СТРУКТУРА, ФАЗОВЫЕ ПРЕВРАЩЕНИЯ И ДИФФУЗИЯ

УДК 532.739.2

ESTIMATION OF EXCESS ENERGIES AND ACTIVITY COEFFICIENTS FOR THE PENTERNARY Ni-Cr-Co-Al-Mo SYSTEM AND ITS SUBSYSTEMS

© 2015 г. A. Dogan*, H. Arslan*, and T. Dogan**

*Kahramanmaras Sutcuimam University, Science and Art Faculty, Department of Physics, Avsar Campus

46100 Kahramanmaras, Turkey **Cukurova University Engineering Faculty, Balcali, Adana, Turkey e-mail: hseyin_arslan@yahoo.com Поступила в редакцию 23.07.2013 г.; в окончательном варианте — 22.10.2013 г.

Using different prediction methods, such as General Solution Model, Kohler, Muggianu, the excess energy and activities of molybdenum for the sections in the penternary Ni—Cr—Co—Al—Mo system with mole ratios xNi/xMo = 1, xCr/xMo = 1, xCo/xMo = 1 and xAl/xMo = r = 0.5 and 1 were thermodynamically investigated at a temperature of2000 K while the excess energy and activities of Bi for the section in the ternary Bi—Ga—Sb system with mole ratio xGa/xSb = 1/9 were thermodynamically investigated at a temperature of1073 K. In case of r = 0.5 and 1 in the alloys Ni—Cr—Co—Al—Mo, it was found that a positive deviation in the activity coefficient was determined as molybdenum composition increased. Moreover, in the calculations performed in Chou's GSM model, the obtained values for excess Gibbs energies are uniform with negative values in the whole composition range of bismuth at 1073 K and showing minimum values of about —2.2 kj/mol at the mole ratio xCr/xCo = 1/9 in the alloy Bi-Ga-Sb.

Keywords: geometrical model, Redlich-Kister coefficients, excess Gibbs energy, activity coefficient, thermo-dynamic model.

DOI: 10.7868/S0015323014060059

INTRODUCTION

Generally, thermodynamic properties of alloys can be obtained by experimental measurements. However, it is not always possible to perform experimental measurement for multi-component alloys due to technological difficulties and the expenses and time consumption. A necessity of handling thermodynamic data on multi-component systems arose nearly several decades ago and still remains a rather topical problem.

To these aim, a relatively simple model is employed in this study that may be demanded in conditions of severe shortage of thermodynamic data concerning subsystems of the system studied, just up to the binary subsystems.

To meet the requirement of thermodynamic data, modeling the thermodynamic of complex melts has attracted many scientists' attention over the past decades

[1-9].

There are various methods of processing thermody-namical data and performing thermodynamic calculations in the case of multi-component systems. To all appearances, the method CALPHAD is still the most widely used one today. One of its main features is that it allows to compile a thermodynamic description of multi-component systems based on the descriptions of the systems of lower order. Moreover, the calculations can be carried out even when the data on corresponding subsystems is incomplete. In such a case, one can attempt to estimate deficient thermodynamic parameters via attraction into

consideration of one or another theoretical model in order to minimize the inaccuracy of calculations. Thus, for instance, in the works [1-5], for performance of calculations of the solubility of carbonitrides in multi-component steels the parameters of interaction of the elements in the non-metal and the metal sublattice of the phases with a bcc structure were estimated in terms of the model that had been proposed in [6] and improved in [7].

This study aims to calculate the excess energy and activity coefficient for multi-component systems such as penternary Ni-Cr-Co-Al—Mo system and Bi-Ga-Sb ternary system. In this penternary system, the activity calculation has not been reported up to now. To this purpose, Gibbs energy is utilized in the alloys and its subsystems. However, in case of acute deficiency of thermodynamic data, particularly when of all corresponding subsystems there are available data regarding only binary ones satisfactory results of ther-modynamic calculations can be obtained via other methods such as Chou's general solution model [8].

