A thin film is clamped at the periphery to form a circular freestanding diaphragm before a uniform pressure, p, is applied to inflate it into a blister. The bulging membrane adheres to a rigid constraining plate with height, w_{0}, from the nondeformed membrane. Increasing pressure expands the contact circle of radius, c. Depressurization causes shrinkage of the contact and “pull-off” or spontaneous detachment from the plate. Simultaneous measurement of (p, w_{0}, c) allows one to determine the adhesion energy, *γ*. A solid mechanics model is constructed based on small strain and linear elasticity, which shows a characteristic loading–unloading hysteresis. The results are consistent with a large deformation model in the literature.

## Introduction

Thin film adhesion has significant impacts in nanotechnology and life sciences. Dannenberg [1] introduced the classical blister test where a uniform pressure drives an axisymmetric delamination of a coating from a rigid substrate. The work of adhesion is then determined by the contact radius as a function of the applied pressure. A notorious shortcoming of the method is the catastrophic crack propagation at onset of delamination. A number of alternative methods are available in the literature, including the expansion of a fixed mass of gas at the film–substrate interface when the sample is exposed to an external vacuum [2] or elevated temperature [3]. Dillard and coworkers [4–7] introduced the constrained blister test by restricting the blister height using a rigid planar plate, thus stabilizing the delamination under a constant pressure. A “modified” constrained blister [8,9] pressurizes a freestanding membrane clamped at the periphery until the membrane makes adhesive contact with the constraining plate (Fig. 1). The modified test belongs to “confined delamination” where the contact circle is bounded by the diaphragm dimension [10]. Plaut et al. [9] extended the work to include short-range to long-range intersurface attraction based on linear elasticity. Xu and Liechti [11] adopted the modified method to investigate structured acrylate layers on a polyethylene terephthalate carrier film. Flory et al. [12] experimentally investigated the adhesion of a pressurized elastomeric film on a planar substrate. The measurement was later analyzed by Long et al. [13,14] based on large deformation using rubber hyperelasticity.

In this paper, a solid-mechanics model for the constrained blister is based on small strain approximation, linear elasticity, and zero-range surface force. The loading–pressurization and unloading–depressurization process are investigated, along with the critical events such as “pull-off” or spontaneous detachment of film from the substrate. The interrelations between the measurable quantities of blister height, applied pressure, and contact radius are derived. These functions will be useful to the experimentalists. Rigorous comparison with the existing theoretical models are made in Sec. 3.

## Theoretical Model

Figure 1 shows a linear elastic, thin, freestanding, planar membrane with radius, ** a**, thickness,

**, elastic modulus,**

*h***, and Poisson's ratio,**

*E***, clamped at its periphery. It is initially free of residual stress and possesses a negligible flexural rigidity. The deformation is therefore dominated by membrane stretching with negligible plate bending. A rigid plate is placed at a distance,**

*v***, from the substrate to restrict the blister height. A uniform pressure,**

*w*_{0}**, is applied to form a free bulging blister. Further increase in**

*p***causes the membrane to adhere to the constraining plate, making a contact circle with radius,**

*p***, contact angle,**

*c***θ**, and deformed annular profile of

**(**

*w***). A force**

*r***exerting on the plate maintains the desirable**

*F***. Upon depressurization, simultaneous measurements of**

*w*_{0}**and**

*p***allow the adhesion energy,**

*c***γ**, to be deduced. A thermodynamic energy balance based on small elastic strain approximation is used to derive the adhesion–detachment trajectory. Table 1 lists all the physical (bold) and normalized (plain) variables to be used hereafter.

