From nano to macro: Introduction to atomistic
modeling techniques
IAP 2007
xxx
Nanomechanics of hierarchical biological
materials (cont’d)
Lecture 7
Markus J. Buehler
Outline
1.
2.
3.
4.
5.
6.
7.
8.
Introduction to Mechanics of Materials
Basic concepts of mechanics, stress and strain, deformation, strength and
fracture
Monday Jan 8, 09-10:30am
Introduction to Classical Molecular Dynamics
Introduction into the molecular dynamics simulation; numerical techniques
Tuesday Jan 9, 09-10:30am
Mechanics of Ductile Materials
Dislocations; crystal structures; deformation of metals
Tuesday Jan 16, 09-10:30am
The Cauchy-Born rule
Calculation of elastic properties of atomic lattices
Friday Jan 19, 09-10:30am
Dynamic Fracture of Brittle Materials
Nonlinear elasticity in dynamic fracture, geometric confinement, interfaces
Wednesday Jan 17, 09-10:30am
Mechanics of biological materials
Monday Jan. 22, 09-10:30am
Introduction to The Problem Set
Atomistic modeling of fracture of a nanocrystal of copper.
Wednesday Jan 22, 09-10:30am
Size Effects in Deformation of Materials
Size 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
x-end-to-end distance
Entropic Regime
Energetic Regime
Figure by MIT OCW.
S = c − kb r
3
2
b =
2nl 2
2 2
© 2007 Markus J. Buehler, CEE/MIT
Entropic elasticity: Derivation
Freely jointed Gaussian chain with n links and length l each
(same for all chains in rubber)
end-to-end
3
2
r distance of
S = c − kb 2 r 2
where b =
2
2nl
chain
(
)
(
)
(
)
ΔS = − kb 2 ∑ λ12 − 1 x 2 + λ22 − 1 y 2 + λ32 − 1 z 2
Nb
r1
λ 1r 1
b
r2
a
Figure by MIT OCW.
λ 2b
λ 2r 2
λ 1a
© 2007 Markus J. Buehler, CEE/MIT
Entropic elasticity: Derivation
The length < rb > in the unstressed state is equal to the mean
square length of totally free chains.
2
It can be shown that
rRMS = n ⋅ l =< < rb2 >
< r >= n ⋅ l
2
b
2
1
< x >=< y >=< z >= n ⋅ l = 2
2b
2
[(
2
) (
2
2
1
3
) (
)]
ΔS = −kN b / 2 λ − 1 + λ − 1 + λ − 1
2
1
2
2
2
3
U = −TΔS = N bkT (λ + λ + λ − 3)
1
2
(
2
1
2
2
2
3
U = C λ +λ +λ −3
2
1
2
2
2
3
)
No explicit dep.
on b any more
C = E/6
σ = ( E / 3)(λ2 − 1 / λ )
© 2007 Markus J. Buehler, CEE/MIT
Persistence length
ξp
t(s) tangent slope
y
l
θi
r
o
Figure by MIT OCW.
x
ξp = l/2
s
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
© 2007 Markus J. Buehler, CEE/MIT
Worm-like chain model
Freely-jointed rigid
rods
Image removed due to copyright restrictions.
DNA 4-plat electron micrograph
(Cozzarelli, Berkeley)
Continuously
flexible ropes
Worm like chain model
© 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
x << L
„
A more accurate model (without derivation) is the Worm-like
chain model (WLC) that can be derived from the KratkyPorod energy expression (see D. Boal, Ch. 2)
„
A numerical, approximate solution of the WLC model:
⎞
1
1
kT ⎛ 1
⎜⎜
F=
− + x / L ⎟⎟
2
ξ p ⎝ 4 (1 − x / L ) 4
⎠
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-pleated
sheet.
„
Tertiary structure: Tertiary structure is the
full 3-dimensional folded structure of the
polypeptide chain.
„
Quartenary Structure: Quartenary
structure 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
Images removed due to copyright restrictions.
Goal: Understand the
scale-specific
properties and crossscale interactions
Macroscopic
properties of collagen
depend on the finer
scales
Material properties
are scale-dependent
(Buehler, JMR, 2006)
© 2007 Markus J. Buehler, CEE/MIT
Elasticity of tropocollagen molecules
⎞
1
1
kT ⎛ 1
⎜⎜
F=
− + x / L ⎟⎟
2
ξ p ⎝ 4 (1 − x / L ) 4
⎠
14
12
Experimental data
Theoretical model
Force (pN)
10
8
6
4
2
0
-2
0
50
100
150
200
250
300
350
Extension (nm)
The force-extension curve for stretching a single type II collagen molecule.
