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Adhesion and Debonding of Soft Elastic Films: Crack Patterns, Metastable Pathways, and Forces Jayati Sarkar,† Ashutosh Sharma,*,† and Vijay Shenoy*,‡ Department of Chemical Engineering, Indian Institute of Technology Kanpur, UP 208 016, India, and Material Research Centre, Indian Institute of Science, Bangalore 560 012, India Received July 31, 2004. In Final Form: October 14, 2004 We study the phenomenon of debonding in a thin soft elastic film sandwiched between two rigid plates as one of the plates is brought into intimate contact and then pulled away from contact proximity by application of a normal force. Nonlinear simulations based on minimization of total energy (composed of stabilizing elastic strain energy and destabilizing adhesive interaction energy) are employed to address the problems of contact hysteresis, cavitation, crack morphology, variation of contact area, snap-off distance, pull-off force, work done, and energy loss. Below a critical distance (dc) upon approach, simulations show the formation of columnar structures and nonrandom, regularly arranged nanocavities at the soft interface at a length scale of ∼3h (h being the thickness of the film). The persistence of such instability upon withdrawal (distance .dc) indicates a contact hysteresis, which is caused by an energy barrier that separates the metastable states of the patterned configuration and the global minimum state of the flat film. The energy and the pull-off force are found to be nonequilibrium and nonunique properties depending on the initial contact, defects, noise, etc. Three broad pathways of debonding leading to adhesive failure of the interface, depending on the stiffness of the film, step size of withdrawal, and the imposed noise, are identified: a catastrophic column collapse mode, a peeling mode involving a continuous decrease in the contact area, and a column splitting mode. The first two modes are caused by a very high stress concentration near the cavity edges. These metastable patterned configurations engender pull-off forces that are orders of magnitude smaller than that required to separate two flat surfaces from contact.
1. Introduction Interfaces can undergo morphological instabilities when subjected to external forces. In debonding of pressure sensitive adhesives,1-8 failure of interfaces mainly occurs either through simple interfacial fracture or through cavitation leading to fibrillation. The presence of surface roughness6 and defects7 can, in some cases, account for the nucleation of random cavities in adhesives where viscoelastic effects enhance cavity formation through a rate-dependent continuous nucleation mechanism.7 Whether the fibrillation or bulk cavitation leads to failure depends on interfacial fracture toughness, stress required for cavity formation, and the amount of lateral confinement. The compliance of the film is also found to play an important role in determining the bond strength of denture soft lining materials.9 The interfaces of weak polymers, however, behave differently. Debonding in weak polymer networks or in the peeling process of living cells under shear flow10,11 leads to either complete pull out of polymer chains or cleavage of molecular bonds. Several models in * To whom correspondence may be addressed. E-mail: ashutos@ iitk.ac.in;
[email protected]. † Indian Institute of Technology Kanpur. ‡ Indian Institute of Science. (1) Lin, Y. Y.; Hui. C.-Y.; Conway, H. D. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 2769. (2) Creton, C.; Hooker, J.; Shull, K. R. Langmuir 2001, 17, 4948. (3) Crosby, A. J.; Shull, K, R.; Lakrout, H.; Creton, C. J. Appl. Phys. 2000, 88, 2956. (4) Wan, K. T. J. Adhes. 2001, 75, 369. (5) Gent, A. N.; Petrich, R. P. Proc. R. Soc. London., Ser. A 1969, 310, 433. (6) Gay, C.; Leibler, L. Phys. Rev. Lett. 1999, 82, 936. (7) Chikina, I.; Gay, C. Phys. Rev. Lett. 2000, 85, 4546. (8) Brown, K.; Hooker. J. C.; Creton, C. Macromol. Mater. Eng. 2002, 287, 163. (9) McCabe, J. F.; Carrick, T. E.; Kamohara, H. Biomaterials 2002, 23, 1347.
the literature have been proposed regarding the rupture of such molecular bonds.12 Type I or type II surface cracks are formed, by opening up and healing of the nanoscale junctions depending on the nature of force applied. Crack formation and healing are also intimately related to the understanding of the microscopic origin of friction.13-15 Contact instability also leads to the development of welldefined surface patterns such as columnar structures in polymer systems16 and nanowires in metals.17 Contact instabilities in purely elastic films show many interesting features that are intimately related to the phenomena of adhesion, debonding, interfacial cracking, and friction. For example, when a curved glass plate is brought in contact proximity of an elastic film, the contact line undulates and evolves into a set of uniformly spaced fingers.18-22 The surface of a soft thin incompressible elastic film bonded to a rigid substrate was found to lose planarity and undergo surface roughening when brought in contact with a plane contactor.23 (10) Kogan, L.; Hui, C.-Y.; Ruina. A. Macromolecules 1996, 29, 4090. (11) Dec´ave´, E.; Garrivier, D.; Bre´chet, Y.; Bruckert, F.; Fourcade, B. Phys. Rev. Lett. 2002, 89, 108101. (12) Seifert, U. Phys. Rev. Lett. 2000, 84, 2750. (13) Kessler, D. A. Nature 2001, 413, 260. (14) Gerde, E.; Marder, M. Nature 2001, 413, 285. (15) Budakian, R.; Putterman, S. J. Phys. Rev. B 2002, 65, 235429. (16) Scha¨ffer, E.; Thurn-Albrecht, T.; Russel, T. P.; Steiner, U. Nature 2000, 403, 874. (17) Landman, U.; Luedtke, W. D.; Gao, J. Langmuir 1996, 12, 4514. (18) Ghatak, A.; Chaudhury, M. K.; Shenoy. V.; Sharma, A. Phys. Rev. Lett. 2000, 85, 4329. (19) Ghatak, A.; Chaudhury, M. K. Langmuir 2003, 19, 2621. (20) Ghatak, A.; Mahadevan, L.; Chung, J. Y.; Chaudhury, M. K.; Shenoy, V. Proc. R. Soc. London., Ser. A 2004, 460, 2725. (21) Shull, K. R.; Flanigan, C. M.; Crosby, A. J. Phys. Rev. Lett. 2000, 84, 3057. (22) Webber, R. E.; Shull, K. R.; Roos, A.; Creton, C. Phys. Rev. E 2003, 68, 021805. (23) Mo¨nch, W.; Herminghaus, S. Euro. Lett. 2001, 53, 525.
