TAPPING 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}]}}; 

    \(opt1 = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        Frame -> True, GridLines -> Automatic, 
        DefaultFont -> {"Times-Roman", 8}}; \)\)

Operating Conditions

Operation
       = {abm -> 1 nano, k -> 20., f0 -> 100 kilo, 
        Q -> 100, set -> 75 nano};

w0 = 2 π f0 /. Operation;

TipSample
       = {Rtip -> 10 nano, E1 -> ESi, E2 -> ESi, 
        nu1 -> 0.3, nu2 -> 0.3, sigma -> 0.34 nano, 
        A -> 0.1 atto};

fresonance = f0 /. Operation;

fminus = f0 (1 - (0.35/Q)) /. Operation;

fplus = f0 (1 + (0.35/Q)) /. Operation;

Frequency = fminus; fandw
       = {f -> Frequency, w -> 2 π Frequency, 
        w0 -> 2 π f0 /. Operation};

case = TipSample ⋃ Operation ⋃ fandw;

Tip-Sample Interaction

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] /. case;

Clear[FLJ]; FLJ[s_] = -(∂_s) WLJ[s];

sFzero = 
      Solve[FLJ[s] == 0, s][[6, 1, 2
        ]]; Print["sFzero = ", N[(sFzero/nano)], " [nm]"]

Fig9x2 = 
      Plot[FLJ[s nano]/nano, {s, (sFzero/nano) /. case, 2}, 
        FrameLabel -> {"s in nm", "FLJ in nN"}, Evaluate[opt]];

Indentation Repulsive Force

kappa1 = ((1 - (nu1^2))/(π E1)); kappa2 = ((1 - (nu2^2))/(π E2)); kappaeff = kappa1 + kappa2;

ks = N[(4/(3 π kappaeff))] /. case; Print["ks = ", ks nano, " [nN/m^2]"]

Total Tip-Sample Force

Clear[Ftotal, s, kk]; Ftotal[s_, kk_] := 
      If[s < sFzero, kk (Sqrt[Rtip]) Abs[s - sFzero]^(3/2) /. case, FLJ[s]];

tableI = 
      Table[ListPlot[
          Table[{(s/nano), Ftotal[s, kk]/nano}, {s, 0.1 nano, 0.5 nano, 
              0.01 nano}], PlotJoined -> True, 
          FrameLabel -> {"s in nm", "F in nN"}, 
          DisplayFunction -> Identity, Evaluate[opt1]], {kk, ks, 10 ks, 
          2 ks}];

Fig9x3 = Show[tableI, DisplayFunction -> $DisplayFunction];

Clear[Force]; Force[s_] := Ftotal[s, ks]

Equation of Motion

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

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;

General 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[x, fun, phifree]; fun[x_] := 
      If[x == 1, -(π/2), 
        ArcTan[(1/(Q ((1/x) - x)))] /. case]; phifree[x_] := -If[x < 1, fun[x], π + fun[x]]

Clear[x, zfree0]; zfree0[x_] := 
      ((abm Q)/(Sqrt[(x^2) + (Q^2) (((1 - (x^2)))^2)])) /. case

Clear[zfree]; x = (w/w0) /. case; zfree[t_] := 
      set + zfree0[x] (1 - Exp[-((w t)/(2 Q))]) 
            Sin[w t + phifree[x]] /. case;

Transient Regime

Fig9x4 = 
      Plot[Evaluate[{z[t cycle]/nano, zfree[t cycle]/nano}], {t, 0, 10}, 
        FrameLabel -> {"cycles", "z and zfree in nm"}, 
        Evaluate[opt]];

Steady-State Regime

Fig9x5 = 
      Plot[{z[t cycle]/nano, zfree[t cycle]/nano}, {t, nn - 2, nn}, 
        FrameLabel -> {"cycles", "z and za in nm"}, 
        Evaluate[opt]];

