## Abstract

The apparent fracture/adhesion energy of an interface with periodic cohesive interactions is of general interest to understanding adhesion via periodic adhesion patches (e.g. between micro- and nanostructured surfaces). There are two important length scales for this class of problems: one corresponds to the period of cohesive interaction and the other is the size of the cohesive zone near the tip of a crack along the interface. By theoretical considerations and numerical simulations, we show that the apparent fracture/adhesion energy depends on the ratio between the period of cohesive interaction and the cohesive zone size: it is equal to the average cohesive energy of the interface if the former is much smaller than the latter but becomes the peak value of the local cohesive energy when the opposite is true. This prediction has been confirmed by numerical simulations on the peeling of a thin-film/strip adhering on a substrate via periodic discrete adhesion patches. Our analysis also provides explanations for a recent molecular dynamics simulation which showed that the apparent adhesion energy of a single-stranded DNA (ssDNA) adhering on a graphite sheet is equal to the peak, rather than the average, value of the van der Waals interaction energy between the ssDNA and the substrate.

## 1. Introduction

When two materials adhere to each other via periodic adhesion patches or microstructures, the local cohesive interaction is not homogeneous but varies periodically along their interface. This scenario occurred in recent molecular dynamics (MD) simulations by Shi *et al*. (2005) on the peeling of a short single-stranded DNA (ssDNA) molecule containing eight adenine bases adhering on a graphite substrate (figure 1*a*). The interaction between the ssDNA and the graphite substrate is not homogeneous but varies periodically along the ssDNA–graphite interface. In an attempt to apply a theoretical model of thin-film peeling to explain their simulation results, Shi *et al*. (2005) found that, to their surprise, the apparent fracture/adhesion energy of the ssDNA–graphite interface is not the average, but rather the peak value, of the ssDNA–graphite interaction energy. A satisfactory explanation of this phenomenon remains elusive to this date.

With rapid advances in micro- and nanofabrication techniques, various micro- and nanostructured surfaces can now be fabricated and assembled for interesting applications. For example, figure 1*b* shows buckled thin layers of GaAs adhering to an elastomeric substrate by John Rogers and co-workers (Sun *et al*. 2006) for applications in stretchable electronics (Khang *et al*. 2006; Jiang *et al*. 2007). The adhesion occurs at a periodic distance of 100 μm on the surface of the substrate while the thickness of the GaAs layer is only approximately 0.1 μm. What is the apparent fracture/adhesion energy of such an interface?

This paper is aimed as an attempt to answer the above question. In §2, we form the basic concept by studying the fundamental problem of two semi-infinite elastic solids adhering to each other via different forms of periodic cohesive laws along the interface. In §3, we consider the more specialized problem of an elastic thin strip adhering to a substrate via periodic adhesion patches, which is partly motivated by the problem of ssDNA–graphite adhesion of Shi *et al*. (2005). Numerical simulations will be carried out to validate the theoretical predictions. Figure 1*c* shows schematically our numerical model on the thin-film peeling problem, which will be used to investigate how the apparent fracture/adhesion energy of the film–substrate interface depends on the film thickness and the period *l*_{p} of adhesion patches.

## 2. Apparent fracture/adhesion energy of an interface joining two infinite elastic solids with a locally varying cohesive interaction law along the prospective fracture path

Figure 2*a* shows a semi-infinite crack along an interface joining two semi-infinite elastic solids with a periodically varying cohesive interaction law along the prospective fracture path. For simplicity, the two solids are assumed to have the same elastic properties with Young's modulus *E* and Poisson's ratio *ν*. The crack is under tensile loading and prescribed to propagate along the *X*-axis that is aligned with the interface. The stress field far from the crack tip is characterized by a mode I stress intensity factor *K*_{app}. Locally, the cohesive energy varies periodically along the prospective fracture path. The apparent fracture/adhesion energy of the interface, defined as the critical far-field energy release rate for sustained crack growth (delamination), will be studied via a cohesive zone model (Dugdale 1960; Barenblatt 1962) with a spatially varying cohesive interaction law. To facilitate the analysis, we also introduce a moving coordinate *x* which follows the tip of the cohesive zone as it develops along the interface. In other words, *x* coincides with the fixed coordinate *X* initially but it translates along the interface together with the tip of the cohesive zone.

