CC BY
29
Ga = xA0°Àa + xBG0'a + RT ( xA ln xA + xB ln xB )+
+Ge
UDC 546.863.22
DIRECT AND INVERSE PROBLEMS OF THERMODYNAMICS PHASE EQUILIBRIA IN INORGANIC SYSTEMS
A.N.Mamedov, E.R.Tagiev
M.Nagiyev Institute of Catalysis and Inorganic Chemistry, NAS of Azerbaijan
[email protected] Received 30.03.2016
The current researches were reviewed on the thermodynamic calculations and computer design phase diagrams of binary and multicomponent inorganic systems. Mathematical apparatus for solving direct and inverse problems of thermodynamics of phase equilibrium was analyzed. Calculation methods have been approbated on a large number of metal, semiconductor, oxide and salt systems. The works on 3D modeling of phase diagrams and thermodynamic functions, calculation of phase equilibria for alloy na-noparticle in contact with a solid nanowire have been analyzed. The works on the calculation of phase diagrams based on first-principles have been reviewed.
Keywords: thermodynamic calculations, phase diagram, inorganic systems, 3D modeling.
Introduction
The phase diagram is a reflection of the thermodynamic state of the system. Therefore, coordinates (temperature, composition) and parameters of thermodynamic equilibrium phases are interconnected. The thermodynamic equations of phase diagrams are based on the equality of the chemical potentials of the components in the equilibrium phases. In the presence of complete thermodynamic data for all possible phases can calculate of the coordinates (temperature-composition T-x) phase diagram. If this is referred to direct problem of thermodynamics of phase equilibrium, the calculation of the thermo-dynamic properties based on the state of the phase diagrams can be called inverse problem. Frequently these problems are solved jointly. Mathematical apparatus of phase equilibrium thermodynamics has been systematized by Kaufman and Bernstein [1], is widely developed and used for computer calculation of phase diagrams of one- and multicomponent systems. Create a package licensed program: Thermo-Calc. Software System (Figure 1). At the same time, relatively simple methods are developed for thermodynamic calculations and solving of the inverse problem [2].
Thermodynamic modeling
For an alloy system A-B, in which X represents the atom fraction of B and which exhibits two competing phases a and P, the free energies of each phase may be represented by the following equations [3]:
(1)
GP = xAG0'p + xBG0'p + RT ( xA ln xA+xB ln xB ) + +Gexc'P. (2)
In (1) and (2) GA,a and GA,p are the free energies of the a and P modifications of the pure element A, while GBa and GBP are the
free energies of the a and P modifications of the pure element B. The excess free energies of mixing of the a and P phases are expressed as Gexc,a and Gexc,P and x - is the atom fraction.
For the liquid solution A and B the equation is written in the form:
Gliq = xAG0M^v
_l_^rexc,liq
xaGa + xbGb + RT ( xA ln xA+xB ln xB )+
(3)
G^a, G^P, G%a, GB,p ; G^ and GBJiq - the molar free energies of pure A and B are taken from the SGTE pure element database. The Gibbs free energy of pure elements as a function of temperature G0(T) = Gi (T) - HfER (GHSER) is represented by Equation (4):
G0(T )=a+bT+cT ln(T )+adT2 +eT + fT3 + +iT 4 + jT7 +kT ~9. (4)
Fig. 1. Flowchart of the CALPHAD method (Matvei Zinkevich. Theoretical and Computational Materials Science. IMPRS-1st Summer School in Stuttgart, June 2-6, 2003).
The g°(t) data are referred to the constant enthalpy value of the Standard Element
Reference HfEK at 298.15 K and 1 atm. as recommended by Scientific Thermodata Europe (SGTE) [4, 5]. The free energies of multicom-ponent solution may be represented by the following equation:
Gm (T)=2 xG(T)+RT2 X ln(x . (5)
i i r-Tii p • exs •
The excess free energies G in equations (1)-(3) and (5) consists of partial molar quantities of components and related to the ther-modynamic activity and activity coefficient:
Gexs =£ x&T =2 xRT ln(y i )=
= ^ xRTln(a /x )•
(6)
If the solution is formed from non-molecular inorganic compounds, such as ApBq,
A-rCf or more compounds, equations (5) and (6) are written in the form:
Gm (T )=£ xG (T)+RT 2 x ln f (x ^, (7)
i i
Gexs =2 xi Gexs =2 xRT ln(Yi )=2 xRTx
x
Xln[(^ /f(x )], (8)
f(xi ) - is function for converting the thermodynamic equations of phase equilibrium applied to the systems of non-molecular compounds [6-14].
