212

8 Property

Property calculation has been added into Pandat

TM

and is characterized in

three categories, thermodynamic property, physical property and kinetic

property. The database format and the equations for these properties are

described below.

8.1 Thermodynamic Property

The Thermodynamic Property function is used to calculate thermodynamic

properties, such as Gibbs energy, enthalpy, entropy, chemical potential,

activity, etc. It is under the PanPhaseDiagram module. User can access this

function through the menu (Property → Thermodynamic Property) after the

thermodynamic database is loaded. A popup window will allow user to input

calculation conditions as shown in Figure 8.1.

Figure 8.1 Thermodynamic Property calculation dialog

The calculation is set as a line calculation with the desired thermodynamic

properties output. User may select the thermodynamic properties as needed by

213

checking the boxes in front of each property as shown in Figure 8.1 and specify

the reference state for each element. The selected property will be calculated

and graph plotted as shown in Figure 8.2. If two or more thermodynamic

properties are selected, they will be plotted separately. Experienced user can

also get these results directly from the Line Calculation in the

PanPhaseDiagram module with proper syntax using self-defined table.

Figure 8.2 Thermodynamic Property calculation results

214

8.2 Physical Property

Physical property calculation implemented in Pandat

TM

allows user to calculate

the molar weight, molar volume, density, surface tension and viscosity. The

interface is similar to the Thermodynamic property calculation as shown in

Figure 8.3. User can set up the calculation condition as a line calculation and

select the properties to be calculated. The default graph is the calculated

property as shown in Figure 8.4 as an example. Again, if two or more

properties are selected, all of them will be plotted one by one.

The detailed calculation method for each property and the format of the

database file are described in the following sections.

Figure 8.3 Physical property calculation dialog

215

Figure 8.4 Physical property calculation results

8.2.1 Molar Weight and Phase Weight Fraction

The molar weight for a phase is calculated from the atomic weights of the

elements involved in the phase. The unit is kg/mol-atoms. Since atomic

weights of elements have been included in Pandat, no input is required in the

database TDB file.

For example, an Fcc phase in Al-Cu binary system with

0.9

Al

x =

and

0.1

Cu

x =

,

its molecular weight is

-3 -3

0.9 26.982 10 0.1 63.546 10

fcc

Al Al Cu Cu

MW x M x M= + = +

-3

30.6384 10 ( / )kg mol atoms= −

(8.1)

8.2.2 Molar Volume and Phase Volume Fraction

The molar volume can be calculated by Pandat

TM

if the model parameters are

properly defined in the database file. The molar volume of a pure component is

described as a function of temperature and pressure, and the excess molar

216

volume of a phase is described in the format similar to that of excess Gibbs

energy. The unit of molar volume is

3

/m mol atoms−

. In the database (.TDB) file,

the format of molar volume is:

PARAMETER Vm(Fcc,Al;0) 298.15 +9.7743e-006*exp(6.91213e-005*T+1.62267e-

011*T**3+0.413484*T**(-1)); 3000 N !

PARAMETER Vm(Fcc,Fe;0) 298.15 +6.72092e-006*exp(6.97895e-005*T); 3000 N !

PARAMETER Vm(Fcc,Al,Fe:Va;0) 298.15 -3e-6; 3000 N !

The calculation of molar volume for a phase is

()

i ij

ok

m i m i j m i j

V xV x x V x x= + −

(8.2)

where

ij

m

V

is the parameter for the excess molar volume of this phase.

Molar volume of a system with phase mixture is calculated by

mm

V f V

=

(8.3)

where

f

and

m

V

are the molar fraction and molar volume for phase

.

Note that, if a phase is described by a compound-energy-formalism (CEF) with

multi-sublattices, the molar volume is automatically described as a function of

T, P, and site fraction (y) following the format of the Gibbs energy. This will

introduce many end members in the molar volume and may lead to instability

in the calculated molar volume. In reality, molar volume should be a function

of mole fraction (x) instead of site fraction (y), it is thus recommended to define

the molar volume of a phase separately from its Gibbs energy if the phase is

described by the CEF model. This can be realized through the user-defined

molar volume property (section 3.3.8).

Using the Al-Ni binary system as an example, type “VARIABLE_X” is defined as

property “V

m

”:

Type_Definition v GES AMEND_PHASE_DESCRIPTION * VARIABLE_X Vm !