Tapping Phase Diagram

Fig9x6 = 
      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

Fig9x7 = 
      ParametricPlot[{z[t cycle]/nano, v[t cycle]}, {t, nn - 3, nn}, 
        AspectRatio -> 1, 
        FrameLabel -> {"z in nm", "v in m over sec"}, 
        Evaluate[opt]];

Numerical Value of the Phase Shift

Clear[vfree]; \(\(vfree[t_] = ∂\_{t, 1}zfree[t]; \)\)

Fig9x8 = 
      Plot[Evaluate[{v[t cycle], vfree[t cycle] /. case}], {t, nn - 2, nn}, 
        FrameLabel -> {"cycles", "v in m over s"}, 
        Evaluate[opt]];

tfree = 
      FindRoot[vfree[t cycle] == 0, {t, nn - .2, nn - .8}]
          [[1, 2]]; Print["tfree = ", tfree, " [cycle]"]

ttap = 
      FindRoot[v[t cycle] == 0, {t, nn - .2, nn - .8}]
          [[1, 2]]; Print["ttap = ", ttap, " [cycle]"]

deltaphi = ttap - tfree; Print["deltaphi = ", deltaphi 360, " [deg]"]

Summary of Results

List of Key Results

Maximum Indentation, Force, and Pressure

tmin = 
      FindRoot[v[t cycle] == 0, {t, nn - 1.5, nn - 1}]
          [[1, 2]]; tmax = 
      FindRoot[v[t cycle] == 0, {t, nn - 1, nn - .5}]
          [[1, 2]];

zave = Abs[(1/2) (z[tmax cycle] + z[tmin cycle])]; Anew = Abs[(1/2) (z[tmin cycle] - z[tmax cycle])];

dmax = If[z[tmin cycle] < sFzero, z[tmin cycle] - sFzero, 0];

Fmax = N[Force[z[tmin cycle]]];

rmax = If[dmax < 0, N[(((((Rtip Fmax)/ks)))^(1/3))] /. case, 0];

Pmax = 
      If[dmax < 0, 
        N[((3 ((((ks/Rtip)))^(2/3)) (Fmax^(1/3)))/(2 π))] /. 
          case, 0];

Operating Parameters

Print["Rtip    = ", (Rtip/nano) /. case, " [nm]"]Print["Hamaker = ", (A/atto) /. case, " [aJ]"]Print["sigma   = ", (sigma/nano) /. case, " [nm]"]Print["abm     = ", (abm/nano) /. case, " [nm]"]Print["Q       = ", Q /. case]Print["f0      = ", N[(f0/kilo)] /. case, " [kHz]"]Print["set     = ", (set/nano) /. case, " [nm]"]

Key Results

Print["f       = ", N[(f/kilo)] /. case, " [kHz]"]Print["ampl    = ", zfree0[x]/nano, "->", (Anew/nano), 
      " [nm]"]Print["zave    = ", (set/nano) /. case, "->", (zave/nano), 
      " [nm]"]Print["phase   = ", N[0], "->", N[deltaphi 360], " [deg]"]
      Print["dmax    = ", (dmax/nano), " [nm]"]Print["Fmax    = ", (Fmax/nano), " [nN]"]Print["rmax    = ", N[(rmax/nano)], " [nm]"]Print["Pmax    = ", N[(Pmax/giga)], " [GPa]"]

Four Figures

Displacement

zGraphics = 
      Plot[z[t cycle]/nano, {t, nn - 1, nn}, 
        FrameLabel -> {"cycles", "z in nm"}, 
        PlotLabel -> "a", DisplayFunction -> Identity, 
        Evaluate[opt]];

Velocity

vGraphics = 
      Plot[v[t cycle], {t, nn - 1, nn}, 
        FrameLabel -> {"cycles", "v in m over sec"}, 
        PlotLabel -> "b", DisplayFunction -> Identity, 
        Evaluate[opt]];

leftshift = .2; Fig9x9 = 
      Show[Graphics[{Rectangle[{0, 0}, {1, 1}, zGraphics], 
            Rectangle[{1 - leftshift, 0}, {2 - leftshift, 1}, vGraphics]}], 
        AspectRatio -> .(5/GoldenRatio)];

