xxx

Markus J. Buehler

Nanomechanics of hierarchical biological materials (cont’d)

Lecture 7

From nano to macro: Introduction to atomistic modeling techniques

IAP 2007

© 2007 Markus J. Buehler, CEE/MIT

Outline

1. Introduction to Mechanics of MaterialsBasic concepts of mechanics, stress and strain, deformation, strength and fractureMonday Jan 8, 09-10:30am

2. Introduction to Classical Molecular DynamicsIntroduction into the molecular dynamics simulation; numerical techniquesTuesday Jan 9, 09-10:30am

3. Mechanics of Ductile MaterialsDislocations; crystal structures; deformation of metals Tuesday Jan 16, 09-10:30am

4. The Cauchy-Born ruleCalculation of elastic properties of atomic latticesFriday Jan 19, 09-10:30am

5. Dynamic Fracture of Brittle MaterialsNonlinear elasticity in dynamic fracture, geometric confinement, interfacesWednesday Jan 17, 09-10:30am

6. Mechanics of biological materialsMonday Jan. 22, 09-10:30am

7. Introduction to The Problem SetAtomistic modeling of fracture of a nanocrystal of copper. Wednesday Jan 22, 09-10:30am

8. Size Effects in Deformation of MaterialsSize effects in deformation of materials: Is smaller stronger?Friday Jan 26, 09-10:30am

© 2007 Markus J. Buehler, CEE/MIT

Entropic change as a function of stretch

22rkbcS −=

22

23nl

b =

x-end-to-end distance

Entropic Regime

Energetic Regime

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

22rkbcS −=

Freely jointed Gaussian chain with n links and length l each (same for all chains in rubber)

22

23nl

b =where rend-to-end distance of chain

( ) ( ) ( )∑ −+−+−−=ΔbN

zyxkbS 223

222

221

2 111 λλλ

Entropic elasticity: Derivation

a

b

r1

r2

λ1r1

λ2b

λ1a

λ2r2

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

><=<⋅= 2bRMS rlnr

The length in the unstressed state is equal to the mean square length of totally free chains.

It can be shown that

>< 2br

22 lnrb ⋅>=<

22

31222

21b

lnzyx =⋅>=>=<>=<<

( ) ( ) ( )[ ]1112/ 23

22

21 −+−+−−=Δ λλλbkNS No explicit dep.

on b any more

( )323

22

212

1 −++=Δ−= λλλkTNSTU b

Entropic elasticity: Derivation

( )323

22

21 −++= λλλCU )/1)(3/( 2 λλσ −= E

6/EC =

© 2007 Markus J. Buehler, CEE/MIT

Persistence length

ξp

s

t(s) tangent slope

The length at which a filament is capable of bending significantly in independent directions, at a given temperature. This is defined by a autocorrelation function which gives the characteristic distance along the contour over which the tangent vectors t(s) become uncorrelated

ξp = l/2o x

l

y

r

θi

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Worm-like chain model

Freely-jointed rigid rods

Continuouslyflexible ropes

Worm like chain model

DNA 4-plat electron micrograph(Cozzarelli, Berkeley)

Image removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Worm-like chain model

This spring constant is only valid for small deformations from a highly convoluted molecule, with length far from its contour length

A more accurate model (without derivation) is the Worm-like chain model (WLC) that can be derived from the Kratky-Porod energy expression (see D. Boal, Ch. 2)

A numerical, approximate solution of the WLC model:

Lx <<

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−= Lx

LxkTFp

/41

/11

41

2ξ

Marko and Siggia, 1995

© 2007 Markus J. Buehler, CEE/MIT

Outline and content (Lecture 7)

Topic: Nanostructure of biological materials (proteins, molecules, composites of organic-inorganic components..); deformation mechanisms; size effects – fracture at nanoscale, adhesion

