Interplay between theory and experimentin AFM nanomechanical studies of polymers

8/7/2019 Interplay between theory and experimentin AFM nanomechanical studies of polymers

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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


0.03 nm


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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Interplay between theory and experimentin AFM nanomechanical studies of polymers