Friction, Adhesion, and Deformation: Dynamic Measurements with the Atomic Force

Microscope

Phil AttardIan Wark Research Institute, University of South Australia, Mawson Lakes SA 5095 Australia

(J. Adhesion Sci. Technol. 16, 753–791 (2002).)

Running title: Friction, Adhesion, and Deformation

Abstract. A selection of recent experimental and theoret-ical work involving the atomic force microscope is reviewed,with the focus being upon dynamic measurements. Four top-ics are covered: calibration techniques for the friction forcemicroscope, quantitative measurements of friction and the ef-fect of adhesion, measurement and theory for the deformationand adhesion of viscoelastic particles, and the interaction andadhesion of hydrophobic surfaces due to bridging nanobub-bles.

I. INTRODUCTION

The atomic force microscope (AFM) [1] is commonlyused to image surfaces and to study the interaction andadhesion of particles. The wide-spread adoption of theAFM is due to its ease of use, the molecular-level infor-mation that it provides, and the variety of surfaces thatcan be studied in a broad range of environments. In addi-tion, the computer interface allows flexible control of thedevice and the automated acquisition of large volumes ofdata, it facilitates multiple repeat experiments to checkreproducibility and to minimise statistical error, and itenables detailed data analysis. This computer controlopens up the possibility of real-time monitoring of ex-periments and the exploration of time-dependent effects.The AFM is well-suited to studying the latter, whereasthe original surface force apparatus [2] and its variants[3,4] either lack automated data acquisition or suffer frominertial and other artefacts that must be accounted inthe quantitative interpretation of dynamic force measure-ments [5,6].

The distinction between equilibrium and non-equilibrium forces is quite important. To some extent,the primary concern with the AFM has been, (and shouldbe), to ensure that the experiments are carried out slowlyenough that equilibrium is established at each instant sothat the measured forces are comparable to those mea-sured statically. Beyond that, an exciting field of re-search exploits the dynamic capabilities of the AFM tomeasure non-equilibrium phenomena in a controlled fash-ion. We review two examples from our laboratory thatshow the utility of dynamic AFM measurements for non-equilibrium systems. Results and quantitative analysesare presented for the deformation, interaction, and adhe-sion of viscoelastic droplets, (§IV), and for the interactionand adhesion of spreading, bridging nanobubbles, (§V).

The most obvious technique that utilises the dynamic

capability of the AFM is the measurement of friction,which is also called friction force microscopy or lateralforce microscopy. Since the original work of Mate el al.[7] the fields of friction force mapping, (sometimes calledchemical imaging), and of nanotribology, have growngreatly, (see, for example, papers in refs [8,9]). This re-search has been severely limited by the lack of a quanti-tative calibration method for the AFM. This deficiencyhas been rectified quite recently by two techniques thatyield the torsional spring constant of the cantilever andthe voltage response of the lateral photodiode to can-tilever twist [10–12]. This review begins by summarisingthe limitations of previous calibration technique and bydetailing the procedures involved in the newer quantita-tive methods, (§II). The results that we have obtainedin our laboratory [13] for the quantitative dependence offriction on adhesion in a system with electric double layerinteractions are then reviewed (§III).

II. CALIBRATION OF THE FRICTION FORCE

MICROSCOPE

A. Critical Review

In order to use the AFM various calibrations have tobe performed. The lateral movement of the piezo is of-ten calibrated using model substrates. The expansionfactor that relates the applied voltage to the distance thepiezo expands in the vertical direction normal to the sub-strate, ∆z, can be measured from the interference fringesdue to the reflection of the laser from the cantilever andthe substrate. The normal spring constant of the can-tilever kx can be obtained gravitationally, thermally, orby resonance techniques [14–16]. The normal photodiodesensitivity factor, α0, relates the measured vertical differ-ential photodiode voltage ∆Vvert to the vertical deflectionof the cantilever, ∆x, which in the constant complianceregime is equal to the piezo movement, ∆x = ∆z. Forthe quantitative measurement of friction, in addition tothese one has to obtain the torsional spring constant ofthe cantilever, kθ, and the lateral photodiode sensitivityfactor, β, which relates the measured lateral differentialphotodiode voltage, ∆Vlat, to the twist angle of the can-tilever, ∆θ.

Unfortunately, almost all lateral calibration techniquesthat have been used to date are approximate in one wayor another, and the measurements of friction that utilise

1

them must be regarded as qualitative rather than quan-titative. Briefly, a critical review of the literature revealsthat in most cases [17–21] the torsional spring constantis calculated, not measured, using an analytic approx-imation [22] that idealises the actual geometry of thecantilever. In addition it ignores the effects of coatingsand thickness variations, which in the case of the normalspring constant can alter the value by an order of mag-nitude. The lateral sensitivity factor, which relates thephoto-diode voltage to the twist angle, has also been ob-tained by assuming it to be proportional to the verticalsensitivity [18], by modelling the beam path and profile[19], and by assuming that the tip is pinned during theinitial part of the friction loop [17,23]. Slippage and de-formation makes the latter method inaccurate, and oth-ers have attempted to improve the method by invokingcertain simple models of friction and deformation [20,21].Measurements of friction parallel to the long-axis of thecantilever using the normal spring constant and sensitiv-ity [24,25], erroneously neglect the bending moment ofthe cantilever [6,21]. Toikka et al. [23] attempted to usegravity acting on an attached lever, but the torque theyapplied can be shown to give negligible cantilever twist[12], and it appears that what they measured was in factphoto-diode saturation. And finally, the commonly usedcalibration method of Ogletree et al. [26] is restrictedby the need for a specialised terraced substrate and anultra sharp tip. For the calibration this method makestwo assumptions about the friction law, namely that fric-tion is a linear function of the applied load, and that itvanishes when the applied load is the negative of the ad-hesion. Counter-examples showing non-linear behaviourare known [13,27], and obviously, any fundamental studyof friction should test quantitatively such an assumptionrather than invoke it for the calibration. That none ofthe previous calibration methods are satisfactory is con-firmed by the fact that many FFM papers give friction interms of volts rather than Newtons [27–29]. Almost allfriction force maps are similarly uncalibrated and the im-ages are given in terms of volts rather than the physicalfriction coefficient.

Feiler et al. [12] have developed a direct technique thatsimultaneously measures the cantilever spring constantand the lateral sensitivity of the photo-diode. That par-ticular method is discussed in detail below.

Meurk et al. [10] have given a method for directly cal-ibrating the lateral sensitivity of the photo-diode. Ba-sically the angle of a reflective substrate is varied withrespect to the laser beam. In some AFM scanners thereis a stepper motor that facilitates the tilt of the head.From the geometry and the amount of movement the de-gree of tilt, ∆θ, can be calculated. The change in thelateral photo-diode voltage, ∆Vlat, is linear in the tiltangle and the ratio of the two gives the lateral sensitivityof the AFM.

The torsional spring constant of the cantilever can beobtained directly by the technique developed by Bog-danovic et al. [11]. Here a protuberance (eg. an upturned

tipped cantilever) is glued to the substrate and force mea-surements are performed against it with the protuber-ance in contact off-set from the central axis of the tiplessforce measuring cantilever. The latter consequently si-multaneously deflects and twists. Recording the normaland lateral photo-diode voltages in the constant compli-ance regime at several different lateral off-sets allows thespring constant divided by the lateral sensitivity to beobtained. Combined with the method of Meurk et al.[10], this allows a full calibration of the AFM. (In prin-ciple one can also obtain the lateral sensitivity with thismethod. However, the small leverage and high torsionalspring constant, makes it impractical to do so.)

B. Quantitative Calibration Technique

Fig. 1 Attard

J. Adh. Sci. Technol.

∆x

∆z

∆θ

FIG. 1. Rectangular cantilever with attached fibre andsphere. When the substrate is moved a distance ∆z, the can-tilever deflects a distance ∆x and twists an amount ∆θ. Thecorresponding changes in the differential photo-diode volt-ages, ∆Vvert and ∆Vlat, are measured.

We now describe in detail a one-step method that si-multaneously measures both the lateral photo-diode sen-sitivity and the torsional spring constant of the cantileverthat has been developed in our laboratory [12]. A glassfibre 50-200µm in length is glued perpendicular to thelong-axis of the cantilever and parallel to the substrate.To ensure that the substrate pushes on the end of thefibre, a colloid sphere is attached at its tip, (see Fig. 1).Using the well-known colloid probe attachment proce-dure of Ducker et al. [30], an epoxy resin is used to at-tach the sphere and a heat-setting adhesive is used to at-tach the fibre. This allows the fibre to be later removedand the cantilever used for friction measurements (ie. themethod is non-destructive). Attaching the sphere is con-venient but not essential; other ways to ensure that it isthe end of the fibre that touches the substrate includegluing the fibre to the cantilever at a slight angle, havinga ledge or colloid probe on the substrate, or perform-ing the measurement with the head or substrate tilted asmall amount, (eg. by using the stepper motor).