The excess free energies in the absence of reliable experimental data is based on model representations of solutions, for example Red-lich-Kister polynomial in the form [15, 16]:
G = xAxB 2 L^B (xA xB) ;
m=0
m LLlqB = a + bT •
n
For the regular solution model
gexs = t x x
g LA-B xA xB • m^Liq
(10)
In (7), (8) ' LAB (m = 0, 1, 2, ...) is the
binary interaction parameter. A and B are the coefficients to be evaluated on the basis of available experimental data.
The thermodynamic functions of formation of compounds and solutions from the pure components. If in the equations (1)-(8) to deduct from the left side of the free energy of pure substances, get the equations for the ther-modynamic functions of formation of compounds and solutions from the pure components. For example:
aGm = Gm (T )-£ xG (T )=RT 2 x, ln( x,) +
i i
+Gmxc. (11)
The partial molar free energies of A
and B in an alloy containing x atom fraction B are defined as
GA = G - XIX • (12)
G B = G + (1 - x) H . (13)
For the partial free energy of mixing
dAG
AGa =AG - x -
dx
AGb =AG + (1 - x)
dAG dx
(14)
(15)
It should be noted that the partial molar free energy component, is its chemical potentials:
G, = ^ and AG, = A|u, = ^ . (16)
The partial free energy of a given component is the same for each phase under equilibrium conditions (Figure 2). The condition can be expressed by a set of nonlinear equations such as
ga= gp . (17)
The equilibrium for a system is represented by the condition that the molar free energy is minimum (Figures 2-4).
G
Ga = Ga
v \ n! » )
\ a
a a+ß ß
Gb = Gb
A
atom fraction
B
Fig. 2. Common tangent construction to define equilibrium between two phases.
Mol. fraction
Fig. 3. Free energy surfaces of competing phases in ternary system and application of
—a —ß
the common tangent principle [3]; G B = G b
Mol. fraction
Fig. 4. Schematic representation of the free energy surface for miscibility gaps at two binary edges [3].
Some of the most interesting works on the calculation of phase diagrams of various types. In [15] thermodynamic assessments of the Eu-Te and Te-Yb binary systems were carried out by using the CALculation of PHase Diagrams (CALPHAD) method based on the available experimental data including thermodynamic properties and phase equilibria. Reasonable models were constructed for all the phases of the two systems. The liquid phases were described by the substitutional solution model with the Redlich-Kister polynomial. The three intermetallic compounds, Eu4Te7, Eu3Te7 and YbTe in the two systems, were treated as stoichiometric pha-ses, while the non-stoichiometric phase (EuTe), which has an homogeneity range, was treated by a two-sublattice model following the schema: (Eu,Te)o.5(Te)o.5. A consistent set of thermodynamic parameters leading to reasonable agreement between the calculated results and experimental data was obtained. Figure 5 shows the calculated Eu-Te phase diagram compared with the experimental data. The phase diagram agrees well with the experimental data.
In [17] melting in the La2O3-Y2O3-Al2O3 system was investigated using high temperature DTA and microstructures were studied by electron microscopy (SEM/EDX). The experimental results are consistent with calculations based on liquid phase description without ternary interactions and solid phases with limited mutual solubi-
lity of La and Y. The thermodynamic description was used to calculate vertical sections at 20 mol. % Al2O3 and at fixed ratio of La2O3/Y2O3=1 as well as solidus surface. The thermodynamic functions for phases were accepted from work [18]. Compound energy formalism [19] taking into account crystal structure and site occupancies in the phases was applied to describe solid solutions in the system. Liquid phase was described by partially ionic liquid model. Calculation of phase diagram was performed using Thermo-calc program set [20]. Liquidus surface for the La2O3-Y2O3-Al2O3 system was calculated in work taking into account solubility of La2O3 in YAG (Y1-xLax)3Al5O12, YAP (Y^Lax)AlO3 and YAM (Y1-xLax) or YAM-4 Al2O3-4AhO9 phases as well as Y2O3 solubility in LaAP (La1-xYx)AlO3 phase. Solubility of Al2O3 in LaYP (La1-xYx)(Y1-^Lay)O3 phase and solubility of Y2O3 in the P-alumina structure LaAl11O18 were not accounted because they are too small according to experimental data.
