Thermodynamic Models of Condensed Phases and Their Application in Materials Science Текст научной статьи по специальности «Физика»

DOI: 10.17277/amt.2018.01.pp.041-046

Thermodynamic Models of Condensed Phases and Their Application in Materials Science

B.B. Khina

Physical-Technical Institute of the National Academy of Sciences of Belarus, 10, Kuprevicha St., Minsk, 220141, Belarus

Corresponding author: Tel.: +375 17 369 86 99. E-mail: [email protected]

Abstract

The review describes the main directions of thermodynamic modeling in materials science and gives a brief description of popular software products. The models used for the description of thermodynamic properties of pure substances and condensed solutions are presented. The regular solution model, which is used for multicomponent melts and disordered substitutional solid solutions, is described along with the Hillert-Staffanson model used for ordered substitutional solid solutions, intermetallics and interstitial phases. The use of the semi-empirical Miedema model for the estimation of the formation enthalpy of complex compounds, whose thermodynamic data are not available in literature, and for metastable (amorphous) phases is described. The method for calculating the excess entropy of mixing of amorphous phases is presented.

Keywords

Thermodynamic modeling; solution models; regular solution; Hillert-Staffanson model; semi-empirical Miedema model; excess entropy of mixing.

Introduction

Thermodynamic modeling (TM) is widely used in modern materials science in the development of promising materials and methods for their production [1, 2]. It makes it possible to significantly reduce the amount of expensive experiments and to assess possible mechanisms of interaction. The main areas of TM can include the following:

1) calculation of the equilibrium composition of a multiphase multicomponent system for the production of new materials under isothermal conditions, for example, in reaction sintering, the interaction of solids with gases (oxidation, nitration), and metallic melts with slags, etc.;

2) calculation of the equilibrium composition and adiabatic temperature of the interaction of multicomponent systems in self-propagating high-temperature synthesis (SHS) of new materials;

3) construction of equilibrium diagrams of the state of binary and multicomponent systems.

Various software products are used in TM. For example, for the TM heterogeneous systems under different conditions (isobaric-isothermal, adiabatic, etc.) the ASTRA-4 program [3] and its Windows version of Terra [2] are used, for the calculation of

© B.B. Khina, 2018

SHS processes the specialized program ISMAN-THERMO [4] is used, for the calculation of phase equilibrium diagrams - expensive programs MTDATA, PANDAT, Thermo-Calc, etc. are used.

In modern TM methods, the CALPHAD approach (CALculation of Phase Diagrams) is used to describe the thermodynamic (TD) properties of phases (compounds and solid solutions based on pure components) [5, 6]. For disordered solid solutions of substitution and melts, a model of regular solutions is used [5], and for ordered solutions and implantation phases, the Hillert-Staffanson model [7, 8] is used. The semiempirical Miedema model is used to estimate the TD parameters of metastable phases (for example, amorphous phases) and multicomponent compounds, data for which are not available in the literature [9].

In this paper, the basic models of solutions are described and the ways of their use are shown for the TM synthesis of promising materials.

Thermodynamic properties of compounds

Traditionally, TD parameters of elements and compounds are described in the following form [10]:

Cp(T) = c0 + cT + c2T 2 + c-2T + c-3T (1)

H0 — AH 29g + j CPS (T )dT + AHtr + j CpJ (T )dT ; (2)

298 Ttr

S0 = S098 + J Cpf1 dT + AStr + j CpTT) dT;

298

AStr =AHjTtr ;

GO _ zj0 ttO0

t — Ht — 1St .

(3)

(4)

whre Cp is heat capacity at constant pressure, T is Temperature, HT is changes in the enthalpy in the formation of a given substance at temperature T, AH298 and SU are standard enthalpy of substance formation at T = 298 K and its standard entropy, Ttr and AHtr are temperature and enthalpy of phase transition (for example, melting), AStr is change in

entropy during a phase transition, GT is change in Gibbs energy at temperature T; lower indices 5 and l denote a solid and liquid states.

Here, all the values are determined to 1 mole per formula unit, e.g., for the compound Ti5Si3, containing 5 moles of Ti and 3 moles of Si.

In the thermodynamic database (TDBD) of the ASTRA-4 and Terra programs, which are used for TM, the characteristics of the substances are specified in a different form. The enthalpy of H is the quantity

formula for the substance, the quantities AH298 and

S098 , and the coefficients appearing in the heat capacity expression (1), which must be found in the reference books. If the TD data for the substance are found in the works on the calculation of the phase diagrams by the CALPHAD method, they are usually given as the temperature dependence of the Gibbs energy G(T). In this case, to determine the value

AH298 , we must use the fundamental thermodynamic relations

S — —

dG dT

P,Nj

H — G + TS — G — T\ — |

■dT J p,n ,

(10)

where P is pressure, Ni is the number of moles of the i-ro substance.

