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Effects of surface tension on the adhesive contact of a rigid sphere to a compliant substrate† Xuejuan Xu,a Anand Jagotab and Chung-Yuen Hui*c In problems of indentation of an elastic half-space by a rigid sphere, the effects of surface tension outside the contact zone are not accounted for by classical theories of contact mechanics. However surface tension plays a dominant role in determining the mechanics of this adhesive contact when the halfspace becomes very compliant and the sphere is very small. Using a finite element method (FEM), we
Received 27th January 2014 Accepted 17th March 2014
present a numerical solution of such a problem, showing the transition between the classical Johnson– Kendall–Roberts (JKR) deformation and a liquid-like deformation in the absence of external load and
DOI: 10.1039/c4sm00216d
gravity. The numerical model is in good agreement with previous experiments [R. W. Style et al., Nat.
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Commun., 2013, 4, 2728].
4 E ; E is the 3 1 n2 Young's modulus and n is the Poisson’s ratio of the elastic halfspace. For incompressible materials, K ¼ 16m/3, where m is the where Wad is the work of adhesion, K ¼
Introduction A standard problem of central importance in the study of adhesion and contact mechanics is the mechanics of interactions between solid spheres.1–3 Consider a rigid sphere of radius R brought into contact with an elastic half-space as shown in Fig. 1(a). (As is well-known in contact mechanics, the general problem of two smooth ellipsoidal surfaces can be mapped onto this problem by substituting R with an effective radius.1–3) The sphere attains an equilibrium contact area A with the half-space even in the absence of external load due to the adhesion between the solid surfaces. The standard model for this process balances marginal adhesion energy that drives the increase of contact area, with the marginal increase in strain energy that resists the increase of contact area. As long as the deformation is small, specically, when the indentation depth d and contact radius a are small in comparison with R, the solution is usually well described by the Johnson–Kendall–Roberts (JKR)2 theory for so materials. The JKR theory predicts that the equilibrium contact radius a and indentation depth d in the absence of external load are given by:1,3 1=3 a 6pWad ¼ (1) R KR d ¼ R
2pWad pffiffiffi 3KR
2=3 (2)
a
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA
b
Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA
c
Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853, USA. E-mail: [email protected] † Electronic supplementary information (ESI) available: Interpolation of FEM results and tting with experimental data. See DOI: 10.1039/c4sm00216d
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(a) A rigid sphere is brought into contact with an elastic halfspace. A finite contact area is attained in the absence of external loads. (b) The deformed configuration when the modulus of the half-space is vanishingly small.
Fig. 1
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small strain shear modulus of the half-space. For Wad ¼ 0, the JKR theory reduces to the Hertz theory.2 Johnson1 has pointed out that the adhesive forces between solid surfaces become relatively more important with reduction of the product of contact size and elastic modulus. Muller et al.4 presented a numerical calculation that indicated that JKR theory applies to so solids, large radii of curvature and large energies of adhesion. For the sphere-plane geometry specically, the JKR model is found to be valid in the domain of a large sphere radius and a compliant half-space. In addition, even though the JKR theory is based on the small strain theory, a numerical study by Lin et al.5 using the large deformation elasticity theory has shown that the JKR model is accurate even for moderately large contact radii (half of the radii of the spheres). There are a few studies that account for the effect of surface tension, although in simulations of the time-dependent growth of contacts, it has been shown that the work of adhesion and surface tension can both play a role.6,7 It is therefore interesting to note that recently observed deformations of so substrates, such as plasticized polystyrenes,8 hydrogels9 and silicone gels10 caused by the adhesion of hard microparticles or nanoparticles, deviate greatly from the JKR theory. For example, Chakrabarti and Chaudhury,9 and Style et al.10 have reported that the exponent of the power-law relationship in eqn (1) and (2) between the contact radius (indentation depth) and the radius of the sphere changes from 2/3 (or 1/ 3) to 1 as the sphere reduces in size or the substrates become soer. A similar result was also obtained by Carrillo and Dobrynin11 using molecular dynamics simulations. In these simulations, nanoparticles of different sizes were brought into contact with a rigid half-space, and for so small particles the
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reduction in their height was found to be proportional to their radii. It has been proposed that the departure from JKR scaling represents the increasing inuence and contribution of solid surface stress in resisting deformation.10,11 Salez et al.12 included surface tension in their analytical model of a similar problem where a spherical elastic particle is placed on a rigid substrate. Using a thermodynamic approach, they demonstrated a continuous change of the deformation mechanics from JKR adhesion to wetting with an increasing contribution of surface tension. We therefore expect a transition in the contact mechanics of particles on an elastic foundation from a limit in which deformation is resisted principally by elasticity, to a second limit in which resistance to deformation is provided mainly by surface tension. The JKR theory can be derived by minimizing the total potential energy of a system. In the absence of external loads, the theory considers two energy terms: the stored elastic energy UE of the half-space and the surface energy of the interface, Us. Specically, the JKR model does not account for the role of surface tensions of the solid surfaces in resisting deformation. That is, the work done by the surface tensions upon a change in surface area is neglected in the equation of energy balance. In addition, because surface tension changes the deformation eld, the elastic energy itself is different from that given by the JKR theory. For a very so elastic half-space indented by small spheres, the contribution of surface tension to the energy balance becomes more signicant, resulting in the breakdown of the JKR theory. In the limit where the half-space is so so that it behaves like a liquid, neglecting gravity, the surface of the half-space outside the contact will remain at and the sphere will move until the Young's equation is satised at the line of contact, where
Fig. 2 (a) Deformation of the half-space near the contact region after each loading step in a simulation with s/mR ¼ 1, and P/mR2 ¼ 10. Only a trace of the axisymmetric system in the r–z plane is shown. The thin red layer on the surface after step 1 symbolizes non-zero surface stress. (b) The loading curves correspond to steps 2 and 3. The elastic energy stored in the system is the area between the blue and red lines.
