2.002 Tutorial PresentationProblem 1-Atomic Force

MicroscopyJustin Lai

1. Approximate the cantilever as a beam which interacts with the surface a force F. Give the expression for the deflection of the cantilever as a function of x, distance. What is the deflection at the end of the cantilever (x=L)? Where on the cantilever is the strain the highest?

FL

h

bL

y

x

External force and moment analysis:

F

AyR

AAzM

L xF

yF

zM

= 0

0: FR AyAyRF

0: AzMFL

FLM Az

F

F

AFLL

Internal force and moment analysis:

F

AFL

x

)(xFxx

)(xFxy

)(xM xz

Making a cut in the beam:

xF

yF

BzM

0)(: FxFxyFxFxy )(

0)(: xMFxFL xz

0)(: xFxx

B

FLFxxM xz )()()( xLFxM xz

From analysis of beam bending geometry:

y

E

yE

ydAM

dAyE

M 2

y

Distance from center to neutral axis

0

0

l

ll f

)( y

the moment from each elemental area

dAyE

M 2

dAyI zz2 = second moment of inertia

12

3bhI zz

12

3bhEM

1

2

2

dx

ud

2h

2h

bdyyI zz2

2

2

2

h

h

dyyb

|2/

2/

3

3

h

h

yb

12

3bh

112

3

Ebh

M

112

3

Ebh

M

2

2

3

12

dx

ud

Ebh

M

2

2

3))((

12

dx

udxLF

Ebh

)(12

32

2

xLFEbhdx

ud

FxEbh

FLEbhdx

ud332

2 1212

)()( xLFxM xz

0 0

FxEbh

FLEbhdx

ud332

2 1212

12

33

612cx

Ebh

Fx

Ebh

FL

dx

du

213

32

3

26)( cxcx

Ebh

Fx

Ebh

FLxu

3

3

3

3 26)(

Ebh

FL

Ebh

FLLxu

0dx

du

0)0( u

Boundary conditions for cantilever:

3

34)(Ebh

FLLu

33

23

26)( x

Ebh

Fx

Ebh

FLxu Where is the strain the highest?

We know that at the wall, the moment is the highest as there is the longest lever arm. Moment causes stress. And stress causes strain.

We can also look at the following equation for x = 0.

)()( xLFxM xz

3

3

4L

Ebhk

y

F

Silicon-Nitrogen cantilever with following specifications:

E = 140 GPa

L=100

h = 320nm

B = 15

m

m

36

3969

)10100(4

)10320)(1015)(10140(

m

mmPak

mNk 0172.

2. Derive a form for the spring constant k for a cantilever.

kxF kyF

3

34)(Ebh

FLLy

3. Variations in dimensions of E and of the cantilever sometimes on the order of a few percent or greater occur. Discuss possible sources of these variations. Calculate how much these variations can affect the cantilever k.

For example, say the height changes from 1 to 1.05, a 5% change:

3

3

)1(4

)05.01)(1)(1(

m

mmPak

157.1)05.01( 3 mTherefore, a 5% change in the height leads to a 16% change in the spring constant k.

By looking at the formula, we can see how each variable is directly and indirectly proportional to a given degree to the spring constant.

The main source of error is found in microfabrication. The very nature of creating instruments and tools on such a small scale lends itself easily to variations in dimensions.

3

3

4L

Ebhk

4. If you wanted to measure the mechanical properties of biomolecules (cell surfaces, or pulling on proteins), the forces involved are typically ~ 102pN. In order to measure forces of this magnitude, what restrictions does this put on the material properties and dimensions of the cantilever? Consider the interaction between the cantilever and the surface to be Hooke’s Law-like also.

cantileverk

specimenk

You want the spring constants of both the cantilever and specimen to be on the order of the same magnitude.

Imagine modeling this as two springs in series. Having spring constants of different order of magnitudes will create deformations that are also of different order of magnitudes. Also, the dissimilarity can cause damage in either component.

5. When AFM is utilized in a mode in which the cantilever is oscillated at a frequency above a surface, it is necessary to know what the resonant frequency f0 of a cantilever is. Write an expression for f0 first assuming one has a tip mass m that is much greater than the mass of the cantilever, thus mcantilever is negligible.

m

kf 02

20 L

Ectf

m

kf

21

0

How does the evaluation change if the mcantilever is not negligible?