17Measurement of

Adhesion and Pull-OffForces with the AFM

17.1 IntroductionImportance of Adhesion Measurements • Review of Adhesion Properties

17.2 Experimental Procedures to Measure Adhesion in AFM and ApplicationsTip Properties • Surface Topography • Force–Distance Curves • Influence of Pull-Off Force on Tapping Mode • Pulsed-Force Mode

17.3 Summary and Outlook

17.1 Introduction

The key to the successful operation of an AFM (Binnig et al., 1986) in the materials sensitive regime(Burnham and Colton, 1989; Miyamoto et al., 1990; Mizes et al., 1991) is the measurement of theinteraction forces between the tip and the sample surface. The tip would ideally consist of only one atom,which is brought in the vicinity of the sample surface. A crude estimation shows that the interactionforces between the AFM tip and the sample surface should be smaller than about 10–7 N for bulk materialsand preferably well below 10–9N for organic macromolecules. On the other hand, there are indicationsthat the measured values, especially of the pull-off force, are considerably off from the theoreticallyexpected values. The reasons are manifold: the shape and size of the tip is not well known (Godowskiet al., 1995; Lekka et al., 1997; Ramirez-Aguilar and Rowlen, 1998). The composition of the surfaces ofthe tip and the sample might differ strongly from their bulk values. Another possibility is that thecontinuum mechanical models (Johnson, 1992; 1996; Johnson et al., 1971; Maugis, 2001) usuallyemployed to analyze the data might fail.

Pull-off force measurements (Creuzet et al., 1992; Mizes et al., 1991; Weisenhorn et al., 1992), oftencalled adhesion measurements, have been carried out for some years. The number of published papersshows that the method has become more and more popular. The theories of contact mechanics that areused in these investigations date back to the time when the only available test bodies were macroscopic.The remarkable precision with which these theories work in the sub-mm regime was first questioned byStalder and Dürig (Dürig and Stalder, 1997; Stalder and Dürig, 1996). This chapter will discuss themeasurement of adhesion with the AFM and the data interpretation with respect to macroscopic theories.The discussion of limiting cases will pinpoint possible limitations.

Othmar MartiUniversity of Ulm

17.1.1 Importance of Adhesion Measurements

The tip in an AFM is in contact, in intermittent contact, or close to the sample surface. The interactionof the tip and the sample therefore depends on the surface properties of both bodies. The physics of theinteraction therefore dominates the imaging process in an AFM. Moreover it strongly affects the mea-surement of other quantities, such as the tapping phase (Sarid et al., 1998).

The adhesion properties of sample surfaces are important in many applications. Adhesive tapes, forinstance, require a high adhesive force almost independent of the opposite material (Jiaa et al., 1994;Koinkar and Bhushan, 1996a). Magnetic tapes and computer hard disks, on the other hand, work best ifthe adhesion with the read and write heads is minimal. Micromechanical devices (MEMS) require a carefulcontrol of adhesion (Bhushan, 1996). Bearings should have low adhesion, parts where two materials arejoined should have a higher adhesion. As these examples have shown, if it is necessary to control adhesionproperties, it is also necessary to measure adhesion which is often correlated with surface energies.

Figure 17.1 shows the classical way to measure adhesive properties. A drop of a liquid with a knownsurface energy is put on the sample. As Figure 17.1 shows, there has to be a balance of forces parallel tothe sample surface. This results in the drop having a contact angle Θ with the sample. The equilibriumis reached when

(17.1)

Here the subscript S denotes surface energy of the sample with respect to the ambient medium, SLdenotes the surface energy of the sample and the liquid, and L the surface energy of the liquid withrespect to the ambient. This measurement method works reliably, but it is almost impossible to charac-terize areas with micrometer diameters. Therefore it is necessary to use a microscopic tool such as theAFM for these measurements.

17.1.2 Review of Adhesion Properties

Adhesion is one of the main forces holding bodies of different materials together. As outlined above,adhesion is measured using test bodies. This chapter summarizes the literature (Dürig and Stalder, 1997;Johnson, 1992, 1996, 1997; Maugis, 2001), as it applies to AFM.

The most common test body is the sphere. Figure 17.2 shows a sketch of such an interaction. Thespherical body (radius R) is in contact with the sample with a circular region with radius Rc. A force Fis applied to the body. A positive force means that the body is pressed against the surface. A negativeforce means that the adhesion keeps the contact, although the force is trying to separate sample and testbody. The indentation of the spherical test body into the sample is called δ.

17.1.2.1 Hertz

The laws of indentation and adhesion are highly nonlinear. Hertz was the first to formulate the laws ofinteraction of a spherical test body with a planar surface (Hertz, 1881). Figure 17.2 shows the definitions

FIGURE 17.1 Contact angle measurement. Surface energies are often measured by the contact angle method. Adrop of liquid is placed on the sample. The surface energies of the liquid vs. the gas, of sample surface vs. the gas,and of the sample surface vs. the liquid have to balance tangentially, resulting in the equation given above.

γ γ γS SL L= + cosΘ

for the equations. Hertz formulated his theory of indentation without any adhesive or other long-rangeforces. The contact radius Rc is then

(17.2)

(17.3)

where F is the applied force, R the radius of the spherical test body, E the Young’s modulus, and ν thePoisson number. Figure 17.3 shows the function. The indentation depth δ is then given by

(17.4)

FIGURE 17.2 Definition of the quantities for the indentation of a spherical test body. The sphere has the radiusR. The radius of the contact area is RC . The indentation depth is δ.

FIGURE 17.3 Curves from different contact mechanics theories. The thin line shows the Hertz theory; the fat solidline, the main branch of the JKR theory; the faint fat line is the second branch of the JKR theory; the long-dashedline is the DMT theory; and the dotted line is the yield stress line. The values were calculated for polypropylene, asshown in Table 17.1.

R DRFc3 =

DE

= −2

3

1 2ν

δ32 2

= D F

R

Equation 17.4 shows that there is a nonlinear law connecting the indentation depth and the applied force.Also the indentation depth is inversely proportional to the tip radius. The force, although depending ina nonlinear fashion on the indentation, is a single-valued function. Therefore any interaction will be wellbehaved with no hysteresis. The contact stiffness kc is then given by

(17.5)

This equation is the basis for the conversion of the indentation δ and the applied force F to the contactradius Rc.

17.1.2.2 DMT

The Derjaguin Muller–Toporov–Theory (Derjaguin et al., 1975) is closely related to the Hertz theory. Inaddition to the Hertz theory, DMT takes into account adhesion inside the junction. This leads to theequation

(17.6)

The indentation depth is then given by the same Equation 17.4 as for the Hertz case. Figure 17.3 showsthis curve. The force as a function of the contact radius is given by

(17.7)

17.1.2.3 JKR

One of the most popular models for contact mechanics which includes adhesion effects is the model byJohnson, Kendall, and Roberts (Johnson et al., 1971). The adhesion is included by defining the pull-offforce F0

(17.8)

where ∆γ is the surface energy difference between the spherical test body and the sample surface. Theminimum contact area is

(17.9)

The applied force F and the contact radius Rc are connected by the following equation

(17.10)

Figure 17.3 shows this function. The branch with the + sign is the stable one, whereas the other can notbe probed by a force controlled setup. The equation can be solved for F. One then obtains

(17.11)

kF FR

D

R

Dcc= =

=δ 2

313

R FDR R Dc3 22 %π γ

FR

DRRc= − π

3

2 ∆γ

F R0

3

2 %π γ

R DRF0 0=

R DR F FF

Fc3

00

2 1 1= + ± +

FR

DRF Fc= −

−

3

0

2

0

17.1.2.4 Maugis

The theory of Maugis (2001) gives a smooth interpolation between the two extreme cases shown above:the DMT theory and the JKR theory. Generally, the real sample behaves in the intermediate range of thetwo theories. Chapter 4 gives a detailed account of this theory.

