### Thermodynamics: Multicomponent Systems

The following is an excerpt from Dewar - Characterization and Evaluation of Aged 20Cr32Ni1Nb Stainless Steels, and references a stainless steel used in this thesis. The content of this article can also be applied to a general case, and should not be limited to to the presented example.

### 1. ThermoCalc Introduction

Potential and molar phase diagrams are the centerpiece of materials science, and are used as a visual
representation of a materials system, providing critical information on how the system reacts to
changes in composition, temperature, pressure, or volume. Phase diagrams also give insight into
the stability of a phase, and transformation reactions that occur from crossing monovariant, or
invariant phase boundaries. Although phase diagrams are a useful tool for visualizing binary, or
ternary systems, there is no real method of visualization of a multicomponent system other than to
reduce the dimensionality of the system to a pseudo-binary, or isoplethal diagram. While isopleths
are a useful representation of multicomponent system, unless the specific alloy composition has
previously been calculated, the researcher will have to do the thermodynamic calculations on their own.
For large systems this can become very tedious trying to provide interaction parameters for each
constituent interaction in each sublattice of each phase of the system. Thankfully a lot of work in the
past two decades has gone into building large databases to compile such data. CALPHAD and
ThermoCalc are the forerunners in compiling thermodynamic data, and providing computational
tools to simply, and efficiently be able to analyze predicted equilibria for specific material systems.

ThermoCalc is a tremendously powerful tool for metallurgists, and materials scientists, as its simplicity does not
require the user to have much understanding of material thermodynamics, but can provide a vast amount of
information, and insight into the alloy the user is dealing with. Whether there is sparse literature on the alloy,
or the user is looking to tweak certain variables to modify the the microstructure, ThermoCalc can vastly reduce
the time and money needed for experimentation, and may altogether eliminate the trail by error approach
utilized in the past.

In the present work, ThermoCalc was used to analyze how additions of both nitrogen, and titanium can affect
the equilibrium microstructure of a 2032Nb alloy, and which chemistry provides the most optimal
microstructure. With isoplethal sectioning only one component can be an independent variable while the others
remain constant. In a proper design matrix all possible permutations and combinations must be
encompassed. In a system with seven elements (chromium, nickel, niobium, silicon, carbon, manganese, and
nitrogen/titanium), hundreds of phase diagrams would need to be analyzed in order to determine the effects
each element, and their interactions have on phase stability, phase solubility, and the driving force of a phase. In
the following sections a method for analyzing, and optimizing the composition of a multicomponent system is
proposed with the use of ThermoCalc as a subroutine. The next section will discuss a proposed
Gibbs energy model for calculating equilibrium for a 2032Nb alloy, and a basic outline of how to
use the ThermoCalc console program will be provided. Afterwords, a proposed methodology for
compiling the data output by ThermoCalc will be presented, as well as ways of representing the data,
to ultimately draw conclusions on how composition of the alloy affects the systems equilibrium
microstructure.

### 2. Multicomponent Modeling of the 2032Nb System

From literature it is proposed the the 2032Nb system is primarily an austenitic solid solution of iron, chromium,
and nickel. (Nb,Ti)(C,N) carbides are the major precipitates during solidification, while in some cases
intradendritic M7Cc carbides are known to precipitate at higher carbon compositions [1–3]. During long-term
aging interdendritic M23C6 is known to precipitate, and a transformation of NbC to either M6C or G-phase will
occur depending primarily of the carbon concentration , and silicon concentrations [4–6]. Nitrogen additions are
proposed to also facilitate the precipitation of another intermetallic phase, Z-Phase [7–9]. With the major
components of the system being chromium, nickel, silicon, niobium, carbon an manganese, the proposed
equilibrium microstructure of the system will contain, austenite solid solution, NbC, M23C6, M7C3, and
G-Phase. Adding nitrogen to the system, Z-phase, and Nb(C,N) should be added to proposed
microstructure. Adding titanium to the microstructure TiC and M6C should be added to the proposed
microstructure. Defining which phases comprise the system to be analyzed is important for defining
Gibbs energy models, and for determining the accuracy, and validity of the ThermoCalc results.

Crystal structure information for each of these phases is critical for modeling Gibbs energy, as
statistical thermodynamics dominates the entropy of the system, and determining the ordering
and interactions of the constituents on each sublattice of the phase contributes significantly to
the overall energy of a phase. Table 1 outlines basic crystallographic information for each of the
phases in the proposed system, suchas the space groups, and number of sites for each phase. In the
next sections, Wyckoff positions and the variable parameters will be outlined for the appropriate
phases.