One of the simplest cohesive laws is (Dugdale 1960)(2.1)where *σ*_{c} is the traction and *δ*_{c} is the local separation of two surfaces; *σ*_{0} is the strength and *δ*_{0} is the extent of cohesive interaction. The cohesive energy associated with this interaction law is simply *γ*=*σ*_{0}*δ*_{0}.

We consider several different forms of periodically varying cohesive interactions along the interface. In §2*a*, we let the cohesive strength *σ*_{0} vary as a sinusoidal function of *X* while keeping the interaction range *δ*_{0} constant. In §2*b*, we let *δ*_{0} vary as a sinusoidal function of *X* while keeping *σ*_{0} constant. In §2*c*, we let both *σ*_{0} and *δ*_{0} vary with *X* while keeping the local cohesive energy *γ*=*σ*_{0}*δ*_{0} constant.

### (a) Periodically varying cohesive strength

Let us first consider the case when *δ*_{0} is kept constant and *σ*_{0} varies aswhere is the average cohesive strength along the interface; *c*_{σ} is the amplitude; *l*_{p} is the wavelength; and *α* is the phase angle of the cohesive strength variation. For this interaction law, the average cohesive energy along the interface is . Upon loading, a cohesive zone develops at the crack tip to a length of *l*_{c} while the surface separation at the left end of the cohesive zone is equal to . In the moving coordinate *x*, the cohesive traction is given by(2.2)

In order to derive the apparent fracture/adhesion energy for the system, we follow Dugdale (1960) in eliminating stress singularity at the end of the cohesive zone by letting(2.3)where *K*_{cohesive} is the stress intensity factor induced by cohesive forces within the cohesive zone. According to Tada *et al*. (2000), for a distributed crack-face traction given in equation (2.2), *K*_{cohesive} can be calculated from(2.4)On the other hand, the surface separation (i.e. the crack opening displacement) at the beginning of the cohesive zone is given by(2.5)where is the contribution due to the far-field applied loading and is the contribution due to the cohesive forces. It is well known in fracture mechanics that (e.g. Tada *et al*. 2000)(2.6)where *E*′ is equal to *E* under plane stress and (*E*/(1−*ν*^{2})) under plane strain, and(2.7)When , the crack opening displacement at the beginning of the cohesive zone will be at the maximum interaction range, signalling imminent crack propagation. From equations (2.3)–(2.7), the cohesive zone size *l*_{c} can be determined as a function of the phase angle *α* from(2.8)In equation (2.8), we have introduced two normalized length scales,(2.9)where(2.10)corresponds to the ‘reference cohesive zone size’ when the cohesive interaction is uniform along the fracture path/interface.

For a given *α*, we can determine from equation (2.8) which, together with equations (2.1)–(2.4), leads to the far-field stress intensity factor as a function of *α*, i.e. . The apparent fracture/adhesion energy of the interface is defined as the maximum of the far-field energy release rate(2.11)Let us first consider two extreme cases.

When the period of the cohesive law is much larger than the cohesive zone size, i.e. , it can be shown that(2.12)In deriving this result, we have used equation (2.8) to find and equations (2.3) and (2.4) to find . The far-field energy release rate is , which then yields equation (2.12). In this case, the far-field energy release rate is governed by the local cohesive energy at the crack tip. Therefore, the apparent fracture/adhesion energy of the interface is equal to the peak value of cohesive energy along the prospective fracture path.

When the period of the cohesive law is much smaller than the cohesive zone size, i.e. , we have(2.13)In this case, the far-field energy release rate and the apparent fracture/adhesion energy is equal to the average cohesive energy of the interface.

For intermediate values of , the far-field energy release rate *G*(*α*) can be numerically evaluated. We plot the behaviours of *G*(*α*) for *c*_{σ}=1 and in figure 3*a*. The cases and 0.1 behave similar to the two limiting cases discussed above. For , *G*(*α*) varies between the two limiting solutions. We note that, in general, *G*(*α*) deviates from both the local cohesive energy and its average along the interface.

The apparent fracture/adhesion energy of the interface is plotted as a function of in figure 3*b*. The data follow an empirical expression,(2.14)where *m*>0 and is a polynomial function of . In equation (2.14), the two limiting solutions for are automatically satisfied. Numerical fitting indicates *m*=1 and . The resulting fitting curve for *G*_{c} is plotted as the solid line in figure 3*b*.

