M ap p in g S t ab ilit y: Bin ar y P h ase Diag r am s
22.14 – In tro to N u c le a r M a te ria l s F e br ua r y 19 & 24, 2015
Im ag es fro m E n g i n e e r i n g M a t e r i a l s S c i e n c e ,
M ilton O hr i n g unle ss othe r w ise note d
M ajo r S t ep s
• P has e di agr am s
• R eadi ng phas e di agr am s
• T her m od y nam i cs
• F r ee ener gy
• F r ee ener gy di agr am s
• C ons t r uct i ng phas e di agr am s f r om f r ee ener gy di agr am s
P h ase Diag r am : E xam p le
L i qui d r e g i o n
α r e gi on
Tw o - pha s e r e gi on s
β r e gi on
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h ase Diag r am M icr o st r u ct u r
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P h ase S ep ar at io n in Real L if e
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E xam p les: Co m p let e S o lu b ilit y
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
B in ar y p h ase d iagr am s f r om A S M H an d b ook s, V ol. 3 ( availa b l e at ve r a. m it . e d u )
E xam p les: M iscib ilit y G ap
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
E xam p les: In t er m et allics
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
E xam p le P h ase Diag r am s
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
E xam p les: Eu t ect ic, E ver yt h in g !
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ® .
F e-C P h ase Diag r am
B as i s f or s t eel m aki ng
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M os t i m por t an t one t o r em em ber & under s t and !
F e-C P h ase Diag r am
A 1: T em per at u r e at w hi ch phas e t ra n s fo rm a t i o n fro m α - γ begi ns
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A 3: T em per at u r e at w hi ch i t ends
F e-C P h ase Diag r am
C om pos i t i on det er m i nes endi ng m i cr os t r uct u r e
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M i cr os t r uct ur e det er m i nes s t eel pr oper t i e s
Read in g P h ase Diag r am s
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Read in g P h ase Diag r am s: T h e L ever Ru le
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
Read in g P h ase Diag r am s: T h e L ever Ru le
𝐶 0 = 𝑂𝑣 𝑒𝑟𝑎 𝑙𝑙 𝐶𝑜 𝑚 𝑝 𝑜 𝑠𝑖𝑡 𝑖𝑜 𝑛
𝑓 𝛽
𝑓 𝛼
𝐶 𝛼 = 𝐶 𝑜𝑚𝑝𝑜𝑠 𝑖 𝑡 𝑖 𝑜𝑛 𝑜 𝑓 𝛼
𝐶 𝛽 = 𝐶𝑜 𝑚 𝑝 𝑜 𝑠𝑖𝑡 𝑖𝑜 𝑛 𝑜 𝑓 𝛽
𝑓 𝛼 = 𝑃ℎ 𝑎 𝑠 𝑒 𝑓 𝑟 𝑎 𝑐 𝑡 𝑖𝑜 𝑛 𝑜 𝑓 𝛼
𝐶 𝛽 − 𝐶 0
=
𝐶 𝛽 − 𝐶 𝛼
𝐶 𝛼 𝐶 𝟎𝟎 𝐶 𝛽
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
𝑓 𝛽 = 𝑃 ℎ 𝑎𝑠𝑒 𝑓 𝑟 𝑎 𝑐 𝑡 𝑖𝑜 𝑛 𝑜 𝑓 𝛽
= 𝐶 0 − 𝐶 𝛼
𝐶 𝛽 − 𝐶 𝛼
Read in g P h ase Diag r am s: T h e L ever Ru le
𝐶 0 = 𝑓 𝛼 𝐶 𝛼 + 1 − 𝑓 𝛼 𝐶 𝛽
𝑓 𝛽
𝑓 𝐴
H ow m uc h of e a c h pha s e ex i st s at t h e sp eci f i ed t em p er at u r e?
𝐶 𝐴 𝐶 𝟎𝟎 𝐶 𝛽
W ha t a r e t he c om pos i t i ons o f each p h ase?
