Appl Phys A (2012) 108:13–18DOI 10.1007/s00339-012-6982-7

I N V I T E D PA P E R

Evaluation of adhesive and elastic properties of materialsby depth-sensing indentation of spheres

Feodor M. Borodich · Boris A. Galanov ·Stanislav N. Gorb · Mikhail Y. Prostov ·Yuriy I. Prostov · Maria M. Suarez-Alvarez

Received: 6 May 2012 / Accepted: 7 May 2012 / Published online: 16 May 2012© Springer-Verlag 2012

Abstract Work of adhesion is the crucial material parame-ter for application of theories of adhesive contact. It is usu-ally determined by experimental techniques based on thedirect measurements of pull-off force of a sphere. Thesemeasurements are unstable due to instability of the load-displacement diagrams at tension, and they can be greatlyaffected by roughness of contacting solids. We show howthe values of work of adhesion and elastic contact modulusof materials may be quantified using a new indirect approach(the Borodich–Galanov (BG) method) based on an inverseanalysis of a stable region of the force-displacements curveobtained from the depth-sensing indentation of a sphere intoan elastic sample. Using numerical simulations it is shownthat the BG method is simple and robust. The crucial differ-ence between the proposed method and the standard directexperimental techniques is that the BG method may be ap-plied only to compressive parts of the force-displacementscurves. Finally, the work of adhesion and the elastic modulus

F.M. Borodich (�) · M.M. Suarez-AlvarezSchool of Engineering, Cardiff University, Cardiff CF24 3AA,UKe-mail: BorodichFM@cardiff.ac.ukFax: +44-29-20874716

B.A. GalanovInstitute for Problems in Materials Science, Kiev 03142, Ukraine

S.N. GorbZoological Institute of the University of Kiel, Kiel, Germany

M.Y. ProstovFaculty of Mechanics and Mathematics, Moscow StateUniversity, Moscow 119991, Russia

Y.I. ProstovMoscow State Technical University of Radioengineering,Electronics and Automation, Moscow 119454, Russia

of soft polymer (polyvinylsiloxane) samples are extractedfrom experimental load-displacement diagrams.

1 Introduction

At macroscale adhesive interactions between surfaces haveusually a negligible effect on contacting bodies. However,these interactions are very important at micro/nano scalesand small things are very sticky due to these interactions.Hence, one needs often to take into account the molecu-lar adhesion between solids at these scales. In addition, tostudy many problems of modern nanotechnology, e.g. inter-actions between modern pharmaceutical powders and prob-lems of biology of cells, one needs to know not only adhe-sive but also the elastic characteristics of contacting objects.In other words, to give quantitative predictions of the adhe-sive forces, one needs to estimate both the work of adhesion(w) and the effective contact elastic modulus (E∗) of con-tacting materials [1].

Note that it is rather difficult to determine experimentallythe values of the work of adhesion for contacting solids and,therefore, w is not a well-known quantity for many mod-ern materials [2]. Various methods were introduced to de-termine the surface energy of a sample. For example, it wasproposed to measure contact angle for several liquids andto employ the Young–Dupre and Dupre equations [3, 4].However, these equations were derived for liquids, while itis known that the breach of adhesive connection betweensolids, as a rule, goes in a non-equilibrium way and the tech-niques based on the transfer of these equations to solids israther questionable [5].

Another very popular approach is based on the use of themaximum critical force (it is also referred to as the pull-offforce) of an adhesive sphere that is directly measured from

14 F.M. Borodich et al.

a series of experiments [6, 7]. However, this approach needsan extended number of experiments in order to evaluate themaximum value of the pull-off force and it has also someadditional drawbacks [8].

The recently proposed model-independent method of ex-traction of adhesion energy from indentation experiments[8] assumes that the force, displacement of the indenter andthe contact area are measured simultaneously. Performingsuch measurements is a very difficult task.

On the other hand, it has been recently suggested to usean indirect approach (the BG method) for estimations ofboth mechanical and adhesive properties of contacting mate-rials. Reinforcing the general declaration that such a methodis possible [9], in this paper we prove that the BG methodfor study the above elastic and adhesive characteristics isfast and robust. In addition, we apply the BG method to ex-perimental curves for polyvinylsiloxane (PVS) samples.

