Quasichemical Interstitial-Solutions: the carbon-carbon interaction energy

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Quasichemical Interstitial-Solutionsthe
carbon-carbon interaction energy
Application of a first-order Quasichemical
theory to Transformations in Steels, Metal
Science, Vol. 16, 1982, pp. 167-169 H.K.D.H.
Bhadeshia Quasichemical Model for Interstitial
Solutions Materials Science and Technology, Vol.
14, 1998, pp. 273-276 H. K. D. H. Bhadeshia
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QUASICHEMICAL SOLUTIONS

• 1935 Guggenheim
• 1966 Aaronson, Domian Pound
  ?? positive and ?? negative, wrong Z
• 1978 Shiflet, Aaronson Bradley
• 1981 Shiflet, Aaronson Bradley
• used McLellan Dunn theory
• ??? ??? negative, wrong Z
• 1982 Bhadeshia, ?? positive and large,
• oscillations, correct Z

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QUASICHEMICAL SOLUTIONS
1989 Mou Aaronson correct Z, ?? positive and
large, oscillations ???negative but with wrong
model Summary Considerable evidence for strong
repulsion between carbon atoms in ferrite. ?? is
large and positive, but why are there
oscillations in its value? Generally agreed that
McLellan Dunn model is the best quasichemical
model for interstitial solutions.
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Interstices in ferrite
closest approach 0.143 nm ?? 150,000 J/mol
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Interstices in austenite
closest approach 0.252 nm ?? 1825 J/mol
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In graphite (Cromwell, 1958), short-range
repulsive C-C component U 1856 exp-35.75r
ev giving 0.23 eV in austenite and 11.0 eV
for C-C in ferrite Actual (Bhadeshia,
1982) 0.086 eV in austenite 1.54 eV in ferrite

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Enthalpy of mixing Ideal solution ??M????
Regular solution ??M ???? Quasichemical ??M
??? ENTROPY of mixing Ideal random
distribution of atoms Regular assumes random
distribution pAB???x?x? ?SM????k?x?lnx????x?lnx??
Quasichemical non-random, pAB??
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Partition Function
g1
N number of atoms N1 number in high energy
state N0 number in ground state
E1
g0
N0 g0
N1 g1exp-E1/kT


g0 g1exp-E1/kT
g0 g1exp-E1/kT
N
N
?????i gi exp-Ei / kT F -kT ln ?
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Nu carbon atoms N iron atoms ?iNFe octahedral
sites
Number of Fe-C and Fe-Fe pairs invariant.
Therefore, only difference is C-vacancy and C-C
pairs. u - u0
u - u
Let number of u - u0 pairs W ? where ? is
unknown u - u 0.5 W(Nu - ?? u0 - u0 0.5
W(?iNFe- Nu - ??
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The closest distance of approach of carbon in
ferrite is 0.143 nm They are further than
this even in cementite ! There is evidence that
at 0.127 nm they would bond covalently as in
some transition metal carbides. but there are no
molecules of carbon in ferrite !
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Conclusions

• Better behaved set of equations for quasichemical
  interstitial solution
• Correct at inifinite repulsion, dilute solution
  limits, or zero repulsion limit
• Consistent formulation of zeroth and first
  approximation.
• ?? positive and large