### (b) Periodically varying cohesive interaction range

In this case, we keep *σ*_{0} constant and vary the interaction range *δ*_{0} of the cohesive law as(2.15)where is the average interaction range; *c*_{δ} is the amplitude; *l*_{p} is the wavelength; and *α* is the phase angle of the cohesive range variation. Compared with the previous case, the average cohesive energy of the interface is still , but it is the interaction range, rather than cohesive strength, that varies along the interface.

The surface separation or crack opening displacement can again be expressed as a linear superposition of the contribution from the far-field loading and that due to the cohesive forces. In the moving coordinate *x*,(2.16)where(2.17)and(2.18)

In the case of large variations of interaction range within (−*l*_{c}, 0), since the cohesive forces depend on *δ*_{0}(*x*),(2.19)the cohesive zone can be ‘fragmented’ in the sense that the cohesive forces may vanish within the apparent cohesive zone (−*l*_{c}, 0). Using equations (2.3), (2.4) and (2.15)–(2.19), the crack opening profile can be determined using an explicit iteration scheme. First, a very small initial value is set for the cohesive zone, which is assumed to be continuous in evaluating *K*_{cohesive} from the first part of equation (2.4), *K*_{app} from equation (2.3) and the crack opening profile from equations (2.16)–(2.18). A check is then performed to see which parts of the cohesive zone are actually out of the interaction range and, accordingly, cohesive forces are set to zero in these regions. Next, a small increment in cohesive zone size is enforced and the aforementioned procedure is repeated. The iteration process stops when the crack opening displacement at the left end of the apparent cohesive zone reaches the local interaction range.

Now consider the two extreme cases of and ∞. When , the period of cohesive variation is infinitely larger than the cohesive zone size. In this limit,(2.20)When , the period of cohesive variation is vanishingly small compared with the cohesive zone size. In this limit,(2.21)Numerical results can be obtained in general for finite . Figure 4*a* shows the crack surface profile upon loading for , with . In this case, the cohesive zone is much smaller than the period of the cohesive law and the far-field energy release rate *G* varies with the local cohesive energy at the crack tip. Accordingly, the apparent fracture/adhesion energy is nearly the peak value of the local cohesive energy, i.e. .

Figure 4*b* shows the profile of fragmented cohesive zone near the crack tip upon loading for , with . In this case, the far-field energy release rate *G* and the apparent fracture/adhesion energy are close to the average cohesive energy .

Figure 5 plots the apparent fracture/adhesion energy *G*_{c} as a function of . In this case, we find that the fitting formula of equation (2.14) is still valid with parameters *m*=0.5 and .

### (c) Varying cohesive strength and interaction range under constant cohesive energy

To consider a special case of varying cohesive law under constant cohesive energy, we keep constant while letting both *σ*_{0}(*X*) and *δ*_{0}(*X*) vary as(2.22)and(2.23)In this case, the cohesive zone can also be fragmented according to the local interaction range. For given *δ*_{0}(*X*) and *σ*_{0}(*X*), the fragmented cohesive zone can be determined following an iterative procedure similar to that described in §2*b*.

In the extreme case , the cohesive zone can be periodically fragmented depending on the cohesive law. The critical far-field energy release rate and the apparent fracture/adhesion energy are equal to the average value of the cohesive energy along the interface. In the other limit , the cohesive zone is small compared with the period of cohesive law variation and the apparent fracture/adhesion energy will be equal to the local cohesive energy, which is also .

As an example of , figure 6*a* shows the calculated cohesive zone when , *c*_{0}=0.5 and *α*=*π*/2. Indeed, the cohesive zone is continuous and the global energy release rate *G* is equal to the local cohesive energy. Figure 6*b* shows an example of , with , *c*_{0}=0.5 and *α*=*π*/2. Here we see that the cohesive zone is periodically fragmented and the apparent fracture/adhesion energy *G*_{c} is found to be the average cohesive energy . A J-integral analysis will confirm that the global energy release rate will be equal to the average cohesive energy even though the cohesive zone is fragmented. In fact, the cohesive zone is generally fragmented for the present form of the cohesive law.