Phase | Structure | Type | Space Group | Formula | Lattice Parameter (Å) | Atoms per cell | Ref. |
---|---|---|---|---|---|---|---|

γ-Fe | Simple FCC | cF4 | Fm3m | - | 3.57 | - | [6],[10] |

Nb(C,N) | FCC (NaCl) | cF8 | Fm3m | (Nb,Ni,Fe,Cr)(C,N) | 4.41 | 8 | [6], [10], [11] |

NbC | FCC (NaCl) | cF8 | Fm3m | (Nb,Ni,Fe,Cr)C | 4.43-4.47 | 8 | [6], [10], [11] |

NbN | FCC (NaCl) | cF8 | Fm3m | (Nb,Ni,Fe,Cr)N | 4.38 | 8 | [6], [10], [11] |

M6C | Diamond FCC | - | Fd3m | (Cr,Fe,Nb,Ni,Si)6C3 | 10.95-11.28 | - | [12] |

M7C3 | Orthorhombic | oP40 | Pnma | (Cr,Fe,Mn)7(C)3 | a = 4.526 b = 7.010 c = 12.142 | 40 | [13] |

M23C6 | Complex Cubic (D84) | cF116 | Fm3m | (Cr,Ni,Fe,Nb)23C6 | 10.57-10.68 | 116 | [6], [10], [12], [14] |

G-Phase | Complex Cubic (D8a) | cF116 | Fm3m | (Ni,Fe)16Si7(Cr,Mn,Nb)6 | 11.2 | 116 | [6], [5], [15] |

Z-Phase | Tetragonal BCC | tP6 | P4/nmm | NbCrN | a = 3.04 c = 7.39 | 6 | [12], [10], [7] |

The next section will go through how ThermoCalc calculated equilibrium, and maps phase diagrams, and the subsequent sections will outline the Gibbs energy models for each of the proposed phases.

#### 2.1. Global Minimization

In a binary, or ternary system, equilibrium can be easily identified graphically by drawing the common tangent between the minimized Gibbs energy curves for each phase mathematically expressed as

| (1) |

This common tangent line, or plane can be formulated as

Stating that the chemical potential for each component must be equal for all phases. In general terms this can be written as

| (2) |

Gibbs phase rule can be derived from Eq. 2 as, F = c- 2 + p where F is the degrees of freedom, c is the
number of components, p is the number of phases, and the +2 represents temperature, and pressure variability.

For sublattice modeling commonly used in multicomponent systems, species are typically represented as
constituent fractions of constituent i in sublattice s of phase α, y_{i}^{(α,s)}, instead of mole fractions, x_{i}, with the
relationship

| (3) |

where b_{ij} is the stoichiometric factor of component i in constituent j, j represents the summations
of all components, and a^{(s)} are the fraction of sites on sublattice s. The constituent fraction can
also be defined as y_{i}^{s} = N_{i}^{s}∕N^{s}, “where N_{i}^{s} is the number of sites occupied by constituent i on
sublattice s and N^{s} is the total number of sites on sublattice s” [16]. With G_{m}^{α} given as a function of
site fractions for multicomponent systems, a Lagrange-multiplier method consisting of a set of
non-linear equations is used to calculate equilibrium instead of Eq. 2. The constraints are:

where b_{k,i}^{(α,s)} are the stoichiometric numbers of component i in species k on sublattice s of phase α, ν_{k}^{(α,s)} are
the charges of ionized species k on sublattice l of phase α, and μ_{i}, π^{(α,s)} and π_{e}^{(α)} are the Lagrange multipliers
for Eq. 4,Eq. 5 and Eq. 6 respectively. Eq. 7, and Eq. 8 are derived by first
multiplying the respective Lagrange multipliers to Eq. 4,Eq. 5 and Eq. 6, adding them to
the total Gibbs energy in Eq. 1 and taking their derivatives with respect to y_{k}^{(α,s)}, and N_{i}. The
unknowns to this set of non-linear equations are m^{α}, y_{i}^{(α,s)}, μ_{i}, π^{(α,s)} and π_{e}^{(α)}, as well as T, p, and N_{i}. A
solutions to this set of equations can be found by employing the Newton-Raphson method (for more information
on this method read [16]).

