NONCONTACT MODE

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::spell,General::spell1];

Needs["Graphics`Arrow`"];

tera  = 10^12  ; micro = 10^-6;
giga  = 10^9   ; nano  = 10^-9; 
mega  = 10^6   ; pico  = 10^-12;
kilo  = 10^3   ; femto = 10^-15; 
milli = 10.^-3 ; atto  = 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. Pi 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[{.0200}]}};

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 Pi f,w0->2 Pi f0}/.Operation;
case  = Union[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_] = -D[WLJ[s],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 [nm]","FLJ [nN]"},
Evaluate[opt]];

Lennard-Jones Force Derivative

Clear[F1];
F1[s_] = D[FLJ[s],s];

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

sF1max = Solve[D[F1[s],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 [nm]","F1 [N/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/Q x2[t] - 
w0^2 (x1[t]-set-abm Sin[2 Pi f t])+
(w0^2)/k F[x1[t]]/.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/(2 Q)t])*
Sin[w0 t-Pi/2]/.case;

Transient Regime

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

Steady-State Regime

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

Displacement-Velocity Phase Diagram

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

Numerical Value of the Phase Shift

vbm[t_] := abm w0 Cos[2 Pi t]/.case;

Fig8x8 =
Plot[{
v[t cycle],
Q vbm[t]/.case},
{t,(nn-2),nn}, 
FrameLabel ->{"cycles","v [m/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 =
D[d[t],{t,2}] +
w0/Q D[d[t],t] +
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/.Sqrt[x_^2]->x];
Print["A10 = ",A10]

$F_1$-Dependent Phase Shift

B1=ComplexExpand[A1];
phi2=ArcTan[-I B1[[1]]/B1[[2]]];
Print["phi2 = ",phi2]

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

Clear[phi3];
phi3[F1_] := Series[Tan[phi20+Pi/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 [N/m]","A/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 [N/m]","A/A0"},
Evaluate[opt]];

Approximate $F_1$-Dependent Phase Shift

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

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

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

Print["phi3 = ",N[phi3[set] 180/Pi/.case]," [deg]"]

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