For intermediate values of *l*_{p}, the apparent fracture/adhesion energy can actually be higher than the local adhesion energy. Figure 7 shows that the maximum of can reach as high as 1.6 for the case of *c*_{0}=0.5, and *α*=*π*/2. Although it may not be obvious how the apparent fracture/adhesion energy can be higher than the local cohesive energy, this does not violate a J-integral analysis. By applying the J-integral along the interface, it can be shown that the far-field energy release rate is related to the local cohesive law as(2.24)For a homogeneous cohesive law independent of *x*, the far-field fracture energy is equal to the local cohesive energy . In the present case of a non-homogeneous cohesive law , this is no longer true. In fact, by replacing cohesive bonds near the beginning of a cohesive zone with bonds of the same cohesive energy but a larger cohesive interaction range and those near the tip of a cohesive zone with cohesive bonds of the same cohesive energy but a larger cohesive strength can bring the current apparent fracture/adhesion energy to a higher value.

## 3. Thin-film peeling: effect of film thickness

For the thin-film peeling problem, the film thickness *H* becomes an important length scale of the problem. In this section, we show that a very small film thickness, e.g. , leads to severe constraint on the actual cohesive zone size.

We consider a thin elastic beam adhering on a rigid substrate (figure 2*b*). The simple cohesive law described by equation (2.1) is adopted to describe the adhesive interaction between the beam and the substrate. For simplicity, the interface is assumed to be free of shear traction and only the normal component of the cohesive force is considered. The cohesive strength and interaction range are considered to be constant equal to and , respectively.

Under a uniform distribution of the cohesive force in the cohesive zone, the stress intensity factor induced by the cohesive forces can be calculated according to a formula given in Tada *et al*. (2000) as(3.1)According to equation (2.3), the stress intensity factor induced by cohesive forces must balance that due to the applied loading, indicating(3.2)It follows from equations (3.1) and (3.2) that:(3.3)Equation (3.3) suggests that the cohesive zone size in thin-film peeling can be highly constrained by the film thickness.

We have conducted numerical simulations of thin-film peeling off a rigid substrate. The structure under study is shown in figure 1*c*. The bonded part of the film is periodically attached to the substrate with period *l*_{p} while the free part of the film is subjected to a vertical peeling force (90° peeling). The peeling force can be determined from the critical energy release rate associated with the peeling process.

In the bonded part of the film on the substrate, the attachment regions are periodically spaced at a spacing equal to the size of the attachment patch. The adhesion energy within each adhesive patch is 1.0 J m^{−2}. Abaqus/Standard is employed to simulate the 90° peeling process. Two-dimensional plain-strain solid elements and cohesive elements are used to model the film and adhesive interaction, respectively. The constitutive response of a cohesive element is defined in terms of the traction–separation law. Failure of the cohesive element is through progressive degradation of the material stiffness after its nominal/equivalent stress reaches 0.02 MPa. Evolution of the failure is based on an isotropic dependence of adhesion energy on the mode mixity ratio. A displacement boundary condition is enforced at the end of the peeling arm to model the peeling process and the method of load–deflection analysis in Abaqus is employed for stable simulation.

At first, *H* and are fixed while the period of adhesion patches, *l*_{p}, varies to see the corresponding change of the peeling force. The parameters for the film are taken to be *E*=0.45 MPa, *ν*=0.3 and *H*=100 μm. We estimate from equation (2.10). Four different periods of adhesion patches, *l*_{p}=83, 125, 167 and 10 000 μm, are considered in the simulation. Figure 8*a* shows the variation of the peeling force as the film is being peeled off the substrate. The horizontal axis is the relative displacement at the end of the peeling arm. Only a portion of the peeling history is presented here due to periodicity. In figure 8*a*, curves correspond to *l*_{p}=83, 125, 167 and 10 000 μm. As *l*_{p} becomes larger, the variation of the global energy release rate becomes stronger. For the case *l*_{p}=83 μm, there is almost no variation in energy release rate while there is 100% variation for the case *l*_{p}=10 000 μm. The apparent fracture/adhesion energy is presented in table 1. For *l*_{p}=83 μm, the apparent fracture/adhesion energy is equal to the average of cohesive energy along the film–substrate interface. As the period of adhesion patches becomes larger, the apparent fracture/adhesion energy also becomes larger. For *l*_{p}=10 000 μm, it is equal to the peak, rather than the average of cohesive energy along the interface.