17.1.2.5 Effect of Tensile Stress

Adhesion measurement in AFM is closely related to the measurement of pull-off forces. The interactionsof the tip with the sample as well as the deformation of the sample before the separation influence themeasurement. The speed of separation does influence materials made of long molecules. Therefore welimit our discussion to the case of tensile stress on the sample, the situation just before the breaking ofthe adhesive junction with the tip. Compressive stress and the dwell time in the indentation regime affectthe contact area. This section is intended to raise the reader’s awareness of the fact that the deformationof the sample may not be neglected.

When the spherical test body is pulling on the sample, the tensile stress might exceed the yield stressof the junction (Dürig and Stalder, 1997). The tensile yield stress H relates to the force Fyield with thefollowing equation

(17.12)

where Rc is the radius of the contact area, which is assumed to be circular. The JKR or the DMT theoryrelates Rc to the applied force. Assuming that the starting force is set at a constant value such as is thecase in the pulsed force mode (Krotil et al., 1999; Rosa-Zeiser et al., 1997) the starting contact area iskept constant. Since the two force laws (Equations 17.12 and 17.11 or 17.6) have a different dependenceon the contact radius, only the law which has the lower contact radius Rc will prevail. Hence there willbe a crossover force Fmax which is given by

(17.13)

This equation always has a solution for the DMT theory. By equating Equations 17.13 and 17.7 oneobtains an implicit equation for the maximum force:

(17.14)

The crossing of the two curves is given by the solution of the above equation. This value in the sametime is the maximum negative force sustainable by the junction.

For the JKR theory we can do a similar calculation. The point where the two curves cross is given by

(17.15)

This is an implicit equation for the force F. Due to its nonlinear behavior it must be solved numerically.Large tensile yield stresses mean that the curve Equation 17.12 crosses curve Equation 17.10 on thenegative branch. This is the situation where the finite tensile yield stress of the junction is not relevant.However, if the curve Equation 17.12 crosses curve Equation 17.10 exactly at F0 we have the limiting case.In that case the equation can be solved.

(17.16)

F Ryield c= − πσ 2

F F F Ryield cmax = = ( )

F D FR R3 3 3 2 22

2= −π + π[ ]σ γ∆

F R D R F FF

Fcmax maxmax= − π = − π + ± +

σ σ20

0

2

323

23 2 1 1

− = −πF D R F0 0

23

23

23σ

From this equation one can calculate the maximum sustainable force for a tip with the radius R

(17.17)

Likewise one can calculate for a given force the critical tensile yield stress H.

(17.18)

For H > Hcrit the contact mechanics shows full JKR behavior. However, for softer materials this is nolonger true. One can also solve Equation 17.18 for the radius Rcrit of the spherical test body.

(17.19)

Solving Equation 17.19 for the critical tip radius Rcrit, one obtains

(17.20)

For a tip radius R < Rcrit, the junction becomes unstable before the JKR limit is reached. Hence for certainmaterials there will always be a deviation from JKR behavior.

Table 17.1 shows results for selected materials, mainly from the polymers group. The critical tip radiusRcrit gives the radius of curvature which must be exceeded to observe the JKR behavior to the end.Table 17.1 clearly demonstrates that the pull-off force is a distinct function of the tensile yield stress σ.The measured pull-off forces are as small as 10% of what one would have detected without the yieldbarrier. The “max force before break” or the “max force before yield” values show that the JKR theoryis applicable without any corrections, provided the pull-off force is below the value shown.

The simple JKR theory Equation 17.8 gives for the surface energy

(17.21)

TABLE 17.1 Critical Radius for Different Materials

Quantity Symbol PP HDPE LDPE PVC PS PMMA Nylon 6 V2A Steel

Young’s modulus E 1.4 GPa 1 GPa 0.2 GPa 2.6 GPa 3.4 GPa 3.2 GPa 1.9 GPa 195GpaPoisson ratio ν 0.43 0.47 0.49 0.42 0.38 0.40 0.44 0.28Yield stress σ 32 MPa 30 MPa 8 MPa 48 MPa 50 MPaRatio JKR αJKR 35% 36% 28% 35% 42%Ratio DMT αDMT 16% 19% 13% 16% 22%Max force before yield Fmax 1.5 nN 2.3 nN 1 nN 1.5 nN 3.1 nNCritical radius (20) Rcrit 31 µm 21 µm 46 µm 31 µm 15 µm

Break stress B 33 MPa 30 MPa 10 MPa 50 MPa 50 MPa 65 MPa 75 MPa 700MPaRatio JKR αJKR 34% 36% 37% 33% 28% 39% 57% 28%Ratio DMT αDMT 16% 19% 20% 15% 13% 18% 29% 13%Max force before break Fmax 1.7 nN 2.3 nN 2 nN 1.7 nN 1.1 nN 2.6 nN 10 nN 1.1 nNCritical radius (20) Rcrit,B 28 µm 21 µm 24 µm 27 µm 43 µm 18 µm 4.5 µm 45 µm

The critical radius is calculated both for the yield and the break stress. The ratio values were calculated with R = 100 nmand ∆γ = 1 N/m. They give the pull-off force as a percentage of the pull-off force one would measure if DMT or JKR wereapplicable references: polymers: van Kevele, 1976; steel: Vogel, 1995.

F D Rmax = π3 3 2 2σ

σ γ γcrit

F

D R

R

D R D R=

π=

ππ =

π1 10

2 23

32

2 23

32

2 23

∆ ∆

RF

D

R

Dcritcrit=

π=

ππ

03 2 3

32

3 2 3σγ

σ∆

RDcrit =

π3

2 2 2 3

∆γσ

∆γ = −π2

3

F

Rmax

The DMT theory Equation 17.7, on the other hand, predicts

(17.22)

If one includes the effects of the finite tensile yield stress σ, one must use Equation 17.15. Whereas itis not possible to find a simple solution for the pull-off force Fmax, one can easily solve Equation 17.15for F0 and hence for ∆γ

(17.23)

Likewise one can also solve for the DMT value of the surface energy using Equation 17.14

(17.24)

Hence, even though the tensile yield stress reduces the pull-off forces in an AFM experiment, there isan analytical way to correct for these problems. Curves calculated with Equations 17.23 and 17.24 areshown in Figure 17.4. The surface energy for samples obeying the DMT laws is always below the correctvalue and has to be corrected. The measurement of the surface energy for a JKR-type sample is correctwhen

(17.25)

FIGURE 17.4 Correction factors to compensate for the effect of the finite yield stress of materials. Shown is acalculation for a tip radius of 100 nm and the sample material polypropylene (see Table 17.1) The squares show theresult for the DMT theory; the diamonds represent the JKR theory.