The essence of the method is that pushing on the tipof the fibre with a force F produces a torque τ = FL,where L is the length of the fibre. The cantilever simul-taneously deflects, ∆x = F/kx, and twists, ∆θ = τ/kθ.The deflection, and hence the force and torque, is ob-tained from the differential vertical photo-diode voltage∆x = α0∆Vvert, where the bare sensitivity factor, α0, ismeasured in the constant compliance regime without theattached fibre. The actual sensitivity factor with the at-

2

tached fibre αL is greater than this because only part ofthe piezo movement goes into deflecting the cantilever,∆x < ∆z, (the rest is soaked up by the twist). The barevertical sensitivity factor has to be measured in a sepa-rate experiment and depends upon the positions of thelaser, the photo-diode, and the cantilever mount. Withpractice, it is possible to obtain better than 10% repro-ducibility in this quantity between different experimentsand after remounting the cantilever. The best way toensure this is to to maximise the total vertical signal andto minimise the differential lateral signal each time.

4/21/01 2:30 PMCalib.xls Fig2

Fig. 2Attard

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-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2

Vertical Voltage

Lat

eral

Vol

tage

FIG. 2. Lateral differential photo-diode voltage as a func-tion of the vertical voltage. Both were measured for a can-tilever with an attached fibre over the whole approach regimeof a single force measurement. The data are from Ref. [12].

The calibration factor of primary interest is the onethat relates the differential lateral photo-diode voltageto an applied torque, τ = γ∆Vlat. This is given by

γ =τ

∆Vlat

=kx∆xL

∆Vlat

= kxα0L∆Vvert

∆Vlat. (1)

This equation predicts a linear relationship between thetwo photo-diode signals, which, as can be seen in Fig. 2,is indeed the case. The slope of this line, combined withthe measured values for the vertical spring constant, thebare vertical sensitivity factor, and the length of the fi-bre, gives the factor that converts the differential lateralphoto-diode voltage to the applied torque in general (ie.independent of the attached fibre). Figure 3 shows thelateral sensitivity factor obtained using a number of dif-ferent fibres. That the same value is obtained each timeshows that it is an intrinsic property of the cantileverand AFM set-up. It also confirms that remounting thecantilever does not preclude reproducible results beingobtained.

4/21/01 2:31 PMCalib.xls Fig3

Fig. 3Attard

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

70 90 110 130 150 170

Length (µm)

γ (p

Nm

/V)

FIG. 3. Lateral sensitivity factor for different fibres. Thecantilevers were taken from the same batch. Circles indicatefibres with an end-attached sphere, diamonds indicate barefibres, filled symbols are for approach, and open symbols arefor retraction. The data are from Ref. [12].

We found that the calibration procedure was straight-forward and relatively robust. The method was less suc-cessful whenever there was significant adhesion betweenthe substrate and the tip of the fibre or the attachedsphere. We minimised such adhesion by using silica sur-faces and conducting the calibration in water at naturalpH.

It is possible to verify independently the procedure byobtaining the sensitivity factor that relates the change inangle to the change in the lateral photo-diode signal, andcomparing this with the value obtained by the methodof Meurk et al. [10]. From the slope of the constant com-pliance region of the force curve with the attached fibre,one can obtain the constants

αL =∆z

∆Vvert, and βL =

∆z

∆Vlat, (2)

for the vertical and lateral deflections, respectively. Withthese the lateral sensitivity can be shown to be given by[12]

∆θ

∆Vlat=βL(1 − α0/αL)

L. (3)

A value of 3×10−4rad/V was obtained using our method[12], compared to 1.7 × 10−4rad/V using the method ofMeurk et al. [10].

The torsional spring constant itself is given by [12]

kθ =−kxL2

1 − αL/α0. (4)

A value of 2 × 10−9N m was obtained using our method[12], compared to 1.2 × 10−9N m calculated from themethod of Neumeister and Ducker [22].

3

III. ADHESION AND FRICTION

A. Intrinsic Force

One of the oldest ideas concerning the nature of fric-tion is embodied in Amontons’ law, which states thatthe friction force f is proportional to the applied load L,f = µL, where µ is the coefficient of friction. For thecase of adhesive surfaces, where a negative load needs tobe applied to separate them, it is known that there canbe substantial friction even when the load is zero. HenceAmontons’ law may be slightly modified

f =

{

µ(L+A), L ≥ −A0, L < −A, (5)

where A > 0 is the adhesion, which is the greatest ten-sion that the surfaces can sustain. This modified versionreflects the plausible idea that friction only occurs whenthe surfaces are in contact. Amontons’ law raises severalquestions: Is friction a linear function of load? Is theonly role of adhesion to shift the effective load? Whatis the law for non-adhesive surfaces? Is friction zero forsurfaces not in contact? And what does contact mean ona molecular scale?

The AFM is an ideal tool to test the fundamental na-ture of friction, and we set out to answer quantitativelythese and other questions [13]. We chose a system thatwould allow us to change the adhesion in a controlledmanner so that as far as possible all other variables werekept constant. We used a titanium dioxide substrate,(root mean square roughness of 0.3nm), and a silicondioxide colloid probe, (root mean square roughness of0.8nm, 7µm diameter). The measurements were carriedout in aqueous electrolyte, (10−3M KNO3), as a functionof pH. The SiO2 is negatively charged at practically allpH, (its point of zero charge is ≈ pH 2), whereas TiO2 ispositively charged at low pH and negatively charged athigh pH, (its point of zero charge is ≈ pH 4.5). Hence atlow pH the attractive double layer interaction betweenthe surfaces causes them to be adhesive, and at high pHthey repel each other and do not adhere.

There have been several other AFM studies of frictionbetween surfaces with electrical double layer interactions[27,31,32]. In some cases an applied voltage has beenused to modify the adhesion, but the friction coefficientsand force laws have all been qualitative in the sense of thepreceding section. A critical discussion of these results isgiven in Ref. [13].

The load, which is the applied normal force, is shownin Figs 4 and 5 as a function of separation for variouspH. It can be seen that the surfaces do indeed interactwith an electric double layer interaction, and that thepH controls the sign and the magnitude of the force law.For pH 4 and 5 the attractive double layer interactiongives an adhesion of A = 10.5 and 4.4 nN, respectively.However at higher pH the surfaces do not adhere.

Fig 4Attard

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-4

-2

0

2

4

6

8

0 10 20 30 40 50Separation, h (nm)

Forc

e (n

N) 0.1

1

10

0 10 20 30 40

FIG. 4. Force on approach as a function of separation. Thesubstrate is TiO2, the 7µm diameter colloid probe is SiO2,and the background electrolyte is 1mM KNO3. From topto bottom the curves correspond to pH = 8, 7, 6, 5, and4. The inset shows constant potential (ψSiO2

= −50mV andψTiO2

= −43mV) and constant charge fits to the pH = 8 caseon a log scale [33].

Fig 5Attard

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-12

-8

-4

0

4

8

0 10 20 30 40 50 60

Separation, h (nm)

Forc

e (n

N)

01234

0 5 10 15

FIG. 5. Same as preceding figure on retraction. The insetmagnifies the three highest pH at small separations [33].

In view of Eq. (5), we are motivated to define thedetachment force Fdetach as the minimum applied forcenecessary to keep the surfaces in contact [13]. For non-adhesive surfaces this is a positive quantity, and for ad-hesive surfaces it is negative, (in fact it is the negative ofthe adhesion). The detachment force at pH = 6, 7, and 8was Fdetach = 1.4, 2.6, and 3.5 nN, respectively, (Fig. 5).In view of the close relationship between adhesion andthe detachment force, one may define an intrinsic force,

Fintrinsic = L− Fdetach, (6)

which may be thought of as the force in excess of thatwhen the surfaces are just in contact. In this lan-guage, Amontons’ law generalised to non-adhesive sur-faces would read f = µFintrinsic.

We measured friction as a function of applied load atvarious pH. This was done in the usual fashion [7] bymoving the substrate back and forth in the direction per-pendicular to the long axis of the cantilever and recordingfriction loops. The length of the scan in each directionwas 0.5µm, and the velocity was 1µm/s. The lateral cal-ibration factor, obtained as detailed above [12], was usedto convert (half) the voltage difference between the two

4

arms of the friction loop to the applied torque τ . Thefriction force was obtained as f = τ/2r, where r = 7µmis the radius of the colloid probe. The applied load wasfixed by using the set-point feature of the AFM (ie. thevertical deflection signal was held constant during thefriction loop).

Fig 6Attard

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0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6 7 8

F applied (nN)

Frict

ion

Forc

e (n

N)

ph4ph5ph6ph7ph8

FIG. 6. Friction force as a function of applied load [33].

Friction is plotted as a function of applied load in Fig 6.In general friction increases with increasing load. At agiven applied load, friction is also larger the lower the pH.Since the adhesion increases with decreasing pH, one mayrestate this fact as the higher the adhesion the higherthe friction at a given applied load. Moreover, frictionis non-zero at zero loads for adhesive surfaces. For non-adhesive surfaces, friction is zero for small but non-zeroapplied loads.

Fig 7Attard

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0

5

10

15

20

25

30

35

40

-5 0 5 10 15 20 25 30 35

F intrinsic (nN)

Frict

ion

Forc

e (n

N)

pH4pH5pH6pH7pH8pH5.5

FIG. 7. Friction force as a function of intrinsic load [33].

The quantitative behaviour of friction with pH is notobvious when plotted as a function of applied load. Butwhen plotted against intrinsic load, Fig. 7, the utility ofthe detachment force becomes evident. The functionalform of the friction force law is fundamentally indepen-dent of pH, and all the measurements lie on a single uni-versal curve. In other words, the major role of pH is todetermine the adhesion, (or more precisely the detach-ment force). Once this parameter has been experimen-tally determined from a normal force measurement at agiven pH, the friction at that pH may be predicted fromthe friction measured at any other pH merely by shiftingthe load by the detachment force.

