How to Measure Forces when the Atomic Force Microscope shows Non-Linear Compliance

2.6 MB·English

by  Phil Attard

How to Measure Forces when the Atomic Force Microscope shows Non-Linear
Compliance
Phil Attard
(Dated: 16 November,2012. [email protected])
A spreadsheet algorithm is given for the atomic force microscope that accounts for non-linear
behavior in the deflection of the cantilever and in the photo-diode response. In addition, the data
analysisalgorithm takesintoaccount cantilevertilt,friction incontact,and base-lineartifacts such
asdrift,virtualdeflection,andnon-zeroforce. Theseareimportantforaccurateforcemeasurement
and also for calibration of the cantilever spring constant. The zero of separation is determined
3 automatically,avoidinghumaninterventionorbias. Themethodisillustratedbyanalyzingmeasured
1 data for thesilica-silica drainage force and slip length.
0
2
n I. INTRODUCTION 10 N)600
Ja Scanning probe microscopy has revolutionised surface 8 ce (n400
or
2 science by enabling the production of images of surfaces V)V) 6 F
1 with molecular resolution. In one technique a topo- e (e ( 200
dd
graphicmapisproducedbasedontheheightadjustment diodio 4 00
] oo
ft ooffathpeieczaon-tdirleivveerrepqruoibreeddtuorimngaiantraaisntearcsocnasnt.anTthdeeoflreicgtiinoanl PhotPhot 2 -50 0 50
o Separation (nm)
s atomicforcemicroscopeusedtunnelingcurrentstodetect 0
. the deflection of the cantilever.1 This was soon modified
t
a tousealightlevertodetectthedeflection,whichhadthe -2
m advantage of not requiring vacuum conditions.2,3 It also 1200 1300 1400 1500 1600 1700 1800
- allowedquitelargedeflectionstobemeasured,anddiffer-
Piezo-drive (nm)
d
entimagingtechniquestobedeveloped,suchasconstant
n
o height imaging. For such techniques to be quantitative, FIG. 1: The raw photo-diode voltage versus the piezo-drive
c the light lever signal (in volts) has to be converted into displacement. The solid curve is measured extension data,
[ the deflection of the cantilever (in nanometers). the dashed line is the tangent to the contact region at first

This issue of quantitatively calibrating the light lever contact, and the dotted line uses the average slope in con-
2
received added impetus with the further modification of tact. The inset shows the analysed force versus separation,
v
9 theatomicforcemicroscopeinwhathascometobecalled with the solid curve resulting from the non-linear analysis
1 colloid probe force microscopy.4 In this technique a col- (see text) and the dashed and dotted curves resulting from
the conventional linear analysis using the first contact slope
0 loid sphere of measured radius (R 10–20µm) is glued
3 to the end of the cantilever spring≈instead of the sharp and the average slope, respectively. The source of the mea-
. sured data is Ref. 6; a summary of the experimental details
2 tip used for imaging. The object is to measure the so-
is given in Ref. 7.
1 called surface force between the substrate and the probe
2 as a function of separation. The surface force is just the
1
spring constant times the cantilever deflection, and so if Here V is the photo-diode voltage and z is the piezo-
p
:
v the light lever is properly calibrated, the measurement drive position. Letting Vb be the base-line voltage far
i may be performed with molecular resolution. from contact, the surface force is
X
ar beTamhereliflgehcttilnegveorffittsheelfbiascuksoufatllhyemcaandteilefvroemrsparninogpotinctaol F(zp)=−k0β−1[V(zp)−Vb], (2)
a split photo-diode. A change in angle of the cantilever, where k is the cantilever spring constant. The force
0
due, for example, to a change in the force on the probe, goes to zero at large separations, which is the base-line
causes the light beam to move across the face of the region. This result invokes the fact that in hard con-
photo-diode. The consequent change in voltage differ- tact the change in tip position is equal and opposite to
encebetweenthetwohalvesismeasuredandtakentobe the change in piezo-drive position,5 ∆z = ∆z , and
t p
proportional to the change in cantilever angle. By using the force on the cantilever is F = k z . (T−he simple
0 t
the piezo-drive to press the cantilever against the hard presentation given here ignores a number of important
substrate,theproportionalityconstantisobtainedasthe linear effects such as base-line drift, friction in contact,
slope of the voltage versus distance signal. cantilever tilt, and non-negligible base-line force. These
To make this clear mathematically, let the measured and non-linear effects will be included in the more so-
constant compliance slope be phisticated linear and non-linear analysis below.) The
separation is
∆V
β . (1)
≡ ∆zp(cid:12)contact h = zp+zt+const
(cid:12)
(cid:12)
(cid:12)
2
= z +k−1Fext(z )+const, (3) photo-diode voltage and the change in cantilever angle.
p 0 p
Since the change in voltage difference in the split photo-
where the constant is chosen so that h=0 in contact.
diodedependsonlyonthatpartoftheopticalbeamcur-
Itoughttobe clearfromthe abovehowimportantthe
rently crossing the boundary (assuming uniform sensi-
calibration factor β, which is the constant compliance
tivity of the photo-diode; any spatial variability in the
slope,istothequantitativemeasurementofsurfaceforces
sensitivity willcontributefurther to non-linearities),any
with the atomic force microscope. However it is not un-
variability in the spatial intensity or width of the beam
usual for the photo-diode voltage versus piezo-drive po-
(e.g. circular or elliptical cross-section, Gaussean inten-
sition curve to be non-linear in the contact region. A
sity distribution) will give non-linear effects.
typicalexampleis showninFig.1. Itis emphasisedthat
thedatainthefigurewereobtainedforhardsurfacesand
sothecurvatureevidentisnotduetoelasticdeformation
ofthe probe orsubstrate. Acurvedconstantcompliance
region such as that in Fig. 1 creates several problems.