In higher order systems the common tangent is now characterized as a hyperplane of multidimensional space, where a global minimization procedure is carried out. To ensure that this global minimization procedure chooses the correct starting value for the Newton-Raphson calculation to give the global equilibrium, and not a metastable equilibrium, the Gibbs energy curve for all phases are approximated to be equal to the solution phase at the set composition, by dividing it into a dense grid of compounds and then finding the hyperplane that best represents equilibrium. These compounds along the hyperplane are then identified with which stable phase they are from. If multiple compounds are inside the same phase designated by ThermoCalc, the program will automatically create a new composition set. For example if two compounds are inside the FCC_A1 phase they will be split into FCC_A1#1, and FCC_A1#2 and given separate constitutions. In the 2032Nb system the matrix phase is austenite and designated as FCC_A1#1, while any NbC, or TiC phase which is also of FCC structure, will be designated as FCC_A1#2.

#### 2.2. Driving force, phase stability, and mapping procedures

Other useful thermodynamic properties that can give insight into how certain phases behave are the driving
force of a phase, which relates to the precipitation rate and kinetics of a phase, and the phases stability range.
Hillert, 2007 [17] makes the comparison of the driving force and stability to a rotating body where each of these
variables can be expressed in terms of its potential energy, where driving force = -dE∕dθ, and stability
= 2d^{2}E∕dθ^{2} for angle θ. Figure 1 taken from Hillert, 2007 [17], shows how the driving force, and stability of an
ellipical mechanical analogue, and a square analogue can determine if an equilibrium is stable or unstable.
For the square its stability is always negative so the equilibrium of the square is always unstable.
Whereas for the ellipse the stability is positive when θ = 0. When the driving force is positive and
increasing a stable, or metastable equilibrium is being reached. When the driving force becomes
negative the system is moving away from equilibrium. Hillert further went on to derive the stability
condition

| (9) |

Eq. 9 can be expressed as the derivative of any potential variable with respect to its conjugate variable, however relating stability to the chemical potential of a phase is the most beneficial form for determining a phases stability. ThermoCalc uses Eq. 9 to determine which phase are stable, and which are unstable when calculating the equilibrium of a system. The stability range of a phase can then be determined where the stability condition crosses zero, which can be useful when analyzing Gibbs energy curves.

The driving force of a metastable phase can be determined through the distance between the stable tangent plane, and the tangent plane parallel to the metastable phase illustrated in Figure 2. It can be seen that the FCC phase has a larger driving force than σ phase.

mapping searches for all equilibria that are monovariant or invariant, which show up as either lines or points on
the final isoplethal phase diagram. ThermoCalc starts the mapping procedures with an “initial equilibrium”
provided by the user at a specific T,P, and x_{i}. Newton-Raphson calculations are employed to solve equations
Eq. 4-Eq. 8, where the stable set of phases is determined at each iteration by calculating their
driving forces, where ThermoCalc enters stable phases, and suspends metastable phase based off of the stability
criteria in Eq. 9. Equilibrium for each set of stable phases is calculated in a stepwise manner
incrementing the predetermined x-axis variable until an invariant point is reached, whereby a new set of stable
phases are entered into equilibrium. At invariant equilibrium c + 1 monovariant equilibria are known to stem
from this point, where ThermoCalc chooses the next monvariant equilibria to trace, and stores the
others to be traced later. This procedure of tracing is continued until the axis limits have been
reached.

#### 2.3. Gibbs free energy models

Modeling the appropriate Gibbs energy equation for each phase is primarily attributed to how the constituents are bonded, and their configuration in each phase. For phases where the constituents are randomly mixed, and disordered, an ideal substitutional model which is most common of ideal gas phases, and the substitutional regular solution model most common for liquid phases and solution phases should be considered. For crystalline solids with different sublattices and Long Range Ordering (LRO) effects, a more complicated model called the Compound-energy formalism (CEF) should be considered where each compounds or end-member has its own Gibbs energy of formation [16]. In general terms the total Gibbs energy of a phase is expressed as

| (10) |

where ^{srf}G_{m}^{α} represents the reference state of the unreacted mixture of constituents of a phase, ^{cnf}S_{m}^{α} is the
configurational entropy of a phase based on the number of possible arrangements of constituents mathematically
represented as S = k ⋅ln(W), and ^{E}G_{m}^{α} is the excess Gibbs energy term. The configurational entropy for the
constituents in a phase will be assumed to undergo random mixing for each of the sublattices of a
phase.