These experiments show that for this system frictionis not a linear function of load (ie. the friction coefficientµ = df/dL is not independent of load). There is a notice-able curvature in the plot, with friction increasing morerapidly at higher loads. The loads that have been ap-plied here are relatively weak, (the average pressure inthe contact region (see below) is less than about 10MPa,and the peak pressure is less than about 100MPa [13]),and it is not clear what will happen at higher loads thanthese.

Whilst it is not implausible that the friction should bezero for negative intrinsic forces in all cases, (this corre-sponds to the surfaces being out of contact), it is a littlesurprising that for positive intrinsic forces the increasein friction is the same in all cases. After all, not onlyis the adhesion and the normal force law different at thedifferent pH, but also the surface chemistry varies due tothe different amount of ion binding that occurs. The factthat the latter has almost no effect on friction is perhapsnot unexpected since over the range of pH studied, forTiO2 only about 1% of the surface sites are convertedfrom H+ at low pH to OH− at high pH, and for SiO2

the change is about 10% [34]. Nevertheless it is not im-mediately obvious why surfaces with different adhesionsdisplay quantitatively the same friction for the same in-trinsic force.

B. Elastic Deformation

In order to investigate this question further we carriedout elastic deformation calculations of the sphere andsubstrate under the experimental conditions [13]. Elas-tic deformation has long been thought to play a dominantrole in friction of macroscopic bodies, mainly in the con-text of using contact mechanics to account for asperityflattening [35,36]. We however were in a position to gobeyond contact theories such as JKR [37] or DMT [38].We used the soft-contact algorithm of Attard and Parker[39,40] and invoked the actual experimentally measuredforce law, which is of course of non-zero range. The al-gorithm self-consistently calculates the surface shape ofthe elastically deformed bodies due to the local pressure,which in turn depends upon the local separation of thedeformed bodies. In this way we obtain the actual surfaceshape and the actual pressure profile, whereas contactmechanics assumes simplified and non-physical forms forboth. We fitted a smooth curve to the measured forcelaw at the different pH, and using the Derjaguin approx-imation, differentiated this to obtain the pressure as afunction of surface separation. The latter is required bythe algorithm [39,40], as is discussed in the following sec-tion. The calculations presented in Ref. [13] are the firstelastic deformation calculations using an actual experi-mentally measured force law. For the present calculationsthere was no hysteresis between the loading and unload-ing cycles. (The hysteresis observed in the original papers

5

[39,40] for soft adhesive bodies has since been attributedto a non-equilibrium viscoelastic effect [41,42]; see §IV.)

friction.xls fig84/21/01 2:44 PM

Fig 8Attard

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

r (µm)

h(r)

(nm

)

FIG. 8. Calculated surface profiles for an applied loadof 5nN. From top to bottom, the pH is 8, 7, 6, 5,and 4, and in each case the measured force law hasbeen used in the calculations. Young’s modulus for SiO2,E/(1 − ν2) = 7.7 × 1010N/m2, has also been used. The bot-tom dashed curve is for an applied load of 720nN for the pH4case. The abscissa is the distance from the central axis in mi-crons and the ordinate is the local separation in nanometres.The data are from Ref. [13].

Figure 8 shows the resultant surface shape at an ap-plied load of 5nN. This load is greater than all the detach-ment forces, and in all cases the surfaces showed non-zerofriction. It can be seen that very little surface flatteninghas occurred, and that the surfaces at different pH areeffectively displaced parallel to each other.

Also included in Fig. 8 is a high load (720nN) case,which shows substantial flattening. However there is nota well defined contact region, and there is certainly not asharp change in the surface profile to demark contact de-spite the fact that these calculations are for the adhesivepH4 surfaces.

The fitted force law includes a Lennard-Jones soft re-pulsion with length scale 0.5nm [13], and one could definecontact as local separations smaller than this. Such an ar-bitrary definition is somewhat problematic, particularlysince the curves at 5nN load, which are not in contact bythe definition, display non-zero friction. In view of thisdiscussion of the meaning of contact for systems with re-alistic surface forces of non-zero range, the inapplicabilityof simple contact theories such as Hertz, JKR, or DMTis clear. One might also conclude that the experimentalverification or refutation of Amontons’ second law, (fora given load friction is independent of contact area), atthe molecular level will be difficult.

4/21/01 2:44 PMfriction.xls fig9

Fig 9Attard

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

r (µm)

h(r)

(nm

)

FIG. 9. Calculated surface profiles for an intrinsic force of15nN. From top to bottom, the virtually indistinguishablecurves correspond to a pH of 8, 7, 6, 5, and 4, respectively.The data are from Ref. [13].

Figure 9 shows the surface shapes at the different pHat an intrinsic load of 15nN, which corresponds to an ap-plied load of 5nN for the pH4 case. The change fromFig. 8 is quite dramatic, and one can see that the pro-files have coalesced. In other words, surfaces at a givenintrinsic load have the same shape and local surface sep-aration. Given that friction is also a universal functionof intrinsic load, (Fig. 7), one may conclude that frictionis a function of the local separation and independent ofthe force law. In so far as the short-range interactionsbetween the atoms on the two surfaces can be expectedto be independent of pH, one can say that these are theinteractions that determine friction. Friction occurs be-tween two bodies when energy can be transferred fromone to another, which means that they have to be closeenough for the interaction between atoms on the two sur-faces to be comparable to the thermal energy [13]. Oneconcludes that the only role of adhesion in friction is todecrease the amount of applied load that is necessary tobring the surfaces to a given separation.

IV. VISCOELASTIC DEFORMATION AND

ADHESION

A. Viscoelastic Theory

The shape of the deformed surfaces given above wereobtained by solving the equations of continuum elastic-ity theory in the semi-infinite half-space approximation[39,43]

u(r) =−2

πE

∫

dsp(h(s))

|r − s| . (7)

Here the elasticity parameter E is given in terms of theYoung’s moduli and the Poisson’s ratios of the two bod-ies, 2/E = (1−ν2

1)/E1 +(1−ν22)/E2, r and s are the lat-

eral distance from the central axis, and p(h) is the pres-sure between two infinite planar walls at a separation of

6

h. The total deformation normal to the surfaces at eachposition is u(r), and hence the local separation betweenthe two bodies is h(r) = h0(r)−u(r). Here the local sepa-ration of the undeformed surfaces is h0(r) = h0 + r2/2R,where R−1 = R−1

1 + R−12 is the effective radius of the

interacting bodies; in general the Ri are related to theprinciple radii of curvature of each body [44].

For contact theories such as Hertz, JKR, or DMT, thepressure pc(r) that appears in the integrand of Eq. (7) isa specified function of radius that when integrated givesu(r) = r2/2R, which corresponds to a flat contact re-gion, h(r) = 0. In contrast for realistic force laws thathave non-zero range, such as van der Waals, electric dou-ble layer, or the actual measured p(h) discussed above,the integral must be evaluated numerically. Because inthis case the local separation depends upon the defor-mation, Eq. (7) represents a non-linear integral equationthat must be solved by iteration for each nominal sepa-ration h0.

An efficient algorithm for the solution of the non-contact elastic equation has been given by the author[39,41], and it has been used to analyse a variety of forcelaws [13,39–42]. Other workers have also calculated theelastic deformation of the solids using realistic surfaceforces of finite range [45–52]. There have of course beena large number of experimental studies to measure theinteraction of deformable solids. These include AFMmeasurements [53–63] , as well as results obtained withthe surface force apparatus and the JKR device [64–73].These studies in general show that the adhesion and in-teraction is hysteretic and time-dependent, particularlyfor highly deformable solids with high surface energies.Such behaviour is characteristic of viscoelastic materials.Maugis and Barquins have given a review of viscoelasticadhesion experiments, which they attempt to interpretin quasi-JKR terms, introducing a somewhat ill-definedtime-dependent surface energy [74].

A proper theoretical treatment of the deformation andadhesion of viscoelastic materials involves replacing theelasticity parameter, which gives the instantaneous re-sponse to the pressure, by the creep compliance func-tion, which gives the response to past pressure changes.In this way the prior history of the sample is accountedfor. Hence the generalisation of the elastic half-spaceequation involves a time convolution integral [75,76],

u(r, t) − u(r, t0)

=

∫ t

t0

dt′−2

πE(t− t′)

∫

dsp(h(s, t′))

|r − s| . (8)

Here p(h(s, t)) is the time rate of change of the pres-sure. The particles are assumed stationary up to time t0,and, if interacting or in contact, have at that time fixeddeformation corresponding to static elastic equilibrium,u(r, t0) = u∞(r). This expression is essentially equiva-lent to that used by a number of authors [77–80], withthe difference being that the latter have treated contactproblems, with p(h(s, t)) replaced by a specified analytic

pc(s, t), whereas here p(h(s, t)) is determined by the phys-ical force law and the past rate of change of separation.