Read in g P h ase Diag r am s
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Read in g P h ase Diag r am s
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Read in g P h ase Diag r am s
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T h er m o d yn am ics Review
G i bbs F r e e E ne r gy
P r essu r e
T em p er at u r e
V ol um e
I nt e r na l E ne r gy
???
E nt r opy
F r ee En er g y
E a c h c om pone nt i h as a tab u l at e d , sp eci f i c G i b b s f r ee en er g y ( ΔG i ) – L o w e r ΔG i in d ic a t e s h ig h e r s ta b ilit y
Courtesy of UC Davis ChemWiki. License: CC by-nc-sa.
h t t p : //c h e m w ik i. u c d a v is . e d u /A n a ly t ic a l_ C h e m is t r y /E le c t r o ch em i s t ry / E l e ct r o ch e m i s t ry _ a n d _ T h erm o d y n a m i cs
F r ee En er g y o f M ixt u r es
T w o par t s :
– A t om i c (m ol e ) fra c t i ons of fre e e ne r gi e s
– Fre e e ne r gy from m i xi ng
f r a c t i on of c om pone nt A
M ol e
G i bbs f r e e e ne r gy of c om pone nt B
O b t ain in g Δ G mi x
O b t ain in g Δ H mi x
L et ' s t ake a s ys t em of at om s ( A & B ) w hi ch t ot al s one m ol e ( N av ), w i t h m ol e f r act i ons X A & X B :
E A - A
E ne r gy of a n A - A bond = E A- A E ne r gy of a n B - B bond = E B - B E ne r gy of a n A - B bond = E A- B
O b t ain in g Δ H mi x
State 2
‘B’ A toms
‘ A ’ A toms
State 1
O nl y A - A a nd B - B bonds
A - A , B - B , a nd A - B bonds
Image by MIT OpenCourseWare.
O b t ain in g Δ H mi x
State 1
State 2
‘ A ’ A toms ‘B’ A toms
z is the coordination
number (here, z=4)
Under which conditions
will H mix =0?
>0?
<0?
Image by MIT OpenCourseWare.
O b t ain in g Δ H mi x
B ond num ber s bef or e m i xi ng:
# 𝐴 − 𝐴
= 𝑧 𝑁 𝑎 𝑣 𝑋𝑋 𝐴
2
# 𝐵 − 𝐵
= 𝑧 𝑁 𝑎 𝑣 𝑋𝑋 𝐵
2
B ond num ber s af t er m i xi ng:
𝑋𝑋 𝐴
𝑋𝑋 𝐵
𝑧 𝑁 𝑎 𝑣 2 𝑧 𝑁 𝑎 𝑣 2
# 𝐴 − 𝐴 =
2 # 𝐵 − 𝐵 =
# 𝐴 − 𝐵 = 𝑧 𝑁 𝑎 𝑣 𝑋𝑋 𝐴 𝑋𝑋 𝐵
2
z = coor di na t i on num ber ( # bonds / at om )
O b t ain in g Δ H mi x
N ext s t eps :
– O bt a i n e ne r gy be fore & a ft e r m i xi ng
– Subt ra c t t o ge t c ha nge i n m i xi ng e nt ha l py
O b t ain in g Δ S mi x
E xam i ne avai l abl e num ber of m i cr os t at es . . .
A
or
B
U s e t he B ol t zm ann equat i on :
O b t ain in g Δ G mi x
𝐸 𝐴 −𝐴 + 𝐸 𝐵 −𝐵
𝐸 𝐴 −𝐵 − 2
Δ 𝐻 𝑚𝑖𝑥
= 𝑧 𝑁 𝑎𝑣
𝑋𝑋 𝐴
𝑋𝑋 𝐵
Dr aw in g F r ee En er g y Diag r am s o f O n e P h ase
W hat happens t o t he f r ee ener gy w hen. . .
|
Δ H m ix < 0 Δ S m ix < 0 |
Δ H m ix < 0 Δ S m ix = 0 |
Δ H m ix < 0 Δ S m ix > 0 |
|
Δ H m ix = 0 Δ S m ix < 0 |
Δ H m ix = 0 Δ S m ix = 0 |
Δ H m ix = 0 Δ S m ix > 0 |
|
Δ H m ix > 0 Δ S m ix < 0 |
Δ H m ix < 0 Δ S m ix = 0 |
Δ H m ix > 0 Δ S m ix > 0 |
Dr aw in g F r ee En er g y Diag r am s o f O n e P h ase
W hat about . . .