##### 2.3.1. Model for liquid phase & austenite solid solution phase

The iron liquid phase can be described as a substitutional solution with Redlich-Kister excess binary contributions, where the general Gibbs energy formula is derived as

The binary interaction parameter term is extended to the multicomponent system with binary excess
contributions from each constituent pair and calculated via Eq. 13. This equation is
also known as the Redlich-Kister (RK) power series. ^{ν}L_{ij} is an experimental parameter that is
temperature dependent and can be expressed linearly by ^{ν}L_{ij} = ^{ν}a_{ij} + ^{ν}b_{ij}T. ν is generally ≤ 3,
where a subregular solution model is where ν = 2, and a subsubregular model is where ν = 3. The
regular-solution model describes every constituent composing the liquid to have an equal probability of
occupying any site in the unit cell of the phase. Constituent interactions in this model are limited
to only binary interactions, where higher-order interactions are disregarded, and assumed to be
insignificant. The i, and j indices represent the components of the system (Cr,Ni,Nb,Si,C,Mn, and
N/Ti).

##### 2.3.2. General compound-energy formalism model

For carbide, and intermetallic phases the constituents of the phase are ordered and grouped into specific sublattices that will effect the configurational entropy, and the enthalpy of mixing of phase where the substitutional regular solutions model is no longer valid. To address the Gibbs energy of sublattice models Hillert and Staffanson (1970) derived the compound-energy formalism (CEF). The general CEF model expands Eq. 10 out, and defines its individual parts as

where I_{0} is the constituent array of zeroth order which specifies one constituent in each sublattice(eg. i:j:k),
P_{I0} is the product of the constituent fractions of I_{0} (eg. y'_{i}y''_{j}y'''_{k}), ^{o}G_{I0} is the Gibbs energy of
formation of compound I_{0}, and L_{I1} and L_{I2} are the interaction parameters of the first order, and
second order component arrays respectively. The zeroth order constituent array I_{0} represents the
energy contribution from each end-member compound in a phase as an individual, and unmixed
species. The excess Gibbs energy term represents the energy contribution from the mixing for these
end-members, where the final composition of a phase may contain multiple constituents in a phases
sublattice. If only binary interactions between two constituents in a sublattice are significant, the
binary excess model is suitable for describing the excess Gibbs energy term where it is formulated
as

| (17) |

Where L_{ij} is calculated from the RK power series. In most cases even with more than two constituents in a
sublattice a binary excess model is sufficient. Extrapolations from ternary to binary interactions can be made
through models such as the Toop method, and Kohler method discussed further in Lukas et.al. [16]. However, if
a ternary interaction parameter must be addressed the following ternary excess model can be used

_{ij}is calculated from the RK power series. Eq. 19 is a composition dependent parameter , where if the composition lies in the center of the ternary constitutional triangle the ternary term will have the largest contribution. lastly, the ν

_{i}terms should be equally to unity.

##### 2.3.3. Model for (Nb,Ti)C, and Nb(C,N)

(Nb,Ti,Fe,Cr)_{1}(C,N)_{1} has an FCC crystal structure for the substitutional sublattice containing
niobium, and titanium, while the carbon and nitrogen interstitial sublattice occupies octahedral
sites in the unit cell which can be inferred from Figure 3. In the CEF model it is proposed to
add vacancy constituents to the interstitial sublattice, where the crystal structure now becomes
(Nb,Ti,Fe,Cr)_{1}(V a,C,N)_{1}. For the vacancy constituent its chemical potential is regarded to be
equal to zero. Expressing the site fraction of vacancies in the interstitial sublattice is expressed
as

| (20) |

The resulting sublattice model is (Nb,Ti,Fe,Cr)_{1}(V a,C)_{1}, and (Nb,Fe,Cr)_{1}(V a,C,N)_{1} with the additions of
titanium and nitrogen to the system respectively. The general formulation of a two sublattice CEF model for
(Nb,Ti)(C,N) is

| (21) |

where the a_{i} terms can be replaced with 0.5 which described the site occupancy of each of the two sublattices.
Eq. 21 can be expanded out as

A visual representation of a (Nb,Ti)_{1}(C,V a)_{1} model is shown in Figure 4, where the Gibbs energy of formation
is calculated for each end-member, and the interaction parameters are calculated for each interaction between
constituents.