10.1021/la048061o CCC: $30.25 © 2005 American Chemical Society Published on Web 01/22/2005
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The surface instability that gives rise to such surface inhomogeneities in soft thin elastic films was theoretically interpreted24-26 to be a result of the interplay of the interaction energy present between the film and the contactor and the elastic energy of the thin film. The physical nature of instability in these elastic films was found to differ from that present in other solid films,27-30 where surface diffusion, plasticity, or viscoelasticity caused the inception of instabilities. The analysis24-26 showed that the elastic instabilities have a dominant wavelength that increases linearly with the thickness of the film and is independent of the nature of interactions present at the interface, which is in contrast to the behavior of viscous liquid films.16,31-33 In essence, the patterned or the corrugated state of the film has the lowest energy when the separation distance (d) is below a critical distance (dc ∼ 10 nm) whereas for d > dc, the flat film has the lowest energy.24,25 Thus, although the linear stability analysis correctly identifies the conditions (d < dc) for the onset of interfacial cavitation by small amplitude deformations when a rigid surface is brought in contact proximity of a thin elastic film, it cannot address the morphology and mechanics of a finite amplitude debonding process when d > dc. An important question is why does the instability pattern persists upon withdrawal at distances much greater than where it was initiated. The asymmetry of the approach and the withdrawal phases or “adhesion-debonding contact hysteresis” remain an open question. Of course contact hysteresis is expected and observed in time-dependent experiments where viscoelasticity, change in interfacial properties, etc., play important roles.34-37 However, the physical origin of adhesion-debonding hysteresis in purely elastic films with contact properties still remains unexplained. Other important issues related to debonding are as follows: What is the maximum distance to which the contactor can be pulled after which the film loses adhesive contact? What are the patterns and morphological pathways of failure during debonding? What is the maximum force required to debond the film completely? Why are the observed forces much less than the theoretical estimates made with the assumption of flat films? We present the analysis of the complete process of contact and debonding of an elastic film between two rigid plates. The basic approach to address the above issues has been reported in a recent letter.38 The paper is organized as follows. The following section contains the description of the theoretical model. The next section shows the results of simulations related to forcedistance relationship, pull-off force, morphological pathways of debonding, energy loss, and snap-off distance. The origins of adhesion-debonding hysteresis are also discussed. A simplified model is also presented to aid the (24) Shenoy, V.; Sharma, A. Phys. Rev. Lett. 2001, 86, 119. (25) Shenoy. V.; Sharma, A. J. Mech. Phys. Solids 2002. 50, 1155. (26) Ru, C. Q. J. Appl. Mech-T Asme. 2002, 69, 97. (27) Asaro, R. J.; Tiller, W. A. Metall. Trans. 1972, 3, 1789. (28) Grinfield, M. J. Nonlinear Sci. 1993, 3, 35. (29) Srolovitz, D. Acta Metall. 1989, 37, 621. (30) Ramirez, J. C. Int. J. Solids Struct. 1989, 25, 579. (31) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998 282, 916. (32) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463. (33) Reiter, G.; Khanna, R.; Sharma, A. Phys. Rev. Lett. 2000, 84, 1432. (34) Chaudhury, M. K.; Owen, M. J. J. Phys. Chem. 1993 , 97, 5722. (35) Kim, S.; Choi, G. Y.; Ulman, A.; Fleischer, C. Langmuir 1997, 13, 6850. (36) Silberzan, P.; Perutz, S.; Kramer, E. J. Langmuir 1994, 10, 2466. (37) Attard, P. J. Phys. Chem. B 2000, 104, 10635. (38) Sarkar, J.; Shenoy, V.; Sharma, A. Phys. Rev. Lett. 2004, 93, 018302.
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Figure 1. Schematic of a soft thin elastic film, of thickness h, bonded to a rigid substrate, separated distance d away from an external contactor. The Cartesian coordinate system (x1, x2) used in the analysis is shown. The dashed line shows the possible configuration of the elastic film.
physical understanding of the quantitative results obtained from simulations. The key findings of this study are summarized in the concluding section. 2. Model Description Figure 1 shows the schematic diagram of a soft, thin, incompressible, laterally unconfined elastic film bonded to a rigid substrate. The film of length L has a thickness h and shear modulus µ, and is under the influence of an external contactor situated at a distance d away from the undistorted free surface. Surface interactions such as electrostatic or long-range van der Waals (VDW) present between the contactor and the film surface act as destabilizing attractive forces which favor deformation of the film. Depending on the strength of the interaction present, the film develops periodic displacements (as shown schematically by the broken lines in the figure) when the separation distance is below a critical distance dc. The value of this critical separation distance varies from a few nanometers (in the case of VDW interactions) to micrometer range (for the longer range electrostatic interactions). If u denotes the displacement vector, the displacement of the film in the direction normal to the free surface is represented by its Fourier series as N-1
u2(x1,0) )
∑ an cos(knx1)
(1)
n)0
where an is the amplitude and kn ()2πn/L) is the wavenumber of the nth mode of deformation. The length of the film along the direction perpendicular to the (x1 x2) plane, i.e., along the depth of the paper, is considered to be much larger than the other two dimensions. Hence the film is assumed to undergo plane strain deformations. In the present study, we take the surface interaction potential to consist of an attractive van der Waals component along with a short-range Born repulsion, represented by
U(d - u‚n) ) -
B A + (2) 2 12π(d - u‚n) (d - u‚n)8
where A is the Hamaker constant (of the order of 10-20 J) and B is the coefficient of the Born repulsion set from the following two conditions: (a) The excess force (per unit area) is zero at the equilibrium separation distance (de) (U′(de) ) 0) and is repulsive at closer distances. (b) The interaction potential at the equilibrium separation distance is equivalent to the adhesive energy at contact, or U(de) ) ∆G. The potential as a function of the separation
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gap distance ξ ()d - u‚n) and the first derivative of the potential with respect to ξ are shown in Figure 2. Later in the paper, we show that the exact form of the potential is not of importance and the physics of the problem is governed largely by the magnitude of the adhesive energy and the slope of the force curve at equilibrium separation distance. To illustrate this clearly, we also consider in some simulations a more general potential of the form
U2(d - u‚n) ) -
A' B′ + + 12π(d - u‚n)2 (d - u‚n)8
[
Us exp
]
(de - d + u‚n) (3) lp
where lp is a decay length and Us is the coefficient of an additional short-range non-VDW attraction.39 This additional near contact potential is due to the specific interactions such as the acid-base interactions.39 Thus, Us is also related to the non-VDW component of the energy of adhesion. To compare results based on potential eqs 2 and 3, the key parameters of these two potentials are made identical near de by the following conditions:
energy of adhesion U(de) ) U2(de) ) ∆G
(4)
U′(de) ) U2′(de) ) 0
U′′(de) ) U2′′(de)
(6)
The values of A′, B′, and lp are determined through (4), (5), and (6) when Us is assigned a value of -1.6 × 10-3 J/m2. The surface instability promoted by the attractive forces is resisted by the elasticity of the film. The stored elastic energy in an incompressible elastic film is
∫V W() dV
(7)
where W is the strain energy density defined as
µ (:) 2
(8)
and is the strain tensor defined as the symmetric part of ∇u. Thus, the total potential energy per unit depth of the film, which is the sum of the elastic energy and the interaction potential, is
Π(a1, ..., an) ) πµ
2
∑
n)0
(sinh(2knh) - 2knh)
∫0
L
+
N-1
U(
a2kL(1 + cosh(2kh) + 2k2h2) 2(sinh(2kh) - 2kh) LAd (10) 12π(d2 - a2)3/2
(5)
equal slope of force curve at (de)
2 2 2 L N-1 nan kn(1 + cosh(2knh) + 2kn h )
Π(a) ) πµ
Minimization of expression 10, with respect to a (,d) leads to
condition for equilibrium
W() )
deformation of the film surface. (A detailed derivation of the elastic part of the potential can be obtained, from the conditions imposed at the film-contactor (rigid bonding) and film-substrate interface (surface traction), through the equations given in the appendix of, ref 40 with ν ) 1/2.) The plane strain elasticity calculation pursued here treats the film in a fully two-dimensional (2D) formalism. At a given gap thickness, d, the stable equilibrium film morphology is the one which minimizes the total energy given by eq 9. To perform a linear stability analysis (LSA), the potential is expanded in a power series about the reference state of the undeformed film24,25 and this homogeneous solution is perturbed with small bifurcation modes. The results of the linear stability analysis can also be obtained directly by minimization of the energy in eq 9 by assuming the amplitude of perturbation to be very small and a single Fourier mode to be present. The use of the potential in eq 2 with only the van der Waals part (the Born repulsion part is not taken into consideration for simplicity) and a single Fourier mode leads to
∑ d - an cos(knx1)) dx1
(9)
n)0
where an values are the Fourier components of the (39) van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927.
