are known, it is possible to fix the fourth: oGA:C = oGA:D + oGB:C – oGB:D

For intermediate phases with homogeneity range, often exist „ideal“ composition (each sublattice is occupied by single constituents only).

For modeling of homogeneity range, one needs information about kind of defects (vacancies, interstitials, anti-site atoms). If homogeneity range is small (or little

information available), start with simplification to stoichiometric compound.

When there are many end members of a phase, little or no thermodynamic information and no data on actual occupancy – assume the same entropy and heat capacity for all end members (using Kopp-Neumann rule).

Setting an end-member parameter equal to zero is a very bad estimate.

Coefficients in the Redlich-Kister seriesWhat is maximal power in equation: Lij = k

=0 (xi – xj). Lij ?

The simplest case is dilute solution starting from one of the pure components (A) – Henry‘s law

Henrian solution is described as a regular or quasiregular solution with a single coefficient oLA,B.

For dilute solution, the sum oGB – HBSER + oLA,B is significant only and should be adjusted

according to experimental data (regular or quasiregular).

Measured solubilities in extended temperature range or solubility at one temperature and enthalpy of solution allow adjustment of two coefficients oLA,B = ao + a1.T. Otherwise, only one coefficient should be adjusted.

Proportionality a1 = ao / Tmelt (Tanaka‘s rule) may be used.

If for one solution phase, enough experimental information for adjustment of two or three RK coefficients is available, one may find it necessary to use the same number of coefficients for other solution phases to obtain good description.

Decissions to be made during assessment

First step of optimization: to get a set of parameters that can roughly reproduce the main feature of the data.

Second step of optimization: fine-tuning the parameters to selected data.

Steps to obtain a first set of parameters

It is required usually, that one can calculate the experimental equilibrium in order to compare the measured quantity with the corresponding quantity calculated from the model.

Some experience is wellcome.

(If all parameterrs are initially zero, that is usually not possible.)

Contradiction beween parameters

Experimental data of the same kind

First, systematic errors, impure samples, bad calibration etc. have to be checked and wrong data excluded.

If the difference remain – assessor must decide to exclude some of the data (otherwise the mean value will be used by least-squares method)

This selection may have to be reconsidered later when the fit to other kind of data may have clarified that originaly excluded data are better than the originaly included one.

Contradiction beween parameters

Experimental data of different kinds

When some set of experimental data cannot be fitted simultaneously with the other data sets – suspected data set can be changed one by one and checked , whether the fit of the remaining dataset improves. Which data set finaly should be excluded is up to the judgment of assessor.

Before the excluding a set of data, one should analyze the original paper carefully and test it by using physical rules as e.g.Gibbs-Konovalov rule, Tanaka‘s rule etc.

There are cases in which different kinds of data provide the same information – e.g. entropy may be obtained by combining the enthalpy and liquidus data and chemical potential and enthalpy data, which may be contradictory (entropy is overdefined). One should question the assumed model, but sometimes this question is difficult to answer.

Weighting of experiments

In the least-squares method, the difference between calculated Fi and measured Li values times a weigthing factor pi is minimized:

(Fi(Ci,xki) – Li) . pi = vi

i=1n vi

2 = Min.

The minimization can be affected by selecting different weightings for experimental data.

The uncertainty assigned to each piece of experimental data is the first step.

The uncertainty should never been adjusted during an optimization, reasonable value provided by experimentalist should be used.

Agreement with experiment may be improved by changing the model. Example: assymetrical miscibility gap requires at least two RK coefficients. No weighting can improve the fit when only one regular parameter is used.

Weighting of experiments – cont.

If model and model parameters are reasonable – the most important experiments may increase the weighting of data.

It improves the fit of those data and the fit of remaining data will be worse.

Example: If one has much thermochemical data – one may also adjust the weightings to give equal importance to few phase-diagram points (increase their weightings).

Phases appearing or missing surprisinglyWhen terminal phase appears to be stable in parts of phase diagram

where they are not stable – it may be at temperatures far above its

real stability range – it may be sufficient to set a breakpoint in the description of Gibbs energy of one or more end members of the phase and continue with constant Cp. High powers of T in the expression for Cp may lead to unphysical Cp description.

If it is not sufficient – Cp may be allowed to decrease to Kopp-Neumann rule value. Example: Fe-Cr-W system - phase.

Possible appearance of a phase at wrong place may be caused by Gibbs energy of a particular end member becoming too negative or the coefficient of RK series having too negative value.

This problem can be solved by calculating driving force: phase at wrong place must have negative driving force at the stable equilibrium at this place.

Reasonable values of parameters Certain limits of the model parameters should be considered. In most cases only ao and a1 coefficients of Gibbs energy expression are

optimized, related to enthalpy and entropy, respectively.

Reasonable limits for enthalpy and entropy:

Enthalpy absolute value should be less than few times 100 kJ mol-1 (per mole of atoms)

Entropy should be always positive and less than 100 Jmol-1 K-1. This should apply to each coefficient in RK series

Checking results of an optimization

A well-optimized set of parameters for Gibbs energies of the system should be able to reproduce the available experiments in best possible way,

New experimental evidence may show that the description should be modified

Criteria for well established description of experimental evidence:

1. Visual check of the agreement between experimental data and calculated values.

2. Sum of squares of errors from least-squares fit is important, (but not unique) criterion.

3. Extrapolation to higher-order systems. If extrapolation of binary data to ternary system is wrong – ternary information may improve binary description.

4. Analyze plausibility of the values of the parameters found by least-squares fit.

(Use of Kopp-Neumann rule, plausibility of sign of coefficient, Tanaka‘s rule)

5. Removing non-significant digits. Safe rounding: select the parameter with the highest relative standard deviation, set it to a rounded value, and reoptimize the remaining parameters. Repeat the procedure.

6. Check that S298 and Cp of all phases are within reasonable limits (0).

Publishing an assessed system(Hints for reviewers.)

1. A list of experimental data used and of those that were not used.

2. A complete report of the model parameters. Indicate which parameters are missing and where to find them (e.g.in commercial database).

(Provide parameters in format on a computer file. This facilitates work of reviewer to check them.)

3. A table of invariant equilibria, including azeotropic maxima and minima.

4. Crystal-structure information and lattice parameters

5. Standard enthalpies of formation.

6. A report about metastable phases.

7. Diagrams with calculated and experimental data together. All experimental data should be plotted (even dataset excluded from assessment – may be separately).

8. The range of validity of the description.

Which is the best method for an assessment?

Assessment may be done by searching the literature (proper for creating of multicomponent database of real materials).

It is also possible to start by describing the models, experiments come later or will be found in literature.

One can start also with the experimental data, followed by selection of models.

There are good features in all methods – reasonable results can be obtained

making a combination of them.

Questions for learning

1. Describe procedure of modeling of intermetallic phases including

crystal-structure information

2. Describe procedure of modeling of intermetallic phases including

thermodynamic information

3. Describe procedure of modeling of stoichiometric phases, carbides and nitrides, Hume-Rothery phases, semiconductors, ionic crystalline phases and phases with miscibility gap in assessment

4. Describe procedure of weighting of experimental data for assessment

5. Describe procedure of checking the assessment results