Further, using the formulas (6) - (8), the resulting expression should be reduced to the form (9), whence it is possible to determine the parameters f1 - f7. Then in the TDBD the formula of the substance, the

coefficients f1 -f and the value AH098 are introduced.

The TM in ASTRA-4 and Terra programs make it is possible to set up to two regular condensed solutions, the TD parameters of which are entered by the user. The calculation of these parameters will be described below.

H (T ) — H (T0) — j Cp (T)dT, T0 — 298 K . (5)

To

As the enthalpy, the "total enthalpy" I is used [2, 3]:

I — AH298 + H (T ) — H (298),

(6)

and instead of G, the reduced Gibbs energy is used:

G* = S - HIT, (7)

which is connected with the "ordinary" Gibbs energy G by the formula

G — —TG* +AH2098 — I — TS,

(8)

and is approximated in the following form:

G* — fi + /2lnx + /3x 2 + /4x 1 + /4x + f x2 + f x3,

x — 10—4 T .

(9)

When using ASTRA-4 and Terra programs, if a compound is missing in the TDBD, it can be added there additionally. In this case, it is necessary to set the

The model of regular solutions in the CALPHAD approach

In the framework of the CALPHAD-approach (CALculation of Phase Diagrams), all phases, including purely stoichiometric ones, are considered as solutions, and the TD functions are not determined for a formula unit, but for 1 mole of a solution (for example, not for Ti5Si3, but for Ti5/8Si3/8). For a liquid phase and disordered solid substitution solutions, where the atoms are located at lattice sites chaotically, a regular solution model is used [5, 6]:

G<9 = YJxiG? + Gl^x + Gl; G<?x = H*; (11)

H 9 —X xtH?+ Ht

(12)

where xi is mole fraction of the i-th component, cp is phase designation, Gi and Hi are the Gibbs energy and the enthalpy of the i-th component in a given phase state, H^x is excess mixing enthalpy associated

tr

with the interaction of components, Gjix id is the ideal mixing energy (entropy term),

Ghx,id = -TSid, Sid = -RS xln x ,

(13)

where R is universal gas constant, Sid is ideal, or configuration mixing entropy.

For multicomponent solutions, the excess mixing enthalpy is found as follows:

Hex = S XiXjLîj + S XiXjXkLîjk + S xixjxkxlLjkl,

i, j i, j,k i, j,k,l

i ф j ф k ф l. (14)

Here Lj, Lj и j are paired (i-j), triple (i-j-k)

and quadruple (i-j-k-l) parameters of interaction of components in solution j. Paired parameters are found using the Redlich-Kister-Muggian polynomials

Ljj =S j - Xj)n, i Ф j, n > 0, (15)

and triple parameters are in the form

LjkXl + JLj

i Ф j Ф k .

LLjjk = °Lfjk + 'Lfjkxi + iL4jkXj + kLfjkXk,

(16)

For coefficients nLjj, mLjk , m = 0, i, j, k, u Ljkl,

i t j t k t l, a linear temperature dependence: L = A + BT is usually used.

In the CALPHAD approach, the Gibbs energy of

the i-th component Gj in the phase state j is determined in the following form:

Gj = at + btT + c TT ln T +

+ XdiT + HSER + Gjpres + Gjmag . (17)

Here Gjpres u Gj are the quantities

describing the contributions of pressure and magnetic ordering to the total Gibbs energy of the -th element in the phase state j, at, b, cj u din are numerical parameters, where n is integer, value HfER = Hi(T = 298) - Hi(T = 0) is correction for the

transition from T = 298 K to T = 0 K as to the standard reference temperature of thermodynamic quantities.

The values of the parameters in expression (17) are given in the SGTE database (Scientific Group Thermodata Europe) [11] for all metallic and nonmetallic elements (with the exception of gases) in different phase states, including hypothetical ones.

The influence of pressure and magnetic ordering (ferromagnetic and paramagnetic) on the Gibbs energy is determined by the formulas [10]:

Gpres = AP(1 + aj + a{T2 /2 + a2T3 /3 + «3T-1); (18) Gmag = RTln(B0 + 1)g(t), T= t/T* ; (19)

g =

1 -

79т-1 474/ -

140p 497

+471 (p-1 - 1)x

/1П-7 V '

х(б + т9/135 + т17б00)

ID при т< 1,

(т-710 + Х-17315 + х-271500)/D при т > 1, D = 518/1125 + 11692(p_1 -1)15975. (20)

Here A and ak, k = 0-3 are numerical parameters, which are given in the database SGTE, B0 is average magnetic moment per atom, T corresponds to the Curie point TC for ferromagnetic substances and the Neel point Tn for paramagnetic substances, parameter p = 0.40 for bcc lattice and 0.28 for other lattices [10].

