Atomistic Simulation of Shock Waves:

From Simple Crystals to ComplexQuasicrystals

Habilitationsschrift zur Erlangungder Lehrbefugnis fur das Fach Theoretische Physik

an der Universitat Stuttgart

vorgelegt vonJohannes Rothaus Grobottwar

Institut fur Theoretische und Angewandte PhysikUniversitat Stuttgart

2005

i

To Antje,Jonathan,

Stephan,and Althea

A shock wave in the wild west

From [25]

ii

Parts of this work have been published already:

J.Roth, R.Schilling, H.-R.Trebin, Stability of monoatomi c and diatomicquasicrystals and the inuence of noise,Phys.Rev.B 41 (1990) 2735.J.Roth, Comment on Formation of a dodecagonal phase in a simplemonatomic liquid, Phys.Rev.Lett. 49 (1997) 4042.J.Roth, F. Gahler, Self-diusion in dodecagonal quasicrystals,Eur.Phys.J.B 6 (1998) 425.M.Hohl, J.Roth, H.-R.Trebin, Correlation functions and th e dynamicalstructure factor of quasicrystals, Eur.Phys.J.B 71 (2000) 595.J.Roth, The uid-solid transition of Dzugutovs potential , Eur.Phys.J.B14 (2000) 449.J.Roth, Jumps in icosahedral quasicrystals,Eur.Phys.J.B 15 (2000) 7.J.Roth, Shock waves in quasicrystals,Mat.Sci.Eng.A 294-296 (2000)753.J.Roth, A.Denton, Solid phase structures for the Dzugutov pair poten-tial, Phys.Rev.E 61 (2000) 6845.G.Schaaf, J.Roth, H.-R.Trebin, R.Mikulla, Numerical simu lation of dis-location motion in three-dimensional icosahedral quasicrystals, Phil.Mag.A 80 (2000) 1657.J.Roth, Shock waves in quasicrystals,Ferroelectrics 250 (2001) 365.J.Roth, Large-scale molecular dynamics simulations of shock waves inLaves-crystals and icosahedral quasicrystals,AIP Conf.Proc. 620, (2002)378.G.Schaaf, J.Roth, H.-R.Trebin, Dislocation motion in icosahedral qua-sicrystals at elevated temperatures: Numerical simulation, Phil.Mag.83, (2003) 2449.J.Roth, Shock waves and solitary waves in bcc crystals,AIP Conf.Proc.706 (2004) 302.J.Roth, Shock Waves in Complex Binary Solids: Cubic Laves Crystals,Quasicrystals, and Amorphous Solid,Phys.Rev.B 71 (2005) 064102.J.Roth, Shock Waves in Materials with Dzugutov-Potential I nteraction,Phys.Rev.B 72 (2005) 014125.J.Roth, ! -Phase and Solitary Waves Induced by Shock Compression ofBCC Crystals, Phys.Rev.B 72 (2005) 014126.

Contents

List of Symbols vii

Deutsche Zusammenfassung ix

Introduction xix

I Shock Wave Physics, Shock Wave Simulations,Structures and Interactions 1

1 Shock Wave Physics 31.1 Why study shock waves in solids? . . . . . . . . . . . . . 31.2 The denition of shock waves . . . . . . . . . . . . . . . 41.3 Denition of the basic observables . . . . . . . . . . . . 51.4 The equation of state . . . . . . . . . . . . . . . . . . . . 61.5 The Rankine-Hugoniot equations . . . . . . . . . . . . . 71.6 Two wave structure from material rigidity . . . . . . . . 91.7 Dierent representations of a stressed state . . . . . . . 121.8 Yield strength of a solid and the Hugoniot elastic limit

(HEL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 Velocity of sound . . . . . . . . . . . . . . . . . . . . . . 161.10 Dependency of the shock wave behavior on the dimen-

sionality of the simulation . . . . . . . . . . . . . . . . . 171.11 General literature about shock waves . . . . . . . . . . . 18

1.11.1 Special literature for the present thesis . . . . . . 19

2 Simulation of Shock Waves in Solids 212.1 Shock wave simulations: State of the art . . . . . . . . . 21

2.1.1 The Hugoniostat . . . . . . . . . . . . . . . . . . 252.2 Shock wave generation . . . . . . . . . . . . . . . . . . . 26

2.2.1 Impact simulations . . . . . . . . . . . . . . . . . 262.2.2 Symmetric impact simulations . . . . . . . . . . 272.2.3 Momentum mirror . . . . . . . . . . . . . . . . . 282.2.4 Other methods . . . . . . . . . . . . . . . . . . . 29

2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . 302.4 Molecular dynamics simulations . . . . . . . . . . . . . . 30

iv Contents

3 Structures and Potentials 333.1 Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Tiling models . . . . . . . . . . . . . . . . . . . . 343.1.2 Defects in quasicrystals . . . . . . . . . . . . . . 363.1.3 New defect types . . . . . . . . . . . . . . . . . . 37

3.2 Quasicrystal models . . . . . . . . . . . . . . . . . . . . 383.2.1 Tetrahedrally close-packed phases . . . . . . . . . 393.2.2 The cubic Laves phase and diatomic icosahedral

models . . . . . . . . . . . . . . . . . . . . . . . . 403.2.3 Other applications . . . . . . . . . . . . . . . . . 433.2.4 Square-triangle-phases and monatomic dodecago-

nal models . . . . . . . . . . . . . . . . . . . . . 433.3 Shock wave experiments on quasicrystals and Laves crystals 483.4 The Dzugutov potential . . . . . . . . . . . . . . . . . . 48

3.4.1 Description of the potential . . . . . . . . . . . . 493.4.2 Other applications . . . . . . . . . . . . . . . . . 503.4.3 The phase diagram of the Dzugutov potential . . 503.4.4 Iron phase diagram . . . . . . . . . . . . . . . . . 52

3.5 The ! -phase . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 The bcc-lattice anomaly . . . . . . . . . . . . . . . . . . 56

II Simulations of Complex Structures 57

4 Shock Waves in the Cubic Laves Phase and in BinaryIcosahedral Quasicrystals 594.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Models, interactions, simulation details . . . . . . . . . . 61

4.3.1 The structure models . . . . . . . . . . . . . . . 614.3.2 The interaction . . . . . . . . . . . . . . . . . . . 614.3.3 Preparation of the samples and simulation details 62

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.1 Elasticity and anisotropy of the sound waves . . 634.4.2 Pressure proles and steadiness of the proles . . 644.4.3 The Hugoniot relation us-up in general . . . . . . 674.4.4 Orientation dependency of theus-up-relation . . 69

Contents v

4.4.5 Dependency of pressure and stress on the shockstrength . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.6 Analysis of defects . . . . . . . . . . . . . . . . . 744.4.7 Laves crystal . . . . . . . . . . . . . . . . . . . . 754.4.8 Quasicrystal models . . . . . . . . . . . . . . . . 784.4.9 Amorphous solid . . . . . . . . . . . . . . . . . . 82

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6 Pictures of the diatomic simulations . . . . . . . . . . . 86

5 Materials with Dzugutov Potential Interactions underHeavy Load: I. Shock Waves 995.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Generation of the shock waves and simulation setup . . 101

5.3.1 Orientation of the samples . . . . . . . . . . . . . 1015.3.2 Preparation of the samples and simulation details1025.3.3 Analysis tools for the shocked structures . . . . . 102

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4.1 Elastic properties and sound velocities . . . . . . 1055.4.2 The Hugoniot relation: us vs. up . . . . . . . . . 1055.4.3 Description of the structural results . . . . . . . 1095.4.4 The four-fold direction . . . . . . . . . . . . . . . 1095.4.5 The other directions and structures . . . . . . . . 1105.4.6 Special phenomena for shock waves along the three-

fold direction of bcc . . . . . . . . . . . . . . . . 1155.4.7 Results for an amorphous structure . . . . . . . . 116

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.5.1 Comparison to shock wave simulations of iron . . 1185.5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . 120

6 Materials with Dzugutov Potential Interactions underHeavy Load: II. The ! -Phase and Solitary Waves 1216.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3 Generation of the shock waves and orientation of the sam-

ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Results for shock waves along the three-fold directions. 124

vi Contents

6.4.1 Evolution of the hydrostatic pressure prole withincreasing piston velocity . . . . . . . . . . . . . 126

6.4.2 Properties of the! -phase . . . . . . . . . . . . . 1286.4.3 Description of the internal structure of the soli-

tary wave peak . . . . . . . . . . . . . . . . . . . 1306.4.4 Decay with width . . . . . . . . . . . . . . . . . . 1366.4.5 Solitary waves in other phases and along dierent

directions . . . . . . . . . . . . . . . . . . . . . . 1396.5 Shock waves and solitary waves along arbitrary directions 1396.6 Ramping up the velocity . . . . . . . . . . . . . . . . . . 1446.7 Theoretical explanation of the occurance of the! -phase 144

6.7.1 Geometrical description of the! -phase . . . . . . 1446.7.2 The stability of the ! -phase in Dzugutov materials1476.7.3 Phonon dispersion and softening . . . . . . . . . 1486.7.4 Evaluation of the Landau theory . . . . . . . . . 152

6.8 Theoretical explanations of the solitary waves . . . . . . 1556.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.9.1 Experiments . . . . . . . . . . . . . . . . . . . . 1576.9.2 Comparison to one-dimensional solitons . . . . . 158

Final Remarks 161

Future Perspectives 163

Appendix 165A The Mie-Gruneisen-Debye-theory . . . . . . . . . . . . . 165B Shock waves in one-dimensional crystals and quasicrystals167C Ablation and generation of shock waves with lasers . . . 167

Bibliography 171

Acknowledgment 195

List of SymbolsThis list contains the more frequently used symbols. A more detailedexplantation may be found in the sections where the symbols are intro-duced.