Examples: Deformation of collagen, vimentin, …: Protein mechanics

Material covered: Covalent bonding and models, chemical complexity, molecular potentials: CHARMM and DREIDING

Important lesson: Models for bonding in proteins, entropic vs. energetic elasticity; complexity of biological materials (multi-functional, nanostructured, precise arrangement, ..); small-scale fracture/adhesion: smaller is stronger

Historical perspective: AFM, single molecule mechanics; Griffith concept and adhesion strength, size effects

© 2007 Markus J. Buehler, CEE/MIT

Proteins

An important building block in biological systems are proteins

Proteins are made up of amino acids

20 amino acids carrying different side groups (R)

Amino acids linked by the amide bond via condensation

Proteins have four levels of structural organization: primary, secondary, tertiary and quaternary

© 2007 Markus J. Buehler, CEE/MIT

Protein structure

Primary structure: Sequence of amino acids

Secondary structure: Protein secondary structure refers to certain common repeating structures found in proteins. There are two types of secondary structures: alpha-helix and beta-pleatedsheet.

Tertiary structure: Tertiary structure is the full 3-dimensional folded structure of the polypeptide chain.

Quartenary Structure: Quartenarystructure is only present if there is more than one polypeptide chain. With multiple polypeptide chains, quartenary structure is their interconnections and organization.

A A S X D X S L V E V H X X

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

20 natural amino acids

Images removed due to copyright restrictions.Table of amino acid chemical structures.See similar image: http://web.mit.edu/esgbio/www/lm/proteins/aa/aminoacids.gif.

© 2007 Markus J. Buehler, CEE/MIT

Hierarchical structure of collagen

Collagen features hierarchical structure

Goal: Understand the scale-specific properties and cross-scale interactions

Macroscopic properties of collagen depend on the finer scales

Material properties are scale-dependent

(Buehler, JMR, 2006)

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Elasticity of tropocollagen molecules

Sun, 2004

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−= Lx

LxkTFp

/41

/11

41

2ξ

0-2

0

50

The force-extension curve for stretching a single type II collagen molecule.The data were fitted to Marko-Siggia entropic elasticity model. The moleculelength and persistence length of this sample is 300 and 7.6 nm, respectively.

2

Forc

e (p

N)

4

6

8

10

12

14

100 150

Extension (nm)

200 250 300 350

Experimental dataTheoretical model

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Modeling organic chemistry

Covalent bonds (directional)Electrostatic interactionsH-bondsvdW interactions

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.htmlhttp://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htm

Model for covalent bonds

Bonding between atoms described as combination of various terms, describing the angular, stretching etc. contributions

Image removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIThttp://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html

Model for covalent bonds

Courtesy of the EMBnetEducation & Training Committee.Used with permission. Images created for the CHARMMtutorial by Dr. Dmitry Kuznetsov(Swiss Institute of Bioinformatics)for the EMBnetEducation & Training committee (http://www.embnet.org)

© 2007 Markus J. Buehler, CEE/MIT

Bond Energy versus Bond length

0

100

200

300

400

0.5 1 1.5 2 2.5

Bond length, Å

Pote

nti

al Energ

y, kcal/

mol

Single Bond

Double Bond

Triple Bond

Chemical type Kbond bo

C−C 100 kcal/mole/Å2 1.5 Å

C=C 200 kcal/mole/Å2 1.3 Å

C≡C 400 kcal/mole/Å2 1.2 Å

Vbond = Kb b − bo( )2

Review: CHARMM potential

http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.htmlhttp://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htm

Different types of C-C bonding represented by different choices of b0and kb;

Need to retype when chemical environment changes

© 2007 Markus J. Buehler, CEE/MIT

http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html

Review: CHARMM potential

Nonbonding interactions

vdW (dispersive)

Coulomb (electrostatic)

H-bonding

Image removed for copyright restrictions.

See the graph on this page:

© 2007 Markus J. Buehler, CEE/MIT

DREIDING potential

Figure by MIT OCW.