An algorithm has been developed for solving the fullnon-contact problem for the case that the creep compli-ance function has exponential form [75]

1

E(t)=

1

E∞

+E∞ − E0

E∞E0e−t/τ . (9)

Here E0 and E∞ are the short- and long-time elasticityparameters, respectively, and τ is the relaxation time.The algorithm can be generalised to more complex ma-terials with multiple relaxation times [75]. The presentthree parameter model is perhaps the simplest model ofviscoelastic materials, although an alternative three pa-rameter expression, E(t)−1 = C0 + C1t

m, 0 < m < 1,has also been studied as a model for liquid-like materials[79–81].

With the exponential creep compliance function, dif-ferentiation of the deformation yields [75]

u(r, t) =−1

τ[u(r, t) − u∞(r, t)]

− 2

πE0

∫

dsp(h(s, t))

|r − s| , (10)

where u∞ is the static deformation that would occur inthe long-time limit if the pressure profile were fixed at itscurrent value,

u∞(r, t) =−2

πE∞

∫

dsp(h(s, t))

|r − s| . (11)

The rate of change of the pressure is

p(h(r, t)) = p′(h(r, t))[

h0(t) − u(r, t)]

, (12)

where h0(t) is the specified drive trajectory. Accordingly,Eq. (10) represents a linear integral equation for the rateof change of deformation. It can be solved using thesame algorithm that has been developed for the elasticproblem [39,41]. It is then a simple matter to solve thedifferential equation for the deformation by simple timestepping along the trajectory, u(r, t + ∆t) = u(r, t) +∆tu(r, t).

The algorithm has been used to obtain results for anelectric double layer repulsion [75] and for a van derWaals attraction [76]. The latter is

p(h) =A

6πh3

[

z60

h6− 1

]

, (13)

where A is the Hamaker constant, and z0 characterisesthe length scale of the soft-wall repulsion. Fig. 10 showsthe shape of viscoelastic spheres during their interaction.The total time spent on the loading branch is ten timesthe relaxation time, so that one expects to see viscoelas-tic effects. At the largest separation prior to approach

7

the surfaces are undeformed. Prior to contact on ap-proach they bulge toward each other under the influenceof the van der Waals attraction. There is a relativelyrapid jump into contact, and initially a fast spreading ofthe flattened contact region, which continues to grow asthe particles are driven further together. At the edge ofthe contact region there is a noticeable rounding of thesurface profiles on the approach branch. Following thereversal of the motion, (unloading), the surfaces becomeextended as they are pulled apart, and there is a sharpertransition between contact and non-contact than on theloading branch. It should be noted, however, that evenin this case the slopes at the edge of the contact regionare not discontinuous as predicted by the JKR theory.Following the turning point, the surfaces are effectivelypinned in contact for a time, and then the contact regionbegins to recede. After the surfaces jump apart there re-mains a memory of the stretching that occurred duringunloading, and for a time comparable to the relaxationtime of the material, the deformed separation is smalleron the unloading branch out of contact than at the cor-responding position upon loading.

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r (µm)

h(r,

t) (

nm)

FIG. 10. Surface profiles for adhesive viscoelastic spheres.The profiles are plotted every millisecond, or every 2nm fromh0 =10nm (top) to -10nm (bottom). The drive speed is|h0| = 2µm/s and the Hamaker constant is A = 10−19J, withz0 = 0.5nm and R = 10µm. The viscoelastic parameters areE0 = 1010N m−2, E∞ = 109N m−2, and τ = 1ms. Theright hand panel is for loading and the left hand panel is forunloading. The data are from Ref. [76].

This hysteresis in surface shape is reflected in the dif-ference in force versus nominal separation curves on theloading and unloading branches, Fig. (11). On approach,prior to contact a given attraction occurs at larger nom-inal separation, for slower driving speeds. In these casesthere is an increased bulge leading to smaller actual sepa-rations, a consequence of the fact that viscoelastic mate-rials soften over longer time-scales. The jump of the sur-faces into contact is reflected in a sharp decrease in theforce. Once in contact the force increases and the nomi-nal separation becomes negative, which is a reflection ofthe deformation and growth of the flattened contact re-gion under increasing load. The faster the particles aredriven together, the steeper is the slope of the force curve,

as one might expect since this corresponds to effectivelystiffer materials.

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Forc

e F(

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πR (

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-10

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0

5

10

-1 0 1 2

FIG. 11. Interaction forces for adhesive viscoelasticspheres. From inside to outside the hysteresis loops corre-spond to driving velocities of |h0| = 1, 2, and 5 µm/s, usingthe viscoelastic parameters of Fig. 10. The crosses representthe static equilibrium elastic result for E∞ = 109N m−2. In-

set. Loading curves near initial contact. The circles representthe static equilibrium elastic result for E0 = 1010N m−2, andthe bold curve is the force for rigid particles. The data arefrom Ref. [76].

Following the reversal of the direction of motion inFig. 11, a small increase in the nominal separation givesa large decrease in the applied load, which causes the un-loading branch to lie beneath the loading branch. Thisbehaviour is reflected in the surface profiles, (Fig. 10),where on the loading branch, increasing the load causesthe contact area to grow, whereas immediately followingthe turning point, decreasing the load stretches the sur-faces at fixed contact area. The hysteresis in the forcecurves manifests the fact that a certain energy has tobe put into the system to move the surfaces a nominaldistance on loading, and less energy is recovered fromthe system in moving the same distance on unloading.This is precisely what one expects from a viscoelasticsystem. The size of the hysteresis loop increases withdrive speed. As the speed is decreased, both loops ap-pear to coalesce on the long-time elastic result, whichcorresponds to static equilibrium, Eq. (7).

Figure 11 also shows that the adhesion, which is themaximum tension on the force loop, increases with drivevelocity. Because the position is here controlled, we areable to calculate the trajectory past the force minimumand beyond the jump out of contact. In an experimentthat controlled the load, the force minimum would be thelast point measured in contact. The position of the min-imum force moves to smaller, (more negative), nominalseparations as the velocity is increased. It can be seenthat the adhesion of the viscoelastic particles is signifi-cantly greater than that of elastic particles.

The velocity dependence of the adhesion is exploredin more detail in Fig. 12. As the velocity is decreased,the curves asymptote to the static equilibrium elastic re-sult, calculated from Eq. (7). It should be noted that

8

the latter is not given by the JKR prediction, which as acontact approximation that neglects the range of the vander Waals interaction, is not exact. It can be seen thatfor elastic materials, the JKR approximation is more ac-curate for particles with larger surface energies. As thevelocity increases, and the system is given less time toequilibrate, viscoelastic effects become more evident andthe adhesion increases. For the present parameters, atspeeds greater than about 10µm/s, there occurs a no-ticeable dependence of the normalised adhesion on thesurface energy, with higher energy particles showing less(normalised) adhesion. The actual adhesion increaseswith surface energy at all driving velocities. This sug-gests that at very high speeds the adhesion will be inde-pendent of the surface energy.

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Speed (µm/s)

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3 πγR

FIG. 12. Adhesion. The maximum tension normalised bythe JKR elastic adhesion as a function of drive velocity (loga-rithmic scale). The parameters are as in Fig. 10, except thatthe Hamaker constant is A = 1, 5, and 10 ×10−20J, (the sur-face energy is γ ≡ A/16πz2

0 = 0.80, 3.98, and 7.96mN/m), forthe dotted, dashed, and solid curves, respectively. The dataare from Ref. [76].

B. Central Deformation Approximation

For the case of elastic particles, a relatively accurateanalytic approximation for the elastic integral has beendeveloped to treat the pre-contact situation [39]. Theelastic central deformation approximation (CDA) con-sists of replacing the deformation u(r) everywhere by itsvalue on the central axis, u(0). An analogous approxi-mation can be made for the viscoelastic case, and resultsin the form of an analytic differential equation have beenpresented for the van der Waals attraction used above[76], and for an electric double layer repulsion [75]. Thelatter has the form

p(h) = Pe−κh. (14)

In this case, the analytic approximation for the centraldeformation u(t) ≡ u(0, t) is [75]

u(t) =f(t)h0(t) − [u(t) − u∞(t)] /τ

1 + f(t), (15)

where f(t) ≡√

8πκRP 2/E20 exp−κ[h0(t) − u(t)]. and

u∞(t) = −E0f(t)/E∞κ. For a given trajectory h0(t),the deformation u(t) is readily obtained from the pre-ceding equation for u(t) by simple time-stepping. Theforce in this approximation is essentially as given byDerjaguin, except of course that the actual deformedseparation is used rather than the nominal separationthat would be appropriate for rigid particles. That is,F (t) = 2πRκ−1P exp−κ[h0(t) − u(t)].

This central deformation approximation is testedagainst the exact results for the pre-contact deformationof a viscoelastic sphere being driven toward a substrate inFig. 13. The deformation is negative, which correspondsto flattening of the particles under their mutual repul-sion. It may be seen that differential equation is quanti-tatively accurate for the deformation. It correctly showsthat at a given position h0, the deformation is greater atthe slower driving speed because the soft component ofthe elasticity has more time to take effect. Conversely,the force is greater at the faster driving speed becausethe surface separation of the effectively stiffer material issmaller at a given position (not shown).

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Cen

tral

Def

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atio

n u(

t) (

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0 1 2 3 4

h0(t) (nm)

ForceF/2πR (N/m)

10-2

10-3

10-4

FIG. 13. Pre-contact flattening for repulsive forces. Thesymbols represent the exact calculation, and the solid curvesare the central deformation approximation, Eq. (15). Theparameters are as in Fig. 10, with P = 107N m−2 andκ−1 = 1nm being used in the pressure law, Eq. (14). Aconstant driving velocity of h0 = 5 (upper) and of 1µm s−1

(lower) is used. The inset shows the corresponding forces nor-malised by the radius for h0 = 1µm s−1, with the bold curverepresenting the infinitely rigid case (no deformation). Thedata are from Ref. [75].

