CC BY
60
DOI: 10.17277/amt.2017.01.pp.036-043
Thermodynamics of Multicomponent Amorphous Alloys: Theories and Experiment Comparison
B.B. Khina1*, G.G. Goranskiy2
1 Physico-Technical Institute, National Academy of Sciences of Belarus, 10, Kuprevich Street, Minsk, 220141, Belarus 2 Scientific and Technological Park "Polytechnic", Belarusian National Technical University, 65, Nezavisimosti Avenue, Minsk, 220027, Belarus
* Corresponding author. Tel.: + 375 (293) 02 93 87. E-mail: khina_brs@mail.ru
Abstract
On the basis of the existing thermodynamic theories of the amorphous matter (semi-empirical Miedema model and Shao theory), the integral (enthalpy and Gibbs energy) and partial molar characteristics of a multicomponent iron-based amorphous alloy are determined. It is demonstrated that for quaternary alloy Fe - 7.3 % Si - 14.2 % B - 8.26 % Ni these approaches give substantially different values of Gibbs energy and enthalpy at elevated temperatures, however, the difference between the values calculated by these two models becomes insignificant at a room temperature. For the first time, the chemical potential and partial molar enthalpy of iron (the base element of the amorphous phase) are compared with the data obtained from electrochemical measurements. It is demonstrated that the existing thermodynamic models give incorrect description of the partial molar parameters of the components of amorphous phase.
Keywords
Amorphous alloys; chemical potential; enthalpy; Gibbs energy; Miedema model; partial molar enthalpy; Shao theory.
© B.B. Khina, G.G. Goranskiy, 2017
Introduction
The amorphous alloys or metallic glasses have unique properties, in particular, high and stable elasticity, good corrosion resistance in various media, low coefficient of friction and high wear resistance, and thus are widely used as materials and coatings. Bulk amorphous alloys are produced by quenching the metal melt (rapid cooling up to room temperature) by melt spinning technique - extrusion of the molten alloy jet on the rotating water-cooled drum or passing it between two rollers [1]. Powders of amorphous alloys are prepared by mechanical alloying (MA) -intensively milling a powder mixture of the initial components (metals and nonmetals) in vibration and planetary ball mills, attritors and other similar devices [2]. There is also a technique for producing thick amorphous ribbons as a result of solid phase diffusion in a binary multi-layer thin-film systems of the type (Fe, Co, Ni) - (Ti, Zr, Hf), Au-La, Cu-Zr, etc. under long-duration annealing at a temperature below the point of crystallization onset of the amorphous alloy [3].
To produce parts made of amorphous alloys, ribbons or wire obtained after spinning and powders after MA are compacted by isostatic pressing or sintering under pressure at a temperature below crystallization point. Amorphous powders are used for spraying coatings onto the fast wearing parts of machines.
To develop new multicomponent amorphous alloys and define their field of temperature stability, it is important to know their thermodynamic characteristics. The enthalpy of formation of amorphous alloys is usually assessed by semiempirical Miedema model (A.R. Miedema), which is used to analyze the amorphization ability of supercooled melts under rapid quenching [4]. However, for triple amorphous alloys, this model gives a great contrast to the experimental values of enthalpy Hexp. Thus, according to [5], relative deviation Oedema- Hexpl / Hexp for Fe-Ni-V alloys is 18 %, for Fe-Ni-Zr system - 14 %, for Ni-Cu-Al -60 %, for Cu-Ag-Au - from 14 to 82 %, for Au-Sb-Zn - from 17 to 29 %, for Y-Cu-Mg -from 14 to 43 %, for Pb-Sn-Sb - from 29 to 115 %,
for Pb-Sn-Zn - from 44 to 57 %, for the alloys of Cu-Pd-Si - from 24 to 50 %.
However, there is a simple electrochemical method for measuring the chemical potential difference of the base metal between a single-phase amorphous alloy and a pure metal electrode (A^,M) by instantaneous fixing of electromotive force (IFEMF) [6, 7]. It has been used by us for multicomponent amorphous-crystalline iron-based alloys, and a theoretical method to determine the chemical potential of iron in the amorphous phase has been developed on the basis of the results of experimental measurements of the multiphase alloy [8].