2kh(1 + cosh(2kh) + 2k2h2) ) 2khS(kh) ) (sinh(2kh) - 2kh) Ah -hY ) (11) µ 2µπd4 The rightmost term of the equation denotes the ratio of the interaction stiffness -Y ) (-U′′) to the elastic stiffness µ/h of the film for the VDW potential. The objective of the LSA is to find the minimum value of -hY/µ for which instability is possible at the film surface, or in other words to find the nontrivial solutions of kh for which the left term in eq 11 is minimum with respect to kh. This minimization yields
4kh + 16(kh)3 + 4(kh + 2(kh)3) cosh(2kh) 2(1 + 2(kh)2) sinh(2kh) - sinh(4kh) ) 0 (12) and kh ) 2.12 is a solution of this equation. Inserting this value of kh ) 2.12 in eq 11, one gets (-hYc/µ) ) 6.22. These key results are same as those obtained from the analytical linear stability model in refs 24 and 25 and denote that (a) inhomogeneous surface deformations become possible only when the attractive force between the contactor and the film exceeds a critical value -Y g 6.22 µ/h (d e dc (critical separation distance)) and (b) the dominant wavelength of the instability depends linearly on the film thickness, λ ) 2π/k ) 2.96h, but is independent of the specific nature of interactions present between the film and the top contactor which are in direct conformity with the experiments.18,21,23 It may be noted that the spontaneous roughening predicted near contact using the analysis presented here is most relevant for soft films of shear modulus less than about 100 MPa. For stiffer films, the maximum stiffness of the van der Waals attractive force (including its repulsive cutoff at very short distances, (40) Sarkar, J.; Shenoy, V.; Sharma, A. Phys. Rev. E 2003, 67, 031607.
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Figure 2) cannot exceed the elastic stiffness and the film thus remains stable. A necessary condition (but not sufficient) for instability from eq 11 is (-Y) > 2kSµ or λ > (4πµS/(-Y)). This condition for a thick film, h f ∞, S(kh) f coth(kh) f 1, reduces to λ > (4πµ/(-Y)), as discussed in a recent publication on the instability of elastic half-spaces (eq 2.32 of ref 41). However, in addition to this necessary condition, the sufficient condition for instability is that the stiffness of the interaction force must exceed the elastic stiffness,24,25 namely, (-Y) > 6.22µ/h. The critical wavenumber at the onset of instability, (-Y) > 6.22µ/h, is thus always obtained as hkc ) 2.12 regardless of the film thickness as shown by the above analysis.24,25 The linear stability analysis, however, does not provide the amplitude of deformations and thus cannot predict the evolving morphology of the film when the contactor is withdrawn. For this purpose, the equilibrium shape of the film is determined based on the following analysis which minimizes the total energy, composed of the elastic stored energy and the adhesive energy of the system. In the simulations, a conjugate gradient scheme was used for the minimization of eq 9. The stresses that develop at the film can be determined from the Fourier coefficients as40
σ22(x1,0) ) 2µ
N-1
ankn cos(knx1)(1 + cosh(2knh) + 2kn2 h2)
n)0
(sinh(2knh) - 2knh)
∑
(13)
The force per unit area (F) that the contactor plate must exert to hold the film in equilibrium at a given separation distance is equal to the average stress developed in the film.
F)
1 L
∫0L σ22(x1,0) dx1
(14)
The force F (per unit area) is a function of the separation distance (d) to which the contactor plate is withdrawn. The work done (per unit area), W, by the plate in stretching the film from critical separation distance, dc, to any separation distance, d, is obtained from the area under the force curve. Thus
W)
∫dd F dy c
(15)
The total work done (per unit area) in debonding the film is obtained from eq 15 when d is replaced by the snap-off distance dp where the film loses contact with the contactor. 3. Results and Discussions The position of the contactor as well as the phase of approach and retraction cycles plays an important role in determining the surface morphology of the elastic film. During the approach phase, the flat film configuration is stable when the contactor is at a gap distance (d) greater than the critical separation distance (dc). However, as the critical separation distance is approached, the intersurface attractive force (-U′′ ) 6.22 µ/h) becomes strong enough to induce inhomogeneous deformation. The inset at the bottom of Figure 4A shows the film morphology when the contactor plate first approaches the critical gap distance dc. The film loses its planarity, deforms periodically, and (41) Hui. C.-Y.; Glassmaker. N. J.; Tang. T.; Jagota. A. J. R. Soc. London Interface 2004, DOI: 10.1098/rsif.2004.0005.
Figure 2. (A) The interaction energy per unit area as defined in (2). (B) The interaction force per unit area as a function of the gap distance ξ for A ) 10-20 J.
forms zones of contact with the top plate. The bridges thus formed at the film surface have the structure of columns with intervening cavities. The column spacing is approximately 3h in accord with the results of the linear stability analysis. The robustness of the energy minimum thus isolated was confirmed by small random perturbation of the equilibrium configuration followed by the energy minimization. Experiments,23 as well as our simulation results to be presented in the next section, reveal that the deformations in the form of columns and contact zones do persist for a finite distance d . dc as the plate is pulled away from this critical separation distance by the application of an external force. At a certain snap-off distance, the columns detach from the contactor to restore the planar film configuration. An interesting question that arises is why do these surface morphologies persist at a distance d . dc as the plate is withdrawn, even though the flat film configuration was found to be stable at such separation distances when the contactor was made to approach the film? We next explore the physical origin of this “adhesiondebonding contact hysteresis”. 3.1. Adhesion-Debonding. The physical origin of the contact hysteresis (at d . dc) is explained by the following simple analysis which is also substantiated by detailed simulation results. For a single Fourier mode, u2(x1,0) ) ak cos kx1, the total energy (per unit length of the film) is shown in Figure 3 for two different values of the gap thickness d above and below the critical distance dc. The patterned configuration with hk ≈ 2.12 (λ ∼ 3h) has the lower energy for d < dc compared to the flat film as is also predicted from the linear stability analysis. However, when the contactor is at a distance d > dc, the flat film configuration has the lower energy compared to the patterned state which now forms a local minimum, which is a metastable state. The global minimum (flat film) and the local minimum (patterned state) are separated by a large energy barrier (Figure 3). It is due to the presence of this energy barrier that the patterned state formed during the approach is unable to relax to its flat film configuration and persists in its metastable configuration upon withdrawal. The real situation, however, is far more complex than the one considered here, as the patterned structure contains many Fourier modes as stated in eq 1, leading to a multiplicity of metastable states of varying energies. Thus, during the pull-off, the energy input in the form of stretching of the columns helps the system to “hop” through a succession of metastable states before attaining the global minimum state of the flat film. Thus, the whole process of pull-off is strongly “path dependent” and is influenced by the parameters such as withdrawal stepsize, noise, initial level of contact, etc. To understand different metastable pathways of debonding, simulations were carried out to quantify the evolving surface profile of the elastic film at each stage
Soft Elastic Films
Figure 3. Energy landscape as a function of wavenumber. The plot shows the presence of an energy barrier separating the flat film and patterned state for large values of d /de. The inset shows a magnified view of the minimum in energy at kh ) 2.12 for d < dc. The physical parameters used are µ ) 0.1 MPa, h ) 10 µm, A ) 10-20 J, and ∆G ) 1.0 mJ/m2.
