203

Tribology Letters, Vol. 26, No. 3, June 2007 (
2007) DOI: 10.1007/s11249-006-9184-7

On the DMT theory J. A. Greenwood* Department of Engineering, University of Cambridge, Trumpington St, Cambridge, CB2 1PZ, UK

Received 20 July 2006; accepted 21 November 2006; published online 26 January 2007

The frequent claim that the Tabor parameter l governs the transition from the DMT theory to the JKR theory is investigated. The change from the simple surface force law r
A/h3 of the DMT theory to the Lennard–Jones law r
A/h3–B/h9 of the MDT theory and the numerical solutions is noted, and the ‘adhesive force’ is evaluated for both laws. Except in the limit of zero Tabor parameter, when the Derjaguin theories reduce to the rigid-sphere model, the predictions are consistently worse than assuming the sphere to be rigid. A ‘semi-rigid’ sphere model is proposed, which correctly describes the asymptotic behaviour as l fi 0, but leaves a considerable gap before the JKR theory can be applied. KEY WORDS: Tabor parameter, DMT, JKR, Adhesion

1. Introduction Bradley [1] showed by pairwise integration of the law of force between molecules that there will be an adhesive force between two rigid spheres equal to 2pRDc, where 1/R = 1/R1 + 1/R2 and Dc is the surface energy c1 + c2. [More generally we have Dc = c1 + c2 )c12.] Subsequently Derjaguin [2] obtained the same answer very much more readily by using the Derjaguin approximation: that the force between elements of curved, inclined surfaces is the same as that between elements of plane, parallel surfaces. Derjaguin noted the agreement with Bradley, and obtained the general result that if the law of force between two planes separated by h is r(h), then the attractive force between two R 1 spheres with a minimum gap h0 will be Tðh0 Þ ¼ 2pR h0 rðhÞdh. The same result follows readily from Bradley’s analysis, and may be referred to as the Bradley–Derjaguin law. [There are too many ‘Derjaguin’ equations for it to be called Derjaguin’s law]. Derjaguin [2] also considered adhesion between elastic, deformable spheres, and found the pull-off force to be halved, to pRDc: he recognised that in the limit of zero contact area, there is a discordance (eine gewisse Unstimmigkeit) between these two values. Apparently in this analysis he regarded the ‘‘adhesive force’’ – the total contribution of the surface forces outside the contact – to be a constant. In 1971 Johnson, Kendall and Roberts [3] put forward an alternative theory of adhesion between spheres (JKR theory), which involves only the surface energy Dc with no reference to a surface force law: it predicts a pull-off force of (3/2)pRDc – midway between the two *To whom correspondence should be addressed. E-mail: [email protected]

Derjaguin values. Once again, although derived using linear elasticity, the answer is independent of the elastic modulus, and so appears to apply when the modulus is infinitely high, and therefore to conflict with Bradley’s value for a rigid solid. It seems possible that the JKR theory, with its essential feature that to form a contact of area pa2 an energy pa2Dc is needed, provided the hint to resolve Derjaguin’s discordance. Certainly 2 years later Derjaguin and his collaborators put forward the DMT theory [4]. This has the same basic postulate as the original 1934 theory, reduced from the original elliptical geometry to the simpler circular contact: that the shape is that of a Hertzian contact, not affected by the surface forces. Now the surface energy term pa2Dc is included in the energy evaluation, but now in addition the ‘‘adhesive force’’ is calculated from the known gap shape around a Hertzian contact using a specific surface force law:
e 3 rðhÞ ¼ 2Dc : the pull-off force now takes the rigid e h sphere value. This resolved the original discordance; but left a new conflict between the JKR and DMT values, both obtained for an elastic solid. The resolution was provided by a brilliant contribution by Tabor [5], who pointed out that the JKR theory predicts that the contact forms a ‘neck’, so that the gap width outside the contact very quickly becomes large. [The point is perhaps more readily understood using Maugis’ observation [6] that the JKR theory is in effect a fracture mechanics solution and the ‘neck’ simply the tip of a standard Griffith crack]. The JKR theory will apply when the gap width exceeds the range of action of the surface forces. By choosing the neck height at pull-off as a representative value, Tabor obtained a condition for the JKR 2 theory to be valid. This may be written ðRDc2 =E Þ1=3 e (where e is a measure of the range of action of the surface 1023-8883/07/0600-0203/0
2007 Springer Science+Business Media, LLC