Atom

H_H__HBH_bB_3B_2C_3C_RC_2C_1N_3N_RN_2N_1O_3O_RO_2O_1F_A13

0.3300.3300.5100.8800.7900.7700.7000.6700.6020.7020.6500.6150.5560.6600.6600.5600.5280.6111.047

180.0180.090.0109.471120.0109.471120.0120.0180.0106.7120.0120.0180.0104.51120.0120.0180.0180.0109.471

0.9370.8901.0400.9971.2101.2101.2101.2101.1671.3901.3731.4321.2801.3601.8601.9401.2851.330

109.47193.392.1180.0109.471109.47192.190.6180.0109.471109.47191.690.3180.090.090.090.0109.471

Si3P_3S_3ClGa3Ge3As3Se3BrIn3Sn3Sb3Te3I_NaCaFeZn

2K = 700 (kcal /mol) /A

KIJ (1) = 700 (kcal /mol) /A

n = 1Bonds

Geometric Valence Parameters for DREIDING

Valence Force Constants for DREIDING

Bond radius Bond angle, deg

Bond angle, degAtomRI,0 A

o

o

2o

2K = 1400 (kcal /mol) /An = 2D = 70 kcal /molD = 140 kcal /molD = 210 kcal /mol

o

2K = 2100 (kcal /mol) /An = 3o

K = 100 (kcal /mol) /rad2Angles

Bond radiusRI,0 A

o

© 2007 Markus J. Buehler, CEE/MIT

UFF “Universal Force Field”

• Can handle complete periodic table

• Force constants derived using general rules of element, hybridization and connectivity

Features:

• Atom types=elements

• Chemistry based rulesfor determination of force constants

Pauling-type bond order correction

Rappé et al.

© 2007 Markus J. Buehler, CEE/MIT

Common empirical force fields

Class I (experiment derived, simple form)CHARMMCHARMm (Accelrys)AMBEROPLS/AMBER/SchrödingerECEPP (free energy force field)GROMOS

Class II (more complex, derived from QM)CFF95 (Biosym/Accelrys)MM3MMFF94 (CHARMM, Macromodel…)UFF, DREIDING

http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.htmlhttp://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htmhttp://amber.scripps.edu/

Harmonic terms;Derived from vibrationalspectroscopy, gas-phase molecular structuresVery system-specific

Include anharmonic termsDerived from QM, more general

Image removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Hydrogen bondinge.g. between O and H in H2OBetween N and O in proteins…

Alpha helix and beta sheets

Images removed due to copyright restrictions.

Image removed due to copyright restrictions.

Image removed due to copyright restrictions.See: http://www.columbia.edu/cu/biology/courses/c2005/images/3levelpro.4.p.jpg

© 2007 Markus J. Buehler, CEE/MIT

Unfolding of alpha helix structure

00

4,000

8,000 00 0.2 0.4

500

1,000

1,500

Forc

e (p

N)

12,000

50 100

Strain (%)

150 200

v = 65 m/sv = 45 m/sv = 25 m/sv = 7.5 m/sv = 1 m/smodelmodel 0.1 nm/s

Figure by MIT OCW.

I

IIa

IIb

III

Figure by MIT OCW.

Source: Ackbarow, T., and M. J. Buehler. "Superelasticity, Energy Dissipation and Strain Hardening of Vimentin Coiled-coil Intermediate Filaments." Accepted for publication in: J Materials Science (in press), DOI 10.1007/s10853-007-1719-2 (2007).

© 2007 Markus J. Buehler, CEE/MIT

Unfolding of alpha helix structure

01.E-08 1.E-06 1.E-04

Pulling speed in m/s

1.E-02 1.E+00 1.E+02

500

1,000

Forc

e at

AP

in p

N

1,500

2,000

2,500

v0 = 0.161 m/s

Simulation dataExperimental dataHomogeneous rupture (theory)

Figure by MIT OCW.