of the debonding process beginning from the critical separation distance, where the inhomogeneities first appear, and ending at the snap-off distance where the film completely loses contact with the contactor. Simulations were performed with different step sizes, different amounts of noise, and different initial contact configurations. The Fourier coefficients in eq 1 were initially assigned small random values to start the search. The optimum values of the Fourier coefficients were obtained through the minimization of the energy by the help of the conjugate gradient method. The gap distance was then incremented by a step size s. The same procedure of minimization was carried forth at each new position of the contactor. To study the path dependence of the pull-off process, two different courses of pull-off were investigated. In the first case the optimized coefficients obtained from one step became the initial choices for the solution to be obtained at the next step of withdrawal. In the second case, these Fourier coefficients were multiplied with (1 + r), where r is a small random number between - and , before proceeding to the next stage of withdrawal. These random perturbations or disturbances are incorporated in the analysis to take into account the noise or defects that may be present in the experiments. As is shown by our simulations, a patterned elastic film clinging to a surface being withdrawn has a myriad of metastable states or morphologies corresponding to local minima of the total energy. Sampling of these local minima during debonding, starting from one of them, is thus an essential part of the description of debonding. As shown in the next section, introduction of small nanometer size steps and small amplitude noise in simulations allows the system to sample a larger part of the energy landscape for nearby locally stable minima. In contrast, for infinitesimal step sizes without any noise, the film continues to be trapped in the initial local energy minimum without any opportunity to escape to a nearby metastable state. The failure in such case is catastrophic by buildup of extremely high stresses that eventually initiate unstable crack propagation at large separation distances. The effects of noise and step size are thus entirely similar in that both of these allow local equilibration of the system at every step of pull-off. Both of these have been retained in simulations together or separately to clearly bring out this similarity. In earlier studies of debonding,6,7 noise in the form of pre-existing defects, nanocavities, and surface roughness of the contactor and the film was thought to be responsible for microscopic fibrillation. These, together
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Figure 4. (A) The bottom figure denotes the surface profile of a film when the contactor first approaches the critical separation distance from the undeformed surface of the film (denoted by the level 0 in the y-axis) during contact. The parameters used in the present case are A ) 10-20 J, h ) 10.0 µm, µ ) 0.5 MPa, and ) 0. The other profiles show that as the separation distance increases in a step size of s ) 0.1 nm, the columns elongate with the contactor without any change in contact area. The columns fail catastrophically only at snap-off distance. (B) The stresses that develop at the film surface, as the contactor plate is withdrawn at the positions shown in (A). The stress singularities at the cavity edges increase linearly with the separation distance, and at snap-off distance ∼∆G/de.
Figure 5. Variation of the average stress F, the fractional contact area R, and the wavelength λ of the patterns formed, with the separation distance when the simulations were carried out without any noise and at a step size of withdrawal s ) 0.1 nm. The other parameters used in the present case are the same as those in Figure 4.
with a finite rate of withdrawal and mechanical/thermal disturbances, certainly remain the source of small noise, which should make transitions among the nearby metastable states possible. Our ongoing simulations with rough contactors (to be published) indeed show similar features as introduction of noise. 3.2. Mechanisms of Debonding. We found the following three distinct mechanisms of debonding from the simulations. (1) Catastrophic Column Collapse Mode. The first mode of debonding is more prevalent when the stepsize of withdrawal is very small (s j 0.3 nm) and the level of noise is negligible ( ) 0). In this mode the structures or columns formed during contact simply elongate as the contactor is withdrawn, with no change in the number of columns or contact area (see Figure 4A and Figure 5). As a result, the column spacing remains constant at ∼3h during withdrawal as seen in Figure 5. In the absence of any disturbance, the film is trapped in the original energy minimum and the edges of the columns remain pinned to the contactor. High stress concentration develops at the cavity edges, increasing linearly with the snap-off distance (as shown in Figure 4B) without any intermediate relaxation. Finally, when the stress at the cavity edges
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Figure 6. Variation of change in elastic energy from the patterned state at dc (∆ΠE), change in interfacial energy (∆U), work done in stretching the film (W), and dissipated or lost energy (L) as a function of separation distance for a film with h ) 10 µm, µ ) 0.5 MPa, A ) 10-20 J, and s ) 0.1 nm.
reaches a value comparable to the maximum adhesive strength between the elastic film and the contactor, all the columns snap back catastrophically. The elastic stress that develops at each point on the surface of the film at each stage of debonding (independent of the mode of debonding) was found to be equal in magnitude to the interfacial adhesive stress that develops at the same point due to the presence of the contactor. This balance of elastic and adhesive stresses at every point on the surface shows that the profiles obtained are indeed equilibrium profiles. The local stability of these equilibrium profiles was confirmed by introduction of very small perturbations. Though in this particular mode of debonding, the stress at the edges of the columns is very high, the maximum average stress or the pull-off force (per unit area) (Fmax) is an order of magnitude less than the maximum adhesive force or the maximum force required to debond two flat films from total contact. For example, the value of Fmax for the case shown in Figure 5 is 9.5 MPa, whereas for the same Hamaker constant, the value of maximum adhesive force Umax′ is 80 MPa as evident from Figure 2. This catastrophic column collapse mode is thus more prominently observed in more compliant films (high ∆Gh/µde2) that can undergo a significant amount of stretching. The total work done W in stretching the columnar configuration to any separation distance d without any change in the interfacial area is found to be the same as that of the change in elastic energy, ∆ΠE, between the configuration at d and the patterned base state at dc (refer to Figure 6). Since there is no change in the area of contact during the pull-off, there is no change in the interfacial energy ∆U (as shown in the inset of Figure 6). Thus, the loss in energy L during withdrawal is exactly zero until the final snapoff occurs. At this point, the elastic energy and the interfacial energy of the system are zero as the columns no longer exist. Thus, the changes in both the elastic and interfacial energy at debonding represent the negative of the energy of the patterned state at dc. The total work done in stretching the film to the snap-off distance is now totally lost. The energy loss, L ) W - (∆ΠE + ∆U), in the case of catastrophic debonding is thus confined to the event of snap-off (Figure 6). (2) Peeling Mode. When the step-size of withdrawal in simulations is larger than about 1 nm, the energy minimization algorithm is able to seek deeper local energy minima than the pinned solutions corresponding to constant contact areas. In this family of solutions, the columns grow slender (Figure 7A) and the area of contact recedes sideways as if the film is peeling from the sides. However, the number of columns remains the same with approximately constant spacing (∼3h) during debonding (Figure 8). This mode of debonding is termed as peeling
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Figure 7. (A) The peeling mode of debonding. The profiles show that as the separation distance increases the columns become slender and the area of contact recedes from the sides resulting in peeling. The parameters are the same as for the film in Figure 4 with step size of withdrawal s ) 2.0 nm. (B) The stresses that develop at the film surface as the contactor plate is withdrawn at the positions shown in (A). The stresses do not accumulate to very high values as in Figure 4B because of intermittent release.
Figure 8. Variation of the average stress, fractional area of contact, and wavelength of the evolving pattern with separation distance for the simulations as shown in Figure 7. In the peeling regime the area of contact decreases linearly with the separation distance.
Figure 9. Curves 1, 2, and 3 correspond to step size of 0.1, 1, and 2 nm, respectively, for the parametric values of the film as denoted in Figure 4. Curves 1 and 3 correspond to catastrophic column collapse and continuous peeling modes of failure, and curve 2 is an intermediate.
mode owing to the lateral reduction in contact area and the lateral expansion of cavities. Buildup of the high stress concentration near the cavity edges (Figure 7B) engenders peeling which in turn prevents excessive increase in stresses. The higher step size forces the structure to hop through a succession of metastable states of lower energies accompanied by intermittent release of stresses. This leads to a continuous decrease in contact area and average stress as seen from Figure 8. For intermediate step sizes (curve 2 of Figure 9), the initial phase of pulling reproduces the features of small step size results (pinning) followed by the large step size
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Figure 10. (A) Variation of change in elastic energy from the patterned state at dc(∆ΠE), change in interfacial energy (∆U), work done in stretching the film (W), and dissipated or lost energy (L) as a function of separation distance for a film with parameters same as that in Figure 6 with a step size s ) 1 nm. (B) Dissipated energy as a function of separation distance for a film with parameters as in (A), with and without noise.