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J. A. Greenwood/On the DMT theory

forces). More conveniently, the Tabor parameter may be 2 defined as l ¼ ðRDc2 =E Þ1=3 =e: large values of l will give the JKR value, Tc = 1.5pRDc; small values the DMT value Tc = 2pRDc. Thus for the small, relatively high modulus particles of interest to Derjaguin, the Tabor parameter is small and the DMT value is correct: for the larger, more elastic contacts of the original JKR work, the JKR value is correct. It is difficult to deduce from Tabor’s analysis what the critical value of the Tabor parameter should be. The neck height is a very ill-defined quantity: the gap at the contact edge varies continuously from zero to infinity: which part is the neck? Equally, the ‘range of action of the surface forces’ is hard to quantify when the forces merely decrease as h)3: The vital contribution was the recognition that this combination of physical variables would be important. Tabor’s argument was triumphantly confirmed when Derjaguin and his colleagues (MYD) [7] performed a full numerical analysis using the Lennard–Jones of force between planes h

law  i 8Dc e 3 e 9 rðhÞ ¼ 3e h  h and produced the complete dependence of the pull-off force on the Tabor parameter, (figure 1). As increases, the pull-off force decreases smoothly from the DMT (or rigid sphere) value to the JKR value. [Note that the parameter e now acquires a definite meaning. First, it is the equilibrium gap width h at which the surface force vanishes, but that is perhaps more a definition of h than of e. More usefully, the surface force attains its maximum value when h = 31/6e, i.e. after the gap increase of 0.201e from the equilibrium separation. And perhaps most definitely (and the only interpretation for the simple force law of the DMT theory) it defines the rate at which the surface forces decay as the gap increases. Unfortunately, this is not the end of the story. In 1983, Muller, Derjaguin and Toporov (MDT) [8] realised that the DMT evaluation of the adhesive force might be wrong. They replaced the
e 3simple law of force of the DMT paper, rðhÞ ¼ 2Dc e h , by the Lennard– Jones law as used in the full numerical analysis by MYD, and presented two alternative methods of evaluating the adhesive force. With the new law of force, the energy (‘thermodynamic’) method as used in the DMT paper again finds the contribution of the surface forces outside the contact reaches its maximum when the approach a and the contact radius a (a = a2/R) vanish, so that the pull-off force is correctly Tc = 2pRDc and occurs with zero contact area. But when the alternative (simpler!) method of directly integrating the surface forces is used, the contribution rises as the approach increases from zero, opposing the increase due to the pressures inside the contact. The pull-off force is still Tc = 2pRDc and occurs with zero contact area – but only for l < 0.24. For larger values of the Tabor parameter, the pull-off force exceeds the Bradley value Tc = 2pRDc, and occurs at a small positive value of the

approach a and of the contact radius. (A particularly clear account of DMT’s error and its correction is given by Pashley [9]). This completely contradicts the MYD numerical solution, and the subsequent more complete ones (Greenwood [10]; Feng [11, 12]), in which Tc < 2pRDc for all l > 0, and always occurs with a < 0. It must be admitted that in the nebulous region between classical elasticity and full molecular dynamical calculations of solid behaviour, the meaning of the ‘approach a’ is particularly nebulous. There is no problem in the Hertz theory: there a = a2/R is an explicit statement of how far the indenter has moved from the position of first contact. Now, the best we can do is to imagine surfaces at a distance e/2 from the outermost atomic ‘‘planes’’ of the isolated bodies: a: then becomes the penetration of these imaginary surfaces. With the Lennard–Jones law of force, where the true ‘range of action of the surface forces’ is not e but infinity, a = 0 certainly does not correspond to the position of zero force between the bodies. The experimentalist must usually be content to deal with changes in a! This paper investigates in more detail how the predictions of the DMT and MDT theories (specifically, the correct, ‘force method’, evaluations of the Derjaguin postulate) compare with numerical solutions, and whether the frequent statement that Tabor’s parameter governs ‘‘the transition from DMT theory to JKR theory’’ can be justified. But one more confusing element should be added: did MDT realise that the DMT method of evaluation is wrong? They certainly never say this. Their paper repeats the complete DMT analysis (the ‘ thermodynamic method’) before introducing the ‘force method’ with the words ‘the adhesive force can also be calculated in a different way’: the title ‘On two methods of calculation...’ is non-committal: and the paper merely concludes that ’the force method should be preferred’. It is clear that Maugis [13] did not read this as a confession of error: he comments merely that the thermodynamic method produces a decreasing adhesive force whereas the force method produces an increasing one – before offering his own, admirable, approximation of taking it to be constant (equal to 2pRDc), and it is this approximation by Maugis (referred to by Greenwood [10] as the DMT-M theory), which is most frequently quoted as DMT’s contribution! The first explicit statement that the DMT ‘thermodynamic’ answer is wrong and that the force answer is the correct one (and why) is given by Pashley [9], suggesting that the proper name for the theory should perhaps be the Derjaguin–Pashley (DP) theory, correctly recording the two contributions, Derjaguin [2] and Pashley [9].