2 Depth-sensing indentation and adhesive contact

Currently depth-sensing indentation (DSI) techniques, thatis, the continuously monitoring of the P –δ curve where P

is the applied load and δ is the displacement (the approach ofthe distant points of the indenter and the sample), are widelyused in materials science [10]. For example, the indentationby sharp pyramidal indenters is used to estimate the elas-tic modulus of a tested material by measuring the slope S

of the unloading branch of the diagram and employing thesemi-empirical BASh (Bulychev–Alekhin–Shorshorov) for-mula or its modifications [11, 12]

S = dP

dδ= C

2√

A√π

E∗ (1)

where A is the area of the contact region and C is a con-stant depending on the boundary conditions of the con-tact [13]. However, strictly speaking, the assumptions ofthe Hertz contact theory are not valid for contact betweena sharp indenter and a plastically deformed surface of thematerial sample having an inhomogeneous field of residualstresses. Although, originally both depth-sensing nanoin-denters introduced by Kalei [14] and atomic force micro-scopes (AFM) introduced by Binnig et al. [15] were basedon the use of sharp pyramidal probes, currently the DSI tech-niques with spherical probes are also widely used.

For adhesive contact of smooth spheres, models arewell established. These models include the JKR (Johnson–Kendall–Roberts), DMT (Derjaguin–Muller–Toporov), andMaugis models [1, 16]. The extension of the JKR approachto axisymmetric solids of arbitrary monomial shape f (r) =Bdrd [17–19] shows that the shape of the probe can consid-erably affect the value of the force of adhesion. Here f is theshape of the probe, r the radial coordinate, d is the degree

of the monom (parabola) and Bd is a constant. In addition,the real probes are not ideally sharp [20], hence the exactshape of a sharp probe is often unknown. Further, only thetechniques based on DSI of spherical probes are consideredand developed. Various devices such as nanoindenters, AFMand other devices with spherical probes attached to the endof cantilever beams are widely used to study non-traditionalmaterials such as polymers, pharmaceutical and biologicalmaterials [21, 22]. Measurements of the pull-off force maybe used for estimations of w in the framework of theoreti-cal models of adhesive contact, e.g. it follows from the JKRmodel that measuring directly the adherence force Padh be-tween a sphere of radius R and an elastic half-space one cancalculate

w = −2

3

Padh

πR. (2)

Although the above mentioned techniques for determi-nation of w and E∗ are very popular, they have a num-ber of drawbacks [6, 12]. In particular, the Padh values ob-tained by direct measurements have poor reproducibility be-cause the tensile (adhesive) part of the load-displacement di-agram may be greatly influenced by surface roughness andthe spring stiffness of the measuring device. Indeed, in thecase of a solitary contact, even slight damage of the contactdue to the presence of contamination or surface irregulari-ties will immediately lead to contact breakage, similar to thecrack propagation in bulk material [23]. Hence, one needs tohave a number of measurements to estimate w properly us-ing (2). On the other hand, it is known that the non-adhesivecontact problems have some features of chaotic systems: thetrend of the compressive P –δ curve (the global character-istic of the solution) is independent of fine distinctions be-tween functions describing roughness, while the stress field(the local characteristic) is sensitive to small perturbationsof the punch shape [24]. One can expect that the same ob-servation is valid for adhesive contact, i.e. the compressivebranch of the adhesive P –δ curve is not sensitive to smallsurface roughness. Therefore, the classic models derived forcontact of smooth spheres are applicable and, in turn, we canuse the indirect BG method for estimations of both mechan-ical and adhesive properties of contacting materials [9].