While the general solution model has been successfully applied to various kinds of scientific and technological topics by many researchers including calculation of thermodynamic properties [9-14], construction of phase diagrams [15-18], much attention has been recently paid to the high order systems to fulfill the requirements of both research and practical applications [8, 19-23]. In the present study, three different thermodynamic prediction methods were used in or-

der to investigate the penternary and ternary alloys for high temperature solution: Chou, as the general solution method (GSM) in question; symmetric Kohler; and the Muggianu methods [9—14].

Fundamental Procedure for the Solution Model and Activity Coefficients Analysis

The second type of formalism which is known as the Redlich—Kister polynomial (R— K) may be expressed as follows:

GE = XXj X oljx - Xj)k,

(1)

k=0

+ w,4GE4 + w25ge5 + w34ge4 + w35ge + w45ge

(2)

'24G24 + yy 25G25 + rr 34G34 + rr 35G35 + W45G45,

where Wy represents the weight probability of each corresponding binary composition point and can be calculated via:

W- = ^^

XiXi

(3)

Here Xj corresponds to the mole fractions of components in the alloy system. Using Eqs. (1)—(3), the excess Gibbs energy is

n

Ge = xyxQf2(2X1(12) - 1)k +

k=0

n

+ Qk3(2X1(13) - 1)k +

k=0

n n

+ x1x4 ^ ^14(2X 1(14) - 1) + x1x5^ Q!5(2X1(15) - 1) +

k=0 k=0 (4) n n

+ x2x3^ ^23(2X2(23) - 1) + x2x4^ ^24(2X2(24) - 1) +

k=0 n

k=0 n

+ X2XsXi ^25(2X2(25) - 1) + X3X4X ^4(2X3(34) - 1) +

k =0 k =0 n n

+ X3X5^ ^35(2X3(35) — 1) + X4X5X ^45(2X4(45) — 1) ■

Note that, X,, Xy values are different for different models and these are defined in Kohler model [24] as

Xi

Xi(ij) = X + X and for Muggianu model [25, 26]

Xi(ij) = Xi

1 - (Xi + Xj ) 2 ■

(5)

(6)

where Qj called the Redlich—Kister, is a parameter independent of composition and only relies on temperature. GE is the excess Gibbs energy of mixing and Xh Xj indicates the mole fractions of component i and j in the i—j binary system, respectively. Equation (1) can be reduced to the regular solution model if the value of n is 0 or sub-regular solution model if n is 1. n takes a higher value when the binary system is out of these two solution models.

When the general solution model is extended to a penternary system, the excess Gibbs energy of mixing can be expressed as follows:

ge = w12ge2 + w13ge3 + wuge + w15ge + w23ge3 +

In Chou's general solution model, Xj in Eq. (4), which is the relationship between compositions of components in a multicomponent system and the selected compositions of i and j in the ij binary system, can be expressed in the following way:

Xi (ij) - Xi + X Xk^

k

i(ij)■

(7)

k=1 k *i,j

Here the coefficient represents the similarity coefficient of component k to component i in ij system, and is defined as

^m =

1 +

n( ji, jk) n(ij, ik)

(8)

where n (ij, ik) is the function, called " deviation sum of squares", related to the excess Gibbs energy of ij and ik binaries, and is given by

n(ij, ik) = J (GE - gE )dXj

(9)

thus substituting Eq. (9) into Eq. (8), the similarity coefficients can be obtained. It should be pointed out that the parameters appearing in above equations prove the following relation

ok = (-i)k ok

j >•

(10)

This relation can be proved easily via Eq. (1).

As for the activity coefficients, recently for multi-component systems a few articles published on this subject [22, 23, 27—34]. The activity coefficient of molybdenum in a molten Ni—Cr—Co—Al—Mo penternary alloy can be expressed by the following equation:

PTE n^i r-iE dGE dGE

Gmo = RT ln YMo = G - Xcr---Xco-

ÔX,

- X

Al

dG_

ÔX

+ (1 - XMo)

Al

Cr

ÔG1

ÔX

ÔX,

Co

(11)

Mo

k=0

k=0

Using Chou's general solution model, the partial excess Gibbs energy of the component Mo in penter-

m

0

nary alloy mentioned above were found by substituting Eqs. (12)—(16) into Eq. (11)

3G1 dx.

