Peeling off an adhesive layer with spatially varying topography and shear modulus

Inspired by recent experiments on hierarchically structured adhesives, we analyze here the effect of spatial variation in surface topography and shear modulus of an elastomeric adhesive on its ability to adhere strongly to a flexible contactor. The undulation of surface and modulus both were assumed to be periodic with periodicity, which is either identical or different for the two parameters; for identical periodicity, the phase lag between the respective undulations is also systematically varied. Calculations show that during continuous lifting of the flexible contactor from complete initial contact, the interfacial crack between the two adherents does not propagate continuously but intermittently, with crack arrest and initiation at the vicinity of minimum thickness and modulus of the layer; the torque required to initiate an arrested crack increases significantly over that required to propagate it on a smooth adhesive surface. The adhesion strength estimated from the corresponding force vs displacement plot is calculated to be higher than that achieved on a smooth and featureless adhesive surface. For in-phase variation in topography and shear modulus of the layer, the adhesive strength is found to be higher than for nonzero phase lag between the two parameters. The adhesion strength is found to diminish also for nonidentical periodicity between modulus and surface undulation. We have derived a scaling law for relating adhesion strength to several of these parameters.

Figure 1

Schematic depicting a layer of adhesive having spatially varying surface topography and shear modulus.

Figure 2

The plots (a) and (c) represent, respectively, in-phase and out-of-phase variation of thickness $h$ and shear modulus $\mu$ of the adhesive layer as a function of distance $x$ measured from the contact line. The figures (b) and (d) represent typical data of peeling torque $M$ plotted as a function of $x$ for cases (a) and (c), respectively. These data are generated for adhesive films with ${\delta }_h=0.005$, ${\delta }_{\mu }=0.5,$ and $K_h=K_{\mu }=1.0$, and $0.005$, $-0.5,$ and $1.0$, respectively. The calculations are carried out for a film of thickness $h_0=500$ μm, shear modulus ${\mu }_0={10}^6$ N/m${}^2$, flexural rigidity of contactor $D=0.02$ Nm, and critical crack opening stress ${\sigma }_c=5\times {10}^4$ N/m${}^2$. The dashed lines show the typical location where the crack becomes arrested.

Figure 3

(a,b) Typical profiles of the flexible adherent and the adhesive surface in the vicinity of where the contact line becomes arrested. Figures (a) and (b) correspond to in-phase and out-of-phase variation of $h$ and $\mu ,$ respectively. The distances A–B and A’–B’ represent the distance travelled by the contact line at the arrest phase. (c) The curvature of the flexible adherent, obtained from the data in figures (a) and (b), are plotted with respect to dimensionless wave number $K_h=K_{\mu }$ of variation of $h$ and $\mu$. The open and filled symbols represent in-phase and out-of-phase variation in these quantities, respectively.

Figure 4

(a) Effect of amplitudes of oscillation of thickness and modulus, ${\delta }_h$ and ${\delta }_{\mu },$ on lifting load $F$ exerted on the contactor are demonstrated by plotting $F$ against the vertical displacement $\Delta$ of the hanging end of the plate. Curves 1–3 represent, respectively, an adhesive layer with uniform thickness and modulus, ${\delta }_h=0.0$ and ${\delta }_{\mu }=0.0$; one of uniform $h$ but $\mu$ varying sinusoidally with ${\delta }_{\mu }=0.9$ and $K_{\mu }=1.25$; and an adhesive layer of uniform $\mu$ but $h$ varying sinusoidally with ${\delta }_h=0.005$ and $K_h=1.0$. Curve 4 represents an adhesive layer with $h$ and $\mu$ both varying out of phase with ${\delta }_h=0.005$, ${\delta }_{\mu }=0.9$, and $K_h=K_{\mu }=1.25$. The shaded area at the inset of the figure shows the energy required to lift the flexible plate in a typical load-displacement experiment. (b) Effect of in-phase and out-of-phase variation of $h$, and $\mu$ is demonstrated by plotting the lift-off torque $M$ against lift-off height $\Delta$ for two different sets of ${\delta }_h$ and ${\delta }_{\mu }$. Curves 1 and 2 represent films with in-phase variation in $h$ and $\mu$ with dimensionless wave number $K_{\mu }=K_h=1.0$, amplitude of oscillations of adhesive thickness ${\delta }_h=0.005,$ and shear modulus ${\delta }_{\mu }=0.75$ and $0.25,$ respectively. Curves 3 and 4 represent the data similar to the above cases with $K_{\mu }=K_h=1.0$, ${\delta }_h=0.005$, but for out-of-phase variation in $h$ and $\mu$, ${\delta }_{\mu }=0.25$ and $0.75,$ respectively. These calculations are all carried out for $q^{-1}=7.7\times {10}^{-4}$m and critical stress ${\sigma }_c=5\times {10}^4$ N/m${}^2$.