217

which means any phase with the Type_Definetion “v” will use mole fractions as

the variables of Vm. Here are examples to define the molar volume parameters

for the Liquid and L12_FCC phases:

Phase Liquid %v 1 1 !

Parameter Vm(Liquid,Al;0) 298.15 +V_Al_Liquid; 3000 N !

Parameter Vm(Liquid,Ni;0) 298.15 +V_Ni_liquid; 3000 N !

Phase L12_FCC %v 2 0.75 0.25 !

Parameter Vm(L12_FCC,Al;0) 298.15 +0.935*V_Al_fcc; 3000 N !

Parameter Vm(L12_FCC,Ni;0) 298.15 +0.935*V_Ni_fcc; 3000 N !

In this case, the molar volume of L12_FCC is calculated as:

0.935* * 0.935* *

fcc fcc

m Al Al Ni Ni

V x V x V=+

(8.4)

although the L12_FCC phase is described by CEF model with two sublattices.

Please refer to the example #28 in the example book for detail information.

8.2.3 Density

Density is calculated from molar volume and the molar weight. It requires the

molar volume parameters in database. The unit for density is

3

/kg m

. Density of

a phase is defined as

(8.5)

The density of a system with phase mixture is calculated from the molar weight

and the molar volume of the mixture:

(8.6)

218

8.2.4 Viscosity

A function to calculate viscosity of liquid phase is added into current Pandat

TM

.

The model used for describing the viscosity of liquid phase is semi-empirical

relation presented in the paper by Seetharaman and Du [1994See],

=

RT

G

A

*

exp

(8.7)

with

V

hN

A =

(8.8)

where V is the molar volume of the melt, h the Plank’s constant, N the

Avogadro’s number.

*

G

is the Gibbs energy of activation, and can be

calculated by

++=

jimix

mo

ii

xxRTGGxG 3

*

(8.9)

where x

i

and x

j

are the molar fractions of component i and j,

o

i

G

, Gibbs energy

of activation of component i, and

mix

m

G

, Gibbs energy of mixing. The

calculation of molar volume is given in above sections. The parameter for the

activation energy of a pure component in database TDB file has the format

Parameter ActivationEnergy(Liquid,Al;0) 298.15 15051+13.519*T; 2000 N !

8.2.5 Surface Tension

Current Pandat has another new function to calculate surface tension of liquid

phase. The model for calculating surface tension of liquid is the semi-empirical

relationship proposed by Yeum et al. [1989Yeu] to estimate the surface

tensions of binary alloys based on the model of Butler [1932But].

i

i

i

i

a

a

S

RT

,

ln+=

(8.10)

where

i

,

,

i

a

and

i

a

are the surface tension, activity at the surface and activity in

the bulk of component i. And

i

S

is the surface monolayer area,

219

3/23/1

ii

VbNS =

(8.11)

where b is geometric factor, N the Avogadro’s number, and V

i

molar volume of

component i. This approach was extended to calculate surface tensions of

multicomponent liquid alloys [1997Zha]. The calculation of molar volume is the

same as given in section 8.2.2. The ratio of the coordination number for the

surface atoms to that for the atoms in the bulk phase, b, is represented by a

parameter beta, described in database (.tdb) file in the format as

Parameter Beta(Liquid,Al;0) 298.15 0.83; 2000 N !

220

8.3 Kinetic Property

Diffusivity related properties can be calculated from Pandat

TM

as long as the

corresponding species mobility parameters are available in the database. User

can input the calculation condition through the interface shown in Figure 8.5

and select the desired properties for output. The default graph is the selected

property as shown in Figure 8.6. If two or more properties are selected, they

will be plotted separately.

Details on the kinetic models for the multicomponent diffusion are referred to

the literature [1982Ågr, 1992And]. Only the key equations related to the

properties of mobility, tracer diffusivity and chemical diffusivity are given in the

below sections.