∆γ = −π1

2

F

Rmax

∆γσ σ

=π

−π

−π

−

−2

3

132

32

2

DF F

DRFmax max

max

∆γσ

=π

− + −π

1

2

12

32F

R R D

Fmax max

∆γ σ< π2

32 3 2D R

is measured. Equivalently, one might state that the surface energy is correct according to JKR if

(17.26)

You find a calculation of this force in Table 17.1. What does it mean if the contact mechanics equationssay that tensile stress will be the failure mechanism of the tip–sample junction?

• When the pull-off force is larger than predicted by the yield stress argument, one can concludethat continuums theories such as JKR, DMT, or Hertz do not adequately describe the physics ofthe separation. It was shown by several authors (Cross et al., 1998; Landman et al., 1992; Rubioet al., 1996; Stalder and Dürig, 1996b) both theoretically and experimentally that the tensile stresson the junction shortly before the separation from the tip can exceed the limits set by continuumsmechanical theories. Hence a comparison of the actual data with the Dürig (Dürig and Stalder,1997) theory discussed above gives a criterion for the applicability of standard theories.

• In an intermediate regime of AFM tip radii of 50 nm, it is often the case that the yield stress isreached before the crazing mechanism usually responsible for the detachment of the tip and thesample sets in. As a consequence, one expects (and also finds in certain cases) that the tips arecontaminated by the sample.

17.2 Experimental Procedures to Measure Adhesion in AFM and Applications

The forces of an AFM tip in contact with the sample should in principle be given by the contact mechanics.The tip of the AFM, however, is usually small. Hence the ideas outlined above show that one needs toknow the shape and the surface properties of the tip. The ideas above were deduced assuming a flatsurface. Real sample topographies are usually rough. Hence one has to calculate the effect of steps onthe pull-off force image. This is done using a simple theory based on continuum mechanics. Finally, wediscuss force–distance curves, the influence of pull-off force on the tapping mode, and pulsed force modetechniques for AFM. The measurement in the AFM are further complicated by the compliance of thecantilever necessary to measure forces.

17.2.1 Tip Properties

Cantilevers are made from a wide range of materials (Akamine et al., 1990; Grütter et al., 1990; Marti,1998; Pitsch et al., 1989; Wolter et al., 1991). Most cantilevers are made of Si and of Si3N4. The realizablethickness depends on the fabrication process and the material properties. Grown materials such as Si3N4

can be made thinner than those fabricated out of the bulk.Cantilevers come basically in two flavors. Straight dashboard-like types are preferentially used for

lateral force measurements and noncontact modes. Their properties are rather easy to calculate. Triangularshaped cantilevers are easier to align, but harder to handle numerically. They are usually made of siliconnitride. Their response to lateral forces is more complicated.

Whereas triangular cantilevers must be calculated using finite element methods, one can get a goodestimate of the normal force compliance of the straight ones using analytical methods. Using the equationfor straight cantilevers

(17.27)

and observing that the length of the two joined cantilever beams in a triangular cantilever are l2eff = l2 +

(w/2)2, where w is the width of the base of the cantilever, one gets for the compliance

F D Rmax < π3 3 2 2σ

kF

z

Eb hN = =

4

3

l

(17.28)

The radius of curvature of silicon nitride cantilevers is limited to about 30 to 50 nm, because of themanufacturing process. The imperfections of the etch pits and the filled in silicon nitride limit thesharpness. Silicon nitride tips can be sharpened during the production by thermal oxidation (Akamineand Quate, 1992). Instead of directly depositing silicon nitride on the wafers with the pyramidal etchpits, an oxide layer is deposited first. Then the silicon nitride is added. When the oxide is removed withbuffered oxide etch, a sharpening effect is observed. Details of the process are described by the inventors(Akamine and Quate, 1992). A second method is to grow in an electron microscope a so-called “supertip”on top of the silicon nitride. It is well known that in scanning electron microscopes with a base pressureof more than 10-10 mbar, hydrocarbon residues are present. These residues are cracked at the surface ofthe sample by the electron beam, leaving carbon in a presumed amorphous state on the surface. It isknown that prolonged imaging in such an instrument degrades the surface. If the electron beam is notscanned, but stays at the same place, one can build up tips with a diameter comparable to the electronbeam diameter and with a height determined by the dwell time. These tips are extremely sharp and canreach radii of curvature of a few nanometers. Therefore, they allow imaging with a very high resolution.In addition they enable the microscope to image the bottoms of small crevasses and ditches on samples.Unprocessed silicon nitride tips are not able to do this, since their sides enclose an angle of 90°, due tothe crystal structure of the silicon.

Alternatives to silicon nitride cantilevers are those made of silicon. The basic manufacturing idea isthe same as for silicon nitride. Masks determine the shape of the cantilevers. Processes from the micro-electronics fabrication are used. Since the thickness of the cantilevers is determined by etching and notby growth, wafers have to be more precise than for the manufacturing of the silicon nitride cantilevers.The manufacturing process of silicon cantilevers guarantees that the tip asperity has a well defined radiusof curvature of 2 to 5 nm.

Since the thickness of the silicon cantilever is determined by etching, it cannot be made as thin as thesilicon nitride cantilever. The lower limit is typically 1 µm. Therefore the stiffness of silicon cantileversis higher, ranging from 1 to 100 N/m. Since the material is a single crystal, unlike the silicon nitride, ithas a very high quality of resonance. Values exceeding 100,000 have been observed in vacuum. Thereforesilicon cantilevers are often used for noncontact or tapping mode experiments. The cantilevers have twodrawbacks when working in contact mode. First, they have a very high affinity to organic materials. Theyoften destroy such samples. Second, their index of refraction matches that of water rather closely. Siliconcantilevers have a very poor reflectivity in aqueous environments.

Occasionally cantilevers are made with tungsten wire (Marti et al., 1987) or thin metal foils, with tipsof diamond (Marti et al., 1988) or other materials glued to them.

The radius of curvature of a tip is not well defined. As outlined above and in the literature, the tipshape and the tip radius of curvature are often the results of uncontrollable processes. The initial variationof the tip radius is further increased by wear during use. Therefore the tip radius is one of the least knownparameters in an AFM experiment. In many experiments one does not pay attention to the tip shape orradius. To obtain quantitative results it is imperative to inspect the tip before and after the AFM exper-iment. The size and shape of the tip influence the measured adhesion values (Bhushan and Sundararajan,1998). Silicon nitride cantilevers were used to probe the adhesion forces vs. a silicon (100) sample. It wasfound that tip sizes in the order of 1 to 10 µm affect the adhesive forces. At high humidity, capillaryforces increase the adhesion considerably.