The BG method is based on an inverse analysis of allpoints at a bounded interval of the force-distance curve ob-tained for a spherical indenter. It is assumed that this branchis described by a known functional expression

F

(P

Pc

,δ

δc

)= 0, Pc ≥ 0, δc ≥ 0, (3)

where Pc and δc are characteristic scales of the contact prob-lem for the force and the displacement, respectively, at lowloads and small displacements. The scales can be chosen ar-

Evaluation of adhesive properties of materials by depth-sensing indentation 15

bitrary using the problem governing parameters R, w, andE∗. Here we use the Maugis notations

Pc = 3

2πwR > 0, δc = 3

4

(π2w2R

(E∗)2

)1/3

> 0. (4)

If (4) are taken then the functional expression (3) has theform [16]

F

(P

Pc

,δ

δc

)≡ P

Pc

− 1√3

(δ

δc

)3/2

+ 4

3= 0 (5)

and

F

(P

Pc

,δ

δc

)≡ (3χ − 1)

(1 + χ

9

)1/3

− δ

δc

= 0, (6)

where χ =√

1 + PPc

for the DMT and JKR models, re-

spectively. The problem is to find the scale characteristicsPc and δc using only the experimental points of the cho-sen interval of the stable unloading branch of P –δ curve. If(Pi, δi), i = 1, . . . ,N are respectively experimental valuesof the compressing load P ≥ 0 and corresponding values ofthe displacement δ ≥ 0 then there is the following system ofnon-linear equations for determining the two unknown val-ues Pc and δc

F

(Pi

Pc

,δi

δc

)= 0, Pc ≥ 0, δc ≥ 0. (7)

The system (7) is overdetermined for N > 2 and therefore,it is possible that there exists no solution in the classic sensebecause it is a non-linear ill-posed problem. Contrary to thedirect methods based on the use of just few reference points,the BG method uses all points of a stable interval of the P –δ

curve [9] to solve the overdetermined system. If Pc, δc havebeen found then w and E∗ can be obtained using (4)

w = 2Pc

3πR, E∗ = Pc

4

√3

Rδ3c

. (8)

To underline the crucial difference between common directapproaches and the non-direct BG approach, the latter ap-proach is applied in the next paragraph only to compressive(positive) values of the load.

3 Robustness of the BG method

Let us show that the BG method is very robust. For this pur-pose, one can use the exact P –δ graphs contaminated bysome noise. For example, one can take a part of the JKRP –δ graph and then superimpose some noise to the data

Pi + (nP )i, δi + (nδ)i,

where Pi and δi are discrete points of the exact theoreticalpoints, while (nP )i and (nδ)i are the random noise values tothe force and displacement data, respectively.

Fig. 1 The theoretical P –δ JKR curve whose compressive part is con-taminated by normally distributed noise 10 % of Pc and 0.1 % of δc

values

This idea can be represented as the following numericalalgorithm for checking the validity of the BG method thatcan be described as follows:

(i) Prescribe E∗ and w of a material and R of an indenter;(ii) Calculate Pc and δc by (4);

(iii) Plot the P –δ graph for these Pc and δc according toan appropriate classic model (JKR (6) or DMT (5));

(iv) Take a part of the P –δ graph and add to it some Gaus-sian noise;

(v) Take only the compressive part of the disturbed data;(vi) Solving overdetermined problem, calculate the esti-

mates Pc and δc;(vii) Calculate estimations E∗ and w by (8);

(viii) Compare the initial values E∗ and w and their esti-mates E∗ and w, and calculate the error of the BGmethod.

The application of the BG method shows that the ob-tained estimates of the elastic modulus and work of adhe-sion have very small error even for rather contaminated data.For example, let us consider a sphere of radius R = 3 mmand a material with the given values w = 5.66 × 10−2 N/mand E∗ = 1.218 MPa. The corresponding scaling param-eters of the problem are respectively Pc = 800 µN andδc = 3.0 µm. Results obtained using our method from data(N = 500) for a corresponding truncated JKR curve whosecompressive part is contaminated by normally distributednoise 10 % of Pc and 0.1 % of δc (Fig. 1) were respectivelyPc = 852 µN and δc = 3.098 µm, w = 6.02 × 10−2 N/mand E∗ = 1.235 MPa, i.e. the errors are 6.5 % for w and1.4 % for E∗.