At a minimum, the calibration factor β is not unique;
the slope depends upon where in the contact region it
is measured, which introduces quantitative uncertainty
into the linear analysis, (compare, for example, the re-
sult that uses the slope at first contact (dashed line and
1. Contents
curve) with the result that uses the average slope (dot-
ted line and curve)). Ambiguity also arises because the
contactbehaviordiffersbetweenextensionandretraction
(not shown). Worse, a non-linear contact region contra- For the case of the rectangular cantilever, a complete
dicts the fundamental assumption of a linear response linearanalysisisgivenin II.In IIB,theeffectsofbase-
§ §
and raises questions about the conventional linear anal- linedrift,virtualdeflection,cantilevertilt,frictionincon-
ysis that is used to quantify surface force measurements. tact,andnon-negligiblebase-lineforceareaccountedfor.
The inset to Fig. 1 graphically illustrates the prob- Theseareoftenneglectedintheconventionallinearanal-
lems with the linear analysis. It can be seen that it ysis. A new result is an algorithm for determining the
gives grosslyunphysicalnon-zeroseparations in the con- zero of separation, IIB5. The effective spring constant
§
tact region. In this case the surfaces are known to be that must be used in the linear analysis and its relation
rigidsotheresultsisunambiguouslyunphysical. Inother to the cantilever spring constant is given in IIB6. In
§
cases, where the surfaces are either of unknown rigidity IIIA are given the non-linear equations for a rectan-
§
or known to be soft, this artefact of the linear analy- gular cantilever that determine the deflection, deflection
sis would be misinterpreted as the elastic deformation angle,verticalposition,andappliedforce,takingintoac-
of the material. Not only the elasticity but any useful counttiltandfriction. In IIIBanalgorithmsuitablefor
§
physical information in the contact region is precluded spreadsheet use is given for the analysis of experimental
by the linear analysis. As will be demonstrated below, data (i.e. the conversion from raw voltage versus piezo-
thenon-linearanalysiscanbeusedtoobtainreliableval- drive position to force versus separation) in the case of
ues for properties like the friction coefficient, roughness, non-linear behavior. The non-linearities can arise either
and topography of the contact region. fromthenon-linearcantileverdeflectionorthenon-linear
It is not only in contact that the linear analysis can photo-dioderesponse,orboth. Forthenon-linearphoto-
fail. If a large surface force is present, then the linear diodecase,thenon-linearityischaracterizedbythemea-
analysis of the data introduces quantitative errors into surementitselfanditisnotnecessarytoknowthedetails
the values of the surface force in the non-contact region. of the source of the non-linearity. This is fortunate be-
This can be a particular problem if one requires reliable cause unlike rectangular cantilevers, these vary between
and accurate surface force measurements, or if one seeks differentmodelsoftheatomicforcemicroscope. In IIIC
§
smallchangesinforceswithcontrolparameters,orifone thecaseofalinearphoto-diodeandnon-linearcantilever
needs to quantify second order effects. In all these cases is explorednumerically for the case shown in Fig. 1, and
the linear analysis can be unsuitable, depending upon itisconcludedthatthenon-linearcantileverdeflectionis
the extent of the non-linearity and the magnitude of the insufficienttoaccountforthemeasurednon-lineareffects.
forces. In IIID, summarizedare the equations for a linear can-
§
The are two possible physical origins of the non- tileverandanon-linearphoto-diode,whicharesomewhat
linearity displayed in Fig. 1. The first possibility is that simpler than the dual non-linear case. In IV, these are
§
the cantileverdeflection becomes non-linearoverthe rel- appliedtomeasuredatomicforcemicroscopedataforthe
atively large range of the contact region. By cantilever drainage force at severaldrive velocities. Results for the
non-linearity is meant that the four relevant quantities slip length and the drainage adhesion are obtained. The
(tip position, deflection, angle deflection, and force) are quantitative and qualitative differences between the lin-
not linearly proportionalto eachother. The second pos- ear and the non-linear analysis of the experimental data
sibilityisanon-linearrelationshipbetweenthechangein are shown.
3
II. LINEAR ANALYSIS In this paper the elastic parameter B will be used, as it
is an intrinsic property of the cantilever. Usually (but
A. Horizontal Cantilever, No Friction not always; see IIB6 below) any quoted or measured
§
spring constant is the horizontal, free spring constant in
the above sense, and so this equation can be used to
1. Deflection
convert from k to B.
0
For free deflection, τ = 0, the angular deflection is
The bending of a cantilever beam under the influence
linearly proportionalto the deflection,
of fores and torques is one of the classic problems of the
theory of elasticity. A beam of length L , and with a
0 3
deflection x and angular deflection θ has stored elastic θ = x. (10)
2L
energy8 0
2B This particular proportionality constant only holds for
U(x,θ)= 3x2 3xL θ+L2θ2 . (4)
L3 − 0 0 the free, horizontal cantilever. For the free tilted can-
0
tilever, the two remain linearly proportional to each
(cid:2) (cid:3)
The elastic parameter B EI depends upon Youngs other, but a different constant applies, as is derived be-
≡
modulus andthe geometricsecondmomentofthe beam; low.
it will be related to the spring constant of the cantilever
The linear proportionality of deflection angle θ and
beam below. Differentiating with respect to the deflec- deflection x underlies the linear analysis of atomic force
tion gives the force exerted on the end of the beam,
microscope. The light lever is assumed to give a change
in voltage that is linearly proportional to the deflection
∂U(x,θ) 2B
F ≡ ∂x = L30 [6x−3L0θ]. (5) aanngdlef,orγce≡, ∆arVe/a∆llθl.inSeianrcleytphreodpeoflrteicotnioanl taongelaec,hdeofltehcetriobny,
anddifferentiatingwithrespectto anglegivesthe torque the abovetwoequations,thenonecangetthe forcefrom
the measured change in voltage,
∂U(x,θ) 2B
τ = 3L x+2L2θ . (6)
≡ ∂θ L3 − 0 0 2k L 2k L
0 0 0 0 0
F =k x= θ = ∆V. (11)
These assume that the beam(cid:2)is in equilibrium(cid:3) with the 0 3 3γ
applied forces and torques.