Source: Ackbarow, T., and M. J. Buehler. "Superelasticity, Energy Dissipation and Strain Hardening of Vimentin Coiled-coilIntermediate Filaments." Accepted for publication in: J Materials Science (in press), DOI 10.1007/s10853-007-1719-2 (2007).

© 2007 Markus J. Buehler, CEE/MIT

Composed almost solely of protein, such as Spidroin (MaSp) I & IISemi-crystalline polymerAmorphous phase: Rubber-like chains, Gly-rich matrixCrystals: H-bonded β-sheets of poly-Ala sequences

Spider Silk

Keten and Buehler, 2007

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Unfolding of beta sheet

Titin I27 domain: Very resistant to unfolding due to parallel H-bonded strands

Keten and Buehler, 2007

Image removed due to copyright restrictions.

00

1000

2000

3000

4000

5000

6000

50 100Displacement (A)

Forc

e (p

N)

150 200 250 300

Force - Displacement Curve

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

y = 0.8068x

y = 0.4478x

y = 0.3171x

y = 0.1485x

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Displacement (Angstrom)

Forc

e (p

N)

Three-point bending test: Tropocollagen molecule

Displacement dForce Fappl

Buehler and Wong, 2007

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Three-point bending test: Tropocollagen molecule

MD: Calculate bending stiffness; consider different deformation rates

Result: Bending stiffness at zero deformation rate (extrapolation)

Yields: Persistence length – between 3 nm and 25 nm (experiment: 7 nm)Buehler and Wong, 2007

0 0.05 0.10.00E+00

2.00E-29

4.00E-29

6.00E-29

Ben

ding

Stif

fnes

s (N

m2)

8.00E-29

1.00E-28

1.20E-28

MD resultsLinear (MD results)

Deformation Rate m/secFigure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Stretching experiment: Tropocollagen molecule

0.5

Forc

e (p

N)

0

4

8

12

0.6 0.7 0.8 0.9

x = 280 nm

1

Experiment (type II TC)Experiment (type I TC)MDWLC

Reduced Extension (x/L)

Figure by MIT OCW. After Buehler and Wong.

© 2007 Markus J. Buehler, CEE/MIT

Fracture at ultra small scalesSize effects

© 2007 Markus J. Buehler, CEE/MIT

Nano-scale fracture

Failure mechanism of ultra small brittle single crystals as a function of material sizeProperties of adhesion systems as a function of material size: Is Griffith’s model for crack nucleation still valid at nanoscale?

σ

Griffith

“Macro”

h>>hcrit

σ

Griffith

“Nano”

h~hcrit

Stress

© 2007 Markus J. Buehler, CEE/MIT

• Inglis (~1910): Stress infinite close to a elliptical inclusion once shape is crack-like

“Inglis paradox”: Why does crack not extend, despite infinitely large stress at even small applied load?• Resolved by Griffith (~ 1950): Thermodynamic view of fracture

G = 2γ

“Griffith paradox”: Fracture at small length scales? Critical applied stress for fracture infinite in small (nano-)dimensions (ξ=O(nm))!

Considered here

Review: Two paradoxons of classical fracture theories

ξ

σ→∞σ→∞

σ∞→∞

σ∞

σ∞

σ∞→∞

Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004

Infinite peak stress

Infinite bulk stress

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ρσσ a

yy 21*0

© 2007 Markus J. Buehler, CEE/MIT

Thin strip geometry

Change in potential energy: Create a “relaxed” elementfrom a “strained” element, per unit crack advance

,...),( aWW PP σ=

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

)1/(21

21

2

2)1(

νσ

σεφ−

==EP

BaE

WP~

2)1( 22

)1( ξνσ −

=

Strain energy density:

Strain energy:

(plane strain)

