A very common method used to characterize adhesive interactions in soft materials is that of Johnson, Kendall, and Roberts (JKR).1 The method consists of bringing a compliant material, typically a hemisphere with a radius R, into contact with a rigid surface, as represented in Figure 1(a). If the materials adhere to one another, the radius of the circular contact region is larger than it would be in the absence of adhesion. The specific value obtained for the contact radius is determined by a balance of adhesive forces and elastic restoring forces that oppose this increase in contact area.2 Sensitivity to small adhesive forces is maximized for highly compliant systems, for which the elastic restoring forces are not too large. Fundamental limitations arise from the geometry itself, however, since the volume of the elastically deformed region scales as the cube of the contact radius.

One way to enhance the sensitivity to adhesion is to use membrane geometry, where a much smaller volume of the material is deformed.3–6 In our approach to this problem, we use the membrane geometry illustrated in Figure 1(b). Pressure is used to inflate a circular membrane of radius R_{m}, so that it comes into contact with a rigid surface displaced from the initial location of the membrane by a distance δ_{m}. Adhesion between the membrane and the substrate causes the membrane to spread over the surface, increasing the contact radius, a. Theoretical background used to interpret these adhesion experiments is described in Sections Membrane Inflation—Noncontact Case, Membrane Contact: Asymmetric Laplace Equation, and Analytic Approximation for Small Contact Angles. We have demonstrated the use of this technique with several different membranes, with different values of membrane stiffness. We have also utilized a quartz crystal microbalance as the substrate, which provides additional information about the nature of the contact.7 Details needed to interpret information about the viscoelastic properties of the membrane and on the nature of the contact between the quartz/membrane provided by the quartz crystal resonator have been provided elsewhere,7 and are summarized briefly in Quartz Crystal Resonator Section. To illustrate the application of the membrane contact technique, we describe the behavior of three model systems: a low modulus silicone gel, an acrylic thermoplastic elastomer, and a styrenic thermoplastic elastomer.

BACKGROUND

Membrane Inflation—Noncontact Case

The undeformed membrane is circular, with a radius of R_{m} and a thickness of h. Deformation of the membrane is assumed to be dominated by stretching (as opposed to bending). We assume that the deformed membrane that is represented in Figure 2(a) forms a spherical cap with a radius of curvature of R. To simplify some of the expressions describing the membrane deformation, we define a normalized displacement , and express areas and volumes in terms of A_{0}, the area of the undeformed membrane

(1)

The following expression for the surface area, A_{m}, enclosed volume, V_{m}, and radius of curvature are exact for a spherical cap.

(2)

(3)

(4)

The state of biaxial strain in the curved sample, which we denote as ε_{m}, is given by the following expression:

(5)

The following standard relationship is used to relate the pressure difference across the membrane (P) to the biaxial membrane tension, T

(6)

The membrane tension is related to the overall membrane deformation energy through the following expression:

(7)

where T_{0} is the pretension in the undeformed membrane (typically arising from the surface tension associated with the membrane surfaces), and U_{e} is the elastic component of the membrane deformation energy. The overall membrane tension is obtained from the differentiation of eq 7

(8)

A variety of strain energy functions can be used to describe the elastic behavior of the membrane.8^{–}11 As one example, we use the simplest result from rubber elasticity theory for the affine deformation of a crosslinked rubber, a model that is often referred to as a “Neo-Hookean” model. The elastic strain energy function in this case is given by the following expression:

(9)

where E is Young's modulus of the membrane and λ_{θ} and λ_{h} are the respective extension ratios describing the deformation in the circumferential and thickness directions. From the symmetry of the problem, we have , and from incompressibility (λ_{r}λ_{θ}λ_{h} = 1) we have . We obtain the following for the membrane tension from the combination of eqs 8 and 9:

(10)

Equation 10 is a general result for the membrane tension of a Neo-Hookean material in a state of equibiaxial tension. For the specific geometry of a flat circular membrane that is inflated into a spherical cap, we use eq 5 for A_{0}/A_{m} to obtain the following:

(11)

The corresponding pressure difference across the membrane is obtained by substituting this expression for the membrane tension into eq 6

(12)

Equation 12 is the relevant constitutive equation for a Neo-Hookean material in our inflation geometry, since it relates the two experimentally measured quantities (P and ) to one another.