The data were fitted to Marko-Siggia entropic elasticity model. The molecule
length and persistence length of this sample is 300 and 7.6 nm, respectively.
Figure by MIT OCW.
Sun, 2004
© 2007 Markus J. Buehler, CEE/MIT
Modeling organic chemistry
Images removed due to copyright restrictions.
Covalent bonds (directional)
Electrostatic interactions
H-bonds
vdW interactions
© 2007 Markus J. Buehler, CEE/MIT
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.
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
http://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htm
© 2007 Markus J. Buehler, CEE/MIT
Model for covalent bonds
Courtesy of the EMBnetEducation & Training Committee.
Used with permission. Images created for the CHARMM
tutorial by Dr. Dmitry Kuznetsov(Swiss Institute of Bioinformatics)
for the EMBnetEducation & Training committee (http://www.embnet.org)
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
© 2007 Markus J. Buehler, CEE/MIT
Review: CHARMM potential
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
Different types of C-C
bonding represented by
different choices of b0
and kb;
Bond Energy versus Bond length
Need to retype when
chemical environment
changes
Potential Energy, kcal/mol
400
300
Single Bond
Double Bond
200
Triple Bond
Vbond = K b (b − bo )
2
100
0
0.5
1
1.5
2
2.5
Bond length, Å
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
http://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htm
© 2007 Markus J. Buehler, CEE/MIT
Review: CHARMM potential
Image removed for copyright restrictions.
See the graph on this page:
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
Nonbonding interactions
vdW (dispersive)
Coulomb (electrostatic)
H-bonding
© 2007 Markus J. Buehler, CEE/MIT
DREIDING potential
Geometric Valence Parameters for DREIDING
Atom
H_
H__ HB
H_ b
B_ 3
B_ 2
C_ 3
C_ R
C_ 2
C_ 1
N_ 3
N_ R
N_ 2
N_ 1
O_ 3
O_ R
O_ 2
O_ 1
F_
A13
Bond radius
o
RI0, A
Bond
angle, deg
0.330
0.330
0.510
0.880
0.790
0.770
0.700
0.670
0.602
0.702
0.650
0.615
0.556
0.660
0.660
0.560
0.528
0.611
1.047
180.0
180.0
90.0
109.471
120.0
109.471
120.0
120.0
180.0
106.7
120.0
120.0
180.0
104.51
120.0
120.0
180.0
180.0
109.471
Atom
Bond radius
o
RI0, A
Bond
angle, deg
Si3
P_ 3
S_ 3
Cl
Ga3
Ge3
As3
Se3
Br
In3
Sn3
Sb3
Te3
I_
Na
Ca
Fe
Zn
0.937
0.890
1.040
0.997
1.210
1.210
1.210
1.210
1.167
1.390
1.373
1.432
1.280
1.360
1.860
1.940
1.285
1.330
109.471
93.3
92.1
180.0
109.471
109.471
92.1
90.6
180.0
109.471
109.471
91.6
90.3
180.0
90.0
90.0
90.0
109.471
o
KIJ (1) = 700 (kcal / mol) /A2
Valence Force Constants for DREIDING
Bonds
n=1
K = 700 (kcal / mol) /A 2
n=2
K = 1400 (kcal / mol) /A
o
o
2
o
n=3
K = 2100 (kcal / mol) /A2
Angles
K = 100 (kcal / mol) /rad2
D = 70 kcal / mol
D = 140 kcal / mol
D = 210 kcal / mol
Figure by MIT OCW.
© 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
Pauling-type bond order correction
• Chemistry based rules
for determination of
force constants
Rappé et al.
© 2007 Markus J. Buehler, CEE/MIT
Common empirical force fields
Class I (experiment derived, simple form)
„ CHARMM
„ CHARMm (Accelrys)
„ AMBER
„ OPLS/AMBER/Schrödinger
„ ECEPP (free energy force field)
„ GROMOS
Harmonic terms;
Derived from
vibrational
spectroscopy, gasphase molecular
structures
Very system-specific
Class II (more complex, derived from QM)
„ CFF95 (Biosym/Accelrys)
„ MM3
„ MMFF94 (CHARMM, Macromodel…)
„ UFF, DREIDING
Include anharmonic terms
Derived from QM, more
general
Image removed due to copyright restrictions.
