The document summarizes Adam Cadien's research on liquid polymorphism in cerium and germanium using density functional theory simulations. Key findings include: 1) Discovering a new liquid-liquid phase transition in cerium through simulations that confirmed experimental results. 2) Predicting evidence of multiple liquid phases in germanium through the first ab initio study of liquid germanium. 3) Forming the first nearly hyperuniform glassy structure of a semiconductor in germanium and analyzing the unique properties and phase transitions of the different liquid and amorphous phases of germanium.

1. The Structure and Dynamics of Monatomic
Liquid Polymorphs - Case Studies of Cerium and
Germanium
Adam Cadien
Committee Members
Howard Sheng
Estela Blaisten-Barojas
Dimitrios Papaconstantopoulos
Amarda Shehu
School of Physics, Astronomy and Computational Science
George Mason University
Fairfax, Virginia 22030
acadien@gmu.edu
April 24, 2015

2. Preamble

3. Clarification
Atoms = Adams

4. Crystalline Polymorphism
Example: Carbon
, 1/42

5. Origins of Liquid Structure
Geometric Approach to the Structure of Liquids J. D. Bernal, 183 Nature (1959)
Coordination of Randomly Packed Spheres J. D. Bernal, 188 Nature (1960), 2/42

6. Polyamorphism
Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata and
K. Funakoshi; Nature, 403, 170 (2000)
A. Cadien, Q. Hu, Y. Meng, Y. Cheng, M. Chen, J. Shu, H. Mao, H.
Sheng, PRL, 110 2013, 3/42

7. A Rare Phenomenon?
Pure materials that exhibit liquid polymorphism.
, 4/42

8. Applications
, 5/42

9. Research Contribution
1. Discovered a new Liquid-Liquid Phase transtion in
Cerium.
Experimental findings confirmed through simulation.
2. Predicted the existence of the first monatomic liquid
critical point.
3. First Ab-Initio study of Liquid Germanium.
4. Found strong evidence of multiple liquid phases in
Germanium.
5. Achieved the first Nearly Hyperuniform glassy structure of
a semiconductor in Ab-Initio simulation.
, 6/42

10. Open Questions - Objectives for Studying Germanium
Glasses
How many unique amorphous structures does Germanium
form?
Is there unknown order in the glass structure?
Hyperuniformity?
How stable are these phases?
How do glasses form, how do they melt?
Liquids
Does Germanium have multiple liquid phases?
Under what conditions are they (meta)stable?
What are their unique properties?
What is the thermodynamic justification for polyamorphism?
Is multiple liquid phases linked to the glass transition?
, 7/42

11. Germanium
Germanium Phase Diagram
dP
dT
=
∆s
∆V
Negative Melt Curve
(dP/dT) at low pressures
Liquid and
Diamond(cF8) phase are
drastically different
materials
Is there a 2nd liquid that is similar to cF8?
S. Sastry, C. A. Angell, Nature Materials 2, pp739-743 (2003), 8/42

12. Simulation Method
Density Functional Theory (DFT); The trade off:
Ab initio is predictive without empirical data.
Scales horribly: roughly O(N3)
Simulation can access
short time scales
Model potentials can
be misleading
Sacrifice time for
accuracy
J. Glosli, F. Ree, PRL 82, 4659 (1999)
C. Wu, J. Glosli, G. Galli, F.Ree, PRL 89, 135701 (2002), 9/42

13. Structural Analysis
, 10/42

14. Structural Analysis
Coordination: Number of neighbors within rcut
ncn = 4πρ
rmin
0
rg(r)dr
Tetrahedral Order Parameter: Geometric property
Si
ang = 1 −
3
8
3
j=1
4
k=j+1
cos Ψjk +
1
3
2
Bond Orientation Parameter: Spherical Harmonics
Qi
l =
4π
2l + 1
l
m=−l
| ¯Qlm |
1
2
where, ¯Qlm =
1
Nb
Nb
j=1
Ylm(θ(rj ), φ(rj ))
, 11/42