Next, we investigate the effect of film thickness *H* while keeping *l*_{p} and fixed. The material parameters for the thin film are taken as *E*=1.0 MPa and *ν*=0.3. According to equation (2.10), . The period of adhesion patches is fixed at *l*_{p}=125 μm. Two different values of the film thickness, *H*=50 and 100 μm, are considered in the simulation. Figure 8*b* shows the variation of the peeling force as the film is being peeled off the substrate. As the film thickness is reduced, the variation of the global energy release rate becomes stronger. The corresponding apparent fracture/adhesion energy of the system is also given in table 2: for *H*=100 μm, it is found to be around the average value of cohesive energy along the interface; for *H*=50 μm, it is 50% higher than the average value. These parameters have been selected so that the effects of *H* and *l*_{p} on the apparent fracture/adhesion energy can be clearly seen.

Our model provides an explanation of the MD simulation results by Shi *et al*. (2005) for the peeling of a ssDNA molecule with eight adenine nucleic bases off a graphite substrate. In that simulation, every other nucleic base was found to be almost perfectly attached to the substrate while the neighbour base was not attached due to geometrical constraints of the chain. In the MD simulation, the peeling force was found to vary significantly as one base after another was peeled off the substrate. Using a generalized elastic strip model, Shi *et al*. (2005) found that the apparent adhesion energy of the system is the peak, rather than the average, value of cohesive energy along the interface.

The ssDNA peeling problem of Shi *et al*. (2005) can be considered as an example of the strip-peeling problem discussed above. Consider the ssDNA chain as a thin strip with effective thickness *H*=0.343 nm, which is the distance between neighbouring graphite sheets and Young's modulus 1 GPa. The adhesive interaction between the strip and the substrate is modelled by the simple cohesive law in equation (2.1), with the interaction range taken to be 0.5 nm (van der Waals force). The adhesion patch between the nucleic base of ssDNA and the substrate has a width of 0.4 nm and an adhesion energy of 4 GPa. The spacing between two adhering patches is taken to be 0.2 nm. With these parameters, the cohesive zone size is estimated to be 0.35 nm according to equation (3.3), which is small compared with the period of 0.6 nm of adhesion patches for ssDNA on the substrate. Therefore, the ssDNA/substrate system studied by Shi *et al*. (2005) satisfies the condition , in which case the apparent fracture/adhesion energy is the peak, rather than the average, value of cohesive energy along the interface. For ssDNA on the substrate, although the variation of interfacial cohesive energy occurs over the distance of several angstroms, one still cannot use its average to estimate the work of adhesion for the molecular chain because the cohesive zone size is smaller than the dimension of a single nuclei base. This is consistent with the observation of Shi *et al*. (2005).

## 4. Conclusion

In this paper, we have shown that, for fracture/adhesion energy along an interface with periodically varying cohesive energy, there exist two important length scales. The first length scale is the period of cohesive energy and the second one is the cohesive zone size near the tip of a crack. If the former is larger than the latter, the apparent fracture/adhesion energy is determined by the peak, rather than the average, value of cohesive energy along the interface. Only when the former is much smaller than the latter can the interface be homogenized with a constant adhesion energy. For intermediate values, the apparent adhesion/fracture energy depends on the detailed distribution of cohesive energy along the interface. In peeling of a thin-film adhering on a substrate, the thickness of the thin film can influence the cohesive zone size. For very thin films, the film thickness can greatly influence the cohesive zone size, causing it to be much smaller than the corresponding value in a bulk solid.

We have conducted numerical simulations of peeling a thin-film/strip adhering on a substrate via periodic adhesion patches to confirm the theoretical model and predictions. In these simulations, we vary the period of adhesion patches relative to the cohesive zone size that depends on the thin-film thickness among other material and loading parameters. We observe that the apparent fracture/adhesion energy can be tailored to vary between the average and the peak value of the local cohesive energy along the interface. Our analysis presents a theoretical explanation for the MD simulations by Shi *et al*. (2005), showing that the apparent adhesion energy of ssDNA on a graphite substrate is determined by the peak, rather than the average value, of the interaction energy along the interface.

## Acknowledgments

A part of this work was supported by a postdoctoral fellowship from the Max Planck Society during 2005–2006.

## Footnotes

- Received September 28, 2007.
- Accepted November 23, 2007.

- © 2007 The Royal Society