NONCONTACT MODE

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::"spell", General::"spell1"]; Needs["Graphics`Arrow`"]; tera = (10^12); micro = (1/(10^6)); giga = (10^9); nano = (1/(10^9)); mega = (10^6); pico = (1/(10^12)); kilo = (10^3); femto = (1/(10^15)); milli = (1/(10.^3)); atto = (1/(10^18)); NSub
       = {c -> 2.99792458 (10^8), e -> (1.6021773/(10^19)), 
        h -> (6.6260755/(10.^34)), hbar -> (1.05457/(10.^34)), 
        K -> (1.380658/(10^23)), me -> (9.109389/(10^31)), 
        mu0 -> ((4. π)/(10^7)), eps0 -> (8.85419/(10^12))}; ESi = 0.179 tera; rhoSi = 2330; nuSi = 0.28; ESiO2 = 0.06 tera; rhoSiO2 = 2200; nuSiO2 = 0.3; ESi3N4 = 0.15 tera; rhoSi3N4 = 3100; nuSi3N4 = 0.24; EAu = 0.08 tera; rhoAu = 19300; nuAu = 0.42; ENi = 0.207 tera; rhoNi = 8900; nuNi = 0.41; EW = 0.345 tera; rhoW = 19300; nuW = 0.28; EPt = 0.17 tera; rhoPt = 21400; nuPt = 0.39; EIr = 0.52 tera; rhoIr = 22500; nuIr = 0.3; EHOPG = 0.2 tera; rhoHOPG = 2250; nuHOPG = 0.3; EPol = 0.001 tera; rhoPol = 1000; nuPol = 0.3; xx = 0.8; EPtIr = xx 0.8 EPt + (1 - xx) EIr; rhoPtIr = xx rhoPt + (1 - xx) rhoIr; nuPtIr = 0.3; opt
       = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        Frame -> True, GridLines -> Automatic, 
        DefaultFont -> {"Times-Roman", 8}, 
        PlotStyle 
          -> {Dashing[{.0001}], Dashing[{.0200}], Dashing[{.0200}], 
            Dashing[{.0200}], Dashing[{.0200}], Dashing[{.0001}]}};

Operating Conditions

Operation
       = {k -> 10., Q -> 20., abm -> .1 nano, 
        set -> 3. nano, f0 -> 100 kilo, f -> 100 kilo};

TipSample
       = {Rtip -> 10 nano, sigma -> .34 nano, 
        A -> 1. atto};

case1
       = {w -> 2 π f, w0 -> 2 π f0} /. Operation; case = TipSample ⋃ Operation ⋃ case1;

Tip-Sample Interaction

Lennard-Jones Potential

Lennard-Jones Potential

Clear[Wb, WLJ]; Wa = ((Rtip A)/(6 sigma)); Wb[s_] := (1/210) ((((sigma/s)))^7) - (sigma/s); WLJ[s_] := Wa Wb[s];

Lennard-Jones Force

Clear[FLJ]; FLJ[s_] = -(∂_s) WLJ[s];

sFzero = 
      Solve[FLJ[s] == 0, s][[2, 1, 2
          ]] /. case; Print["sFzero = ", N[(sFzero/nano)], " [nm]"]

Fig8x2 = 
      Plot[FLJ[s nano]/nano /. case, {s, (sFzero/nano), 2}, 
        FrameLabel -> {"s in nm", "FLJ in nN"}, Evaluate[opt]];

Lennard-Jones Force Derivative

Clear[F1]; F1[s_] = (∂_s) FLJ[s];

sF1zero = 
      Solve[F1[s] == 0, s][[2, 1, 2
          ]] /. case; Print["sF1zero = ", N[(sF1zero/nano)], " [nm]"]

sF1max = 
      Solve[(∂_s) F1[s] == 0, s][[2, 1, 2
          ]] /. case; Print["sF1max = ", N[(sF1max/nano)], " [N/m]"]

F1max = F1[sF1max] /. case; Print["F1max = ", N[F1max], " [N/m]"]

AGraphics = 
      Plot[F1[s nano] /. case, {s, (sF1zero/nano), 2}, 
        FrameLabel -> {"s in nm", "F1 in N over m"}, 
        DisplayFunction -> Identity, Evaluate[opt]];

BGraphics = 
      Graphics[Line[{{(sF1zero/nano), k /. case}, {2, k /. case}}]];

Fig8x3 = 
      Show[{AGraphics, BGraphics}, DisplayFunction -> $DisplayFunction];

Numerical Solution

Equation of Motion

nn = 2 Q /. case; cycle = (1/f) /. case; stop = nn cycle;

Clear[Force]; Force[s_] := FLJ[s] /. case

Clear[z, v, x1, x2]; eq1=x1'[t]==x2[t];eq2=x2'[t]==-((w0 x2[t])/Q) - 
              (w0^2) (x1[t] - set - abm Sin[2 π f t]) + 
              (((w0^2) F[x1[t]])/k)/.case;

Numerical Solution

Clear[z, v, x1, x2]; {z, v} = {x1, x2} /. 
        NDSolve[{eq1, eq2, x1[0] == set /. case, x2[0] == 0} //. 
              F -> Force, {x1, x2}, {t, 0, stop}, 
            MaxSteps -> 100000][[1
          ]];

