1. 1/12 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 Simplified model and new phase diagram

2. 2/12 Finite Temperature Alloy Phase Diagram Boron-Carbon (Okamoto, 1992) Rhombohedral 3 rd law violated ? 19.2% < 20% C (Eckbom)

3. Electronic Density Functional Theory Al 3+ Born-Oppenheimer approximation Wavefunction  (N) (r 1, r 2,..., r N ) Schrödinger: H  (N) = E  (N) Transform  (N) to N coupled 1-body problems for  i (r) - (double counting) Approximate V eff [  (r)] in Generalized Gradient Approx. FCC Aluminum, one unit cell Hohenberg-Kohn/Kohn-Sham: 3/12

4. 4/12 First Principles Enthalpies of Boron-Carbon Variants include: CBC/CBB chains; B 12 /B 11 C/B 10 C 2 icosahedra; Rotations of icosahedra  ʹ -boron graphite B 13 C 2 Rhombohedral B 4 C = B 12 C 3 Monoclinic h(x)

5. 5/12 B 13 C 2 B 12 (ico) + CBC (chain) Rhombohedral Pearson hR15 B 4 C == B 12 C 3 B 11 C (ico) + CBC (chain) Monoclinic Pearson mC30 Polar Carbon B 12 (ico) C-B-C chain B 11 C (ico)

6. 6/12 Partition Functions and Free Energies Helmholtz Gibbs Semi-Grand

7. 7/12  ʹ -boron B 13 C 2 Rhomb. B 4 C=B 12 C 3 Mono.

8. 8/12 Specific Heat at  =0 B4CB4C “B 13 C 2 ”

9. 9/12 Free Energy for rhombohedral “B 13 C 2 ” Composition: y B = excess B per CBC chain0  y B  1 y C = # C per B 12 icosahedron0  y C  1 N B = 13+y B -y C N C = 2+y C -y B x C = N C /15 Entropy: S (chain) /k B = y B ln 2 – y B ln y B – (1-y B ) ln (1-y B ) S (ico) /k B = y C ln 6 – y C ln y C – (1-y C ) ln (1-y C ) Landau Free Energy: G(y B,y C,T) = G(0,0) +  y B –  y C –T {S (chain) +S (ico) } G(x C ) = min G(y B,y C ; x C ) y B,y C

10. 10/12 Free Energy at T=2500K

11. 11/12 0 K 1000 K 2000 K 3000 K “B 13 C 2 ” “B 4 C” “B 13 C 2 + graphite” 600 K

12. 12/12 Conclusions Boron-carbide has two low temperature phases “B 13 C 2 ” (Rhombohedral) “B 4 C” (Monoclinic) Only “B 13 C 2 ” survives to high temperature, even though “B 4 C” has lower enthalpy! The phase “B 13 C 2 ” has a broad composition range, falling slightly short of B 4 C. First principles thermodynamics is feasible and useful.