Variables and Parameter

a half the edge length of the tiles

b in-plane lattice constant of the ! -phase

b0 equilibrium in-plane lattice constant of the ! -phase

c arbitrary velocity of sound

c11, c12 elastic constants of an isotropic material

cb bulk velocity of sound: cl =p

K=

cl longitudinal velocity of sound: cl =p

F=

ct transversal velocity of sound: cl =p

G=

cx velocity of sound for structure or direction x

d perpendicular lattice constant of the ! -phase

d0 equilibrium perpendicular lattice constant of the ! -phase

Epot potential energy

Ex potential energy of structure x

F uniaxial elastic constant

G shear modulus

K bulk compressibility

kT temperature in energy units

k0 reciprocal lattice constant along the three-fold axis of bcc

P hydrostatic pressure: P = ( Pxx + Pyy + Pzz )=3

P0 unit of pressure and stress:P0 = =a3

Pii uniaxial pressure (stress) along directioni

rab distance between an atom of typea and an atom of type b

rC cut-o radius

r s relative strength vd=veS shear pressure (stress):S = Pxx (Pyy + Pzz )=2

viii List of Symbols

Se shear pressure (stress) at the end of the sample

Sp peak shear pressure (stress)

t time

t0 unit of time: t0 = ap

m=

T temperature

t r time to ramp up the piston velocity

u unspecied shock front velocities

ui velocity of the yer plate

up piston velocity

up;e eective piston velocity

us in a closer sense: shock wave velocity,

frequently used for all types of wave velocities

v0 unit of velocity: v0 =p

=m

vd potential minimum for diagonal interactions

ve potential minimum for edge interactions

w position of the B and C layers with respect to the latticeconstant d in bcc and the ! -phase

Y yield stress

energy unit

ij strain tensor

t time interval

, Lame constants

Poisson ratio

density

Reduced Units

Reduced units are used throughout the thesis. Lengths are given in a,and energies in . All other units are converted into a, and the massm. Thus there are the following relations: t0 = a

pm= , v0 =

p=m ,

and P0 = =a3

Deutsche ZusammenfassungUbersichtDiese Habilitationsschrift handelt von Molekulardynamik simulationenan geordneten Festkorpern. Im ersten Teil werden die Grundlagen derStowellenphysik kurz zusammengefasst, gefolgt von einemUberblickuber die Simulation von Stowellen in Festkorpern. Ein we iterer Ab-schnitt enthalt die Beschreibung der untersuchten Strukt uren, Wechsel-wirkungen und relevanten Phasendiagramme.Im zweiten Teil der Arbeit folgt die Beschreibung der Ergebnisse: Imersten Kapitel werden binar ikosaedrische Quasikristalle und verwandteLaves-Phasen behandelt, im zweiten monoatomar zwolfzahlige Quasi-kristalle, eng verwandte tetraedrisch dicht gepackte Phasen und ku-bisch innenzentrierte Kristalle. Das dritte Kapitel hande lt von der in-termediaren ! -Phase und solitaren Wellen, die in den innenzentriertenKristallen bei Stowellen entlang von dreizahligen Symmetrierichtungenauftreten.In allen Fallen gibt es drei Bereiche unterschiedlichen Materialverhal-tens: bei schwachen Stowellen ndet man elastische Deformationen,in einem mittleren Bereich elastische und plastische Deformation bzw.Phasenubergange. Im Bereich starker Stowellen werden die Ausgangsstruk-turen amorphisiert.

Stowellenphysik in FestkorpernStowellen treten uberall im Universum und in allen Zusta nden derMaterie auf. Man ndet sie sowohl in der Bugwelle der Heliosphareoder in Supernovaexplosionen als auch bei der Sonolumineszenz undbeim Uberschallknall.Stowellen existieren auch in Festkorpern und ermoglich en dort die Un-tersuchung von Materiezustanden, die mit anderen Methoden nicht er-reicht werden konnen:

die Bestimmung der Zustandsgleichung bei sehr hohen Druckenund Temperaturen wie zum Beispiel im Erdinnern,

die Untersuchung des mechanischen Verhaltens von Festkorpernbei plotzlichen Lastwechseln,

die Analyse dynamischer Phasenubergange, die an den Stowellen-fronten auftreten.

x Deutsche Zusammenfassung

In Simulationen ermoglichen sie die Untersuchung von ausgedehn-ten Defekten wie Versetzungen und Stapelfehler, die sonst kunst-lich eingebaut werden mussten. Dieses Vorgehen spielt insbeson-dere bei komplexen Kristallstrukturen und Quasikristallen eineRolle.

Eine Stowelle (Abs.1.2) ist deniert als eine sich fortbewegende Un-stetigkeit der thermodynamischen Groen. Stabile Stowe llen propa-gieren mit Uberschallgeschwindigkeite bezuglich des Ausgangsmediumsund langsamer als der Schall im komprimierten Medium. In Gasen undFlussigkeiten stellen diese Denitionen wegen der fehlenden Ruckstell-krafte kein Problem dar, wohl aber in Festkorpern. Hier ve rsteht manunter einer Stowelle eine permanente Anderung von Atompositionen,das heit es muss Plastizitat oder Phasenubergange auftreten. In dieserArbeit haben wir uns allerdings nicht an diese rigorose Denition gehal-ten, sondern alle abrupten Anderungen thermodynamischer Groen alsStofronten bezeichnet.

Grundlage der Stowellenphysik sind die Rankine-Hugoniot-Gleichung-en (Abs.1.5). Sie stellen nichts anderes dar als eine Umformulierungder Erhaltung von Masse, Impuls und Energie uber die Stofront hin-aus. Ursache ist die unterschiedliche Materiegeschwindigkeit auf beidenSeiten der Stofront.

Im Festkorper (Abs. 1.6) konnen wegen der endlichen Festigkeit mehrereWellen entsehen. bei schwachen Stowellen ndet nur eine elastischeVerformung statt. Ist die Welle starker als die elastische Hugoniot-grenze (HEL), so folgt nach der elastischen Welle eine plastische odereine Phasenumwandlung. bei starken Stowellen sind elastische undplastische Front nicht mehr unterscheidbar. Im Bereich deselastisch-plastischen Schocks ist die Stowelle nicht stationar.

In den Simulationen spielt auch die Dimensionalitat der Stowelle einewichtige Rolle (Abs.1.10): Ist die Temperatur zu niedrig oder der An-fangszustand nicht aqulibriert, so fehlt die Koppelung der Bewegunglongitudinal und transversal zur Stowelle. Das System verhalt sich reineindimensional, es tritt keine Thermalisierung eine lineares Anwachsender Stofrontdicke und starke Solitonen auf. Dieses Verhalten ist nichtzu verwechseln mit dem in Kapitel6 berichteten Auftreten von solitarenWellen.

Deutsche Zusammenfassung xi

Simulation von Stowellen in Festkorpern

Simulationen von Stowellen in Festkorpern (Abs. 2.1) haben vor etwavierzig Jahren begonnen. Wegen der begrenzten Rechnerleistung wur-den Vereinfachungen gemacht, die dazu fuhrten, dass die Strukturensich wie im vorigen Abschnitt beschrieben eindimensional verhielten,was zu der (falschen) Ansicht fuhrte, dass es in Festkorpern gar keineStowellen gabe. Im Laufe der Zeit konnte gezeigt werden, dass Sto-wellen in Festkorpern existieren und zu Plastizitat durc h Abscherungfuhren. Im Jahre 1998 gelang schlielich der Durchbruch mit der erstenSimulation eines achenzentriert kubischen Kupferkrist alls mit mehrerenMillionen Atomen. Erstmls konnten einzelne Stapelfehler isoliert wer-den. Seither hat man eine Vielzahl neuer und unerwarteter Phanomenewie nichtstationare Stowellen, verschiedene Phasenubergange und soli-tare Wellen gefunden und konnte beispielsweise Modelle der Entstehungvon Plastizitat durch Versetzungen uberprufen.Fur die Simulation von Stowellen in Festkorpern werden S tandard-Molekulardynamikprogramme eingesetzt (Abs.2.4). Der Ausgangszu-stand sollte gut aquilibriert sein, was mit Hilfe von isoth ermen undisotherm-isobaren Ensembles erreicht wird. Die eigentlichen Simulatio-nen werden mikrokanonisch durchgefuhrt.Zur Erzeugung der Stowellen werden mehrere Methoden verwendet(Abs.2.2): zwei Klotze werden mit konstanter Geschwindigkeit aufeinan-der zubewegt, sobald sie sich treen, entstehen im Zentrum zwei Sto-wellenfronten. Man kann das Verfahren modizieren indem man einender Klotze durch einen Spiegel ersetzt, der alle Teilchenimpulse umkehrt.Quer zur Stowellenrichtung werden periodische Randbedingungen ver-wendet um ein Auseinanderbrechen der Probe zu verhindern.

Strukturen und Potentiale

Ein Ausgangspunkt dieser Arbeit war die Frage, ob sich die Auswirkungvon Stowellen in aperiodisch geordneten Quasikristallenund in Krist-allen unterscheidet. In Quasikristallen sind beispielsweise zusatzlichelokalisierte und ausgedehnte Defektmoden moglich (Abs.3.1).Die Simulation von Stowellen stellt hohe Anforderungen an simulier-bare Modellstrukturen: es sollten vergleichbare quasikristalline und krist-alline Strukturen existieren, die Modelle sollten moglichst stabil sein,die Ergebnisse, insbesondere eventuelle Defekte, gut analysierbar wer-

xii Deutsche Zusammenfassung

den konnen und brauchbare Wechselwirkungen sollten bekannt sein.

Alle untersuchten Modelle sind tetraedrisch dicht gepackt (Abs.3.2).Simuliert wurden binar ikosaedrische Quasikristalle wiesie experimentellals (AlCu)Li und (AlZn)Mg bekannt sind und vereinfacht als D eko-ration von Rhomboedern dargestellt werden konnen. Da bisher keinerealistischen Wechselwirkungen bekannt waren, wurden Lennard-Jones-Potentiale verwendet. Die Potentialparametern wurden an die Geome-trie angepasst. Als kristallinen Vergleichsstruktur wurde die kubischeLaves-Phase untersucht, die zum Beispiel bei MgCu2 auftritt.

Stabile monoatomare ikosaedrischen Quasikristallmodelle sind nicht be-kannt. Die einzige Struktur, die sich fur Stowellensimul ationen eignet,ist ein zwolfzahliger Quasikristall. Hier existiert ein Modell, das ausdekorierten Drei- und Vierecksprismen besteht und in der dritten Di-mension periodisch ist. Der Quasikristall wird durch das Dzugutov-Potential (Abs. 3.4) stabilisiert und wurde bei Abkuhlsimulationen auseiner Schmelze entdeckt. Das Minimum des Dzugutov-Potentials gle-icht dem Lennard-Jones-Potential, aber es existiert noch ein Maximum,das verhindern soll, dass sich bei Abkuhlsimulationen a chenzentriertkubische Kristalle bilden. Die Prismen lassen sich zu vielen kristalli-nen und aperiodischen tetraedrisch dicht gepackten Phasenzusammen-bauen und durch rhombische und hexagonale Prismen erganzen. Eineder kristallinen Phasen, die sogenannte -Phase, kann als Approximantverstanden werden und ist stabiler als der Quasikristall.

