Tutorial 1

1. An Inertial slider is a friction based drive used in AFMs and is a promising drive for MEMS.  One version of the drive is as shown in the figure

Write down the equation of motion for the slider considering the friction between the sapphire ball and alumina plate? What are the conditions for,

• Sapphire and alumina to stick and move as one unit?
• Sapphire slips against alumina?

Assume what ever you need as a practical engineer would do.

If a quadratic voltage pulse followed by sharp drop to zero voltage is applied between the top and bottom face of the piezo, trace the displacement of the slider.  (Displacement of the piezo =  d₅₁ Voltage; d₅₁ = 1 nm/volt )

2. The surface roughness can be modeled as a fractal using the Weierstrass-Mandelbrot Function,

G is a scaling constant and D is the fractal dimension and = 1/λ is the frequency mode corresponding to the reciprocal of the wavelength (λ) of roughness and nl is the lower cut off frequency and upper summation limit can be cut off to nh a higher cutoff frequency.

Derive the traditional roughness parameters like Ra, Rz etc from the above equation.

Can you simulate a surface profile numerically using the above function?

3. Derive the constants a and b of the Leonard Jones potential (given by V=-a/r⁶ + b/r¹²) as a function the experimentally determinable parameters (like the Elastic modulus and equilibrium inter-atomic distance r₀ )
4. Assuming that the atoms in the metal can be modeled as spring and mass system derive the coefficient of friction between two atoms as shown in figure. The Spring stiffness can be assumed as E/r_o. E – elastic modulus and r_o The equilibrium distance between atoms. Assume Leonard jones potential between atoms B and D and that there is no interaction between B and C And B and D.