Figure 11. Profile shows the initial columns of the film at critical separation distance, split as the contactor plate is withdrawn, when a certain amount of noise is introduced, increasing the number of columns and decreasing the area of contact. At later stages of debonding, some of the columns no longer persist but snap back while others continue to be in contact. The column collapse mode also leads to catastrophic decrease in the area of contact. The parameters used for the simulation in the present case are A ) 10-19J, h ) 10.0 µm, µ ) 0.5 MPa, ) 0.01, and s ) 1 nm.
behavior (peeling). The escape from the initial high barrier state occurs only after some stretching of columns leading to the release of pent-up elastic energy. The ascending branch (“elastic branch”) of the average-stress curve, Figure 9, reflects the linear increase of elastic stresses in the columns without any change in the contact area. The initiation of peeling limits Fmax, after which the average elastic stress declines (“release branch”) with further increase in the separation distance and a concurrent reduction in the contact area. The release branch of the average-stress curve is realizable only in displacement controlled experiments, where an increase in the separation distance requires a lower amount of load. It is to be noted that for “peeling mode” the distance dmax corresponding to the maximum average stress Fmax is much lower than the snap-off distance dp. Thus an increase in stepsize, at zero level of noise, helps the transformation of the failure mode from “pinning” to “peeling”. As may be anticipated and shown later, this transition to peeling is more readily obtained for less compliant films. Comparison of Figure 6 and Figure 10A for small and large step sizes shows that the ascending, constant contact area,“elastic” branch of the force curve is identical in both cases as long as it exists. The maximum force and the distance (dmax) obtained at the end of this branch are however lower at larger step sizes indicating a more facile transition to the peeling mode at higher step sizes. In the elastic pinned branch of the force curve, the total work done W in stretching the film is same as the change in the elastic energy ∆ΠE. However, peeling is initiated after the attainment of Fmax at dmax. Thus, the elastic energy storage capacity of the columns is reduced because of the reduction in their area of contact. Thus, it is evident from Figure 10A that though in the “pinning branch” the change in stored energy increases with separation distance, the opposite occurs on the peeling branch where the columns are in a relaxed metastable state. However, the work done is still increasing on the peeling branch with the separation distance (W )L ∫dcd F dy). It is apparent that the work done is greater than the energy that the columns can store. Part of this work goes in increasing interfacial debonding by reducing the contact, which is reflected in the increase in the interfacial energy ∆U in the peeling mode (refer to Figure 10A). The balance of the work not used for creating a new interface is lost or dissipated in the process according to the energy balance, L ) W - (∆ΠE + ∆U). The mechanism of loss may be understood as follows. Even for a separation distance
greater than dmax, there are two types of metastable states possible, “pinned” and “peeled”. The former is manifest in complete absence of noise and vanishingly small step sizes. In such a case, there is no change in the contact area and thus the work done increases only the elastic energy. The “peeled”, relaxed state is of lower elastic energy ΠE, as well as of lower overall (elastic plus interfacial) energy. The elastic energy difference between the pinned and the peeled states is reflected in the energy loss. This is akin to the energy loss for the pinned morphology at the final moment of its catastrophic debonding when all the stored energy is lost by quenching in the lowest energy state of a flat, completely relaxed film. Transition to a “peeled” metastable state may thus be viewed as a minicatastrophe. Basically the energy loss on the unstable branch is related to relaxation of elastic energy which can only be dissipated as heat, vibrations, and sound. (3) Splitting and Column Collapse Mode. The last pathway of debonding is witnessed in simulations when a certain amount of noise is introduced to perturb the Fourier amplitudes obtained in the simulation step previous to the current separation distance. Similar to the case of large step sizes, this strategy also helps in starting from initial conditions away from the solution branch of catastrophic column collapse. Although the peeling mode still remains a prevalent mode of debonding for small amounts of noise () and specially for stiff films, other modes of debonding, namely, splitting and column collapse modes also become manifest for relatively large noise amplitudes and for more compliant films that can be stretched to larger separations. Figure 11 shows that the contact zones no longer remain continuous but develop blisters or cavities within, which grow further with withdrawal. In essence small nuclei or defects introduced by noise within the contact zones also start to grow by peeling, thus reducing the area of contact. As a consequence, the decrease in the area of contact no longer maintains a linear relationship with the separation distance (Figure 12) as seen in the sideways peeling of the columns. Instead, the contact area decreases in a stepwise fashion with the separation distance as the column split occurs. Whenever the columns split, there is an abrupt increase in the number of columns and a subsequent decrease in the average width of the columns reducing the contact area. Simulations showed that column splitting, when it occurred, coincided nearly with the occurrence of maximum average stress at dmax. This indicates the same mechanism for column splitting as for the onset
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Figure 12. Variation of the average stress, fractional area of contact, and length scale of pattern formed with separation distance for the splitting and column collapse mode as shown in Figure 11. The area of contact decreases in a stepwise fashion with the separation distance.
Figure 13. Variation of average stress with separation distance for various levels of noise ( ) 0.0, 0.001, 0.01), for A ) 10-19 J, h ) 0.1 µm, µ ) 0.1 MPa, and s ) 1 nm.
of peeling mode. This is indicated later in Figure 16. In the later stages of debonding, some of the columns that bridge the gap no longer remain in contact but snap back, whereas the other columns still continue to be in contact (Figure 11). Thus, the wavelength remains constant at ∼3h until the maximum average stress is reached after which the wavelength decreases as the columns split, leading to a higher number of columns. At still higher separations, the column collapse mode again increases the wavelength as shown in Figure 12. Column splitting, when it occurs, results in precipitous decrease in the average stress, usually followed by a regime of more nearly constant average stress as in Figure 13. Continuous peeling from the sides of the split columns prevents the buildup of elastic stresses in the constant average stress regime. This provides a microscopic understanding of experimental observations of constant average stress regime which becomes more prominent on rough surfaces that allow cavity initiation within the contact zones.6,42 The effects of increase in step size and noise both have similar effects in bringing down the pull-off force (Fmax) required to separate the films. Both the effects are also found to decrease the work required for debonding. In the case of “catastrophic column collapse mode” (marked by smaller step sizes) the work required for debonding is highest, followed by the “peeling mode” of failure (marked by higher step sizes) as can be compared from Figure 6 and Figure 10A. Similar to the effect of step size, the increased amount of noise also decreases the work required for debonding (as shown quantitatively later in Figure 22). The introduction of noise leads to stepwise reduction in contact area by stick-slip or pinning-peeling of the interface. Thus, the stored elastic (42) Chiche; A., Pareige, P.; Creton, C. C. R. Acad. Sci IV, Phys. 2000 1, 1197.
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Figure 14. (A) Variation of the area of contact with the separation distance for two cases, one when pull-off process starts at the critical distance from slight contact (Rinitial ∼ 0.5) and the other when the contactor plate is first pressed to intimate contact (Rinitial ∼ 0.8) (bonding phase) and then withdrawn (debonding phase). The pull-off results show that though the Rinitial differs considerably, the contact area is almost equal at larger distances. The parameters used are A ) 10-20 J, h ) 5.0 µm, µ ) 0.1 MPa, and a step size of 1 nm with no noise. (B) Variation of F/µ starting with different initial fractional contact area, Rinitial, with other parameters the same as in (A).