Ifthemeasuredgradientofthephoto-diodesignalincon-
Inverting these equations gives the standard expres-
tact is β ∆V/∆z = ∆V/∆x, then
sions for the deflection and the angle in terms of the ≡ p −
applied force and torque,8
2L
0
γ = − β. (12)
x= 1 2L3F +L2τ , (7) 3
2B 3 0 0
(cid:20) (cid:21) Hence F = k β−1∆V. This is the conventional lin-
0
and −
ear analysis for extracting the force from of atomic force
1 microscope measurements. It ought be clear it assumes
θ = L2F +2L τ . (8)
2B 0 0 a horizontal cantilever with no torque, neither of which
(cid:2) (cid:3) assumption holds in practice.
2. Spring Constant
3. Spring Constant Calibration
Onehastobealittlecautiousaboutassigningaspring
constant. The above equations refer to a free, horizontal
One of the most important issues in measuring forces
cantilever beam, and now the spring constant for such
with the atomic force microscopeis the determinationof
a beam with normal force and zero torque (free deflec-
the spring constant. This is usually the largest source
tion) will be given. It is emphasized that this cannot be
of systematic error. The common thermal calibration
applied to the atomic force microscope without modifi-
procedure9 that is often built into the software of the
cation because in that case the cantilever beam is tilted,
atomic force microscope gives erroneous results, with a
the force is not entirely normal to the beam, and there
systematic overestimate of the spring constant of 15%–
arenon-zerotorquesduetothisandduetofriction. This
30%.9,10 The source of the error in the derivation has
case will be handled shortly.
been identified and a more reliable thermal calibration
For the horizontal beam with zero torque, τ = 0, the
method has been given.10 (The correct thermal calibra-
deflection is linearly proportional to the applied force,
x=[L3/3B]F,fromwhichonecanidentifythecantilever tion formula for the cantilever spring constant is given
0 in Eq. (35) below.) An even more accurate and reliable
spring constant as
wayofdeterminingthe springconstantistousethe long
k 3B/L3. (9) range hydrodynamic drainage force.11–13 In general the
0 ≡ 0
4
z with the constant zext calculated to give zero separation
q p 0
x 0 atfirst contact,as defined below. In contact, ∆h=0, so
q L that ∆z = ∆z .5
0 t p
−
Trigonometric functions of the tilt angle occur fre-
R
quently below andso itis convenientto define fixedcon-
L stants
2
R
C cosθ , and S sinθ . (15)
z F 0 0 0 0
t z ≡ ≡
F
y Note that in the present geometry for the atomic force
h
microscope,S θ 0.2,whichissmallinmagnitude
0 0
≈ ≈−
andnegativeinsign. (Allangleshereandthroughoutare
◦
measured in radians; in degrees, θ 11 .)
0
FIG. 2: Cantilever geometry in the atomic force microscope ≈−
(not to scale).
2. Friction Force
drainageforceis knownexactly,atleastinthe largesep-
In this work the lateral force will be taken to be due
aration regime
to friction(in contact)andit willbe takento be linearly
proportional to the load,
6πηR2z˙
p
F (h)= − , (13)
drain
h µF , extension, contact,
z
F = µF , retraction, contact, (16)
whereηistheviscosity. Itispermissibletousethepiezo- y  − z
0, non-contact.
drivevelocityz˙ ratherthantherateofchangeofchange 
p
of separation in this because the deflection is small and
This is known as Amontons law. As drawn in Fig. 2, on
its rate of change is negligible at large separations. The
extension z˙ < 0, and so in contact on extension z˙ >
correctspringconstantgivesagreementbetweenthisand p t
0. This means that due to the tilt angle, y˙ < 0 and
the measured force at long range.
F > 0. Since in contact F > 0 (at least sufficiently
It shouldbe noted thatif this methodof calibrationis y z
far into contact), this accounts for the sign of the first
usedinconjunctionwiththe conventionalanalysisofthe
equality. (In general the friction coefficient is positive.)
measureddata (linear calibration,horizontalcantilever),
The opposite occurs on retraction (y˙ > 0 and F < 0),
then the spring constant that results is the effective y
the second equality. Out of contact there is no friction.
spring constant k rather than the intrinsic cantilever
eff The assumption that the friction force is linearly pro-
spring constantk . These are defined in the full analysis
0 portionaltotheloadisasignificantone. Thisiscertainly
forthetiltedcantileverwithfrictionthatistreatednext.
theclassicalmodeloffriction,atleastatthemacroscopic
level. There is evidence in the atomic force microscope
literature for16 and against17 such an assumption. The
B. Tilted Cantilever with Friction
formerdata is perhaps the mostconvincing asfour inde-
pendent measurements were made (two different colloid
1. Model probes, two different friction measurement methods).
Immediatelyaftertheinitialcontactonextension,and
Following earlier work,14,15 the cantilever and probe at the beginning ofthe retractionbranchin contact (the
in the atomic force microscope is modeled as in Fig. 2. turn point), the probe is not moving at uniform velocity
Thekeyfeaturesarethefixedtiltangleofthecantilever, and so the assumed form for the friction force will pro-
θ < 0, such that the total angle of the cantilever is the duceartifactsatthesepointsinthe analyzeddata. Also,
0
sumofthisandthedeflectionangle,θ =θ +θ,andthe when the force at either first or last contact is non-zero,
tot 0
rigid lever arm, L (θ ), which connects the cantilever the model gives a discontinuity in the friction force and
2 tot
tothe pointofapplicationofthe normalsurfaceforceF consequently a discontinuity in the surface force that is
z
and the lateral friction force F , if present. The force F also an artefact of the simple model.
