The Structure and Dynamics of MonatomicLiquid Polymorphs - Case Studies of Cerium and

Germanium

Adam Cadien

Committee MembersHoward Sheng

Estela Blaisten-BarojasDimitrios Papaconstantopoulos

Amarda Shehu

School of Physics, Astronomy and Computational ScienceGeorge Mason UniversityFairfax, Virginia 22030

acadien@gmu.edu

April 24, 2015

Preamble

Clarification

Atoms 6= Adams

Crystalline Polymorphism

Example: Carbon

, 1/42

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

Polyamorphism

Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata andK. 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

A Rare Phenomenon?

Pure materials that exhibit liquid polymorphism.

, 4/42

Applications

, 5/42

Research Contribution

1. Discovered a new Liquid-Liquid Phase transtion inCerium.

I Experimental findings confirmed through simulation.

2. Predicted the existence of the first monatomic liquidcritical point.

3. First Ab-Initio study of Liquid Germanium.

4. Found strong evidence of multiple liquid phases inGermanium.

5. Achieved the first Nearly Hyperuniform glassy structure ofa semiconductor in Ab-Initio simulation.

, 6/42

Open Questions - Objectives for Studying Germanium

GlassesI How many unique amorphous structures does Germanium

form?I Is there unknown order in the glass structure?I Hyperuniformity?I How stable are these phases?

I How do glasses form, how do they melt?

LiquidsI Does Germanium have multiple liquid phases?

I Under what conditions are they (meta)stable?I What are their unique properties?

I What is the thermodynamic justification for polyamorphism?

I Is multiple liquid phases linked to the glass transition?

, 7/42

Germanium

Germanium Phase Diagram

dP

dT=

∆s

∆V

I Negative Melt Curve(dP/dT) at low pressures

I Liquid andDiamond(cF8) phase aredrastically differentmaterials

Is there a 2nd liquid that is similar to cF8?

S. Sastry, C. A. Angell, Nature Materials 2, pp739-743 (2003), 8/42

Simulation Method

Density Functional Theory (DFT); The trade off:

I Ab initio is predictive without empirical data.

I Scales horribly: roughly O(N3)

I Simulation can accessshort time scales

I Model potentials canbe misleading

I Sacrifice time foraccuracy

J. Glosli, F. Ree, PRL 82, 4659 (1999)C. Wu, J. Glosli, G. Galli, F.Ree, PRL 89, 135701 (2002), 9/42

Structural Analysis

, 10/42

Structural Analysis

Coordination: Number of neighbors within rcut

ncn = 4πρ

∫ rmin

0rg(r)dr

Tetrahedral Order Parameter: Geometric property

S iang = 1− 3

8

3∑j=1

4∑k=j+1

cos

(Ψjk +

1

3

)2

Bond Orientation Parameter: Spherical Harmonics

Q il =

(4π

2l + 1

l∑m=−l

| Qlm |

) 12

where, Qlm =1

Nb

Nb∑j=1

Ylm(θ(~rj), φ(~rj))

, 11/42

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

Microstructure

24A3/atom - Low DensityAmorphous (LDA)

Coordination = 4Bond Order Q3 = 0.60

Tetrahedrality = 0.9Bond Length = 2.51A

20A3/atom - High DensityAmorphous (HDA)

Coordination = 6Bond Order Q3 = 0.35

Tetrahedrality = 0.6Bond Length = 2.72A

Tetrahedrons → Shorter Bonds & Fewer Neighbors → Lower Density

, 13/42

Quench Rate Dependence

How does the material respond to different cooling rates?

, 14/42

LDA Annealing

288 atoms, NVT.

Nearly Hyperuniform Network?, 15/42

Hyperuniform Structures: Ideal Glass?

Uniformity measured by structure factor1: S(Q → 0) = 〈N2〉−〈N〉2〈N〉

1J. 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

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

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± 0.009S(Q → 0)WWW = 0.073± 0.010

1J. 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

HDA - Hyperuniform LDA Transition

Mimic Experimental Compression

I Each point - NVT

I Relax each pointvia CJ opt.

I Shrink volume -compression

I Expand volume -decompression

I Test reversibility

I 0K Transition point.

, 19/42

HULDA Transition Pressure

Calculated transition at 6.4GPa. Agrees with experiment (6GPa)1.

