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With: Will Huhn (Physics @ Carnegie Mellon )

Thermodynamics from First Principles: Low Temperature Phase Transition Predicted in the Compound B 13 C 2 /B 4 C. With: Will Huhn (Physics @ Carnegie Mellon ). Outline: Thermodynamics from first principles why? how? Predicted new phase of boron carbide two low temperature phases

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With: Will Huhn (Physics @ Carnegie Mellon )

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  1. 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

  2. 19.2% < 20% C Rhombohedral 3rd law violated ? Finite Temperature Alloy Phase Diagram Boron-Carbon (Okamoto, 1992) (Eckbom)

  3. Electronic Density Functional Theory • Born-Oppenheimer approximation • Wavefunction (N)(r1, r2, ..., rN) • Schrödinger: H(N) = E (N) Al3+ Al3+ Al3+ Al3+ Hohenberg-Kohn/Kohn-Sham: FCC Aluminum, one unit cell Transform (N) to N coupled 1-body problems for i(r) - (double counting) Approximate Veff[(r)] in Generalized Gradient Approx.

  4. First Principles Enthalpies of Boron-Carbon ʹ-boron graphite h(x) B13C2 Rhombohedral B4C = B12C3 Monoclinic Variants include: CBC/CBB chains; B12/B11C/B10C2 icosahedra; Rotations of icosahedra

  5. Polar Carbon B12(ico) B11C(ico) C-B-C chain B13C2 B12(ico) + CBC(chain) Rhombohedral Pearson hR15 B4C == B12C3 B11C(ico) + CBC(chain) Monoclinic Pearson mC30

  6. Partition Functions and Free Energies Helmholtz Gibbs Semi-Grand

  7. ʹ-boron B13C2 Rhomb. B4C=B12C3 Mono.

  8. Specific Heat at =0 B4C “B13C2”

  9. Free Energy for rhombohedral “B13C2” Composition: yB = excess B per CBC chain 0  yB  1 yC = # C per B12 icosahedron 0  yC  1 NB = 13+yB-yC NC = 2+yC-yBxC = NC/15 Entropy: S(chain)/kB = yBln 2 – yBlnyB – (1-yB) ln (1-yB) S(ico)/kB = yCln 6 – yClnyC – (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

  10. Free Energy at T=2500K

  11. 3000 K 2000 K “B13C2” “B13C2 + graphite” 1000 K 600 K “B4C” 0 K

  12. 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.

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