energy decreases further and the loss or energy dissipation increases as shown in Figure 10B leading to early snapoff. Due to the presence of the interaction potential, the stresses in the contact zones, including the edge of the columns, are not as severe as near a crack tip. The normal stress across a column in our case varies as σ ∼ (1 - (r/ a)2)-n, where r is the distance measured from the center of the column and 2a is the width of the column. The exponent n was found to be 0.1 for small values of separation distance d, and increased very slowly with increasing d to a maximum of 0.2. This exponent is much smaller than 0.5, which is found ahead of a crack tip in an elastic solid.21 The reduction of the exponent is brought about by the adhesive interaction with the contactor. In the presence of noise, the exponent is further reduced because of intermittent relaxation of stresses by peeling of contact zones. Thus, noise acts to stabilize the crack in newly found, neighboring metastable states. Indeed, in the absence of any noise, an excessive buildup of stresses leads to unstable, catastrophic crack growth. Debonding experiments are usually carried out first by bringing the top plate in intimate contact, thus increasing the initial fractional area of contact Rinitial, rather than pulling from the critical distance where the instability first sets in. Figure 14A shows interesting differences as well as similarities when the process of debonding is started from different initial states of contact characterized by different initial fractional areas of contact at the start of withdrawal phase. When withdrawal is initiated from more intimate contact (Rinitial g 0.8), the constant contact area or pinned regime is less prominent compared to the case of superficial initial contact (Rinitial ∼ 0.5) and peeling begins at lower separation distances. Although the maximum pull-off force increases somewhat with increased initial contact area, the descending branch of the force-distance curves are almost independent of the initial contact area at distances larger than dmax (Figure 14B). Finally the snap-off distance remains unaltered regardless of the initial contact area. 3.3. An Elementary Model for Snap-Off. A simple model for capturing the essential features of debonding is now presented. A film of thickness h, length l, with interaction potential U is considered as shown in Figure 15. The fraction of the film in “contact” with the contactor
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which leads to the condition that elastic force is equal to the adhesive force
(d -h ξ)(1 -R R) ) RU′(ξ)
E
(21)
which in turn is equal to the external force F. Further
(
d-ξ δΠ 1 ) Eh δR 2 h(1 - R)
)
2
+ U(ξ) ) 0
(22)
The solution of (20) and (22) yields that at contact the value of u for a given value of d is very close to d - de or (ξ ) de) where de is the equilibrium separation distance. Thus Figure 15. (Case 1) Schematic of a soft thin elastic film of thickness h and length l, as considered in the elementary model. The portion in contact is Rl and the displacement of that portion is u. The nominal gap distance is d. (Case 2) Schematic of an undeformed film.
is denoted by Rl and the displacement of that part of the film surface is u. The part of the film not in contact with the contactor moves down by an amount (R/(1 - R))u to conserve the total volume of the film. This model approximates the columnar structure of the film seen in simulations. The elastic energy in the film per unit depth is given by
ΠE ) Rl
1 1 R u2 u2 Eh + (1 - R)l Eh ) 2 h 2 1-Rh u2 R 1 (16) l Eh 2 h 1-R
()
(
)
( )(
)
The interaction energy per unit depth for a nominal gap thickness d is given by
(
ΠU ) RlU(d - u) + (1 - R)lU d +
R u (17) 1-R
)
Thus the total potential energy per unit area, which is the sum of the elastic and interaction energy, is thus given by
Π)
u2 R 1 Eh + RU(d - u) + 2 h 1-R
( )(
)
(
(1 - R)U d +
R u (18) 1-R
)
To further simplify this model, the part of the interaction arising from the film not in contact, i.e., the section of the film that has moved down, is neglected. The interaction energy is strongly dependent on the intersurface separation distance, and the separation distances at these places, not in contact, are higher than the critical distance to make their effect felt. Thus the energy per unit area as a function of the separation distance ξ ) d - u and fractional contact area R, for a given nominal distance d is given by
Π)
d-ξ2 R 1 Eh + RU(ξ) 2 h 1-R
(
)(
)
(19)
For a given value of d, the task is to find the values of ξ and R that minimize the total energy (19). The condition taken on ξ gives
d-ξ R δΠ )-E + RU′(ξ) ) 0 δξ h 1-R
(
)(
)
(20)
R(d) ≈ 1 -
(
E 2hU(de)
)
1/2
(d - de)
(23)
Noting that U(de) ) ∆G and E ) 3µ (incompressible elastic material)
3µ dR )2h∆G d(d)
(
1/2
)
(24)
This implies that the rate of change of area with separation distance varies as (h∆G/µ)-1/2. The condition that the contact area becomes zero at snap-off leads to the condition
( )
dp 2∆Gh ≈ de 3µde2
1/2
(25)
The approximate scaling ratio (25) is in close agreement with full numerical solutions of (21) and (22), which is shown in Figure 17 along with the full simulation results for snap-off distance. As discussed, the optimized value of the separation distance (ξopt) is very close to de (as obtained from U′(ξ) ) (6µU(ξ)/h)1/2 through (20) and (22)), and the maximum force obtained is very small compared to U′(ξmax) (the maximum in interaction force). The dimensionless maximum force Fmh/µde, obtained from this elementary model is related to the other parameters through the relation (∆Gh/µde2)1/2 as evident from Figure 20. The maximum interaction force can be obtained when the film surface does not deform (u ) 0), as shown by case 2 in Figure 15. The film on debonding goes from “total” contact (R ) 1) to “no” contact (R ) 0). In that case the total energy per unit area is given by
Π ) Fξ + U(ξ)
(26)
Minimization with respect to ξ leads to F (the external force) ) -U′(ξ). The maximum occurs at ξmax and has the value of ∆G/de (maximum adhesive value) as evident from Figure 2 (can also be obtained analytically from (2)). This is the case we refer as the case of debonding of two flat films from total contact or, the “DLVO limit”. In case 1, when debonding occurs with an initial patterned configuration, the film prefers to debond via peeling mode (a constantly decreasing contact area) described by the energetics given in (19). The force of detachment in this case can be determined as 0.02 MPa for a film with A ) 10-20 J, h ) 10.0 µm, µ ) 0.5 MPa, and s ) 2 nm. This is 3 orders of magnitude smaller compared to 80 MPa, the maximum adhesive force as seen from Figure 2.
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Figure 16. Ratio of the distance characteristic of pull-off force and the cavitation distance as a function of the compliance of the film.
The present simplified model models the peeling mode of debonding, where the number of columns remains the same, but the columns grow slender and thus the area of contact decreases as the debonding proceeds. The model also captures the small pull-off force (compared to maximum force) required to completely debond the film. The main drawback of the model is that it neglects the elastic column-column interactions which play a crucial role in determining the morphologies during debonding. 3.4. Scaling Relations Obtained from Simulation Results. In this section, we present scaling results for the maximum force, the distance characteristic of this maximum force, the snap-off distance, and the work done to detach the film from the contactor as obtained through numerical simulations. For higher step sizes of withdrawal and low levels of noise, once the maximum force is reached the columns begin to peel from sides of the contactor. The peeling rate is uniform for these films. With the introduction of noise, the rate of reduction in area, however, no longer remains a constant. Whenever a column splits or collapses, there is a considerable amount of decrease in the contact area. Interestingly, in the case of column splitting, the distance (dmax) at which the maximum average stress occurs is the same as the distance at which cavitation first occurs (dcav). This is shown in Figure 16. The maximum snap-off distance where the film loses complete contact with the contactor is obtained for a wide range of parameters h ∼ 0.1-10 µm, A ∼ 10-19-10-21 J, and µ ∼ 0.1-10 MPa, Rinitial ∼ 0.5-0.95, step sizes (s ∼ 0.01-2.0 nm), noise amplitude ( ∼ 0.001-0.01), and ∆G ∼ 1-100 mJ/m2. The dependence of the snap-off distance dp on ∆G, µ, and h is represented by a master curve shown in Figure 17 of the form
dp/de ≈ Cdp
( ) ∆Gh µde2
p
(27)
A positive value of p as seen from the simulations indicates that stiff films debond faster, due to adhesive and deformational limitations, and that there are two distinct regimes: one for the small step sizes without noise and another for higher step sizes and noise. For very small step sizes, the exponent p is close to 1 for cases with ) 0 and small step sizes (s < 0.3 nm) (Figure 17). As shown later, p ∼ 1 also implies that the debonding stress is ∼∆G/ de, which is of course the scaling for debonding of flat surfaces. This is because the contact area is constant in this case until the catastrophic column collapse occurs. Increase in the level of noise and step size both help the system to cascade through metastable states leading to faster debonding. This leads to lower values of the exponent p as shown in Figure 17 for such cases and as also seen commonly in tack-test experiments. As evident
Figure 17. The snap-off distance obtained from the simulation results for simple model for s ) 2 nm and full simulations for various amount of noise (: 0.0, 0.001, 0.01), different stepsize of withdrawal (s: 0.01, 0.1, 0.3, 1.0, 2.0 nm) and for different parametric values of h ∼ 0.1-10 µm, A ∼ 10-19-10-21 J, and µ ∼ 0.1-10 MPa; p has the same meaning as in eq 27.