It should be noted that in the literature the theory has never been compared to experiment for the partial molar quantities of amorphous alloys (chemical potential and partial molar enthalpy of components). Moreover, based on the Miedema model and other thermodynamic theories of amorphous state, these quantities have not been previously calculated.
In connection with the above, the aim of this study is to compare the data on the chemical potential
of iron in the amorphous phase and its partial
molar enthalpy hFam) obtained from experiments with
the predictions of the existing theories. For this it is necessary first of all to determine the theoretical values of the partial molar elements in the amorphous phase.
Materials and Methods
The compositions of the alloys which were used in experimental measurements and theoretical calculations are the following (wt %.): Fe - 7.3 % Si -14.2 % B - 8.26 % Ni (alloy 1) h Fe - 0.32 % Si -4.8 % B - 6.68 % Ni - 2.42 % Co - 8.88 % Cr -6.42 % Mo (alloy 2). After quenching from liquid state by spinning technique the complete amorphization in them was not achieved: the proportion of the amorphous phase was 78 % in alloy 1 and 82 % in alloy 2. Therefore, they were subjected to MA in the attritor, which resulted in dissolution of crystalline inclusions (iron borides and intermetallides) in an amorphous matrix, whereby the proportion of the amorphous phase increased up to 98 % in both alloys [8]. Electrochemical measurements of A^,Fe value by IFEMF method for multiphase amorphous-crystalline alloys data at different MA times were performed in an electrochemical cell using the solution of potassium chloride and iron sulfate in an anhydrous alcohol as an electrolyte [6, 7]
-a-Fe | KCl, Fe^SO^, C^OH | <Fe>aUoy+,
where a-Fe is a cathode made of pure ARMCO-iron, <Fe)alloy+ is an anode the role of which is played by iron in amorphous-crystalline alloy.
On the basis of the measured A^,Fe values and previously developed theory [8] there were calculated
^Fam} and 4am) values for alloys 1 and 2 which
appeared to be dependent on the MA time; the latter may be caused by a change in atomic-cluster structure of the amorphous phase as a result of periodic plastic deformation under MA.
Theoretical Study
Thermodynamic Miedema model for amorphous phase
In assessing the enthalpy of crystalline, liquid and amorphous phases, for which there is no data in the reference literature, as well as for predicting the range of compositions of amorfizing alloys the semi-empirical Miedema model is used, which was developed for binary alloys [9-11] and generalized for ternary and multicomponent systems [4]. Under this model, the change in enthalpy AHchem in the formation of the phase of the elements is expressed as
N
AHchem = I X,x] (xjAHitm + XsAHtm)), (1)
j >i=1
here xi and x. are mole fractions of components i and j, i ^], N is the number of components, x] is atoms concentration of i-th kind at the border of Wigner-Seitz cells (Wigner-Seitz), AH^1]1 is enthalpy of dissolution of element i in j.
For each pair i-j values xj and xj are defined as
Xi = xiVi
2/3
/( xV23 + xV2/3
] ]
).
xj = 1 xi,
(2)
where Vi is molar volume of i -th element, the values of which for some substances are adjusted according to the type of crystal lattice [12].
Parameter f included in formula (1) has the form
f = 1+Y( X + X- )2, (3)
where for the amorphous state y = 5 [10].
Quantities AH^nj included in expression (1) are determined in the form [12]
A ijchem _ AHi in j =
2PV
i] i
2 / 3
.-1/3
„-1/3
+ Q(3 - n]3 P
(i
R_
P
* \2 ■9*.) e-
(4)
where e is an elementary electric charge (electron charge), 9* is electronegativity parameter for 7-th component, n7 is electron density parameter at the boundary of the Wigner-Seitz cell, P7j and R/P7j are numerical parameters, depending on the nature of the pair of elements 7-j, value Q/P7j = 9.4 for all substances.
