Free energy driving force for Austenite-Fe2B boride transformation

Fe2B boride is a stoichiometric compound, in which there is no solubility or the solubility is

extremely small in this compound, and therefore this kind of compound is also known as ‘line

compound’. The free energy of a line compound, by definition, is represented by a single value at a

given temperature. Modeling the growth of stoichiometric compounds via the phase-field approach

poses serious numerical challenges since the derivative of the free energy does not exist [1]. One of

the common approaches to solve this problem is by approximating the free energy of a line

compound with a parabolic function of composition, which is based on Calphad method. In the

following, two approaches to solve the free energy driving force of Fe2B boride will be given, the

first is based on Calphad method, and the second is based on phase-diagram calculation.

1. Calphad method

We consider a system of austenite solid solution, which is an FCC solid solution, and the sub-lattice

is in the form of interstitial solid solution. Following two sub-lattice model of calphad method [2],

(Fe)1(B,◊)c is used as the interstitial solid solution model. The index c represents the relative number

of sites in the interstitial sublattice and depends on the crystal structure, which is 1 for FCC.

Following, two-sublattice model, molar free energy of austenite, can be described as follow

(1)inserting the element of summation into the calculation, then we obtain the molar free energy of

austenite as follow,

(2)where, y is the site fraction, which is related to the molar fraction, XB, as ,

where a is equal to c since we consider the FCC austenite of Fe. is the reference energy

term for boron in austenite and is the reference energy term for vacancy in austenite.

is obtained from the optimization of SGTE database by Van Rompaey et al[2]) as follow

. (3)

and is the enthalpy at reference state for iron and boron respectively and are obtained

from SGTE database for pure elements [3]. On the other hand, the reference energy state for vacancy

in austenite, , is obtained from Per Gustafson [4] and defined as

.

(4)

On the other hand, according to the optimization thermo-chemical data base of Fe2B

boride from ref. [2], the molar free energy of boride is obtained as follow,

.

(5)

Calculation of Eq.(2) produces a free energy curve for austenite, , as shown in the Fig. (1),

whereas considering stoichiometric compound of Fe2B boride at 0.33 molar fraction of boron,

calculation of Eq. (5) results one value of molar free energy of boride, , as shown by the

arrow in the Fig. (1). The solid straight line in this figure corresponds to the common tangent line of

curve from the point where the maximum solubility of boron in austenite exist. Therefore we

found two problems from this observation:

- common tangent line of which will never meet the position of value, due to

the contour of curve and position of value

- value which is independent on the boron concentration, and thus has no derivative

on the concentraton

Fig.1 Free energy curve for austenite and the position of free energy value for Fe2B boride

The similar problem is also found in other case of stoichiometric compound-solid solution

transformation as for the case of -’ transformation in Al-Cu alloy as depicted in the Fig.2 [1]. In

order to solve the problem, they approach the free energy driving force with the parabolic function

0 0.1 0.2 0.3 0.4 0.5

-200000

-160000

-120000

-80000

-40000

0Free energy (J/m

ol)

Molar fraction of boron

m

oG

BFeoG 2

approach of composition, where the results of common tangent curve after this approach is given in

the Fig.3. The similar approach is also conducted in the previous study of the present case by

following the below equation for the parabolic function approach of composition for molar free

energy of Fe2B and austenite respectively,

(6)

(7)

where F and b is the evaluated constant in the previous works. The common tangent curve calculated

using this approach is depicted in the Fig. 4, where Fig.4(a) and (b) for the variations of Fe 2B and

austenite free energy driving force curve respectively.

Fig.2 Common tangent curve for -’, in Al-Cu alloy calculated from thermo-chemical database [1]

Fig.3 Common tangent curve for -’, in Al-Cu alloy, where parabolic function of composition is introduced (a) for

the variations of curve (b) for the variations of ’curve [1]

(a) (b)

Fig.4 Common tangent curve for -Fe2B, in Fe-B alloy, where parabolic function of composition is introduced (a) for

the variations of curve (b) for the variations of Fe2B curve

2. Based on phase-diagram calculations

The second approach to determine the free energy driving force for PF simulations is as described by

Militzer et al [5], which is based on the phase-diagram calculations. Here, the phase boundaries of

the evaluated phases are linearized, i.e. for the case of to transformation in the Fe-C alloy [5],

the phase boundary of and (Fig.5) is approached with the following equations,

(8)

(9)

Fig. 5 Schematic diagram of phase boundaries - in Fe-C alloy which is described by the linearized method

1000K

1150K

T(K)

Concentration of carbon (weight%)0.0025

Linearized by Eq.(7)

Linearized by Eq.(6)

(a) (b)

where and are the equilibrium concentration of carbon in a and g respectively.

The free energy driving force, G, is then calculated with the following equation(10)

where S is the entropy difference which is taken from the thermo-chemical database, and T is the

undercooling, which is obtained from the following equation,

(11)

where k is the partition factor and governed by , and T is the actual temperature.

Adopting the above method, the similar approached can be applied for the present case of -Fe2B

boride transformation. At first, we examined Fig.6 which showing the phase boundary of phase

and Fe2B phase. From this figure, we understand that line boundary no 1 and 2 will be the examined

phase boundary for and Fe2B phase respectively. Line 1 is then linearized with the following

equation,

(12)

Fig.6 Phase boundaries of -Fe2B phase adopted from ref (2)

Fe2B line

1468K

1184.6K

Fe2B

x10-4

x10-5

Molar fraction of boron

line line

where is the boron equilibrium concentration in phase, and 1.88E6 is the slope, m, of line 1,

which is obtained from the following calculation,

(13)

Considering the linier condition of Fe2B line, and constants equilibrium concentration of Fe2B

compound at 0.33, then the temperature boundary of Fe2B phase is considered as similar with the

actual temperature of the concerned case.

The undercooling temperature is then defined as follow,

(14)

where TFe2B is equal with T.

The free energy driving force is calculated using Eq.(10), where S is obtained from ref.(6).

References

1. S.Y. Hu, J.Murray, H. Weiland, Z.K. Liu and L.Q. Chen: Comp. Coupl. of Phase Diag. and

Thermochem. 31 (2007) 303-312

2. T.V. Rompaey, K.C. Hari Kumar, P.Wollants: J. Alloy Compd. 334 (2002) pp. 173-181

3. A T Dinsdale: CALPHAD 15(4) (1991) pp. 317-425

4. Per Gustafson: Scandinavian J. Metall. 14 (1985) pp. 259-267.

5. M. Militzer, M.G. Mecozzi, J. Sietsma and S. Van der Zwaag: Acta materialia 54(2006)

3961-3972

6. M. Palumbo, G. Cacciamani, E. Bosco and M. Baricco: Intermetallics 11 (2003) 1293-1299