y
and torque τ on the cantilever treated in the preceding All the following results will be givenexplicitly for ex-
section are a function of these two forces, the lever arm, tension in contact. The retraction results may be ob-
and the tilt angle, as will now be derived. tained by the replacementµ µ, and the non-contact
⇒−
Note that in the figure the piezo-drive is connected to results may be obtained by the replacement µ 0. The
⇒
the base of the cantilever, so that extension corresponds superscripts‘ext’(contact,µ>0),‘ret’(contact, µ<0),
to z˙ < 0 and retraction corresponds to z˙ > 0.5 The and‘nc’(non-contact,µ=0)willoftenbeusedtodenote
p p
separation between the surfaces is each of the three cases. For the voltage, the piezo-drive
position,andthe surfaceforce,whichispossiblyvelocity
h z +z +zext, (14) dependent, the superscripts ‘ext’ and ‘ret’ will be used
≡ p t 0
5
also in the non-contact situation. The subscript ‘c’ de- 4. Linear Analysis of Measured Data
notesaquantityincontact,andthesubscript‘b’denotes
a quantity in the base-line region far from contact. Themeasureddataintheatomicforcemicroscopecon-
sists of the raw photo-diode voltage V˜(t) and the piezo-
drive position z (t). These are both a function of time,
p
and so the voltage may equivalently be regarded as a
3. Linear Cantilever Equations
function of position, V˜(z ). One has in fact two sets
p
of data, one for extend, V˜ext(z ), and one for retract,
In Eqs (45) and (46) below, non-linear expressionsare p
V˜ret(z ). Onlythe equationsforextensionwillbe shown
derivedforthedeflection,theangulardeflection,andthe p
explicitly here.
force. In the linear regime one can simply make the re-
Thetildeonthevoltageisusedtodenotetherawmea-
placement θ θ to obtain
tot 0
⇒ sured voltage. The raw voltage contains contributions
fromthe changein angle ofthe cantilever andfromvari-
L3 L2L
x= 0 [C +µS ]+ 0 2 [S µC ] F , (17) ousartifactsthatincludeaconstantvoltageoff-set,ther-
0 0 0 0 z
(cid:26)3B 2B − (cid:27) mal drift, and virtual deflection.12,13 Two further physi-
caleffects havetobe carefully accountedfor,namely the
and
dragforceonthecantileverandthelongrangeasymptote
ofthesurfaceforce. Thedeflectionangleofthecantilever
L2 L L
θ = 0 [C +µS ]+ 0 2 [S µC ] F duetothesurfaceforcesiswhatisdesiredtoextractfrom
0 0 0 0 z
2B B −
(cid:26) (cid:27) the measured voltage. The notation V(zp) will be used
EextF . (18) to denote the measured voltage that has been corrected
z
≡
for these various artifacts and forces.
Herethelengthoftheleverarmintheundeflectedstateis Inthebase-lineregion,wherethesurfacesarefarfrom
L L (θ )= R√2+2cosθ . In the linear regime, the contact,thesurfaceforceissmallandinmanycasesneg-
2 2 0 0
forc≡e, deflection, and deflection angle are linearly pro- ligible. Hence almost all of the measured voltage in this
portional to each other. In particular it is convenient region is due to the artifacts just mentioned. These in
todefine the proportionalityconstantbetweendeflection general are linear functions of position, and one can de-
and deflection angle from fine the measured base-line voltage on extension as
V˜ext(z )=V˜ext+β˜ext[z z ], (21)
x = 2L30[C0+µS0]+3L20L2[S0−µC0]θ b p b b p− pb
≡ D3Lex20t[θC.0+µS0]+6L0L2[S0−µC0] (19) adnV˜dexstim(zipla)/rldyzpforre.trTahcetiocno.effiTchieenbtassfeo-rlintheissloapreeoisbβt˜abeixnte≡d
zpb
by a linear fit t(cid:12)o the measured data in an interval about
IngeneralL2 L0,andthisandtheaboveresultscould (cid:12)
≪ the fixed posit(cid:12)ion z in the base-line region. Once the
be expanded to linear order in L /L . There is no great pb
2 0
coefficients are determined, this linear fit is applied to
advantage in doing this.
the whole measuredregime,not just the base-lineregion
The vertical position of the tip depends upon the de-
(becausethe artifactsthatit removesapply tothe whole
flection, the deflection angle, and the length of the lever
regime). With this the corrected voltage on extension is
arm, Eq. (40) below. The linearized form of Eq. (39) for
the lengthofthe leverarmis L (θ )=L R2S θ/L .
For the case of a tipped cantil2evetro,tL2(θt2ot−) is eq0ual t2o Vext(zp)=V˜ext(zp)−V˜bext(zp). (22)
the length of the tip, and there is no dependence on the
This is zero in the base-line region.
deflection angle (i.e. the term in R2 may be set to zero).
This expression removes from the raw signal not only
Linearising Eq. (40) below for the vertical tip position
the artifactsmentionedabovebutalsothe constantdrag
yields force. (In some cases the drag force is not constant.12,13
This effect, which can be important for cantilevers with
z = C x+ L S R2S02θ0 +L C θ θ alowspringconstant,isnotincludedinthepresentanal-
t 0 2 0 2 0 0
− L ysis.) It also removes the linear extrapolation of the
(cid:20) 2 (cid:21)
R2S2θ asymptote of surface force,
= DextC +L S 0 0 +L C θ θ
0 2 0 2 0 0
θ(cid:26)/αext. − L2 (cid:27) (20) Fbext(zp)=Fbext+Fbext′[zp−zpb]. (23)
≡ Here the constant force is Fext = Fext(h ) and the
b b
The proportionality constant is α dθ/dz . It has a derivative is Fext′ = dFext(h )/dh , where the separa-
differentvalue for eachofthe threec≡asesµ>t0 (contact, tion is h = zb + zext. (Thbis nebglects the deflection
b pb 0
extension), µ < 0 (contact, retraction), and µ = 0 (non- of the cantilever, which should be negligible; if not, add
contact). to the separation z Fext/k .) In almost all cases
tb ≈ b 0
6
this extrapolated surface force from the base-line region With γ having been obtained from the measured con-
is negligible. In those cases where it isn’t, it has to be tactslopeandthecalculatedrateofchangeoftipposition
added back, as will be done shortly. with angle, Eq. (26), one can now give the surface force
Thecontactregioniswheretheseparationbetweenthe as a function of separation. From Eq. (18), the angle
surfaces is zero, andthe tip movesequaland opposite to deflection is
the piezo-drive, ∆z = ∆z .5 In the linear regime, the
slopeisconstantantdth−isisaplsocalledtheconstantcom- θext(zp)=γ−1Vext(zp)+EncFbext(zp). (28)
pliance regime. It does not matter whether one fits the
raw data or the corrected data because the two contact Here the contribution of the linear extrapolation of the
slopes are related by asymptote of the surface force, Fbext(zp), which is as-
sumed a known function and which was removed from
∆Vext ∆V˜ext the raw voltage signal, has been added back to give the
βcext ≡ ∆zc , β˜cext ≡ ∆zc , βcext =β˜cext−β˜bext. full deformation angle. Note that it is the non-contact
p p value of the conversion factor, Enc = E(µ = 0) that is
(24)
used here. Inserting this into Eq. (18) gives the surface
Ifapositivevoltagecorrespondstoarepulsiveforce,then
force Fext(z ). Explicitly in terms of the voltage it is
the slope ought to be negative. p
The light lever measures the angle of the cantilever.
