Force Distance Curves by AFM

2020-02-27 131浏览

  • 1.AFM Force-Distance Curves -- a brief introduction Sh. Liu
  • 2.Contact AFM & Force Curve
  • 3.Principle of AFM
  • 4.Principle of Force-Distance Curve No Forces Onset of Attractive Repulsive Attractive Relieve of forces
  • 5.Modeled Force Curve
  • 6.Classical Force Curve Force Repulsion • Tip-Sample interaction Contraction Distance Jump-to/off-Contact O distance DefSens = 1 Slope which contribute to ??????c • Approach • Retract Displacement Contact Non-Contact Region Region Zero-Line • Tip-Sample interatomic force expressed by Lennard-Jones：. • ?????? ?????? = − ?????? ????????????1 + ?????? ????????????2 • The Attractive force generally follows a law − ??????−?????? with ?????? ≥ 3 not ?????? = 7. • The repulsive part is much more complex than modelled.
  • 7.Force-Displacement to Force-Distance? ??????c ?????? ?????? ??????sAssumption:a. Sample deformation can ONLY be described by Hooke’s Law. b. No additional force beyond TipSample interaction. c. System stays (pre-)stationary at each measuring point. d. Substrate is stiff enough while cantilever is soft enough. Apparent force given by Hooke’s Law(a): ??????App = −??????c??????c Displacement and Distance combined by(b): ?????? = ?????? − (??????c + ??????s) While ???????????? and ???????????? can be determined by(b, c): ??????s = ??????c ??????s ??????cHence:?????? = ?????? − ????????????c ??????ℎ?????????????????? ?????? = 1 + ??????c ??????s ≈ 1 Furthermore(d): ???????????? ??????c ≪ ??????s ?????? = ?????? − ?????? ∙ ??????App ??????c ≈ ?????? − ??????App ??????c
  • 8.Force-Displacement to Force-Distance? ??????c ?????? ?????? ??????s Force-Displacement ↓ Force- Distance Suppose the tip and sample deformation can ONLY be described by Hooke’sLaw:??????c ??????c = 1 2 ??????c ??????c 2 ??????s ??????s = 1 2 ??????s ??????s 2 By forcing the systempre-stationary:????????????tot ?????? ??????s = ????????????tot ?????? ??????c = 0 So weobtain:??????s = ??????c ??????s ??????c And the force can be written as (L-J attractive part is considered only) : ?????? = − ????????????cs ???????????? = − ?????? ????????????Hence:??????c??????c = ?????? ?????? − ????????????c ?????? ??????ℎ?????????????????? ?????? = 1 + ??????c ??????s
  • 9.Dependence on kcantileverLennard-Jones:?????? ?????? ?????? ?????? = − ????????????1 + ????????????2Mathematics:In order to satisfy (pre-)stationary:??????2??????tot ?????? ??????c 2 > 0 Wehave:??????c ?????? > ???????????? 1 ?????? − ????????????c ??????−1 Jump-to-contact happenwhen:???????????? ≥ ??????c Cantilever deflection (??????c)jtc and tipsample distance ??????jtc bedetermined:??????+1 ?????? ??????c jtc = ???????????? ????????????c ??????jtc = ???????????? ??????c jtc ??????ℎ?????????????????? ?????? = 1 + ??????c ??????s Therefore, using cantilevers with highkc:• the jump-to-contact first increases with kc and then, for higher kc, disappears. • The jump-off-contact always decreases and the total hysteresis diminishes with kc. • If kc is greater than the greatest value of the tip-sample force gradient, hysteresis and jumps disappear.
  • 10.Dependence on kcantilever Force-displacement curves (broken lines , approach and retract) obtained with three cantilevers of different elastic constant for kc >> ksample. The continuous line is the tip-sample interaction, modeled with a LennardJones interaction with A = 10 -77 Jm6, B = 10-134 Jm12. Dependence of jump-to-contact and jump-off-contact distances on the elastic constant of the cantilever. The tip sample interaction has been modeled with a LennardJones interaction with A = 10 -77 Jm6, B = 10-134 Jm12. • The softer the tip, the larger the difference between Approach & Retract curves. • Approach and retract curves will doubling if kc large enough. • The Jump-to/off-contact happens in angstrom(Å ) scales.
  • 11.Theory of Zero-Line • Zero lines are mainly effected by optical interference • Detected signal is a mixture of reflected signal and scatter signal. • The hysteresis due to the viscosity of medium. ??????v ?????? = ?????? ???????????? ???????????? 2fv iΘ cra1ntilever r2 D s Large hysteresis will appear if approach rate improper
  • 12.Theory of Non-Contact-Region • Jump-to-contact happen only when cantilever elastic constant is smaller than the maximum value of tip-sample forcegradient:???????????? ≥ ??????c • Jump-to-contact is always present at an atomic scale. • Substrate may wetting tip in Jumpoff-contact procedure. kc ~ 5000 N/m Ni tip Force (nN) Separation (Å) Gold substrate Reprint from [2]
  • 13.Theory of Contact-Region When the sample is pressed against tip and ?????? = 0: −?????? = ??????c??????c = ??????c??????s ??????c+??????s ?????? = ?????????????????????????????? 1 11 ??????ℎ?????????????????? ??????mix = ??????c + ??????s However, ks changes with the contactradius:E∗ ??????s = 2?????? 1 − ??????2 Where ?????? is the contact radius, E∗ is reduced Young modulus and ?????? is Poisson ratio. • Most simple solution was given by Hertz andSneddon:– ??????ad = 0 – ?????? = 3 ???????????? ??????∗ – ??????0 = 0 – ?????? = ?????? ????????????∗ – ?????? ?????? = 3??????∗?????? 1−??????2 2??????R = 3?????? 1−??????2 2????????????2 – 1 ??????∗ = 3 (1−??????2 4 ?????? + 1−??????i2) ??????i • ??????ad adhesion force; ?????? contact radius, ??????0 contact radius at zero load; ?????? the deformation of the spherical tip, ?????? pressure. ?????? and ??????i are Young modulus of the flat surface and the tip.
  • 14.Theory of Contact-Region DMT’ssolution:'>solution: