Electronic structure calculations: methods and applications

George C. SchatzNorthwestern University

Materials Fracture and Degradation

Steven Mielke, Diego Troya, LiPeng Sun, Jeff Paci, Ted Belytschko, Sulin Zhang, Roopam Khare and George C. Schatz

Northwestern University

Also thanks to:Rod Ruoff, Horacio Espinosa– Northwestern

Peter Zapol, Orlando Auciello-ANLRoberto Car - Princeton

Using Electronic Structure Theory to Model Mechanical Properties of Nanomaterials

•Nanotubes, rods and other nanomaterials provide the simplest systems for which mechanical properties (stress/strain behavior) can be measured.

•These materials provide an excellent opportunity to learn about the influence of defects and chemical functionalization.

•They are sometimes amenable to study using electronic structure theory methods, thus providing a platform for connecting fundamental theory with experiment.

Electronic Structure Theory will be used to:

•Establish shape of stress/strain curves, and their sensitivity to nanotube structure.

•Interpret experiments, establish theoretical limits.

•Examine the role of defects and chemical functionalization on fracture behavior.

•Integrate single nanotube results with bulk results.

Using electronic structure theory to describe fracture in nanosystems is a big challenge

•Minimum size systems to model structure typically contain >100 atoms, and real systems are usually much larger.

•The quality of theory needs to be carefully considered: bonds are being broken, open-shell effects can be important, both finite cluster and periodic boundary conditions need to be considered.

•Finite temperature effects might be important, but it is impossible to do useful MD calculations with most electronic structure models. Multiple pathways to fracture are possible.

Carbon Nanotubes (defects, chemical functionalization)

Ultrananocrystalline Diamond Films (grain boundary fracture, doping effects)

Model Systems

Electronic structure methods

DFT (PBE): SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms): a self-consistent DFT program. Highest accuracy of the methods we have studied, but computational effort is a serious problem.

PM3: Semiempirical method which is reasonably close to DFT for carbon-based nanostructures. Largely restricted to finite cluster calculations.

MSINDO: Semiempirical method that can be used both for clusters and for periodic boundary conditions. For carbon-based nanostructures it is less accurate than PM3.

SCC-DFTB (density functional-based tight-binding with self-consistent charges): an approximate DFT method.

Other methods

For systems with carbon and hydrogens, can use Tersoff-Brenner (reactive bond-order) potential for MM calculations.

Mixed MM/CM studies (and ultimately QM/MM/CM) to extend from a few atoms to a continuum description.

Carbon nanotube fracture

Carbon nanotubes are likely the strongest known materials

Their superior mechanical properties (resisting more than 1 order of magnitude largertensile loads than reinforced steel) and their lightweight nature (six times lessthan steel) make them perfect candidates for reinforcement materials in nanocomposites

Fracture of Carbon Nanotubes MM calculations with empirical force fields

T. Belytschko, S. P. Xiao, G. C. Schatz and R. Ruoff, Phys. Rev. B 65, 235430/1-/8 (2002).

(Multiwall tubes)

(single wall tube results)

Carbon nanotube fracture

l0

l/2l/2

strain= l/ l0

Determination of stress vs strain curve for [5,5] tube with Stone-Wales defect (170 carbon atoms)

Strain = 0.255 (just before fracture)

0 0.05 0.1 0.15 0.2 0.25strain

0

2

4

6

8

E-E

0 / a.

u.

PM3 results:

Strain = 0.26 (just after fracture)

0 0.05 0.1 0.15 0.2 0.25strain

0

2

4

6

8

E-E

0 / a.

u.

[5,5] tube [10,0] tube

Stress-Strain Curves: undefected tubes

One and Two-Atom Vacancies

Fracture for one/two atom vacancies

Two-atom vacancy (sym) [5,5] tube

One-atom vacancy (asym), [10,0] tube

Fracture for Large Hole DefectsHole Defect

Slit Defect

MM results

Chem vap depArc discharge

Undefected

CNTs

have fracture stress >100 GPa

Measurements of near-ultimate strength for multiwalled carbon nanotubes and irradiation-induced

crosslinking improvements,Bei Peng, Mark Locascio, Peter Zapol, Shuyou Li, Steven L. Mielke, George

C. Schatz and Horacio D. Espinosa, Nature Nanotech, 3, 626-631 (2008)

Stress-strain results show fracture stress in excess of 100 GPa

Calculations shows significant load transfer in the cross-linked tube

Polymer/Carbon Nanotube Composites

Ramanathan, T.; Liu, H.; Brinson, L. C..J. Polymer Sci B (2005), 43(17), 2269-2279.

Graphite and Graphite Oxide (GO)

Graphite Oxide d = 0.71 nm

Graphite

B. Brodie, 1855

C:O:H is 2:1:0.2

Graphene-Based Composites

• TEGO - Thermally exfoliated GO: 30% of carbon is lost as CO/CO2 during heating to 1500K

TEGO Properties:•

Individual sheets (2600 m2/g)•

Readily dispersed in polymers•

Retain inherent mechanical, thermal, electrical properties of graphene

•

Dramatically improved composite properties

GO

TEGO

TEGO Nanocomposites

Remarkable increase in Tg

(30C) with 0.05% loading of TEGO!

Thin sheets wrinkled in situ

interaction with polymer

(Cate Brinson, Northwestern)

TEGO-PMMA Mechanical PropertiesN

orm

aliz

ed v

alue

sPMMA/1wt% Nanoinclusion

Ramanathan, T.; Abdala, A. A.; Stankovich, S.; Dikin, D. A.; Herrera-Alonso, M.; Piner, R. D.; Adamson, D. H.; Schniepp, H. C.; Chen, X.; Ruoff, R. S.; Nguyen, S. T.; Aksay, I. A.; Prud'Homme, R. K.; Brinson, L. C.. Functionalized graphene sheets for polymer nanocomposites. Nature Nanotechnology (2008), 3(6), 327- 331.