2. The DMT adhesive force Although it is now clear that the energy method of evaluating the adhesive force is wrong (an energy

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J. A. Greenwood/On the DMT theory 1 Greenwood Feng

0.9

T

max

/ 2πR∆γ

0.95

0.85

0.8

0.75 −2 10

−1

0

10

10

1

10

µ

Figure 1. Dependence of the pull-off force on the Tabor parameter, Tc(l) Values from Greenwood (1997) and Feng (2000) are indistinguishable.

method using an arbitrary mode of deformation can only yield an approximate value for the force), it appears that the correct evaluation using the force method has never been performed for the law of force used by
e 3 DMT, rðhÞ ¼ 2Dc . This may be related to Maugis’ e h comment [13] that it is not physically consistent for the stress to be zero at the edge r = a) of the Hertzian contact but equal to the theoretical stress 2Dc/e. at r = a+: of course with the Lennard–Jones law the discontinuity disappears. For small values of the approach a, the force evaluation needs care. Accurate values are found by integrating the difference between the surface forces for the gap with a rigid sphere and those for the Hertzian gap, ie evaluating f(a*), where the adhesive force is  F  F=2pRDc ¼ 1 þ fða Þ and a* = a/e (table 1). The total force is then T  T=2pRDc ¼ 1 þ fða Þ ð2=3pÞða =lÞ3=2 . Figure 2 shows the result. The pull-off force now always exceeds the rigid sphere value, although negligibly so until l exceeds 0.1. But for l = 1 it is seen that on the DMT theory, the pull-off force is almost Tc = (2.30)pRDc compared with the value for l = 1 from the full numerical solutions of Tc = (1.67)pRDc. The corresponding result using the Lennard–Jones law as used in the MDT theory is less embarrassing. As related above, the pull-off force remains at the rigid sphere value until l < 0.24 (Pashley [9]): it then rises slowly so that for l = 1 it is Tc = (2.06)pRDc – not far from the rigid sphere value, but again very different from the numerical solution. [The very different values for the two laws in Table 1 need comment. The Hertz gap is always less than the gap for a rigid sphere. For the simple surface force law r
C/h3, this means that the

forces are always greater. For the Lennard–Jones law, the forces are smaller for the rising part of the curve, greater for the falling part, and on integration, the decrease and increase largely cancel, so that the answer is relatively close to the answer for a rigid sphere.]

3. The area of contact It will be convenient throughout this paper to regard the contact as occurring between an elastic, deformable sphere and a rigid half-space. The equations apply equally, of course, to contact between two elastic spheres, with plane strain moduli E¢1 ” E1 /(1)m12), E¢2 ” E2/(1)m22) and radii R1, R2, provided we substitute 1/E* = 1/E¢1 + 1/E¢2 and 1/R = 1/R1 + 1/R2. The essential postulate of Derjaguin’s [2] theory is that the sphere develops a flat just as in the Hertz theory of non-adhesive contact. Figure 3 demonstrates that for moderate values of the Tabor parameter (l = 3) such a flat does appear to form: an experimenter observing such a shape would surely have no difficulty in identifying the area of contact. For larger values of l the ‘flat’ becomes even flatter. Nonetheless, there is of course no flat. Interaction between the two bodies is governed by the surface force law: a ‘flat’ implies a constant gap between the two, and a constant gap implies a constant pressure. When the pressure varies in an approximately Hertzian manner, the gap cannot be constant: it merely approximates very, very closely to one. For low values of the Tabor parameter relevant to the DMT (or DP) theory, figure 4 shows that the situation is very different. Where is the contact? Feng [12] evades the