For small strains (λ_{r} close to 1), the membrane tension can be expressed in terms of the area modulus, K_{a}:

(13)

From linear elasticity with a plane stress boundary condition, K_{a} is given by the following expression:

(14)

where ν is Poisson's ratio for the membrane. For an incompressible material, K_{a} = Eh, and the elastic terms in eqs 11 and 13 differ by more than 10% when exceeds 0.23. In our experiments, at the point where the membrane comes into contact with the substrate is equal to 0.29, where nonlinear terms in the expression for the membrane elasticity represent a 17% correction to the linear term.

The quantity is the value of the normalized displacement for which the elastic contribution to the tension is equal to the pretension. For we can ignore the pretension, and include only the elastic term. Elastic effects can be neglected in the opposite extreme, where .

Membrane Contact: Asymmetric Laplace Equation

The method used in this work is an adaptation of the method presented by Timoshenko and Woinowsky-Krieger.12 It is based on the axisymmetric Laplace equation obtained by equating the external pressure with the Laplace pressure associated with the membrane curvature. Equivalent approaches include the work of Rooks et al., who studied meniscus formation at wire/substrate interfaces in the presence of liquid solder,13 and the very recent work of Heinrich and Ounkomol, who studied the force-deflection relationship for pipette-aspirated vesicles used as force transducers in biophysics experiments.14 The shape of the interface is described by the relationship between the local height, z, of the membrane as a function of the radial position, r. The origin of the coordinate system used to define z and r is located at the substrate surface, as illustrated in Figure 2(b). The local slope of the interface is specified by the angle ϕ, with . The boundary conditions for the shape of this interface are as follows:

(15)

Here θ is the contact angle that the membrane makes with the substrate, and δ_{m} is the separation between the substrate and the tube to which the membrane is attached.

The total pressure difference across the membrane has the following form:

(16)

where P is the difference between the pressure inside and outside the membrane at the contact plane (z = 0), is the difference in densities of the media on either side of the membrane, g is the gravitational acceleration, and and are the principal radii of curvature, measured from the inside of the membrane. The term involving accounts for the hydrostatic pressure associated with the presence of fluids with density on either side of the membrane. Our sign convention for is that this quantity is positive when the external fluid has a higher density than the internal fluid. This is the case in all of our contact experiments, where, the internal fluid is air and the external fluid is water.

The expression for the pressure can be written in a more useful form by using the expressions for the radii of curvature.13R_{1}, which corresponds to the radius of curvature in the r–z plane, is given by the following expression:

(17)

R_{2} corresponds to the radius of curvature in the plane perpendicular the r–z plane, and is given as follows:

(18)

It is also useful to define the contour, s, which describes the shape of the membrane. The shape of the membrane is most conveniently described by the values of r and z at each value of s. Incremental changes in s, z, and r are related to one another through ϕ, which also varies along the membrane

(19)

Equations 16–19 can be combined to give the following expressions for the evolution of ϕ along the contour.

(20)

The numerical procedure for obtaining the complete membrane profile proceeds as follows. We begin with a particular combination of a, P, and T, and use eq 20 to calculate the evolution of ϕ with s. The initial conditions for s = 0 are that ϕ is equal to an assumed contact angle, θ, z is equal to 0, and r is equal to a. At successive iterations, s is increased by a small amount, and eq 20 is used to recalculate ϕ for this new value of s. Equation 19 is then used to obtain the corresponding values of r and z. If the correct contact angle is not used for the initial condition, there will be no solution for any values of s that are consistent with the boundary condition at the other side of the membrane, i.e., z = δ_{m} at r = R_{m}. The correct contact angle is obtained by iterating θ until this boundary condition is met. Conversely, the value of the membrane tension can be iterated at a fixed value of the contact angle. In the example described below in Quartz Crystal Resonator Section, for example, a nonadhesive membrane with θ = 0 is used, and the membrane tension is iterated until the membrane boundary conditions are met.