Other TD functions (enthalpy, entropy, heat capacity) of pure substances in the phase state j can be found using formulas (10).

The chemical potential, or the partial molar free energy of the i-th component of the solution j is found as follows

j = dGj

=

dN

(21)

pT, N., j *i

and for a substitution solution or a melt is usually expressed as

+ + RT ln x,

(22)

where p0,j is the standard value of the chemical potential of the i-th component of the solution, Ahj t is excess partial enthalpy of mixing; for the pure component, i.e. if xi = 1, its chemical potential coincides with the Gibbs energy in a given phase state:

p0,j = Gij.

Since in formulas (11) - (16) the integral TD values (G and H) are expressed in terms of mole fractions xi, rather than the number of moles Ni, then

the partial molar quantities zj = pj, hj, Ahj. ^ can be

n

n

determined from the following expression, in which the 1st component (solution base) is excluded,:

k dZ

zP= Zp + K - Xj ), (23)

j=2 dxj

where Sj- is the Kronecker symbol (5y = 1 if i = j,

&ij = 0 if i t j), K is the number of components.

For the TM using ASTRA-r and Terra programs, taking into account the formation of a condensed solution, the coefficients of the polynomial describing the excess partial enthalpy of mixing components

AhPXcj =(a + bT + cxt + dTxi )(1 - xt )2 , (24)

where a, b, c u d are numerical parameters.

It is clear from the comparison of expressions (24) and (11), (12), (22) with allowance for (23) that the formula (24) used in these programs is correct only for binary solutions. For multicomponent solutions, it is approximate, since it does not take into account the pair and triple interaction of atoms of different varieties - otherwise the formula (24) would contain the concentrations of other components.

The Hillert-Staffanson Model

This model is used for compounds (carbides, nitrides, borides, etc.) and solid interstitial solutions, ordered solid substitution solutions and intermetallides, i.e. phases whose structure can be represented as two or more sublattices embedded in each other [7, 8]. The specific form of the model depends on the crystal structure of the phase, i.e. from the number of sublattices, and the possibility of placing atoms of a particular sort not only in "one's own", but also in the "alien" sublattice. As in the previous case, the Gibbs energy is found for 1 mole of the solution.

Consider a four-component phase (Ay AByB )a x x (Cy' Dy'D )b with two sublattices, where the atoms A and B are located in the first, and the atoms C and D -in the second sublattice; y/, i = A, B and yj", j = C, D are the proportion of nodes in the first () and the second (")sublattices occupied by atoms of a given kind (these quantities are used in place of atomic fractions xi), yA + yB' = 1, yc' + yD" = 1, a + b=1. For such a phase, the model has the form [5-8]

G = Gsrf + Gmix,id + Hex, H = Hsrf + Hex , (25)

where G5rf (surface of reference) and H5rf describe the free energy and enthalpy of the mixture of sublattices

for the case when each of them is filled only by atoms of one kind, Gmix,id is the ideal energy of mixing atoms in each of the sublattices, which is summed over all sublattices (this is the entropy term), Hex is the excess enthalpy of mixing associated with the interaction of atoms belonging to different sublattices. Values Gsf

and H5rf have the meaning of the "standard state" in relation to the situation when the phase is strictly stoichiometric and the excess mixing energy is absent

(Hex = 0).

The parameters in the formula (25) have the form

GSrf = yAyCGA:C + yAyDGA:D +

+ yByCGB:C + yByDGB:D; Hsrf = yAyCHA:C + yAyDHA:D +

+yByCHsC + yByDHBD. (26)

Here, values Gi:j and Hi:j have the meaning of the Gibbs energy and the enthalpy of the "pairwise" interaction between the 1st sublattice only filled with atoms i (i = A, B), with a second sublattice filled only with atoms j (j = C, D). They are found as

Gi:j = fG, Gj) + P; Hf = f(Hi, Hj) + P;

P = A + BT; i, j = A, B, C, D , (27)

where Gk and Hk are the Gibbs energy and the enthalpy of atoms k, k = A, B, C, D, with a crystal lattice corresponding to their standard phase state at room temperature, P is the so-called "lattice stability parameter" describing the change in Gibbs energy (actually enthalpy) in the hypothetical phase transformation of a pure element from its stable state to a state with a lattice corresponding to a given compound.

Values Gmix,id (entropy term) and Hex have the

form

G = — TS ■

^Jmix,id ~ 1 Jid ■>

Sid = —R[a(yA ln yA + yB ln yB) + b(yC ln yC + yD ln y'D)],

(28)

Hex = yAyCyDLA:C,D + yByCy'D>LB:C,D + yAyByCLA, B:C +

+ yAyByDLA,B:D + yAyByCyDLA,B:C,D • (29)

Here LA:C,D is enthalpy of the "triple" interaction of atoms A located in the 1st sublattice with the second sublattice containing atoms C and D; LA,B:C is the energy of the "triple" interaction of atoms C located in

the second sublatticed with the first sublattice containing atoms A and B. Value Lb.cd is similar to La.C,D, while Lab.D is similar to Lab.C ; parameter Lab.CD has the meaning of the energy of the "quadruple" interaction of atoms A and B in the first sublattice with atoms C and D in the second sublattice. The following form of the record of the lower indices is used: the comma separates the atoms that are in the same sublattice, and the colon separates the sublattices themselves.

