Multi-Scale Numerical Modelling of Phase Transition

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radial distribution function, Al lattice, 0 K "RADIAL"

60

g(r)

50 40 30 20 10 0

0

0.5

1 1.5 2 r/atomic diameter

2.5

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1.8

Al lattice solidification from 2010 K 10 10 K "RADIAL"

1.6 1.2

4

1

g(r)

g(r)

"RADIAL"

5

1.4

0.8 0.6

3 2

0.4

1

0.2 0

Al lattice melting from 10 K to 2010 K

6

0

0.5


2.5

0

3

0

0.5

liquid Al at 2010 K

2.5

g(r)

1

g(r)

"RADIAL"

3

1.2 0.8 0.6

2 1.5

0.4

1

0.2

0.5

0

3

Al lattice at 10 K

3.5

"RADIAL"

1.4

2.5

a

b

1.6


0

0.5


2.5

0

3

0

c

0.5


2.5

3

d

radial distribution function for Al

,# 6 3 5

/ 4 ) 0 4 0 ; Æ " 0 : 0 0 4 4 ) / " ; 4 ) 4 / 5 0 4 ) 2 0 4 ) 2$% " / 0 ) 2$% 0 0 4

Al at equilibrium 2010K

Al at 10K b

a

Interatomic potential used: RTS

Al at equilibrium, 10K c

7# " 5

4 2$% " 4 ) / ! ! ! " 2$% = 0 " 4 4 @

4 $ ' 14 %0 *2! 8 / !1 !!!! " 8 !1 +! !1 +!!!!

" / +! !1 !!!! " 4 3 / !$% !$% 4 9 +! !1

!1 +!!!! / !1 !!!! 0 +

Al lattice solidification from 2010 K 10 10 K

Al lattice melting from 10 K to 2010 k

-1400

"gib" "potential"

-1250

gibbs free energy and potential energy (ev)


-1200

-1300 -1350 -1400 -1450 -1500 -1550 -1600 -1650 40000 42000 44000 46000 48000 50000 52000 54000 56000 58000 60000 time-step

"gib" "potential"

-1450 -1500 -1550 -1600 -1650 -1700 -1750

0

2000

4000

6000

8000 10000 12000 14000 16000 18000 20000 time-step

b

a

liquid Al at 2010 K

-1565 "gib" "potential"

gibbs frre energy and potential energy (ev)


-2000

-1800

-1600

-1400

-1200

-1000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 time-step

Al lattice at 10 K "gib" "potential"

-1570 -1575 -1580 -1585 -1590 -1595

-1600 60000 61000 62000 63000 64000 65000 66000 67000 68000 69000 70000 time-step

c

d

gibbs free energy and interatomic potential for Al

2# 6 ) 5

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Cu lattice melting from 10 K to 3010 K

9

"RADIAL"

8

3

6

2.5

5

g(r)

g(r)

"RADIAL"

3.5

7

4

2 1.5

3 2

1

1

0.5

0

Cu at 10 K

4

0

0.5


2.5

0

3

0

0.5


b

3

a

Cu at 3010 K

0.8

2.5

Cu solidification from 3010 K to 10 K

2.5

"RADIAL"

"RADIAL"

0.7

2

0.6 1.5

0.4

g(r)

g(r)

0.5 0.3

1

0.2 0.5

0.1 0

0

0.5


2.5

3

0

0

0.5

c

1

1.5 r/atomic diameter

2

2.5

3

d

radial distribution function for Cu

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4 ) 5 . 0 0 4 " 5 80 / 0

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Cu at equilibrium 3010K

Cu at 10K b

a

Interatomic potential used: RTS

Cu at equilibrium, 10K c

# " 8

Cu lattice solidification from 3010 K to 10 K

-1000 gibbs free energy and potential energy (ev)

"gib" "potential"

-1100 -1200 -1300 -1400 -1500 -1600 -1700 50000

55000

60000

65000 time-step

70000

75000

80000

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variation of the Al shaer viscosity with temperature

