1. Modeling Atomic Force Microscopy Helen Tsai Group 10
2.002 Tutorial Problem 1
Modeling Atomic Force Microscopy
Helen Tsai
Group 10

2. Atomic Force Microscopy (AFM)
Imaging Tool
Use of dragging or tapping
Use of laser
Modeled as a cantilever

3. Approximate AFM as cantilever beam which interacts with a surface with a force F. Give the expression for the deflection of the cantilever as a function of x, distance.
First make cut in the middle of the beam and find the moment equation.

4. Moment at cut:
-Mxz(x) - (L-x)F = 0
Mxz(x) = -(L-x)F
Use EI dv2 = M
dx2
EI dv2 = -F(L-x) = -FL +Fx

5. EI dv = Fx2 –FLx +C1 dx
EI v = Fx3 – FLx2 + C1x +C2
Using Boundary Conditions of
v(0)= dv = 0 dx
We find that C1 and C2 equal 0.
v (x) = 1 (Fx3 - FLx2) EI
What is the deflection at the end of the cantilever?

6. v (L) = 1 (FL3 - FL3) EI
= -FL3
3EI
Therefore, the the strain on the cantilever beam is the highest at the wall.
Moment is highest at the wall, which means that the stress is highest at the wall. ( = My/I) Stress is related to strain by the equation:  = E . So if stress is greatest at the wall, then so is the strain.

7. Derive a form of the spring constant k of the cantilever based on material properties (E).
v(L) = -FL3
3EI
F = 3EI v(L) L3
This is in the same form as F = kx
Therefore, kL = 3EI

8. Suppose cantilever made of SiN with properties of:
E = 140 GPa
= 3100 kg/m3
L = 100 m
h = 320 nm
b = 15 m
What is the spring constant?
kL = 3EI We know that I = bh3 L
therefore, kL = Ebh3
4L3
Plugging in values, kL = N/m

9. What are possible sources of variations in the cantilever and how much do these variations affect the cantilever k?
kL = Ebh3
4L3
kL will change proportionally to b.
If there is an increase in h, then kL will increase proportionally by the change in h cubed.
kL is inversely proportional to L3.

10. Biomolecular Testing - (cell surfaces, or pulling on proteins.
To measure force of cell surface, AFM must emit a comparable force to that of the resistance of the cell surface. Stiffness has to be the same.
Modeling the cantilever with most of its mass at the tip:
Use  n = sqrt(kL/m) m = mass of tip
 n = 2 fo
kL = Ebh3
4L3
we find that
resonant frequency: fo = h(sqrt(Ebh))/(2Lsqrt(Lm))

11. Modeling cantilever with its mass in the cantilever:
resonant frequency: fo = ct(sqrt(E/P)) / L2
c = shape dependent constant