### Mechanical Response With a Fixed Contact Circle.

**<**

*c***≤**

*r***) inclining at a small angle**

*a***ψ**≈ ∂

*w**/*∂

**to the horizon. Balancing vertical forces**

*r***σ**is the average membrane stress in the annulus. Rearrangement of Eq. (1) yields

**/(**

*F**π*

*a*^{2}

**) =**

*p**F/p*. Note that Φ ≤ 1 if delamination at all occurs. When the membrane is in full contact with the substrate with

*w*

_{0}= 0, Φ = 1. When

*θ*= 0, the load is fully supported by the applied pressure, and Eq. (1) requires Φ =

*c*

^{2}. At

*r = c,*the debonding angle at the contact edge becomes

*r**,*and width,

*d*

**, is stretched to an elongated length of**

*r**d*

**× sec**

*r***ψ**. For

**ψ**≈ 0, the linear engineering strain is approximated by

**ε**

_{linear}= sec

**ψ**− 1 ≈

**ψ**

^{2}/2 ≈ (

*d*

*w**/d*

**)**

*r*^{2}/2. The average linear strain over the annulus (

**<**

*c***<**

*r***) is given by**

*a***σ**=

**.**

*E***ε**, is given by

which is a cubic relation, *p* ∝ *w*_{0}^{3}, for a constant *c*.

### Loading by Pressurization.

The constraining plate is fixed at *w*_{0} = 1. A uniform pressure is applied to drive a blister. Figure 2 shows *c*(*p*), *θ*(*p*), and *F*(*p*), while Fig. 3 shows the corresponding changing blister profile as a function of applied pressure. Initial loading proceeds along path OA where the blister is yet to make contact with the plate and *c* = 0 always. An elastic strain gradually builds up upon loading. A free expanding blister is governed by *p* = 12*w*_{0}^{3} by substituting *c* = 0 and Φ = 0 in Eq. (8). At A, the applied pressure reaches a critical threshold *p*_{A} = 12, and the blister makes a point contact with the plate. Further increase in *p* proceeds along path AB, flattens the blister at the plate, expands the contact circle, and raises force pressing against the plate to ensure mechanical equilibrium. In the absence of interfacial adhesion (*γ* = 0), *F = p**c*^{2} or Φ = *c*^{2}, *θ* = 0, and the mechanical response *c*(*p*) follows Eq. (8) with *c* > 0. As the already strained membrane makes contact with the plate, the inner rim of the annulus moves into the contact, and a residual strain ε_{0} is locked up at the interface. Increase in *c* stretches the shrinking freestanding annulus further. The residual stress *σ*_{0}(*r*) is a monotonic increasing function of *r* and continuous at *r* = *c*, and reaches a maximum equal to the freestanding annular stress, *σ* = *σ*_{0} at *r* = *c*. No slippage is assumed at the membrane–plate interface. Depressurization deflates the blister along BAO reversibly, and the elastic energy due to the locked up strain returns to the freestanding annulus. No hysteresis is expected in the loading–unloading process. To account for interfacial adhesion (*γ* > 0), the intersurface attraction is taken to have zero range such that adhesion occurs only when the membrane makes intimate contact with the plate. As the contact expands, the contact angle remains zero with *θ* = 0, as the blister is supported by the applied pressure. The resultant force on the plate is found by a simple vertical force balance. The mechanical response *c*(*p*) traces the same path AB in Fig. 2 during loading as if no adhesion is present. Depressurization leads a different path, resulting in a hysteresis.

### Delamination by Depressurization.

**ε**is the strain in the freestanding annulus, and

**ε**the residual stress at the contact edge. The first term corresponds to the potential energy due to the applied pressure, and the second term is the elastic energy due to the annular strain subtracted by the locked-up strain. It is apparent that

_{0}**ε**has a negative contribution to

_{0}**, as the annular membrane moves into the contact circle locking up the stored elastic energy. Mechanical equilibrium is established when the energy balance**

*G***=**

*G***γ**is satisfied with

**γ**the interfacial adhesion energy. Equation (9) can be recast as

with $f1(c,\Phi )=1121/3\xd7(c2\u2212\Phi )22c2\xd7(1+c2\u22124\Phi +2\Phi 2\zeta \u22122)\u22121/3$

and $f2(c,\Phi )=14\xd7181/3\xd7(1+c2\u22124\Phi +2\Phi 2\zeta \u22122)1/3$

The constitutive relation, *p*(*c*, *w*_{0}), for a fixed *γ* can be found by solving Eq. (10) in a self-consistent manner. Upon pressurization along OAB, **θ** = 0, **ε** = **ε _{0}**, and

**= 0. The deformed membrane is here fully supported by**

*G**p*.