Figure 8.5 Kinetic Property calculation dialog

221

Figure 8.6 Calculated diffusivity in the Al-Ni system

8.3.1 Atomic Mobility

In order to simulate diffusivity related properties such as chemical diffusivities

of components, atomic mobility data of species in phases are required and

stored in the database. Mobility of species k is related to its activation energy

(

k

Q

) by

/

k

Q RT

kk

M M e

−

=

(8.12)

where

k

M

is a frequency factor, R is the gas constant and T the temperature in

Kelvin. Define

k

MQ

as

ln ln

k k k k

MQ RT M RT M Q= = −

(8.13)

Then, atomic mobility can be calculated from

k

MQ

.

k

MQ

is a function of

composition, temperature and pressure and can be expressed as Redlich-Kister

type of polynomial expansion as for the excess Gibbs energy [1982Ågr]. Each

polynomial coefficient is stored in database. For example, the coefficient for

222

term contributed to Al from Ni in Fcc phase (

,Ni

Al

MQ

) is described in TDB file as

follows,

Parameter MQ(Fcc&Al,Ni;0) 298.15 -285517+R*T*Ln(0.0007933); 6000 N !

In Pandat, atomic mobility of species can be obtained through table operation

with field of “M(*@*)”. For example, M(Al@Fcc) represents the atomic mobility

of Al in Fcc phase.

8.3.2 Tracer Diffusivity

Tracer diffusivity of a species k is directly related to its atomic mobility by

*

kk

D RTM=

(8.14)

where R is the gas constant and T the temperature. Tracer diffusivity can be

obtained in Pandat

TM

from table with field of “DT(*@*)”. For example, the tracer

diffusivity of Al in Fcc can be extracted from calculation result with the table

field of “DT(Al@Fcc)”. The natural logarithm of the tracer diffusivity is available

with “logDT(*@*)”.

8.3.3 Chemical Diffusivity

Chemical diffusivity of species k,

n

kj

D

, could be calculated by

n

kj kj kn

D D D=−

(when j is substitutional) (8.15)

n

kj kj

DD=

(when j is interstitial) (8.16)

and

()

ii

kj ik k i i ik i Va i

jj

i S i S

D u u M u y M

uu

= − +

(8.17)

where S represents the set of the substitutional species,

ik

.is the Kronecker

delta, and

i

is the chemical potential of species i.

k

u

is defined as

223

k

k

j

iS

x

u

x

=

(8.18)

Chemical diffusivity of species k in phase p can be obtained in Pandat

TM

with

the table field of “DC(k,j@p:n)”, where j and n are the gradient species and the

reference species, respectively. Its corresponding natural logarithm is

“logDC(k,j@p:n)”.

8.4 User defined properties

Pandat allows user to define any property of a phase or a system following

some simple syntax rules. The user-defined-property database can be added

to the original database, either TDB or PDB files, through Append database

function. (Please refer to Section 3.3.9 for the detailed description on Append

database function). Users can also add the user-defined-property parameters

into their home-developed TDB files to develop a combined database for User-

defined property calculations.

Three methods are implemented in Pandat to add User-defined-properties,

depending on the nature of the properties. Key words Phase_Property;

System_Property and Property are used in the syntax respectively.

Phase Property is used to define a property of a phase with a similar

expression which describes the Gibbs energy of a disordered solution phase.

Let U be the user-defined phase property and it is expressed as:

1

1 1 1

()

c c c

o k k

i i i j i j ij

i i j i k

U xU x x x x L

−

= = = +

= + −

(8.19)

where

i

x

is the molar fraction of component i and

0

i

U

is the property of the

pure component i,

k

ij

L

is the k

th

order interaction parameter between

components i and j. The syntax used in the TDB file are:

Type_Definition z PHASE_PROPERTY U 1 !

Type_Definition v GES AMEND_PHASE_DESCRIPTION * VARIABLE_X U !

224

In this definition, “v” is the identifier, “*” means any phase, and VARIABLE_X is

the key word indicating X as the variable. The meaning of this definition is that

any phase with the identifier “v” will use mole fractions (x) as the variables for

property U.

System Property is used to define a property of a system with more than one

phase. For a property in a system with multi-phase mixture, the property of the

system is the weighted average of that of each phase. By default, the arithmetic

mean is applied to a user defined Phase Property in a multi-phase system. For

example, the user defines Phase Property U in a system with and β two-

phase mixture. The property of U of this system is calculated by

U f U f U

=+

(8.20)

If the simple arithmetic mean does not apply, more complicated expression can

be defined by User through Command System Property. For example, the

value of the system property can be calculated through the function:

0

()

i

i

i

U f U f U f f M f f

+

=

= + − −

(8.21)

where

U

+

is the user defined property of U in α+β two phase region.

U

and

U

is the property U in α and β phase respectively; f

α

and f

β

are the phase

fraction of α and β phase respectively. M

i

are the i

th

order of the additional

parameters which are used to describe additional effects on the user defined

property U.