The inset of Fig. 13 compares the load on a viscoelasticsphere to that on an undeformable one at a given posi-tion. It can be seen that the load required to move the de-formable particle a nominal amount (the drive distance)is less than that required for a rigid particle because thesurface separation between deformed particles is greaterthan that between undeformed particles. The agreementbetween the central deformation approximation Eq. (15)and the exact calculations in the inset confirms the valid-ity of the elastic Derjaguin approximation. As the lattershows, the major effect of deformation on the force arisesfrom the change in surface separation rather than from

9

any increase in contact area due to flattening.

C. Deformation and Adhesion Measurements

The AFM is an ideal tool for the study of viscoelasticeffects because of its real-time acquisition of data duringcontrolled dynamic measurements. The data that aredirectly obtainable are the force as a function of drivedistance for both loading and unloading, and the adhe-sion. Detailed analysis of this data using the elastic andviscoelastic theories described above should allow the ex-traction of the amount of deformation, and the values ofthe elastic parameters and relaxation times.

In our laboratory we have recently commenced a re-search program of quantitative AFM measurement andanalysis of the interaction, deformation, and adhesion ofviscoelastic particles [82]. We use an emulsion polymeri-sation process to make poly-dimethylsiloxane (PDMS)droplets [83,84]. The deformability ranges from liquid-to solid-like, and is controlled by the ratio of trimer tomonomer cross-linker used in the synthesis. Dependingupon the conditions, micron-sized droplets form, and aretransferred to the AFM on a hydrophobic glass slideto which they are allowed to adhere. A 7µm silicacolloid probe is attached to the cantilever; the well-defined and known geometry and surface chemistry ofthe probe enables a quantitative analysis of the measure-ment. The zeta potential of the droplets is measuredby electrophoresis [85]. The surface chemistry of thedroplets is very similar to that of the silica probe; atpH9.6 the zeta potentials are -46 and -62 mV, respec-tively.

There have been a number of previous AFM studies ofdeformable solid surfaces [53–63]. In addition, the AFMhas been applied to air bubbles [86–89] and to oil droplets[90–93]. Measurements of such systems raise two imme-diate issues: the determination of the normal sensitivityfactor, which relates the measured vertical photo-diodevoltage to the deflection of the cantilever, and the deter-mination of the zero of separation. Two further issues ofanalysis arise: the conversion of the nominal separationto the actual separation, (ie. the determination of the de-formation), and the relation of the material and surfaceproperties of the substrate to the measured interaction.

One can perform the vertical calibration by a priormeasurement on a hard substrate in the constant com-pliance regime. We performed this calibration in situ bysimply moving off the droplet and pressing against thesubstrate [82]. If this is not possible, (because eitherthe drop is macroscopic or because a deformable probeis attached to the cantilever), then one can perform thecalibration on another cantilever provided that one takescare with the remounting and alignment of the laser, asdescribed in §II above and in Ref. [12].

The matter of determination of the zero of separationcan only be done if the force law is known. At large

separations the deformation is always negligible becausehere the force is weak. In practical terms of course itis a matter of whether or not one has the instrumentalresolution to measure weak enough forces, and this is de-termined by the ratio of the cantilever spring constant tothe deformability of the substrate or particle. Assumingthat this regime is accessible, then at large separationsthe measured force must equal that between rigid parti-cles. If the latter is known, then this fact can be used toshift the experimental data so that it coincides with theknown force law at large separations. When this is done,the drive distance, which has arbitrary zero, is convertedto a nominal separation, which is the separation betweenrigid particles. This procedure is now illustrated, as is themethod of calculating the deformation of the particles,which allows the conversion of the nominal separation tothe actual separation.

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Drive Distance (nm)

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-100 -50 0 50

Nominal Separation (nm)

Force (nN)

FIG. 14. AFM measurement of the force between a PDMSdroplet (-46mV) and a silica sphere (-62mV) in 1mM KNO3 atpH9.8. The drive speed is 1.2 µm/s, and the drive distance iswith respect to an arbitrary zero. The flat force extrema arisefrom photo-diode saturation. Inset. Force on a logarithmicplot. The zero of the nominal separation is determined byshifting the data to coincide with the electric double layerforce at large separation calculated using the measured zetapotentials. The straight line is the linear Poisson-Boltzmannlaw for rigid particles, and the partly obscured curve is theelastic central deformation approximation, Eq. (18), with afitted elasticity parameter, E∞ = 7 × 105J m−3. The CDAis shown dashed for h0 < −19nm, which, for a pure doublelayer interaction, is the point of actual contact, h = 0. Thedata are from Ref. [82].

Figure 14 shows the force between a silica sphere, (di-ameter 7µm), and a solid-like PDMS droplet, (diameter1.2µm, 50% trimer), measured as a function of the drivedistance [82]. After the initial zero force regime, one cansee the electric double layer repulsion due to the inter-action of the two negatively charged surfaces. At a forceof around 20nN there is a jump into contact due to avan der Waals attraction, followed by a soft complianceregime. The latter is characterised by a finite slope anda non-zero curvature. Upon reversing the direction, (ig-noring the instrumental saturation at about 35nN force),the soft compliance is again evident, with the change in

10

slope indicating hysteresis. The adhesion of the surfacescontributes to this hysteresis, and they do not jump apartuntil being driven a distance of several hundred nanome-tres from the point of maximum load. (Again the instru-mental saturation at about -35nN is ignored.)

The analysis of the data is illustrated in the inset toFig. 14. The zero of separation is established by shiftingthe measured data horizontally to coincide with the lin-ear Poisson-Boltzmann law at large separations. It canbe seen that over a limited regime the data is indeedlinear on the log plot, with a slope corresponding to theexpected Debye length. The relatively short range of thisregime is due to a combination of the large deformabilityof the PDMS droplet and the stiffness of the cantilever,k = 0.58N/m, chosen in order to measure larger appliedloads and as much of the adhesion as possible. The dataat the largest separations are only just significant com-pared to the resolution of the AFM; that the data appar-ently begins to decay faster than the Debye length at theextremity of the range exhibited is due to contaminationby interference fringes.

The linear Poisson-Boltzmann law used here is givenby F (h0) = 2πRκ−1

D P0e−κDh0 , where κ−1

D = 9.6nm isthe Debye screening length, h0 is the nominal separation(between rigid particles), and R = 0.6µm is the radius ofthe PDMS droplet. In linear Poisson-Boltzmann theory,the pre-factor in the pressure law, Eq. (14), is given by

P0 = 2ε0εrκ2Dψ1ψ2, (16)

where ε0 = 8.854 × 10−12 is the permittivity of freespace, εr = 78 is the dielectric constant of water, andψ1 = −46mV and ψ2 = −62mV are the surface potentialsof the PDMS and the silica sphere, respectively, whichare measured independently by electrophoresis [85]. Inpractice an effective surface potential is used, whichessentially converts this into the non-linear Poisson-Boltzmann law in the asymptotic regime [94,95]. One re-places ψ by 4γkBT/q, where q = 1.6×10−19 is the chargeon the monovalent electrolyte ions, kB = 1.38 × 10−23 isBoltzmann’s constant, T = 300K is the temperature,and

γ =eqψ/2kBT − 1

eqψ/2kBT + 1. (17)

As discussed above following Eq. (15), the centraldeformation approximation (CDA) for elastic particlesgives for the pre-contact deformation [39]

u = −√

8πR/κDE2P0e−κD [h0−u]

≡ −ωe−κD [h0−u]. (18)

Although this can be solved by iteration to obtain thedeformation u for any nominal separation h0, for the pur-poses of plotting it is easier to specify h and to calculatedirectly the corresponding u and h0. The resultant forceis F (h0) = 2πRκ−1

D P0e−κDh, where the actual separation

is h = h0 − u.

The inset of Fig. (14) compares this elastic CDA withthe measured data using a fitted elasticity of E∞ = 7 ×105N/m2. At large separations in the weak force regimeit coincides with the rigid particle result, but due to theextreme softness of the particles, the force increases muchless rapidly than the linear Poisson-Boltzmann predicts.The CDA predicts that the surfaces come into actual con-tact, (h = 0), at a nominal separation of h0 = −19nm,and the theory is continued past this point as a dashedline. There is a noticeable increase in the steepness ofthe data beyond this point, which suggests that the forceis no longer a pure double layer interaction. The agree-ment between the approximation and the measurementsis quite good, which confirms the utility of the formerand the role of deformation in the latter.

The CDA shows becomes relatively linear on the logplot at negative nominal separations, as do the measure-ments. Effectively, the Debye length has been renor-malised due to the elasticity of the substrate. It isstraightforward to obtain from Eq. (18) an expressionfor the CDA decay length in this regime. The limitingforce is given by

F (h0) = 2πRκ−1D P ′

0e−κh0 , (19)

where the decay length is

κ =κD

1 + ωκD, (20)

and the renormalised pressure coefficient is

P ′

0 = P0e−κω. (21)

The length ω was defined above and the regime of validityof this result is −ω < h0 � κ−1

D .The amount of deformation is substantial, being of the

order of 100nm at the largest applied loads, compared toa particle diameter of 1200nm. It is possible that the turnup in the force just prior to the van der Waals jump couldbe due to the contribution from the underlying rigid sub-strate at these large deformations. Alternatively, thereis some evidence that this is instead due to a steric re-pulsion due to extended polymer chains; see above andbelow.