1O. Shimomura, S. Minomura, N. Sakai, K. Asaumi, K. Tamura, J. Fukushima,and H. Endo. Philo. Mag., 29 pp547558 (1974), 20/42

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

Glasses Study Summary

Phase Coord. Bond Len. Symmetry Conduction

LDA ∼ 4 2.51A tetrahedral semimetalHDA ∼ 6 2.72A octahedral metalHDL 6-9 2.69A octahedral/random metal

I Quenching HDL forms LDA somewhere near 650K at24A3/atom

I LDA becomes nearly hyperuniform through annealing or slowquenching

I Is the Low Density phase amorphous or liquid?

I HDA is difficult to form and likely contains crystal fragments

I LDA transition to HDA at 6GPa

, 22/42

Metastable Phase Diagram

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

Accessing LDL

288 atom HULDA state, canonical (NVT) dynamics at fixedtemperature.

730K 770K

, 24/42

Structural Evolution 730K

, 25/42

Structural Evolution 770K

, 26/42

Dynamics - RMSD

RMSD =√〈|ri(t)− ri(0)|2〉

, 27/42

Dynamics: RMSD

Ballistic

SingleHop

Stationary

, 28/42

Hopping

Glass or Liquid?

, 29/42

van Hove FunctionGs(r , t) =

⟨1N

∑i<N δ(r − ri(t) + ri(0))

⟩The probability an atom has moved a distance r , in time t.

, 30/42

van Hove FunctionGs(r , t) =

⟨1N

∑i<N δ(r − ri(t) + ri(0))

⟩The probability an atom has moved a distance r , in time t.

, 31/42

van Hove FunctionGs(r , t) =

⟨1N

∑i<N δ(r − ri(t) + ri(0))

⟩The probability an atom has moved a distance r , in time t.

, 32/42

Intermediate Scattering FunctionFs(k , t) =

∫Gs(r , t)e−ik·rdr

I Inaccurate at longer timescales due to insufficientdata

I Decay occurs near 50ns →10 years of simulation

KWW Fit:

ISFs(t) = A ∗ e(−tτ )

β

Fit the coefficients (A,β,τ)

W. Kob and H. Anderson, PRL, 73 1376 (1994), 33/42

Intermediate Scattering Function

I 770K LDL relax in∼50ns

I 770K HDL relaxin ∼0.5ns

, 34/42

Dynamic Heterogeneity

a

aL. Berthier PRL 99 060604 (2007)

Donati, Glotzer, Poole, Kob, Plimpton, PRE, 60 3107 (1999), 35/42

HDL A simple liquid

, 36/42

Potential Energy Landscape

I Barrier between LDL & HDL at high temperatures.I No Barrier between HDL and LDA at low temperatures.I Large barriers between local minima in LDA.

, 37/42

Summary

I Discovered that LDA tends towardsHyperuniform structure

I analyzed through long wavelength limit of S(Q).

I First demonstration of liquid polymorphism inpure Ge through Ab-Initio simulation.

I Revealed the relaxation mechanism in LDLI using high fidelity ab-initio simulationI formation of defect droplet in LDA/LDL

I A clear picture of the PEL of the amorphousphases of Ge is developed.

, 38/42

Moving Forward

Open Questions:

I Pressure minimum in the supercooled liquid?

I Liquid-Liquid critical point or Void transition?

I Spinodal transition?

Progression:

I Challenge the Aptekar-Ponyatovsky 2 phase model

I Case studies wtih DFT: Silicon, Gallium, Arsenic, Antimony

I Extend case studies to multicomponent systems: Water, Silica

I More experimental data to compare against, driven bysimulation.

, 39/42

Publications and Code

Publications

I “Liquid Polyamorphism in Supercooled Germanium”, A.Cadien & H. Sheng - in preparation

I “Polymorphic phase transition mechanism of compressedcoesite”, Q.Y. Hu, J.-F. Shu, A. Cadien, Y. Meng, W.G.Yang, H.W. Sheng, H.-K. Mao, Nat. Comm 6 6630 (2015)

I “First-order liquid-liquid phase transition in Cerium”, A.Cadien, QY Hu, Y Meng, YQ Cheng, MW Chen, JF Shu, HKMao, HW Sheng, PRL, 110 12 (2013)

I “Highly optimized embedded-atom-method potentials forfourteen fcc metals”, H. W. Sheng, M. J. Kramer, A. Cadien,T. Fujita, M. W. Chen, PRB, 83 134118 (2011)

CodeAll analysis code developed by Adam Cadien and available at;https://github.com/acadien/matcalc

, 40/42

Thank You!

Questions?

, 41/42

Yatta!

, 42/42