206

J. A. Greenwood/On the DMT theory Table 1. Adhesive force for simple and Lennard–Jones surface force laws.

a*

0.0005

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

fDMT/a* fMDT /a*

0.9619 0.0374

0.9469 0.0510

0.0265 0.0684

0.8883 0.0980

0.8487 0.1246

0.7980 0.1531

0.7126 0.1878

0.6347 0.2062

0.5481 0.2133

0.4283 0.2021

0.3407 0.1802

geometric problem by considering only the pressure distribution, and argues that the assumption of ‘traditional contact mechanics’, that contact is where there is a positive pressure, is the natural one to adopt. I believe this reading of ‘traditional contact mechanics’ to be oversimplified. The traditional assumption is that contact is where there is surface traction. Only as a tacit subclause do we add ‘and of course tensile stresses cannot exist’. This is perhaps a philosophical dispute rather than a scientific one – but clearly there is a problem. A detectable attraction between two bodies exists before they ‘jump into contact’: a smaller one exists after they ‘jump out of contact’. This should, of course, be reworded. Properly, a detectable attraction exists before they jump to a position where a much larger attraction operates: on unloading they jump from a position of moderate attraction to a position of rather low attraction. The process can be clearly depicted on a load/approach graph with no reference to contact (see figure 5 below): this is perhaps the correct scientific treatment. The moral of these remarks is that before we ask whether Derjaguin’s postulate that the contact area obeys the Hertz equations is correct we need to ask what a contact area is! For low l there appears to be no geometrical indication: and it is hard to see an experimenter enthusing over a definition based on

pressures. On the other hand, it is presumably over the region of substantial surface forces that any resistance to sliding would develop. It is, of course, known that friction under negative loads can be measured – though conceivably it could arise in a central region of positive pressure inside an outer tensile region – which would justify the use of the p > 0 definition. It is therefore with great doubts of its significance that figure 6 is offered, comparing the ‘‘contact radii’’ as found by the numerical solutions with the Derjaguin prediction from Hertz theory, that a2 = Ra, all for l = 0.1. For the numerical solutions, both the obvious definitions of contact radius are used: (1) radius at which the surface force vanishes, p = 0 and (2) radius for maximum surface force r(rmax). The basic failure of Derjaguin’s postulate is that contact does not in fact begin at a = 0 but considerably earlier. In addition, the growth of the contact area is poorly predicted. 4. The force–approach relation Figure 7 compares the force–approach curves (for l = 0.1) as found by the force method for both the simple (DMT) and the Lennard–Jones (MDT) laws of force. In accordance with Derjaguin’s postulate, for positive separations (approach a < 0), the Bradley–Derjaguin rigid

1.2

1.15 DMT

1.1

1

MDT

T

max

/ 2πR∆γ

1.05

0.95

0.9 exact

0.85

0.8 −2 10

−1

10

µ

Figure 2. Pull-off force on DMT and MDT theories.

0

10

207

J. A. Greenwood/On the DMT theory 10 9 8

µ=3 H(0) = - 0.01

7

H = h/ε − 1

6 5 4 3 r(σmax)

2 r(p=0)

1 0

−3

−2

−1

0

1

2

3

r* = r / β

Figure 3. Contact shape for a moderate value (l = 3) of the Tabor parameter. Contact radii on the two definitions are shown. The ‘flat’ is only a very good approximation to a flat. For larger l the ‘flat’ becomes flatter. 1.8 1.6 1.4

µ = 0.1

H = h/ε − 1

1.2

H(0) = −0.07

1 0.8 0.6 r(σmax)

0.4 r(p=0)

0.2 0 −0.2 −6

−4

−2

0

2

4

6

r* = r / β

Figure 4. Contact shape near pull-off for a low value (l = 0.1) of the Tabor parameter. Where is the contact area?

sphere theory should be used. For the law h
Lennard–Jones 
e 9 i e 3 of force between planes, rðhÞ ¼ 8Dc  where e is h h 3e spacing, this becomes Tðh0 Þ ¼ 2pRDc hthe
interatomic
 i 4 e 2 1 e 8 where h0 = )a + e. For the simple 3 h 3 h (DMT) law of force the law (not shown in figure 7) is T ¼ 2pRDcðe=hÞ2 More oddly, perhaps, the rigid sphere theory is extended to positive values of the approach. How can a

rigid sphere penetrate a rigid half-space? The answer is that ‘‘penetration’’ here means simply a reduction of the gap below the equilibrium value h = e, and this certainly occurs in the numerical solutions in addition to the elastic deformation: why then jib at it here? The figure brings out what has already been referred to: that if the numerical solutions are indeed correct, then the Derjaguin theory is a poor way of predicting the behaviour. For positive values of the approach, or

208

J. A. Greenwood/On the DMT theory 1 0.9 0.8 0.7

T* = T / 2πR∆γ

µ = 0.1

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

−α* = −α / ε

Figure 5. Exact force–approach curve for l = 0.1 (from Greenwood 1997) The dashed lines show jumping-on and jumping-off for an apparatus of finite stiffness.