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Fig. 3 (a) s/mR ¼ 0. Plots of indentation depth (left) and contact radius (right) against the work of adhesion. (b) s/mR ¼ 50. Plots of indentation depth (left) and contact radius (right) against the ratio of work of adhesion to surface tension.
gsg + s cos q ¼ gsa
(3)
where the g's are the surface energies, the subscripts s, g, and a stand for rigid sphere, compliant substrate (gel) and air respectively, s is the surface tension on the exposed gel surface and q is the contact angle as indicated in Fig. 1(b), which shows the deformed conguration in this limit. Eqn (3) can be rewritten in a more revealing form:
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s(1 + cos q) ¼ W 0ad
(4)
The le hand side of eqn (4) is the energy release rate due to a peeling of a thin membrane at (p q) degrees,13,14 and the right hand side is the work of adhesion of the sphere/solid interface, where the superscript zero denotes the particular case where the half-space has a vanishing modulus. We have assumed that there is no sliding in the contact region, i.e., the
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Fig. 4 (a) Indentation depth d/R is plotted against Wad/mR at Wad/s ¼ 1. (b) Deformed surface profiles in simulations of the same contact area at:
s/mR ¼ 0; (c) s/mR ¼ 1; (d) s/mR ¼ 50.
surface tension of the region in contact is trapped in the same state as the region outside the contact. In this work we simulate contact between a rigid sphere and an elastic half-space from the elasticity-dominated to the surface-tension-dominated limit using a large deformation nite element model (FEM), which incorporates surface tension in addition to elasticity. Specically, we study the signicant departure from JKR that occurs when the dimensionless parameter s/mR is large. In the surface-tension-dominated regime, elastic deformation is very large and the use of small strain theory is difficult to justify. This means that the nonlinear theory of elasticity, which accounts for geometrical and material nonlinearity, is needed to determine the deformation. Consistent with Style et al.,10 we assume that the surface tension is constant and isotropic. We show that our numerical results accurately capture the recent experiments reported by Style et al.10 In addition, we compare our numerical results with an approximate analytical expression derived by Style et al.10 and show that this expression is an excellent approximation to describe the new theory.
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Dimensional analysis and the two limits In the following, we shall assume that the elastic half-space is incompressible, isotropic and homogeneous. For simplicity, the nonlinear elastic behaviour is given by the classical neo-Hookean model15 with the small strain shear modulus m. Dimensional analysis shows that the normalized indentation depth d/R and contact radius a/R (see Fig. 1(a)) can be expressed as functions of two dimensionless parameters (Wad/mR, Wad/s), d Wad Wad ¼ f1 ; (5) R mR s a Wad Wad ; ¼ f2 mR s R
(6)
Note that the surface tension of the sphere does not enter into the analysis since the sphere is rigid.
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Fig. 5 Experimental results (scattered points) of Style et al.10 are fitted by interpolating the results of FEM simulations (solid lines). The corresponding fitted values of work of adhesion and surface tension are provided in the legend. Note that the blue line in the right figure should be linear by theory; the nonlinearity is due to the limitations of interpolation.20
Normalized indentation depth (left) and contact radius (right) are plotted against normalized work of adhesion showing good agreement between the FEM results (stars) and eqn (12) and (13) (solid lines).
Fig. 6
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In the elasticity-dominated regime, the surface tension is negligibly small, s/mR / 0, and the deformation is completely determined by Wad/mR. In this regime, for small contacts, eqn (5) and (6) reduce to the JKR theory given by (1) and (2). In the other limit where the half-space becomes liquid-like, its shear modulus vanishes and s/mR / N. Using the geometric relationship d/R ¼ cos q + 1 and eqn (4), the condition of contact line equilibrium results in d Wad ¼ s R a ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Wad 1 1 s
(7)
(8)
i.e. the deformation depends only on Wad/s. Thus, the behaviours of functions f1 and f2 in the surface-tension-dominated limit are completely determined. Notably, for eqn (8) to apply, the work of adhesion cannot be more than twice the surface tension. In this limit, both the contact radius and the indentation depth are directly proportional to the radius of the sphere, consistent with recent experimental ndings.10 In the intermediate regime, where both Wad/s and Wad/mR are important, f1 and f2 cannot be determined in closed form. To study the transition from JKR to the surface-tension-dominated limit, we carried out a nite element analysis using ABAQUS®.
