Leture: Cantilever Mehanis The Fore Sensor Ron Reifenberger Birk Nanotehnology Center Purdue University 0
Defletion vs. z Week 4 Overview Fore Spetrosopy Piezos Cantilever Instrumentation Controller Calibration Laser & Photodiode Contat Stati Cantilever Topographi Imaging Adhesion AFM Loal Material Property Map Theory Elastiity Analysis Software Fore Sensing and Control Tip-Sample Interation Cantilever Dynamis Controller
The Loaded Miroantilever L In general w(,t) w t F q θ www.olympus.o.jp q is the defletion of the tip at =L, w(l,t). L Notation: antilever thikness, t width, w If the antilever is made from a material with a Young s modulus E, what is the defletion shape w(,t), the bending stiffness k (spring onstant), and the slope at =L when a point fore F is applied? Is defletion proportional to F?
antilever thikness, t width, w ; length, L Young s modulus, E y The problem: What is y()? z L F Torque: τ = r F = r F sinθ( zˆ ) if r F, then τ = r F Bernoulli Euler assumption ( ) If torque refers to bending of a rigid objet, it is often alled a moment M rather than a torque τ. This distintion makes it lear we are disussing bending rather than twisting. The magnitude of the bending moment at point is M = F L ( ) r = L Use y() instead of w(,t) In equilibrium, all internal moments developed at by the antilever due to bending must balane the applied moment M at point.
Internal moments develop beause of internal stresses neutral ais y Internal springs between atoms are etended when antilever is bent f f Internal springs between atoms are ompressed when antilever is bent In equilibrium, torques balane torques. This means eah internal spring applies a moment around the neutral ais that tends to restore the antilever to its original position. The horizontal spring fores (f) are thus translated into a vertial fore that tends to restore the antilever to its unbent shape. 5
BEFORE da AFTER (urvature greatly eaggerated) dy N.A. A B y w t df dl A B l dy d = ydθ = R dθ d y = R t / for any given y, Hook ' s Law says : df d = E = da da = w dy A B y df = E w dy R applied moment = internal moment : ( ) y E R + t + t y Ewt df R t t M = F L = y = E w dy = applied F L urvature of beam at κ = = R Ew t ( ) R dθ R A B t dθ -t /
urved line dl Review - defining the urvature of a line dy d dθ If urved line is a irle : κ π = = π R R y ( in rad ) R hange in angle of tangent line hange in ar length m dθ hange in angle of tangent line dθ κ = = hange in ar length d dy dy tan θ = θ = atan d d ( d ) = ( d) + ( dy) d dy d y dθ d dy d = = d atan = d d d d dy dy + + d d dy d = ( d) + ( dy) = d + d d = d dy + d d y dθ κ = d = d dy + d R dθ d d d
Defletion (y) vs. : dy if the urvature is small, then d d y F L = R d Ew t ( ) ( ) F L y( ) = d y = d Ewt F L = Ewt 6 when = L q yl ( ) = 4L Ew t F Ew t F = kq k = 4L spring onstant Slope (Θ) vs. : dy F Θ ( ) = = L d Ewt when = L Θ ( L) = 6L Ew t F F = Θ( L) = kl q Θ ( L) = L Standard results in terms of area moment, I [units: m 4 ]: + t t L EI L ; ( ) ; ; ( ) I w y dy = w t q y L = F k = Θ L = F EI L EI
Notation In this leture In general w(,t) y y() F q=y(l) Neutral Ais Converting to antilever oordinates: y() = w() y(l) = w(l) q
Profile of antilever y ( in nm) 500 400 00 00 00 E= GPa; L=00 μm; t = μm; w = 50 μm; F=5 nn F=5nN Θ 0 0 50 00 50 00 (in µm)
Linear and Angular Defletion Plots 4L q yl ( ) = F Ew t F = kq k = Ew t 4L Θ ( L) = 6L Ew t F F = Θ( L) = kl Cantilever Defletion (nm) 00 0 0. k=0.4 N/m k=.0 N/m k=.0 N/m 0 0 40 60 80 00 Applied Fore (nn) Angular Defletion (μrads) 0,000,000 00 0 k=0.4 N/m k=.0 N/m k=.0 N/m L=00 μm 0 0 40 60 80 00 Applied Fore (nn)
Impliations Resonant Frequeny Work Done = Elasti Energy Stored U()=½k spring onstant k (N/m) +Fore m m =0 final f = π k m final F restore = -k final Caution: when omparing to antilever, note that eah part of the antilever is displaed by a different amount. Need to introdue an effetive mass to desribe this effet.
f Stati model k Ew t = = π m π 4Lm where m and m eff ant eff = 0.4m = ρlw t ant F FL FL q = θ = EI EI EI Ewt k = = L 4L L q eff θ Summary Dynami model (Euler-Bernoulli Equation) 4 w(, t) w(, t) EI +ρ w t = 0 4 t f f EI Et π ρa π ρ i = βi = βi First eigenmode (i=) βi EI = = π L ρa π (.876) Ew t 4L Will be disussed in Part II of this ourse m
Up Net: Tip Approah to the Substrate