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.
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.
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.|
|Nb(C,N)||FCC (NaCl)||cF8||Fm3m||(Nb,Ni,Fe,Cr)(C,N)||4.41||8||, , |
|NbC||FCC (NaCl)||cF8||Fm3m||(Nb,Ni,Fe,Cr)C||4.43-4.47||8||, , |
|NbN||FCC (NaCl)||cF8||Fm3m||(Nb,Ni,Fe,Cr)N||4.38||8||, , |
|M7C3||Orthorhombic||oP40||Pnma||(Cr,Fe,Mn)7(C)3||a = 4.526 b = 7.010 c = 12.142||40|||
|M23C6||Complex Cubic (D84)||cF116||Fm3m||(Cr,Ni,Fe,Nb)23C6||10.57-10.68||116||, , , |
|G-Phase||Complex Cubic (D8a)||cF116||Fm3m||(Ni,Fe)16Si7(Cr,Mn,Nb)6||11.2||116||, , |
|Z-Phase||Tetragonal BCC||tP6||P4/nmm||NbCrN||a = 3.04 c = 7.39||6||, , |
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.
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
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
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, yi(α,s), instead of mole fractions, xi, with the relationship
where bij 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 yis = Nis∕Ns, “where Nis is the number of sites occupied by constituent i on sublattice s and Ns is the total number of sites on sublattice s” . With Gmα 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 bk,i(α,s) are the stoichiometric numbers of component i in species k on sublattice s of phase α, νk(α,s) are the charges of ionized species k on sublattice l of phase α, and μi, π(α,s) and πe(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 yk(α,s), and Ni. The unknowns to this set of non-linear equations are mα, yi(α,s), μi, π(α,s) and πe(α), as well as T, p, and Ni. A solutions to this set of equations can be found by employing the Newton-Raphson method (for more information on this method read ).
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.
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  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 = 2d2E∕dθ2 for angle θ. Figure 1 taken from Hillert, 2007 , 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
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 xi. 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.
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 . In general terms the total Gibbs energy of a phase is expressed as
where srfGmα represents the reference state of the unreacted mixture of constituents of a phase, cnfSmα is the configurational entropy of a phase based on the number of possible arrangements of constituents mathematically represented as S = k ⋅ln(W), and EGmα 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.
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. νLij is an experimental parameter that is temperature dependent and can be expressed linearly by νLij = νaij + νbijT. 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).
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 I0 is the constituent array of zeroth order which specifies one constituent in each sublattice(eg. i:j:k), PI0 is the product of the constituent fractions of I0 (eg. y'iy''jy'''k), oGI0 is the Gibbs energy of formation of compound I0, and LI1 and LI2 are the interaction parameters of the first order, and second order component arrays respectively. The zeroth order constituent array I0 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
Where Lij 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. . However, if a ternary interaction parameter must be addressed the following ternary excess model can be usedLij 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 νi terms should be equally to unity.
(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
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
where the ai 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.
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)7C3. 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.
M23C6 consists of three sublattices, therefore the general form for the Gibbs energy expression will be
G-Phase has the sublattice model (Fe,Ni)16Si7(Cr,Mn,Nb,Ti)6, and a crystal structure represented in Figure 6 which was composed with information from Holman et al., 2008 . 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)16Si7(Cr,Nb,Ti)6. The model for three sublattices described in Eq. 28, and Eq. 29 will be used to formulate its Gibbs energy.
Z-Phase has the sublattice model (Nb)1Cr1(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 EGm = 0 since no mixing of constituents is involved, and the surface of reference term and the configurational entropy term are described as
The filled thermodynamic properties for Z-phase can be obtained from Denielsen and Hald, 2007 .
 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.
 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.
 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.
 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.
 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.
 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.
 Minitab, . Technical support document rank deficiency; 2010. http://www.minitab.com/support/documentation/Answers /RankDeficiency.pdf.
- 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
- 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)23C6.. 8
- 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
- Independent repetition of a treatment in a factorial experiment. 26
- response variable
- The dependent variable of a factorial experiment, or a regression model.. 24
- A computational thermodynamics program that can calculate equilibrium phase diagrams for multicomponent systems, as well as Scheil simulations, and various thermodynamic properties (Cp, ΔHm, ΔGm etc...). 3
- A specific level of a factor in a factorial design.. 24
- the effect of the ith level of factor ‘B’
- random error component
- for all instances of ...
- in a set ...
- Overall mean effect
- Chemical potential of component or end-member i
- the effect of the ith level of factor ‘C’
- the effect of the ith level of factor ‘A’
- total Gibbs energy; G = ∑ αmα ·Gmα
- partial Gibbs energy of component i in phase α; Giα = T,P,Nj
- integral molar Gibbs energy of a phase
- constituent array of order i
- interaction parameter of compound I
- fraction of a phase
- moles of component i
- gas constant, 8.314Jmol-1K-1
- coefficient of multiple determination
- molar entropy of a phase
- Temperature (K)
- total mol fraction of component i; xi = ∑ αmα · xiα
- mole fraction of component i in phase α
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