BaV ~ξ=

σ

ε

Thin strip geometry

)1/( 2νσε−

=E

© 2007 Markus J. Buehler, CEE/MIT

BaE

WP~

2)1( 22

)1( ξνσ −

=0)2( =PW

Thin strip geometry

aBE

WWW PPP ξνσ

2)1( 22

)1()2( −−=−=

EG

2)1( 22 νξσ −

=

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Fracture of thin strip geometryTheoretical considerations

EG

2)1( 22 νξσ −

= 2γ = G GriffithE Young’s modulusν Poisson ratio, and σ Stress far ahead of the crack tip

σ

σ

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004

ξ.. size of material

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Fracture of thin strip geometryTheoretical considerations

)1(4

22 νσγξ−

=th

crE

)1(4

2νξγσ−

=E

fσ ∞ for ξ 0 Impossible: σmax=σth

Stress for spontaneous crack propagation

Length scale ξcr at σth cross-over

Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004

ξ.. size of material

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Transition from Griffith-governed failure to maximum strength of material

ξ

σf

σthTheoretical strength

Griffith

ξcr

2max

~σγξ E

cr

Breakdown of Griffith at ultra small scales

- Griffith theory breaks down below a critical length scale

- Replace Griffith concept of energy release by failure at homogeneous stress

© 2007 Markus J. Buehler, CEE/MIT

1== σε

6/10 2=r

0

20

2kr=μ 3/1=ν

φργ Δ= AbN

μ3/8=E

794.0/1 20 ≈= rAρ

4=bN 1≈Δφ

)1(4

22 νσγ−

=th

crEh

Choose E and γ such that length scale is in a regime easily accessible to MD

Atomistic model

2000 )(

21)( rrkar −+=φ

Bulk (harmonic, FCC)

Interface (LJ) “dispersive-glue interactions”

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

612

4)(rr

r σσεφ

0.5720 =k587.1≈a

3.9≈thσ

r

“repulsion”

“attraction”

φ

© 2007 Markus J. Buehler, CEE/MIT

At critical nanometer-length scale, structures become insensitive to flaws: Transition from Griffith governed failure to failure at theoretical strength, independent of presence of crack!!

Griffith-governed failure

Atomistic simulation resultsFailure at theor. strength

Atomistic simulation indicates:

(Buehler et al., MRS Proceedings, 2004; Gao, Ji, Buehler, MCB, 2004)

)1(4

22 νσγξ−

=th

crE

)1(4

2νγσ−

=h

Ef

thf σσ =

00 1 2 3 4 5

0.2

0.4

0.6

0.8

1

Stre

ss σ

0/σ c

r

1

23

hcr/h

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Stress distribution ahead of crack

0.40 0.1 0.2

x*/hcr

0.3 0.4

0.5

0.6

0.7

Stre

ss σ

yy/σ

th

0.8

0.9

1

1.1

1

(1): Griffith (2): Transition (3): Flaw tolerance

2

3

(3): Maximum Stress Independent of ξ

hcr/h = 0.72

hcr/h =1.03

hcr/h =1.36

hcr/h =2.67

hcr/h =3.33

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

( )( ) 22114

th

scr τνν+

νγ=ξ−μ

Shear loading

Keten and Buehler, 2007

Image removed due to copyright restrictions.

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Summary: Small-scale structures for strength optimization & flaw tolerance

Fracture strength is insensitive to structure size.

Fracture strength is sensitive to structural size.

There is no stress concentration at flaws. Material fails at theoretical

strength.

Material fails by stress concentration at flaws.

Material becomes insensitive to flaws.Material is sensitive to flaws.

h < hcrh > hcr

2maxσ

γEhcr ∝

(Gao et al., 2004; Gao, Ji, Buehler, MCB, 2004)

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Can this concept explain the design of biocomposites in bone?

Characteristic size: 10..100 nm

(Gao et al., 2003, 2004)

2max 1J/mGPa,100,25.0,

30==≈≈ γνσ EE

Estimate for biominerals:

022.0* ≈Ψ nm30≈crh

Image removed due to copyright restrictions.