Figure 5

(a,b) The effect of amplitude of modulation of thickness and modulus ${\delta }_h$ and ${\delta }_{\mu }$ on adhesion strength is demonstrated by plotting the maximum crack initiation torque $M_{max}$ against ${\delta }_h^{3/4}$ and ${\left(1-{\delta }_{\mu }\right)}^{-1/6},$ respectively. ${\delta }_{\mu }$ is taken as positive and negative, respectively, for in-phase (○) and out-of-phase (•) variation in $h$ and $\mu$. (c) The effect of wave number, $K_{\mu }=K_h,$ is shown in (c). Whereas calculations in (a) are carried out for ${\delta }_{\mu }=0.9$, in (b) the data correspond to ${\delta }_h=0.005$; for both figures, the wave number is held constant at $K_{\mu }=K_h=1.25$. To obtain the data in figure (c), ${\delta }_{\mu }$ and ${\delta }_h$ are held constant at $0.9$ and $0.005,$ respectively. For all these cases, the critical stress at the line of contact of the adhesive and the adherent is maintained at ${\sigma }_c=5\times {10}^4$ N/m${}^2$.

Figure 6

(a) Excess torque $M_{max}{|}_{\mathrm{excess}}$ is plotted against $\xi$ for different values of critical stress ${\sigma }_c$. All data can be fitted to a single straight line $M_{max}{|}_{\mathrm{excess}}\left(\mathrm{Nm}/\mathrm{m}\right)=2.6\times {10}^7\xi \left({\mathrm{m}}^2\right)$. (b) Excess adhesion strength obtained for in-phase variation of thickness and modulus of the adhesive is plotted against the quantity $\zeta ={\left(a\xi \right)}^2/2D$. The error bar represents the standard deviation in data for different combinations of $K_h=K_{\mu }=K$, ${\delta }_h,$ and ${\delta }_{\mu }$.

Figure 7

(a) Results from calculations in which the phase lag between oscillations in thickness and modulus is varied systematically from $-\pi <{\varphi }_l<\pi$. Both $h$ and $\mu$ are considered sinusoidal with wave number $K_{\mu }=K_h=1.25$, amplitude of oscillation ${\delta }_{\mu }=0.9,$ and ${\delta }_h=0.005$. Curves 1–4 represent, respectively, ${\varphi }_l=0$, $-\pi /2$, $\pi /2,$ and $\pi$. (b) $M_{max}$ data calculated for different ${\varphi }_l$ values. In all these cases, the critical stress at the line of contact of the adhesive and the adherent is maintained constant at ${\sigma }_c=5\times {10}^4$ N/m${}^2$.

Figure 8

The lift-off torque $M$ is calculated for an adhesive layer with different wave numbers of oscillations of $h$ and $\mu$, plotted against displacement $\Delta$ of the free end of the plate. Here $h$ remains sinusoidal with wave number $K_h=1.25$ and amplitude of oscillation ${\delta }_h=0.005,$ and $\mu$ varies with ${\delta }_{\mu }=0.9$ and range of $K_{\mu }=$ 0.625–2.0. Dotted lines 1 and 2 represent, respectively, the trace of maximum torque $M_{max}$ for in-phase and out-of-phase variation in $h$ and $\mu$ with equal wave number, $K_h=K_{\mu }=1.25.$ Curves 3–5 represent oscillation in $\mu$ with $K_{\mu }=0.625,$ $1.0,$ and $2.0,$ respectively, and dotted line 6 represents $K_{\mu }=1.2$. Circles on the dotted lines 1, 2, and 6 represent the location of peak values of torque in the respective cases. These calculations are all carried out for critical stress ${\sigma }_c=5\times {10}^4$ N/m${}^2$.