Non-statistical thermodynamic optimization: an extravagance or a useful tool?

Cong (Leo) Dai

2

Outline

• Thermodynamic modeling

• CALPHAD method

• Principles and examples

• Advantages and deficiencies

• Kantorovich idea

• Conclusions and future efforts

3Thermodynamic modeling

PhasSage, www.factsage.com PhasSage; http://honghua66.en.ecplaza.net; Solutionizing Heat Treatment Furnace (27982050)

A

B C

4

What is needed to use thermodynamic modeling?

• Analytical descriptions of the Gibbs energies of stoichiometric compounds and solutions• Publications including

tables of thermochemical data

• Databases

• A program minimizing the Gibbs energy of a system• Thermo-Calc• FactSage• Pandat• MALT2

Reliability of the analytical representations of the Gibbs energies?

5Phase diagrams of the Fe–Nb system

TCFE2 database TCFE6 database

BA

6Constructing the Gibbs energies

7CALPHAD technique is not unique

• Step 1: a model

The coin is a disc• Step 2: identify unknown model’s

parameters

Radius r, thickness h, density ; they all must be positive

• Step 3: collect all available experimental data

r, h, d, l, a, V, m, • Step 4: solve an optimization problem

8Finding r, h and

2 2 2 2

1 1 1 1

2 2 2

1 1 1

2

1

2 2 2

2 2

m

0,

i

0

n

, 0

, ,

h d lr

a V m

n n nni i i i

i i i ii i i i

n n ni i i

i i ii i i

ni

i i

r r h r

r r h h r d r l

r h d l

a V m

a V m

h

h

h

r

r

Non-linear least squares problem with linear constraints

9Almost the same happens in CALPHAD

• Experimental data1. Enthalpies of mixing at 1150°C

2. (Ag)+L / L liquidus temperatures

3. L / L+(Cu) liquidus temperatures

4. Chemical potential of Cu in (Ag)

5. Chemical potential of Ag in (Cu)

u

Ag

L

C

Ag AgA

Cu CuCuex Cu C

L LLex L L

gex Ag

u Cu0

A Ag

L0

g0

1

,

1

,

1

,

n

i ii

i

n

i

i

i

i

n

i

G x x

G x x C x T

C T

G x x C x T

x

����������������������������

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����������������������������

10Building the Gibbs energies of the phases

2exp calc L

9

exp1

2 2exp calc L exp calc L

2 5

exp exp1 1

2exp calc Ag expCu Cu Ag A

expCu

L

L Ag

Ag

CuL

, ,

, , , , , ,

, ,

i i i

i i

i i i i i i

i ii i

H H x T

H

T T x T T T x T

T T

x T

C

C C CC

C

��������������

����������������������������

2

calc Cug

expAg

Cu

CuAgL, ,

min , ,x T

C CC

C

���������������������������� ����������������������������

A traditional CALPHAD method rests on a statistical foundation

11

Statistical description of data

1

2

1

2

1

Mean: ,

( ) Variance: ,

1

( ) Standard error of the mean:

( 1)

n

ii

n

ii

n

ii

xx

n

x x

n

x x

n n

• Measurement is non-repeated

• Measurement is insufficient

• Measurement errors are affected by random errors and systematical errors

12

Kantorovich idea(1962)

Instead of minimizing the sum of squared derivations, simultaneous inequalities should be used, which can be solved by linear programming methods.

Kantorovich, L.V. Sib. Mat. Zh., 1962, vol.3, No.5, p. 701.

13

Interval data

• All possible outcomes of an experiment belong to a finite interval!

• Any value in the same interval data is equally acceptable!

Non-statistical approach14

1. Error analysis

2. Postulate a model

3. Solve inequalities

i iy a b x

i i iy y y

i i iy a b x y

15

Data from Kawakami

Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351.

Reach the required temperature

Remove the porcelain tube

Measure the temperature variation

16

Mixing Enthalpy of liquid Mg-Ag alloys

( )E

E EP

GH G T

T

Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351.

A B

17

Mixing Enthalpy of liquid Mg-Ag alloys

No feasible region.

)21()1(

)21()1(

MgMgMgE

MgMgMgE

xbaxxH

xbaxxG

axxH

axxG

MgMgE

MgMgE

)1(

)1(

18

Data from Gran

J. Gran et al./ CALPHAD 36 (2012 89-93)

Heat the furnace

Hold at a predetermined temperature

Quenched in an Argon-stream

Analyzed by ICP-AES analysis(Inductively Coupled Plasma-

Atomic Emission Spectrometry)

Activity of Mg in liquid Mg-Ag alloys19

ln( ) ln( )Mg MgE

MgMgMg

a aRT G RT

xx

J. Gran et al./ CALPHAD 36 (2012 89-93)

(1 )E

E EMg Mg

Mg

GG G x

x

20

Activity of Mg in liquid Mg-Ag alloys

axG

axxG

MgEMg

MgMgE

2)1(

)1(

bxxaxG

xbaxxG

MgMgMgEMg

MgMgMgE

)41()1()1(

)21()1(

22

No feasible region.

Comparison Future work!

Conclusions21

• CALPHAD technique is successful but not

always satisfactory.

• A non-statistical approach is established and

applied to thermodynamic optimization.

Future efforts22

• Comparisons in liquid Mg-Ag alloys.

• Rules of assigning interval data.

• Non-statistical approach applies to other

interesting thermodynamic assessments.

23

Acknowledgements

Supervisor:

- Dr. Dmitri Malakhov

Thanks for your attention!