The Miedema model uses specific value dimensions, which like in the original [12], are given in the English notation: [V7] = cm3, [R/Pj] = V 2e,
[n,] = d.u., [9*] = V, [Q/Pj] = V 2e(d.u.)-23, [Pj = = V-1cm-2(d.u.)-13, where V = Vol t, e is electron charge, cm = cm, d.u. (density unit) is electron density unit: 1 d.u. » 6-1022 cm-3 [13]. Then from formula (1.74) it is
seen that factor e at item (9* - cp*) 2 is written only to
meet the balance of the dimensions. In expression
(1.74) value AH/*™1 has dimension eV (electron-volt),
i.e. it is determined per 1 atom and for translation into J/mol we use the conversion factor: 1 J/mol = = 1.602M0-19 • Na = 96.494 • 103 eV, where NA is Avogadro number. Values of all the parameters included in formula (4) for different pairs of elements are given in [12, 14].
Since for amorphous phases, the standard state for pure components is a liquid one [9], the integral enthalpy is written as
N
H am =X xlH]rq + AH,
chem •
(5)
i=1
Here H,iq is enthalpy of a pure 7-th component in liquid state, which is defined as a polynomial by temperature degrees
H^ = ai - CiT -£(n - 1)diT + H
SER
(6)
where H,SER = Hl (T = 298) - Hl (T = 0) is a standard value (SER = standard element reference); values of coefficients al, cl, dl n and the number of members n in the polynomial are given in the database (DB) SGTE [15].
To calculate Gibbs energy of the amorphous phase
(7)
G = H — TS
"am _ n am 1 Jam
you need to know its entropy Sam.
As noted above, the standard state for the pure components in this situation is a liquid one, thus the entropy of the amorphous phase can be written as
N N
Sam =1 + S.+ Sid, Sid =-Rl X ln Xi, (8)
i=1 i=1
where Sc is entropy of atomic size mismatch (mismatch entropy), inherent only in amorphous bodies, Sid is perfect mixing entropy.
In expression (8), standard entropy of the 7-th element in liquid state S'iq is determined by the DB SGTE [15] as a polynomial
Sfq = -b, -c, -c, lnT-Xnd/ J"'1, (9)
n
and the entropy of atomic size mismatch Sc is related to the atoms diameters d by the following formulas [16, 17]:
Sc= R { 1.5 (Z2 -1) y +1.5 (Z-1)2 J2 -
-[- 1)(Z-3)2 + ln Z](1 - J3 )}, (10)
y1
N
= (1/)K + dj )(( - dj )2
y>i=1
N
xix j ,
J2 =
c2 didj (di- dj)2
j>i=1
XiXj, (11)
N
y3 =c2/ c2, Cn =X x,d,n, n = 2,3 7=1
where Z = 1/(1 - for dense chaotic packing of spherical particles = 0.64.
Thus, formulas (1) - (11) provide a complete description of the integral thermodynamic parameters of the amorphous phase (enthalpy, Gibbs free energy and entropy) within the semiempirical Miedema model.
Thermodynam7c model of stab7l7z7ng the amorphous phase
In his works [18-20] G. Shao proposed the theory of stabilizing the amorphous phase in reference to the melt, which uses an analogy with the thermodynamic description of the magnetic transition in CALPHAD approach used to calculate the alloy state diagrams.
A change in Gibbs energy AG]
liq ^am
in the conversion
of supercooled liquid (liq) into amorphous phase (am) is expressed as
AGHq^am = RT ln(1 + a)f (t), T = T/Tg. (12)
In formula (12) a is a stabilization parameter characterizing the increase in the thermodynamic stability of amorphous alloy compared with the supercooled liquid at T < Tg, where Tg is temperature of glass transition or Kauzmann temperature (W. Kauzmann) which depends on the composition; for pure metals Tg = (0.25-0.64)Tm, where Tm is a melting point.
Function f(t) is defined as
f _ Jl + «it-1 + а2т3 + а3т9 + a4T15 при t< 1; ( )
f _ Ь —5 7 —15 , —25 i (13)
[61t 5 + b2t 15 + b3T 25 при t> 1;
a1 = — 0.9917, a2 = — 0.1117, a3 = — 4.966-10—3, a4 = — 1.11710-3, b1 = — 0.1054, b2 = —3.3474-10—3, b3 = —7.0296-10—4.