Once the membrane profile is obtained, the total membrane area, A_{m}, and the enclosed membrane volume, V_{m}, are calculated from the following expressions:

(21)

(22)

The relationship between the membrane tension and the membrane area places an additional constraint on the membrane tension, T. If the constitutive relationship between T and A_{m} is known, a self consistent solution to the problem is obtained for single, well-defined value of the contact angle. The energy release rate is then obtained from the following general expression:

(23)

An implicit assumption in eq 23 is that the tension in the contacting and noncontacting regions of the membrane is identical, i.e., the membrane contact is assumed to be frictionless. If the tension in the contacting and noncontacting portions of the membrane are not equal, additional energy released during peeling needs to be accounted for in the energy balance. The limiting case where T = 0 in the contacting region, has been described by Kendall.15 The net result is the inclusion of an additional elastic term in the expression for the energy release rate.

Analytic Approximation for Small Contact Angles

The expressions from the previous section can be used to obtain the energy release rate from the measured pressure and contact radius, for given values of the geometrical parameters. If the magnitude of dz/dr is small, the equations for the membrane shape can be linearized, enabling us to make the connection to related small-strain treatments of the membrane contact problem.6, 16 If the hydrostatic pressure term involving the density difference between the two phases is ignored, the membrane and angle of contact, θ, are given by the following expressions:

(24)

(25)

Additionally, when using the small angle approximation for θ, eq 23 reduces to the following:

(26)

Two additional geometrical properties are also relevant to the overall nature of the test. The first of these is the total area of the membrane, which in general terms is given by eq 21. For small values of dz/dr, it is helpful to write the expression for the membrane area and enclosed volume in the following forms:

(27)

(28)

The area strain for the contact case, which we refer to as ε_{m}, is given by − 1.

To make the connection to related theories of membrane contact, it is useful to consider the expressions for , θ, and ε_{m} that are obtained in two experimentally relevant situations. The first of these is the case where the pressure difference is reduced to zero after the membranes are brought into contact with one another. The second of these is the situation where the pressure difference remains fixed at the value of the contact pressure required to bring the membrane into contact with the surface. The radius, R_{m}, of the undeformed membrane defines the length scale of the problem. We use this length to define the dimensionless lengths, , and , and the dimensionless membrane area, . A similar dimensionless volume, , is also defined

(29)

Case I: P = Contact Pressure

The pressure required to bring the membrane into contact with the surface is the pressure required to give δ = δ_{m}, which according to eq 6 is equal to , for the small values of that are consistent with our assumption of small dz/dr. The following expression is obtained for the profile in this case

(30)

This profile leads directly to the following expressions for θ and :

(31)

(32)

Because the elastic component of the membrane tension is determined by the strain, we use the membrane strain, ε_{m}, equal to – 1, as a measure of the membrane area. The following expression is obtained from the combination of eqs. 27 and 30:

(33)

Finally, we obtain the following expression for from the combination of eqs 28 and 30:

(34)

For small values of , we can write a in terms of to obtain the following relationship between the membrane contact radius and the energy release rate:

(35)

This expression highlights the sensitivity of our experimental geometry to relatively weak adhesive interactions. With δ_{m} = 1 mm, R_{m} = 3.5 mm, = 10^{−4} J/m^{2} and T = 0.1 N/m, eq 35 gives a = 0.27 mm, a value which is quite easily measurable, in spite of this very low value of the adhesion energy.

Case II: P = 0

If the pressure is reduced to zero after the contact is made, eq 24 reduces to the following:

(36)

The following expressions are obtained for the contact angle, energy release rate, membrane strain, and enclosed volume:

(37)

(38)

(39)

(40)

An expression equivalent to eq 38 has been obtained by Wan and Kogut, using and energy approach similar to that outlined in the appendix.6

Quartz Crystal Resonator

While not a major focus of this work, the ability to use quartz crystal resonators is an important motivating factor for this experimental geometry. The complexity of the experimental arrangement, which includes the crystal mount and the associated electronics, makes it difficult to directly image the membrane contact angle by imaging the membrane from the side [corresponding to the schematic illustration in Fig. 2(b)]. The membrane shape analysis provides a means for obtaining the contact angle from images similar to those shown in Figure 3(b), while still obtaining useful information from the behavior of the quartz crystal resonator. Because of the piezoelectric nature of quartz, an alternating voltage applied through electrodes on each side of the disk generates an acoustic shear wave. The electrical response of the crystal circuit at frequencies close to the crystal resonance is strongly affected by the nature of the shear wave propagation into the contacting material.17 Our apparatus is equipped with a network analyzer that gives the admittance spectrum of the system in the vicinity of one or more of the mechanical resonant frequencies of the quartz crystal. A Lorentzian fit to the real component of the admittance gives the complex resonant frequency ( ) from which two quantities can be extracted. The real component of is the resonant frequency at the nth harmonic (), and the imaginary component of is the bandwidth (Γ_{n}), which is a measure of the energy dissipation. The data reported in this article was obtained at the third harmonic (n = 3, f_{n} = 15 MHz). When the viscoelastic membrane is inflated into contact with the quartz crystal, the resonant frequency decreases and the energy of dissipation increases. For a uniform contact across the entire surface of the quartz, the shift in the complex frequency () is related to the complex acoustic impedance () of the membrane:

(41)

where is the fundamental frequency and is the acoustic impedance of quartz . Equation 41 is based on the assumption that the crystal is uniformly loaded over the active area, A_{0} of the crystal, which is defined by the electrode geometry. However, in our inflation experiment the membrane does not make contact with the entire crystal. Instead, contact is made over an area A which represents a fraction of A_{0}. The sensitivity of the QCM is greatest at the center, and decays in a roughly Gaussian fashion towards the perimeter of the active area.18 Provided that the contact is concentric with the circular electrode, the relationship between and the measured coverage-dependent frequency shift, , is given as follows:19

(42)

In the present article, we focus primarily on the elastic analysis of the membrane geometry. Our discussion of the quartz crystal response is restricted to qualitative differences between the behaviors of the membranes. These values are presented in terms of the initial slope of frequency response, when plotted as a function of the normalized contact area. These values are related to the full contact frequency shift, , and β through the following expression:

(43)

The values of β that we obtain for membrane contact in water are typically close to 2, so the slopes reported in Table 1 are expected to be about four times larger than the full contact values of the corresponding quantities. Details of the analysis of the frequency response of the quartz crystal as applied to these membrane systems has been described in a recent publication.7 For a more general review of quartz crystal resonators, the review article by Johannsmann is an excellent resource.17

Table 1. Characteristics of the Different Membranes Used in the Experiments

H (mm)

E (Pa)

a

It was difficult to obtain reliable data for the dissipation of the dissipation of the acrylic membranes.

Figure 3(a) shows a schematic diagram of the implementation of the pressure measurement system with a quartz crystal resonator. A syringe pump (NE-1000) is used to control the inflation of the membrane. A system of differential pressure transducers (MKS Baratron) is used to record the pressure of the membrane during inflation. The maximum pressure for a given sensor varies from 100 Pa to 0.1 MPa, depending on the specific pressure sensor that is used. When the inflated membrane comes into contact with the electrode surface of the quartz crystal, a circular contact is formed that is imaged through a low-power optical microscope. A typical image of the membrane contact is represented in Figure 3(b). These contact areas are quantified using an image analysis program (ImageJ, National Institutes of Health). A LABVIEW programming interface is used to control the experiment and acquire the data.

In a typical experiment, the membrane is floated onto water before being transferred to the end of a glass, cylindrical expansion chamber. The membrane is then inflated into contact with the quartz crystal at an inflation rate of 2 mL/h. The images and the data from the crystal resonator both provide a signature of membrane/substrate contact, as illustrated in Figure 4. Inflation continues until a maximum specified pressure, and the pump is stopped. After several minutes the pump is used to withdraw air from the expansion chamber, decreasing the pressure and reducing the membrane/substrate contact area.

Sample Preparation

Three model membranes were prepared by spin coating different toluene solutions onto a polished salt crystal: a highly concentrated solution of a crosslinkable silicone (Q3-6575, Dow Corning), a 10 wt % solution of an acrylic triblock copolymer with poly(methyl methacrylate) end blocks and a poly(n-butyl acrylate) midblock (PMMA/PnBA/PMMA), and a 10 wt % solution of a triblock copolymer with polystyrene end blocks and an ethylene/butene midblock (SEBS). Both triblock copolymers had a total molecular weight of about 170,000 g/mol and an end-block weight fraction (combined for both endblocks) of 0.3. After spin casting onto the salt crystal, each membrane was floated onto water and then manually transferred to the end of the cylindrical expansion chamber. Noncontact inflation experiments were performed and the relationship between the inflation pressure and the normalized displacement, , was measured for each of these membranes. Because these noncontact experiments did not require the presence of the quartz crystal, a side-view arrangement for direct visualization of the membrane displacement was easy to implement.