Figure 18. The distance characteristic of the maximum force for various amounts of noise and step size for the same parametric values as in Figure 17: (A) represents results for Rinitial ) 0.5 and (B) for Rinitial ) 0.8.
from the figure, the elementary model gives the value of p as 0.5 close to those obtained in simulations with higher noise and step size. However, it produces a much lower value for snap-off distance compared to the value obtained from full simulation results. This is anticipated because the elementary model reproduces only the “peeling part” of withdrawal and does not account for the “pinning” regime of debonding process. As regards the influence of initial contact area from where the debonding phase is initiated, the snap-off distance (dp) is nearly independent of the initial area since the withdrawal branch away from dmax merges for the same parameters irrespective of the nature of withdrawal as shown in Figure 14B. The distance characteristic of the pull-off force, dmax (Figure 9), is found to be related to the compliance (∆Gh/ µde2) through a power law in analogy to eq 27 (dmax/de ≈ Cdmax(∆Gh/µde2)q) (Figure 18). The impact of noise and increase in step size have similar effects on dmax as in the case of snap-off by distance controlled mode (dp). The two characteristic distances as found from the simulations are linearly related for small step sizes and for noiseless cases as shown in Figure 19. For small step sizes (∼0.3 nm), the two coincide with each other and the maximum average stress occurs at snap-off by catastrophic column collapse. For higher step sizes and noise level, the maximum average stress is attained much before the final snap-off occurs (dmax ∼ 0.5dp). An increase in the initial area of contact leads to the pull-off force Fmax being attained at smaller separation distance as evident from Figure 19.
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Figure 19. The plot shows the linear relationship present between the two characteristic snap-off distances (dmax ) a(dp)b). Figure 21. Variation of force exponent δ with step size (A), initial contact (B), and noise (C) and (D) for initial contact Rinitial ) 0.5 and Rinitial ) 0.8, respectively.
Figure 20. Variation of nondimensional maximum force with compliance of the film. (A) shows the effect of stepsize, for Rinitial ) 0.5 and (B) for Rinitial ) 0.8. (C) and (D) show the effect of noise and initial contact on the maximum force, respectively. The simulation results from the simple model for s ) 2 nm is shown in (A).
The values of maximum average stress obtained from simulations were found to fall on a master curve of the form
( )
Fmaxh ∆Gh ≈ CF µde µde2
δ
(28)
where CF and δ are functions of step size (s), noise level (), and initial contact (Rinitial). For small step sizes (s j 0.3) and without the effect of any noise ( ) 0), the average stress in the film increases linearly with separation distance as evident from Figure 5. Thus at snap-off the maximum average stress Fmax∼ µdp/h, which in turn is proportional to the maximum adhesive force per unit area ∆G/de (obtained from the condition p f 1 in eq 27 for s ∼ 0.3 nm and ) 0). This is shown in Figure 20A,B, where the ratio of the maximum average stress Fmax to the maximum adhesive force (per unit area) ∆G/de, when plotted against the compliance ∆Gh/µde2, is constant. Thus, in this regime, Fmax ∼ ∆G/de with a proportionality factor in the range of 0.075-0.25. This is still about an order of magnitude less than the DLVO force (per unit area) ∆G/ de that is required to separate two flat films from contact, as evident from Figure 20A,B. An increase in the step size (s ∼ 1-2 nm), decreases the maximum average stress by several (∼2) orders of magnitude compared to the small step size results. The maximum force computed from the simple model for a step size of 2 nm shows agreement with the results from simulations. Introduction of noise ( ∼ 0.001-0.01) has a similar effect as that of increased step size in bringing down the maximum average stress (Figure 20C). On the other hand, an increase in the initial contact area (Rinitial ∼ 0.5-0.95) needs a higher pull-off
Figure 22. The dimensionless work done for various (A) step sizes ( ) 0, Rinitial ) 0.5), (B) noise (s ) 1 nm, Rinitial ) 0.5), and (C) initial contact ( ) 0, s ) 2 nm).
force as discussed earlier. Simulations also show that the pull-off force for such cases to be two to four times higher from that of slight contact results as evident from Figure 20D. For small step sizes (s j 0.3) and no disturbance ) 0, the pull-off force tends toward the scaling obtained for flat rigid surfaces, irrespective of the initial level of contact (Rinitial ) 0.5 or higher Rinitial ) 0.8) indicated by δ f 1 (see Figure 21A). Increase in step size (s ∼ 1-2 nm), in the absence of any noise decreases δ to a value of 0.8, for initial contact Rinitial ) 0.5 and a value ∼0.5 for higher initial contact, Rinitial g 0.8 (Figure 21A,B). Noise also brings down the value of δ close to 0.5, as evident from Figure 21C,D, irrespective of the level of initial contact (Rinitial ) 0.5 or Rinitial ) 0.8) and the magnitude of step size (s ∼ 0.3-2 nm). Thus, it is clear from Figure 21 that while very small step size and the complete absence of noise help to achieve the limit of DLVO scaling for flat surfaces, an increase in step size and noise provides a value of δ close to 0.5, as frequently observed in experiments.22,43 This regime is characterized by peeling of contact areas and expansion of intervening cavities. Thus, the elementary model which reproduces the peeling regime gives a value of δ ) 0.5. The above considerations also explain the contention that the surface energy of soft solids as measured from debonding experiments is a nonequilibrium and nonunique property depending on the initial state (slight contact vs intimate contact), defects, vibrations, contactor roughness, etc. Since Fmax/Fmaxflat = CF(µde2/ ∆Gh)1-δ, the discrepancy in the forces between the flat and instability controlled modes of failure increases with decreasing shear modulus, increasing adhesive strength and film thickness. Figure 22 shows the ratio of work done (per unit area) to the adhesive energy in pulling the film to detachment. It can be observed that the dimensionless work done can (43) Kendall, K. J. Phys. D: Appl. Phys. 1971, 4, 1186.
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Figure 23. Variation of (A) the maximum pull-off force, (B) the distance characteristic of the maximum pull-off force, (C) the snap-off distance, and (D) the work done (per unit area), with the compliance of the film. The filled symbols are for de ) 0.1732 nm and the open symbols are for de ) 0.1368 nm. The higher values denote simulations for s ) 1 nm, and the lower values denote simulations for s ) 2 nm.