The viscoelastic nature of the PDMS droplet is clearlyexhibited in Fig. 15, which shows the velocity depen-dence of the interaction. (The hydrodynamic drainageforce is negligible here.) In general the repulsive force ata given drive position increases with increasing drive ve-locity. This is consistent with the notions that underliethe creep compliance function, namely that viscoelasticmaterials are initially stiff and soften over time. Onemay conclude from the data that relaxation processesdecrease the force at a given nominal separation for par-ticles that are being more slowly loaded. The physicalmechanism by which this occurs is the flattening of theparticle, which increases the actual separation and conse-quently decreases the force. Driving more slowly allowstime for this deformation to occur.

11

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F (

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FIG. 15. Velocity dependence of the PDMS loading curve.From top to bottom the velocities are 3, 1, and 0.5 µm/s.The curves are the viscoelastic central deformation ap-proximation using fitted parameters E0 = 5 × 106J m−3,E∞ = 5 × 105J m−3, and τ = 0.03s. The bold curve isthe double layer force between rigid particles. Inset. Forceon a logarithmic scale. The data are from Ref. [82].

The viscoelastic CDA has been fitted to the data inFig. 15. The long-time elasticity, E∞ = 5 × 105N m−2,is a little less than that used in the elastic CDA fit-ted in Fig. 14; evidently the latter incorporates someof the initial stiffness. The fitted short-time elasticity,E0 = 5 × 106N m−2, is substantially greater than theshort time one. At the fastest driving velocity shownthe loading curve approaches that between rigid sur-faces. The relaxation time used in the approximationis τ = 0.03s, and it is sufficient to describe the transi-tion from short- to long-time behaviour observed in theexperiments.

The viscoelastic CDA may be described as semi-quantitative. There are a number of reasons for the ev-ident discrepancies between the theory and the experi-ments. First is the obvious fact that the CDA is an ap-proximation to the full viscoelastic theory. In particularit is not accurate when there is substantial surface flatten-ing, as occurs, for example, in the post-contact regime.Second of course is the simplicity of the three-parameterviscoelastic model. Doubtless there are multiple relax-ation modes in the PDMS droplet, and the model is onlyuseful in so far as one dominates the experiment. Third isthe use of the purely exponential double layer force law.Close to actual contact this is not correct, (due for exam-ple, to the non-linear nature of the Poisson-Boltzmannequation and also to charge regulation effects, such asconstant potential boundary conditions). Despite thesesimplifications the CDA represents a viable approximatetheory that can be used to extract the material parame-ters of viscoelastic materials.

An additional consideration is that close to contactother forces will start to contribute, as discussed in con-nection with the CDA prediction of contact in Fig. 14.In particular, the kink in the data in Fig. 15 at a load of1.5–2nN. is evidence of such a non-electric double layerforce. This and the subsequent steeper gradient of the

measured data likely indicate actual contact of a diffusepolymeric steric layer. (Miklavcic and Marcelja have usedmean-field theory to model the interaction of polyelec-trolytes and obtained a similar initial softening of thedouble layer repulsion followed by a steeper steric inter-action [96].) That this kink occurs at a substantiallylower load than the putative van der Waals jump identi-fied in Fig. 14, and is of different character, supports amodel of the PDMS droplet as a dense core surroundedby a diffuse corona of polymer tails.

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FIG. 16. Hysteresis and adhesion of the PDMS particle.The velocities are |h0| = 4, 2, and 0.5 µm/s, from top tobottom at the point of reversal. The data are from Ref. [82].

Figure 16 shows the velocity dependence of the hys-teresis and and the adhesion of the PDMS droplet. Thearea of the hysteresis loops, which gives the amount ofenergy dissipation, increases with drive speed, which iswhat one would expect for a viscous system. The max-imum load drops with decreasing speed, as predicted bythe viscoelastic theory, Fig. 11. The difference betweenFig. 11 and Fig. 16 is that in the former the turning pointis at a fixed nominal separation, whereas in the latter it isat a fixed drive distance; the nominal separation at fixeddrive distance decreases with speed due to the decreasedcantilever deflection.

The adhesion, which is the minimum load, or, equiv-alently, the maximum tension, also increases with drivespeed. What is also noticeable on the retraction curvesare the long-range attractions that increase with sepa-ration and that appear as discrete steps. These may beattributed to individual bridging polymers, with the flatregions corresponding to the peeling of the polymer fromthe silica sphere segment by segment, and the regionsof increasing force corresponding to the stretching of theindividual polymer chains. Such forces between individ-ual bridging polymers have been explored in other AFMmeasurements [63,97–101]. Between one and three bridg-ing chains can be seen in the individual force curves inFig. 16. The force due to the longest bridging polymeris remarkably independent of velocity.

12

V. BRIDGING NANOBUBBLE DYNAMICS

A. Experimental Evidence

In 1972 Blake and Kitchener [102] found that bub-bles ruptured at inexplicably large separations from hy-drophobic surfaces, but it took a decade before the ex-istence of a long range attraction between such surfaceswas confirmed by direct force measurements [103–105].The force appeared to be universally present betweenhydrophobic surfaces, (ie. those on which water dropletshad a high contact angle), and was much stronger thanthe van der Waals attraction, which was the only otherknown attractive force between identical surfaces. It pro-duced extremely large adhesions, and it had a measur-able range of hundreds of nanometres [106,107], which isorders of magnitude larger than most surface forces.

The broad features of this unusual force were repro-duced in a number of laboratories and many efforts weremade to explain its origin. The earliest attempt at aquantitative theory suggested that the surfaces coupledby correlated electrostatic fluctuations, with the conse-quence that the decay length of the attraction shouldbe half the Debye length [108]. This idea was subse-quently taken up and developed by a number of au-thors [109–112]. Although several experiments appearto show the predicted dependence on the electrolyte con-centration, [104,105,113], the vast majority are insensi-tive to the concentration or valence of the electrolyte[107,114–117]. One must conclude that the proposedelectrostatic mechanism is not in general the origin forthe measured hydrophobic attraction. It had also beenproposed that surface induced structure in the water wasresponsible for the long range interaction [118]. Thispoly-structural theory is contradicted by the evidencefrom computer simulations, which show that the struc-ture induced by surfaces propagates less than about 1nminto the water [119,120]. Further, the fact that the solvo-phobic force measured in non-hydrogen bonding organicliquids is almost identical to that measured in water hasalso been taken as evidence against the theory [121]. Fi-nally, vapour cavities had been observed between thehydrophobic surfaces when they were in contact [122],and a theory for the force in terms of separation-inducedspinodal cavitation has been developed [123–125]. It isdifficult to design an experimental test of this theory.

In 1994 Parker [107] explored the phenomenon withhis MASIF device [3,4], a type of AFM that uses macro-scopic surfaces, (radii 2mm), and like that instrumentelectronically collects large volumes of data at high reso-lution. Some of this data is reproduced in Fig. 17, wherethe extreme range and strength of the attraction is evi-dent. The steps in the force at large separations had notpreviously been seen with the surface forces apparatus,because of its low resolution and few data points. (Theyare also difficult to see with the AFM because the low in-ertia and weak spring constant of the cantilever leads to

a rapid jump into contact with no data available betweenthe onset of the attraction and the jump; but see below).These steps in the data provided the key to understand-ing the physical origin of the force. It was proposed thatthere were sub-microscopic bubbles present on the hy-drophobic surfaces, and that each step represented theinstant of attachment of a bubble on one surface to theother surface [107,126]. These bridging bubbles spreadalong the surfaces and give rise to the measured force.An attractive feature of the ‘nanobubble’ theory is thatthe range of the interaction between hydrophobic surfacesis set by the height of the bubbles on the isolated surface,and there is no need to invoke any new long-range forceto account for the data. The fact that calculations of theforce due to multiple bridging bubbles were in quantita-tive agreement with the measured data provided strongsupport for the proposed physical origin [107].

4/21/01 3:26 PMdynbbub.xls Fig17

Fig. 17Attard

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-8

-6

-4

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0

0 50 100 150 200

Separation (nm)

F/R

(m

N/m

)

-1

-0.8

-0.6

-0.4

-0.2

0

100 150 200 250

FIG. 17. Force measured between hydrophobic glass sur-faces in water, (MASIF, R = 2.1mm). Three separate ap-proach curves are shown. Inset. Magnification at largeseparations showing steps in the data. The data are fromRef. [107].

Further support for the notion that nanobubbles pre-existed on the hydrophobic surfaces and that their bridg-ing were responsible for the measured attractions sub-sequently came from de-aeration experiments, whichshowed that the force tends to be more short-rangedwhen measured in de-aerated water [116,127]. Wood andSharma [127] showed that the force was also of shorterrange when measured between surfaces that had neverbeen exposed to the atmosphere, which suggests that thebubbles attach to defects in the surfaces when they weretaken through the air-water interface.