In [21] thermodynamic consequences of sketching rectilineal phase boundaries are analyzed. It is shown that although thermodynamics does not prohibit such boundaries, they result in peculiar temperature dependencies of components lattice stabilities. An idiosyncrasy of these functions and restrictions imposed by them are worth keeping in mind if it is intended to pencile T-x section on which all phase boundaries are straight lines.
400-
722.69 К
Fig. 5. Calculated phase diagram of the Eu-Te system compared with experimental data [15].
Calculation of phase equilibria for alloy nanoparticle in contact with a solid nan-owire [22]. We will analyze in more detail this interesting study. In [22] phase equilibria in a system constituted of an alloy nanoparticle in contact with a solid nanowire has been modelled based on the minimization of a Gibbs free energy function. The Gibbs free energy consists of a bulk, surface and interface contribution. The bulk contribution is taken from CALPHAD thermodynamic databases and the surface properties from the literature. The effect of particle size and surface and interfacial properties on the liquidus line of the Au-Ge and In-Si systems has been studied. The results are compared to the bulk phase diagram and phase equilibria calculated for nano-systems with different geometries.
The geometry and the dimensions of the systems in the initial and the final state are depicted in detail in Figure 6.
a b
Fig. 6 . System in the initial state (a); the particle is a
hemisphere, of radius RPart and the wire a cylinder of
radius Rwire and of length lWWire [22]; System in the final state (b ); the particle is a hemisphere of radius Rpart, the wire a cylinder of radius Rwire and of length lwire [22].
Both in the initial and in the final state, the nanowire and the nanoparticle are supposed to have a circular cross-section, although real
systems adopt faceted geometries, according to the Wulff construction. The nanoparticle is assumed to be a truncated sphere, with a contact angle p on top of the nanowire. In the initial
state, P; is determined based on experimental
observation. In the final state, P follows from
' 1 f
the calculated volume of the liquid nanoparticle [3]. The final contact angle is thus determined by geometric considerations.
The Gibbs free energy of small systems can be written as
^system _^bulk _^surfase
(18)
Gbulk - is the contribution from the bulk of the
material and G
surfase
- is the contribution of the
different surfaces or interfaces present in the system Au-Ge.
The total Gibbs free energy of the system in the initial state is the sum of the bulk and the surface - interface contributions:
^system_q
^surface
4< ^rt)2" 2<rt 4
*Yau+ysG°e+<ire YA
,u,Ge'
(19)
where y Al, Y °0l
are the surface energy density
of solid Au and Ge, y AUG is surface energy density of the interface between solid Au and Ge.
The Gibbs free energy of the final state is the sum of the bulk and the surface and interface contributions:
^system _^bulk ^ ^surface
+x°q(g°q - g°
«iiq [ XAuGu - ghser) +
) + RT ( x>x^ + x°>x°q) +
I vli^liq , v t fV'q _ v11^v
Au xGe + Lv ( xAu xGe )
■[4n(Rpart)2
art h] Y^lGe + 2nRw1relw1re YGe + nRwire YAu,Ge, (20)
where Y AU Y Gq are the surface energy density
of liquid Au and Ge, YAu,Ge is surface energy density of the interface between liquid Au and Ge, Y '¿G is surface energy density of the interface between solid Au and liquid Ge.
v
The transition temperature between the initial and the finale state cannot be called eu-tectic temperature since there are no three phases in the equilibrium. Furthermore, eutectic points may not exist for nanosystems. However, the transition temperature of the nano systems can be compared to the bulk eutectic temperature and the minimum semiconductor solubility in the liquid can be compared to the bulk eutec-tic composition. This comparison is relevant for the application to nanowires growth. The transition temperature between the final and the initial state is obtained by comparing the Gibbs free energy of these two states:
AG = GZT (XT"11, T) - G—, (21)
where G^em(X£equil,T) is the minimum Gibbs free energy of the final state at the temperature T. As long as AG is positive, the initial state is the stable state, when it becomes negative, the final state is the stable state.
Two liquidus lines are compared to the bulk liquidus line in Figure 7.