Finite element model In our FEM model, the sphere is analytically rigid and the elastic half-space is modelled using axisymmetric quadratic elements with a neo-Hookean strain energy density function W¼
m ðI1 3Þ 2
(9)
where I1 ¼ l12 + l22 + l32 is the rst invariant of the le Cauchy– Green deformation tensor and the li's are the principal stretch ratios. A set of user-dened axisymmetric surface tension elements16 were attached to the half-space surface, transmitting surface tractions caused by the curvature of the deformed surface to the material. We employed a similar loading scheme to the original JKR model in our FEM simulations. The system consisting of the rigid sphere and the elastic half-space is loaded in three steps (Fig. 2(a)): (1) surface tension is applied incrementally to the surface elements and stays at the nal value s in the following two steps. No deformation occurs in this step and it is performed rst as a matter of convenience. (2) A vertical point force P is applied incrementally on the north pole of the rigid sphere, pushing it into the half-space, which results in a nite contact radius a. (3) The point force P is reduced to zero incrementally with the contact radius a held xed, i.e. no separation of previously contacting regions is allowed in this step. The elastic energy is calculated as the net work done in the last two steps. The loading cycle is then repeated with different contact radii. We assume a no-slip boundary condition at the contact. The two surfaces in contact are not allowed to slide with respect to
4630 | Soft Matter, 2014, 10, 4625–4632
each other. Under this boundary condition, we only need to specify the surface tension on the substrate surface outside the contact. (Without no-slip boundary conditions we would need to introduce a new parameter into the model – the interfacial surface tension in the contacting region – and the surfacetension-dominated limit would be different.) The last two steps allow us to calculate the energy release rate.17 In step 2, the work done by the applied force on the halfÐd space to make an indentation dP is 0 P Pload ðd0 Þdd0 : Upon unloading, the total strain energy is ðd ð dP Pload ðd0 Þdd0 þ Punload ðd0 Þdd0 (10) G¼ 0
dP
where Pload(d0 ) and Punload(d0 ) denote the loading and unloading curves (Fig. 2(b)) in steps 2 and 3 respectively. The energy release rate G, with respect to the contact area, is calculated by numerically evaluating G¼
vG vA
(11)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi using the central difference method. A ¼ 2pRðR R2 a2 Þ is the area in contact (for a small deformation, A z pa2). Since the surface tension is present throughout the loading and unloading steps, the surface and elastic energy induced by surface tension is accounted for in the computation of the energy release rate. The equilibrium contact radius a and indentation depth d are determined by the energy balance condition Wad ¼ G. It is useful to consider the results as shown in eqn (5) and (6), with a/R and d/R as functions of (Wad/mR, Wad/s), because in the elasticity-dominated limit the behaviour depends only on Wad/mR, whereas in the surface-tension-dominated limit it depends only on Wad/s. However, in our numerical procedure, we control s/mR and a/R, and each FEM simulation yields a value for Wad/mR. These three numbers are then used to create triads of (a/R, Wad/mR, and Wad/s) and (d/R, Wad/mR, and Wad/s), which create the two numerically-dened surfaces we desire.
Finite element results A. Elasticity-dominated and surface-tension-dominated limits The FEM results at s/mR ¼ 0 (Fig. 3(a)) are in good agreement with the JKR model, except for large contacts where the small strain assumptions of JKR theory are no longer valid. Fig. 3(b) also shows that at s/mR ¼ 50, the FEM results are very close to the analytical solutions in the surface-tension-dominated limits of eqn (7) and (8). B. Transition from the elasticity-dominated limit to the surface-tension-dominated limit The transition from the elasticity-dominated limit to the surface-tension-dominated limit occurs with increasing s/mR. Fig. 4(a) plots the transition by varying s/mR from 0 to 50 in the simulation at constant Wad/s ¼ 1. In the elasticity-dominated limit, s/mR / 0, the FEM results approach the JKR theory in blue. As s/mR increases, d/R increases asymptotically to the surface-tension-dominated limit in green.
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The deformed surface prole near the contact line at different surface tensions can be used to picture the transition. Fig. 4(b–d) show the same sphere making contacts of the same area with three half-spaces at increasing s/mR. From le to right, the peak at the contact line gets smoothed out as s/mR increases. At s/mR ¼ 50, where surface tension dominates, the surface outside the contact becomes almost at, and the half-space behaves like a liquid in the absence of gravity. C.
Comparison with experiments
Style et al.10 recently studied contacts between glass spheres and compliant silicone substrates with Young's moduli ranging from 3 to 500 kPa. They observed signicant deviations of the substrate's deformation from the predictions of JKR theory for their soer gels. Because the glass spheres in their study were at the micron scale, gravity loading was neglected. We tted their experimental results using our FEM models, and estimated the values of work of adhesion and surface tension for four silicone gels of known values of Young's moduli. Details of our tting procedure are provided in the ESI.† The tted results are shown in Fig. 5. For the stiffest substrate (green), where E ¼ 500 kPa, s/mR is relatively small and the deformation is in the elasticity-dominated limit. The soest substrate (blue) deforms like a liquid as the plot of a versus R is almost linear. Our tting estimates that the works of adhesion are 52–64 mN m1 and the values for surface tension are in the range 32–43 mN m1, both slightly lower than the values suggested by Style et al.10 (with a work of adhesion of 71 mN m1 and a work of surface tension of 45 mN m1). Note that there is a discrepancy between the experimental results and their numerical t in the plot of d against R for the soest gel (blue). The translation in log scale suggests that the numerical t differs from the experimental results by a constant factor.
Approximate formula for deformation Style et al.10 proposed the following approximate relationship between the normalized indentation depth, work of adhesion and surface tension: 3=2 16 d s d Wad pffiffiffi þ 2p ¼0 2p (12) mR mR R 3 R
Eqn (12) reproduces the JKR result at s/mR ¼ 0 and yields the result in the surface-tension-dominated limit as s/mR / N. In Fig. 6 (le) we veried eqn (12) (solid lines) against our FEM results (stars). It shows that eqn (12) is very accurate even at intermediate values of s/mR. We propose the following phenomenological interpolated relationship between the normalized indentation depth and the contact radius, 2
a2 ¼ 2d d þ
1 2
2pðs=mRÞ þ 1=ðd þ d Þ
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(13)
¯ ¼ a/R and d ¼ d/R. Eqn (13) reduces to the JKR theory as where a s/mR / 0, and it also gives the correct geometric relationship in the surface-tension-dominated limit as s/mR / N. Using eqn ¯ is plotted against Wad/mR (solid lines) and (12) and (13), a compared to the FEM results (stars) in Fig. 6 (right). As mentioned earlier, the JKR theory breaks down when the contact radius is large (Fig. 3(a)). The discrepancy between the FEM results (stars) and this approximation (solid lines) of eqn (12) and (13) is also relatively large at small values of s/mR when a/R is large. However, for the cases where surface tension is important, the proposed approximation gives very close estimates even at large scale contacts.