Clear[zfree]; zfree[t_] := 
      set + abm Q (1 - Exp[-((w0 t)/(2 Q))]) 
            Sin[w0 t - (π/2)] /. case;

Transient Regime

Fig8x4 = 
      Plot[{z[t cycle]/nano, zfree[t cycle]/nano}, {t, 0, 5}, 
        FrameLabel -> {"cycles", "z in nm and za in nm"}, 
        Evaluate[opt]];

Steady-State Regime

Fig8x5 = 
      Plot[{z[t cycle]/nano, zfree[t cycle]/nano}, {t, nn - 2, nn}, 
        FrameLabel -> {"cycles", "z and za in nm"}, 
        Evaluate[opt]];

Bimorph-Cantilever Phase Diagram

Fig8x6 = 
      ParametricPlot[{z[t cycle]/nano, zfree[t cycle]/nano}, {t, nn - 2, 
          nn}, AspectRatio -> 1, 
        FrameLabel -> {"z in nm", "za in nm"}, Evaluate[opt]];

Displacement-Velocity Phase Diagram

Fig8x7 = 
      ParametricPlot[{z[t cycle]/nano, v[t cycle]}, {t, nn - 2, nn}, 
        AspectRatio -> 1, 
        FrameLabel -> {"z in nm", "v in m over sec"}, 
        Evaluate[opt]];

Numerical Value of the Phase Shift

vbm[t_] := abm w0 Cos[2 π t] /. case;

Fig8x8 = 
      Plot[{v[t cycle], Q vbm[t] /. case}, {t, nn - 2, nn}, 
        FrameLabel -> {"cycles", "v in m over s"}, 
        Evaluate[opt]];

t0 = nn - .75; Print["t0 = ", t0, " [cycle]"]

\(t1 = 
      \(FindRoot[v[t\ cycle] == 0, {t, nn - 0.6, nn - 0.25}]
          \)[[1, 2]]; 

    Print["t1 = ", t1, " [cycle]"]\)

phi1 = t0 - t1; Print["phi1 = ", phi1 360, " [deg]"]

Approximate Analytical Solution

Equation of Motion

Analytical Solution

Clear[k1, w1, dF]; k1[dF_] := k - dF; w1[dF_] := w0 (Sqrt[k1[dF]/k]);

Clear[d, A1]; d[t_] := A1 Exp[I w t]

\(\(eq3 = 
      ∂\_{t, 2}d[t] + \(w0\ ∂\_t d[t]\)\/Q + 
          w1[dF]\^2\ d[t] == abm\ w0\^2\ Exp[I\ w\ t]; \)\)

Steady-State Regime

Clear[A1]; sol1 = Solve[eq3, {A1}]; A1 = A1 /. sol1[[1]]; Print["A1 = ", A1]

A02 = ComplexExpand[Abs[A1]]; \(\(A10 = Simplify[A02 /. \@x_\^2 -> x]; \)\)Print["A10 = ", A10]

$F_1$-Dependent Phase Shift

\(B1 = ComplexExpand[A1]; 

    phi2 = ArcTan[
        \(-\(\(I\ 
                B1[[2]]\)\/\(B1
                  [[1]] + B1[\([3]\)] + 
                B1[\([4]\)]\)\)\)]; 
Print["phi2 = ", Simplify[phi2]]\)

phi20 = phi2 /. w -> w0; Print["phi20 = ", phi20]

Clear[phi3]; phi3[F1_] := Series[Tan[phi20 + (π/2)], {F1, 0, 1}]Print["phi3 = ", phi3[F1]]

$F_1$-Dependent Amplitude

A11 = 
      (A10/(abm Q)) /. k1 -> k - dF /. w -> w0 /. case;

Fig8x9 = 
      Plot[Evaluate[A11], {dF, -.1 k /. case, .1 k /. case}, 
        PlotRange -> {0, 1}, 
        FrameLabel -> {"F1 in N over m", "A over A0"}, 
        Evaluate[opt]];

wm1 = w0 (1 - (0.35/Q)) /. case; wm2 = w0 (1 + (0.35/Q)) /. case;

A22 = 
      Abs[Re[(A10/(abm Q)) /. k1 -> k - dF /. w -> wm1 /. 
            case]]; A23 = 
      Abs[Re[(A10/(abm Q)) /. k1 -> k - dF /. w -> wm2 /. 
            case]];

Fig8x10 = 
      Plot[Evaluate[{A22, A23}], {dF, -.1 k /. case, .1 k /. case}, 
        FrameLabel -> {"F1 in N over m", "A over A0"}, 
        Evaluate[opt]];

Approximate $F_1$-Dependent Phase Shift

Clear[phi3]; phi3[s_] := -((Q F1[s])/k) /. case

Fig8x11 = 
      Plot[((180 phi3[s nano])/π), {s, (set/nano) - 1 /. case, 
          (set/nano) + 1 /. case}, 
        FrameLabel -> {"s in nm", "phi3 in deg"}, 
        Evaluate[opt]];

Print["set = ", (set/nano) /. case, " [nm]"]

Print["phi3 = ", N[((phi3[set] 180)/π) /. case], " [deg]"]

Print["phi1 = ", phi1 360 + 90, " [deg]"]