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Thermodynamics from First Principles: Low Temperature Phase Transition Predicted

in the Compound B13C2/B4C

With: Will Huhn (Physics @ Carnegie Mellon)

Outline:• Thermodynamics from first principles

why? how?• Predicted new phase of boron carbide

two low temperature phases• Simplified model and new phase diagram

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Finite Temperature Alloy Phase DiagramBoron-Carbon (Okamoto, 1992)

Rhombohedral

3rd law violated ?

19.2% < 20% C(Eckbom)

Electronic Density Functional Theory

Al3+

Al3+ Al3+

Al3+• Born-Oppenheimer approximation• Wavefunction (N)(r1, r2, ..., rN)• Schrödinger: H(N) = E (N)

Transform (N) to N coupled 1-body problems for i(r)

N

iiiieffi EV

1

2 021 r - (double counting)

Approximate Veff [(r)] in Generalized Gradient Approx.

FCC Aluminum, one unit cell Hohenberg-Kohn/Kohn-Sham:

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First Principles Enthalpies of Boron-Carbon

Variants include: CBC/CBB chains; B12/B11C/B10C2 icosahedra;

Rotations of icosahedra

ʹ-boron graphite

B13C2

Rhombohedral B4C = B12C3 Monoclinic

h(x)

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B13C2

B12(ico) + CBC(chain)

RhombohedralPearson hR15

B4C == B12C3

B11C(ico) + CBC(chain)

MonoclinicPearson mC30

Polar CarbonB12(ico)

C-B-Cchain

B11C(ico)

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Partition Functions and Free Energies

Helmholtz

Gibbs

Semi-Grand

HBCBTkE

cBH QTkTVNNFeTVNNQ B ln,,,,,, /

GBCBCBHTkPV

CBG QTkTPNNGTVNNQedVTPNNQ B ln),,,(,,,,,, /

SBN

CCGTkN

S QTkTPNYTPNNNQeTPNQC

BC ln,,,,,,,,, /

TSUF

CB

CBCCBB

CCBB

NNNNN

NNPVFG

NNGY BC

• •

• •

• •

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ʹ-boron

B13C2

Rhomb. B4C=B12C3 Mono.

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Specific Heat at =0

B4C “B13C2”

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Free Energy for rhombohedral “B13C2”

Composition:yB = excess B per CBC chain 0 yB 1yC = # C per B12 icosahedron 0 yC 1

NB = 13+yB-yC NC = 2+yC-yB xC = NC/15

Entropy:S(chain)/kB = yB ln 2 – yB ln yB – (1-yB) ln (1-yB)

S(ico)/kB = yC ln 6 – yC ln yC – (1-yC) ln (1-yC)

Landau Free Energy:G(yB,yC,T) = G(0,0) + yB – yC –T {S(chain)+S(ico)}

G(xC) = min G(yB,yC; xC) yB,yC

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Free Energy at T=2500K

11/120 K

1000 K

2000 K

3000 K

“B13C2”

“B4C”

“B13C2 + graphite”

600 K

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Conclusions

• Boron-carbide has two low temperature phases• “B13C2” (Rhombohedral)• “B4C” (Monoclinic)

• Only “B13C2” survives to high temperature, even though “B4C” has lower enthalpy!

• The phase “B13C2” has a broad composition range, falling slightly short of B4C.

• First principles thermodynamics is feasible and useful.