Conclusions For sufficiently compliant materials and small particles, surface tension can play a dominant role in determining the mechanics of adhesive contact between two surfaces. We present a numerical study of the transition between the classical JKR deformation and the surface-tension-dominated deformation of a compliant elastic half-space indented by a rigid sphere in the absence of an external load and gravity. Using a newly developed surface tension element, we were able to simulate very large contacts, which are usually observed when very small spheres (micron scale) and very so (less than 100 kPa) substrates are brought into contact. The indentation depth and contact radius due to adhesive contact without applied external load are characterized completely by two dimensionless parameters (Wad/mR, Wad/s). There are two theories for deformable solid surfaces in adhesive contact, the JKR theory and the one presented by Derjaguin, Muller and Toporov18 (the DMT theory). The validity of these two theories for specic cases can be examined by RWad 2 19 calculating the Tabor parameter mTabor ¼ ; where E*2 z0 3 2 E* ¼ E/(1 n ) ¼ 4m is the effective elastic modulus and z0 is the range of interactions between molecules. In the experiments of Style et al.,10 the value of the Tabor parameter was in the order of 104,21 and hence we replaced explicit adhesive interactions with the energy balance method as in the JKR theory. However, our FEM can be extended to include explicit adhesive interactions, e.g. to investigate contacts of spheres with stiffer substrates.6,7 We further note that the differences between the DMT and JKR theories are relatively minor compared to the qualitative change when surface tension replaces elasticity as the dominant term resisting adhesive contact. Although in the JKR analysis, there are no specied boundary conditions at the contact, we employed a no-slip boundary condition during the loading and unloading process in our FEM simulations. This was consistent with our analysis at the surface-tension-dominated limit analogous to Kendall's peel test. An extra parameter that could potentially affect the phenomenon, namely interfacial tension, is also rendered irrelevant by this no-slip condition. In our FEM model, surface elements were applied on the entire surface of the half-space. However due to the no-slip boundary condition in the contact zone, surface traction applied to surface elements within the
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contact was balanced by the boundary constraint. Therefore surface tension was only applied effectively at the exposed surface of the half-space. There is no doubt that this no-slip boundary condition is not appropriate for all situations. However, it is also easy to incorporate frictionless boundary conditions into our FEM model. The largest deformation achieved in our FEM simulations was at a/R ¼ 1. The deformation limit is imposed by the meshing of the FEM model. However larger deformations (i.e. passing the point of a/R ¼ 1), can be achieved with some adaptive nite element remeshing techniques. We presume that in the experiments of Style et al.,10 the contact area increases monotonically and that the work of adhesion is the work released upon contact growth. However, this remains an assumption as it is possible that in the process of depositing particles onto the substrate, contact areas may decrease following an initial increase. In that case, the work of adhesion may not be a unique quantity and the effects of contact angle hysteresis become important.22 Lastly, we neglected gravity in our study because the spheres were very small and the indentations were mainly a result of the action of adhesive forces between the two solid surfaces. In the surface-tension-dominated limit, because there is no external force, the net component of the surface tension for any sections outside the contact must not have a vertical component, and the surface must be at. However if the sphere is sufficiently large, gravity also plays a part in causing the deformation. Especially in the surface-tension-dominated limit, the surface outside the contact is no longer at, because gravity induces a gradient in pressure differences as one moves away from the contact line. According to the Young–Laplace equation, this will result in a changing surface curvature.
Acknowledgements We thank Robert W. Style and Eric R. Dufresne for providing their experimental data and helpful discussions and comments. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46463.
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Notes and references 1 K. L. Johnson, Tribol. Int., 1998, 31(8), 413–418. 2 K. L. Johnson, K. Kendall and A. D. Roberts, Proc. R. Soc. London, Ser. A, 1971, 324(1558), 301–313. 3 D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer, 2000, vol. 130. 4 V. M. Muller, V. S. Yushchenko and B. V. Derjaguin, J. Colloid Interface Sci., 1980, 77(1), 91–101. 5 Y. Y. Lin and H. Y. Chen, J. Polym. Sci., Part B: Polym. Phys., 2006, 44(19), 2912–2922. 6 A. Jagota, C. Argento and S. Mazur, J. Appl. Phys., 1998, 83(1), 250–259. 7 Y. Y. Lin, C. Y. Hui and A. Jagota, J. Colloid Interface Sci., 2001, 237, 267–282. 8 D. S. Rimai, D. J. Quesnel and A. A. Busnaina, Colloids Surf., A, 2000, 165, 3–10. 9 A. Chakrabarti and M. K. Chaudhury, Langmuir, 2013, 29(23), 6926–6935. 10 R. W. Style, C. Hyland, R. Boltyanskiy, J. S. Wettlaufer and E. R. Dufresne, Nat. Commun., 2013, 4, 2728, DOI: 10.1038/ ncomms3728. 11 J. M. Y. Carrillo and A. V. Dobrynin, Langmuir, 2012, 28(29), 10881–10890. ´. Rapha¨ 12 T. Salez, M. Benzaquen and E el, So Matter, 2013, 9(45), 10699–10704. 13 R. Long, K. R. Shull and C. Y. Hui, J. Mech. Phys. Solids, 2010, 58(9), 1225–1242. 14 K. Kendall, J. Phys. D: Appl. Phys., 1975, 8(13), 1449. 15 L. R. G. Treloar, Proc. Phys. Soc., 1948, 60(2), 135–144. 16 X. Xu, A. Jagota, S. M. Peng, D. Luo, M. Wu and C. Y. Hui, Langmuir, 2013, 29(27), 8665–8674. 17 S. Vajpayee, C. Y. Hui and A. Jagota, Langmuir, 2008, 24, 9401–9409. 18 B. V. Derjaguin, V. M. Muller and Y. P. Toporov, J. Colloid Interface Sci., 1975, 53(2), 314–326. 19 D. Tabor, J. Colloid Interface Sci., 1976, 58(1), 2–13. 20 D. T. Sandwell, Geophys. Res. Lett., 1987, 14(2), 139–142. 21 J. N. Israelachvili, Intermolecular and surface forces: revised third edition, Academic Press, 2011. 22 O. Pitois and X. Chateau, Langmuir, 2002, 25, 9751–9756.
