APPENDIX (LECTURE #2) :

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1 APPENDIX (LECTURE #) : I. Review / Suary of Cantilever Bea Theory II. Suary of Haronic Motion III. Liits of orce Detection IV. Excerpts fro Vibrations and Waves, A.P. rench, W. W. Norton and Copany, 97

2 I. Review / Suary of Cantilever Bea Theory fro. [] A cantilevered bea is one that is fixed at one end and free at the opposite end, as shown in igure. fixed end t x b L free end igure. Noenclature for a cantilevered bea with rectangular cross section ; L=length or span (), b=width (), t=height or thickness (), I=oent of inertia of cross-sectional area ( 4 ), E=Young s (elastic) odulus (Pa=N/ ), and EI=flexural odulus (N ) Consider the case where a concentrated force is applied in the downwards direction at the free end of a cantilever (igure.). y z M=-L O V=- x free-body diagra igure. A loaded, cantilevered bea and corresponding free-body diagra A free-body diagra of the bea shows that a reactant shear force, V, and a reactant bending oent M, ust exist in order to aintain static equilibriu. By taking the conditions for equilibriu one finds that : y = = V + V = - ()

3 M o = = M + L M = -L () No atter where a transverse cut is taken along the bea and a free-body diagra constructed, the agnitude of the shear force, V, is found to be constant and equal to throughout the length of the bea: Since V(x) = = constant () dm V(x)= - dx, the oent, M(x), varies linearly fro a axiu of zero at the free end to a iniu of -L at the wall. Hence, M(x) is linear and equal to : M(x) = -(L-x) (4) Equations () and (4) are shown graphically in igure. V(x) M(x) - -L -(L-x) igure. Shear and bending oent diagras for the cantilevered bea given in igure. The equation for the slope of the y-displaceent curve, θ(x), is defined as follows: θ(x)= x M(x)dx EI (5)

4 Substituting equation (4) into equation (5) we obtain : x θ(x)= - (L - x)dx = - (L - x)dx EI EI x θ(x) = - ( Lx) - + C EI The integration constant, C, can be obtained fro the boundary condition that the slope of y-displaceent curve, θ(x), ust be zero at the wall (x=): x (6) θ ( ) ==- ( L) - +C C = EI x θ ( x ) =- ( Lx) - EI The equation for the y-displaceent curve or elastic curve, y(x), can be found as follows: x ( ) θ ( ) y x = x dx (8) Substituting equation (7) into equation (8) we obtain : (7) x x y( x) = - ( Lx) - dx EI Lx x y( x) = - C EI 6 + (9) The integration constant, C, can be obtained fro the boundary condition that the y-displaceent y(x) ust be zero at the wall (x=) : 4

5 L y( x ) ==- - +C C = EI 6 Lx x y( x ) =- EI 6 () The axiu deflection occurs at the free end of the cantilever and can be found by substituting x=l into equation () : L yax ( x= L) = () EI Equations () and () are shown graphically in igure 4. y(x) y(x)=-/ei[(lx /)-(x /6)] y ax =-L /EI igure 4. Elastic curve of cantilevered bea By rearranging equation (), one can obtain the applied load as a function of the deflection at the end of the bea: EI = L y ax () Here, we see that the applied force is directly proportional to the displaceent at the end of the bea and hence, the cantilever can be represented by a linear elastic, Hookean spring (igure 5.): =kδ () 5

6 where δ=y ax is the axiu deflection at the end of the cantilever (force spectroscopy notation), and k is the cantilever spring constant : EI k = (4) L = k y(x) δ=y ax y ax =-L /EI igure 5. Representation of cantilevered bea by a linear elastic, Hookean spring Hence, k is a function only of the bea diensions and the elastic odulus. Typically, V-shaped cantilevers are used for high-resolution force spectroscopy experients (igure 6.). side view b θ t cantilever probe tip top view L L θ d b d igure 6. Diensions of a V-shaped cantilever bea 6

7 Table I. displays approxiate forulas for the k of V-shaped cantilevers. Table I. orulas for the k of V-shaped cantilevers []. Reference Cantilever Spring Constant, k % error [] 5 Et d b + L 4 L [].5Et d 6 L [4] 4 Et d d + L b [4] 4 Et d d cosθ ( cos ) + θ L b References : [] Mechanics of Materials, D. Roylance, John Wiley and Sons, Inc [] T. R. Albrecht, S. Akaine, T. E. Carver, and C.. Quate, J. Vac. Sci. Tech. A8, 86 (99). [] H.-J. Butt, P. Siedle, K. Siefert, K. endler, T. Seeger, E. Baberg, A. L. Weisenhorn, K. Goldie, and A. Engel, J. Microscopy 69, 75 (99). [4] J. E. Sader, Rev. Sci. Instru. 66 (9), 458 (995). 7