\ Bulk
\
\
Y /¿'¿s Liquidus-Mïiamond
/
Diamond-rfcc
1W1,1 1 -
D 02 0.4 0 5 0.9 1
Au Xg* Ge
Fig. 7. Calculated liquidus line of the Gerich side of the Au-Ge system for different wire radii [22].
The transition temperature is lower in nanosystems compared to the bulk eutectic temperature. It equals 523 K in the 10 nm system and 416 K in the 5 nm system whereas the bulk eutectic temperature equals 629 K. The minimum Ge solubility in the droplet also decreases, namely from 29 at.% Ge at the eutectic point for the bulk system to 24 at.% Ge slightly above the transition temperature in the 10 nm
system and to 20 at.% Ge in the 5 nm system. One can also note that at a given temperature, the Ge solubility in the liquid is more important in nanosystems than in the bulk system. Researches on phase diagrams of nanosystems are also carried out in the works [23-26].
A solution-based thermodynamic description of the ternary Ni-Al-Mo system is developed in [27], incorporating first-principles calculations and reported modeling of the binary Ni-Al, Ni-Mo and Al-Mo systems. To search for the configurations with the lowest energies of the N phase, the Alloy Theoretic Automated Toolkit (ATAT) was employed and combined with VASP [28-30]. The liquid, bcc and y-fcc phases are modeled as random atomic solutions, and the y-Ni3Al phase is modeled by describing the ordering within the fcc structure using two sublattices.
An improvement of the thermodynamic description of the Fe-B, Fe-Si-B, Nd-Fe-B, Fe-Si-B-P system by means of Calphad method has been carried out in works [31-37] considering not only the equilibria involving the stable Fe2B phase but also the metastable ternary equilibria in which the Fe3B phase occurs. Furthermore, the glass transition is introduced in the Calphad framework as a second-order one using the tools provided by the Hillert-Jarl formalism of the ferromagnetic transitions not yet applied to Fe-B and Fe-Si-B. The assessments have been made using data available in the literature regarding both the amorphous and crystalline phases. The results improve the previous ones for the glassy phase while keeping the agreement with experimental data concerning stable equilibria.
3D-modeling of phase diagrams
Phase complex of ternary systems in form computer 3D-model in T-x-y coordinates has the goods on traditional methods presentation of multicomponent systems, because it describes phase transitions and equilibria to the full, it's informative method of record retention about system based experimental data. Based T-x-y-model we can construct required poly-thermal and isothermal sections, liquidus surface isotherms, calculate material phase balance
of ternary system, make a forecast unstudied systems [38]. In [39] is constructed 3D-model of the phase diagram system LiF-KI-Li2Cr04. This system is stable triangle of quaternary reciprocal system Li,K||F,I,CrO4. Each phase equilibria volume is painted in its colour for visual expression (Figure 8).
In [40] 3D models of T-x-y diagrams are designed first as virtual models (prototypes), which formally correspond to their geometrical construction only. Then the prototype is filling with experimental data and is converted into the model of real system. But if data do not have single-valued interpretation, then it is possible to construct all possible versions of phase diagram. For instance, liquidus of the system V-Zr-Cr=A-B-C with the compound V2Zr=R and two Ri, R2 (or three Ri, R3, R2) polymorphous modifications of the compound ZrCr2 consists of surfaces of primary crystallization of Zr, solid solutions V(Cr) and R(R2) - the compound V2Zr=R with the low-temperature modification R2 plus the high-temperature modification R1 (Figure 9).
In [41] for determining and modeling these properties thermodynamic equations were used which were solved based on thermody-namic information obtained from the phase diagram of the binaryboundary systems by using a
300'C
Fig. 8. Solid state 3D-model of phase complex of the ternary system LiF-KI-Li2CrO4 with polythermal section [39].
limited number of experimental data for the ternary system. The calculation and visualization of immiscibility and crystallization surfaces of phases in the above state systems were performed by the program grafikus.ru/plot3d. In Figure 10 the liquidus surface of Ge in Cu-Tl-Ge system is presented. At the calculations the limited number of experimental data for the liquidus surface of Ge was used.