The main properties of an AFM tip are its surface energy and its radius. The surfaces of most cantileversare made of silicon or silicon nitride. Since cantilevers are usually stored in air, their surfaces are oftencovered by a thin oxide layer. The surface energy of silicon ranges between 1 and 1.4 N/m (Burdorf,1999). That of silicon oxide might be as high as 3.5 N/m. Silicon nitride, on the other hand, has a surfaceenergy of 0.7 N/m.

kEbh w

N = +

3

22

2 4

32

l

17.2.2 Surface Topography

It was mentioned above that the models usually used to describe adhesion do not include effects of thetopography (Bhushan, 1999; Koinkar et al., 1996). Therefore a model investigation was done at theUniversity of Ulm. A model based on continuum mechanics and taking into account only interactionsdue to the shape of the tip and surface topography was established (Stifter et al., 1998). Interactionsbetween the atoms of the tip and the sample were summed up (Figure 17.5).

In the model, atoms are represented by a spherically symmetric potential that cannot account forchemical bonding. The interaction is composed of the attractive van der Waals force and the repulsivePauli force. Both are combined in the Lennard–Jones potential

(17.29)

or

(17.30)

Here r0 is the position of the potential minimum and ε the depth of the potential minimum. The twopotential parameters a and b are defined by

(17.31)

The vector→r consists of a contribution of the tip–sample distance

→z and the local positions of the tip

and sample,→rs and

→rt , respectively. Equation 17.30 is used for the Lennard–Jones potentials because the

two parts of the potential — attractive and repulsive — can be treated separately. Integrating theLennard–Jones potential over the volume of the tip and the volume of the sample gives the interactionpotential between the tip and sample. The integration is performed in two steps, first on the tip and thenon the sample, to simplify changes in the surface shape. To simulate an SFM operating in the constantforce mode, the force acting in the direction perpendicular to the sample surface is calculated. The

FIGURE 17.5 Geometry of the model calculation. The distance→r between a point in the tip and a point in the

sample is a combination of the distance→z between the tip and the sample and the positions of the point in the tip

→rt

and of the point in the sample→rS.

V r V r V rr

r

r

ratom attr rep

r r r( ) = ( ) + ( ) =

−

ε 0

12

0

6

2

V ra

r

b

ratom

r( ) = − +6 12

a r b r= =2 06

012ε ε and

coordinate in this direction is the z axis. The x and y axes are in the plane of the sample surface. Figure 17.5shows the coordinate system used for these calculations. The derivative of the potential with respect toz gives the force between the tip and the sample. For a (x, y) position on the sample, the z-feedback ofthe SFM determines the value the tip has to be moved to realize the preset force. The calculation worksin the same way. For a given (x, y) position and a fixed force value Ffix , the surface distance ztopo wascalculated. The pull-off force in the experiment is the maximal negative force appearing as the tip isremoved from the surface. This corresponds to the minimum of the force Fmax curve vs. z (see Figure 17.6).

A commonly used model for the tip is a sphere with radius R. The sample is represented as a plane.This model was solved in the literature (Stifter et al., 1998). Then the model can be expanded to includethe effects of a surface step. The step represents a sharp topographical change. Only forces along a lineperpendicular to the step have to be calculated.

To get the force for the step, it was necessary to make two volume integrations. To simplify thecalculation, the attractive and the repulsive parts of the Lennard–Jones potential (Equation 17.30) aretreated separately. The first integration — over the volume of the tip — can be done analytically. Theresult of the integration is the potential between an atom in the surface and a sphere with radius R. Thesecond integration is split into two parts: the infinite plane is handled analytically, and the terrace (a semi-infinite slab) is treated numerically.

The step changes the interaction of the sample with the tip. Because of the curvature, the tip canapproach closer to the surface. Its interaction changes. Figure 17.7 shows the influence of the materialon the appearance of a step. The figure, calculated with the model outlined above, using a tip radius of10 nm and a step height of 1 nm, nicely demonstrates that, within this theoretical framework whichtreats surfaces as infinitely stiff, the surface potential, given by parameters A and B, determines theadhesion. For a materials pairing with equal adhesive properties, a characteristic rim of higher adhesionappears. This feature is also present when the terrace has a higher adhesion.

The conclusion from Figure 17.7 is that adhesion forces or pull-off forces are only meaningful at theflat parts of the sample. Figure 17.8 is a measurement by AFM of the adhesion across a step. Themeasurement confirms the theoretical model.

17.2.3 Force–Distance Curves

The easiest way to measure pull-off force properties of a sample surface is to perform force–distancecurves. Force–distance curves are obtained by slowly (1 nm/s to 1 µm/s) lowering the tip onto the sample.

FIGURE 17.6 Principle of the measurement of force–distance curves. Far away from the sample surface the canti-lever spring is not deflected from the zero position (right side of the image). When the tip approaches the sample,an instability for soft cantilevers occurs. The tip snaps to the sample (“snap on peak”). The cantilever is deflectedtoward the sample (middle of the image), indicating a tensile stress at the tip–sample junction. Finally, at closeapproach the tip penetrates slightly into the sample. The cantilever spring is deflected away from the sample. Whenretracting the cantilever from the sample, the tip–sample junction breaks at a value larger than the snap-on peak.This force we call the pull-off force.

FIGURE 17.7 Adhesion at a step as a function of the material composition. The top part of the image shows thetopography. The bottom part depicts the calculated adhesion trace across the step. A Lennard–Jones-type model wasused. The three curves are calculated for the binding energy ε and binding distance R0 indicated in the figure. Thetighter the tip is bound to the sample, the larger are the artifacts at the steps. The parameters for the sample awayfrom the step correspond to Curve 2.

FIGURE 17.8 Pull-off force of a graphite step. The image size is 210 × 300 nm. The data were measured in a 1 mMsolution of NaClO4. The simulations were calculated with a tip radius of 100 nm and a step size of 1.55 nm (multistep). Details of the calculation can be found in Stifter et al. (1998).

Figure 17.6 shows the principle of such an experiment. Far away from the sample, no force will act onthe cantilever. When coming close to the sample, the nonlinear interaction of the tip with the samplewill result in an, at least, double well potential. This potential has two stable positions. Which one willbe realized depends on the history. At the point where the tip snaps to the surface, it jumps from oneminimum to the second one. Upon further approaching the tip, the sample becomes indented. The slopeof the force distance curve is now mainly given by the compliance kt of the cantilever. Assume that thesample has a compliance ks and the tip kt

(17.32)

Hence the effective compliance of a soft cantilever and a stiff sample is almost entirely given by the softcantilever.

The effect of the cantilever can be compensated for. If one knows the sensitivity of the detection system(which deflection is necessary to measure a certain force), one can correct the slopes by subtracting thedeflection of the cantilever from the position one imposed on the cantilever from the outside(Figure 17.9).

(17.33)

The true z-position ztrue is calculated from the measured one zmeasured with Equation 17.33. The force F isdetermined by the AFM calibration, the compliance of the cantilever k has to be determined indepen-dently, or, as a first guess, to be taken from the manufacturer’s data sheets. The corrected curve shouldthen be the real force interaction.