The parameters of the "triple" interaction are determined by the Redlich-Kister-Mugian polynomial (15), which in this model is written in terms of the

values y' and y". Thus, the expression for Lab.C has the form

La,BcC =1 nLA,B:.c(y'A - yB ), n > 0, (30)

n

and the parameter of the "quadruple" interaction is described by the formula

LA,B:C,D = °LA,B:C,D + LA,B:C,D (yA - yB ) +

The value of AHchem is calculated by the formula

2La,B:C,D (yC - yD ) •

(31)

The four-component phase model with two sublattices (25) - (31) is written in a general form. For binary compounds, it has a simpler form and is related to the phase structure.

The Miedema's Semi-Empirical Model

It is used in the evaluation of TD parameters of complex compounds for which there is no data in the literature, as well as metastable (crystalline and amorphous) phases. The enthalpy changes when a phase is formed from components in a given phase state, i.e. excess enthalpy of mixing, is found as

Hex = AHchem + AHel.

(32)

where AHchem is associated with the chemical interaction, and AH1 - with elastic interaction of atoms.

In the Miedema's model, the quantity AH el exists only for solid crystalline phases, and AHchem exists for any phases [9]. However, the calculations performed by the author for various compounds showed that taking into account the parameter Hel leads to a significant deviation of the value Hex from experimental data and predictions of other models. Therefore in most cases this quantity is not taken into account.

AH

chem

K , ^xixj (

x, ( (+xs ah^ )

• (33)

j >i=1

Here, Xi is molar concentration of i-th component, xS is the concentration of atoms of the i-th type at the boundary of the Wigner-Seitz cells AHC^ is enthalpy of dissolution of the element i in j, i ^ j, the parameter fij has the form

fj =1 + y(( + x) J,

(34)

where y = 5 for the amorphous phase, 0 for disordered solid solutions and 8 for ordered phases (solid solutions and intermetallides) [12].

Values xS and xsj for each pair i-j are found by the formula

Xs = x,V,2/7(x,V,2/3 + XjVj2/3), xs = 1 - xs, (35)

where Vi is molar volume of the i-th element, which for some substances is corrected taking into account the type of crystal lattice [13].

Quantities AH^™ which are expressed in eV, are found as [13, 14]

AH,

chem i in j

2P V

,j ,

2 / 3

n-1/3 + n-1/3

(( V* J2 e + Q(3 - nj3 )2 - P^

P

Pj

(36)

where e is elementary electric charge, is the

electronegativity parameter for the i-th component, ni is electron density parameter at the Wigner-Seitz cell boundary, Pij and R/Pij are numerical parameters that depend on the nature of the pair of elements i-j, value of Q/Pij = 9.4 for all substances. Parameter values for various phase components are given in the works [9, 13, 14].

The Miedema's model uses specific dimensions, which are given here in the English notation:

[Vi] = cm3, [R/Pij] = V2e, [nf] = d.u., [ ] = V,

[Q/Pj = V2e(d.u.)-23, [Py] = V-1cm-2(d.u.)-1/3, where V = Volt, e is electron charge, cm ^ centimeter, d.u. (density unit) « 6-1022 cm-3 is electron density unit. Then in formula (36) the factor e is written only for the balance of dimensions, i.e., e = 1.

+

X

x

In amorphous phases, along with the ideal entropy of mixing Sid (see formula (13)), there is excess

entropy SCT, associated with the difference in atomic dimensions [15, 16]:

s,= R{2 (Z2 — 1) + 2 (Z — 1)2 y2 —

2(Z — 1)(Z — 3) + lnz](1 — y3) , (37)

where Z = 1/(1 - ^ = 0.64 is packing factor for dense chaotic packing of spherical particles, and the parameters y1, y2 are y3 are calculated as

y =— L ( + dj \di- dj f

-3 j>i=1

к

--

xixj;

У 2 =—- L didj (di - dj f

-3 j >i=1 3к

XiXj ;

(38)

Уз =-f, -n =L xidi"' n = 2,3 -3

i =1

Here di and dj are diameters of atoms of the i-th and j-th elements, i t j (Goldschmidt diameters are usually used).

The application of the above solution models for TM expands the fields of application of TM, increases the accuracy of calculations and clarifies their physical meaning, which can be used in constructing more complex physicochemical models for the synthesis of new materials.

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