8 II. Suary of Haronic Oscillators (*reference : Vibrations and Waves, A. P. rench, W. W. Norton and Copany, NY 97.) II.A. ree Vibrations Basic Physics Equations : δ(t)=displaceent() v(t)=velocity(/s)=dδ(t)/dt=δ'(t) a(t)=acceleration(/s )= d δ(t)/dt =δ''(t) (t)=force(n)=a(t) where : =ass(g) U(δ)=potential energy(n)= (δ)dδ Type of Haronic Motion : Siple Haronic Motion (SHM) : ν=natural or resonant frequency (Hz= oscillation/s=s-) ϖ=natural or resonant angular frequency=πν (rad/s-) δ =displaceent aplitude () φ=phase constant ϖt+φ=phase s=spring recovery force k=spring constant (N/) Daped Haronic Motion (DHM) : β=daping (viscosity) coefficient d=dashpot or dissipative force ϖ ο =natural or resonant angular frequency for a daped syste (rad/s-) Q=quality factor Model Scheatic : s=-kδ(t) s=-kδ(t) δ δ δ ο +δ δ ο +δ fixed oscillating fixed oscillating d=-βδ (t) Equations of Motion : a= s δ''(t)+kδ(t)= a= s + d δ''(t)+βδ'(t)+ kδ(t)= Solutions to Equations of Motion : δ(t)=δ cos(ϖ ο t-φ) ϖ ο =k/ δ δ +δ φ δ(t)= δ e -β t/ cos(ϖ ο t-φ) ϖ ο = [(k/)-(β /4 )] δ δ +δ Q =k/β δ e -βt/ +δ e -βt/ ω ο t ω ο t 8

9 II.B. orced Vibrations Type of Haronic Motion : Driven Haronic Motion (DHM) : ϖ= frequency of applied force oscillation (rad/s-) ϖ= ω o resonance occurs; axiu aplitude of oscillations, δ ο + Model Scheatic : forced oscillation : a(t)=cos(ωt-φ) δ Equations of Motion : a= s - a δ''(t)+kδ(t)= a (t) Solutions to Equations of Motion : δ(t)=δ cos(ϖt-φ) δ (ω)= /(k-ω ) δ o /k - o /k Driven / Daped Haronic Motion (DDHM) : ϖ= frequency of applied force oscillation for daped syste (rad/s-) ο + s=-kδ(t) δ ο +δ oscillating forced oscillation : a(t)=cos(ω t-φ) d=-βδ (t) a= s + d - a δ''(t)+βδ'(t)+ kδ(t)= a (t) δ(t)=δ cos(ϖ t-φ) δ (ω )= /(k-ω ) δ ω o = k/ ω δ (ax) =Qo/k(-/4Q ) / high Q δ δ ο +δ oscillating o /k low Q ω o = k/ ω 9

10 III. Liits of orce Detection [-4] The lower bound of force detection of any force spectroscopy easureent is deterined either by the resolution or theral fluctuations of the transducer., Transducer Resolution. Previously, we have shown that a highresolution force transducer can be represented by a linear elastic, Hookean spring (equation ()). Let s assue that the iniu detectable displaceent is a one-ato deflection (δ in =. n). Substituting this value into equation () we obtain the iniu detectable force, in : in = (. n)k (5) Theral Oscillations. In the absence of any externally applied forces, a force transducer in equilibriu with its surroundings will fluctuate due to the - nonzero theral energy at roo teperature, k B T = 4. N, where k B B is - the Boltzann constant =.8 J/K and T is the absolute teperature (roo teperature 95K). If we odel the force transducer as a onediensional, free haronic oscillator as shown in igure 7. s =-kδ(t) cantilever δ δ o +δ igure 7. Theral oscillation of a free cantilever bea By neglecting higher odes of oscillation and aking use of the equipartition theoru, the average root-ean-square (RMS) aplitude of the displaceent oscillation, <δ > /, can be derived as follows.

11 The potential energy of a force transducer is δ U = (δ)dδ (6) Substituting Hooke s law for a free, one-diensional haronic oscillator (equation ()) into equation (7) and integrating gives δ U = kδdδ U =½kδ (7) The equipartition theoru states that if a syste is in theral equilibriu, every independent quadratic ter in the total energy has a ean value equal to ½k B T. B Hence, U = ½k δ = ½k B BT (8) where δ is the aplitude of the displaceent oscillation (igure 7.). Rearranging equation () and solving for δ we obtain = k B T k δ (9) where : <> denotes a statistical echanical average over tie. Substituting eq. (9) into Hooke s Law, equation (), gives the equation for the RMS aplitude fluctuations in force: = kbt k () A ore precise forulation can be derived for a daped haronic oscillator [5] : 4kBTkB w ' Q = () o where B is the easured bandwidth (s - ), Q is the quality factor =(k) / /β, is the ass (Ns /), β is the daping coefficent (Ns/), w o is the resonant frequency for a daped syste (s - ), and k is the transducer spring constant (N/).

12 References : [] E. Evans, K. Ritchie, and R. Merkel, Biophys. J. 995, 68, 58. [] Nanosystes : Molecular Machinergy, Manufacturing, and Coputation, K. Eric Drexler, John Wiley and Sons, 99. [] J. L. Hutter, Bechhoefer, J. Rev. Sci. Instru. 99, 64, 868. [4] H.-J. Butt, P. Siedle, K. Seifert, K. endler, T. Seeger, E. Baburg, A. L. Weisenhorn, K. Goldie, and A. Engel J. Microsc. 99, 69, [5] D. Sarid, Scanning orce Microscopy, Oxford University Press, p. 48