Die Suche nach Grundzustanden fur das Dzugutov-Potential (Abs.3.4.3)zeigt jedoch, dass im Gegensatz zu den beim Abkuhlen entstehendenStrukturen eine ganz andere Phase, namlich ein kubisch innenzentrierterKristall, bei T = 0 und P = 0 stabiler als alle anderen Phasen ist.

Bei den Stowellensimulationen wurde eine weitere Phase beobachtet,die ! -Phase (Abs.3.5). Bekannt ist sie heutzutage als die Struktur vonMgB2, sie tritt auch bei den Elementen Ti, Zr und Hf und bei vielenLegierungen auf. Die ! -Phase entsteht aus innenzentriert kubischenKristallen, wenn zwei der drei Gitterebenen parallel zur dreizahligenAchse zusammenfallen. Diese Phasenumwandlung wird verursacht durcheine allgemeine Instabilitat innenzentriert kubischer K ristalle bei Ver-formung entlang dieser Richtung.

Deutsche Zusammenfassung xiii

Stowellen in kubischen Laves-Phasen und binarikosaedrischen Quasikristallen

Vergleicht man das elastische Verhalten von Kristall und Quasikristall(Abs.4.4.1), so stellt man wie erwartet fest, dass der Kristall anisotropist, wahrend sich der Quasikristall isotrop verhalt.Eine gute Ubersicht (Abs.4.4.3) uber die Ergebnisse erhalt man ausdem Hugoniotdiagramm, das ist die Auftragung der Geschwindigkeitender Wellenfronten, auch Pistongeschwindigkeit genannt, gegen die Ge-schwindigkeit, mit der die Klotze sich bewegen. Man beobachtet dreiBereiche, die allerdings oft nicht scharf abgegrenzt sind:ein elastischerBereich, ein Bereich zweier Wellenfronten (elastisch-plastisch) und einrein plastischen Bereich. Auch bei den Geschwindigkeiten der plastis-chen Wellen ndet man den Unterschied in der Anisotropie zwischenKristall und Quasikristall. In der Arbeit wird insbesonder e der Bereichmittelstarker elastisch-plastischer Stowellen genaueruntersucht. Beisehr starken Stowellen tritt vollstandige Fragmentieru ng auf, ein Ver-halten, das vollstandig unabhangig von der Ausgangsstruktur und de-shalb fur die Untersuchung von Defektstrukturen nicht besonders inter-essant ist.Bei den Laves-Kristallen (Abs.4.4.7) zerbricht der Ausgangskristall inKristallite, die gegen einander verdreht sind. Die Grenzender Kristallitelaufen diagonal durch den Kristall und sind bis zu zehn Atomabstandendick. Die entstehenden Strukturen lassen sich besser untersuchen, wennder Kristall nach dem Durchlaufen der Stowelle abgeschreckt wird,denn dann verschwinden die Verdrehungen. Das Zerbrechen setzt direktan der elastische Hugoniotgrenze (HEL) ein. Mit steigenderStowellen-intensitat wird die Fragmentierung immer feiner und geht k ontinuierlichin den Bereich starker Stowellen uber.Bei den Quasikristallen (Abs.4.4.8) ndet man grundsatzlich ein analo-ges Verhalten, allerdings treten hier zusatzliche Defekte auf. Verwen-det man das vereinfachte Quaskristallmodell, so beobachtet man Ring-prozesse bei denen funf Atome auf alternative Positionen springen. Die-ses Verhalten ist schon ohne Stowellen vorhanden, wird aber durchdie Stowellen intensiviert. Mit steigender Stowellenst arke degener-ieren die Ringprozesse erst zu Ketten, dann zu Netzen, und durchziehenschlielich das ganze Material. Letzten Endes fuhrt dieses Verhalten zueiner Ausschmierung der elastischen Hugoniotgrenze und zueinem kon-

xiv Deutsche Zusammenfassung

tinuierlichen Ubergang zwischen elastisch und elastisch-plastisch. Ver-wendet man ein verbessertes Quasikristallmodell, so verschwinden dieRingprozesse und es gibt nur noch einzelne Sprunge der Atome. DieStabilitat der Proben wird erhoht und das Verhalten dem de s Laves-Kristalls ahnlicher.Zusammenfassend kann man sagen, dass sich die hier simulierten binarenKristalle und Quasikristalle weitgehend wie ionische Materialien verhal-ten. Erklaren kann man dies durch die Wahl der Potentialpar ameter.Bei den Quasikristallen wurden zusatzliche Defekte gefunden, die zueiner Schwachung des Materials fuhren. Da die Ringprozesse aber nichtzu einer Anderung der Zellstruktur der Quasikristalle fuhren, kan n mansie nicht als quasikristallspezische Defekte auassen. Sie konnen auchin anderen komplexen Kristallstrukturen mit partieller Be setzung vonGitterplatzen aufteten.

Stowellen in Materialien, die uber Dzugutov-PotentialewechselwirkenAuch bei den monoatomaren Strukturen wurde zunachst das elastischeVerhalten untersucht. Die Quasikristalle sind in der quasiperiodischenEbene isotrop, senkrecht dazu aber nicht. Alle anderen Strukturen sindwie erwartet anisotrop.Die Ergebnisse sollen wieder anhand des Hugoniotdiagramms(Abs.5.4.2)dargestellt werden. Auf den ersten Blick sieht es so aus, alsob es hiernur den elastisch-plastischen und den plastischen Bereichgabe. Diesliegt am Phasendiagramm: schon bei schwachen Stowellen tritt einUbergang von den Ausgangsstrukturen zu dichten Kugelpackungen auf,also den Phasen, die nach Konstruktruktion des Dzugutov-Potentialseigentlich ungunstig sein sollten. Deshalb taucht der elastische Bereichim Phasendiagramm gar nicht auf. Da das Potential sehr kurzreich-weitig ist, ist die Stapelfolge in den dichtgepackten Phasen zufallig undnicht rein kubisch achenzentriert oder hexagonal dicht g epackt.Es wurden viele Ausgangsstrukturen und -orientierungen untersucht.Die detaillierten Ergebnisse konnen hier nicht im Detail geschildertwerden (Abs.5.4.3). Ausgezeichnet ist die vierzahlige Richtung bei in-nenzentrierten Ausgangskristallen: hier ndet der Phasenubergang in-nerhalb ein- bis zwei Atomlagen statt und fuhrt zu fast defektfreienEinkristallen oder Zwillingen. Mit zunehmender Stowellenintensitattreten aber mehr und mehr Defekte auf. Bei anderen Ausgangsstruk-

Deutsche Zusammenfassung xv

turen und -orientierungen verschieben oder drehen sich oftdie Atomebe-nen senkrecht zur Stowellenrichtung. Die Ausgangsordnung bricht inmehreren Stufen zusammen, und es entstehen dicht gepackte Kristallemit vielen Defekten, die nur langsam ausheilen. Verwendet man denImpulsspiegel in der Simulation, so wirkt dieser als Keim fur fast defek-tfreie Kristallschichten. Fur die Analyse der entstandenen Strukturenwurden vor allem radiale und Winkelverteilungsfunktionen eingesetzt.Spezielle Phanomene beobachten man bei Quasikristallen und Approx-imanten, wenn die Stowellenrichtung in der Grundebene derStrukturliegt (Abs.5.4.5): In diesem Fall treten zwischen der elastischen und derplastischen Stowellenfront Flips auf, das heit die Zellstruktur andertsich. Zusatzlich zu den Dreiecks- und Vierecksprismen ndet man rhom-bische und hexagonale Prismen, die sich als Verallgemeinerung der Aus-gangszellen verstehen lassen und bei Flips in Quasikristallen auftretenmussen.Die Simulationen mit Dzugutov-Pontentialen lassen sich uberraschen-derweise qualitativ mit Stowellensimulation an Eisen vergleichen, beidenen spezielle Embdedded-Atom-Wechselwirkungen eingesetzt wurden(Abs.5.5.1). Ursache ist die qualitative Ahnlichkeit des Phasendia-gramms von Eisen und dem Dzgutov-Potential. So ndet man fur dieinnenzentriert kubischen Ausgangsstrukturen ahnliche Endstrukturen.Auch solitare Wellen, die im nachsten Abschnitt detailli ert beschiebenwerden, sind beim Eisen beobachtet worden.Zumsannefassend kann man sagen, dass sich das Dzugutov-Potenialnicht sehr gut fur Stowellensimulationen von Quasikrist allen eignet,da die Stabilitat verglichen zu gering ist. Im Gegensatz zum binarenFall wurden hier aber eindeutig quasikristallspezische Defekte, namlichZellips, aufgefunden. In beiden Fallen zeigt sich, dass Auftreten derDefekte durch die Deformation der Probe verstarkt wird.

Die ! -Phase und solitare WellenInnenzentrierte kubische Kristalle besitzen einen inharente Instabilitatentlang der dreizahligen Achse, was durch das Verschwinden der Summeder k-Vektoren im reziproken Raum bei zwei Dritteln der Zonengrenzeverursacht wird. Dies fuhrt bei vielen Materialien zu einem Phasen-ubergang in die ! -Phase.In unserem Fall ist die ! -Phase nur in einem kleinen Kompressionsbere-ich stabil (Abs.6.7.2), so dass eine Hin- und Rucktransformation zwi-

xvi Deutsche Zusammenfassung

schen ! -Phase und innenzentriert kubischer Phase stattndet solangedie Phasenumwandlung sich langsamer als die Schallgeschwindigkeitausbreitet (Abs.6.4). Wird die Schallgeschwindigkeit uberschritten, soentstehen solitare Wellen, die im Innern die ! -Phase enthalten. Erhohtman die Temperatur bei der Simulation, so verschwinden die solitarenWellen nach und nach.Bei sehr groen Proben mit einem Querschnitt von 40x40 Atomabstan-den beobachtet man auch bei niedrigen Temperaturen ein Abklingender solitaren Wellen (Abs. 6.4.4). Betrachtet man die Probe als eineAnsammlung von Atomketten parallel zur Stowellenrichtun g, so ndetman, dass die Amplitude der Stowellen entlang der Ketten nicht ab-nimmt, sondern die Korrelation zwischen benachbarten Ketten verlorengeht. Bewegt man sich mit dem Maximum des Drucks in der solitarenWelle mit so beobachtet man, dass diese Front zunachst achist, dannzu uktuieren anfangt, und am Ende sich einige langwellige Modenaufschaukeln, die gerade in den Querschnitt der Simulationszelle passen.Nach solitaren Wellen wurde auch in anderen Symmetrierichtungengesucht (Abs.6.4.5). Dabei wurde festgestellt, das es in Spiegelebenenund entlang der vierzahligen Richtungen keine solitaren Wellen gibt.Im ubrigen hangt das Auftreten stark von der Methode der St owellen-erzeugung ab. Um den Einuss der unendlich starken Beschleunigungam Anfang der Simulation zu verringern, wurden Simulationen mitlangsam ansteigenden Pistongeschwindigkeiten durchgefuhrt. Die so-litaren Wellen treten dennoch auf. Sie konnten auch bei Stowellen-simulationen in Eisen mit realistischen Wechselwirkungenbeobachtetwerden und waren dort sogar stabiler als in den hier vorgestellten Sim-ulationen.Fur die Beschreibung der ! -Phase und der solitaren Wellen existiertein phanomenologisches Landau-Ginzburg-Modell (Abs.6.7), das allebeobachteten Phanomene voraussagt. Aus diesem Grund wurde dieAnwendbarkeit des Modells untersucht (Abs.6.7.4). Leider wurde fest-gestellt, dass die Vorraussetzungen fur die Landau-Ginzburg-Beschrei-bung nicht gegeben sind. Auch die Dispersionsrelation der Phononen(Abs.6.7.3) wurde bestimmt und ein Weichwerden der longitudinalenMode gefunden. Allerdings liegt dies nicht bei zwei Dritteln der Zo-nengrenze sondern fast bei der Halfte, solange sich die Probe in derinnenzentrierten Phase bendet. Wird sie aber uniaxial komprimiert,so wandert die Instabilitat zum erwarteten k-Wert.