1
Tishteokceaylitboraatneatlyhzeinligghattolmeviecrfobrycemmeaicsruorsicnogpethfeorpcreodpaotra- Fext(zp)= En1cγVext(zp)+Fbext(zp), h>0, (29)
tionality constant between angle and photo-diode volt-  EextγVext(zp)+Fbext(zp), h=0.
age,
From Eq. (20) the separation is
∆V
γ . (25)
≡ ∆θ 1
hext(z ) = z + θext(z )+zext (30)
p p αnc p 0
This is the same on extension and retraction, and it is
1 Enc
the same in contact and out of contact. This expression = z + Vext(z )+ Fext(z )+zext.
p αncγ p αnc b p 0
assumesalinearphoto-diode,butnotnecessarilyalinear
cantilever.
Theconstantzext iscalculatedsothathext =0whenthe
The value of this conversion factor follows from the 0
surfaces first come into contact (see next). The separa-
measured slope in contact, Eq. (24), and the linear pro-
tionequationisnormallyusedexplicitlyforh>0. These
portionality between angular deflection and tip position,
three equations are written explicitly for extension; for
Eq. (20). Evaluating these on extension in contact one
retractionincontactchangethesuperscript‘ext’to‘ret’,
has
including Eext E(µ) Eret E( µ).
≡ ⇒ ≡ −
∆V ∆z ∆z
γext = p t = βext/αext. (26)
∆z ∆z ∆θ − c
p t
5. Zero of Separation
One has a similar result for retraction in contact γret =
βret/αret. Since this has to be a property of the light Theconstantzext remainstobe determined. The con-
− c 0
lever,the value of γ cannotdepend upon whether or not ventionalway ofestablishing the zeroof separationis by
the surfaces are in contact, or whether the measurement eye,whichistosaytheforcecurveisshiftedhorizontally
is made on extension or on retraction. Hence one must until itlooks ‘right’. The problemwith this is thatthere
have γext =γret, or is often ambiguities in identifying first contact, particu-
larlywhen one has a steeply repulsive surface forceprior
βext α(µ) to contact. Also the constant that gives h = 0 at first
c = , (27)
βret α( µ) contactmaybedifferenttotheconstantthatgivesh=0
c −
for most of the contact region, even when the correct
sinceαext =α(µ)andαret =α( µ). Thelefthandsideis friction coefficient is used, as will be demonstrated by
−
a measuredquantity, and the righthand side is a known explicit data below. Finally, choosingthe zero ofsepara-
non-linear function of µ, Eq. (20). There exist sophis- tion by eye introduces a psychological element into the
ticated algorithms for solving such non-linear equations, analysis and the potential for personal bias that would
with perhaps the mostcommonif notthe mostpowerful bebestremovedbyhavingamathematicalalgorithmfor
being to guess the solution. (This can be turned into a finding contact.
quadratic equation for µ if one expands D(µ) to lead- Itmaybeobjectedthatidentifyingthecontactandthe
ing order in L /L . There is a small loss of accuracy in base-lineregionsalreadyintroduce some formof psycho-
2 0
suchanexpansion,whichisnotcompensatedbytheeven logical bias into the analysis of the experimental data.
smaller gainofanexplicit analytic solution.) This result However, it turns out that the various fits are not very
provides a way of measuring the friction coefficient. sensitivetothechoiceoftheregionusedforthefit,within
7
reason, and the result do not vary significantly with dif- equatingthe cantileverdeflectionto the verticalposition
ferent choices. The zero of separation, however, feeds of the tip, x z . The relationship between the mea-
t
≡
directly into the final result, and a difference of as small suredphoto-diodevoltageandthe verticaltippositionis
as 1nm can quantitatively effect the values of parame- given by the calibration factor obtained from the slope
ters that one is trying to measure (e.g. the slip length of the contact region. In this case, the effective spring
in drainage flow can be of the same order), and it can constant that gives the non-contact force is
evenqualitativelyeffectthephysicalinterpretationofthe
data. k Fz =αnc/Enc. (34)
The strategy is to define zext so that the separation is eff ≡ z
0 t
exactly zero at first contact, (and to define zret so that
0
h=0atlastcontact). ‘First’(or‘last’)contactisdefined The constants Enc E(µ = 0) and αnc α(µ = 0) are
≡ ≡
to mean the point at which the extrapolated base-line defined in Eqs (18) and (20), respectively.
voltageintersectsthe extrapolatedcontactvoltage. This The difference between the cantilever spring constant
definition is precise and unambiguous, it is able to be k0 = 3B/L30 and the effective spring constant keff can
calculatedmathematically,anditisphysicallyreasonable be substantial. For the case analyzed in detail below
◦
and in accord with one intuitive understanding of the (L0 = 110µm, R = 10.1µm, θ0 = 11 ), the cantilever
−
meaning of contact. springconstantisk0 =1.37N/mandtheeffectivespring
It should be understood that there are conceptual constant is keff =1.68N/m.
problemswiththemeaningof‘separation’atthemolecu- In using the equations for the tilted cantilever to con-
larlevel. Theseparationasdefinedhere,h z +z +z , vert measured atomic force microscope data to force,
p t 0
≡
is not precisely zero over the whole contact region. It one should use the cantilever spring constant. In using
rather measures the difference between changes in the the equationsforthe horizontalcantilever(simple spring
piezo-drivepositionandchangesinthe tipposition. Itis model, the conventional approach) to convert measured
positive when there is a protuberance on the substrate, atomic force microscope data to force, one should use
and it is negative when there is a depression. It is also the effective spring constant. In calculating a theoreti-
negative when compression of a deformable surface oc- cal force curve modeled with the cantilever as a simple
curs. Hence the separation h in contact really gives a spring, one should also use the effective spring constant.