(14)
Since
H _ G + TS _ G — T J — l dT
(15)
P, N1
q _ — T
then by differentiating (12) taking into account (13) and (14), we obtain the expression for a change in enthalpy during glass transition of supercooled melt:
= RT ln (1 + a)q (т); (16)
df (т) I о1т-1 - 3а2т3 -9а3т9 -15a4т15 при т < 1;
дТ [561т-5 +15b2 т-15 + 25Ь3т-25 при т> 1.
(17)
In Shao theory [18-20] the stabilization parameter a included in formulas (12) and (16) is described as a function of composition similarly to the Gibbs energy of a multi-component solution in CALPHAD approach [21]:
a = Z xa +Z xXj Л + Z xixjxk A$k;
i, j, k
1, J
(18)
aj _ZnЛ (—xj )", n >0;
n
Л jk _ X 1 Ajk + xj J A,jk + Xk k A,jk, 1 * J * k,
where parameters Aj and Ajk, i ^ j ^ k which are expressed by the Redl-Kister-Muggianu polynomial, characterize binary a ternary interaction of atoms.
Glass transition temperature Tg of a multi-component phase is determined in the same way:
Tg = Z xiTg, i +Z xixj Qj + Z xixjxk %;
l, j
1, J, k
(19)
Qj _Z n ( — x, )n, n > 0;
n
^ jk _ X i'Q jk + xj J Q jk + Xk k ^ijk, 1 * j * k,
where 7g i is Kauzmann temperature for pure i -th component.
Thus, Gibbs energy and enthalpy of the amorphous phase are related to the corresponding values for the melt at the same temperature:
^am = Gliq + AGliq^am, H am = Hl + AH liq^am. (20)
To use Shao theory it is necessary to determine Gibbs energy Gliq and enthalpy Hliq of liquid phase in (20). Within the CALPHAD approach [21] they are described as follows:
Gliq _ Z X1G1 + Gid + Hex; Gid _ —TSid; 1_1
k
Hliq _Z XiH14 + Hex, 1_1
(21)
where G/iq and Hliq are Gibbs energy and enthalpy of the i-th element in the liquid state, Hex is excess enthalpy of mixing
N N
Hex = Z XiXjLj + Z XiXjXkLijk, i * j *k, (22)
J >1 _1
k > j >1 _1
The parameters of pair Lij and triple Lijk interaction in the melt are described using the Redlich-Kister-Muggianu polynomials
Lj _Z % (x — Xj )n, 1 * j, n > 0; (23)
n
Ljk _ 0Ljk + 1 LjkX, + JLjkXj + kL,jkXk, 1 * j * k. (24)
Partial molar quantiti es of components
To determine the partial quantities of the components of the alloy (of chemical potential ^(am)
and partial molar enthalpy h (am) ) the following formula is used
N
Z(am) _ Zam + Z(5j — Xj)
cX,
(25)
J=2 -j
where Zam is integral thermodynamic parameter, zam = Gam, Ham, ^am, (Gibbs energy, enthalpy and entropy per 1 mol of the solution), z(am) is the corresponding partial molar value for the i-th component in this phase, Sj is Kronecker symbol: Sj = 1 at i = J, Sj = 0 at i * J. Here it is assumed that number i = 1 is referred to alloy base (in this case - Fe).
Thus, to determine the partial molar characteristic of any component of the amorphous phase it is necessary to perform differentiation of the corresponding integral parameter by concentrations of all components except the first one (Fe) taking into
N
account that x1 = XFe = 1 - Z Xi . At the same time for
i=2
Shao model the derivatives of values AGliq^am (see formula (12)) and AHliq^am (see expression (16)) by the concentration of the 7-th component have the following form:
SAG,
liq ^am
ÔXj
ÔAH,
liq ^am
dXi
—f ^ +rtln (1+a)p, p = f ; 1 + a ôx7 dXj
~-—q ^ + RT ln (1 + a)r, r = .