Deutsche Zusammenfassung xvii

Solitare Wellen wurden auch in achenzentriert kubische n Kristallenentlang der zweizahligen Achse beobachtet (Abs.6.8). Auffallig ist dabei,dass es sich ebenfalls um eine Richtung handelt, in der die Atome denkurzesten Abstand haben. Auch bei den zwolfzahligen Quasikristallenund Approximanten ndet man Ansatze von solitaren Wellen senkrechtzur Grundebene. Hier sind sie aber nicht stabil, da die Atomketten mitkurzem Atomabstand sozusagen ausgedunnt sind. Es stellt sich deshalbdie Frage, ob solitare Wellen auch in einfach kubischen Kristallen ent-lang der vierzahligen Achse existieren. Nun sind aber einfach kubischeKristalle instabil. Dieses Problem wurde gelost, indem Wechselwirkun-gen nicht nur zwischen nachsten sondern auch zwischen zweitnachstenNachbarn eingefuhrt wurden. Es konnten tatsachlich soli tare Wellenentlang der vier- und zweizahligen Richtung erzeugt werden je nachdem, ob die kurzeren oder langeren Wechselwirkungen starker waren.Zusammenfassend fuhren diese Beobachtungen zu einem einfachen Mo-dell: solitare Wellen in einer bestimmten Richtung treten auf, wenn dieAtome ununterbrochene Ketten mit kurzesten Atomabstand en bilden.Die solitare Welle stellt einen Stopuls wie bei harten Kug eln dar, derdie Atomkette entlang lauft und von eindimensionalen Kett en harterKugeln wohlbekannt ist. Die atomaren Ketten mussen dicht gepacktsein, da sonst die Korrelation zwischen benachbarten Ketten verlorengeht und die solitare Welle verschwindet.Die Frage einer analytischen Beschreibung der solitaren Wellen wurdeebenfalls diskutiert. Der Zerfall der solitaren Wellen zeigt, dass einedreidimensionale Beschreibung notwendig ist. Da solche nichtlinearenGleichungen aber nur numerisch losbar sind, wurde dieser Weg bishernicht weiter verfolgt.

Amorphe StrukturenBei den diatomaren und bei den monoatomaren Simulationen wurdenauch das Verhalten von amorphen Ausgangsstrukturen mit denselbenWechselwirkungen und Zusammensetzungen wie bei den Quasikristallenund Kristallen untersucht (Abs. 4.4.9und 5.4.7). Die amorphen Phasenwurden durch Abkuhlen einer Schmelze gewonnen.Erwartet wurde, dass es hier keine elastische Hugoniotgrenze gibt, unddementsprechend keinen Bereich mit elastisch-plastischer Welle, son-dern dass sich das Material verhalt wie eine Flussigkeit. Bei den binarenStrukturen ist dies tatsachlich der Fall. Die Stowelleng eschwindigkeit

xviii Deutsche Zusammenfassung

ist im ganzen untersuchten Bereich eine lineare Funktion der Geschwin-digkeit, mit der die Klotze bewegt werden.Die Stowellengeschwindigkeiten bei der monoatomaren amorphen Phaseweichen von der Linearitat ab: bei Pistongeschwindigkeiten bis etwa15% der Schallgeschwindigkeit hat man nichtstationare dispersive Wellen,die sich etwa mit Schallgeschwindigkeit bewegen. Erst dannsetzt daslineare Verhalten ein. Eine Ursache fur den quasielastischen Bereuchist nicht bekannt.

Schlubemerkungen und AusblickIn dieser Arbeit konnten erste Ergebnisse von Stowellensimulationenin komplexen geordneten Strukturen berichtet werden. Bei den Quasi-kristallen konnten tatsachlich neuartige Plastizitats moden beobachtetwerden, die nun im Detail in Gleichgewichtssimulationen weiter unter-sucht werden sollten.Bisher wurden Modellpotentiale verwendet. Inzwischen sind auch furQuasikristalle spezische Wechselwirkungen berechnet worden. Zusam-men mit verbesserten Modellen sind nun auch realistische Simulationenmoglich.Fur manche Phanomene wie dem Ausheilen der Defekte und demEr-reichen stationaren Stowellenzustande sind noch wesentlich groereProben und langere Simulationszeiten notwendig. Hier seiauch nochauf die Entwicklung des Hugoniostaten hingewiesen, eines Ensembleszur Gleichgewichtssimulation des Zustands hinter der Stowellenfront.Spekulativ ist auch immer noch die Beobachtung der solitaren Wellen.In den Simulationen wurden sie unter verschiedenen Umstanden beob-achtet und ihre Existenz nach allen Seiten abgesichert. Jetzt warenexperimentelle Untersuchungen sehr hilfreich.Bisher wurden Stowellen nur in dreidimensionalen Quasikristallen stu-diert und der zweidimensionale Fall vernachlassigt, obwohl ein wohlun-tersuchtes binares Modell existiert und die Analyse der Ergebnisse vieleinfacher sein sollte. Diese Simulationen sollen nachgeholt werden.Eine Zukunftsvision stellt die Simulation der Laserabtragung dar. Hiertreten Stowellen, Risse und Versetzungen auf. Zur Simulation wer-den mehrere Millionen Atome benotigt. Eine besondere Rolle spieltdas Elektronengas bei der Warmeleitfahigkeit. Sein Ein uss muss mitniten Elementen modelliert werden, was bedeutet, dass Multiskalen-simulationen notwendig werden.

IntroductionWith the advent of large massively parallel supercomputersit has be-come possible to carry out multi-million atom molecular dynamics sim-ulations. One of the most spectacular applications is the atomisticscale study of defects generated by shock waves in three dimensions.Before it was not possible to resolve the details of the defects [101],and the simulations were limited to four-fold symmetry directions offcc crystals. Thus it occurred as a surprise when Germann et al. [72]found that even simulations of one of the simplest systems, fcc-crystalswith Lennard-Jones interactions, yielded quite complicated phenomenaif shocked along the two- and three-fold directions, for example marten-sitic phase transitions and non-steady wave fronts. The phenomena arestill partially unexplained [ 206]. One of the surprises included dier-ent kinds of solitary wave trains along the three-fold direction. Mean-while non-steady solitary waves have been observed in bcc-iron alongthe three-fold direction by Kadau [118] and by the present author in asimple cubic structure along the four-fold direction.Atomistic simulations of shock waves have spread out in dierent di-rections: other crystal structures like diamond for example [56,237] arestudied, pre-existing defects [101,26] are built in, and poly-crystals [116]are simulated. In the present thesis I added yet another direction: theextension to binary crystal structures and to even more complex aperi-odic structures, namely the quasicrystals. The wealth of behavior ob-served indicates that I may have scratched up to now only the surfaceof possible phenomena.The other central topic of this thesis is the study of solitary wavesrelated to shock waves. Holian and Straub [102,204,97] have treated thesubject in one dimension and have generalized it to three dimensions.Today two kinds of solitons have to be distinguished: one-dimensionallocked-in solitons and non-steady solitary waves. My simulations for therst time have revealed solitary waves which require a three-dimensionaltreatment since they develop transverse modulations. For the case ofsolitary waves in bcc-crystal shocked along the three-folddirection Icould show that they are the super-sonic continuation of a sub-sonicintermediate phase transition to a hexagonal! -phase.The habilitation thesis is devoted to two research communities which

xx Introduction

have nearly no overlap: the community which studies shock compressionof solids, and the people working on quasicrystals. To make the thesisas comprehensible as possible for both groups I have included sectionson the fundamentals of shock wave physics and on the foundations ofquasicrystals. Each section may seem superuous for the other group,but then I ask people to apologize and simply skip the section.I will start in Part I with a repetition of the general basics o f shockwaves in condensed matter. The essential equations required to under-stand the behavior of shock waves are given and the dierencebetweenshocks in uids and solids are specied. An overview of the literatureon shock compression of solids follows. The next topic is thestate of theart of large-scale shock wave simulations. I address the methods to gen-erate shock waves in simulations and shortly characterize our generalpurpose molecular-dynamics code IMD suitable for massively-parallelsimulations.In a short overview the crystallographic characteristics of quasicrystalsare exposed and the path to atomistic structures and new kinds of de-fects is drawn. The structure models investigated in the simulationsare explained subsequently. A description of the special Dzugutov po-tential applied in the simulation and a discussion of its phase diagramfollow. Part I ends with the presentation of experimental results relateddirectly or indirectly to the model materials studied here.The starting point of my research was to compare the behaviorof pe-riodic and aperiodic materials under the inuence of shock waves. Dueto the rather limited number of suitable models I had not many choices.The outcome of this attempt is the central part of this thesis exposedin Part II. It is subdivided into three chapters: a diatomic c ase, amonatomic case, and the solitary wave phenomena together with the! -phase.If the most frequent icosahedral quasicrystals are to be studied one hasto resort to binary models. It turned out that this is a huge ta sk sinceup to now large scale shock wave studies do not exist even for ordinarycrystals. The results on diatomic quasicrystals are given in the rstchapter.If arbitrary crystal symmetries are permitted a monatomic dodecagonalmodel endowed with the special Dzugutov potential [53] can be applied.The results are the content of the next chapter. The ground state ofthis model, however, is not a quasicrystal but an ordinary bcc-phase.