15. Forming Amorphous Germanium
Quenching: Cool the material fast enough to avoid crystallization.
Canonical Dynamics (NVT), 288 atoms.
Takes ∼26 hours each across 256 cores.
, 12/42

16. Microstructure
24˚A3
/atom - Low Density
Amorphous (LDA)
Coordination = 4
Bond Order Q3 = 0.60
Tetrahedrality = 0.9
Bond Length = 2.51˚A
20˚A3
/atom - High Density
Amorphous (HDA)
Coordination = 6
Bond Order Q3 = 0.35
Tetrahedrality = 0.6
Bond Length = 2.72˚A
Tetrahedrons → Shorter Bonds & Fewer Neighbors → Lower Density
, 13/42

17. Quench Rate Dependence
How does the material respond to different cooling rates?
, 14/42

18. LDA Annealing
288 atoms, NVT.
Nearly Hyperuniform Network?, 15/42

19. Hyperuniform Structures: Ideal Glass?
Uniformity measured by structure factor1: S(Q → 0) =
N2
− N 2
N
1
J. Hansen and I. McDonald, “Theory of Simple Liquids”, 1986
S. Torquato and F. H. Stillinger, Physical Review E, 68, 041113 1-25 (2003)., 16/42

20. Structure Factor at Long Wavelengths
Approach Q → 0 from another dimension, R. Shape function:
α(r; R) = (1 − r/2R)2
(1 + r/4R)
S(Q → 0, R) = 1 +
∞
0
e−iqr
rG(r)α(r; R)dr
S. Torquato and F. Stillinger, PRE, 68, 041113 (2003)
A. de Graff and M. Thorpe, Acta Cryst, A66, pp22-31 (2010), 17/42

21. Hyperuniform Structures: Ideal Glass?
Uniformity measured by structure factor1: S(Q → 0) =
N2
− N 2
N
S(Q → 0) = 0 S(Q → 0)HULDA = 0.068 ± 0.009
S(Q → 0)WWW = 0.073 ± 0.010
1
J. Hansen and I. McDonald, “Theory of Simple Liquids”, 1986
S. Torquato and F. H. Stillinger, Physical Review E, 68, 041113 1-25 (2003)., 18/42

22. HDA - Hyperuniform LDA Transition
Mimic Experimental Compression
Each point - NVT
Relax each point
via CJ opt.
Shrink volume -
compression
Expand volume -
decompression
Test reversibility
0K Transition point.
, 19/42

23. HULDA Transition Pressure
Calculated transition at 6.4GPa. Agrees with experiment (6GPa)1.
1
O. Shimomura, S. Minomura, N. Sakai, K. Asaumi, K. Tamura, J. Fukushima,
and H. Endo. Philo. Mag., 29 pp547558 (1974), 20/42

24. Electronic Density of States
Histogram of band energies.
288 atom samples, 4x4x4 K-Points. E-Fermi is set to 0.0eV.
G. Kresse and J. Hafner, PRB, 47 pp558-561 (1993)
N. Bernstein and J. Mehl and D. Papaconstantopoulos, PRB, 66 075212 (2002), 21/42

25. Glasses Study Summary
Phase Coord. Bond Len. Symmetry Conduction
LDA ∼ 4 2.51˚A tetrahedral semimetal
HDA ∼ 6 2.72˚A octahedral metal
HDL 6-9 2.69˚A octahedral/random metal
Quenching HDL forms LDA somewhere near 650K at
24˚A3/atom
LDA becomes nearly hyperuniform through annealing or slow
quenching
Is the Low Density phase amorphous or liquid?
HDA is difficult to form and likely contains crystal fragments
LDA transition to HDA at 6GPa
, 22/42