— (xNi xCr )§1 2xNi xCr^12

Cr

xCo^2 + 2xNixCo(Si(13) O^D x Al^3 +

+ 2xNixAl(^14* — 1)^l4 — xMoS4 +

+ 2х№хМо(^(Ц15 — 1)^(5> +

+ XCo^5 + 2xCrXCo(1 — ^(2)23)^23 + + x Al^6 + 2xCrXAl(1 — ^(U^i + XMo^7 +

+ 2xCr xMo(1 — ^2(25) ^25 +

+ 2xCoxAl(^3(34) — ^3(34))^34 + + 2xCoxMo(^3(35) — ^3()35))^35 +

+ 2xai xMo(^4(45) — ^4(45))^45>

dGE dx,

- xCr§1 + 2xNiX Cr(^1(12) 1)^12 +

Co

+ (xNi xCo)$2 2x Ni XCo^1 3 XAl^3 +

+ 2xNi xAl(^((14) — 1)^( 4 — XMo^4 + + 2xNixMo(^({15) — 1)^( 5) + XCr^5 — 2xCrXCo^2()23) X (I3)

X Q2 3 + 2xCrXAl(^2(24) — + 2xCrXMo(^2(25) —

(12)

2xNi XMo^ 1 5 + 2xCr XCo(^2(23) ^23)РУ +

+ 2xCr XAl (^2(24) ^24)M4 + XCr^7

(15)

— 2xCr xMo^2(25)^25 + 2xCoXAl(^3(34) — ^3(34))^34

+ xCo§9 — 2xCoXMo^3 (35)^3 5 +

+ XAlS 10 — 2x Al XMo^4(45)^45>

where

51 =^(02).

52 =nj?

53 = q(0)

84 =Q(°5)

85 =Q2r

(0) 24 '

87 = n2S)-

Я - r»(0)

O8 — D34

8б = 0

89 — D

" ^(2(2X1(12) - 1),

- Q(13)(2Хщ3) - 1),

- Q(j|(2X1(14) - 1),

- Q(15)(2X1(15) - ^

- Q23(2X2(23) — 1),

- Q24(2X2(24) - 1),

- Q21^(2X2(25) - 1),

■ D34)(2X3(34) - 1),

- D315(2X3(35) - 1)

(0)

and 810 — D45

■ D45(2X4(45) 1).

Xj (j) terms in Eq. (16) are defined in Eq. (7).

(16)

^2(25))^25 + xAl58 + 2xCoXAl(1 ^3(34))^34 + XMo59 +

+ 2xCoXMo(1 — ^3(335) + 2xAl XMo(^4(45) — ^4(45))^45>

= -xCr&1 + 2xNixCr(£,1a2) - 1)^(;2 -

dx

Al

xCoS2 + 2xNixCo(^(3) 1)^(3' + (xNi xAl)S3

(1)

2xNix Al^ 1 4) xMo5 4 + 2xNixMo(^1a5) +

(4)

■>(1)

+ 2xCrXCo(^2(23) — ^2(23))^23 + XCr56 —

— 2xCrXAl^2()24)^2.4 +

+ Z^Cr-KMo^^) — ^2с25))^У + xCo58 —

— 2xCoXAl^3(34)^34 + 2xCoXMo(^3(35) — ^3(34))^35

+ xMo510 + 2xAlXMo(1 — 4-()45)iiCS,

d- = -xcA + 2xNixCr(^1(12) - O^K -

dxMo

- xCoS2 + 2xNixCo(^1(13) - O^D - xAlS3 +

+ 2xNixAl(^1(14) - O^K + (xNi - xMo)S4 -

(14)

RESULTS AND DISCUSSION

Some elements of the system in the present study possess a considerably higher temperature of melting. On that for those proportions of elements that the

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