Mineral platelet

Tension zone

High shearzones

Protein matrix

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Autumn et al., PNAS, 2002

Adhesion of Geckos

200nmImages remove due to copyright restrictions.

Courtesy of National Academy of Sciences, U.S.A. Used with permission.Source: Autumn, Kellar, Metin Sitti, Yiching A. Liang, Anne M. Peattie, Wendy R. Hansen, Simon Sponberg, Thomas W. Kenny, Ronald Fearing, Jacob N. Israelachvili, and Robert J. Full. "Evidence for van der Waals adhesion in gecko setae." PNAS 99 (2002): 12252-12256.(c) National Academy of Sciences, U.S.A.

© 2007 Markus J. Buehler, CEE/MIT

Adhesion at small length scales

• Schematic of the model used for studies of adhesion: The model represents a cylindrical Gecko spatula with radius attached to a rigid substrate.

• A circumferential crack represents flaws for example resulting from surface roughness. The parameter denotes the dimension of the crack.

-At very small length scales, nanometer design results in optimal adhesion strength, independent of flaws and shape (Gao et al., 2004)

-Since F ~ gR (JKR model), increase line length of surface by contact splitting(Arzt et al., 2003)

Strategies to increase adhesion strength

Image removed due to copyright restrictions.

Elastic

Rigid Substrate

x

y Rigid Substrate

Flaw due to surface roughness

ξ

αξ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Equivalence of adhesion and fracture problem

2Rcr

Rigid

Soft

Rigid

Soft

Similar:Cracks in homogeneous material

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Equivalence of adhesion and fracture problem

22

'8'σπ

ER

EKG crI ==

γγ Δ== 2G

Adhesion energy

Energy release rate 2

8σπ

crI RK =

σ

Soft

Rigid

2Rcr

∆γ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Theoretical considerationsAdhesion problem as fracture problem

( )αππ 12 FaaPKI = γΔ=

*2

2

EKI

2

*2

thcr

ERσγβ Δ

=

2

*

thRE

σγψ Δ

=( ))(2 2

1 απαβ F=

)1( 2* vEE −=

nmRcr 225~

Typical parameters for Gecko spatula

Function (tabulated)

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Continuum and atomistic model

Three-dimensional model

Cylindrical attachment device

Harmonic

LJ

LJ: Autumn et al. have shown dispersive interactions govern adhesion of attachment in Gecko

2000 )(

21)( rrkar −+=φ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

612

4)(rr

r σσεφ

Elastic

Rigid Substrate

x

y Rigid Substrate

Flaw due to surface roughness

ξ

αξ

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Stress close to detachment as a function of adhesion punch size

Smaller size leads to homogeneous stress distribution

RRcr /

Has major impact on adhesion strength:At small scaleno stress magnification

Figure removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Vary E and γ in scaling law

2

*8

thcr

ERσ

γπ

Δ=

• Results agree with predictions by scaling law

• Variations in Young’s modulus or γ may also lead to optimal adhesion

RRcr /

The ratio

governs adhesion strength

Figure removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Adhesion strength as a function of size

00 1 2 3 4 5

0.2

0.4

0.6

0.8

1

Stre

ss σ

0/σ c

r

1

23

hcr/h

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Optimal surface shape

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−+

+−−

−=rrrr

ERz th

11ln1ln

)1/(2 2

2νπσψ

( )⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−+

+−−

−=rrrr

ERz th

11ln1ln

)1/(2 2

2νπσψ

( )( )

( )∑∞

= ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−+

−⎟⎠⎞

⎜⎝⎛

−+++

++⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−1

2

2

12122

12122

1212

n nnlnn

rnrnlnrn

nrnln

λλλ

λλλ

λλ

( )( )

( )⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−+

−⎟⎠⎞

⎜⎝⎛

−−+−

−+⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−

− ∑∞

=12

2

1212ln2

1212ln2

1212ln

n nnn

rnrnrn

nrn

λλλ

λλλ

λλ

Single punch

Periodic array of punches

Derivation: Concept of superposition to negate the singular stress

PBCs

Concept: Shape parameter ψ

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Optimal shape predicted bycontinuum theory & shape parameter ψ

The shape function defining the surface shape change as a function of the shape parameter ψ. For ψ=1, the optimal shape is reached and stress

concentrations are predicted to disappear.