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
http://www.pharmacy.umaryland.edu/faculty/amackere/force_fields.htm
http://amber.scripps.edu/
© 2007 Markus J. Buehler, CEE/MIT
Alpha helix and beta sheets
Hydrogen bonding
e.g. between O and H in H2O
Between N and O in proteins…
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
I
IIb
IIa
III
Figure by MIT OCW.
1,500
Force (pN)
12,000
1,000
500
0
8,000
0
0.2
v = 65 m/s
v = 45 m/s
v = 25 m/s
v = 7.5 m/s
v = 1 m/s
model
model 0.1 nm/s
0.4
4,000
0
0
50
100
150
200
Strain (%)
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
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).
Unfolding of alpha helix structure
Simulation data
Force at AP in pN
2,500
Experimental data
Homogeneous rupture (theory)
2,000
1,500
1,000
500
0
1.E-08
v0 = 0.161 m/s
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
Pulling speed in m/s
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
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).
Spider Silk
„
„
„
„
Composed almost solely of protein, such
as Spidroin (MaSp) I & II
Semi-crystalline polymer
Amorphous phase: Rubber-like chains,
Gly-rich matrix
Crystals: H-bonded β-sheets of poly-Ala
sequences
Images removed due to copyright restrictions.
Keten and Buehler, 2007
© 2007 Markus J. Buehler, CEE/MIT
Unfolding of beta sheet
Titin I27 domain: Very
resistant to unfolding
due to parallel Hbonded strands
Force - Displacement Curve
6000
Force (pN)
5000
4000
3000
2000
1000
0
0
50
100
150
200
250
300
Displacement (A)
Figure by MIT OCW.
Keten and Buehler, 2007
Image removed due to copyright restrictions.
© 2007 Markus J. Buehler, CEE/MIT
Three-point bending test:
Tropocollagen molecule
3.5
y = 0.8068x
3
Force (pN)
2.5
2
Figure by MIT OCW.
Displacement d
Force Fappl
1.5
y = 0.4478x
y = 0.3171x
1
y = 0.1485x
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Displacement (Angstrom)
Buehler and Wong, 2007
© 2007 Markus J. Buehler, CEE/MIT
Three-point bending test:
Tropocollagen molecule
Bending Stiffness (Nm2)
1.20E-28
1.00E-28
8.00E-29
6.00E-29
4.00E-29
MD results
Linear (MD results)
2.00E-29
0.00E+00
0
0.05
Deformation Rate m/sec
0.1
Figure by MIT OCW.
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
© 2007 Markus J. Buehler, CEE/MIT
Stretching experiment: Tropocollagen molecule
Experiment (type II TC)
Experiment (type I TC)
MD
WLC
Force (pN)
12
8
4
0
x = 280 nm
0.5
0.6
0.7
0.8
0.9
1
Reduced Extension (x/L)
Figure by MIT OCW. After Buehler and Wong.
© 2007 Markus J. Buehler, CEE/MIT
Fracture at ultra small scales
Size effects
© 2007 Markus J. Buehler, CEE/MIT
Nano-scale fracture
„
„
Failure mechanism of ultra small brittle single crystals as a
function of material size
Properties of adhesion systems as a function of material size:
Is Griffith’s model for crack nucleation still valid at nanoscale?
“Nano”
σ
Stress
σ
“Macro”
h>>h
h~h
critcrit
Griffith
© 2007 Markus J. Buehler, CEE/MIT
Review: Two paradoxons of classical
fracture theories
• 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
Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004
Infinite peak stress
σ∞
σ→∞ σ→∞
σ∞
σ yy
⎛
a⎞
⎜
⎟
= σ ⎜1 + 2
ρ ⎟⎠
⎝
*
0
σ∞→∞
ξ
σ∞→∞
Infinite bulk stress
© 2007 Markus J. Buehler, CEE/MIT
Thin strip geometry
"Relaxed"
element
σ
"Strained"
element
a~
δa
a~
ξ
a
σ
Figure by MIT OCW.
Change in potential energy: Create a “relaxed” element
from a “strained” element, per unit crack advance
WP = WP (σ , a,...)
© 2007 Markus J. Buehler, CEE/MIT
Thin strip geometry
Strain energy density:
σ
ε=
σ
E /(1 −ν 2 )
σ
1
1
= σε =
2
2 E /(1 −ν 2 )
2
φP(1)
Strain energy:
WP(1)
V = ξa~B
σ 2 (1 −ν 2 ) ~
=
ξa B
2E
(plane strain)
ε
© 2007 Markus J. Buehler, CEE/MIT
Thin strip geometry
"Relaxed"
element
"Strained"
element
σ
a~
δa
a~
ξ
a
σ
Figure by MIT OCW.