In 1998 Carambassis et al. [117] obtained AFM resultsthat, by virtue of the detail of the force curves, providedsignificant support for nanobubbles as the origin of thelong-range attraction. By using a colloid sphere attachedto the cantilever, they were able to obtain the force dueto a single nanobubble in the contact region, and theirresults were more readily interpretable than the multiplebubble results of Parker et al. [107]. Perhaps the moststriking new feature that appears in Fig. 18 is the shortrange repulsion that appears prior to the jump into con-tact. The data suggests that prior to interaction there ison one surface in one case a nanobubble of height about60nm, and in the other case a nanobubble of height about

13

150nm. The evident repulsion prior to the jump towardcontact is in part a double layer interaction between theliquid-vapour interface and the approaching solid surface.A quantitative theory for the data following the jumphas been made by Attard [128] and is discussed in moredetail below. According to the theory, the jump into con-tact following the initial repulsion is due to the bridgingof the bubble between both surfaces, and the extendedsoft-contact, varying-complience region is a dynamic ef-fect due to its lateral spreading. The results of Caram-bassis et al. [117], have been confirmed by a number ofsimilar AFM measurements [129–132]. These later pa-pers include measurements of forces in de-aerated water,and concur with Sharma and Woods’ earlier conclusionthat the force was on average shorter ranged in this case[127]. Finally, infra-red spectroscopy has been used toshow the presence of gaseous CO2 between aggregatedhydrophobic colloids [133].

dynbbub.xls Fig184/21/01 3:27 PM

Fig. 18Attard

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0 20 40 60 80 100 120 140 160

Separation (nm)

Forc

e (n

N) 0.01

0.1

1

10

50 100 150 200 250

FIG. 18. Force between a silica colloid (R = 10.3µm) andglass surface. Both surfaces were hydrophobed by exposureto silane vapour, and the AFM measurements were performedin 9.5mM (crosses) and 0.19mM (triangles) NaCl at a drivevelocity of 4.5µ/s. Inset. Large separation repulsion on alogarithmic scale. The curve is the calculated hydrodynamicdrainage force. The data are from Ref. [117]

Taken in total, the evidence in support of the existenceof nanobubbles is overwhelming. There is now generalconsensus that they are responsible for the long rangeattractions measured between hydrophobic surfaces, asoriginally proposed by Attard and co-workers [107,126].

B. Theory for Bridging Bubbles

In order to calculate the force due to a bridging bub-ble one must first calculate the bubble shape. This isdone by optimising the appropriate constrained thermo-dynamic potential [134,135]. In this case the externalatmospheric pressure, p0, the temperature, T , the liquid-vapour surface energy γ, and the difference in solid sur-face energies, ∆γ > 0, (the contact angle at equilibriumis θ = cos−1[−∆γ/γ]), are fixed, as is the number ofgas molecules, N . The last condition is important, as

assuming diffusive equilibrium of the gas with the atmo-sphere leads to the prediction that all bubbles are unsta-ble [107,126,128]. The constrained Gibbs free energy foran arbitrary bubble profile z(r) is

G([z]|X,h0)=

p0V −NkBT lnV + γAlv − ∆γAsv, (22)

where kB is Boltzmann’s constant, V [z] is the volumeof the bubble, Alv[z] is the liquid-vapour surface area,Asv[z] is the solid-vapour surface area, X represents thefixed variables listed above, and h0 is the separation ofthe solid surfaces.

The equilibrium bubble profile, z(r), may be obtainedby functional differentiation, which results in the Eular-Lagrange equations and which was the original procedureused to obtain the force due to a bridging bubble [107].Alternatively, the profile may be parameterised by a suit-able polynomial expansion and the optimisation may becarried out with respect to the coefficients, which proce-dure has certain numerical advantages [128]. If the coeffi-cients are denoted by ai, then the dependence of the pro-file on them and upon the separation may by symbolisedas z(r; a, h0). The equilibrium profile z(r) = z(r; a, h0),is the one that minimises the constrained potential andhence the equilibrium coefficients satisfy

∂G([z]|X,h0)

∂ai

∣

∣

∣

∣

a

= 0. (23)

The thermodynamic potential is the minimum value ofthe constrained potential, G(X,h0) ≡ G([z]|X,h0). Theforce between the solids is [128]

F (h0) = −(

∂G(X,h0)

∂h0

)

X

= −(

∂G([z]|X,h0)

∂h0

)

a,X

= ∆p

(

∂V

∂h0

)

a

− γ

(

∂Alv∂h0

)

a

. (24)

Even though the ai depend upon h0, the second line fol-lows from the variational nature of the constrained ther-modynamic potential, as manifest in the preceding equa-tion [134,135].

One advantage of the constrained thermodynamic po-tential approach is that the approach to equilibriumcan be explored by holding particular variables constant.This is illustrated in Fig. 19 where the potential is plot-ted as a function of the contact radius. Minima in thepotential correspond to equilibrium values. Whetherthese minima are local or global determines whetherthat particular size is stable or metastable. It can beseen that there are deep minima at microscopic radii,and more shallow minima at sub-microscopic radii. Mi-croscopic bubbles are absolutely stable at small separa-tions and sub-microscopic bubbles are absolutely stable

14

at large separations, and that there is an overlappingregime at intermediate separations where one branch ismetastable with respect to the other. (All the bridgingbubbles are stable with respect to the hemispherical bub-ble on the isolated surface, which has Gibbs free energyof 50.35pJ.) Hence the bridging bubble is hysteretic; ap-proaching from large separations the bubble is initiallysub-microscopic before jumping to microscopic dimen-sions, and conversely upon retraction, with the reversejump occurring at larger separations.

dynbbub.xls Fig194/21/01 3:27 PM

Fig. 19Attard

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30

40

50

0 0.5 1 1.5 2

Contact Radius (µm)

Con

stra

ined

Gib

bs P

oten

tial (

pJ)

29

30

31

0 0.05 0.1

FIG. 19. Gibbs potential for a bridging bubble as a func-tion of the constrained contact radius. The surface separa-tions are, from bottom to top, h0 = 30, 40, 50, 60, 70, 80 and90 nm. The equilibrium radius, which is given by the min-imum in the potential, is microscopic at small separations,and sub-microscopic at large separations. The liquid-vapoursurface tension is γ = 0.072 Nm−1, the external pressure isp0 = 105 Nm−2, the hydrophobic surfaces are both of radiusR = 20µm and have an equilibrium contact angle of θ = 100◦,and the number of gas molecules is fixed at N = 1.4×105. In-

set. Magnification of the minimum at sub-microscopic radii.The data are from Ref. [41].

4/21/01 3:27 PMdynbbub.xls Fig20

Fig. 20Attard

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0

20

40

60

0 200 400 600 800 1000 1200 1400

r (nm)

z(r;

h 0)

(nm

)

-50

-30

-10

10

30

50

20 40 60

FIG. 20. Equilibrium shape of a bridging bubble. The bub-ble shrinks as the separation increases, from right to left themicroscopic bubbles occur at separations of h0 = 0, 10, 20, 30,40, 50, 60, and 70 nm. The other parameters are as in the pre-ceding figure. Inset. Magnification of the large separation,sub-microscopic bubbles, with, from right to left, h0 = 60,70, 80, 90, and 100 nm. The first two profiles are metastablewith respect to their microscopic counterparts at the sameseparation. The data are from Ref. [41].

Figure 20 shows the equilibrium shape of the bridgingbubble. In accord with the constrained thermodynamicpotential calculations of the preceding figure, one can seethat at small separations the equilibrium bridging bubblehas a microscopic lateral radius, whereas at larger separa-tions it is sub-microscopic. There is a marked distinctionbetween the two sizes. On the isolated surface this bub-ble sits as a hemisphere of radius 50nm, height 41.3nm,and contact radius 49.2nm. Hence it can be seen that atsmall separations the bubble has expanded laterally bymore than a factor of twenty. In general the bubbles areconcave or saddle shaped, which indicates that the in-ternal gas pressure is less than the external atmosphericpressure. However, the departure from cylindrical shapeis relatively small, and it will be shown below that ap-proximating the bubble as a cylinder provides simple butaccurate results for the force due to the bridging bubble.

dynbbub.xls Fig214/21/01 3:28 PM

Fig. 21Attard

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-0.2

0

0 20 40 60 80 100

Separation (nm)

Forc

e ( µ

N)

-0.05

-0.03

-0.01

50 100

FIG. 21. The interaction force due to an unconstrainedbridging bubble, (parameters as in Fig. 19). The attrac-tion is large at small separations where the bubble is micro-scopic, and it is weak at large separations where the bub-ble is sub-microscopic. Note that the jump between thetwo branches occurs at smaller separations on approach,h0 = 52 nm, than on retraction, h0 = 80 nm, which givesrise to hysteresis in the force. The dotted curve that ter-minates at h0 = 76nm is the bridging cylinder approxima-tion, Eq. (25). The horizontal arrow is the classical capillaryadhesion, Eq. (26). Inset. Expansion of the force on thesub-microscopic branch. No bridging bubble with these pa-rameters is stable beyond h0 = 112 nm. The data are fromRef. [41].

The hysteresis due to the local minima in the con-strained thermodynamic potential appears clearly in theforce plot, Fig. 21. The force due to the bridging bub-ble is attractive and monotonically increasing with sep-aration. It is weak on the sub-microscopic branch andmuch stronger on the microscopic branch. The jump onapproach occurs at smaller separations than that on re-traction.