topographic map of the substrate. The zero plane of Finally,inRef.10thecorrectequationsforthethermal
the mapis here defined as the plane passingthroughthe calibrationoftheatomicforcemicroscopecantileverwere
point of first or last contact. given. In that paper the cantilever spring constant was
Let zpexct be the piezo-drive position at first contact, denotedk0 (herealsodenotedk0),andtheeffectiveforce
and let Vext = Vext(zext) be the corrected voltage at measuring spring constant was denoted k (here denoted
c pc
first contact. (In the linear case, it makes no difference keff) andwas givenin terms ofthe cantileverspring con-
to the results what point is selected for zext.) In con- stant in Eq. (17) of Ref. 10. In the present notation, the
pc
tact, Vext(z ) = Vext+βext[z zext]. The position at correct thermal calibration method gives the cantilever
p c c p − pc spring constant as
which the voltage in contact extrapolates to zero, which
is defined as first contact, is
2β L
cb 0
k = − (35)
zext =zext Vext/βext. (31) 0 3[C +2L S /L ]
pcb pc − c c (cid:26) 0 2 0 0
(Thebase-linecorrectedvoltageiszero,andsothisisthe 6k T
B
0.7830
same asthe intersectionofthe base-lineandcontactraw × sπL20fRPDCQ
voltages.) Whenthevoltageiszerotheangulardeflection
2
is C2+(3L S C /L )+3S2L2/L2 .
× 0 2 0 0 0 0 2 0
θext(zext)=EncFext(zext). (32) (cid:27)
pcb b pcb (cid:2) (cid:3)
Here the measured quantities are β =∆V/∆z , which
Inserting this into the equation for the separation, and cb p
is the contact slope evaluated near the base-line voltage,
setting the latter to zero, hext(zext) = 0, gives the shift
pcb (the averageofthe extendandretractvalues), f ,which
constant, R
is the resonance frequency of the first mode in Hz, P ,
DC
Enc which is the direct current power response in V2Hz−1,
zext = zext Fext(zext). (33)
0 − pcb− αnc b pcb andQ,whichisthequalityfactor. Thelengthoftherigid
part at the end of the cantilever, L , defined in earlier
1
analyses10,14,15 hashereandthroughoutbeensettozero.
6. Effective Spring Constant This inserted into Eq. (34) gives the effective spring
constantforusewhenthecantileverismodeledasasim-
The conventional modeling of the atomic force micro- plespring(i.e.tiltneglected),whichisusuallythecasein
scopeisnotonlylinearbutalsoeffectivelytakesthecan- the linear analysis of measured data and the theoretical
tilever to be horizontal. Ignoring the tilt is equivalent to modeling of force-separationcurves.
8
7. Effective Drag Length a spherical probe the dependence on angle is practically
negligible, and L (θ ) can be replaced by L , or even
2 tot 2
The above procedure for analyzing the measured data by 2R.
removes the constant force due to the drag on the can- The vertical position of the tip depends upon the de-
tilever from the extension data, and its equal and op- flection andthe deflection angle. Againsimple geometry
posite value from the retraction data. In some case it shows that
is useful to have available an explicit value for this drag
z =xcosθ +L (θ )sin(θ )θ L S θ . (40)
force. t tot 2 tot tot tot− 2 0 0
Like the drainage force, and unlike the virtual deflec-
Since it is the change in tip position that is important,
tion,thecontributiontothegradientofthebase-linedue
thishasbeenchosentobezerointhenon-deflectedstate.
to thermal drift is equal and opposite on extension and
This equation is the major source of non-linearity in the
retraction.
cantilever.
Withβ˜ext andβ˜ret beingthemeasuredbase-lineslopes
b b TheforceF andtorqueτ onthecantilever,whichwere
of the raw voltage as defined above, and Fext′ = Fret′
b − b treatedin IIA,arealinearlyproportionaltothenormal
being the gradient of the drainage force in the base-line §
forceF andto thelateralforceF ,withthe proportion-
z y
region, then the gradient of the voltage due to thermal
ality constantdepending upon the leverarmandthe tilt
drift is
angle,
ddV˜ztehxt = 21 β˜bext−β˜bret − γbE2nc Fbext′ −Fbret′ . (36) F =Fzcosθtot+Fysinθtot, (41)
p
h i h i and
This is equal and opposite to the gradient on retraction.
With zp,turn being the turn point at the end of the ex- τ =FzL2(θtot)sinθtot FyL2(θtot)cosθtot. (42)
−
tendbranchandthebeginningoftheretractbranch,itis
readily shown that the constant drag force on extension Using the linear model offriction, Fy = µFz, IIB2,
± §
is the force andtorqueare linearlyproportionalto the sur-
face force F . For extension in contact one has
1 z
Fext = Fext+ V˜ext V˜ret (37)
drag − b 2γEnc b − b F =[cosθ +µsinθ ]F , (43)
tot tot z
h i
1 dV˜ext
− 2γEnc dzth 2zp,turn−zpexbt−zprebt . and
p
(cid:2) (cid:3) τ =L (θ )[sinθ µcosθ ]F . (44)
One can define an effective drag length from 2 tot tot− tot z
(Of course for retraction in contact, µ µ, and out
Fderxatg ≡−6πηz˙pextLeff. (38) of contact, µ = 0.) Inserting these int⇒o t−he standard
cantilever equations (7) and (8) gives the deflection and
One should not take L too literally, but it is expected
eff
deflection angle as linearly proportional to the normal
to be somewhat less than the length of the cantilever,
force (on extension in contact),
typically one third to one half of L . It should be in-
0
dependent of the drive velocity, although because it is
L3
derivedfromthe difference in the base-linevalues, it can x = 0 [cosθ +µsinθ ]
tot tot
3B
have relative large errors, on the order of 10–20% (see (cid:26)
results below). L2L (θ )
+ 0 2 tot [sinθ µcosθ ] F , (45)
tot tot z
2B −
(cid:27)
III. NON-LINEAR CANTILEVER OR and
PHOTO-DIODE
L2
θ = 0 [cosθ +µsinθ ]
tot tot
A. Non-Linear Cantilever 2B
(cid:26)
L L (θ )
0 2 tot
+ [sinθ µcosθ ] F . (46)
tot tot z
ForasphericalcolloidprobeofradiusR,simplegeom- B −
(cid:27)
etry, Fig. 2, gives the lever arm as
Because the total angle depends upon the deflection an-
L2(θtot)=R 2+2cos(θ0+θ). (39) gle,θtot =θ0+θ,theserepresentanon-linearrelationship
between the force, deflection, and deflection angle. This
p
In the undeflected state this will be written L last equation is best written by taking the proportional-
2
≡
L (θ ) = R√2+2cosθ . For the case of a tipped can- ity function over to the other side, which gives the force
2 0 0
tilever,L (θ )isequaltothelengthofthetip,andthere as an explicit non-linear function of the deflection angle,
2 tot
isnodependenceonthedeflectionangle. Infact,evenfor F (θ).