WP(1)
WP( 2 ) = 0
WP = W
( 2)
P
−W
(1)
P
=−
σ 2 (1 − ν 2 )
2E
ξaB
σ 2 (1 −ν 2 ) ~
=
ξa B
2E
G=
σ 2ξ (1 −ν 2 )
2E
© 2007 Markus J. Buehler, CEE/MIT
Fracture of thin strip geometry
Theoretical considerations
G=
σ 2ξ (1 −ν 2 )
E Young’s modulus
2γ = G
2E
ν
σ
Griffith
Poisson ratio, and
Stress far ahead of the crack tip
σ
"Relaxed"
element
"Strained"
element
σ
a~
δa
a~
ξ
a
σ
σ
Figure by MIT OCW.
ξ.. size of material
Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004
© 2007 Markus J. Buehler, CEE/MIT
Fracture of thin strip geometry
Theoretical considerations
Stress for spontaneous crack
propagation
4γE
σf =
ξ (1 −ν 2 )
σ Æ ∞ for ξ Æ 0
Impossible: σmax=σth
"Relaxed"
element
σ
Length scale ξcr at σth cross-over
ξ cr =
4γE
σ th2 (1 −ν 2 )
"Strained"
element
a~
δa
a~
ξ
a
ξ.. size of material
σ
Figure by MIT OCW.
Buehler et al., MRS Proceedings, 2004 & MSMSE, 2005; Gao, Ji, Buehler, MCB, 2004
© 2007 Markus J. Buehler, CEE/MIT
Breakdown of Griffith at ultra small scales
Theoretical strength
σf
σth
ξ cr
Griffith
ξcr
γE
~ 2
σ max
ξ
Transition from Griffith-governed failure to maximum strength of material
- 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
Atomistic model
Bulk (harmonic, FCC)
1
φ (r ) = a0 + k 0 ( r − r0 ) 2
2
r02
μ = k0
2
r0 = 21 / 6
a ≈ 1.587
E = 8 / 3μ
k 0 = 572 .0
ν = 1/ 3
4γE
hcr = 2
σ th (1 −ν 2 )
Interface (LJ) “dispersive-glue interactions”
⎛ ⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎞
φ (r ) = 4ε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟
⎜⎝ r ⎠
⎝ r ⎠ ⎟⎠
⎝
γ = N b ρ A Δφ
ρ A = 1 / r02 ≈ 0.794
Nb = 4
Δφ ≈ 1
ε =σ =1
σ th ≈ 9.3
φ
“repulsion”
“attraction”
r
Choose E and γ such that length scale is in a regime easily accessible to MD
© 2007 Markus J. Buehler, CEE/MIT
Atomistic simulation results
σ f = σ th Failure at theor. strength
1
σf =
Stress σ0 /σcr
4γE
0.6
0.4
1
0.2
h(1 −ν 2 )
3
2
0.8
0
0
1
2
3
4
5
hcr /h
Griffith-governed failure
ξ cr =
Atomistic simulation indicates:
4γE
σ th2 (1 −ν 2 )
Figure by MIT OCW.
¾ 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!!
(Buehler et al., MRS Proceedings, 2004; Gao, Ji, Buehler, MCB, 2004)
© 2007 Markus J. Buehler, CEE/MIT
Stress distribution ahead of crack
(3): Maximum Stress Independent of ξ
1.1
3
1
hcr /h = 0.72
Stress σyy /σth
0.9
0.8
hcr /h =1.03
2
hcr /h =1.36
0.7
hcr /h =2.67
0.6
hcr /h =3.33
0.5
0.4
1
0
0.1
0.2
x*/hcr
0.3
0.4
(1): Griffith (2): Transition (3): Flaw tolerance
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
Shear loading
"Relaxed"
element
σ
4γ s μ ν
ξ cr =
(1+ ν )(1 − 2ν )τ th2
"Strained"
element
a~
δa
a~
ξ
a
σ
Image removed due to copyright restrictions.
Figure by MIT OCW.
