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Interplay between theory and experimentin AFM nanomechanical studies of polymers
Sergey Belikov and Sergei Magonov
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2006 Veeco Instruments Inc.
Agenda
Introduction
Simulation of Dynamic AFM Modes
Euler-Bernoulli & Krylov-Bogoljubov-Mitropolsky approachTapping Mode and Frequency Modulation
High-Resolution Imaging of Molecular Lattices: Experiment & Modeling
Compositional Mapping of Model Polymer Blends
Local Mechanical Probing: DvZ (indentation) & AvZ curves
Tapping Mode Curves: Modeling & Experiment
Summary
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Main AFM Functions: High-Resolution ImagingCompositional MappingQuantitative Probing of Materials Properties
Outstanding Technical Issues:
Sensitivity of Optical Detection, Fast Imaging & Mapping, Minimization of Thermal Drift, Efficient Drive of Probe, Imaging under Liquids, Probes
Key Hurdle:
Tip-Sample Forces: Understanding, Measurements & Control
Dynamic AFM & Quantitative Mechanical Data:
Dream or Reality
Introduction
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Simulation of Dynamic AFM
Main features of our approach
1. Euler-Bernoulli with boundary conditions including piezodrive of the cantilever base(rare considered by others)
2. Solution as a composite3. Van der Pole coordinates (amplitude, phase) transformation & separation of fast and
slow variables
4. Application of KBM averaging method
5. Analysis of KBM-derived differential equations
6. Classification of dynamic AFM modes (tapping mode, frequency modulation)
[ ]( )
[ ]( )
+++
+
=
++
+
=
d
xg ydy y x Z F F
N g
d x
ydy y x Z F F N g
C r a
C r a
102
1
02
1
2coscos
1
21
1cos
sincos1
21
1sin
( )111
=
= Qg
x = A sp
d = A 0
phase
F a force in approach
F a force in retract
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( ) ( ) ( )( )( )[ ] ( )( )=
+
+
L x pt x Z H x
t x Z at
t x Z t
t x Z ,
,,)(2
,4
42
2
2
t A sin0
Boundary conditions: ( ) t A Z t Z sin,0 00 += ( ) 0,0 =
t x Z ( ) 0,2
2
=
t L x Z ( ) 0,3
3
=
t L x Z
L Z
Z 0
x
p
= 24
2
sec
mS
EI
a ( ) [ ] N S Z H m1 [ ] N S pP = m1
( ) ( ) ( ) ( ) ( )t xt x zt x zt xU t x Z p ,,,,, +++=
n
n
n
n
QQ 214 2
=
( ) t A Z t xU C sin, 0+Oscillation of the base
Probe Motion in Dynamic AFM
Solution as a compositeThe tip weight
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( ) ( ) ( ) ( )( ) 21
112111
1,,2
l
p
Z t l zt l zt lU H +++=++
( ) t A Z t lU C sin, 0+=
( ) ( ) ( ) ( ) ( )t ll zt l zt lU t x Z p ,,,, 11 +++=
( ) ( ) nt t eat At l z 022101111 coscos, +++=
( )( )
( )
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2
1
11111
21111
sgn;2
l
c
Z S
Z F
+++=++
( )d t d t At A +=++= coscossin 1101where
1
1
21
11 214 QQ
= 111
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Probe Motion in Dynamic AFM
( )
( )
=
++=
++=
y x
gd y y x y x x
y
ygd y x y x x
coscos4cossin,cos1
sincos4sin,cos1
221
11
1
2211
1
( ) ( )( ) ( )
( ) ( )( ) ( ) ( )
=
+
++=
++=
g x
gd x x
x
gd x x x
coscos4
cossin,cos1
sincos4sin,cos1
221
11
2211
1
y= Introducing phase difference(slow variable)
Two fast variables
Averaging over fast variable ( ) gives:
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[ ]( )
[ ]( )
+
++++=
+++=
ggd ydy y x Z F F m x
gd x ydy y x Z F F
m
x
cr a
cr a
0
2211
1
0
2211
21
1
cos4coscos1
2
sin4sincos1
2
KBM Approach
( )( )
1
sgn,, +=
m
Z F c ( ) ( ) ( ) ( )1,;1, +== zF zF zF zF r a
( )
( )
+
++=
+=
ggd ydy y x y x x
gd ydy y x y x x
2
0
22111
1
2
0
22111
1
cos4cossin,cos2
sin4sinsin,cos2
The transition to stationary solutions gives:
Viscoelasticity willbe added!