26. Metastable Phase Diagram
D. Li and D. Herbach, J. Mater. Sci. 32 pp1437 (1997), 23/42

27. Accessing LDL
288 atom HULDA state, canonical (NVT) dynamics at fixed
temperature.
730K 770K
, 24/42

28. Structural Evolution 730K
, 25/42

29. Structural Evolution 770K
, 26/42

30. Dynamics - RMSD
RMSD = |ri (t) − ri (0)|2
, 27/42

31. Dynamics: RMSD
Ballistic
Single
Hop
Stationary
, 28/42

32. Hopping
Glass or Liquid?
, 29/42

33. van Hove Function
Gs(r, t) = 1
N i<N δ(r − ri(t) + ri(0))
The probability an atom has moved a distance r, in time t.
, 30/42

34. van Hove Function
Gs(r, t) = 1
N i<N δ(r − ri(t) + ri(0))
The probability an atom has moved a distance r, in time t.
, 31/42

35. van Hove Function
Gs(r, t) = 1
N i<N δ(r − ri(t) + ri(0))
The probability an atom has moved a distance r, in time t.
, 32/42

36. Intermediate Scattering Function
Fs(k, t) = Gs(r, t)e−ik·r
dr
Inaccurate at longer time
scales due to insufficient
data
Decay occurs near 50ns →
10 years of simulation
KWW Fit:
ISFs(t) = A ∗ e(−t
τ )
β
Fit the coefficients (A,β,τ)
W. Kob and H. Anderson, PRL, 73 1376 (1994), 33/42

37. Intermediate Scattering Function
770K LDL relax in
∼50ns
770K HDL relax
in ∼0.5ns
, 34/42

38. Dynamic Heterogeneity
a
a
L. Berthier PRL 99 060604 (2007)
Donati, Glotzer, Poole, Kob, Plimpton, PRE, 60 3107 (1999), 35/42

39. HDL A simple liquid
, 36/42

40. Potential Energy Landscape
Barrier between LDL & HDL at high temperatures.
No Barrier between HDL and LDA at low temperatures.
Large barriers between local minima in LDA.
, 37/42

41. Summary
Discovered that LDA tends towards
Hyperuniform structure
analyzed through long wavelength limit of S(Q).
First demonstration of liquid polymorphism in
pure Ge through Ab-Initio simulation.
Revealed the relaxation mechanism in LDL
using high fidelity ab-initio simulation
formation of defect droplet in LDA/LDL
A clear picture of the PEL of the amorphous
phases of Ge is developed.
, 38/42

42. Moving Forward
Open Questions:
Pressure minimum in the supercooled liquid?
Liquid-Liquid critical point or Void transition?
Spinodal transition?
Progression:
Challenge the Aptekar-Ponyatovsky 2 phase model
Case studies wtih DFT: Silicon, Gallium, Arsenic, Antimony
Extend case studies to multicomponent systems: Water, Silica
More experimental data to compare against, driven by
simulation.
, 39/42

43. Publications and Code
Publications
“Liquid Polyamorphism in Supercooled Germanium”, A.
Cadien & H. Sheng - in preparation
“Polymorphic phase transition mechanism of compressed
coesite”, Q.Y. Hu, J.-F. Shu, A. Cadien, Y. Meng, W.G.
Yang, H.W. Sheng, H.-K. Mao, Nat. Comm 6 6630 (2015)
“First-order liquid-liquid phase transition in Cerium”, A.
Cadien, QY Hu, Y Meng, YQ Cheng, MW Chen, JF Shu, HK
Mao, HW Sheng, PRL, 110 12 (2013)
“Highly optimized embedded-atom-method potentials for
fourteen fcc metals”, H. W. Sheng, M. J. Kramer, A. Cadien,
T. Fujita, M. W. Chen, PRB, 83 134118 (2011)
Code
All analysis code developed by Adam Cadien and available at;
https://github.com/acadien/matcalc, 40/42

44. Thank You!
Questions?
, 41/42

45. Yatta!
, 42/42