Keten and Buehler, 2007
© 2007 Markus J. Buehler, CEE/MIT
Summary: Small-scale structures for
strength optimization & flaw tolerance
γE
h cr ∝ 2
σ max
"Relaxed"
element
σ
"Strained"
element
a~
δa
ξ
a~
a
σ
Figure by MIT OCW.
h > hcr
h < hcr
Material is sensitive to flaws.
Material becomes insensitive to
flaws.
Material fails by stress concentration
at flaws.
There is no stress concentration at
flaws. Material fails at theoretical
strength.
Fracture strength is sensitive to
structural size.
Fracture strength is insensitive to
structure size.
(Gao et al., 2004; Gao, Ji, Buehler, MCB, 2004)
© 2007 Markus J. Buehler, CEE/MIT
Can this concept explain the design of
biocomposites in bone?
Characteristic size: 10..100 nm
Image removed due to copyright restrictions.
Mineral platelet
Tension
zone
High shear
zones
Protein matrix
Estimate for biominerals:
σ max
Figure by MIT OCW.
E
≈
, ν ≈ 0 . 25 , E = 100 GPa, γ = 1J/m 2
30
Ψ * ≈ 0 . 022 h cr ≈ 30 nm
(Gao et al., 2003, 2004)
© 2007 Markus J. Buehler, CEE/MIT
Adhesion of Geckos
Autumn et al., PNAS, 2002
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.
200nm
Images remove due to copyright restrictions.
© 2007 Markus J. Buehler, CEE/MIT
Adhesion at small length scales
Elastic
ξ
αξ
y
Rigid Substrate
Flaw due to
surface roughness
Image removed due to copyright restrictions.
Rigid Substrate
x
Figure by MIT OCW.
Strategies to increase adhesion strength
-Since F ~ gR (JKR model), increase line length
of surface by contact splitting
(Arzt et al., 2003)
-At very small length scales, nanometer
design results in optimal adhesion strength,
independent of flaws and shape
(Gao et al., 2004)
• 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.
© 2007 Markus J. Buehler, CEE/MIT
Equivalence of adhesion and
fracture problem
2Rcr
Rigid
Rigid
Soft
Soft
Similar:
Cracks in homogeneous material
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
Equivalence of adhesion and
fracture problem
s
2Rcr
Energy release rate
Rigid
Dg
Soft
K I2 π Rcr 2
G=
=
σ
E' 8 E'
KI =
π
8
Rcrσ 2
G = 2γ = Δγ
Adhesion energy
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
Theoretical considerations
Adhesion problem as fracture problem
Function (tabulated)
P
K I = 2 πa F1 (α )
πa
ψ =
ΔγE *
Rσ th2
β = 2 (παF12 (α ) )
E * = E (1 − v 2 )
Rcr = β 2
Figure by MIT OCW.
K I2
= Δγ
2E *
ΔγE *
σ th2
Rcr ~ 225 nm
Typical parameters for Gecko spatula
© 2007 Markus J. Buehler, CEE/MIT
Continuum and atomistic model
Three-dimensional model
Cylindrical attachment device
1
2
φ ( r ) = a0 + k 0 ( r − r0 ) 2
Harmonic
Elastic
ξ
⎛ ⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎞
φ (r ) = 4ε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟
⎜⎝ r ⎠
⎝ r ⎠ ⎟⎠
⎝
LJ
αξ
y
Rigid Substrate
Flaw due to
surface roughness
Rigid Substrate
x
Figure by MIT OCW.
LJ: Autumn et al. have shown dispersive interactions govern
adhesion of attachment in Gecko
© 2007 Markus J. Buehler, CEE/MIT
Stress close to detachment as a function
of adhesion punch size
Rcr / R
Figure removed due to copyright restrictions.
Has major
impact
on adhesion
strength:
At small scale
no stress
magnification
Smaller size leads to homogeneous stress distribution
© 2007 Markus J. Buehler, CEE/MIT
Vary E and γ in scaling law
Rcr =
8 E * Δγ
π σ th2
The ratio
Figure removed due to copyright restrictions.
Rcr / R
governs adhesion strength
• Results agree with predictions by scaling law
• Variations in Young’s modulus or γ may also lead to optimal adhesion
© 2007 Markus J. Buehler, CEE/MIT
Adhesion strength as a function of size
1
0.8
Stress σ0 /σcr
3
2
0.6
0.4
1
0.2
0
0
1
2
3
4
5
hcr /h
Figure by MIT OCW.