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[ ]( )
[ ]( )
+++
+
=
++
+
=
d xg
ydy y x Z F F N g
d x
ydy y x Z F F N g
C r a
C r a
102
1
02
1
2coscos
1
21
1cos
sincos1
21
1sin
KBM Approach
md N 21 =
Stationary solutions:
( )111 == Qg
x = A sp
d = A 0
phase Z c height
Tapping mode (Amplitude modulation) Frequency modulation
g constant (usually 0) constant (usually /2)Curves: AvZ, vZ
( A and are obtained by solving theequations for each Z c)
Curves: r vZ, AvZ,
( r and A are obtained by solvingthe equations for each Z c)
Images: F a and F r depend on surface location ( XY )
Two FM modes
constant excitationconstant amplitude
ZvXY, vXY ( r = sp)
( Z and are obtained by solving theequations for each r )
Height ZvXY, Phase vXY (A =A sp)
( Z and are obtained by solving theequations for fixed Asp)
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( )
+=
=
00 coscos
2cos
sin
ydy y x Z F N
d x
[ ]( )
[ ]( )
=++
+=+
d xg ydy y x Z F F
N
d xg
ydy y x Z F F N
cr a
cr a
0 1
2
10
2coscos1
21sincos
1
( )( )
++=1
1 2
1
11
du
u
uua zF
ak
KBM Approach
J.E. Sader & S.P.Jarvis APL 2004 ,84 , 1801
u = cos y
[ ]( )
[ ]( )
++=
++=
0
0
coscos1
cos
sincos1
sin
ydy y x Z F F N
d
x ydy y x Z F F
N
cr a
cr a
F F F r a ==1 =Tapping mode ( )
2 =Frequency modulation ( )J. Cleveland et al APL 1998 , 72 , 2613
Experimental data ( x(A), , Zc, g ) and use of two equations mighthelp to restore F a and F r in dynamic AFM modes
tsF F F r a
Conservative case
==
Garcia&Perez Surf Sci Rep 2002 , 47 , 197
x = A0=const; 211l Z Sk =
( ) [ ]
d A Ad F
kA f
f Ak d f ts coscos21
,,, 002
00
000 ++=
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15 nm
20 nm
20 nm
20 nm
0.8 nm0.8 nm 0.4 nm
Si probes (5-10 nm)
Carbon spike (~3nm) Diamond probe (~5nm)
T XT Y TX
0.5TX
2T X2T X T Y
Tapping Mode Imaging of Polydiacetylene Crystal
bc
T x
T y
How to explain the presence of 2T x and 0.5T x spacings in AFM images?
15 nm 15 nm
How true is true molecular resolution?
0.49 nm c 1.41 nm
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LabVIEW AFM Simulator
Z
Z1
D
1
2
O
R
O1
O2
R1R2
P1
P2
XX1
X2
= iii PF cos2 / 3
2T
0.5T
Tapping mode: Hertz model
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R = 5 nmR = 0.15 nm R = 100 nm
X
Y
Y
X
Y
X
Y Y
X
X X
Y
Height corrugations ~
0.2 nm
Height corrugations ~
0.03 nm
Height corrugations ~
0.01 nm
Imaging in Light Tapping (A sp=20 nm)
Lattice Pattern: Dependence on Tip Radius
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Asp =19 nmAsp =20 nm A sp =17 nm
Y-bifurcation
X-bifurcation12 peaks 11 peaks 9 peaks 8 peaks
Tip with R = 5 nm
YYY
X X X
X
Y
X
Y
X
Y
Experimental patterns
Lattice Pattern: Imaging at Different Forces
Y-bifurcation
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Lattice w/Defects: Effect of Tip Radius & Force
R = 150 pm, A sp = 19 nm R = 1 nm, A sp = 18 nm R = 5 nm, A sp = 18 nm
X
Y
X
Y
X
Y
X
Y
X
Y
X
YAtomic-scale images change their pattern as tip size and/or tip force increasesthat makes their assignment to real surface structures very difficult.
A presence of single atomic or molecular defects in AFM images does not mean that true atomic-scale resolution in imaging of the surrounding lattice was achieved.
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Sharp Spherical
76nm10 nm
Olympus Team-Nanotec
Imaging with Sharp & Spherical Probes
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2.2 GPa (6.0%)14.9 MPa (8.2%)
EPDM PP
Indentation & Phase Imaging of iPP/EPDM Blend
0.73
0.93
0.50
Asp /A 0
0.33
PPPP EPDMheight phase, sin
0.20
10 m10 m10 m
EPDM PP
I d i f M l il P l h l
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380.5 MPa (4.8%)
Indentation of Multilayer Polyethylene
40.4 MPa (4.0%)
PE-0.86 PE-0.92 Sneddon & Oliver-Pharr Models
( )
( )
=1
0
2
2
2
11
),(
x
dx x f x Ea E akD
Smax
hi hmaxpenetration, nm
F o r c e , n N
1.4 MPa (7.7%)
76 nm
PDMS
(adhesion and viscoelasticity will be added!)
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AFM Nanoindentation : nm-Scale Depth
~ 11 nm
elastic plastic
1 st
1 st
2 nd
2 nd
1st2 nd
3 rd
Elastic and Plastic Deformation of Single Crystals of Alkane C 390H782
d l l
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AFM Nanoindentation : Lateral Resolution
z
A
z
A
250 nN90 nN 1.3 uN Asp /A 0=0.5V/1.0V Asp /A 0=1.0V/2.0V
Force Volume (AvZ curves) of SBS triblock copolymer
100 nm500 nm
z
A
Deflection curves (nanoindentation) Amplitude curves (tapping mode)
Si l ti f AvZ & vZ Curves i T i g M d
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Simulation of AvZ & vZ Curves in Tapping Mode
[ ]( )
[ ]( )
++=
++=
0
0
coscos1
cos
sincos1
sin
ydy y x Z F F N
d x
ydy y x Z F F N
cr a
cr a
Tapping mode
( )
=
2
0
8
0
41
38
z z
z z
zU pp ( ) ( ) z RU zF pprp 2=
MaugisMaterial-related avalanche
Lennard-Jones
Instrument- or environment-
related avalanche
Derjaguin
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Simulation of TM Amplitude & Phase curves
ModelingA sp /A 0
Experiment (Si substrate)Asp /A 0 Asp /A 0
Phase, Phase, Phase,
Saddle-node bifurcation in amplitude/phase coordinates is a
birth or annihilation of stable and unstable stationary points thathappened as Z is changing.
S. L. Lee, S. W. Howell, A. Raman, R. Reifenberger Ultramicroscopy 2003 , 97 , 185
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A
z
Phase,
R1
Conservative case: Tip size effect (R 1
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