For disturbance 5 % of both Pc and δc (Fig. 2), the resultswere respectively Pc = 822 µN, δc = 3.043 µm, w = 5.82 ×10−2 N/m and E∗ = 1.226 MPa, i.e. the errors are 3 % for

16 F.M. Borodich et al.

Fig. 2 The theoretical P –δ JKR curve whose compressive part is con-taminated by normally distributed noise 5 % of Pc and δc values

w and 0.6 % for E∗. The units for displacements and forcein Fig. 1 and Fig. 2 are respectively µm and µN. One canconclude that the method is not only fast, but also robust.

4 Application of the BG method to experimental data

The method was applied to experiments that used a rela-tively large sapphire sphere. The forces and displacementsof a smooth spherical probe (R = 3 mm) contacting aflat polymer surface (Fig. 3) were continuously measured.Polyvinylsiloxane (PVS) was used for experimental studies.PVS is a silicone elastomer that is often used in dentistry asan impression material.

PVSs have also been used for production of gecko andinsect inspired synthetic brushed and mushroom shaped mi-crostructured adhesive surfaces [25]. Physical properties ofPVSs can be modulated by variation of fillers, in particu-lar, they can have various viscosities. The specimens wereprepared in a similar way to the fibrillar specimens testedin [25]. They were produced at room temperature by pour-ing two-compound polymerizing polyvinylsiloxane into thesmooth template lying on a smooth glass support. After thepolymerization process, the cast of the smooth glass was ob-tained. Two PVS samples were used in the experiments. Ascan be seen in Fig. 4, the loading and unloading branchesare very close to each other and, hence the contribution ofviscosity to the P –δ curve of the tested samples was verylow.

The displacement of the sphere attached to a glass can-tilever beam with known spring constant was detected by thefiber-optic sensor. The interacting force between the sphereand the sample is recorded as a force versus time curve. Onecan find a detailed description of the used force tester (TetraGmbH, Germany) in [21]. The tester was used previously

Fig. 3 A sapphire spherical probe and its reflection in the smooth sur-face of the polyvinylsiloxane specimen tested

Fig. 4 The experimental loading–unloading branches of the P –δ

curve. The BG method is applied to compressive points of the unload-ing curve. The solid line is the JKR curve that correspond to the foundvalues of the scaling parameters Pc and δc

for adhesion measurements of the attachment pads of in-sects. All experiments were carried out at room tempera-ture (22–24 ◦C) and at a relative humidity of 47–56 %. Anaccuracy of about 10 µN was achieved for force measure-ments. The spring constant was determined with an accu-racy of approximately ±2.5 N/m. Since the surface asper-ities are squashed during loading, only unloading brancheswere studied, where the classic models of smooth adhesivecontact are applicable.

Evaluation of adhesive properties of materials by depth-sensing indentation 17

Fig. 5 The experimental P –δ curve. The solid line is the JKR curvethat correspond to Pc and δc whose values were found using all exper-imental points of the unloading branch

Since the sphere was attached to the end of the can-tilever beam (the spring constant of the beam was k1 =122 N/m), the real displacement δ of the contacting sphereis δ = δΣ −PΣ/k1 where PΣ and δΣ are the recorded forceand displacement, respectively; while δ = δΣ after the jumpout of the contact point. However, the points near that pointwere not used. Hence, δ = δΣ − PΣ/k1 versus PΣ graphsare presented in Fig. 4. The same experimental points arepresented in Fig. 5.

Calculating the Tabor–Maugis parameter [16, 26, 27],

μ =(

Refw2

(E∗)2z30

)1/3

,

where the effective radius is equal to the radius of the sphere,Ref = R, and z0 is the equilibrium distance between atomsof contacting pair, that is usually between 0.3–0.5 nm, onecan easily check that all experiments were in the range ofapplicability of the JKR model.