##### 2.3.4. Model for M23C6, and M7C3

M23C6 has a sublattice model of (Cr,Fe,Ni)_{20}(Cr,Fe,Nb)_{3}(C,N)_{6}, and M7C3 has a model of
(Cr,Fe,Mn)_{7}C_{3}. The equivalent position parameters as well as their Wyckoff positions are listed in Table 2.
M7C3 will be modeled after the two-sublattice model, Eq. 21, similar to the Nb(C,N) model. If the
the ternary excess model in Eq. 18 is chosen instead of the binary model the excess term for M7C3
will take the form.

| (27) |

M23C6 consists of three sublattices, therefore the general form for the Gibbs energy expression will be

| (28) |

where

##### 2.3.5. Model for G-phase

G-Phase has the sublattice model (Fe,Ni)_{16}Si_{7}(Cr,Mn,Nb,Ti)_{6}, and a crystal structure represented in
Figure 6 which was composed with information from Holman et al., 2008 [15]. It should be noted that G-phase
is currently not implemented in any of the iron databases of ThermoCalc, and was appended from a nickel
database to be used in the study analyzed in this report. Since nickel is a constituent of the first sublattice
interaction parameters, as well as the Gibbs energy reference state may greatly over/under-estimated. Since
the stability of the other carbides and intermetallics in the system are dependent on G-phase the
relativistic effects should not be changed, and only the absolute equilibrium will differ from reality.
Futhermore, appending G-phase from the nickel database causes the manganese constituent to occupy
the majority of the sites on the third sublattice, which from the literature [6, 20, 21] niobium is
suppose to be the dominant occupant. Manganese should then be suspended from the sublattice
model in ThermoCalc, where the model is now (Fe,Ni)_{16}Si_{7}(Cr,Nb,Ti)_{6}. The model for three
sublattices described in Eq. 28, and Eq. 29 will be used to formulate its Gibbs
energy.

##### 2.3.6. Model for Z-phase

Z-Phase has the sublattice model (Nb)_{1}Cr_{1}(N)_{1} and crystal structure shown in Figure 7, which contains only
one constituent for each of the sublattices. This simplifies the Gibbs energy expression greatly where ^{E}G_{m} = 0
since no mixing of constituents is involved, and the surface of reference term and the configurational entropy
term are described as

| (30) |

The filled thermodynamic properties for Z-phase can be obtained from Denielsen and Hald, 2007 [22].

### 3. References

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[2] Berghof-Hasselbcher, E., Gawenda, P., Schorr, M., Schtze, M., Hoffman, J.. Atlas of Microstructures. Materials Technology Institute; 2008.

[3] Nishimoto, K., Saida, K., Inui, M., Takahashi, M.. Changes in microstructure of hp-modified heat-resisting cast alloys with long term aging. repair weld cracking of long term exposed hp-modified heat-resisting cast alloys. (report 2). Quarterly Journal of the Japan Welding Society 2000;18(3):449–458.

[4] Nishimoto, K., Saida, K., Inui, M., Takahashi, M.. Mechanism of hot cracking in haz of repair weldments. repair weld cracking of long term exposed hp-modified heat-resisting cast alloys. (report 3). Quarterly Journal of the Japan Welding Society 2000;18(4):590–599.

[5] Powell, D.J., Pilkington, R., Miller, D.A.. The precipitation characteristics of 20% cr/25% ni—nb stabilised stainless steel. Acta Metall 1988;36(3):713–724.

[6] Chen, Q.Z., Thomas, C.W., Knowles, D.M.. Characterisation of 20cr32ni1nb alloys in as-cast and ex-service conditions by sem, tem and edx. Mater Sci Eng, A 2004;374(1-2):398–408.

[7] Danielsen, H.K., Hald, J.. On the nucleation and dissolution process of z-phase cr(v,nb)n in martensitic 12%cr steels. Mater Sci Eng, A 2009;505(1-2):169–177.

[8] Danielsen, H., Hald, J.. Influence of z-phase on long-term creep stability of martensitic 9-12%cr steels. In: 9th Liege Conference on Materials for Advanced Power Engineering. 2010, p. 310.

[9] Sourmail, T., Bhadeshia, H.. Microstructural evolution in two variants of nf709 at 1023 and 1073 k. Metall Mater Trans A 2005;36(1):23–34.

[10] Erneman, J., Schwind, M., Liu, P., Nilsson, J.O., Andrén, H.O., Ågren, J.. Precipitation reactions caused by nitrogen uptake during service at high temperatures of a niobium stabilised austenitic stainless steel. Acta Mater 2004;52(14):4337–4350.