xxi

So I could extend my exploration to a number of crystalline and qua-sicrystalline phases stabilized by the Dzugutov potential.The peculiar behavior of shock waves along the three-fold axis of thebcc crystals leads to the nal chapter of Part III: non-steady solitarywaves are observed. Since all attempts to eliminate these waves andall alterations of the initial conditions were fruitless I h ave come to theconclusion that these solitary waves are not an artefact butdeserve amore closer look. Results obtained by other researchers conrm thisopinion [118]. The solitary waves are closely related to a bcc-latticeanomaly and an intermediate phase transition. Analytical theories forthe transformation [29,30,181,182] have been examined and it has beenshown that generalizations are necessary.The thesis ends with a consideration of the prospects of shock wavesimulations: its extension to polyatomic materials for example and totechnologically relevant applications like laser ablation.

xxii Introduction

Part I

Shock Wave Physics,Shock Wave Simulations,

Structures andInteractions

Chapter 1

Shock Wave Physics

1.1 Why study shock waves in solids?

The scientic study of shock waves is simply appropriate due to thefact that they are everywhere and occur in all states of matter [123]:from the bow shocks of the sun and the earth, from the explosion ofsupernovae down to the shock waves that ignite thermonuclear fusion,cause sonoluminescence or a sonic boom.If we assume that shock waves exist in solids (which is well knownfrom experiment, and meanwhile from simulations also) thenthere area number of good reasons to study them in computer simulations:

Shock waves permit to compute the equation of state under ex-tremely high pressure and/or high temperature conditions, forexample to look for the metalization of hydrogen.

Shock waves allow to study the mechanical properties of solidsunder sudden load change.

Shock waves can be used to excite plasticity modes without in-troducing defects articially. This can be helpful in the ca se ofcomplicated structures like quasicrystals.

Strong enough shock waves will produce defect structures whichcan be analyzed post mortem. The generation of extended defectsis also possible.

Shock waves frequently lead to phase transitions. Thus theyper-mit to study the dynamics of the transition.

We may in sort say with Jim Asay [8]: Shock waves are useful to studyproperties of materials which are inaccessible to other methods.

4 Shock Wave Physics

1.2 The denition of shock waves

The denition of shock waves depends to some degree on the pointof view and on the application. Although it is possible to dene shockwaves in continuum models rigorously by their mathematicalproperties,namely as an innitesimal jump, such a denition may not be su itablefor physical applications due to the discrete nature of the atomic struc-ture. This is true already for uids where shock fronts always have anite width, but even more for solids due to the rigid nature o f thelattice.Usually shock waves are dened by a moving discontinuity of aphysicalobservable like pressure, stress, density or material velocity [37]. Thereis no problem with this denition in liquids or gases since any valueof the jump of the physical observables at the discontinuity surface isallowed due to the lack of retention forces. Viscous ow can occurand steady or stationary waves are permitted. The shock waves canbe described by the Navier-Stokes equations despite the large gradientswhich are present at the discontinuity interface [100,104].In solids there is a problem with this denition since they possess niteyield strength. If the shock stress is above, then the solid will behave likea uid. But shock waves are possible already at lower shock intensities.Then two waves are observed, namely an elastic precursor wave and aplastic wave.According to Boslough and Asay for example [25] (See also Bethe [19])stable shock waves have to be super-sonic with respect to theunshockedmaterial and sub-sonic with respect to the shocked state.If we use the denition that a shock wave is a wave which causesper-manent rearrangement of the atoms [94] (in contrast to strong elasticwaves), then super-sonic elastic waves which cause transient deforma-tions are excluded, since only plastic waves lead to permanent modi-cations. Another distinction is given by Wallace [217]: In a sound wavedissipation can be neglected, but not for a shock wave where dissipationis essential.We will not be rigorous with our notation, but will call any ty pe ofwave front with an abrupt change of thermodynamic quantities a shockwave [123].Dislocations and slip planes with stacking faults for example are gener-ated by plastic ow and relax the uniaxial stress to hydrostatic compres-

1.3 Denition of the basic observables 5

1 11

Figure 1.1: Denition of the basic variables. Unshocked state: P0 , V0 , E0 ,u0 = 0. Shocked state: P1 , V1 , E1 , u1 = up . The interface between shockedand unshocked moves with the shock wave velocity us . The shocked materialis driven by the piston and moves with velocity up .

sion. In contrast to the uid the stress will not be relaxed completelybut only down to the level of the yield strength of the material. Trueshock waves, which lead to steady proles [101], have to be accompa-nied by dissipative, irreversible ow transverse to the shock direction.The ow is extended due to the dislocations and the stacking faultsfor example [93, 101, 94]. In uids viscous ow is found in the shockfront [ 100,104]. Thus the ow is localized to a small region.

1.3 Denition of the basic observables

The terminology of shock wave physics originates from the generation ofshock waves by a piston in a tube lled with gas. The same expressionsare used even if we have a solid instead of a gas and a yer plateor alaser instead of a piston. The thermodynamic state of the material istypically characterized by the pressureP , the volume V , the internalenergyE , and a velocity u (Fig. 1.1). The temperatures of the shockedand unshocked material do not show up directly in the equation of state(See AppendixA) and are therefore omitted in the gure and anywhereelse. In the unshocked material the observables are indexedwith "0".In the standard experiment the unshocked material is at rest: u0 = 0,and therefore u0 is often neglected. The shocked material is indexedwith "1" and is driven with the piston velocity u1 = up. Sometimesupis called particle velocity since it is the velocity with which the shockedmaterial moves.

6 Shock Wave Physics

Figure 1.2: Change of state in three basic transformation processes: staticisothermal compression, isobaric heating and shock compression.

There is an important dierence between up and us which comes intoplay if there are several shock fronts or if the material is pre-shockedor shocked several times:up is dened in a volume like the state ob-servablesP , V , E whereasus is dened for an interface. In the case ofseveral shock fronts there is aus for each interface and aup for eachvolume in between.In a solid, the pressureP has to be replaced by the stressPxx if xdenotes the direction of the shock wave. In shock wave physics thestressPxx is frequently called Pxx in analogy with the pressureP .

1.4 The equation of state

As noted in the introduction, shock compression oers new possibili-ties to evaluate the equation of state of a material. In the pressure-temperature plane shock compression lies between static compressionand isobaric heating (Fig. 1.2).Now lets consider a uid or a material beyond its yield strength. Ifthe equations of state for isothermal and adiabatic compression in thepressure-volume (P -V ) diagram with shock compression (Fig.1.3) arecompared it is found that the growth of the pressure is the strongest

1.5 The Rankine-Hugoniot equations 7

R

1

0

PA

I

V

H

Figure 1.3: Pressure-volume equation of state of a uid. The system movesalong I in an isothermal and along A in an adiabatic process. The Hugoniotcurve is denoted by H , and the straight Rayleigh line by R. In a shockexperiment the system starts at 0 and jumps to 1.

in shock compression, provided the starting point is the same for allprocesses (here denoted 0).There is a very important point to note: in an isothermal or ad iabaticprocess the system moves along the lines in Fig.1.3. In shock compres-sion the systemjumps from the initial state 0 to the nal state (denoted1). The Hugoniot curve H is now dened as the collection of end-pointsof all experiments or simulations starting at the same point0. Thus theHugoniot curve is not an ordinary path of a thermodynamical process. Ifthe system starts from ambient conditions, the Hugoniot curve is called"principal", otherwise "secondary" or "reshocked". In the isothermalor adiabatic processes the work is the area below the curve. In shockcompression the work is the area below the Rayleigh line (Fig. 1.3).

1.5 The Rankine-Hugoniot equations

The Rankine-Hugoniot equations are the basic equations of shock wavephysics, describing the change of the equation of state across a singleshock front in a simple medium. The equations have been derived sev-eral times (originally in Ref. [150, 109, 110], reprinted in Ref. [151, 107,

8 Shock Wave Physics

108]). The Rankine-Hugoniot equations are truly valid in solids onlyif the stress is higher than the yield strength. Then the solids are ina state of hydrostatic thermal equilibrium, more precisely in a steadystate.The Rankine-Hugoniot equations are the consequence of the conserva-tion of mass, momentum and energy across the shock front interface. If is the density, P the hydrostatic pressure, andE the internal energy1

with index 0 on the one side and index 1 on the other side of the dis-continuity then we may write down the following equations (CompareSec.1.3) with respect to the interface velocity us:

mass conservation: 0us = 1(us up); (1.1)

momentum conservation:

P0 + 0u2s = P1 + 1(us up)2; (1.2)

energy conservation:

E0 + P0= 0 +12

u2s = E1 + P1= 1 +12

(us up)2; (1.3)

where we have already substitutedu0 by us and u1 by us up. Ifthe compressed state is not hydrostatic, thenP has to be replaced bythe uniaxial stress Pxx . Frequently it is assumed that the state 0 isuncompressed andP0 is set to zero in Eqs.1.2 and 1.3.From the conservation equations one can derive the Hugoniotequations:

0= 1 = ( us up)=us (1.4)

P1 = 0usup + P0 (1.5)

E1 E0 =12

(P1 + P0)(V0 V1): (1.6)

If we carry out a simulation with given piston velocity up, initial pressureP0, volume V0, and energyE0, and measure the shock wave velocityus

1More precisely, E denotes the energy density per unit mass (See for example[48]), but to avoid confusion we will stay with the common sloppy labeling "internalenergy".

1.6 Two wave structure from material rigidity 9

then we can derive from the Hugoniot equations the quantities P1, V1,and E1 which means that we are able to determine the equation of state.The equations may be solved forus, up and E1 E0:

us = V0[(P1 P0)(V0 V1)]1=2 (1.7)

up = [( V0 V1)(P1 P0)]1=2 (1.8)

E1 E0 = P1upV0=us u2p=2: (1.9)

Now we can computeup, us and subsequently E1 E0 if we knowthe pressure-temperature equation of state. Apparently there are twoequivalent representations: instead of theP -V -diagram we can drawa us-up-diagram (or a mixture of both). The us-up-diagram is oftencalled Hugoniot diagram.In the shock wave simulations it is sometimes advisable to work in acoordinate system where the unshocked medium 0 moves at velocity ui .Then the Hugoniot equations read [25]:

0= 1 = 1 (up ui )=(us ui ) (1.10)

P1 P0 = 0(us ui )(up ui ) (1.11)

E1 E0 =12

(P1 + P0)(V0 V1) =12

(u1 ui )2: (1.12)

The Rankine-Hugoniot relations may be found in many publications[156, 25, 1, 14]. Duvall and Graham [48] list generalized equations foran arbitrary sequence of multiple shock fronts, Henderson [83] com-pares dierent coordinate systems, and Davis [34] presents a long listrepresentations of the Rankine-Hugoniot equations.