PMMA values:E-2.1 GPa, Ultimate strength -

70 MPaTg

–

105C, Thermal degradation temperature -

295C

(fracture stress)

TEGO/PMMAa-SWNT/PMMA

Dekany

and coworkers, Chem. Mater.18, 2740 (2006)

Structure of Graphite Oxide?(1939)

(1946)

(1969)

(1996)

(1998)

Lerf, Klinowski (1998) Model Dekany (2006) Model

1. Trans linked cyclohexane chairs that is functionalized with tertiary OH, 1,3-ether, ketone, quinone and phenol

2. Ribbons of aromatic rings

1. Aliphatic six-membered rings containing epoxides and hydroxides. Ketones and other C=O bonds are on the edges. No 1,3 ethers.

2. Aromatic regions that give rise to a nearly flat carbon grid.

1.

What is the structure of GO, and how do defects in graphite get propagated into GO?

2.

What are the mechanical properties of GO and TEGO?

3.

What does this teach us about graphite oxidation?

Basic Questions

Monte Carlo-based simulations of graphite oxide formation: Introduction

• Add OH and epoxide groups to graphene.

• Add these functional groups to both sides of the basal plane.

• Build sheets with experimentally-observed stoichiometry (C10 O5 H2 ).

• Use cluster- and PBC- models.

• SCC-DFTB, PBE(DZP) Harris(DZP), simulations.

A graphene sheet.

The algorithm• 1) Add two OH and three epoxide

groups to the basal plane composed of, e.g., 128 carbon atoms with PBCs. Locations chosen at random.

• 2) Geometry-optimize structure, calculate energy, using Metropolis MC to accept or reject structures for further functionalization.

• Steps 1) and 2) are repeated N times.

• Result is N sheets of partially- oxidized GO. A partially-oxidized sheet of GO.

Paci, Jeffrey T.; Belytschko, Ted; Schatz, George C. Journal of Physical Chemistry C (2007), 111(49), 18099-18111.

Final results: C2 OH0.2 stoichiometry

undefectedgraphene

Defect = line of epoxides

Some aromaticity remains for low oxygen/carbon ratio

Requirements: • Planar, cyclic, one p-

orbital/atom perpendicular to the plane of the ring.

• 4n+2 pi electrons, where n is an integer.

Shown here: A carbon to oxygen ratio of 2:0.77.

Aromatic carbons are shaded purple.

Final results

1. Interplanar spacing: 5.8Å (calc) 6.0Å (expt) (vs. graphite = 3.4Å)2. Chemical species (out of 64 oxygens):

Found at edges

Unimportant

Unimportant

3. Hole formation: Does not occur in pristine graphene. Previously existing holes can expand during growth.

O

Paci, Jeffrey T.; Belytschko, Ted; Schatz, George C. Journal of Physical Chemistry C (2007), 111(49), 18099-18111.

Heating GO to 1323K leads to gaseous CO, as well as a mechanism for making

holesOH O

O

O

+ COOH

O

O

Fracture studies: Notched graphene sheet

-593.5

-593.0

-592.5

-592.0

-591.5

-591.0

-590.5

0.00 0.05 0.10 0.15 0.20

Ene

rgy

(Eh)

Strain

Spacecraft surfaces made of polymeric hydrocarbons erode in low Earth orbit (LEO) (~200-700 km).

J. W. Connell, High Perform Polym 12, 43 (2000)

Polymer degradation in LEO

Most abundant species in atmosphere as function of altitude

Minton, in Chemical Dynamics in Extreme Environments, (World Scientific, Singapore, 2001), pp 420.

Roble, in The Upper Mesosphere and Lower Thermosphere:A Review of Experiment and Theory, Geophysical Monograph 87, pp 1 – 21, 1995.

Materials erosion in low earth orbit

•Materials (polymers) on the RAM surfaces of satellites in low earth orbit are degraded by 5eV O(3P) as well as other neutrals, ions, UV, electrons and dust.

•Many mechanisms for erosion have been discussed including intersystem crossing, collision induced dissociation and ion-surface reactions.

O + graphite

Time of Flight / s0 500 1000 1500 2000 2500

-100

-50

0

50

100

150

200

250

N(t)

/ ar

b. u

nits

0

100

200

300

400

500

0

200

400

600

800

J. Zhang, T. K. Minton, High Performance Polymers (2001), 13(3), S467-S481. K. T. Nicholson, T. K. Minton, S. J. Sibener, J Phys Chem B (2005), 109(17), 8476-8480.

CO CO

CO2

Configuration-biased Monte Carlo studies of graphite oxide

Jeffrey T. Paci, Ted Belytschko, George C. Schatz, J. Phys. Chem. C, 111, 18099-111 (2007)

O + graphite simulationsPaci, Jeffrey T.; Upadhyaya, Hari P.; Zhang, Jianming; Schatz, George C.; Minton, Timothy K. Theoretical and Experimental Studies of the Reactions between Hyperthermal O(3P) and Graphite: Graphene-Based Direct Dynamics and Beam-Surface Scattering Approaches. Journal of Physical Chemistry A (2009), 113(16), 4677-4685.

Summary• Molecular dynamics (molecular mechanics)

with quantum forces provides the most accurate approach for describing structures and bond breaking.

• System size (few hundred atoms) and time scales (<10 ps) are the primary limitations.

• Fracture properties of carbon nanotubes and other nanomaterials.

• Structure of graphite oxide.• Degradation of graphite in LEO.