Dr aw in g F r ee En er g y Diag r am s o f O n e P h ase
S t ar t w i t h t he f r ee ener gy of t he t w o s epar at e com pon en t s i n one phas e. . .
A dd i n t he f r ee ener gy of m i xi ng. . . T hen s uper i m p os e al l pos s i bl e phas es .
Dr aw in g F r ee En er g y Diag r am s
I m age : Q . Ji an g, Z . W e n . “T h e r m od yn am i c s O f M at e r i al s .” A vai l ab l e t h r ou gh M I T L i b r ar i e s at h t t p : / / l i n k .s p r i n ge r .c om / b ook / 10.1007/ 978- 3- 642- 14718- 0/ p age / 1 .
Δ G mi x
P lo t G A and G B , s how how H mi x and T Δ S mi x change G to t
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Dr aw in g F r ee En er g y Diag r am s of M u lt ip le P h ases
E xam pl es :
– S o lid /liq u i d so lu tio n
– M isc ib ility g a p
– E u te c tic ( z e r o s o lu b ility )
– E u te c tic ( so m e so lu b ility )
– Int e rm e t a l l i c (orde re d) c om pound
F r ee E n er g y Diag r am s to P h ase
Diag r am s
I m age : Q . Ji an g, Z . W e n . “T h e r m od yn am i c s O f M at e r i al s .” A vai l ab l e t h r ou gh M I T L i b r ar i e s at h t t p : / / l i n k .s p r i n ge r .c om / b ook / 10.1007/ 978- 3- 642- 14718- 0/ p age / 1 .
T 1
T 2
T 3
T 3
T 2
T 1
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Tw o - P h ase Reg io n Cr it er io n
I m age : Q . Ji an g, Z . W e n . “T h e r m od yn am i c s O f M at e r i al s .” A vai l ab l e t h r ou gh M I T L i b r ar i e s at h t t p : / / l i n k .s p r i n ge r .c om / b ook / 10.1007/ 978- 3- 642- 14718- 0/ p age / 1 .
T 2
T 2
𝝏𝝁 𝐴
𝝏𝑛
= 𝝏𝝁 𝐵
𝝏𝑛
C h emi ca l p o t en t i a l ch a n g e i s
t h e s a me a t mi x t u r e( s )
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Ho w w ill t h e p h ases f o r m ?
N ucl eat i on & Gr o wt h
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S pi nodal D ecom pos i t i o n
This image is in the public domain.
W i k i me d i a C o mmo n s
Ho w w ill t h e p h ases f o r m ?
N ucl eat i on & Gr o wt h
S pi nodal D ecom pos i t i o n
P h ase Diag r am : S p in o d al Reg io n
Courtesy of W. Craig Carter. Used with permission.
h t t p : / / p r u f f l e .m i t .e d u / ~c c ar t e r / 3.21/ L e c t u r e _27/
S p in o d al Deco m p o sit io n E n er g y
I m age : Q . Ji an g, Z . W e n . “T h e r m od yn am i c s O f M at e r i al s .” A vai l ab l e t h r ou gh M I T L i b r ar i e s at h t t p : / / l i n k .s p r i n ge r .c om / b ook / 10.1007/ 978- 3- 642- 14718- 0/ p age / 1 .
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In - Class E xam p le: W - Z r
Dr a w t h e Gi b b s f r ee en er g y di a gr a m a t 1000C
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
In - Class E xam p le: W - Z r
α
W
Source: ASM Handbook, Volume 3: Alloy Phase Diagrams. Reprinted with permission of ASM International ®.
Zr W 2
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2 2.14 Materials in Nuclear Engineering
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