1.6 Two wave structure from material rigid-ity

In a solid, both the P -V - and the us-up-Hugoniot diagram look morecomplicated than in a uid. Discontinuities exist which ind icate thatseveral shock fronts are present at a given time.There are two major reasons for multiple shocks [76]: either the shockstrength is beyond the dynamical yield strength of the solid(Sec.1.8),

10 Shock Wave Physics

Figure 1.4: Hugoniot curve of an elastic-plastic solid. The full line is thereaction upon shock compression, the dotted line the path for hydrostaticcompression. From [144].

then an elastic and a plastic wave is present, or there is a phase transi-tion [48], then a plastic wave and a transformation wave are observed.In both cases the us-up-Hugoniot curves look rather similar. In thesimulations the elastic-plastic behavior (Chap. 4) and an elastic-phasetransformation behavior (Chap. 5) have been observed. A high-pressuresolid-solid transformation has not been found in our model systems.If a solid is compressed hydrostatically, its volume shrinks and the pres-sure rises (Fig.1.4) [156]. The hydrostatic curve is smooth as a functionof stress and volume. If the solid is compressed uniaxially stress andpressure will also increase until the maximal resolved shear stress of thematerial is reached. Then it will yield and change from the uniaxiallycompressed state to hydrostatic compression. In theP -V -diagram thisevent is visible through a cusp, called the Hugoniot elasticlimit (HEL).The location of the cusp is estimated in Sec.1.8.Fig. 1.5 shows schematically the typical behavior of a solid with niteyield strength. The left part is the P -V -diagram, the right part containstypical stress proles. The P -V -diagram can be divided into three partswith stressesPa , Pb, and Pc. The borders of the regions are dened bythe points denoted HEL and OD. The latter is the overdrive stressdened as the end-point of the line continuing the P -V -curve under theHEL straight on.

1.6T

wo

wave

structurefrom

material

rigidity11

Pa

PHEL

PHEL

Pc

Pb

Pc

Pb

Pa

POD

Figure 1.5: P -V -diagram (left) and corresponding wave proles (right) of a solid with nite yield strength. Theshort dashed line in the left part is the direct continuation of the P -V -curve below the HEL. From [ 76].

12 Shock Wave Physics

Below the HEL the solid reacts elastically with peak stressPa . We nda single jump of the wave pulse in the diagram on the right. BetweenHEL and OD a two-wave structure is observed. The stress builds upas long as the material has not yielded and an elastic precursor waveis generated with height is PHEL . Subsequently the stress increases inthe plastic precursor part until the nal plastic wave follo ws with peakstress Pb. Ideally there would be two jumps in the stress prole: theelastic precursor and the plastic front. In reality the second jump issmeared out by the plastic precursor in between [216]. Beyond OD isthe overdriven regime. There is still an elastic wave but it is invisiblesince the system jumps immediately to the plastically deformed stateat stressPc.In experiment and in simulations the plateau stressesPa , Pb, and Pccan not last forever. They are released by a so-called rarefaction wavewhich cannot be a shock wave due to the stability criteria of shockwaves [25, 19]. While the shock wave sharpens, the rarefaction wavespreads out in time.If the wave prole is a time-independent function moving at xed veloc-ity, then the wave is called steady. This is true for shocks inthe elasticand overdriven regime as long as the driving pressure has notbeen re-leased. In the case of a two-wave prole (Part b in Fig.1.5) the shockwave is steady only if the velocity of the elastic and the plastic waveare the same which is the exception. Thus the wave is non-steady ingeneral, but the elastic and plastic part taken separately may be steady.Fig. 1.6 nally represents the correspondence between the typicalus-up- Hugoniot diagram and the P -V - or more generallyPxx -V -Hugoniotdiagram of a solid. The information contained in both plots is thesame since they are related by the Rankine-Hugoniot equations. Usuallywe will represent the us-up-Hugoniot since it is easier to obtain in thesimulations.

1.7 Dierent representations of a stressedstate

A typical shock compression experiment involves simultaneous loadingof all points of a planar surface [75], and leads to a uniaxial deformed

1.7 Dierent representations of a stressed state 13

Figure 1.6: Schematic Hugoniot curves of an elastic-plastic solid showing thecorrespondence of theus -up and Pxx -V representations. HEL is the Hugoniotelastic limit, OD the beginning of the overdriven regime. Up per part: the y-axis intercept c0 is the longitudinal velocity of sound, the virtual intercep tof the full line is the bulk velocity of sound. Lower part: the long straightline is the Rayleigh line through the HEL. Below HEL is the ela stic regime.The dots mark the borders of the elastic-plastic regime. Abo ve OD is theoverdriven or plastic regime usually studied in experiment . The open circlemarks a typical elastic-plastic shock. From [ 95].

14 Shock Wave Physics

state. If a material can resist shear deformation then this uniaxiallystressed conguration results in shear stresses, as the longitudinal andlateral stress components are not equal. The stress conguration can bedecomposed into a mean or hydrostatic pressureP and the shear stressS. Every plane in the body except those parallel or perpendicular tothe shock normal is subjected to shear stress.The hydrostatic pressure is given by

P =13

(Pxx + Pyy + Pzz ); (1.13)

the Pii are the uniaxial stresses (in shock wave physics frequentlycalled"uniaxial pressures"), Pij = 0, if i 6= j . For isotropic and homogeneousmaterials Pyy = Pzz . Therefore

P = Pxx 23

(Pxx Pyy ); (1.14)

and the shear stress is maximized on planes lying at 45 degrees to theshock normal:

S =12

(Pxx Pyy ): (1.15)

The stressS is called the maximal resolved shear stress. The uniaxialstresses can be expressed by:

Pxx = P +43

S (1.16)

Pyy = Pzz = P 23

S: (1.17)

1.8 Yield strength of a solid and the Hugo-niot elastic limit (HEL)

For the elastic shock wave, stress and strain at the shock front are givenby [144,156]:

Pxx = ( + 2 ) xx ; (1.18)

Pyy = Pzz = xx ; (1.19)

1.8 Yield strength of a solid and the Hugoniot elastic limit (HEL) 15

where xx denotes the one-dimensional strain component, and and are the Lame constants.If yielding occurs behind the elastic precursor wave, the shock yieldstressY can be given by

Y = 2 xx : (1.20)

This result is obtained by assuming either a maximal shear stress or thevon-Mises criterion for yielding due to the condition of one-dimensionalstrain.The hydrostatic pressure at the wavefront is dened by

P =

+23

xx : (1.21)

The dierence between the Hugoniot curve and the hydrostatic com-pression curve is given by (Fig.1.4):

Pxx = P +43

xx = P +23

Y =

23

+

Y2

+23

Y: (1.22)

This is the value of the stressPxx at the Hugoniot elastic limit (HEL)and the factor 2= 3 + = K is the bulk compressibility modulus.The yield strength includes a rate-dependent parameter anda pressuredependency as well. It is found thatY is an increasing function of hy-drostatic pressure. In general the propagation of elastic-plastic waves isa very complicated phenomenon strongly dependent on material whichcan be modeled only roughly by the simple considerations presentedhere.An alternative description is given by the stress components:

Pxx = ( K +43

) xx ; (1.23)

Pyy = Pzz =

1

xx : (1.24)

K is the bulk compressibility, the Poisson ratio. Then the maximalresolved shear stress can be written as

S =12

(Pxx Pyy ) = xx =

1 22(1 )

Pxx : (1.25)

16 Shock Wave Physics

1.9 Velocity of sound

In the simulation of anisotropic monocrystals the questionarises: Whichis the correct velocity of sound useful to scale the shock velocity us andthe piston velocity up such that the Hugoniot curves for shock waves indierent directions can be compared directly? There exist two positiveconstants c and s in the expansion us = c + sup + : : : , as shock wavesare always super-sonic and the limit forup ! 0 is the velocity of soundc. Because the velocity of sound is related to an elastic constant E andthe density through c =

pE= , the question amounts to the correct

elastic constant.According to Nagayama [144] the velocity c is related to the hydrostaticor adiabatic bulk compressibility K through c2 = K= . It is the velocityof a virtual longitudinal sound wave due to the hydrostatic compression.We call c the bulk sound velocity or hydrodynamic sound speed [189]and denote it by cb. It is related to the ordinary longitudinal velocityof soundcl and the transverse velocity of soundct through

c2b =K

=K + 43 G

43

G

= c2l 43

c2t : (1.26)

G is the shear modulus. These are the usual statements about thescaling velocity c found in many references (See for example [180]).But our simulations show that E should depend on the direction of theshock wave, which is not possible forK . The problem is resolved bythe observation that most of the authors are experimentalists and dealwith data on isotropic polycrystals in the overdriven regime.A more careful approach (like the one by Davison [35]) leads us in thecase of isotropic materials already for weak shock waves to

c := cl =p

( + 2 )=; (1.27)

and for strong shock waves to

c := cb =p

( + 2 =3 )=: (1.28)

For anisotropic materials cl becomes equal top

F= , where in the caseof a principal axes coordinate systemF = c11, and cb gets equal top

(c11 + 2 c12)=3, which is a trace and therefore independent of the co-ordinate system. The elastic constantsc11 and c12 are the componentsof the elastic tensor in Voigt notation.

Dependency of shock wave behavior on dimension 17

The simulation results show that this is indeed the correct answer: Ifscaled with the direction-dependentcl , all curves have the samey-axisintercept in the Hugoniot plot. If unscaled they fall on one curve if upgets large (See also Fig.1.6).

1.10 Dependency of the shock wave behav-ior on the dimensionality of the sim-ulation

The behavior of shock waves in one dimension is completely dierentfrom other dimensions [102,204,97]. The reason is the lack of transversedirections. Plasticity is avoided since the atoms cannot exchange theirpositions. Dissipation and thermalization are also not possible.

We want to stress this point here since it has been a matter of longdisturbances between dierent research groups. Observations related tothe dimensional dependency show up in our simulations and have to bediscussed critically.

In one dimension non-steady waves of growing non-equilibrium materialare observed. There is no plastic ow, only damped non-linear elasticwaves. Thus non-steady proles are not shocks [217] in the proper sense.

In the simulation of Lennard-Jones crystals it has been observed [101]that no plasticity is present below up=c0 = 0 :25 for shock waves alongthe four-fold axis whereup=c0 is the piston velocity scaled with the ve-locity of sound. At this speed the shock strength is approximately equalto theoretical strength of perfect crystals and it is the shock strength re-quired to generate partial dislocations. If extended defects are present,the threshold is reduced down to aboutup=c0 = 0 :1.