z
9
B. Non-Linear Analysis of Measured Data (possibly an extrapolation beyond the region of the fit),
is
The non-linear analysis in this section holds for both
∆V˜ext
sources of non-linearity: the cantilever treated explicitly β˜ext c
in the preceding subsection and the photo-diode non- cb ≡ ∆zp (cid:12)(cid:12)V˜ext
linearity, for which no specific model is given. In later (cid:12) b
(cid:12) 1
sub-sections, one or other of these will be turned off. = (cid:12) . (50)
The raw measured voltage V˜ is a non-linear function ae1xt+2ae2xtV˜bext+3ae3xt(V˜bext)2+...
of the effective total angle, θ˜ = θ +θ θ = θ +
tot tot b bf 0
− The tip position is a linear function of the deflection
θ+θ θ . Notethedistinctionbetweenthephysicalor
b bf
relevan−ttotalangleθ andthe effectivetotalangleθ˜ . angle in the base-line region, and the gradient is
tot tot
The physical contributions are the tilt angle θ , which
0 dθ
is known, and the deflection angle due to the surface αnc (51)
b ≡ dz
force, θ, which is to be obtained. The base-line angle θb t(cid:12)zt=0
is essentially an artifact arising from thermal drift and (cid:12)(cid:12) 1
= (cid:12) ,
virtual deflection, and it includes the angle due to drag DncC +L S +L C θ R2S2θ /L
0 2 0 2 0 0− 0 0 2
force, here assumed constant (but see Refs. 12,13, where
variable drag is shown to occur for weak cantilevers), where Dnc = D(0) = [2L30C0 + 3L20L2S0]/[3L20C0 +
and the angle deflection due to the linearly extrapolated 6L L S ], as follow from Eqs (19) and (20).
0 2 0
asymptote of the surface force, θ . In the non-linear Therawvoltageisthesamefunctionofthetotalangle
bf
case, whether it be the non-linear cantilever or the non- inandoutofcontact,andonextensionandonretraction.
linearphoto-diode,onecannotjustsubtractthebase-line Hence the rate of change of raw voltage with total angle
voltagefromtherawvoltagetoobtainacorrectedvoltage (equivalently, deflection angle) in the base-line region,
that gives θ directly. However,the total angle is a linear γext, can be evaluated in contact on extension at the
b
functionitscomponentparts,andsotheimmediatetasks base-line voltage. One has
are to obtainthe effective total angle fromthe measured
rawvoltage,θ˜ (V˜),andtoobtainthebase-lineangleas ∆V˜
tot γext (52)
a function of the piezo-drive position, θb(zp). b ≡ ∆θ (cid:12)
Doanon-linearfitofthemeasuredphoto-diodevoltage (cid:12)cb,ext
(cid:12)
in the contact region on extension, ∆z (cid:12) ∆z ∆V˜
= t(cid:12) p
Itzispexctb(eV˜st)=noate0xtto+usaee1xttVo˜o+maae2nxtyV˜t2e+rmase3xitnV˜t3h+e .fi.t.. A(4ls7o) = −∆β˜θcebx(cid:12)(cid:12)(cid:12)(cid:12)tc/bα,eebxxtt.∆zt(cid:12)(cid:12)(cid:12)(cid:12)cb,ext ∆zp(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)cb,ext (53)
one should be certain that the fit begins just after first Here αext α (µ) is given by Eq. (20) with µ>0. One
contact,andthatitendsbeforeanyanomaliesassociated can simbilar≡ly obbtain γret. The correct value of µ makes
b
with the turn around point at the end of the extension γext =γret. Actually,the correctµshouldmake∆V˜/∆θ
branch. b b
equal on extend and retract for all voltages.
In the non-linear cantilever case, one regards the an-
With this conversion factor, the angle corresponding
gular deflection θ as the independent variable. From the
to the base-line voltage is
non-linear Eq. (46), one can calculate the surface force
Fcazn(θc)a,lacunldatinestehretidnegfltehcitsioinntxo(tθh)e. nInosne-rltininegarthEeqs.e(i4n5t)oothnee θbext(zp)=EncFbext+(γbext)−1β˜bext[zp−zpexbt]. (54)
non-linear equation Eq. (40) one can calculate the tip This makes the base-line angle at zext equal to that due
pb
position zt(θ). Hence from these non-linear cantilever to the surface force alone; Enc = E(0), Eq. (18), con-
equations, z (θ;µ) is easily calculated. A non-linear fit
t vertsforcetoangleinthelinearregimeoflowforce. The
canbemadetothis,(incontactandonextension,µ>0), change in angle from its value at zext, includes the con-
b
tribution from the linear extrapolation of the asymptote
θext(z )=bextz +bextz2+bextz3+... (48)
c t 1 t 2 t 3 t of the surface force. The latter is
The leading coefficient is be1xt = αext, which was given θext(z )=Enc Fext+Fext′[z zext] . (55)
analyticallyinEq.(20). Onehasθteoxtt,c(zt)=θ0+θcext(zt). bf p b b p− pb
As in the linear case, the measured voltage is fitted to n o
These last two equations apply for all z , not just
a straight line in the base-line region, Eq. (21), p
in the base-line region. The constant factors do not
V˜ext(z )=V˜ext+β˜ext[z z ]. (49) change value in or out of contact. One can see that
b p b b p− pb θext(z ) θext(z ) is the virtual angle without any con-
b p − bf p
Now the raw contact slope from the non-linear fit tributionfromthesurfaceforceasymptote. Henceθ˜ =
tot
Eq.(47),evaluatedatthebase-linevoltageconstant,V˜ext θ + θ θext(z ) θext(z ) is the total effective angle
b 0 − b p − bf p
10
(physical plus virtual) without double counting the sur- case, one defines the plane of zero separation as pass-
face force. ing through the point of first (or last contact), and one
From Eq. (47), z (V˜) is known in contact. The tip defines the point of first (or last contact) as the point of
pc
position is related to the piezo-drive position in contact intersection of the extrapolations of the contact and of
by 0=h z +z +zext, or the base-line raw voltages. In the non-linear case this is
≡ pc tc 0
accomplished as follows.