The syntax for the System Property used in the TDB file is:

System_Property Sys_U 1 !

Parameter L(Sys_U, Alpha, Beta;0) 298.15 M0; 3000 N !

Parameter L(Sys_U, Alpha, Beta;1) 298.15 M1; 3000 N !

Using command Property, user can also define special properties associated

with phases in the original database. Any phase property available from Pandat

225

Table can be used for user-defined property, such as G, H, mu, and ThF.

However, the star symbol in a property, like mu(*), cannot be used.

The syntax for the Property used in the TDB file is:

Property GFcc_GLiq 298.15 G(@Fcc)-G(@Liquid); 6000 N !

8.4.1 User-defined Molar Volume Database

The molar volume (V

m

) is one of the pre-defined properties in Pandat. The

molar volume of a pure component is described as a function of temperature

and pressure, and the excess molar volume of a phase is usually described in

the format similar to that used to describe the excess Gibbs energy of the

phase. Note that, if a phase is described by a compound-energy-formalism

(CEF) with multi-sublattices, the molar volume is automatically described as a

function of T, P, and site fraction (y) following the format of the Gibbs energy.

This will introduce many end members in the molar volume and may lead to

incontinuity in the calculated molar volume. In reality, molar volume should be

a function of mole fraction (x) instead of site fraction (y), it is thus

recommended to define the molar volume of a phase separately from its Gibbs

energy if the phase is described by the CEF model. This can be realized

through the user-defined molar volume property.

In brief, the molar volume of element i with crystal structure φ can be

expressed as:

0

0

( ) exp( 3 )

T

i

T

V T V dT

=

(8.22)

where V

0

is the molar volume under atmospheric pressure at the reference

temperature T

0

. And is the coefficient of linear thermal expansion (CLE). The

volume of a phase with crystal structure φ can then be obtained via the

Redlich-Kister polynomial:

()

ex

i i m

i

V T xV V

=+

(8.23)

226

where x

i

is the mole fraction of element i and

ex

m

V

is the excess molar volume.

Using ternary system as an example, the

ex

m

V

can be expressed by:

1

,,

11

()

cc

ex k k

m i j i j ij i j p i j p

i j i k

V x x x x L x x x L

−

= = +

= − +

(8.24)

The terms

,

k

ij

L

and

,,i j k

L

are the interaction parameters from the constituent

binary and the ternary systems, respectively. The standard unit of molar

volume is m

3

/mol.atom.

The molar volume database also enables us to calculate the density based on

the relationship of

m

M

V

=

(ρ is the mass density, M is the molar mass, and

V

m

is the molar volume).

Here we use the Al-Ni binary system as an example to demonstrate the

calculation of molar volume through user-defined property. Please refer to the

AlNi_Vm.tdb for details. In the beginning of the database file, a Type_Definition

is given as below:

Type_Definition v GES AMEND_PHASE_DESCRIPTION * VARIABLE_X Vm !

In this definition, “v” is the identifier, “*” means any phase, and VARIABLE_X is

the key word indicating X as the variable. The meaning of this definition is that

any phase with the identifier “v” will use mole fractions (x) as the variables for

V

m

. Here are examples to define the molar volume parameters for the Fcc and

L12_FCC phases in the Al-Ni system:

Phase Fcc %(v 1 1 !

Constituent Fcc: Al,Ni:!

Parameter Vm(Fcc,Al;0) 298.15 +V_Al_fcc; 3000 N !

Parameter Vm(Fcc,Ni;0) 298.15 +V_Ni_fcc; 3000 N !

Parameter Vm(Fcc,Al,Ni;0) 298.15 -2.85e-6; 3000 N !

Phase L12_FCC %v 2 0.75 0.25 !

227

Constituent L12_FCC: Al,Ni:Al,Ni:!

Parameter Vm(L12_FCC,Al;0) 298.15 +V_Al_fcc; 3000 N !

Parameter Vm(L12_FCC,Ni;0) 298.15 +V_Ni_fcc; 3000 N !

Parameter Vm(L12_FCC,Al,Ni;0) 298.15 -3.2e-6; 3000 N !