Figure 24. Variation of the ratio of (A) the snap-off distance, (B) the distance characteristic of the maximum pull-off force, (C) the maximum pull-off force, and (D) the work done (per unit area) in stretching the film for (1) potential with short-range non-VDW attraction (eq 3) and (2) potential including van der Waals attraction and Born repulsion (eq 2) with compliance of the film. The simulations are carried for the same parametric values as in Figure 17.
be related to the compliance of the film as
( )
W ∆Gh ≈ CW ∆G µde2
n
(29)
The ratio W/∆G is greater than 1 for smaller step sizes (s < 0.3). When the compliance is higher, this is true also for higher step size (s ) 1 nm; Figure 22A). Introduction of larger step sizes (Figure 22A) and higher noise (Figure 22B) brings down the work done (per unit area). Therefore, the work done for debonding is lesser in the “peeling” mode of failure than the “catastrophic column collapse” mode and decreases further where column splitting and collapse occurs. An increase in the initial contact is found to increase the work done (per unit area) required for debonding the film (Figure 22C). Simulations are also carried out for different values of the equilibrium distances, de ) 0.1368 nm and de ) 0.1732 nm, so as to work with different adhesive energies at contact. The results shown in Figure 23 reveal that de does not affect the intercepts, the coefficients, CF, Cdp, Cdmax, and CW, and the slopes δ, p, n, and q. The increased value of the equilibrium distance effectively decreases the adhesive energy at contact, thereby requiring a lower pulloff force, dmax, snap-off distance, and the work done for debonding. However, the choice of equilibrium distance does not affect the scalings for the above quantities. We now illustrate that the results presented here are independent of the detailed form of the interaction potential considered by presenting results for the second form of the interaction potential, eq 3. In the case of the elementary model, it is seen that the gap distance ξ satisfies U′(ξ) ) (6µU(ξ)/h)1/2 and, as discussed, the optimum value of ξ ≈ de. Thus, in most cases, the part of the film that is in intimate contact with the contactor experiences the interaction potential only in the vicinity of de. Thus, the two interaction potentials (U) whose detailed form may differ but are described by the same set of parameters near de may lead to identical results. Thus the interaction potentials that satisfy U1(de) ) U2(de) and U1′′(de) ) U2′′(de) are likely to predict similar results. Indeed as shown in Figure 24, simulation results of the two potentials, eqs 2 and 3 satisfying conditions (4), (5), and (6) are the same. It is evident that neither the work
done, the snap-off distance, nor the pull-off force or the distance of attainment of pull-off force are affected by the details of the potential. 4. Conclusions The main aim of the study is to investigate physics of debonding at soft interfaces with particular attention to pattern formation and its effect on the snap-off distance and pull-off force. The present study yields several important results in this respect which are summarized below: (1) A soft thin incompressible elastic film undergoes surface roughening when the separation gap with the contactor is below a critical separation distance. The dominant wavelength of surface morphology, at this critical distance, is approximately three times the thickness of the film, in agreement with linear stability analysis.24,25 The dominant wavelength persists to be 3h during withdrawal of the contactor until the maximum force is reached. (2) The linear stability is inadequate to account for the persistence of instability at distances in excess of the critical distance during withdrawal. The present work illustrates that the presence of an energy barrier separating the patterned state and the flat film configuration leads to the persistence of such patterned configuration during withdrawal. The “adhesion-debonding” hysteresis, therefore, arises from the presence of this barrier. (3) The nonlinear analysis shows that the surface takes the form of columns that bridge the gap with the contactor and the presence of intervening cavities. However, it must be noted that the simulations presented here are 2D, which cannot distinguish whether the cavities or the columns form the connected domains. Three-dimensional simulations currently underway in our group are required to address this issue. There are mainly three broad modes of adhesive failure uncovered by nonlinear simulations. In the first mode, the area of contact remains constant as the plate is withdrawn and the edge of columns are pinned to the contactor. Debonding in this case occurs finally by a “catastrophic” failure of the columns when elastic stresses near the column edges cross a threshold. This mode is more probable for more compliant films. In the
Soft Elastic Films
second mode, termed as the peeling mode, the area of contact decreases almost linearly with the separation distance and there is no excessive buildup of stresses near the column edges. The decrease in the maximum pull-off force from the DLVO force is largely due to the peeling effect. Finally, an abrupt change in area (stepwise decrease in area) arising from splitting or cavitation of the bridging columns indicates splitting mode or column collapse mode of debonding. Both are present for compliant films with imposed noise. (4) For the catastrophic mode of failure accompanied by pinning, the work done increases the elastic energy of the film since there is no change in the contact area and interfacial energy. The total work done is dissipated at final snap-off by column collapse. In the peeling mode of failure, at each stage there is a minicatastrophe, the difference in energy of the higher energy “pinned” metastable state and the relaxed “peeled-state” is dissipated. The elastic energy of the columns decreases with peeling leading to energy loss. Introduction of noise increases this dissipation causing early snap-off. This energy dissipation is distinct from the bulk viscoelasticity, plasticity, and bulk cavitation, none of which is considered in our model of a purely elastic film. (5) Formation of cavities engenders extremely high stresses near the column edges leading to the peeling of contact zones at much smaller average pull-off force than that predicted for debonding flat surfaces (Fmaxflat ∼ ∆G/ de). For example, with A ) 10-20 J, h ) 10.0 µm, µ ) 0.5 MPa, and Fmaxflat ) 80 MPa: Fmax ) 9.5 MPa for s ) 0.1 nm, ) 0, Rinitial ) 0.53; Fmax ) 0.1 MPa for s ) 1 nm; Fmax ) 0.04 MPa with ) 0.01; and Fmax ) 0.2 MPa for s ) 1 nm , ) 0, Rinitial ) 0.9 (intimate contact). (6) Simulation results show that the snap-off distance, the pull-off force, and the work done can be described by scaling relations as dp/de ≈ Cdp(∆Gh/µde2)p, Fmaxh/µde ≈ CF(∆Gh/µde2)δ and W/∆G ≈ CW(∆Gh/µde2)n. Small step size without imposed noise helps to achieve the limit of DLVO force (δ f 1 and p f 1). Higher step size and introduction of higher amounts of noise aids the structure to cascade through lower energy barriers resulting in faster debonding and lowering of pull-off force required to debond the film. Increased step size, noise, and initial contact area make δ f 0.5 as observed in experiments. Experi-
Langmuir, Vol. 21, No. 4, 2005 1469
ments on elastic films show the same general features regarding the force-displacement curves. The Fmax value of 2500 Pa as observed in experiments22 to that of 2632 Pa (s ) 1 nm) from simulations for a film with h ) 100 µm, µ ) 100 Pa, and ∆G ) 120 mJ/m2 compares favorably. (7) The detailed form of the potential present between the film and the contactor is not of significance since it does not effect either the snap-off distance, the maximum force, or the distance at which the maximum in force occurs. The snap-off process is governed by the adhesive energy and the stiffness of the adhesive interaction potential at the equilibrium separation. The present work sheds light on several outstanding issues in adhesion and debonding at soft interfaces such as adhesion-debonding hysteresis and the role of pattern formation. It is clearly demonstrated that the patterns that form due to the competition between attractive forces and the elasticity of the soft film governs much of the phenomena associated with adhesion/debonding. In particular, the debonding pathways via pattern formation require much smaller forces than those predicted by simple theories that assume flat surfaces separating in normal direction. In this regard, the patterns are analogous to defects (dislocations) in crystalline solids which bring about a great reduction in the yield stress (like the maximum adhesive force). Further, the study suggests that the determination of adhesive energies from debonding experiments involving a flat punch have to account for the formation of patterns which affects the pull-off force. Acknowledgment. Discussions with M. K. Chaudhury, A. Jagota, and A. Ghatak are gratefully acknowledged. V.S. and A.S. acknowledge DST, India (Nanoscience Program) for support. Supporting Information Available: Bifurcation diagram for different film thicknesses showing that the region of instability is confined between two wavelengths and instability sets in when -Y/µ > 6.22/h. The lower branch of the instability region is marked by 4πµ/(-Y) showing that the necessary condition for instability is given by λ > (4πµ/(-Y)). The material is available free of charge via Internet at http://pubs.acs.org. LA048061O