Depressurization along path BCD in Figs. 2 and 3 does not lead to an immediate shrinkage of the contact area but a reducing blister volume and an increasing *θ* in turn. At C, *θ*_{C} > 0 and 0 < *G*_{C} < *γ*. The coupled adhesion line force at the contact edge and the applied pressure now maintain the contact area (*c* = constant) at the expense of the collapsing blister profile. At *D*, *G* finally reaches *γ* to trigger delamination. Along DHP, the energy balance *G* = *γ* is satisfied, and the contact further diminishes. The residual stress distribution *σ*_{0}(*r*) remains intact within 0 ≤ *r* ≤ *c*, but is no longer continuous at the contact edge, *σ*_{0}(*c*) ≠ *σ*(*c*). At P, (∂*c/∂p*) → ∞ at *c** and *p**. Any further decrease in *p* (< *p**) deviates from the energy balance, leading to “pull-off” when the contact shrinks spontaneously from *c** to 0, and the membrane detaches from the plate. The loading (OAB) and unloading (BCDHP) processes give rise to a hysteresis when energy is dissipated due to the changing debonding angle. In the special case of loading–pressurization to an initial contact radius *c* < *c** along path OAM, subsequent depressurization traverses MN with a constant *c*, before “pull-off” occurs at N without any gradual shrinkage of contact. Loading–unloading hysteresis now follows OAMN.

Figure 4 shows *c*(*p*) for a range of adhesion energy and *w*_{0} = 1. Delamination path DHP is identical to the same path in Figs. 2 and 3. In weak interfaces with *γ* < *γ*^{† }= 9.98, “pull-off” occurs at positive pressure or *p** > 0. In case of *γ*^{†}, “pull-off” occurs at P^{†} with *p** = 0 and *c** = 0.2060. In strong interfaces with *γ* > *γ*^{†}, suction with *p** < 0 is necessary to detach the membrane. The “pull-off” locus, *c**(*p**), along APP^{†}P′, is where all *c*(*p*) curves terminate. Figures 5(a) and 5(b) show the *p**(*γ*) and *c**(*γ*) for a range of *w*_{0}. Strong interface requires smaller *p** to detach the membrane at larger *c**. Should the plate be placed further away from the membrane, a larger *p** is expected. Figure 5(c) shows *c**(*p**) for a range of *w*_{0}. For any *w*_{0}, *p** = 0 always requires *c** = 0.2060.

## Discussion

It is worthwhile to compare the present model with the related literature. There are a number of similarities with the Johnson–Kendall–Roberts (JKR) model [15] for adhesion of elastic solid spheres with radius, *R*. The free membrane blister prior to contacting the constraining plate resembles the geometry of the JKR spheres, and both models assume a planar contact area. According to Maugis' interpretation [16,17], a change in the external load, *P*, from *P*_{1} to *P*_{2} does not instantly lead to the equilibrium configuration along the energy balance (*G* = *γ*), but a more tortuous path. Increasing *P* raises the approach distance, *δ*, yet *a* remains constant. The behavior then strictly follows the Hertz theory with *δ* = *a*^{2}/*R* as if no adhesion is present. Once final load *P*_{2} is reached and held constant, *δ* and *a* will eventually move to the equilibrium configuration. On the other hand, decrease in load reduces *δ* but leaves *a* unchanged. With *P*_{2} held constant, (*δ*, *a*) move to equilibrium. Maugis' model thus leads to a loading–unloading hysteresis. In the present work of membrane adhesion, the predicted behavior is quite consistent with the JKR model. The loading–pressurization expands the contact circle but *θ* = 0 as if *γ* = 0, in reminiscent of the Hertz contact during the JKR-loading. Upon unloading–depressurization, *θ* increases but *a* remains constant until the energy balance is satisfied, in parallel to the JKR-unloading. Other similarities include the “pull-off” instability and the nonzero critical contact radius. A distinct difference is that the “pull-off” force depends only on the sphere radius in JKR, but both radius and film thickness in the present model.