Even though the L12_FCC phase is modeled by CEF model with two sublattices,

four end members, and 13 interaction parameters, its molar volume property

can be described as a function of

like a solution phase through user-defined

property. In other words, the molar volume of L12_Fcc is:

(8.25)

Figure 8.7 shows the calculated Al-Ni binary phase diagram with molar volume

contour lines. Please refer to AlNi_Vm.pbfx and section 3.3.5 for details on the

calculation of contour diagrams.

Figure 8.7 Al-Ni binary phase diagram with the calculated contour lines of

molar volume (cm

3

/mol).

228

The lattice constant, or lattice parameter, refers to the physical dimension of

unit cells in a crystal lattice. For FCC crystal structure, the lattice constant can

be calculated by:

(8.26)

An example is given for the calculation of lattice constant in Ni-Al binary at Ni-

rich corner. The calculated lattice constant agrees with the experimental data

well as is shown in Figure 8.8. Please refer to AlNi_lattice.pbfx for details. It

should point out that the unit of Vm in the above equation is cm

3

/mol.

Figure 8.8 Comparison of calculated and measured lattice constants of the γ

and γ’ phases in Ni-Al binary alloys at room temperature

8.4.2 Thermal Resistivity and Thermal Conductivity

Thermal conductivity of a pure element or a stoichiometric phase at

temperature above 273 K is described as a function of temperature using the

following equation:

12

A BT CT DT

−

= + + +

(8.27)

229

where κ is the thermal conductivity and T is the temperature in Kelvin. This

function can reasonably fit most of the experimental thermal conductivity data

of elements at temperature above 273 K.

The thermal conductivity of solid solution phase can be calculated from

thermal resistivity, which is the reciprocal of thermal conductivity. According

to the Nordheim rule, the thermal resistivity (ρ) of a solid solution phase can be

described by the following Redich–Kister polynomials:

0

()

i

AB A A B B A B i A B

i

x x x x L x x

=

= + + −

(8.28)

where

AB

is the thermal resistivity of the solution phase in the A-B system.

x

j

and ρ

j

are the mole fraction and thermal resistivity of pure elements j,

respectively. L

i

are the i

th

order interaction parameters which are used to

describe the effect of solute elements on the thermal resistivity. In general, the

interaction parameter can be expressed as:

1

i i i i

L a bT cT

−

= + +

(8.29)

where the parameters a

i

, b

i

and c

i

are evaluated based on the experimental

data.

The interface scattering parameters is introduced to describe the effect of the

second phase on the thermal resistivity in two-phase region. The thermal

resistivity of the two-phase region is described as following:

0

()

i

i

i

f f f f M f f

+

=

= + − −

(8.30)

where ρ

α+β

is the thermal resistivity of the alloys in α+β two-phase region, f

p

and

ρ

p

(p = α, β) are the mole fraction and thermal resistivity of the phase p,

respectively. M

i

, which can be considered to be linearly temperature dependent,

is the i

th

interface scattering parameter and and can be evaluated from the

experimental data. The thermal conductivity of alloy system and the value of

each phase can be obtained by using the reciprocal of the thermal resistivity

values through output option.

230

In this example, thermal resistivity of the Al-Mg binary alloys is described

using the user-defined property function.

As shown in the AlMg_ThRss.tdb, the thermal resistivity ThRss property is first

defined as user-defined property since it hasn’t been pre-defined in the current

Pandat software.

Type_Definition z PHASE_PROPERTY ThRss 1 !

In accordance, the following definition is also needed to add this property to the

original database.

Type_Definition e GES AMEND_PHASE_DESCRIPTION * VARIABLE_X ThCond !

As is seen, the thermal resistivity of the Fcc phase or the Hcp phase follows the

same format as that of Gibbs energy for a disordered solution phase.

Parameter ThRss(Liquid,Al;0) 298.15 1/ThCond_Al_Liq; 3000 N !

Parameter ThRss(Liquid,Mg;0) 298.15 1/ThCond_Mg_Liq; 3000 N !

Parameter ThRss(Fcc,Al;0) 298.15 1/ThCond_Al_Fcc; 3000 N !

Parameter ThRss(Fcc,Mg;0) 298.15 1/ThCond_Mg_Hcp; 3000 N !

Parameter ThRss(Fcc,Al,Mg;0) 298.15 0.02566-1.3333e-05*T+14.5*T^(-1);

3000 N !

Parameter ThRss(Hcp,Al;0) 298.15 1/ThCond_Al_Fcc; 3000 N !