As an example, Figure 17.10 shows a measured and corrected force–distance curve on a polystyrenesample. The cantilever compliance and response have been measured with a silicon sample. The values

FIGURE 17.9 Correction of the influence of the cantilever compliance. The cantilever compliance is the reason forthe double minimum potential landscape when the tip interacts with the sample. On a stiff surface, the change offorce when the cantilever base approaches the sample is given by the cantilever compliance. This compliance mustbe subtracted from the measured force–distance curves. On the adhesive part, the cantilever compliance creates amultivalued force curve, giving rise to adhesive hysteresis. The corrected curve has been calculated from the uncor-rected curve. Both have the same underlying potential.

1 1 1

k k keff s t

= +

z zF

ktrue measured= −

obtained in this way were used for the correction. The data clearly show that the simple model outlinedhere is not sufficient. First, the piezo actuators in the AFMs are hampered by hysteretic behavior. Thisbehavior is not easily cast into equations. Therefore a good nano-indentation experiment would need aposition control for z. Second, if plastic deformations occur in the sample (which, according to thereasoning above are almost inevitable), dissipation will separate the approach and the retract curves.

The pull-off force in Figure 17.10 is about 30% of what the JKR theory predicts. This is why the pulloff force has about the same magnitude as the snap-in force. The latter should not depend on the yieldstress of the sample, since there is no contact before the sudden change in force.

17.2.4 Influence of Pull-Off Force on Tapping Mode

A popular imaging mode in scanning force microscopy is the tapping mode (Anczykowski et al., 1996a,b,1998; Sarid et al., 1996; Spatz et al., 1995, 1997; Winkler et al., 1996; Zhong et al., 1993). It was realizedthat adhesion can play an important role in the imaging properties (Evans and Ritchie, 1997; Izrailevet al., 1997; Sarid et al., 1998; Schmitz et al., 1997). In the tapping mode operation (also called dynamicmode operation) the cantilever of the AFM is operated at or near its resonance. The amplitude and phaseof the oscillation are complex functions of the drive amplitude, frequency, and of the interaction potentialbetween the tip and the sample. Experience showed that the phase was very sensitive to changes in thematerials properties. The equation of motion for the cantilever can be formulated as follows:

(17.34)

where z is the position of the end of the cantilever, z0 the drive amplitude, d the separation of the cantileverreference point and the sample. δ is the damping of the cantilever, m its reduced mass, and U(z) theinteraction potential. ω0 is the drive frequency. The instantaneous resonance frequency now changesalong the trajectory of the tip. The resulting amplitude and phase will be an averaged function of thesecond derivative of the potential energy of the tip over a full period.

(17.35)

FIGURE 17.10 Measured force–distance curve. Shown on the horizontal axis is the distance to the sample surface.Zero is just before the position where the approach curve has its snap-on peak. The vertical axis depicts the force.There is a considerable hysteresis in the retract (unloading) curve. The molecules in the polystyrene samples seemto adhere to the tip; the tensile forces can considerably stretch the molecules before their adhesion bonds to the tip fail.

mz zz

U z d z t z z t˙̇ ˙ cos cos+ + ∂∂

− −( )

−( ) =2 02

2 0 0 0 0δ ω ω

˜ ,z t zz

f z U z( ) = ∂∂

( ) ( )

ω0

2

2

The potential U(z) is weighted by some function f(z), which has a complicated dependence on theinteraction details.

From Equations 17.34 and 17.35, it is clear that the compliance of the sample will affect the resonancefrequency. Since the tapping mode is usually performed with a constant resonance frequency, this meansthat the mechanical properties of the sample will show up in the amplitude, and also more prominentlyin the phase. Hard samples will increase the potential on the average, hence increasing the resonancefrequency of the tip–sample system.

Samples which exert a nonvanishing adhesive force on the tip lower the average potential for thetip–sample system. In a mean-field reasoning, the resonance frequency has to go down. The stronger theadhesion, the larger the phase shift of the system. Figure 17.11 shows data from Sarid et al. (1998). Thephase change for a given set point as a function of the adhesion is clearly visible.

A strong adhesive force can create multiwell potential landscapes. The oscillation of the cantilever canthen be confined to one or the other well. Jumps between the wells can occur. This behavior has beenobserved, for instance by Anczykowski et al. (1998) (Figure 17.12). The jumps from one potential wellto the other occur when the original well becomes unstable, i.e., loses its confinement. The result is ahysteresis when the set-point of the microscope is brought closer to the sample surface and then backagain.

It is not trivial to extract meaningful quantitative adhesion data from tapping-mode phase measure-ments. The effects of the finite yield stress of the material certainly have an influence on the depth of thepotential wells. They will reduce the effects of adhesion. On the other hand, adhesion for macromoleculesis always connected with a rearrangement of molecules. This rearrangement has its own time constants.In principle it should be calculated with molecular dynamics methods. Unfortunately these methods arenot yet able to calculate the dynamics on the time scale relevant for a tapping-mode experiment.

17.2.5 Pulsed-Force Mode

A third method to measure pull-off force properties (or better, pull-off forces) is the pulsed-force mode(PFM). A sinusoidal z-modulation is used to continuously acquire force vs. distance curves. The PFM isone of the so-called intermittent contact modes in AFM. The key to the operation of the PFM (and thetapping mode) is the frequency dependence of the elastic moduli of samples. The apparent shear modulusincreases when the frequency of the acquisition of force-distance curves is increased. The time the tip isin contact with the surface can be directly estimated by the modulation frequency. Typically the AFM

FIGURE 17.11 Phase change as a function of the adhesion properties for a polymer with E = 100 GPa. The solidand dashed lines show the theoretical calculation for the tapping-mode phase with and without taking adhesion intoaccount. (From Sarid, D., Hunt, J.P., Workman, R.K., Yao, X., and Peterson, C.A. (1998), The role of adhesion intapping-mode atomic force microscopy, Applied Physics A Materials Science & Processing, 66, S283-S286. Withpermission.)

cantilever in the PFM is brought into contact at a rate of 1 to 10 kHz. Thus, the PFM has a contact timeof about 10 µs (as measured from oscilloscope traces), compared to > 10 ms when measuring force–dis-tance curves and ≈1 µs in tapping mode. Delicate samples can be measured this way, with a resolutionand gentleness comparable to that of the tapping mode.

Electronic circuitry (Krotil et al., 1999; Miyatani et al., 1998; Rosa-Zeiser et al., 1997) creates thesinusoidal modulation frequency, which is applied either to the z-electrode of the scan piezo or to aspecial modulation piezo. The sinusoidal shape is necessary to avoid the excitation of resonances in themicroscope. The cantilever-base moves up and down, as depicted in Figures 17.13 and 17.14. The resultingforce signal, as detected by the AFM head, is a repetitive measurement of force–distance curves. The peakpositive force (1 in Figure 17.14) is detected by a “sample and hold” circuit. It serves as the input of thecontrol loop and is fed into the AFM control electronics, faking a constant force measurement. Thefeedback loop now maintains a constant peak positive force. The PFM allows an exact determination ofthe peak interaction force independent of the operating environment.