The dierences between one-dimensional processes and realshocks aresummarized in Tab. 1.1. The last row indicates that "one-dimensional"does not necessarily mean one spatial dimension. Shock waves in two- orthree-dimensional samples can behave like one-dimensional if the pistonvelocity is too small, or the temperature is too low, or no equilibrationis carried out. Then the transverse coupling is missing.

18 Shock Wave Physics

Table 1.1: Comparison of one-dimensional processes and more-than-one-dimensional shock waves [100,217].

1d behavior 2d/3d behaviorshock waves non-steady elastic steady plasticshock front linear growth nitethicknesssoliton-like yes destroyed

by collisionsthermalized no yesconditions T = 0 T > 0 !!for 3d-LJ-fcc up=c0 0:25 up=c0 > 0:25

1.11 General literature about shock waves

I am not aware of a basic textbook about shock compression in solids.But there are a number of notable books and series which contain thefoundations of the shock compression physics of solids.A complete account of shock waves is given in the "Handbook ofShockWaves" by Ben-Dor et al., especially the rst volume [18] published in2001. It contains a history of shock waves, starting in 1759 and endingin 1945 [123] which is regrettable since the shock wave physics of solidsstarted more or less in 1945. The history of molecular dynamics simula-tions of solids began even later with the paper by Tsai and Beckett [211]in 1966 according to a review by Germann [69].A few important dates may be noted: In 1860 Riemann publishedhistheory of waves of nite amplitudes ( Uber die Fortpanzung ebenerLuftwellen von endlicher Schwingungsweite). Rankine [150] set up thebasic equations of shock waves in 1870, which together with Hugo-niots contributions in 1887 and 1889 [109,110] are known today as theRankine-Hugoniot equations. In 1906 Duhem proved that trueshockwaves exist only in perfect uids with discontinuous fronts according toRiemanns and Hugoniots theory. Gruneisen [80] proposed his famousequation of state pv + G(v) = ( v)e (v specic volume, e specic inter-nal energy, ( v) 6= f (T ) Gruneisen coecient, G(v) related to latticepotential) based on his own work [79] and contributions by Mie [136] in

1.11 General literature about shock waves 19

1926. In 1942 Bethe [19] calculated the stability of shock waves for anarbitrary equation of state.The scientic exchange of shock wave physics has long been hamperedby the Iron Curtain, due to the strategic relevance of shock wave studiesfor nuclear weapons. Therefore many researchers are still working in thenational labs in the US, and the same may be true for Russia.A complete account of shock waves from a Russian point of viewpub-lished originally in 1966 also exists [228].One of the best sources about shock wave physics in solids currentlyavailable is the series "High-Pressure Shock Compression of Solids" fromSpringer Verlag, edited rst by R. A. Graham and now by L. Davi sonand Y. Horie. It is part of the more extended series "High-PressureShock Compression of Condensed Matter" which includes for exampleexplosives and heterogenous materials. Also notable is thebook byJ. N. Johnson and R. Cheret collecting a number of "Classical Papersin Shock Compression Science" which are dicult to obtain.The latest development on the physics of shock waves can be found inthe biannual proceedings of the "Conference of the AmericanPhysicalSociety Topical Group on Shock Compression of Condensed Matter"published in the AIP Conference Proceedings.

1.11.1 Special literature for the present thesis

A number of lists of reviews of shock compression of solids havebeen published by Graham and Davison [76, 77, 38] with a list ofreferences from 1920 to 1979. A chronological bibliographyuntil1993 has been given by Asay and Shahinpoor [9].

The general basics of shock waves may be found in [77,76,75,83,25,34],

and shock waves in solids have been addressed especially in [156,143,75,144].

Mechanical properties, plasticity, dislocations and fragmentationare treated in [28,113,78,15].

An account on ceramic materials is found in [132,133,13],

20 Shock Wave Physics

The Hugoniot elastic limit (HEL), which determines the dyna m-ical yield stress, (HEL) and the dierent wave forms observed inshock compression are discussed in [76,75,1].

The equation of state (EOS) is the topic of [1,192,34].

Hugoniot data from many experiments are shown in [156,133,14,189,209].

Duvall and Graham [48] and Batsanov [13] treat phase transitionscaused by shock waves.

Reviews of simulations of shock waves in solids exist from Holianand co-workers [94,101,99,98,70,95] and from Wallace [217], andfrom their opponents, MacDonald and Tsai [127]. Robertson etal. [158] have addressed energetic solids.

Shock waves in granular materials are reviewed by Nesterenko[145].

Chapter 2

Simulation of Shock Waves inSolids

2.1 Shock wave simulations: State of theart

To review the history of shock waves in general would lead much toofar. The interested reader can nd a general presentation for the timeup to 1945 in [123]. A history of later times does not exist, so onedepends on the reviews mentioned in Sec.1.11. We will present onlycertain aspects related to the main topics of this thesis: the simulationof defect structures and phase transitions in inert materials and thediscussions related to steadiness or stationarity of shockwaves. Severalreviews have been written by Holian [94,99,95] on simulations. Thus wewill not deal with shock waves in liquids, gases, plasmas, and energeticmaterials, although the latter belong to the solids.In the "prehistory" of shock wave simulations only continuum mechanicsand nite element computations have been applied. Therefore no state-ments about the creation of defects or plasticity on an atomic scale werepossible. It was not even clear whether the uid case, represented by theNavier-Stokes equation, could be simulated on an atomic level [104,93]since microscopic dynamics is time-reversible and dissipative processesare lacking.The rst molecular dynamics simulations of shock waves where carriedout by Tsai and Beckett [211] in 1966. They studied one-dimensional1

samples and found no plasticity due to that restriction. Later theyextended their work to two- [210,212] and three-dimensional simulations[213,214], but without new insight. The next important step were thre e-dimensional simulations by Paskin and Dienes [147] in 1972. For therst time they obtained the linear Hugoniot relation us = c0 + sup. The

1The term one-dimensional is used for the dynamical behavior of the systemwhich indicates that the motion of the atoms was restricted t o one dimension. Thespatial dimension of the crystals was three.

22 Simulation of Shock Waves in Solids

Figure 2.1: Simulation result from Holian [ 93]. The diagonal stripes aredefect bands wrapped periodically around the simulation ce ll.

problem with these simulations was that the temperature wasset tozero, therefore a one-dimensional behavior with non-steady waves wasobserved again. The claim now was that steady shock waves could notat all exist in solids.The long-standing debate was nally resolved by Holian and Straub ina number of papers [102,103,204]. They could show that it is necessaryto equilibrate the samples, and that a certain minimal piston velocityis required to induce plasticity and to generate steady shock waves.The question on how dislocations are generated and how this leads toplasticity seemed to be solved when Holian [93] in 1988 carried out sim-ulations of shock waves along the four-fold direction in an fcc crystalequipped with Lennard-Jones potentials. He found steady shock wavesand a plasticity mode realized by stacking faults. The slipping of thematerial caused the relaxation of the uniaxial stress into ahydrody-namically compressed state. The problem with these simulations wasthe small size of the simulation cell which leads to a back-folding of thestacking faults and to a periodic array of defects (Fig.2.1). Thus it stillwas not known what really happens for weak shock waves.The next major step which conrmed the results from 1988 weretherst multi-million atom simulations by Holian and Lomdahl [ 101] in1998. Now the sample was large enough that the interaction ofthestacking faults could be neglected. At stronger shock wavesa whole net-work of independent partial dislocations and stacking faults was found(Fig. 2.2). Another important result was that the threshold of steadyshock waves could be lowered considerably by extended defects.This paper kicked o a wave of new shock wave simulation studies. The

2.1 Shock wave simulations: State of the art 23

Figure 2.2: First multi-million atom simulation of shock waves by Holia nand Lomdahl [101].

question of plasticity appeared to be solved now even for weak shockwaves and the mechanisms probably rather simple. Thus the very com-plicated results found by Germann et al. [72] came as a surprise: Theystudied for the rst time shock waves in fcc crystals along two- andthree-fold directions and obtained non-steady shock waves, oscillatorybehavior, solitary waves, martensitic transformations, and delayed plas-ticity, i.e. uplastic 6= uelastic (Fig. 2.3). A number of detailed studies fol-lowed about the defect structures in fcc crystals, for example by Hirthet al. [89], Maillet et al. [ 129], Tanguy et al. [206], and Germann etal. [73]. The last two papers could for the rst time demonstrate howdislocation loops are created and stacking faults are formed. Up to thenthere had been several models for plasticity, especially mechanisms fordislocation generation (See for example [198, 135, 219, 141], and morerecently [134,74]), some models even requiring super-sonic dislocations.These models had never been tested, but now they can be compared tosimulations. Beyond shock waves, high-speed generation ofdislocationsin fcc materials has been studied by Schitz et al. [187,188].For completeness we mention a Russian group [7,230,231,232,233,234,235]. which also contributed important results to the study of f cc crys-tals with Lennard-Jones potentials, especially to the shock-melting pro-

24 Simulation of Shock Waves in Solids

Figure 2.3: Hugoniot plot of fcc-crystals with Lennard-Jones interact ions.From Germann et al. [ 72].

cess. Ravelo et al. [152] used the constant-stress Hugoniostat to deter-mine the shock melting location in the phase diagram of the Lennard-Jones potential more precisely.Further developments include the simulation of other structures, forexample by Kadau et al. [115, 116, 117] who studied bcc with EAM-potentials for iron, Elert, and Zybin et al. [ 56, 237] who worked ondiamond structures, and the results presented in this thesis. It is nolonger possible to keep track of all new developments [26, 16, 17, 46].The latest account of the present state has been given by Holian [96].All these studies were concerned with monocrystalline samples wherenanosecond time-scales are sucient. Simulations of polycrystallinesamples, which are much closer to application, have also been attempted.But they still represent a major challenge. The results obtained even

2.1 Shock wave simulations: State of the art 25

for the largest samples with up to 24 million atoms have not yielded asteady state [70,95] since time-scales of microseconds are required.