Parameter ThRss(Hcp,Mg;0) 298.15 1/ThCond_Mg_Hcp; 3000 N !

Parameter ThRss(Hcp,Al,Mg;0) 298.15 0.02140-1.3669e-05*T+12.7158*T^(-1);

3000 N !

Parameter ThRss(Hcp,Al,Mg;1) 298.15 0; 3000 N !

Parameter ThRss(Hcp,Al,Mg;2) 298.15 0.14825-7.7706e-05*T+25.3031*T^(-1);

3000 N !

Thermal resistivity of the intermetallic phases with narrow solid solubility rage

in the phase diagrams is treated like that of a stoichiometric compound phase,

i.e., it is composition independent and is described as below:

231

Parameter ThRss(AlMg_Beta,*;0) 298.15 1/42; 6000 N !

Parameter ThRss(AlMg_Eps,*;0) 298.15 1/42; 6000 N !

Parameter ThRss(AlMg_Gamma,*;0) 298.15 -0.03267+2.7412e-05*T+20.722*T^(-1);

6000 N !

In order to describe the thermal resistivity within two-phase region, a system

property, Sys_ThRss, is then defined by the command:

System_Property Sys_ThRss 1 !

Parameter L(Sys_ThRss, Fcc, AlMg_Beta;0) 298.15 0.005; 3000 N !

Parameter L(Sys_ThRss, Hcp, AlMg_Gamma;0) 298.15 0; 3000 N !

Parameter L(Sys_ThRss, Hcp, AlMg_Gamma;1) 298.15 0.01; 3000 N !

After the thermal resistivity has been properly modeled for each phase, the

thermal conductivity of each phase and that of the system can be directly

calculated and outputed by using extra output in Pandat defined as

1/ThRss(@*) and 1/Sys_ThRss, respectively. The comparisons between the

calculated and measured thermal conductivities of the Al-Mg alloys are shown

in Figure 8.9 to Figure 8.11. This example demonstrates that the user-defined

property function is very powerful and flexible to allow users define various

types of properties. The property can be a function of any phase properties that

can be calculated by PanPhaseDiagram module.

232

Figure 8.9 Comparison between the calculated and measured thermal

conductivities in the (Al) solid solution in the Al-Mg binary system

Figure 8.10 Comparison of the calculated and measured thermal conductivities

in the (Mg) solid solution in the Al-Mg binary system

233

Figure 8.11 Calculated thermal conductivities of Mg-Al system at 298 K.

8.4.3 Calculation of T

0

Curve Using User-defined Property

T

0

curve is the trace of a series of points in a two-phase field at which the

Gibbs energy of the two phases are identical. In this example, the original

database is ABC.tdb, and the T

0

-curve property is defined in the Appended

database ABC_T0.tdb. In the ABC_T0.tdb file, the T

0

-curves of Bcc/Liquid and

Fcc/Liquid phases are defined as:

Property GFcc_GLiq 298.15 G(@Fcc)-G(@Liquid); 6000 N !

Property GBcc_GLiq 298.15 G(@Bcc)-G(@Liquid); 6000 N !

Where “Property” is the key word for user-defined property and G(@Bcc),

G(@Fcc), G(Liquid) are the Gibbs free energies of the Bcc, Fcc, and Liquid

phases, respectively. Note that, the Bcc, Fcc and Liquid are defined in the

ABC.tdb. In this particular case, the above “Property” can be directly defined in

the ABC.tdb, i.e., the ABC.tdb and ABC_T0.tdb can be combined into one

database. This example demonstrates that user-defined property can be

234

separated from the original database. This design makes it possible to obtain

user-defined properties even the original database is in encrypted pdb format.

In this example, the GBcc_GLiq property is defined as the Gibbs free energy

difference between the Bcc phase and the Liquid phase, and GFcc_GLiq

property is defined as the Gibbs free energy difference between the Fcc phase

and the Liquid phase.

Using the Contour function (Section 3.3.5), we can calculate contour maps of

these user-defined properties and plot them on the calculated phase diagram.

When we set the calculation condition as GBcc_GLiq=0, the T

0

curve for

Bcc/Liquid is obtained as is shown by the green line in Figure 8.12. Similarly,

by setting GFcc_GLiq=0, we obtain the T

0

curve for Fcc/Liquid as is shown by

the red line in Figure 8.12.