FIGURE 17.12 Influence of multiple potential wells on the tapping mode amplitude and phase behavior, experimentaldata. The cantilever was driven at the harmonic resonance frequency. A bias voltage was applied for the measurement atthe right side to increase the attractive part of the potential. When combining the harmonic potential related to a Hookianspring with the interaction potential for two bodies, e.g., the Lennard–Jones potential or similar potentials, a double-wellpotential landscape is formed. Shown are the amplitude of the tapping-mode oscillation on the top and the phase on thebottom. In every part the sample surface is at zero z-position. (From Anczykowski, B., Cleveland, J.P., Kruger, D., Elings,V., and Fuchs, H. (1998), Analysis of the interaction mechanisms in dynamic mode SFM by means of experimental dataand computer simulation, Appl. Phys. A Materials Science & Processing, 66, S885-S889. With permission.)

FIGURE 17.13 Movement of the tip in the pulsed-Force mode measurement. In a typical setup the cantilever ismounted on a piezo. This piezo moves the cantilever base up and down with a predefined curve form. Most oftena sinusoidal movement is used. As long as the cantilever is free, it is not bent, indicating that no interaction force ispresent. Indentation causes the cantilever to bend upwards, tensile forces before pull-off deflect the cantilever beamdownwards.

The PFM automates the measurement of surface properties at every point on the sample by addi-tionally sampling a few characteristic values of the force–distance curves using additional “sample andhold” circuits. It has a minimal impact on the data acquisition rate. Data taken from the curve are thelocal stiffness, the local pull-off force (adhesion), and local charges (Figure 17.14). The pull-off forceoutput is the difference between the output of a peak-detector for negative forces and a sample andhold measuring the zero base line. These quantities are complementary to what one would get fromtapping mode, where the phase signal is related to the energy dissipation and, sometimes, to the adhesionon the sample surface. As pointed out above, the measured pull-off forces have to be corrected for thefinite yield stress of the samples. If this is done, then reliable values for the surface energies can beextracted.

Figure 17.15 shows the effect of a changing Young’s modulus of the sample surface. The force curveon the hard material is shorter; the pull-off force is less. This is the behavior one would expect from theJKR theory.

As an example, we show a measurement of spherulites in polypropylene (Hild et al., 1998; Marti et al.,1999). The images in Figure 17.16 were acquired in the PFM with a frequency of 1.6 kHz. The left sideshows the topography; the right side, the pull-off force. The amorphous parts in the topography imagesappear to be lower than the crystalline areas. This can be explained by differences in local stiffness. Fora given force, the tip indentation in the softer material is larger than in the harder one. Because of thedifferent indentation at a given applied force, harder parts of a surface appear to be higher. In the pull-off force image, the amorphous parts between the crystalline lamellae exert a high force on the tip (darkcolor). The data on the right side of Figure 17.16 have not been corrected for the yield stress phenomenon.

17.3 Summary and Outlook

The measurement of pull-off forces with the AFM is widespread nowadays. The interaction of the AFMtips and the sample surfaces yields valuable information on the surface energies of the samples. However,in day-to-day work, one is using methods like tapping mode, force–distance curves, and pulsed-force-mode

FIGURE 17.14 Operation of the pulsed force mode measurement (Krotil et al., 1999; Rosa-Zeiser et al., 1997). Atpoint 1 a sample and hold circuit measures the peak indentation force. This force is kept constant by the feedbackcontrol circuits of the microscope. The peak force (Fmax) minus the force measured at 3 (Fis) is a function of the localsample stiffness and the cantilever stiffness kc. The force at 2 is the zero point value. It is constant if there are nolong-range forces. It varies if magnetic, electrostatic, or other forces are present. Finally, the force at 4 measured bya peak picker circuit is the pull-off force.

to get a quick overview of the sample. Usually one is interested in finding differences in the morphologyof the samples. The quantitative nature of the pull-off force measurement is usually not necessary.

However, experiments trying to correlate pull-off force measurements with theories sometimes fail.Usually one is able to range samples correctly with different methods. For polymer samples, often theabsolute surface energies calculated from reasonable assumptions of the tip shape do not agree withmeasurements by contact angles. The idea of Stalder and Dürig (Dürig and Stalder, 1997) that the yieldstress is an important quantity for the determination of surface energies has not been fully exploited yet.From the considerations in this chapter it is clear that probably no AFM, independent of the operatingmode, works in the “adhesion” dominated regime. It rather operates in the yield stress or break stressdominated regime. The theoretical considerations show that the shape of the force–distance curve, if themovement of the z-piezo were linear, and the pull-off force together are sufficient to extract both thesurface energy and the yield (break) stress from the data.

FIGURE 17.15 Signals in the pulsed-force mode measurement. The top trace shows the movement of the piezo,which is the same as that of the mounting point of the cantilever spring. The bottom trace shows a typical force vs.time curve. The left side simulates the behavior on a stiff surface; the right side shows a typical trace obtained on acompliant, nonviscous surface. The measurement system determines the peak force Fmax and a second force value ata fixed time offset, Fis. The difference between these two forces depends on the stiffness of the sample.

FIGURE 17.16 Spherulite of polypropylene. The left image shows the topography; the right image, the pull-offforce measurement. The scan size of both images is 1200 × 1200 nm. The height range in the left image is 35 nm(white corresponds to high lying areas; black to low lying areas). The pull-off force values range from –40 to –90 nN(white corresponds to low pull-off forces).

If the shape of force–distance curves is not known, then one can use the macroscopic yield stress toinfer the surface energy. One wonders why the tips of an AFM do not show substantial contaminationwith the sample material, since the failure of an AFM–sample junction seems to be located entirely withinthe sample. In the case of polymers, the situation is not as simple as one might guess. The macromoleculeshave typical gyration radii of up to 30 nm. Hence a junction might consist of only a few molecules. Theyield-stress limit is an indication of the point up to which continuum mechanics apply. For smaller tips,one will have to take into account the discrete nature of the interaction.

When one has to measure adhesion forces, or more specifically pull-off forces, one has to select themethods according to the priorities of the experiment. If a fast overview on the variations of surfaceproperties and on the spatial distribution of them is desired, tapping-mode phase imaging is the methodof choice. However, its physics is not easily handled. Extensive calculations and, probably, modeling ofthe sample–tip system are required to get quantitative data. The force–distance curves, on the other hand,allow a rather precise measurement of the pull-off forces, at the expense of speed. Imaging is almostimpossible. Force–distance curves give the shape of the interaction, making it comparatively easy tocalculate quantitative data. The pulsed-force mode is a compromise between the previous two measure-ment modes. It only acquires some selected values of the force–distance curve. This is done efficientlyand poses no obstacle to imaging in reasonable time. Data from force–distance curves and from thepulsed-force mode can be corrected for the yield-stress effect.

There is still considerable research necessary to sort out all the effects besides the adhesion forces inpull-off force measurements. Nevertheless, meaningful measurements can be done, provided the exper-iment is carefully planned. Therefore the AFM has a good chance to become a versatile mechanical testingdevice for ultra-small amounts of materials.