2.1.1 The Hugoniostat

One important new development deserves to be mentioned here, al-though it has not been applied in the present thesis. The simulations ofshock waves is a dynamical process, i.e. non-equilibrium molecular dy-namics (NEMD) simulations are required. The time a simulation lastsis dictated by the size of the sample and the propagation timeof thefastest shock front. Sometimes it would be desirable to simulate smallersamples but for longer times in order to observe for example relaxationprocesses of defects or phase transitions.This suggestion can be realized with a new ensemble, the so-called Hugo-niostat which has been developed by several people [200,201,128,129].Closely related to this development is another approach by Reed etal. [153,154].The idea of the Hugoniostat is to equilibrate the sample rst and thento apply modied equations of motions in such a way that the systemmoves on a Hugoniot curveE E0 = 12 (P + P0)(V0 V ). Thus onehas an equilibrium (EMD) simulation and the simulation time and sizeof the sample are decoupled. The method can be combined with otherconstant observables, like volume or pressure and energy ortempera-ture. Versions for Gaussian [200, 201] and Nose-Hoover-type [128, 129]ensembles have been presented. The method of Reed et al. evenpermitsto simulate several phases and metastable states.Since the Hugoniostat methods are very new they still have tobe testedthoroughly. Up to now the methods have been applied in a detailedstudy of the piston velocity dependency of the defect structures gen-erated by a shock wave: A fcc crystal supplied with Lennard-Jonespotentials has been shocked along the four-fold axis [128]. It is notclear whether the defect structures obtained in this way arethe same asfor dynamical simulations [95]. The best implementation of the Hugo-niostat and a comparison of dierent versions can be found in[152].With the Hugoniostat it will be possible to solve some of the problemscaused by long-time relaxation especially in the diatomic simulations(Chap. 4) but maybe also to anneal the transformed crystals in themonatomic simulations (Chap. 5). The problem could be a modication

26 Simulation of Shock Waves in Solids

Figure 2.4: Dierent choices of the coordinate system (From [ 100]). Thepiston velocity is up , the velocity of the shock front us .

of the plasticity or transformation modes. No such diculty is presentfor amorphous materials, and therefore they supply the bestexampleto test the Hugoniostat.

2.2 Shock wave generation

Shock waves in computer simulations can be observed from dierentviewpoints. First of all there are dierent choices of the coordinatesystems. In the rst case the unshocked material is xed (Fig. 2.4,top). In the second case the shocked material is at rest (Fig.2.4, center)which permits to time-average the shocked material. In the third casethe shock front is at rest which allows to average the properties of theshock front, but it requires the a priori knowledge of the shock frontvelocity. The averaging, however, is not possible if there are severalwave fronts with dierent velocities.

2.2.1 Impact simulations

The simplest method to create shock waves is the impact method (Fig.2.5). The setup is similar to the experiment where a yer plate is

2.2 Shock wave generation 27

impact

u

pbc

us

Figure 2.5: Impact simulations. \pbc" means periodic boundary conditi ons.The velocity of the yer plate is uimpact , the speed of the shock waveus .

shot onto a sample. The yer plate has a xed kinetic energy, and isstopped after collision. Therefore it will always lead to a time-dependentbehavior of weakening shock waves. Since the shock front propagatesinto the material, will lead to position dependent damage. The methodworks well for meteorite impact or laser pulses but is not suitable tostudy defects caused by stationary shock waves. Simulations of binaryquasicrystals carried out with the symmetric impact method have beenreported in Refs. [168,166,164,167].If the yer plate is kept moving at constant speed, then the setup isequivalent to a piston compressing a sample at rest at speedup =uimpact , thereby creating a shock wave at speedus.

2.2.2 Symmetric impact simulations

Instead of shooting a thin plate onto a massive sample one canuse twoblocks of equal size. But there is still the problem of non-stationarity: Ifa xed amount of kinetic energy is attributed to each moving block theywill slow down as the energy is transformed into heat, and thepistonvelocity will change. Therefore it is better to keep the mobile blockmoving at constant speed after collision, or { in a symmetricway { tomove both blocks at constant speed (Fig.2.6). In the simulation the twoparts are moved towards each other at constant velocities up. Two

28 Simulation of Shock Waves in Solids

u pu

p

p

pbcpbc

us - u u s - up

Figure 2.6: Symmetric impact simulation: colliding blocks. Notations areas in Fig. 2.5.

shock waves are created at the central plane where the blockscollideand propagate through the compound sample at velocities (us up),where us is the required shock velocity. The advantage of this methodis that we can generate stationary states. A drawback is the movingshock front, therefore an averaging with respect to the shock front isnot possible. Furthermore, we are always simulating two shock wavesat a time which can be helpful sometimes since we can compare theresults of both shock waves, and in the case of non-crystalline materialthey need not be identical.

2.2.3 Momentum mirror

The last disadvantage of the former method can be avoided if one ofthe blocks is replaced by a momentum mirror (Fig. 2.7). When theatoms reach the mirror, their velocity component parallel to the shockdirection is inverted. Instead of keeping the mirror xed it is possiblealso to move it at any desired speed, for example such that theshockfront is stationary. One shortcoming of the mirror method is that itintroduces a perfectly rigid boundary which may cause side eects. Sucheects are clearly visible in the layering of the atoms near the mirror(See Chap.5) and in the orientation dependency of the solitary waves(See Chap.6). In general, and far from the mirror, it has been shownthat the symmetric impact method and the momentum mirror yie ldequivalent results. The shock wave velocities, for example, and the

2.2 Shock wave generation 29

pu

pbc

us p- u

mirror

Figure 2.7: Momentum mirror. Notations are as in Fig. 2.5.

defect structures are identical.

2.2.4 Other methods

In our simulations we have employed the collision and the momentummirror method. For completeness we will also mention other methodswhich have been applied frequently. In the methods presented up to nowthere is always at least one open boundary. This can be avoided in theshrinking-boundary-method, where the periodicity parallel to the shockfront is reduced continuously during simulation such that the materialin between is compressed uniaxially [95].The last method is the so-called "ramjet" method [238] which workslike a conveyer belt: at the one end of the simulation box new materialis generated and equilibrated. It moves continuously into the centralregion, where it is shocked. At the other end of the simulation box theshocked material is discarded. Usually the coordinate system is set upsuch that the shock front is stationary. Then the velocities are given inFig. 2.4, bottom. This method has rst been applied by Klimenko andDremin [226]. The advantage is that in the case of steady waves it ispossible to average the observables over a long time, and onecan getgood statistics even for rather small systems.

30 Simulation of Shock Waves in Solids

2.3 Boundary conditions

Closely related to the shock wave generation methods are theboundaryconditions. Since the shock compression leads to increasing pressure,one has to keep the borders of the simulation box xed or to applyperiodic boundary conditions. Free boundaries are not possible. In theunshocked part of the box open boundaries are also possible,but in theshocked part they would lead to non-steady waves as has been observedby chance when the simulation box was not chosen properly, oreven toan exploding simulation box.For ordered aperiodic materials it is not possible to apply periodicboundary conditions directly. In the case of quasicrystalsthe solutionis well-known: there is an irrational quotient of the two sets of linearlyindependent reciprocal lattice vectors which is replaced by a rationalnumber (Sec.3.1). The quasicrystal is transformed into a rational ap-proximant which is an ordinary crystal with a large unit cell . Thus itis possible again to identify the parallel boundaries of theunit cell andto apply ordinary boundary conditions.

2.4 Molecular dynamics simulations

All the simulations have been carried out with IMD, the I TAP M olecularD ynamics Simulation Package [202,173]. IMD is a simulation programthat supports a large number of serial and parallel computers. Paral-lelization is implemented geometrically by subdividing the simulationbox into cells and distributing them on dierent computing n odes. Theadministration is carried out by the linked-cell-method. T herefore IMDis especially suitable for simulations with short-range interactions, likepair or EAM potentials, three-body potentials and covalent bonding.Long-range interactions are only supported in the serial version. De-pending on the computer available, IMD can be run with MPI and/orOpenMP.A number of thermodynamical ensembles, like NVE, dierent kinds ofNVT, isotropic and anisotropic NPT are available. Special features in-clude modules which support for example crack, shock or heattransportsimulations.Details about the implementation, parallelization, and further develop-

2.4 Molecular dynamics simulations 31

ment of IMD can be found in [202, 173, 81, 168, 20, 184, 67, 179]. Thelatest developments are published onwww.itap.physik.uni-stuttgart.de/~imd.For the shock simulations a NVE ensemble was used. Equilibrationswere performed with the NVT-Nose-Hoover and NPT-Andersen ensem-ble, depending on the volume or pressure to be xed. At low tempera-ture and low pressure the dierences between NVT and NPT equilibra-tion are marginal.Beyond molecular dynamics it is also possible to use IMD to optimizethe potential energy of a structure. For the quenching of theshockedsamples IMD provides the microconvergence (mic) and the global con-vergence (gloc) method. In the rst case an atom is moved if its velocityvector points in the direction of a minimum, otherwise its position iskept xed and the velocity is set to zero. In the second case the globalforce in conguration space is used to gure out whether all atoms aremoved or not. If a sample is close to equilibrium, the gloc method worksmuch better than mic, especially if one tries to remove the kinetic en-ergy. For the shocked structures, however, it was necessaryto reducethe energy with the mic method rst and then to minimize it wit h thegloc method.The simulations presented in this thesis have been run on dierent com-puters lasting from single and double CPU PCs up to massivelyparallelsupercomputers like the Cray T3E.

32 Simulation of Shock Waves in Solids

Chapter 3

Structures and Potentials

3.1 Quasicrystals

This section is devoted to people who are not familiar with quasicrys-tals. Only the major dierences between crystals and quasicrystalsare presented. For a deeper introduction to quasicrystals the textbookby Janot [112] and the collection of reviews from the winter school inAlpe dHuez [88] are recommended. The latest research can be foundin the report by Trebin about the German \Schwerpunktprogra mmQuasikristalle" [207] and the Proceedings of the 8th International Con-ference on Quasicrystals [149].Quasicrystals have been discovered by Shechtman in 1982, but it tookuntil 1984 that the rst publication was accepted [ 190]. The rst sam-ples where only micrometer-sized, and so many people doubted the ex-istence of equilibrium quasicrystals. Today, however, it is possible togrow centimeter-sized perfect single quasicrystals!What is so special about quasicrystals? The starting point is their pe-culiar diraction patterns (Fig. 3.1). It has sharp Bragg peaks whichclearly indicates that quasicrystals possess long-range order. But thesymmetry in the present example is decagonal and it is well-known thatsymmetries other than two-, three-, four-, and six-fold areforbidden incrystals since they are not compatible with periodicity. Thus quasicrys-tals cannot be periodic. Together with the incommensurate crystalsthey form the group of aperiodic structures. In ordinary crystals thereis a set of symmetry related shortest distance vectors between the Braggpeaks and all other vectors are sums of it. In quasicrystals there areat least two such sets of vectors with an irrational quotient of theirlengths. As a consequence there should be Bragg peaks ever