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Effects of surface tension on the adhesive contact of a rigid sphere to a compliant substrate† Xuejuan Xu,a Anand Jagotab and Chung-Yuen Hui*c In problems of indentation of an elastic half-space by a rigid sphere, the effects of surface tension outside the contact zone are not accounted for by classical theories of contact mechanics. However surface tension plays a dominant role in determining the mechanics of this adhesive contact when the halfspace becomes very compliant and the sphere is very small. Using a finite element method (FEM), we
Received 27th January 2014 Accepted 17th March 2014
present a numerical solution of such a problem, showing the transition between the classical Johnson– Kendall–Roberts (JKR) deformation and a liquid-like deformation in the absence of external load and
DOI: 10.1039/c4sm00216d
gravity. The numerical model is in good agreement with previous experiments [R. W. Style et al., Nat.
www.rsc.org/softmatter
Commun., 2013, 4, 2728].
4 E ; E is the 3 1 n2 Young's modulus and n is the Poisson’s ratio of the elastic halfspace. For incompressible materials, K ¼ 16m/3, where m is the where Wad is the work of adhesion, K ¼
Introduction A standard problem of central importance in the study of adhesion and contact mechanics is the mechanics of interactions between solid spheres.1–3 Consider a rigid sphere of radius R brought into contact with an elastic half-space as shown in Fig. 1(a). (As is well-known in contact mechanics, the general problem of two smooth ellipsoidal surfaces can be mapped onto this problem by substituting R with an effective radius.1–3) The sphere attains an equilibrium contact area A with the half-space even in the absence of external load due to the adhesion between the solid surfaces. The standard model for this process balances marginal adhesion energy that drives the increase of contact area, with the marginal increase in strain energy that resists the increase of contact area. As long as the deformation is small, specically, when the indentation depth d and contact radius a are small in comparison with R, the solution is usually well described by the Johnson–Kendall–Roberts (JKR)2 theory for so materials. The JKR theory predicts that the equilibrium contact radius a and indentation depth d in the absence of external load are given by:1,3 1=3 a 6pWad ¼ (1) R KR d ¼ R
2pWad pffiffiffi 3KR
2=3 (2)
a
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA
b
Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA
c
Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853, USA. E-mail: [email protected] † Electronic supplementary information (ESI) available: Interpolation of FEM results and tting with experimental data. See DOI: 10.1039/c4sm00216d
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(a) A rigid sphere is brought into contact with an elastic halfspace. A finite contact area is attained in the absence of external loads. (b) The deformed configuration when the modulus of the half-space is vanishingly small.
Fig. 1
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small strain shear modulus of the half-space. For Wad ¼ 0, the JKR theory reduces to the Hertz theory.2 Johnson1 has pointed out that the adhesive forces between solid surfaces become relatively more important with reduction of the product of contact size and elastic modulus. Muller et al.4 presented a numerical calculation that indicated that JKR theory applies to so solids, large radii of curvature and large energies of adhesion. For the sphere-plane geometry specically, the JKR model is found to be valid in the domain of a large sphere radius and a compliant half-space. In addition, even though the JKR theory is based on the small strain theory, a numerical study by Lin et al.5 using the large deformation elasticity theory has shown that the JKR model is accurate even for moderately large contact radii (half of the radii of the spheres). There are a few studies that account for the effect of surface tension, although in simulations of the time-dependent growth of contacts, it has been shown that the work of adhesion and surface tension can both play a role.6,7 It is therefore interesting to note that recently observed deformations of so substrates, such as plasticized polystyrenes,8 hydrogels9 and silicone gels10 caused by the adhesion of hard microparticles or nanoparticles, deviate greatly from the JKR theory. For example, Chakrabarti and Chaudhury,9 and Style et al.10 have reported that the exponent of the power-law relationship in eqn (1) and (2) between the contact radius (indentation depth) and the radius of the sphere changes from 2/3 (or 1/ 3) to 1 as the sphere reduces in size or the substrates become soer. A similar result was also obtained by Carrillo and Dobrynin11 using molecular dynamics simulations. In these simulations, nanoparticles of different sizes were brought into contact with a rigid half-space, and for so small particles the
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reduction in their height was found to be proportional to their radii. It has been proposed that the departure from JKR scaling represents the increasing inuence and contribution of solid surface stress in resisting deformation.10,11 Salez et al.12 included surface tension in their analytical model of a similar problem where a spherical elastic particle is placed on a rigid substrate. Using a thermodynamic approach, they demonstrated a continuous change of the deformation mechanics from JKR adhesion to wetting with an increasing contribution of surface tension. We therefore expect a transition in the contact mechanics of particles on an elastic foundation from a limit in which deformation is resisted principally by elasticity, to a second limit in which resistance to deformation is provided mainly by surface tension. The JKR theory can be derived by minimizing the total potential energy of a system. In the absence of external loads, the theory considers two energy terms: the stored elastic energy UE of the half-space and the surface energy of the interface, Us. Specically, the JKR model does not account for the role of surface tensions of the solid surfaces in resisting deformation. That is, the work done by the surface tensions upon a change in surface area is neglected in the equation of energy balance. In addition, because surface tension changes the deformation eld, the elastic energy itself is different from that given by the JKR theory. For a very so elastic half-space indented by small spheres, the contribution of surface tension to the energy balance becomes more signicant, resulting in the breakdown of the JKR theory. In the limit where the half-space is so so that it behaves like a liquid, neglecting gravity, the surface of the half-space outside the contact will remain at and the sphere will move until the Young's equation is satised at the line of contact, where
Fig. 2 (a) Deformation of the half-space near the contact region after each loading step in a simulation with s/mR ¼ 1, and P/mR2 ¼ 10. Only a trace of the axisymmetric system in the r–z plane is shown. The thin red layer on the surface after step 1 symbolizes non-zero surface stress. (b) The loading curves correspond to steps 2 and 3. The elastic energy stored in the system is the area between the blue and red lines.