© 2007 Markus J. Buehler, CEE/MIT
Optimal surface shape
Single punch
z = −ψ
2σ th R ⎡
⎛ 1 + r ⎞⎤ Concept:
2
−
r
+
r
ln
1
ln
⎜
⎟⎥
−
r
1
πE /(1 −ν 2 ) ⎢⎣
⎝
⎠⎦ Shape parameter ψ
(
)
Periodic array of punches
z = −ψ
2σ th R ⎧⎡
⎛ 1 + r ⎞⎤
2
r
r
ln
1
ln
−
+
⎜
⎟⎥
⎨
r
1
−
πE /(1 −ν 2 ) ⎩⎢⎣
⎝
⎠⎦
(
)
Images removed due to copyright restrictions.
⎡ ⎛ (2nλ + r )2 − 1 ⎞
⎛ 2nλ + 1 ⎞⎤ PBCs
⎛ 2 nλ + r + 1 ⎞
⎟
(
)
− ∑ ⎢ln⎜⎜
+
+
−
2
n
λ
r
ln
2
n
λ
ln
⎜
⎟
⎜
⎟⎥
2
⎟
+
−
−
2
n
λ
r
1
2
n
λ
1
(
)
−
2
n
λ
1
⎝
⎠
⎝
⎠⎦⎥
n =1 ⎣
⎢ ⎝
⎠
∞
⎡ ⎛ (2nλ − r )2 − 1 ⎞
⎛ 2nλ − r + 1 ⎞
⎛ 2nλ + 1 ⎞⎤ ⎫⎪
⎜
⎟
+ (2nλ − r ) ln⎜
− ∑ ⎢ln⎜
⎟ − 2nλ ln⎜
⎟⎥ ⎬
2
⎟
−
−
2
n
λ
r
1
2
n
λ
1
−
⎝
⎠
⎝
⎠⎥⎦ ⎪⎭
n =1 ⎢
⎣ ⎝ (2nλ ) − 1 ⎠
∞
Derivation: Concept of superposition to negate the
singular stress
© 2007 Markus J. Buehler, CEE/MIT
Optimal shape predicted by
continuum theory & shape parameter ψ
Figure removed due to copyright restrictions.
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.
© 2007 Markus J. Buehler, CEE/MIT
Creating optimal surface shape in
atomistic simulation
"Rigid
Restraint"
Figure by MIT OCW.
Strategy: Displace atoms held rigid to achieve smooth surface shape
© 2007 Markus J. Buehler, CEE/MIT
Stress distribution at varying shape
Optimal shape
Figure removed due to copyright restrictions.
ψ
ψ=1: Optimal shape
© 2007 Markus J. Buehler, CEE/MIT
Robustness of adhesion
Figure removed due to copyright restrictions.
• 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.
© 2007 Markus J. Buehler, CEE/MIT
Discussion and conclusion
„
„
„
We used a systematic atomistic-continuum approach to investigate brittle
fracture and adhesion at ultra small scales
We 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
• Strengthening at small grain size (HallPetch effect)
• Weakening at even smaller grain sizes
after a peak
d
http://me.jhu.edu/~
dwarner/index_file
s/image003.jpg
Images removed due to copyright restrictions.
http://www.sc.doe.gov/bes/IWGN.Worldwide.Study/ch6.pdf
© 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:
2nd source
┴
┴
┴┴
d
Nucleate second source in
other grain (right)
Physical picture: Higher
external stress necessary to
lead to large dislocation
density in pileup
1
σY ~
d
See, e.g. Courtney, Mechanical Behavior of Materials
© 2007 Markus J. Buehler, CEE/MIT
The strongest size: Nano is strong!
Different mechanisms have
been proposed at
nanoscale, including
• GB diffusion (even at low
temperatures) – Wolf et al.
Figure removed due to copyright restrictions.
See p. 15 of http://www.imprs-am.mpg.de/summerschool2003/wolf.pdf
• GB sliding – Schiotz et al.
• GBs as sources for
dislocations – van
Swygenhoven, stable SF
energy / unstable SF energy
(shielding)
Yamakov et al., 2003, Schiotz et al., 2003
© 2007 Markus J. Buehler, CEE/MIT
Typical simulation procedure
1. Pre-processing
(define geometry, build
crystal etc.)
2. Energy relaxation
(minimization)
F=ma
3. Annealing (equilibration
at specific temperature)
Image removed due to copyright restrictions.
4. “Actual” calculation; e.g.
apply loading to crack
5. Analysis
Real challenge:
Questions to ask and what to learn
© 2007 Markus J. Buehler, CEE/MIT