Indentation

clear[ind]; ind[t_] := If[z[t] < sFzero, z[t], sFzero]

indGraphics = 
      Plot[ind[t cycle]/nano, {t, nn - 1, nn}, 
        FrameLabel -> {"cycles", "d in nm"}, 
        PlotLabel -> "a", DisplayFunction -> Identity, 
        Evaluate[opt]];

Force

FGraphics = 
      Plot[Force[z[t cycle]]/nano, {t, nn - 1, nn}, 
        FrameLabel -> {"cycles", "F in nN"}, 
        PlotLabel -> "b", DisplayFunction -> Identity, 
        Evaluate[opt]];

leftshift = .2; Fig9x10 = 
      Show[Graphics[{Rectangle[{0, 0}, {1, 1}, indGraphics], 
            Rectangle[{1 - leftshift, 0}, {2 - leftshift, 1}, FGraphics]}], 
        AspectRatio -> .(5/GoldenRatio)];

Appendix

The following tables sum up the conditions and numerical results for tapping on a soft sample, a polymer, and a hard sample, silicon.  For each, the frequency of operation is at one of the two steepest slopes of the resonance curve, or on resonance.  Common to all cases are the following conditions:

Tapping on a Polymer Sample

R             = 10 [nm]
Hamaker = 0.1 [attoJ]
sigma      = 0.34 [nm]
abm        = 1 [nm]
Q            = 100
f0           = 100. [kHz]
set          = 75 [nm]

f=f0

f       = 100. [kHz]
ampl    = 100->81.5026 [nm]
zave    = 75->75.0657 [nm]
phase   = 0->-35.8166 [deg]
dmax    = -6.62973 [nm]
Fmax    = 78.6542 [nN]
rmax    = 8.14232 [nm]
Pmax    = 0.566457 [GPa]

f=f0(1-0.35/Q)

f       = 99.65 [kHz]
ampl    = 82.1635->77.2123 [nm]
zave    = 75->75.0084 [nm]
phase   = 0->-4.48696 [deg]
dmax    = -2.39673 [nm]
Fmax    = 17.0965 [nN]
rmax    = 4.89564 [nm]
Pmax    = 0.340588 [GPa]

f=f0(1+0.35/Q)

f       = 100.35 [kHz]
ampl    = 81.6843->84.2583 [nm]
zave    = 75->75.139 [nm]
phase   = 0->-68.5406 [deg]
dmax    = -9.31221 [nm]
Fmax    = 130.935 [nN]
rmax    = 9.64998 [nm]
Pmax    = 0.671345 [GPa]

Tapping on a Silicon Sample

f=f0

f       = 100. [kHz]
ampl    = 100->75.6009 [nm]
zave    = 75->75.03 [nm]
phase   = 0->-40.7831 [deg]
dmax    = -0.763788 [nm]
Fmax    = 276.808 [nN]
rmax    = 2.76367 [nm]
Pmax    = 17.3041 [GPa]

f=f0(1-0.35/Q)

f       = 99.65 [kHz]
ampl    = 82.1635->75.2304 [nm]
zave    = 75->75.0026 [nm]
phase   = 0->-7.0997 [deg]
dmax    = -0.420616 [nm]
Fmax    = 113.123 [nN]
rmax    = 2.05089 [nm]
Pmax    = 12.8412 [GPa]

f=f0(1+0.35/Q)

f       = 100.35 [kHz]
ampl    = 81.6843->75.8707 [nm]
zave    = 75->75.0463 [nm]
phase   = 0->-76.9993 [deg]
dmax    = -1.01728 [nm]
Fmax    = 425.482 [nN]
rmax    = 3.18948 [nm]
Pmax    = 19.9702 [GPa]