Acknowledgments

The overview presented here is based on the contributions of many persons. I would like to thank ThomasStifter, Hans-Ulrich Krotil, Sabine Hild, and Charly Imhof for the measurements. I had long discussionswith Martin Pietralla, Sabine Hild, Thomas Stifter, Gerd-Ingo Asbach, and Bharat Bhushan on thissubject. Gerhard Volswinkler built the electronics. This work was funded in part by the Deutsche Fors-chungsgemeinschaft (SFB 239), the Land Baden-Württemberg, and by Witec GmbH Ulm.

References

Akamine, S., Barrett, R.C., and Quate, C.F. (1990), Improved atomic force microscope images usingmicrocantilevers with sharp tips, Appl. Phys. Lett., 57, 316.

Akamine, S. and Quate, C.F. (1992), Low temperature thermal oxidation sharpening of microcast tips,J. Vac. Sci. Technol., B10 (6), 2307-2311.

Anczykowski, B., Cleveland, J.P., Kruger, D., Elings, V., and Fuchs, H. (1998), Analysis of the interactionmechanisms in dynamic mode SFM by means of experimental data and computer simulation,Appl. Phys. A Materials Science & Processing, 66, S885-S889.

Anczykowski, B., Kruger, D., Babcock, K.L., and Fuchs, H. (1996), Basic properties of dynamic forcespectroscopy with the scanning force microscope in experiment and simulation, Ultramicroscopy,66 (3-4), 251-259.

Anczykowski, B., Kruger, D., and Fuchs, H. (1996), Cantilever dynamics in quasinoncontact force micros-copy: spectroscopic aspects, Phys. Rev. B Condensed Matter, 53 (23), 15485-15488.

Bhushan, B. (1996), Nanotribology and nanomechanics of MEMS devices, in Proceedings of the 9th AnnualWorkshop on Micro-Electro-Mechanical Systems, IEEE, New York, 91.

Bhushan, B. (1999), Principles and Applications of Tribology, John Wiley & Sons, New York.Bhushan, B. and Sundararajan, S. (1998), Micro/nanoscale friction and wear mechanisms of thin films

using atomic force and friction force microscopy, Acta Metallurgica, 46 (11), 3793-3804.Binnig, G., Quate, C.F., and Gerber, C. (1986), Atomic force microscope, Phys. Rev. Lett. 56 (9), 930-933.

Burdorf, K. (1999), Silicon-direct-bonding: packaging technology on wafer level, Institute for Microsen-sors, -actuators and -systems (IMSAS), University of Bremen, http://www.imsas.uni-bremen.de/service/sdb.html.

Burnham, N.A. and Colton, R.J. (1989), Measuring the nanomechanical properties and surface forces ofmaterials using an atomic force microscope, J. Vac. Sci. Technol. A (Vacuum, Surfaces, and Films),7 (4), 2906-13.

Creuzet, F., Ryschenkow, G., and Arribart, H. (1992), A new tool for adhesion science: the atomic forcemicroscope, J. Adhes., 40 (1), 15-25.

Cross, G., Schirmeisen, A., Stalder, A., Grütter, P., Tschudy, M., and Dürig, U. (1998), Adhesion interactionbetween atomically defined tip and sample, Phys. Rev. Lett., 80 (21), 4685-4688.

Derjaguin, B.V., Muller, V.M., and Toporov, Y.P. (1975), Effect of contact deformations on the adhesionof particles, J. Colloid Interface Sci., 53, 314-320.

Dürig, U. and Stalder, A. (1997), Adhesion on the nanometer scale, in Micro/Nanotribology and ItsApplications, Bhushan, B. (Ed.), Kluwer, Dordrecht, Vol. E330, 61.

Evans, E. and Ritchie, K. (1997), Dynamic strength of molecular adhesion bonds, Biophys. J., 72 (4),1541-1555.

Godowski, P.J., Maurice, V., and Marcus, P. (1995), Analytical problems in atomic force microscopy:distortion of surface structures during imaging, Chem. Anal., 40, 231-242.

Grütter, P., Rugar, D., Mamin, H.J., Castillo, G., Lambert, S.E., Lin, C.-J., Valletta, R.M., Wolter, O., Bayer,T., and Greschner, J. (1990), Batch fabricated sensors for magnetic force microscopy, Appl. Phys.Lett., 57, 1820.

Hertz, H. (1881), Über die Berührung fester elastischer Körper, J. Reine Angew. Math., 92, 156-171.Hild, S., Rosa, A., and Marti, O. (1998), Deformation induced changes in surface properties of polymers

investigated by scanning force microscopy, in Scanning Probe Microscopy of Polymers, Ratner, B.D.and Tsukruk, V.V. (Eds.), Oxford University Press, Vol. 694, 110.

Izrailev, S., Stepaniants, S., Balsera, M., Oono, Y., and Schulten, K. (1997), Molecular dynamics study ofunbinding of the avidin–biotin complex, Biophys. J., 72 (4), 1568-1581.

Jiaa, C.L., Nguyen, P., Teng, E., and Eltoukhy, A. (1994), The role of surface properties in the mechanicalperformance of thin film rigid disks, Thin Solid Films, 248 (1), 41-6.

Johnson, K.L. (1992), Introduction to contact mechanics: a summary of the principal formulae, inFundamentals of Friction: Macroscopic and Microscopic Processes, Singer, I.L. and Pollock, H.M.(Eds.), Kluwer Academic Publishers, Dordrecht, Vol. 220, 589.

Johnson, K.L. (1996), A continuum mechanics model of adhesion and friction in a single asperity contact,in Micro/Nanotribology and its Applications, Bhushan, B. (Ed.), Kluwer Academic Publishers,Dordrecht.

Johnson, K.L. (1997), Adhesion and friction between a smooth elastic spherical asperity and a planesurface, Proceedings of the Royal Society of London Series A Mathematical Physical and EngineeringSciences, 453 (1956), 163-179.

Johnson, K.L., Kendall, K., and Roberts, A.D. (1971), Surface energy and the contact of elastic solids,Proc. R. Soc. London, Ser. A, 324, 301-313.

Koinkar, V.N. and Bhushan, B. (1996a), Micro/nanoscale studies of boundary layers of liquid lubricantsfor magnetic disks, J. Appl. Phys., 79 (10), 8071-5.

Koinkar, V.N. and Bhushan, B. (1996b), Microtribological studies of unlubricated and lubricated surfacesusing atomic force friction force microscopy, J. Vac. Sci. Technol. A Vacuum Surfaces and Films,14 (4), 2378-2391.

Krotil, H.-U., Stifter, T., Waschipky, H., Weishaupt, K., Hild, S., and Marti, O. (1999), Pulsed forcemode: a new method for the investigation of surface properties, Surface and Interface Analysis,27, 336-340.

Landman, U., Luedtke, W.D., and Ringer, E.M. (1992), Atomistic mechanisms of adhesive contact for-mation and interfacial processes, Wear, 153 (1), 3-30.

Lekka, M., Laidler, P., Gil, D., Cleff, B., and Stachura, Z. (1997), Elastic surface properties studied usingscanning force microscopy, Electron Technology, 30 (2), 173-6.

Marti, O. (1998), AFM instrumentation and tips, in Handbook of Micro/Nanotribology, Bhushan, B.(Ed.), CRC Press, Boca Raton, 81.