Figure removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Creating optimal surface shape in atomistic simulation

Strategy: Displace atoms held rigid to achieve smooth surface shape

"Rigid Restraint"

Figure by MIT OCW.

© 2007 Markus J. Buehler, CEE/MIT

Optimal shape

Stress distribution at varying shape

ψ

ψ=1: Optimal shape

Figure removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Robustness of adhesion

• By finding an optimal surface shape, the singular stress field vanishes.

• However, we find that this strategy does not lead to robust adhesion systems.

• For robustness, shape reduction is a more optimal way since it leads to (i) vanishing stress concentrations, and (ii) tolerance with respect to surface shape changes.

Figure removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Discussion and conclusionWe used a systematic atomistic-continuum approach to investigate brittle fracture and adhesion at ultra small scalesWe find that Griffith’s theory breaks down below a critical length scale

Nanoscale dimensions allow developing extremely strong materials and strong attachment systems: Nano is robust

Small nano-substructures lead to robust, flaw-tolerant materials. In some cases, Nature may use this principle to build strong

structural materials.

Unlike purely continuum mechanics methods, MD simulations can intrinsically handle stress concentrations (singularities) well and provide accurate descriptions of bond breaking

Atomistic based modeling will play a significant role in the future in the area of modeling nano-mechanical phenomena and linking to continuum mechanical theories as exemplified here.

© 2007 Markus J. Buehler, CEE/MIT

Fundamental length scales in nanocrystalline ductile materials

Similar considerations as for brittle materials and adhesion systems apply also to ductile materials

In particular, the deformation mechanics of nanocrystalline materials has received significant attention over the past decade

http://www.sc.doe.gov/bes/IWGN.Worldwide.Study/ch6.pdf

• Strengthening at small grain size (Hall-Petch effect)

• Weakening at even smaller grain sizes after a peak

http://me.jhu.edu/~dwarner/index_files/image003.jpg

d

Images removed due to copyright restrictions.

© 2007 Markus J. Buehler, CEE/MIT

Hall-Petch Behavior

It has been observed that the strength of polycrystalline materials increases if the grain size decreases

The Hall-Petch model explains this by considering a dislocation locking mechanism:

┴┴

┴┴d

Nucleate second source in other grain (right)

Physical picture: Higher external stress necessary to lead to large dislocation density in pileup

dY1~σ

2nd source

See, e.g. Courtney, Mechanical Behavior of Materials

© 2007 Markus J. Buehler, CEE/MIT

The strongest size: Nano is strong!

Yamakov et al., 2003, Schiotz et al., 2003

Different mechanisms have been proposed at nanoscale, including

• GB diffusion (even at low temperatures) – Wolf et al.

• GB sliding – Schiotz et al.

• GBs as sources for dislocations – van Swygenhoven, stable SF energy / unstable SF energy (shielding)

Figure removed due to copyright restrictions.See p. 15 of http://www.imprs-am.mpg.de/summerschool2003/wolf.pdf

© 2007 Markus J. Buehler, CEE/MIT

Typical simulation procedure

1. Pre-processing (define geometry, build crystal etc.)

2. Energy relaxation (minimization)

3. Annealing (equilibration at specific temperature)

4. “Actual” calculation; e.g. apply loading to crack

5. Analysis

Real challenge:Questions to ask and what to learn

F=maImage removed due to copyright restrictions.