1+a dXj dXj
(26)
The Calculation Results and Discussion
When calculating Gibbs energy and enthalpy of stabilizing the amorphous phase by Shao model (see formulas (12), (13), (16) and (17)), component glass transition temperature Tg 7 for Fe, Ni, Co, Si and B, as well as parameters Aj and A7jk appearing in expressions (18) and values Qj and Q7jk, appearing in formulas (19), were taken from [18-20, 22]. Since there are no A h Q values in the literature, for a number of components they were assumed to be equal to zero. Values Tg 7 for Cr and Mo were defined as Tg 7 = 0.25Tm 7, where Tm 7 is the melting point of the 7-th element. To calculate Gibbs energy Gliq and enthalpy Hliq of the multi-component liquid phase (see expressions (20) - (24)), we used the parameter values of pair L7j and triple L7jk interaction of the components of metal melts according to the data [23], as well as the data of numerous works on the calculation of binary and triple equilibrium state diagrams by CALPHAD
method. In this paper we present the results of calculations for alloy 1.
To compare the values of integral and partial molar thermodynamic characteristics of the amorphous phase according to Shao and Miedema theories it was assumed that the composition of the amorphous phase corresponds to the total composition of alloy 1, i.e. it is 100 % amorphous (we ignore the presence of 2 % of crystalline inclusions).
Figure 1 shows temperature dependences of Gibbs energy and enthalpy of the amorphous phase for composition 1; for comparison it shows data for supercooled liquid iron and melt of composition 1 and value Tg.
According to Fig. 1, Gibbs energy and enthalpy of supercooled four-component melt have significantly lower values than those of pure iron, which is explained by the presence of excess enthalpy of mixing Hex (see formula (21)). Miedema model over the entire temperature range gives lower values Gam and Ham than Shao model, however, at room temperature (300 K) the difference in free energy does not exceed 8 kJ/ mol (see Fig. 1 a). According to Shao model, value Gam falls below Gibbs energy of melt Gliq at T < 500 K, i.e. the conversion of supercooled melt into the amorphous phase becomes thermodynamically favorable. Also, according to this model, Ham< Hliq at T < 750 K, and as the temperature falls to Tg = 307 K this value reaches the value calculated by the Miedema model.
The calculation of chemical potential and partial molar enthalpy of iron in the amorphous phase of
20 10 0 -10
ö -20 E
=5 -30 (3 -40 -50 -60 -70
60
400 500 600 700 800 900 1000 1100 1200 T, K
a)
300 400 500 600 700 800 900 1000 T, K
1100 1200
b)
Fig. 1. The dependence of Gibbs energy (a) and enthalpy (b) phases for composition 1 on temperature:
1 - pure liquid iron; 2 - supercooled melt; 3 - amorphous phase by Miedema model; 4 - amorphous phase by Shao model
T, K T, K
a) b)
Fig. 2. Temperature dependences of chemical potential of iron (a) and its partial molar enthalpy (b) in various phases for composition 1:
1 - pure liquid iron; 2 - supercooled melt; 3 - amorphous phase by Miedema model; 4 - amorphous phase by Shao model
composition 1 according to two theories is presented in Fig. 2.
Figure 2 shows that Miedema model gives
significantly higher values of both and hF^
than Shao model. At the same time, for T < 750 K chemical potential of iron in the amorphous phase according to Shao model is somewhat lower than in a supercooled melt, and as it approaches point Tg value
uFam* decreases by 15 kJ/ mol compared with the
value at 500 K (see Fig. 1 a). When temperature decreases the most significant change is observed in
value hFam)) in Shao model (Fig. 1 b): as it approaches Kauzmann temperature Tg it decreases sharply (from 10 kJ/mol at 600 K to 160 kJ/mol), and then it rises sharply up to 118 kJ/mol at 300 K. Such a type of
curves ^Fam) and hFam) for this composition of the
amorphous phase is due to temperature dependence of parameters p and r in equations (26) which arise when differentiating expressions for AGliq^am and AHliq^am in Shao theory.