Also shown in Fig. 21 is the force due to a cylindricalbridging bubble. In this approximation the optimum ra-dius of the cylinder, r(h0), is obtained by minimising theconstrained thermodynamic potential given above. Formicroscopic cylinders, the pressure inside the bubble may

15

be neglected. The inverse formula for the separation asa function of radius can be given explicitly [128]

h0 = 2√

R2 − r2 − 2R+2Rr∆γ − 2r2γ

(rp0 + γ)√R2 − r2

. (25)

The force is F = −πr2p0 − 2πrγ. It can be seen inFig. 21 that the bridging cylinder approximation is quiteaccurate for the force on the microscopic branch.

The adhesion or capillary force due to the bridgingbubble is also of interest. The largest radius occurs atcontact, h0 = 0, and in the bridging cylinder approxima-

tion it is r∗ = (−3γ/2p0)[

1 −√

1 + 8Rp0∆γ/9γ2]

[128].

The capillary adhesion given by F ∗ = −πr∗2p0 − 2πr∗γ.As can be seen in Fig. 21, this result is more accurate forsmall colloids than the classical result

F ∗ = 2πRγ cos θ. (26)

(Both results agree in the limit of large R.)

C. Spreading Bubble

The calculated force in Fig. 21 appears qualitativelydifferent to the measured forces shown in Fig. 18. Al-though the experiments show a definite jump towardcontact, the attraction is about two orders of magni-tude weaker than the calculated adhesion. In addition,the pre-jump repulsion and the soft-contact, varying-compliance region are missing from the calculations.

Obviously, the calculated force due to the bridgingbubble is only relevant after attachment of the bubble tothe approaching surface, and no attempt has been madeto describe the force curve prior to this point. The largeseparation repulsion evident in the inset of Fig. 18 is inpart due to the hydrodynamic drainage force betweenthe colloid and the substrate, F = −6πηR2h0/h, whereη = 10−3kg m−1 s−1 is the viscosity of water. The sharpincrease in the repulsion immediately prior to the jump isprobably a combination of deformation plus an electricaldouble layer repulsion. The decay length of the measuredforce was observed to decrease with increasing electrolyteconcentration, but was about one fifth the Debye lengthin pure water, and about twice the Debye length in 10mMmonovalent electrolyte [117].

The soft-contact, varying-compliance region prior tothe colloid probe coming into hard contact with the sub-strate appears to be a dynamic effect due to the spread-ing of the bubble (ie. surface drying). For the case of aliquid drop on a surface, it is well-known that a grow-ing drop makes a greater angle of contact with the sub-strate than a shrinking one, and that the gap betweenthe advancing and receding angles increasing with in-creasing velocity [136–138]. The existence of hysteresisand dynamic effects indicates that the equilibration ofthree phase contact occurs over macroscopic time scales,

and that the thermodynamic driving force towards equi-librium is small compared to dissipative forces, (see thediscussion of viscoelasticity in §III). Similar contact an-gle hysteresis occurs for a hemispherical bubble in con-tact with a substrate. Hence for the present problem ofa bridging bubble, one expects hysteresis and velocity-dependent effects as the bubble spreads or recedes.

Of course in order to have hysteresis one must havedissipation, and the simplest model is to invoke a dragforce that is proportional to the velocity and length ofthe contact line,

Fd = −2πarcrc. (27)

Here rc is the contact radius, rc is its velocity, and a isthe drag coefficient. The physical origin of the contactline friction is not clear, although two likely contributingmechanisms are viscous dissipation due to hydrodynamicflow in the contact region [137], and jumping of the con-tact line between asperities [136,138]. In the state ofsteady motion of the contact line, the thermodynamicdriving force must exactly balance the drag force,

−∂G(rc|X,h0)

∂rc− 2πarcrc = 0. (28)

The first term is the derivative of the constrained thermo-dynamic potential of a bridging bubble of fixed contactradius rc but otherwise of optimum shape (cf. Fig. 19).This differential equation for the contact radius maybe solved for a given trajectory h0(t) by simple time-stepping [128]. The force between the probe and thesubstrate was taken to be given by Eq. (24).

dynbbub.xls Fig224/21/01 3:28 PM

Fig. 22Attard

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0

5

10

15

20

0 20 40 60 80 100 120 140 160

Separation (nm)

Forc

e (n

N)

FIG. 22. Dynamic force due to a spreading bridging bub-ble. The AFM data is that of Fig. 18 [117] and the curves areEq. (28) using a fitted drag parameter of a = 3.2kN s m−2

[41]. The curve passing through the crosses is for N such thaton the isolated substrate the hemispherical bubble has radiusRb = 75nm and height zb = 62nm, and the curve passingthrough the triangles is for N such that Rb = 200nm andzb = 165nm. The other parameters are as in Fig. 19.

Figure 22 shows that this model of contact line motionis able to quantitatively describe the measured data inthe soft contact regime. The rapid jump toward contactupon bubble attachment, the minimum in the force, and

16

the ever-steepening repulsion are all present in the the-oretical calculations. The origin of the repulsion is thatthe drag on the contact line prevents the bubble growingto its optimum size at a given separation. As the colloidparticle is driven toward the substrate, the consequentcompression of the bubble leads to the repulsive force.

Several simplifications have been made in the modelcalculations. The calculations are for two identicalspheres of radius 20µm, whereas the experimental datais for a sphere of radius 10.3µm interacting with a flatsubstrate. Similarly, the calculations are for a symmet-ric bridging bubble, which is likely a poor approximationto reality immediately following attachment to the ap-proaching surface. Additionally, in the latter regime thevelocity of the contact line is almost certainly changingrapidly, and assuming steady state conditions likely in-troduces errors here. Finally, no attempt has been madeto include the pre-attachment forces in the calculations.The bubble was taken to attach when the separationequalled its height on the isolated surface, which was fit-ted to the data, and the initial contact radius was chosento give zero normal force at this point.

Because of the variability in the measured data, andbecause of the limited number of force curves analysed,one cannot yet claim to have confirmed the drag law (27).Nevertheless it is of interest to compare the fitted dragcoefficient, a = 3.2 × 103N m−1 s−1 with the value of6 × 10−2N m−1 s−1 estimated by de Ruijter et al. [138]from molecular dynamics simulations of a spreading hex-adecane droplet. The large discrepancy between the twomay be due in part to the low viscosity of the simulatedliquid, (two orders of magnitude less than that of water),to the low surface tension, (about one fifth that of wa-ter), and to a low level of coupling between the substrateand the liquid in the simulations. The average speed ofthe contact line in the simulations is about 1m s−1 [138],whereas in the experiments [117] and in the theory [128]the bubble spreads at about 10µm s−1. In both sim-ulations and theory the product of drag coefficient andvelocity is 3–6×10−2N m−1, is of the same order of mag-nitude as the surface tension.

Despite the caveats outlined above, the agreement be-tween theory and experiment supports the notions thatbridging bubbles are responsible for the measured forces,and that it is the motion of the contact line that gives riseto the details of the force curve. Accordingly, the theorycombined with the dynamic force measurements allowsthe phenomenon of dynamic wetting to be followed withmolecular resolution.

VI. SUMMARY AND CONCLUSION

The atomic force microscope is ideally suited to car-rying out dynamic measurements that can elucidate avariety of time-dependent and non-equilibrium phenom-ena. Here three examples have been reviewed: friction,

viscoelasticity, and wetting.In the case of friction a quantitative method of cal-

ibrating the torsional spring constant and the lateralphoto-diode response was described [12]. The methodis direct, non-destructive, and single step. The frictionbetween metal oxide surfaces in aqueous electrolyte wasmeasured as a function of applied load using the pH tocontrol the adhesion [13]. It was found that with thedetachment force used to shift the applied load, frictionbecame a universal function of the intrinsic load indepen-dent of the pH. Elastic deformation calculations furtherrevealed that surfaces with the same intrinsic load wereat the same local separation, which suggests that frictionis mediated by the short-range interactions between theatoms.

A theory for the deformation and adhesion of viscoelas-tic particles interacting with realistic surface forces ofnon-zero range was summarised [75,76]. A triangulardrive trajectory led to hysteretic force loops, with thehysteresis and the adhesion increasing with velocity. Acentral deformation approximation was introduced thatgave accurate analytic results in the pre-contact regime,and that allowed the zero of separation in AFM forcemeasurements to be established. AFM measurements onPDMS droplets were shown to be qualitatively in accordwith the theory, and the viscoelastic material parameterswere extracted from the data by fitting the theory to it[82].

The force between hydrophobic surfaces has been as-cribed to bridging nanobubbles [107], and the soft-contact, varying-compliance region observed in AFMmeasurements has been attributed to the drying of thesurface as the bubble spreads laterally [117]. This is a dy-namic effect that depends upon the drive velocity. Thethermodynamic force due to a bridging bubble has beencalculated, and, assuming steady state conditions and asimple model of contact line friction, a quantitative ac-count of the measured data has been obtained [128].

Acknowledgement. It has been a privilege to work withArchie Carrambassis, Adam Feiler, Graeme Gillies, IanLarson, John Parker, Mark Rutland, and James Tyrrell,and I thank them for their very significant contributionsto the experimental work reviewed here. Discussions withKristen Bremmel, Sonja Engels, and Clive Prestidge havealso been helpful.

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