[11] Pickering, F.B., Keown, S.. Niobium in stainless steels. In: Stuart, H., editor. Niobium, Proceedings of the International Symposium. Warrendale, PA: Met. Soc. AIME; 1981, p. 1113–1142.

[12] Sourmail, T.. Literature review precipitation in creep resistant austenitic stainless steels. Mater Sci Technol 2001;17(January):1–14. URL http://www.thomas-sourmail.org/papers_html/precipitation_review/precipitation_review.pdf.

[13] Xiao, B., Xing, J.D., Feng, J., Zhou, C.T., Li, Y.F., Su, W., et al. A comparative study of cr 7 c 3 , fe 3 c and fe 2 b in cast iron both from ab initio calculations and experiments. Journal of Physics D: Applied Physics 2009;42(11):115415.

[14] Jo, T.S., Lim, J.H., Kim, Y.D.. Dissociation of cr-rich m23c6 carbide in alloy 617 by severe plastic deformation. J Nucl Mater 2010;406(3):360–364.

[15] Holman, K.L., Morosan, E., Casey, P.A., Li, L., Ong, N.P., Klimczuk, T., et al. Crystal structure and physical properties of mg6cu16si7-type m6ni16si7, for m = mg, sc, ti, nb, and ta. Mater Res Bull 2008;43(1):9–15.

[16] Hans Lukas, S.G.F., Sundman, B.. Computational Thermodynamics. Cambridge University Press; 2007.

[17] Hillert, M.. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis. Cambridge University Press; 2007.

[18] Liu, Z.K.. Computational thermodynamics using thermo-calc; 2010. ThemoCalc & Dictra Training Course.

[19] Song, Y.Y.. Thermodynamic study on b and fe substituted cr23c6 using first-principles calculations. Ph.D. thesis; Pohang University of Science and Technology; 2010.

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[22] Danielsen, H.K., Hald, J.. A thermodynamic model of the z-phase cr(v, nb)n. Calphad 2007;31(4):505–514.

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### 4.Glossary *

- constituent
- Element, or species, that occupies a specific sublattice of a specific phase. A phase can also
be considered as a constituent of the total system.. 5, 14
- end-member
- The final chemical formula of a stable or metastable phase whose sublattice(s) are
occupied by single constituents. For example M23C6 is an end member of (Cr,Ni,Fe,Nb)
_{23}C_{6}.. 8 - factor
- The independent variable of a factorial design. 23
- main effect
- How much the change in an individual factor effects the change in the response variable of
a factorial design.. 24
- replicate
- Independent repetition of a treatment in a factorial experiment. 26

- response variable
- The dependent variable of a factorial experiment, or a regression model.. 24
- ThermoCalc
- A computational thermodynamics program that can calculate equilibrium phase diagrams
for multicomponent systems, as well as Scheil simulations, and various thermodynamic properties
(C
_{p}, ΔH_{m}, ΔG_{m}etc...). 3 - treatment
- A specific level of a factor in a factorial design.. 24

### 5. Acronyms *

### 6. Nomenclature *

- β
_{j} - the effect of the ith level of factor ‘B’
- ϵ
_{ijk} - random error component
- ∀
- for all instances of ...
- ∈
- in a set ...
- μ
- Overall mean effect
- μ
_{i} - Chemical potential of component or end-member i
- ψ
_{k} - the effect of the ith level of factor ‘C’
- τ
_{i} - the effect of the ith level of factor ‘A’
- G
- total Gibbs energy; G = ∑
_{α}m^{α}·G_{m}^{α} - G
_{i}^{α} - partial Gibbs energy of component i in phase α; G
_{i}^{α}=_{T,P,Nj} - G
_{m}^{α} - integral molar Gibbs energy of a phase
- I
_{i} - constituent array of order i
- L
_{I} - interaction parameter of compound I
- m
^{α} - fraction of a phase
- N
_{i} - moles of component i
- R
- gas constant, 8.314Jmol
^{-1}K^{-1} - R
^{2} - coefficient of multiple determination
- S
_{m}^{α} - molar entropy of a phase
- T
- Temperature (K)
- x
_{i} - total mol fraction of component i; x
_{i}= ∑_{α}m^{α}· x_{i}^{α} - x
_{i}^{α} - mole fraction of component i in phase α

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