3.5 r(p=0) r(σmax)

3

DMT

a* = a / β

2.5

2

1.5

1

0.5 µ = 0.1

0 −0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

α* = α / ε

Figure 6. Relation between ‘‘contact radii’’ and approach. The numerical solution shows that as an elastic sphere approaches a rigid plane, ‘contact’ occurs long before a rigid sphere would touch the plane – which is when the Derjaguin theories assume contact occurs. a* ” a/b where b3  R2 Dc E

positive values of the load, the rigid sphere approximation is equally poor. 5. The semi-rigid model The DMT and MDT theories postulate that the surface forces may be found from the shape to which the

bodies in contact deform, if there were no surface forces. The surface forces are assumed not to deform the bodies. It is interesting to consider what is almost the opposite postulate: that the surface forces may be found from the undeformed shape of the bodies, but then do deform the bodies. More specifically, that the gap is that for a rigid sphere with the same minimum gap, so that

209

J. A. Greenwood/On the DMT theory

1

exact

0.8

T* = T / 2πR∆γ

DMT

MDT

0.6

µ = 0.2

0.4 Bradley

0.2

0

1

0.5

−0.5

0

−1

−1.5

−2

−2.5

−3

α* = α / ε

Figure 7. Force–approach curves for the DMT and MDT theories (l = 0.1). The DMT theory uses the simple, one-term law of force: the MDT theory the usual Lennard–Jones law. Both theories assume rigid body behaviour for a < 0 .

 hðrÞ ¼ hð0Þ þ r2 2R. Since the surface forces are found from this shape, the total force is still h i Tðh0 Þ ¼ 2pRDc ð4=3Þðe=hÞ2  ð1=3Þðe=hÞ8

as above –

and so the pull-off force necessarily takes the rigidsphere value 2p RDc – but now we do not take h0 = )a + e. Instead, the elastic deformation on the axis, w(0), is calculated from the surface forces, and the approach is found from h0 = )a )w(0) + e.

Figure (8) shows the justification for this approximation. For a = )0.15, close to the maximum tensile force, the surface forces from the numerical solution (for l = 0.2: for l = 0.1 the curves can barely be separated) are compared with those assumed here. The agreement is so good that the natural question is, why is the total force not correct? Part of the answer  is that the correct shape is close to hðrÞ ¼ hð0Þ þ r2 2R, when r is small, but when r is large, the correct shape is close to

exact from rigid shape

0.2

0.15 µ = 0.2

σ (µ ε / ∆γ)

α = −0.15 T* = 6.032

0.1

0.05

0

−0.05

0

1

2

3

4

5

6

7

8

r* = r / β

Figure 8. Surface forces for a rigid sphere compared to the exact values (l = 0.2,a = )0.15). Even close to pull-off, as here, the surface forces can be found accurately from the rigid geometry. For smaller values of l the curves are hard to separate.

210

J. A. Greenwood/On the DMT theory 1 DMT

0.5

MDT

0

αx / µ

−0.5 exact

−1

−1.5 semi−rigid

−2

−2.5 −2 10

−1

0

10

10

µ

Figure 9. Dependence of approach at pull-off ax on the Tabor parameter.

 hðrÞ ¼ a þ e þ r2 2R, so that the distant forces are slightly overestimated. But the figure suggests that the problem is more in the detailed shape at moderate radii. [Attempts to model this using a different curvature fail: the sphere is indeed slightly flattened near r = 0: but the curvature then increases above its original value before falling back to it.]

Accordingly, the deformation on the Raxis is calculated 1 using the well-known result wð0Þ ¼ Effi2 0 pðrÞdr, which pffiffiffiffiffiffiffiffi 3=2 2H1 ½CR4 =H31  C16 =H91  here gives wð0Þ=e ¼ ð16=3Þl p=2 2n where pffiffiffi H1 = h(0)/e + 1 and C2n ¼ 0 cos # d# ¼ ð p=2Þðn  12Þ! = n!. In particular, pffiffiffi for pull-off, h(0) = 0 we have wð0Þ=e ¼ ð16=3Þl3=2 2½C4  C16  ¼ 2:1162 l3=2 and a*/l = 2.1162l1/2 where a* ” a/e.