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Fig. 3 (a) s/mR ¼ 0. Plots of indentation depth (left) and contact radius (right) against the work of adhesion. (b) s/mR ¼ 50. Plots of indentation depth (left) and contact radius (right) against the ratio of work of adhesion to surface tension.
gsg + s cos q ¼ gsa
(3)
where the g's are the surface energies, the subscripts s, g, and a stand for rigid sphere, compliant substrate (gel) and air respectively, s is the surface tension on the exposed gel surface and q is the contact angle as indicated in Fig. 1(b), which shows the deformed conguration in this limit. Eqn (3) can be rewritten in a more revealing form:
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s(1 + cos q) ¼ W 0ad
(4)
The le hand side of eqn (4) is the energy release rate due to a peeling of a thin membrane at (p q) degrees,13,14 and the right hand side is the work of adhesion of the sphere/solid interface, where the superscript zero denotes the particular case where the half-space has a vanishing modulus. We have assumed that there is no sliding in the contact region, i.e., the
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Fig. 4 (a) Indentation depth d/R is plotted against Wad/mR at Wad/s ¼ 1. (b) Deformed surface profiles in simulations of the same contact area at:
s/mR ¼ 0; (c) s/mR ¼ 1; (d) s/mR ¼ 50.
surface tension of the region in contact is trapped in the same state as the region outside the contact. In this work we simulate contact between a rigid sphere and an elastic half-space from the elasticity-dominated to the surface-tension-dominated limit using a large deformation nite element model (FEM), which incorporates surface tension in addition to elasticity. Specically, we study the signicant departure from JKR that occurs when the dimensionless parameter s/mR is large. In the surface-tension-dominated regime, elastic deformation is very large and the use of small strain theory is difficult to justify. This means that the nonlinear theory of elasticity, which accounts for geometrical and material nonlinearity, is needed to determine the deformation. Consistent with Style et al.,10 we assume that the surface tension is constant and isotropic. We show that our numerical results accurately capture the recent experiments reported by Style et al.10 In addition, we compare our numerical results with an approximate analytical expression derived by Style et al.10 and show that this expression is an excellent approximation to describe the new theory.
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Dimensional analysis and the two limits In the following, we shall assume that the elastic half-space is incompressible, isotropic and homogeneous. For simplicity, the nonlinear elastic behaviour is given by the classical neo-Hookean model15 with the small strain shear modulus m. Dimensional analysis shows that the normalized indentation depth d/R and contact radius a/R (see Fig. 1(a)) can be expressed as functions of two dimensionless parameters (Wad/mR, Wad/s), d Wad Wad ¼ f1 ; (5) R mR s a Wad Wad ; ¼ f2 mR s R
(6)
Note that the surface tension of the sphere does not enter into the analysis since the sphere is rigid.
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Fig. 5 Experimental results (scattered points) of Style et al.10 are fitted by interpolating the results of FEM simulations (solid lines). The corresponding fitted values of work of adhesion and surface tension are provided in the legend. Note that the blue line in the right figure should be linear by theory; the nonlinearity is due to the limitations of interpolation.20
Normalized indentation depth (left) and contact radius (right) are plotted against normalized work of adhesion showing good agreement between the FEM results (stars) and eqn (12) and (13) (solid lines).
Fig. 6
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In the elasticity-dominated regime, the surface tension is negligibly small, s/mR / 0, and the deformation is completely determined by Wad/mR. In this regime, for small contacts, eqn (5) and (6) reduce to the JKR theory given by (1) and (2). In the other limit where the half-space becomes liquid-like, its shear modulus vanishes and s/mR / N. Using the geometric relationship d/R ¼ cos q + 1 and eqn (4), the condition of contact line equilibrium results in d Wad ¼ s R a ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Wad 1 1 s
(7)
(8)
i.e. the deformation depends only on Wad/s. Thus, the behaviours of functions f1 and f2 in the surface-tension-dominated limit are completely determined. Notably, for eqn (8) to apply, the work of adhesion cannot be more than twice the surface tension. In this limit, both the contact radius and the indentation depth are directly proportional to the radius of the sphere, consistent with recent experimental ndings.10 In the intermediate regime, where both Wad/s and Wad/mR are important, f1 and f2 cannot be determined in closed form. To study the transition from JKR to the surface-tension-dominated limit, we carried out a nite element analysis using ABAQUS®.