Marti, O., Drake, B., and Hansma, P.K. (1987), Atomic force microscopy of liquid-covered surfaces:atomic resolution images, Appl. Phys. Lett., 51 (7), 484-486.

Marti, O., Ribi, H.O., Drake, B., Albrecht, T.R., Quate, C.F., and Hansma, P.K. (1988), Atomic forcemicroscopy of an organic monolayer, Science, 239, 50-52.

Marti, O., Waschipky, H., Quintus, M., and Hild, S. (1999), Scanning probe microscopy of heterogeneouspolymers, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 154 (1-2), 65-73.

Maugis, D. (2001), Adhesion of solids: mechanical aspects, in Modern Tribology Handbook, Bhushan, B.(Ed.), CRC Press, Boca Raton.

Miyamoto, T., Kaneko, R., and Ando, Y. (1990), Interaction force between thin film disk media and elasticsolids investigated by atomic force microscope, J. Tribol., 112 (3), 567-72.

Miyatani, T., Okamoto, S., Rosa, A., Marti, O., and Fujihira, M. (1998), Surface charge mapping of solidsurfaces in water by pulsed-force-mode atomic force microscopy, Appl. Phys. A Materials Science &Processing, 66, S349-S352.

Mizes, H.A., Loh, K.-G., Miller, R.J.D., Ahuja, S.K., and Grabowski, E.F. (1991b), Submicron probe ofpolymer adhesion with atomic force microscopy: dependence on topography and material inho-mogeneities, Appl. Phys. Lett., 59, 2901-2903.

Pitsch, M., Metz, O., Kohler, H.-H., Heckmann, K., and Strnad, J. (1989), Atomic resolution with a newatomic force tip, Thin Solid Films, 175, 81.

Ramirez-Aguilar, K.A. and Rowlen, K.L. (1998), Tip characterization from AFM images of nanometricspherical particles, Langmuir, 14 (9), 2562-2566.

Rosa-Zeiser, A., Weilandt, E., Hild, S., and Marti, O. (1997), The simultaneous measurement of viscoelas-tic, electrostatic and adhesive properties by SFM: pulsed force mode operation, MeasurementScience and Technology, 8, 1333-1338.

Rubio, G., Agraït, N., and Vieira, S. (1996), Atomic-sized metallic contacts: mechanical properties andelectronic transport, Phys. Rev. Lett., 76 (13), 2302-2305.

Sarid, D., Hunt, J.P., Workman, R.K., Yao, X., and Peterson, C.A. (1998), The role of adhesion in tapping-mode atomic force microscopy, Applied Physics A Materials Science & Processing, 66, S283-S286.

Sarid, D., Ruskell, T.G., Workman, R.K., and Chen, D. (1996), Driven nonlinear atomic force microscopycantilevers: from noncontact to tapping modes of operation, J. Vac. Sci. Technol. B, 14 (2), 864-867.

Schmitz, I., Schreiner, M., Friedbacher, G., and Grasserbauer, M. (1997), Phase imaging as an extensionto tapping mode AFM for the identification of material properties on humidity-sensitive surfaces,Appl. Surf. Sci., 115 (2), 190-198.

Spatz, J.P., Sheiko, S., Möller, M., Winkler, R.G., and Marti, O. (1995), Forces affecting a substrate intapping mode, Nanotechnology, 6, 40-44.

Spatz, J.P., Sheiko, S., Möller, M., Winkler, R.G., Reineker, P., and Marti, O. (1997), Tapping scanningforce microscopy in air — theory and experiment, Langmuir, 13, 4699-4703.

Stalder, A. and Dürig, U. (1996a), Study of plastic flow in ultrasmall Au contacts, J. Vac. Sci. Technol. B,14 (2), 1259-1263.

Stalder, A. and Dürig, U. (1996b), Study of yielding mechanics in nanometer-sized Au contacts, Appl.Phys. Lett., 68 (5), 637-639.

Stifter, T., Weilandt, E., Marti, O., and Hild, S. (1998), Influence of the topography on adhesion measuredby SFM, Appl. Phys. A Materials Science & Processing, 66, S597-S605.

van Kevele, D.W. (1976), Properties of Polymers, Elsevier, Amsterdam, 303.Vogel, H. (1995), Gerthsen: Physik, 18 ed. Springer, Heidelberg.

Weisenhorn, A.L., Maivald, P., Butt, H.-J., and Hansma, P.K. (1992), Measuring adhesion, attraction, andrepulsion between surfaces in liquids with an atomic-force microscope, Physical Review B, 45 (19),11226-11232.

Winkler, R.G., Spatz, J.P., Sheiko, S., Moller, M., Reineker, P., and Marti, O. (1996), Imaging materialproperties by resonant tapping-force microscopy: A model investigation, Phys. Rev. B CondensedMatter, 54 (12), 8908-8912.

Wolter, O., Bayer, T., and Gerschner, J. (1991), Micromachined silicon sensors for scanning force micro-scopy, J. Vac. Sci. Technol., B9, 1353.

Zhong, Q., Inniss, D., Kjoller, K., and Elings, V.B. (1993), Fractured polymer/silica fiber surface studiedby tapping mode atomic force microscopy, Surf. Sci., 290 (1-2), L688-92.

Symbols

Symbol Unit Meaning

~z m Averaged movement of tipa Nm7 Coefficient of attractive part of the Lennard–Jones potentialb m Width of cantileverb Nm13 Coefficient of repulsive part of the Lennard–Jones potentialD

Compliance factor

E Modulus of elasticity, Young’s modulus

F N Force applied to spherical test bodyF(z) — Weight function for averaging the potentialF0 N Pull-off force (JKR)Fmax N Maximum force in the junction before yieldingFyield N Yield forceh m Thickness of cantileverHcrit Critical yield stress

k Cantilever compliance

kc Contact stiffness

keff Effective compliance of the junction

kN Compliance of the cantilever for normal forces

kS Sample compliance

kT Tip compliance

l m Length of cantileverm kg Effective mass of cantileverR m Radius of the spherical test bodyr0 m Distance of potential minimum in the Lennard–Jones potentialR0 m Minimum contact radiusRc m Contact radiusRcrit m Critical radius of spherical test body for yield failureU(z) Nm Interaction potential between tip and sampleVatom Nm Potential of a single atom

m2

N------

N

m2------

N

m2------

Nm----

Nm----

Nm----

Nm----

Nm----

Nm----

Vattr Nm Attractive component of the potential of a single atomVrep Nm Repulsive component of the potential of a single atomw m Base width of a triangular cantileverz M Coordinate perpendicular to sample or cantileverzmeasured M Measured displacement of the cantilever end-pointztrue M True displacement of the cantilever end-point∆γ

Difference in the contact potential

Θ rad Contact angleδ m Indentation depthε Nm Magnitude of the potential in the Lennard–Jones potentialγL

Contact potential of the test liquid vs. air

γS Contact potential of the sample vs. air

ν — Poisson numberσ

Yield stress

σcrit Critical yield stress for a given tip radius

ω0 Angular frequency of excitation

Nm----

Nm----

Nm----

N

m2------

N

m2------

1s---