DMT semi−rigid

MDT

0

10

α(0)

exact

Bradley

−1

10

10−2

−1

10

0

10

µ Figure 10. Dependence of approach at zero load a (0) on the Tabor parameter. For positive loads, even for low l no useful estimate of the load/ approach curve is known.

J. A. Greenwood/On the DMT theory

Also, from T(h0) = (2/3)p RDc [4/H12 )1/H18] it follows that the zero-load point is when H1 = 2)1/3 = 0.7937, so that wc ð0Þ=e ¼ ð16=3Þl3=2 21=3 ½C4  4C16  ¼ 8:6662l3=2 , and the corresponding approach is ac ” )h0 )w(0) + e = 0.2063 + 8.6662l3/2. All models give load/approach curves of the same general shape (figure 7) for low values of l, so the curve can be well described by three values: load and approach at pull-off, and approach at zero load. Values of the pull-off load have already been given in figure 2. Figures (9 and 10) compare the estimates of the approach at pull-off and at zero load on the different theories. (A corresponding ‘‘semi-rigid’’ theory for the law of force of the DMT theory is easily obtained by dropping the C16 terms above, but seems to be of less interest.) For the approach at pull-off, the semi-rigid theory is useful up to l = 0.2: the Derjaguin theories are poor. For the approach at zero load, the semi-rigid theory is useful only up to l = 0.1: but in this range the Derjaguin theories are consistently appalling. For l > 0.1 the theories are equally bad.

6. Conclusion The frequent statement that the Tabor parameter governs the transition from the DMT theory to the JKR theory seems completely unwarranted: the Tabor parameter governs the transition from rigid body behaviour to JKR behaviour. The Ôsemi-rigidÕ theory offered here gives useful predictions for l £ 0.1: that is, it is the correct asymptotic theory for l fi 0 just as the JKR theory is the asymptotic theory for l fi ¥. In particular it describes the basic feature that the tensile surface forces attract the two bodies towards each other! Thus, the ‘separation’ ()a) at pull-off is non-zero. For small values of l the Derjaguin theories grossly over-

211

estimate the approach at which the load is zero. There remains a substantial gap below the lowest value of the Tabor parameter at which the JKR theory can be used.

References [1] R.S. Bradley, The cohesive force between solid surfaces and the surface energy of solids. Phil. Mag. 13 (1932) 853. [2] B.V. Derjaguin, Theorie des Anhaftens kleiner Teilchen. Kolloid Zeitschrift 69 (1934) 155. [3] K.L. Johnson, K. Kendall and A.D. Roberts, Surface energy and the contact of elastic solids. Proc. Roy. Soc London A324 (1971) 301. [4] B.V. Derjaguin, V.M. Muller and Yu.P. Toporov, Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci. 53 (1975) 314. [5] D. Tabor, Surface forces and surface interactions. J. Colloid Interface Sci. 58 (1977) 2. [6] D. Maugis and M. Barquins, Fracture mechanics and the adherence of viscoelastic bodies. J. Phys. D (Appl. Phys.) 11 (1978) 1989. [7] V.M. Muller, V.S. Yuschenko and B.V. Derjaguin, On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. J. Colloid Interface Sci. 77 (1980) 91. [8] V.M. Muller, B.V. Derjaguin and Yu.P. Toporov, On two methods of calculation of the force of sticking of an elastic sphere to a rigid plane. Colloids Surf 7 (1983) 251. [9] M.D. Pashley, Further consideration of the DMT model for elastic contact. Colloids Surf 12 (1984) 69. [10] J.A. Greenwood, Adhesion of elastic spheres. Proc. Roy. Soc A453 (1997) 1277. [11] J.Q. Feng, Contact behaviour of spherical elastic particles: a computational study of particle adhesion and deformations. Colloids Surf A 172 (2000) 175. [12] J.Q. Feng, Adhesive contact of elastically deformable spheres: a computational study of pull-off force and contact radius. J. Colloid Interface Sci. 238 (2001) 318. [13] D. Maugis, Adhesion of spheres: the JKR–DMT transition using a Dugdale model. J. Colloid Interface Sci. 150 (1992) 243.