Note that in the framework of the JKR model, the originof the displacement coordinate is not readily extracted fromthe experiments. On the other hand, this reference point isvery important because the solution of the overdeterminedproblem (7) is quite sensitive to the shifting of the δ values.As it follows from (6) for the JKR model, P(0) = −(8/9)Pc

at δ = 0. The origin was selected as the point where P isequal to 8/9 of the experimentally measured pull-off force.Although the least squares method is not the best one forsolving overdetermined problems, the standard least squaresnumerical procedure was used in the present studies. Figs. 4and 5 show the same experimental P –δ curve, however,the different parts of the curves were used for applicationof the BG method. The solid lines on the figures show theJKR curves that correspond to Pc and δc whose values werefound using either the truncated curve, i.e. only compressive

points of the unloading curve (Fig. 4) or all experimentalpoints of the unloading branch (Fig. 5).

For the first PVS sample, the following values of the workof adhesion

w = {5.57; 5.25; 5.52; 5.15; 5.59; 7.07} × 10−2 J/m2

were obtained for truncated curves. Hence, one has

maxw/minw = 1.5.

The corresponding determined values of the contact modu-lus were

E∗ = {1.40; 1.24; 1.69; 1.90; 1.27; 1.22} MPa,

and therefore,

maxE∗/minE∗ = 1.56.

Evidently, the use of the full stable branch, i.e. the use ofnot only the compressive part of the P –δ curve but also thepoints of the tensile part (Fig. 5) will improve the accuracyof the method. Indeed, for non-truncated curves we obtained

w = {5.41; 5.33; 5.28; 6.05; 4.87; 6.5} × 10−2 J/m2

and

E∗ = {1.36; 1.21; 1.51; 1.37; 1.38; 1.32} MPa,

and therefore,

maxw/minw = 1.33

and

maxE∗/minE∗ = 1.24.

For the second PVS sample, we obtained

w = {4.64; 4.01; 4.98; 5.98; 5.54; 5.60} × 10−2 J/m2

and

E∗ = {2.1; 1.26; 1.27; 1.22; 1.83; 1.19} MPa

for truncated curves; while for non-truncated curves

w = {4.96; 3.37; 4.48; 5.29; 5.42; 5.1} × 10−2 J/m2

and

E∗ = {1.51; 1.39; 1.30; 1.24; 1.42; 1.27} MPa.

Hence, one has

maxw/minw = 1.5

and

maxw/minw = 1.22

for truncated and non-truncated curves, respectively, while

maxE∗/minE∗ = 1.76

and

maxE∗/minE∗ = 1.61

18 F.M. Borodich et al.

for these two cases.Evidently, the extracted values of both characteristics

vary within the same polymer sample. However, the vari-ations of the obtained results may be caused not only by theerrors of the measurements or the method, but also by thephysics of polymers. Indeed, polymer molecules are ratherlong and the contact area may have interacting molecules invarious orientations, i.e. it may contain groups that are per-manently electron-rich or electron-poor. Due to this effect,the seeking values may vary.

5 Conclusion

The recently proposed BG method for indirect determina-tion of adhesive properties of materials [9] was examinedusing both numerical simulations and application to exper-imental data. Although the BG method is targeted mainlyto determination of the work of adhesion, the estimations ofthe contact modulus E∗ by (8) may be a good alternative tothe common BASh approach (1).

It has been proven using both numerical simulations andexperimental testing of PVS samples that the BG method isfast and robust. Even for heavily contaminated data based onJKR P –δ graphs, the error was very small. Hence, the vari-ations of the extracted values of the same polyvinylsilox-ane polymer sample could be caused not by the errors ofthe method, but mainly by inhomogeneity of the physicalproperties of the sample, i.e. it is caused by the physics ofpolymers.

Acknowledgements This work was initiated in the framework ofthe ADHESINT International Network supported by the LeverhulmeTrust. It was also partly supported by the SPP 1420 priority programof the German Science Foundation (DFG) “Biomimetic Materials Re-search: Functionality by Hierarchical Structuring of Materials” (projectGO 995/9-1).

In 2007, one of the authors (FB) delivered the first presentation ofthe BG method. After the presentation, Professor J.R. Willis (DAMTP,University of Cambridge) advised to develop an algorithm for check-ing the validity of the BG method and Professor K.L. Johnson (De-partment of Engineering, University of Cambridge) suggested to thinkabout determination of the zero reference point on displacement axis.The authors are very grateful for these stimulating discussions and ad-vise.

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