These functions are shown in Fig. 3 where, for clarity, you can see functions f(t) (see formula (13))
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
/ A
/
/
i - i i i - § -
i i -
-
T..... _i_
300 400 500
600 700 T, K
800 900 1000
a)
-0.002
-0.004
-0.012
300 400 500
600 700 T, K
b)
800 900 1000
Fig. 3. Temperature dependences of Shao model parameters for amorphous phase 1:
a -f (line 1) and q (line 2) to calculate the integral thermodynamic quantities (Gibbs energy and enthalpy, respectively); b - p (line 1) and r (line 2) to calculate the partial molar quantities (chemical potential and partial enthalpy, respectively)
0
and q(x) (see formula (17)) where t = T/Tg. At the same time, value r included in the expressions for the partial molar enthalpy of the components of the amorphous phase, has a sharp minimum at T = Tg (see Fig. 3 b). The latter leads to the presence of a minimum on temperature dependence hFem) (see Fig. 2 b).
From physical considerations the partial molar characteristics (chemical potential and partial enthalpy) of the base element are unlikely to change greatly in a narrow range of temperature in the region of stability of the amorphous phase. According to experimental data, with a slight change in temperature around the room one (± 10 K) measured value A^,M remained almost unchanged. Therefore, a strong change in the calculated values |j,Fa™) (Fig. 2 a) and
especially hFem) (Fig. 2 b) in a narrow range of
temperature is an artefact due to the fact that Shao model [18-20] structurally is designed to evaluate the integral thermodynamic parameters Gam and Ham and is not suitable for the correct description of the partial molar quantities of elements in a multicomponent amorphous phase.
Figure 4 shows a comparison of the calculated values MFe(am) and hFe(am) for the amorphous phase of alloy 1, obtained according to two models, with the quantities obtained in our previous work [8] on the basis of experimental measurements for different MA
40 20
0 0 E
=5 -20
1 -40
) -60 ?
-80
A
-100 -120 -140
■ 5'
2
/3
0
10
20
30 40 t, min
50
60
Fig. 4. Chemical potential
h(am)
(am)
(lines 1 - 3) and partial molar
enthalpy of iron
AFe (lines 4 - 6) in the amorphous phase of alloy 1 defined on the basis of the experiment for different MA times (lines 1 and 4) [8] and calculated by Miedema model (lines 2 and 5) and Shao model (lines 3 and 6) at a room temperature
times in attritor t. A decrease in the experimental
values |i Faem) (line 1) and (line 4) with MA time
results from both dissolving the crystalline phases and a change in atomic-cluster structure of the amorphous phase under periodic plastic deformation. The latter factor is not taken into account in the thermodynamic Miedema and Shao models, therefore the corresponding estimated partial molar quantities do not change in time.
From Fig. 4 it is seen that theoretical values iF^ and hFam) significantly differ from the values determined on the basis of experimental data, Shao model (line 3 for and line 6 for giving a rather significant deviation from the experiment. Fluctuations of values hFam) according to Shao model
(line 6) are related to the fact that the composition of the amorphous phase and value Tg which depends on it change when crystalline inclusions dissolve. Since for the amorphous phase of alloy 1 Kauzmann temperature is to close to a room temperature, small variations in its composition and value Tg lead to fluctuations in parameter r and consequently to a noticeable change 1in hFem) (by «10 kJ/mol).
Conclusion
Thus, in this study it is shown that the existing thermodynamic theories of amorphous state (Miedema and Shao models) give similar values of integral quantities Gam and Ham for the amorphous alloy with composition 1 at a room temperature, but the values of the partial molar quantities according to these two theories differ considerably. Moreover, the calculated
data demonstrate a large deviation from values iFT"*
and 4?» defined on the basis of electrochemical
measurements, and this error is very significant for Shao model.
This is due to the fact that Miedema and Shao models were developed for the theoretical estimation of integral thermodynamic parameters (enthalpy and Gibbs energy) and are not designed to determine the partial molar quantities of elements in the multicomponent amorphous phase (chemical potentials and partial enthalpy). The development of new thermodynamic models for amorphous solids for physically correct description of their partial molar characteristics is a very complex task and requires further research.
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