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creemore
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Homework Statement
An AFM tip has a spring constant k, a colloidal bead with radius r is glues onto the cantilever tip. Derive an expression and plot the deformation of the cantilever spring, Δx, as a function of distance the tip of the cantilever is moved towards another bead of the same radius.
Homework Equations
The cantilever tip can be treated as a Hookean spring. So Fspring = -kΔx
The non-retarded Van der Waals interaction free energy between two spheres of the same radius is:
W = -AR/(12D)
where D is the distance between the spheres, R is the radius of the spheres, and A is the Hamaker constant.
The Van der Waals force is simply the derivative of the potential w.r.t. D:
Fvan = AR/(12D2)
The Attempt at a Solution
The question seems fairly straight forward. At equilibrium, Fspring = Fv, so
-kΔx = AR/(12D2)
we can say D is the distance the cantilever tip is manual moved (z), plus the distance the spring is stretched because of VDW interaction.
-kΔx = AR/(12(z+x)2)
For some reason, I can't solve for Δx as a function of z. I feel like I'm missing some trivial step, and would appreciate any help with a solution.
Thanks in advance.