Finite element model In our FEM model, the sphere is analytically rigid and the elastic half-space is modelled using axisymmetric quadratic elements with a neo-Hookean strain energy density function W¼
m ðI1 3Þ 2
(9)
where I1 ¼ l12 + l22 + l32 is the rst invariant of the le Cauchy– Green deformation tensor and the li's are the principal stretch ratios. A set of user-dened axisymmetric surface tension elements16 were attached to the half-space surface, transmitting surface tractions caused by the curvature of the deformed surface to the material. We employed a similar loading scheme to the original JKR model in our FEM simulations. The system consisting of the rigid sphere and the elastic half-space is loaded in three steps (Fig. 2(a)): (1) surface tension is applied incrementally to the surface elements and stays at the nal value s in the following two steps. No deformation occurs in this step and it is performed rst as a matter of convenience. (2) A vertical point force P is applied incrementally on the north pole of the rigid sphere, pushing it into the half-space, which results in a nite contact radius a. (3) The point force P is reduced to zero incrementally with the contact radius a held xed, i.e. no separation of previously contacting regions is allowed in this step. The elastic energy is calculated as the net work done in the last two steps. The loading cycle is then repeated with different contact radii. We assume a no-slip boundary condition at the contact. The two surfaces in contact are not allowed to slide with respect to
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each other. Under this boundary condition, we only need to specify the surface tension on the substrate surface outside the contact. (Without no-slip boundary conditions we would need to introduce a new parameter into the model – the interfacial surface tension in the contacting region – and the surfacetension-dominated limit would be different.) The last two steps allow us to calculate the energy release rate.17 In step 2, the work done by the applied force on the halfÐd space to make an indentation dP is 0 P Pload ðd0 Þdd0 : Upon unloading, the total strain energy is ðd ð dP Pload ðd0 Þdd0 þ Punload ðd0 Þdd0 (10) G¼ 0
dP
where Pload(d0 ) and Punload(d0 ) denote the loading and unloading curves (Fig. 2(b)) in steps 2 and 3 respectively. The energy release rate G, with respect to the contact area, is calculated by numerically evaluating G¼
vG vA
(11)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi using the central difference method. A ¼ 2pRðR R2 a2 Þ is the area in contact (for a small deformation, A z pa2). Since the surface tension is present throughout the loading and unloading steps, the surface and elastic energy induced by surface tension is accounted for in the computation of the energy release rate. The equilibrium contact radius a and indentation depth d are determined by the energy balance condition Wad ¼ G. It is useful to consider the results as shown in eqn (5) and (6), with a/R and d/R as functions of (Wad/mR, Wad/s), because in the elasticity-dominated limit the behaviour depends only on Wad/mR, whereas in the surface-tension-dominated limit it depends only on Wad/s. However, in our numerical procedure, we control s/mR and a/R, and each FEM simulation yields a value for Wad/mR. These three numbers are then used to create triads of (a/R, Wad/mR, and Wad/s) and (d/R, Wad/mR, and Wad/s), which create the two numerically-dened surfaces we desire.
Finite element results A. Elasticity-dominated and surface-tension-dominated limits The FEM results at s/mR ¼ 0 (Fig. 3(a)) are in good agreement with the JKR model, except for large contacts where the small strain assumptions of JKR theory are no longer valid. Fig. 3(b) also shows that at s/mR ¼ 50, the FEM results are very close to the analytical solutions in the surface-tension-dominated limits of eqn (7) and (8). B. Transition from the elasticity-dominated limit to the surface-tension-dominated limit The transition from the elasticity-dominated limit to the surface-tension-dominated limit occurs with increasing s/mR. Fig. 4(a) plots the transition by varying s/mR from 0 to 50 in the simulation at constant Wad/s ¼ 1. In the elasticity-dominated limit, s/mR / 0, the FEM results approach the JKR theory in blue. As s/mR increases, d/R increases asymptotically to the surface-tension-dominated limit in green.
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The deformed surface prole near the contact line at different surface tensions can be used to picture the transition. Fig. 4(b–d) show the same sphere making contacts of the same area with three half-spaces at increasing s/mR. From le to right, the peak at the contact line gets smoothed out as s/mR increases. At s/mR ¼ 50, where surface tension dominates, the surface outside the contact becomes almost at, and the half-space behaves like a liquid in the absence of gravity. C.
Comparison with experiments
Style et al.10 recently studied contacts between glass spheres and compliant silicone substrates with Young's moduli ranging from 3 to 500 kPa. They observed signicant deviations of the substrate's deformation from the predictions of JKR theory for their soer gels. Because the glass spheres in their study were at the micron scale, gravity loading was neglected. We tted their experimental results using our FEM models, and estimated the values of work of adhesion and surface tension for four silicone gels of known values of Young's moduli. Details of our tting procedure are provided in the ESI.† The tted results are shown in Fig. 5. For the stiffest substrate (green), where E ¼ 500 kPa, s/mR is relatively small and the deformation is in the elasticity-dominated limit. The soest substrate (blue) deforms like a liquid as the plot of a versus R is almost linear. Our tting estimates that the works of adhesion are 52–64 mN m1 and the values for surface tension are in the range 32–43 mN m1, both slightly lower than the values suggested by Style et al.10 (with a work of adhesion of 71 mN m1 and a work of surface tension of 45 mN m1). Note that there is a discrepancy between the experimental results and their numerical t in the plot of d against R for the soest gel (blue). The translation in log scale suggests that the numerical t differs from the experimental results by a constant factor.
Approximate formula for deformation Style et al.10 proposed the following approximate relationship between the normalized indentation depth, work of adhesion and surface tension: 3=2 16 d s d Wad pffiffiffi þ 2p ¼0 2p (12) mR mR R 3 R
Eqn (12) reproduces the JKR result at s/mR ¼ 0 and yields the result in the surface-tension-dominated limit as s/mR / N. In Fig. 6 (le) we veried eqn (12) (solid lines) against our FEM results (stars). It shows that eqn (12) is very accurate even at intermediate values of s/mR. We propose the following phenomenological interpolated relationship between the normalized indentation depth and the contact radius, 2
a2 ¼ 2d d þ
1 2
2pðs=mRÞ þ 1=ðd þ d Þ
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(13)
¯ ¼ a/R and d ¼ d/R. Eqn (13) reduces to the JKR theory as where a s/mR / 0, and it also gives the correct geometric relationship in the surface-tension-dominated limit as s/mR / N. Using eqn ¯ is plotted against Wad/mR (solid lines) and (12) and (13), a compared to the FEM results (stars) in Fig. 6 (right). As mentioned earlier, the JKR theory breaks down when the contact radius is large (Fig. 3(a)). The discrepancy between the FEM results (stars) and this approximation (solid lines) of eqn (12) and (13) is also relatively large at small values of s/mR when a/R is large. However, for the cases where surface tension is important, the proposed approximation gives very close estimates even at large scale contacts.