1.8

"visal.dat" 1.75

shaer viscosity (gr/sm)

1.7

1.65

1.6

1.55

1.5

1.45

1.4 200

400

600

800 1000 temperature (K)

1200

1400

1600

+# " 4 5 4 4 4 . 4 5 0 . 4 " 4 / $ 7% "

" 4 / 8 + 6 5 4 , 4 " / $ 7,% " K 4 ,!

the variation of the Pb shear viscosity with Temperature

0.0016

exp value=0.0015 0.0015

shear viscosity (Kg/sm)

0.0014

0.0013

0.0012

0.0011

0.001

0.0009 400

450

500

550

600


750

800

850

900

*# " 4 4

variation of Cp with temperature for Al

10

experiment MD simulation

Cp Cal/MolK

8

6

4

2

0 200

400

600

Temperature K

800

1000

1200

# " 4 5

0 " 4 4 ' " 4 / 0 " Æ " 4 4 70 2 ! "

0

.

,

variation of the Al isothermal compressibility with temperature

0.0245

Kt 0.024 0.0235 0.023

Kt (1/GPa)

0.0225 0.022 0.0215 0.021 0.0205 0.02 0.0195 300

400

500

600 700 Temperature (K)

800

900

1000

# " 4 5

variation of Pb isothermal compressibility with temperature

0.0285

Kt 0.028 0.0275 0.027

Kt (1/GPa)

0.0265 0.026 0.0255 0.025 0.0245 0.024 0.0235 0.023 300

400

500

600 temperature (K)

700

800

900

,# " 4

,

variation of the Al bulk modulus with temperature

51

exp=70 50 49

Bulk Modulus (GPa)

48 47 46 45 44 43 42 41 300

400

500


800

900

1000

7# " 4 5 9

variation of Pb bulk modulus with temperature

43

exp=45 42

Bulk modulus (GPa)

41

40

39

38

37

36

35 300

400

500

600 temperature (K)

700

800

900

2# " 4 9

4 4 ; 88 $50 80 0 D0 50 5 %0 5 0

4 5 0 4 Æ

4 / $ 2 % $ 72%0 4

' ( & ) = 8 4

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variation of the shear modulus with temperature

9.6

Al Pb

9.4 9.2

Shear modulus (GPa)

9 8.8 8.6 8.4 8.2 8 7.8 7.6 300

400

500

600

700 temperature (K)

800

900

1000

1100

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Isoentropic compressibility (Ks), Thermal pressure coefficient (Gv)

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0-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90

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+7

variation of the LF with T for Sn 10%w Pb alloy with contact angle 45 Deg

1

10 K/s 2 K/s 1 K/s

0.9 0.8

liquid fraction

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200

250 300 temperature K

350

400

450

500

550

* # ?/ 4 *0 - !

: > ?;+

" . 0 0 " . 5-= $ 227% I+J -X 8 0 0 6 ++0 D 0 2,, I+,J " ";0 0 8 5 0 =" 0 27, I+7J &1 6 A 0 5 = 8 I+2J P5 -

-)3 9 0 8 - C; 5 6 ) - 8 0 5 0 6 +,0 D ,0 7!+ 7 !0 272 I*!J &0 8 " 0 8 0 6 70 7+! 0 27+ I* J 84 5 8 . 0 = ?/ - 0 P A - 0 2, I*J 3 & 0 3? G 0 5 11 10 - 6 " 9 50 P A0 2+ I*+J - 5 0 = " = 0 27 I**J 3) 0 A X 0 " 5 0 22* I*J 8 0 - 0 ) &0 2,*

7

I*J 5 0 5 C/ 0 C4 0 2 I*,J A A 0 3 - 0 P 5 6 +*0 2+ I*7J A A 0 3 - 0 P 5 6 +0 2* I*2J 3 - 0 P 8 ) 0 6 +0 27 I!J P 5 -

- ) 3 9 0 5 0 6 +,0 D ,0 7!+ 7 !0 272 I J P - 4 '0 5 0 - - ) - ) 0 5 0 6 +0 ,2+0 27* IJ P - 4 '0 ) - ) 0 5 0 6 ++0 ++0 27 I+J P - 4 ' ) - ) 0 5 0 6 +*0 7++0 27 I*J D &