In [10] the calculation and 3D modeling of the partial Gibbs free energy and YbTe liqui-dus, as well as the crystallization surface of solid solutions based on (Sb2Te3)y(Bi2Te3)1-y in the ternary YbTe-Sb2Te3-Bi2Te3 system were carried out by using the equations of thermodynamics of non-molecular compounds. The equations were solved on the basis of limited number of experimental points on the phase diagram and published data of boundary systems [43, 44]. The surfaces of the crystallization and partial excess Gibbs energy of YbTe were 3D modeled for high-temperature region (1000-2000) K in the concentrations range 0.22-1.0 mole fraction of YbTe on sections y = T - xYbTe) = 0-1
calculation and modeling of Gibbs energy and crystallization surfaces was performed by the program 0riginLab2015 and Grafikus.ru/ plot3d.
Fig. 9. First variant of the system V-Zr-Cr T-x-y diagram [40].
Fig. 10. The thermodynamic 3D computer modeling of surface crystallization of the Ge [41].
Fig. 11. 3D computer modeling of the liquidus and solidus surfaces of the YbTe-Sb2Te3-Bi2Te3 system equations (22), (23) [10].
In the YbTe-Sb2Te3 and YbTe-Bi2Te3 systems there are regions of solid solutions with xYbTe=0-0.22 and xYbTe=0-0.15 mole fraction of YbTe, respectively [42]. In the YbTe-Sb2Te3-Bi2Te3 system there is a field of crystallization of alloys on the basis (Sb2Te3)y (Bi2Te3)1-y in the concentration range xYbTe=0<x<0.22 and y=0-1. Liquidus and solidus surface in this part of the phase diagrams of YbTe-Sb2Te3-Bi2Te3 system are approximated by the following relationship was performed in Figure 11. T, K(likvidus)= (1326x3-3816x2+3774x-392)y+(-100x2+251x+709)(1-y), (22)
T, K(solidus)=(653x2-1014x+1254)y+(400x2-676x +1135)(1-y)-200x(1-x)2y(1-y) (23)
Conclusion
This review covers a small part of the research on the thermodynamics of phase equilibria. All equations for calculating the phase and database for the thermodynamic functions of pure substances and homogeneous mixtures are not given. At the same time, the presented overview shows that there are a lot of interesting problems of thermodynamics of phase equilibria for inorganic systems.
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QEYRi-ÜZVi SiSTEMLORDO FAZA TARAZLIGI TERMODiNAMiKASININ DÜZ
УЭ OKS MOSOLOLORÍ
A.N.Mammadov, E.RTagiyev
iki va goxcomponentli qeyri-üzvi sistemlarin faza diaqramlannin termodinamiki hesablanmasi va kompüter modella§dirilmasina aid i§larin xülasasi verilmi§dir. Faza tarazligi termodinamikasinin düz va tars masalalarinin hallinin riyazi aparati tahlil edilir. Hesablama üsullan metal, yarimkegirici, oksid va duz sistemlarinda sinaqdan kegirilir. Faza diaqramlannin va termodinamiki funksiyalarin 3D-modella§dirilmasi, arinti nanohissaciklarin nanonaqilla faza tarazligi, ilkin prinsiplar asasinda termodinamiki modella§maya aid i§lar tahlil edilir.
Agar sözlar: termodinamiki hesablamalar, hal diaqrami, qeyri-üzvi sistemhr, 3D modelh§m3.
ПРЯМЫЕ И ОБРАТНЫЕ ЗАДАЧИ ТЕРМОДИНАМИКИ ФАЗОВЫХ РАВНОВЕСИЙ В
НЕОРГАНИЧЕСКИХ СИСТЕМАХ
А.Н.Мамедов, Э.Р.Тагиев
Представлен обзор исследований по термодинамическому расчету и компьютерному моделированию фазовых диаграмм бинарных и многокомпонентных неорганических систем. Анализируется математический аппарат для решения прямых и обратных задач термодинамики фазового равновесия. Методы расчеты апробируются на большом количестве металлических, полупроводниковых, оксидных и солевых систем. Рассматриваются работы по 3Б-моделированию фазовых диаграмм и термодинамических функций, по решению уравнений фазовых равновесий для наночастиц сплавов в контакте с твердой нанопроволокой, исследования по расчету фазовых диаграмм из первопринципных позиций.
Ключевые слова: термодинамические расчеты, фазовая диаграмма, неорганические системы, 3D-моделирование.