For the contact experiments three different types of contacts were utilized, in order to modify the strength of the adhesive interaction in water (Fig. 5). In the first case, the membrane was brought into contact with the bare gold electrode surface of the quartz crystal [Fig. 5(a)]. In the second case, a poly(ethylene glycol) (PEG) layer with a molecular weight 5700 g/mole and a dry thickness, z*, of 8 nm was grafted onto the gold surface to minimize the adhesion [Fig. 5(b)]. This brush layer was obtained by spin coating a thin film of thiol-terminated PEG onto the electrode surface, annealing for 2 h at 75 °C, and rinsing the excess, ungrafted PEG from the surface. In the third case, PEG brushes were attached to both membrane and gold surfaces [Fig. 5(c)]. The gold surface was modified with thiol-terminated PEG as described earlier, using a PEG molecule with a molecular weight of 2000 g/mole. The membrane was modified by spreading a monolayer of polystyrene/PEG diblock copolymer onto a water surface from a dilute solution in chloroform, and transferring this layer onto the surface of the membrane. The diblock copolymer had block lengths of 3800 g/mole for the polystyrene block and 4800 g/mole for the PEG block. The dry thickness of the brushes were 2.5 nm for the brush on the gold surface, and 1.0 nm for the brush on the membrane surface. The diblock copolymer and the shorter PEG-thiol were obtained from Polymer Source, and the longer PEG-thiol was obtained from Nektar Therapeutics.

RESULTS AND DISCUSSION

The three different types of membrane that we utilize in these experiments correspond to three elastic regimes. We use crosslinked silicone gels, which have a low elastic modulus, and a correspondingly low value of the dimensionless quantity Eh/T_{0}. The other two systems are both self-assembled triblock copolymer systems that can easily be produced in membrane form by spin coating onto sodium chloride crystals, floating onto a water bath and then attaching them to the end of the inflation tube. Results for each of the three membrane systems are described below.

Silicone Membranes

The silicone membranes have the lowest elastic moduli of the three different membrane systems that were investigated. The aim here was to develop a system for which the membrane tension was dominated by the pretension in the membrane. Because this pretension results primarily from surface energetics, it is a more controlled and well-known quantity than the elastic contribution to the membrane tension. As illustrated in the following two subsections, the elastic contribution to the membrane tension can be highly dependent on the deformation history of the membrane. This hysteresis in the membrane tension is problematic if the aim of the experiment is to extract the adhesion energy of the membrane by application of eq 23. While relatively small values of the dimensionless quantity Eh/T_{0} can be obtained by use of the silicone membranes, use of these membranes is problematic for other reasons. First, it is very difficult to obtain the idealized geometries illustrated in Figure 1(a) with these membranes. These very compliant membranes tended to adhere to the interior of the glass inflation chamber. The effect of this adhesion on the pressure/displacement relationship for the noncontact inflation experiments is illustrated in Figure 6. At the largest values of δ/R_{m}, the membrane is no longer adhering to the interior of the inflation chamber, and the idealized geometry of Figure 2(a) is realized. By fitting the measured inflation curve to the prediction from eq 12 in this highly inflated regime, we obtain Eh = 0.14 N/m. We have assumed here that the membrane pretension, T_{0}, for this noncontact experiment with air on either side of the membrane, is equal to 0.044 N/m, which is twice the surface energy of poly(dimethyl siloxane).20 The thickness of the silicone membrane was not measured precisely but is estimated to be about 10 μm, giving E ≈ 1.4 × 10^{4} Pa for this material.

Figure 7 shows the contact pressure for the case where the silicone membrane was brought into contact with a brush-coated electrode surface [Fig. 5(b)]. A pressure of 60 Pa was needed to bring the membrane into contact with the electrode surface, at which point A/A_{0} = 0. Here A_{0} is the active area of the quartz surface, defined by the area of the circular gold electrode on the bottom (noncontacting) surface of the quartz crystal. This electrode has a radius of 3.175 mm, which is only slightly smaller than the membrane radius, R_{m}. The active electrode area is the appropriate normalization for the response of the quartz crystal resonator, and is used as our area normalization throughout this article. The silicone membranes are actually relatively poorly crosslinked gels with a high dissipation at the 15 MHz resonant frequency of the crystal. Useful information is not obtained from the quartz crystal response for this reason, a fact that further limits the utility of the silicone membranes. When the dissipation is not as high, as is the case with the styrenic membranes described in Styrenic Triblock Copolymer Membranes Section, the QCM provides a very accurate measure of the thickness of the hydrated brush layer between the membrane and the gold electrode surface.7