0 - X & D 0 - 0 6 0 *,0 27, IJ P - 4 '0 5 ) - ) 0 5 0 6 +0 D 20 *2 *+70 27* IJ & A & = 3 )0 5 0 6 +20 D 20 + *+0 22 I,J A 1 '0 9 )4 3 " 40 5 0 6 +*0 D 0 7+7+!0 27 I7J Y " 0 89 0 = P & " 0 6 * 0 *!*!20 227 I2J P & 0 U & - ?0 - - 0 C = G - 0 22+ I!J 3' & "4'0 5 0 6 +0 D ,0 *7, *2,0 27, I J 8 X A 8 9 0 - C 0 5 , 0 22 22+ 7,

0

IJ 8 5 ) 0 3' A 0 " 50 6 50 P 22 I+J 8 9 8 X A 0 - - C 0 C = G - 0 22 I*J Y " 8 9 0 P 8 ) 0 6 7,0 72+!0 227 IJ 6 0 5 8 0

8 9 0 C/. - - 0 - 70 227 IJ = - 0 8 9 0 9 1 U ?0 5 0 6 *,0 D +0 2, 270 222 I,J 9 8 5 60 P 8 ) 0 4 *0 !20 2,, I7J 3' 8 5 ) 0 5 0 6 * 0 D 0 +*+!0 22+ I2J 3'0 = 340 6 +*0 D +0 272 I,!J = . 5 &0 5 0 6 +0 2, I, J 8 8 0 5 P 3'0 5 0 6 *0 D 0 +2++20 22* I,J 8 5 ) 0 8 8 3'0 =-=P = 0 6 +0 D 0 ,0 22 I,+J 8 5 ) 3'0 5 0 6 * D 0 7, 20 22, I,*J 1 X ? 8 & 0 - - 0 C = G - 0 22 I,J "4'0 P ? 3'0 " 50 6 !50 272 I,J - ) F0 " 50 6 50 5 22!

77

I,,J 8 5 ) 3'0 5 0 4 *0 D ,0 ++*0 22* I,7J - 9 H0 8 5 ==0 C 6 3 6 - 0 22 I,2J C ) 3 D - 0 " 50 6 +50 22 I7!J & 8 P ' 0 " 50 6 ,50 5 22 I7 J 8 - P )0 " 50 6 750 P 22, I7J 8 8 3 ? 0 - - C 0 C = G - 0 22 I7+J 8 3 - 6 3 6 0 " 90 6 +90 G 22 I7*J 8 0 A 54 - 6==0 D U 0 1 8 3 9 0 C 8 P 8 0 22 I7J 8A ) 0 & - 0 - 0 9 0 27 I7J - 3 0 9 " 5 5 8 40 P 8 ) 0 6 * 0 2++0 22* I7,J A A 3 - 0 P 5 0 6 +0 ***0 2* I77J & 3 " 0 ?& 8 0 P 8 #

0 6 !0 +,0 227 I72J 5 ?0 1 ) 4 3 -0 P 8 ) 0 6 70 2 2,0 227 I2!J P 80 8 0 9

& 0 22, I2 J ) C 0 0 277 I2J - 6 0 D @ @0 27!

72

I2+J " D 8 0 F 6 5 - 0 " 0 3 "0 F 4 ) 0 ? 0 227 I2*J 6 3 6 8 3 - 0 ) - 9 - D 0 90 2$%# , 720 22 I2J )5 " 0 "0 F 4 ) 0 22 I2J = 0 A

0 & " 0 A0 22! I2,J )0 F 4 ) 0 222 I27J = 33

2!

Multi-Scale Numerical Modelling of Phase Transition

Multi-Scale Numerical Modelling of Phase Transition

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