In our previous work [10], a circular diaphragm clamped at the periphery adheres to the planar surface of a cylindrical punch. The cross section is essentially the same as Fig. 1, though the confined delamination is driven by a tensile force on the punch rather than a uniform pressure. “Pull-off” is found to be *c** = 0.1945, which is consistent with *c** = 0.2060 in the present work (c.f. Fig. 5(c)). In the punch model, the membrane is initially in full contact with the punch prior to delamination and is therefore free of any residual stress within the contact circle. The present model requires the membrane to be strained within the contact circle.

*w/∂r*)

_{r}_{<}

*= 0, and all elastic energy is stored in the freestanding annulus in both the loading and unloading processes. Williams [8] determined the strain energy release rate due to residual stress within the contact area in the absence of adhesion and discussed the necessity of zero contact angle during loading. The present work accounts for coupled residual stress and adhesion in the contact area. It is possible to modify our model to accommodate for such approximation that the membrane stress is taken to be uniform and continuous at*

_{c}*r*=

*c*. Substituting

**ε = ε**in Eq. (9), the strain energy release rate becomes

_{0}**=**

*G***σ**(1 − cos

*h***θ**) for a finite angle

**θ**, which is essentially the Young–Dupré equation [18]. Equation (10) becomes

Figure 2 shows the loading–unloading process using Eq. (11) as a thin dark dashed curve which almost coincides with the exact calculation for *w*_{0} = 1 and *γ* = 5. Slightly larger *p** and *c** are expected at “pull-off.” It is also worth to mention that models by Plaut et al. [9] and Xu assume loading and unloading to follow the energy curve and does not show the loading–unloading hysteresis as in the present model.

Long et al. [13,14] modeled a hyperelastic blister under large pressure. The membrane is blown into truncated sphere with a large meridian angle as large as 90 deg that makes adhesion contact with the plate. The loading–pressurization assumes *θ* = 0, and unloading–depressurization shows an initial constant contact radius and an increasing *θ*. The 3D stresses and strains in cylindrical coordinates are computed numerically. The “pull-off” parameters are also deduced based an energy balance. The general features and trends of the interrelationships between measurable quantities are consistent with the present model, e.g., monotonic decreasing functions of *p**(*γ*) and *c**(*γ*). It is, however, difficult to compare the two models, since two different normalization schemes are used. The present model presents a limiting extreme of the large deformation model and provides an analytical solution to the experimentalists working with linear elastic membranes under small strain.

As a final remark, the assumption that plate bending is ignored in our theoretical model deserves further discussion. In linear elasticity, bending dominates at small deformation when the blister height is comparable to membrane thickness, or *w*_{0} ∼ ** h**. However, many practical membranes (e.g., biomembranes and lipid bilayers in cells and liposomes) do not strictly follow this rule because of the high degree of flexibility. Such films virtually conform to the substrate topology under membrane stretching in a practical sense. To facilitate description of complex biomembrane without involved solid mechanics, a representative pseudo elastic modulus, or areal expansion modulus, is adopted such that plate bending can be ignored at all times. A strictly plate bending model is mathematically involved. Moreover, practical experiments always operate in a deformation regime governed by membrane stretching. A comprehensive model involving mixed bending-stretching leads to an analytical solution, if at all possible, which will be challenging for experimentalists to adopt for data analysis.

## Conclusion

A linear elastic model is built for a constrained blister test where the clamped membrane adheres, delaminates, and detaches from the constraining plate. The adhesion–delamination process and the relations between the plate-membrane gap, applied pressure, blister height, contact radius, and adhesion energy, as well as the “pull-off” thresholds, are derived.

## Acknowledgment

This work was supported by National Science Foundation Grant No. CMMI # 1232046. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.