Acrylic Triblock Copolymer Membranes

The first of the two triblock copolymer membranes utilized in our experiments were based on acrylic triblock copolymers with poly(methyl methacrylate) endblocks and a poly(n-butyl acrylate) midblock. The acrylic copolymers are chosen because they are excellent models for commercially relevant pressure sensitive adhesive (PSA) systems.21 The membrane experiment provides a convenient means for assessing PSA performance in aqueous environments. The inflation curve (in air) and the contact curve (in water) for these membranes are shown in Figures 8 and 9, respectively. These figures correspond to the silicone curves shown in Figures 6 and 7. From the fit to the noncontact data, we obtain a value of 1.65 N/m for Eh, using T_{0} = 0.07 N/m (approximately twice the surface energy for the acrylic polymer). From the thickness of 1.25 μm we obtain E = 1.3 × 10^{6} Pa.

Substantial hysteresis is observed for the contact and noncontact experiments. The hysteresis in the noncontact experiment (Fig. 8) is a measure of the viscoelastic relaxation in the membrane itself. The origins of the hysteresis in the data from Figure 9 are a bit more complicated. If the membrane does not adhere to the brush surface, then the hysteresis can be attributed exclusively to viscoelastic relaxation in the membrane. The presence of adhesive interactions will increase the observed hysteresis, however, since the energy release rate is a function of the crack velocity, and is larger for an advancing crack (decreasing contact area) than for a receding crack (increasing contact area).2 Because we obtain much better reproducibility for the styrenic triblock copolymer membranes than we do for the acrylic membranes, our most detailed analysis is applied to these styrenic systems, as described in the following subsection.

Styrenic Triblock Copolymer Membranes

A series of noncontact inflation curves for the styrenic triblock copolymer system is shown in Figure 10. These curves are equivalent to the silicone curve shown in Figure 6, and to the acrylic curve shown in Figure 8. The data in Figure 10 were generated by inflating the membrane to δ/R_{m} = 0.2, deflating by decreasing the pressure to zero, reinflating to δ/R_{m} = 0.28, deflating a second time, reinflating to δ/R_{m} = 0.34, and deflating a third time. The data shown in Figure 10 correspond to the three subsequent inflation steps, in addition to the final deflation step. Hysteresis between inflation and deflation is clearly observed, but the subsequent inflation steps trace out a single curve. These data indicate that for the strain regime corresponding to these experiments, there is no permanent creep induced as a result of the inflation. In contrast to the situation for the silicone and arcylic membranes, we were not able to fit the inflation data to eq 12 with values of T_{0} that can be attributed entirely to surface energetics. The solid line in Figure 10 has T_{0} = 0.3 N/m and Eh = 6 N/m. The membrane has a thickness of 1 μm, so we obtain E = 6 × 10^{6} Pa.

The contact pressure for a styrenic membrane in contact with a brush-coated electrode surface is shown in Figure 11(a), and the corresponding shift in the resonant frequency of the quartz crystal circuit is plotted in Figure 11(b). The thickness of this membrane is 1.25 μm, which is somewhat larger than the membrane used to generate the data in Figure 10. Contact with the styrenic membranes produces a large shift in the resonance frequency of the crystal circuit, with correspondingly low dissipation. These membranes are ideally suited for characterization of the brush structure for this reason.7 The effect of the membrane contact on the resonant frequency shift is quantified by the slope of the Δf_{A} versus A/A_{0} at low values of A/A_{0}, represented by the dashed line in Figure 11(b). A similar procedure can be used to quantify the relationship between the contact area and the dissipation, ΔΓ_{A}. These slopes are listed for the three membranes in Table 1, along with the thicknesses and elastic moduli for the three different membranes.