z (V˜)=zext z (V˜). (56) Begin by defining zext as the piezo-drive position at
tc 0 − pc pcb
which the fitted voltage in contact, Eq. (47), equals the
(Strictly, this holds atone particularpositionincontact. constant part of the base-line voltage, V˜ext,
b
See the discussion in the third and fourth paragraphs of
§IIB5.) The constant z0ext will be determined explicitly zpexcbt = ae0xt+ae1xtV˜bext+ae2xt[V˜bext]2+... (60)
in the following subsection; in the mean time it can be
regardedas an arbitraryhorizontalshift that establishes Now a small correction to this will be made such that
contact at zero separation. Using successively Eqs (47), zext,∗ is the piezo-drive position at which the fitted volt-
pcb
(56), and (48), one can calculate the deflection angle for agein contactequals the actualbase-line voltageatthat
any given raw voltage in contact, θ (V˜). In contact, the position, viz. zext,∗ = zext(V˜ext(zext,∗)). Using a Taylor
c pcb pc b pcb
total effective angle as a function of raw voltage is expansion to linear order about V˜ext, one has
b
θ˜teoxtt,c(V˜)=θ0+θcext(V˜)+θbext(zpc(V˜))−θbexft(zpc(V˜)). zext,∗ zext = dzp V˜ext(zext,∗) V˜ext
(57) pcb − pcb dV˜ext b pcb − b
This is the desired relationship between the raw photo- c h i
diode voltage and the total cantilever angle. It is most = (β˜ext)−1β˜ext zext,∗ zext , (61)
cb b pcb − pb
convenient to do a non-linear fit of this (necessarily for
h i
voltages in the contact region) or
θ˜teoxtt,c(V˜)=ce0xt+ce1xtV˜ +ce2xtV˜2+... (58) zext,∗ = β˜cebxtzpexctb−β˜bextzpexbt. (62)
pcb β˜ext β˜ext
cb − b
Although this is explicitly derived for contact on exten-
sion, it ought equal the analogous result for contact on Since in general β˜ext β˜ext , the difference between
retraction, (assuming the correct value of µ, and the va- cb ≫ b
zext,∗ and zext is(cid:12)gene(cid:12)rally(cid:12)sma(cid:12)ll, in many cases negligi-
lidityofthelinearfrictionlaw). Eitherexpressioncanbe pcb pcb (cid:12) (cid:12) (cid:12) (cid:12)
used unchanged for any V˜ measured out of contact and ble. (cid:12) (cid:12) (cid:12) (cid:12)
At this particular position, the voltages extrapolated
on extension or retraction. Hence the left hand side will
be written simply θ˜tot(V˜). Vf˜ro(mzexcto,∗n)ta=ctV˜a(nzdexftr,∗o)m, atnhdehbeanscee-ltinhee acorerreesqpuoanl,diVn˜g∗ e≡f-
At a given measured datum z ,V˜ , in or out of con- c pcb b pcb
tact, on extension, the deflectio{npangl}e is fective total angles must also be equal, θ˜teoxtt,c(V˜∗) =
θ˜ext (V˜∗). The left hand side is
θ(z ,V˜)=θ˜ (V˜) θ θext(z )+θext(z ). (59) tot,b
p tot − 0− b p bf p θ˜ext (V˜∗)=θ +θext(V˜∗)+θext(zext,∗) θext(zext,∗),
The total physical angle is of course θ (z ,V˜) = θ + tot,c 0 c b pcb − bf pcb
tot p 0 (63)
θ(zp,V˜). From this the force F(zp,V˜) follows from and the right hand side is
Eq. (46), and the separation h(z ,V˜)=z +z (z ,V˜)+
z0ext follows from Eqs (40) and (4p5). p t p θ˜teoxtt,b(V˜∗)=θ0+θbext(zpexctb,∗). (64)
Note that in the non-linear analysis, this equation for
Hence at this particular position, the deflection of the
the separationis applied both in and out of contact; the
cantilever due to surface forces is equal to the linearly
separationisnotsimplysettozeroincontact(c.f.Fig.1).
extrapolateddeflectionduetothebase-linesurfaceforce,
Non-zero values of h in the contact region give informa-
tion about the topography and other physical attributes θext(V˜∗)=θext(zext,∗), (65)
of the surfaces, as will be shown below. c bf pcb
with θext(z ) being givenby Eq.(55). In this linear part
bf p
of the curve, the position of the tip at this angle is
1. Zero of Separation
zext,∗ = θext(zext,∗)/αnc
tcb bf pcb b
Theconstantzextremainstobedeterminedinthenon-
linear case. As0discussed in the linear case in §IIB5 = Enc Fbext+Fbext′[zpexctb,∗−zpexbt] /αnbc,(66)
above, there are good reasons for desiring a mathemati- n o
calalgorithmfor determining the zeroofseparationthat with αnc α(0) being given by Eq. (51), which is just
b ≡
avoids guess-work or personal bias. As in the linear Eq. (20) with µ=0,