Conclusions For sufficiently compliant materials and small particles, surface tension can play a dominant role in determining the mechanics of adhesive contact between two surfaces. We present a numerical study of the transition between the classical JKR deformation and the surface-tension-dominated deformation of a compliant elastic half-space indented by a rigid sphere in the absence of an external load and gravity. Using a newly developed surface tension element, we were able to simulate very large contacts, which are usually observed when very small spheres (micron scale) and very so (less than 100 kPa) substrates are brought into contact. The indentation depth and contact radius due to adhesive contact without applied external load are characterized completely by two dimensionless parameters (Wad/mR, Wad/s). There are two theories for deformable solid surfaces in adhesive contact, the JKR theory and the one presented by Derjaguin, Muller and Toporov18 (the DMT theory). The validity of these two theories for specic cases can be examined by RWad 2 19 calculating the Tabor parameter mTabor ¼ ; where E*2 z0 3 2 E* ¼ E/(1 n ) ¼ 4m is the effective elastic modulus and z0 is the range of interactions between molecules. In the experiments of Style et al.,10 the value of the Tabor parameter was in the order of 104,21 and hence we replaced explicit adhesive interactions with the energy balance method as in the JKR theory. However, our FEM can be extended to include explicit adhesive interactions, e.g. to investigate contacts of spheres with stiffer substrates.6,7 We further note that the differences between the DMT and JKR theories are relatively minor compared to the qualitative change when surface tension replaces elasticity as the dominant term resisting adhesive contact. Although in the JKR analysis, there are no specied boundary conditions at the contact, we employed a no-slip boundary condition during the loading and unloading process in our FEM simulations. This was consistent with our analysis at the surface-tension-dominated limit analogous to Kendall's peel test. An extra parameter that could potentially affect the phenomenon, namely interfacial tension, is also rendered irrelevant by this no-slip condition. In our FEM model, surface elements were applied on the entire surface of the half-space. However due to the no-slip boundary condition in the contact zone, surface traction applied to surface elements within the
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contact was balanced by the boundary constraint. Therefore surface tension was only applied effectively at the exposed surface of the half-space. There is no doubt that this no-slip boundary condition is not appropriate for all situations. However, it is also easy to incorporate frictionless boundary conditions into our FEM model. The largest deformation achieved in our FEM simulations was at a/R ¼ 1. The deformation limit is imposed by the meshing of the FEM model. However larger deformations (i.e. passing the point of a/R ¼ 1), can be achieved with some adaptive nite element remeshing techniques. We presume that in the experiments of Style et al.,10 the contact area increases monotonically and that the work of adhesion is the work released upon contact growth. However, this remains an assumption as it is possible that in the process of depositing particles onto the substrate, contact areas may decrease following an initial increase. In that case, the work of adhesion may not be a unique quantity and the effects of contact angle hysteresis become important.22 Lastly, we neglected gravity in our study because the spheres were very small and the indentations were mainly a result of the action of adhesive forces between the two solid surfaces. In the surface-tension-dominated limit, because there is no external force, the net component of the surface tension for any sections outside the contact must not have a vertical component, and the surface must be at. However if the sphere is sufficiently large, gravity also plays a part in causing the deformation. Especially in the surface-tension-dominated limit, the surface outside the contact is no longer at, because gravity induces a gradient in pressure differences as one moves away from the contact line. According to the Young–Laplace equation, this will result in a changing surface curvature.
Acknowledgements We thank Robert W. Style and Eric R. Dufresne for providing their experimental data and helpful discussions and comments. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46463.
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Notes and references 1 K. L. Johnson, Tribol. Int., 1998, 31(8), 413–418. 2 K. L. Johnson, K. Kendall and A. D. Roberts, Proc. R. Soc. London, Ser. A, 1971, 324(1558), 301–313. 3 D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer, 2000, vol. 130. 4 V. M. Muller, V. S. Yushchenko and B. V. Derjaguin, J. Colloid Interface Sci., 1980, 77(1), 91–101. 5 Y. Y. Lin and H. Y. Chen, J. Polym. Sci., Part B: Polym. Phys., 2006, 44(19), 2912–2922. 6 A. Jagota, C. Argento and S. Mazur, J. Appl. Phys., 1998, 83(1), 250–259. 7 Y. Y. Lin, C. Y. Hui and A. Jagota, J. Colloid Interface Sci., 2001, 237, 267–282. 8 D. S. Rimai, D. J. Quesnel and A. A. Busnaina, Colloids Surf., A, 2000, 165, 3–10. 9 A. Chakrabarti and M. K. Chaudhury, Langmuir, 2013, 29(23), 6926–6935. 10 R. W. Style, C. Hyland, R. Boltyanskiy, J. S. Wettlaufer and E. R. Dufresne, Nat. Commun., 2013, 4, 2728, DOI: 10.1038/ ncomms3728. 11 J. M. Y. Carrillo and A. V. Dobrynin, Langmuir, 2012, 28(29), 10881–10890. ´. Rapha¨ 12 T. Salez, M. Benzaquen and E el, So Matter, 2013, 9(45), 10699–10704. 13 R. Long, K. R. Shull and C. Y. Hui, J. Mech. Phys. Solids, 2010, 58(9), 1225–1242. 14 K. Kendall, J. Phys. D: Appl. Phys., 1975, 8(13), 1449. 15 L. R. G. Treloar, Proc. Phys. Soc., 1948, 60(2), 135–144. 16 X. Xu, A. Jagota, S. M. Peng, D. Luo, M. Wu and C. Y. Hui, Langmuir, 2013, 29(27), 8665–8674. 17 S. Vajpayee, C. Y. Hui and A. Jagota, Langmuir, 2008, 24, 9401–9409. 18 B. V. Derjaguin, V. M. Muller and Y. P. Toporov, J. Colloid Interface Sci., 1975, 53(2), 314–326. 19 D. Tabor, J. Colloid Interface Sci., 1976, 58(1), 2–13. 20 D. T. Sandwell, Geophys. Res. Lett., 1987, 14(2), 139–142. 21 J. N. Israelachvili, Intermolecular and surface forces: revised third edition, Academic Press, 2011. 22 O. Pitois and X. Chateau, Langmuir, 2002, 25, 9751–9756.
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