Figure 8.12 Calculated isopleth of A-B-10at.%C with T

0

curves of Bcc/Liquid

and Fcc/Liquid phases

User can run the ABC_T0.pbfx batch file to obtain Figure 8.12. Two points

should be addressed for this batch file:

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• Both the “start” and “stop” values should set to be zero for the contour

mapping as following to get the T

0

line.

<contour name="Contour_T0_Fcc_Liq" property="GFcc_GLiq" start ="0" stop ="0"

step="1"/>

<contour name="Contour_T0_BCC_Liq" property="GBcc_GLiq" start ="0" stop ="0"

step="1"/>

• To obtain the T

0

curve, each phase needs to be considered individually, thus

the equilibrium type is set to be “individual”

<individual_phase value="true"/>

<equilibrium_type type="individual"/>

8.4.4 Calculation of Spinodal Curve Using User-defined

Property

A Spinodal curve is where the determinant of the Hessian of Gibbs free energy

with respect to composition is zero. For a phase with c-components, above

condition is expressed as

=0 (8.31)

where the molar fraction of component 1 is chosen as the dependent variable.

The second derivative of G w.r.t. molar fractions can be calculated from the

thermodynamic factors:

(j,k=2,3,…,c) (8.32)

Example #1: Spinodal curve of Fcc phase in the Al-Zn binary system

In this example, the spinodal curve of the Fcc phase in the Al-Zn binary system

is calculated through user-defined property. A user-defined property d2GdxZn2

for the Fcc phase is defined in AlZn_Spinodal.tdb as:

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Property d2GdxZn2 298.15 ThF(Zn,Zn@Fcc)-ThF(Al,Zn@Fcc)-ThF(Zn,Al@Fcc)+

ThF(Al,Al@Fcc); 6000 N !

where ThF(Zn,Zn@Fcc), ThF(Al,Zn@Fcc), ThF(Zn,Al@Fcc), and ThF(Al,Al@Fcc)

are the thermodynamic factors of Fcc phase. Since the value of d2GdxZn2 is

usually a large number, we define the Hessian function, HSN, as d2GdxZn2

multiplied by a factor 1E-4:

Property HSN 298.15 1e-4*d2GdxZn2; 6000 N !

As shown in the AlZn_Spinodal.pbfx, the AlZn_Spinodal.tdb is appended to the

AlMgZn.tdb. The spinodal line is calculated through contour mapping by using

following conditions:

<contour name="Spinodal" property="HSN" start="0" stop="0" step="1"/>

<equilibrium_type type="individual"/>

The calculated Fcc Spinodal is shown in Figure 8.13 with the stable Al-Zn

binary phase diagram.

Figure 8.13 Calculated spinodal curve of the Fcc phase within the Al-Zn binary

system

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Example #2: Spinodal curve of Fcc phase in the X-Y-Z ternary system

In this example, the original database is XYZ.tdb, and the user-defined HSN

property is defined in XYZ_Spinodal.tdb as described below:

Property d2Gdx2 298.15 ThF(Y,Y@Fcc)-ThF(X,Y@Fcc)-ThF(Y,X@Fcc)+ThF(X,X@Fcc);

6000 N !

Property d2Gdy2 298.15 ThF(Z,Z@Fcc)-ThF(X,Z@Fcc)-ThF(Z,X@Fcc)+ThF(X,X@Fcc);

6000 N !

Property d2Gdxy 298.15 ThF(Y,Z@Fcc)-ThF(X,Z@Fcc)-ThF(Y,X@Fcc)+ThF(X,X@Fcc);

6000 N !

Property HSN 298.15 1e-10*(d2Gdx2 * d2Gdy2 - d2Gdxy * d2Gdxy); 6000 N !

Note that the HSN property within the XYZ ternary system is derived and

described as a function of the thermodynamic factor ThF. A factor of 1E-10 is

used to scale the HSN property since the numerical value of HSN is very big.

Again the spinodal lines are calculated through contour mapping. Details can

be found in XYZ_Isotherm_Spinodal.pbfx and XYZ_Isopleth_Spinodal.pbfx.

Figure 8.14 and Figure 8.15 show the calculated spinodal curves superimposed

on the stable phase diagrams:

Figure 8.14 Calculated isothermal sections of the X-Y-Z system at 800 and

600K with spinodal curves, respectively

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Figure 8.15 Calculated isopleths within the X-Y-Z system for 10 and 20 at.%Z

with spinodal curves, respectively