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

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

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.

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

EI dv = Fx2 –FLx +C1 dx 2 EI v = Fx3 – FLx2 + C1x +C2 6 2 Using Boundary Conditions of v(0)=0 dv = 0 dx We find that C1 and C2 equal 0. v (x) = 1 (Fx3 - FLx2) EI 6 2 What is the deflection at the end of the cantilever?

v (L) = 1 (FL3 - FL3) EI 6 2 = -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.

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

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 L3 12 therefore, kL = Ebh3 4L3 Plugging in values, kL = 0.017 N/m

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.

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))

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