Attachment of the polymer brush to the gold surface results in a surface that is minimally adhesive to the elastomeric membrane. To more completely eliminate adhesive interactions, a second brush can be added to the membrane surface as well, resulting in the contact geometry illustrated in Figure 5(c). The relationship between the pressure and the contact area for this geometry is shown in Figure 12(a). Two successive inflation and deflation curves are shown, giving nearly identical results and illustrating the elastic recovery of the membranes between inflation cycles. By assuming that the membrane contact angle in these experiments is zero, we can obtain the membrane tension and corresponding membrane profiles from the procedure outlined in Background Section. Two representative profiles, corresponding to extreme values for the contact area, are plotted in Figure 12(b). From these profiles we obtain the membrane strain, ε_{m}. The relationship between the membrane tension and membrane strain that we obtain in this way is shown in Figure 13. The strain at the initial contact point is equal to (δ_{m}/R_{m})^{2}, which for our values of δ_{m} (1 mm) and R_{m} (3.5 mm) is equal to 0.082. Lower strains are accessed by the non-contact experiment, and the corresponding data shown in Figure 10. The values of the tension corresponding to the values of Eh and T_{0} obtained from this noncontact experiment are plotted as the solid line in Figure 13. Together, the contact and noncontact experiments give a complete picture of the constitutive relationship for the membrane strains ranging from zero to 0.35.

Finally, we note that the membrane technique can be used to characterize relatively large adhesive forces as well. If the membrane is brought into contact with a bare surface of the gold electrode, we obtain the results shown in Figure 14. In this case a substantial negative pressure (1500 Pa) is needed in order to detach the membrane from the electrode surface. To quantify the observed adhesion, it would be beneficial to work with membranes that have a more quantifiable membrane tension. A positive feature of the styrenic membranes is that they are completely elastic, with a very reproducible relationship between the tension and the membrane strain. The drawback of these membranes is that the hysteresis between inflation and deflation stages makes it difficult to quantify the membrane tension, in situations where the contact angle is not already known.

CONCLUSIONS

The focus of this article has been on the description of a technique for quantifying the adhesion of elastomeric membranes to flat surfaces. The analysis is based on the numerical solution of the axisymmetric Laplace equation. In some limiting cases, useful analytic results are also obtained. In general, there are two unknowns in the problem: the membrane contact angle and the membrane tension. If one of these quantities is known, the analysis enables the other quantity to be determined from the measured parameters. The most quantitative application of this technique was in the generation of the membrane tension data shown in Figure 13. These data were obtained by designing an experiment where the contact angle between the membrane and the substrate was known to be zero. These results highlight the favorable features of the SEBS membranes, which include excellent mechanical stability, ease of formation and surface modification, minimal damping of the quartz crystal resonance, and complete elastic recovery.

Despite these advantages, the substantial hysteresis between loading and unloading cycles makes it very difficult to determine contact angles, and hence the adhesion energy, because of uncertainties in the actual value of the membrane tension. For model studies designed to investigate the molecular origins of adhesion in aqueous systems, membranes that lack the toughness of SEBS, but which are more ideally elastic, are expected to be better candidates. The analysis described in this paper provides the basis for interpreting these types of experiments.

Acknowledgements

This work was supported by grants from the Human Frontier Science Program, NIH (R01 DE14193), and NSF (DMR-0525645). We acknowledge a variety of helpful comments from Dr. C.F. Creton.

APPENDIX: ALTERNATIVE DERIVATION OF THE ENERGY RELEASE RATE

An energy balance can be used to determine the pressure dependence of the energy release rate, which is a measure of the driving force for detachment of the membrane from the surface of the material. The approach is conceptually similar to the original analysis of adhesive contact of spheres, dating back to the classic work of Johnson et al.1, and applied more recently to adhesive membranes by Shanahan22 and Wan and Kogut.6 The relevant quantities in our case is the membrane deformation energy U, and the work done by the pressure difference in changing the enclosed volume of the membrane. The energy release rate can be written in the following form:

(A1)

where A_{c} is the membrane/substrate contact area, assumed to remain circular to preserve the axial symmetry of the problem. For a fixed value of the pressure, and with the relationship between U and T specified in eq 8, is given by the following relationship:(A2)

(A2)

In terms of the variables defined in Analytic Approximation for Small Contact Angles Section, we can write the expression for in the following form: (A3)

(A3)

The expressions for the energy release rate in the previous sections are consistent with eq. A3. For example, if P is equal to the contact pressure, eq 33 is used for ε_{m}, and eq 34 is used for V_{m}, then application of eq A3 results in the expression for the energy release rate given by eq 32. Similarly, taking P = 0 and using eq 40 for V_{m} results in eq 38 for .