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Langmuir 1999, 15, 4966-4974
Adhesive and Elastic Properties of Thin Gel Layers Cynthia M. Flanigan and Kenneth R. Shull* Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208 Received August 18, 1998. In Final Form: April 26, 1999 We employ linear elastic fracture mechanics to describe the adhesive and frictional properties of thin, elastic gel layers against a glass, hemispherical indenter. A triblock copolymer with poly (methyl methacrylate) endblocks and a poly (n-butyl acrylate) midblock is diluted with a selective solvent for the center block in order to form a low modulus, thermally reversible gel that behaves as an elastic solid at room temperature. Through simultaneous measurements of the circular contact area, normal displacement, and applied load, we show that the fracture mechanics expressions provide an accurate means to describe the adhesive behavior of the highly compliant polymer gels. By modifying the surface properties of the indenter, we show that the relationship between load and displacement is sensitive to the response of shear forces at the gel/indenter interface, whereas the relationship between the contact area and displacement is unaffected by this response. In addition, we demonstrate the utility of our methodology for quantifying adhesion in situations where the applied load cannot be measured directly.
Introduction A variety of tests, including peel-,1-3 blister-,4,5 probe tack-6,7 and JKR (Johnson, Kendall, and Roberts)-based axisymmetric adhesion experiments,8-13 have been developed to study adhesion in elastomers over the past few decades. Our work utilizes the axisymmetric test geometry shown in Figure 1 to investigate the adhesive and frictional response of films with thicknesses in the range of 100 µm. As progress develops in expanding the use of these types of films in commercial applications, we need to develop a methodology to quantify both the mechanical and adhesive nature of these confined layers. The traditional forms of testing adhesives are not necessarily well suited for analyzing the properties of thin layers. Since these films cannot be treated as a semiinfinite elastic half-space, many of the assumptions of previous fracture mechanics treatments cannot be utilized. Specifically, several researchers have shown that the contact radius relative to the thickness of the film is an important parameter characterizing the significance of confinement effects on the elastic properties of the film.14-18 These observations lead us to two basic questions. First, how do we obtain quantitative information about the elastic and (1) Gent, A. N.; Kinloch, A. J. J. Polym. Sci. 1971, 9, 659-668. (2) Gent, A. N. Rubber Chem. Technol. 1982, 55, 525-535. (3) Gent, A. N.; Hamed, G. R. Rubber Chem. Technol. 1982, 55, 483493. (4) Chu, Y. Z.; Durning, C. J. J. Appl. Polym. Sci. 1992, 45, 11511164. (5) Gent, A.; Lewandowski, L. J. Appl. Polym. Sci. 1987, 33, 15671577. (6) Barquins, M.; Maugis, D. J. Adhes. 1981, 13, 53-65. (7) Creton, C.; Leibler, L. J. Polym. Sci. 1996, 34, 545-554. (8) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (9) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 10131025. (10) Brown, H. R. Macromolecules 1993, 26, 1666. (11) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989. (12) Silberzan, P.; Perutz, S.; Kramer, E. J.; Chaudhury, M. K. Langmuir 1994, 10, 2466-2470. (13) Deruelle, M.; Leger, L.; Tirrell, M. Macromolecules 1995, 28, 7419. (14) Shull, K. R.; Crosby, A. J. Eng. Mater. Technol. 1997, 119, 211215. (15) Mowery, C. L.; Crosby, A. J.; Ahn, D.; Shull, K. R. Langmuir 1997, 13, 6101-6107.
Figure 1. Schematic representation of the geometry employed during the adhesion test. A rigid indenter of radius R is pressed against a thin, gel layer of thickness h, under an applied load, P. The contact area between these two bodies is circular, with a radius of a.
adhesive properties of a thin layer? The second question concerns the effect of frictional shear forces that develop because of the lateral expansion of a nearly incompressible thin film. For several years, researchers have studied the interplay between friction and adhesion in a material in intimate contact with a polymer. The relationship between these two factors has typically been associated with shear experiments, where one surface slides across another.19,20 In our work, we discuss the effects of friction as they are related to thin elastic films loaded in a compressive manner. Axisymmetric adhesion tests are now widely used for quantifying the adhesive properties of elastomers. A convenient aspect of these tests is that the surface properties of both the indenter and film layer may be tailored for a specific application. Typically, the elastic layer is formed from a polymer melt with an elastic modulus between 105 and 106 Pa. We are interested in the use of polymer gels with elastic moduli ranging from below 102 Pa to greater than 104 Pa, depending on the concentration of polymer in the gel. Polymer gels are an (16) Shull, K. R.; Ahn, D.; Mowery, C. L. Langmuir 1997, 13, 17991804. (17) Ganghoffer, J.-F.; Gent, A. N. J. Adhes. 1995, 48, 75-84. (18) Ganghoffer, J. F.; Schultz, J. J. Adhes. 1996, 55, 285. (19) Klein, J.; Kumacheva, E.; Mahaiu, D.; Perahia, D.; Fetters, L. J. Nature 1994, 370, 634-636. (20) Klein, J. Polym. Annu. Rev. Mater. Sci. 1996, 26, 581-612.
10.1021/la9810556 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/08/1999
Properties of Thin Gel Layers
interesting class of materials since they are able to incorporate a large amount of solvent molecules into their structures while still exhibiting a solidlike behavior. While several researchers have provided great insight into the structure and gelation mechanisms of these materials,21-29 very little has been presented to analyze the adhesive nature of these weak, swollen polymer networks on a molecular level. Several important advantages are associated with the use of low-modulus materials in adhesion experiments. Because the deformation is determined by a balance between adhesive interactions and elastic restoring forces, these materials are very sensitive to relatively weak adhesive interactions. Also, because the forces required to measurably deform the gel are small, experiments can be designed for probing interactions with individual biological cells or living tissues. Suppose, for example, that the indenter in Figure 1 is coated with a monolayer of cells. One advantage of using a soft gel as the elastic layer is that the cells can be brought into intimate contact with this layer without destroying the cells. Hence, there is a natural bridge for applying new or refined techniques in elastomer adhesion to other research areas such as tissue engineering. In a review by Hammer and Tirrell30 on biological adhesion, the authors suggested the possibility of using a contact mechanics approach to quantify the adhesive interactions between a cell and a polymer surface. We believe that it may be possible to measure the adhesive forces between a layer of cells and a soft material, such as a polymer gel, by applying a modified analysis of the JKR theory in which only the displacement and contact area between the two bodies need to be measured. This technique would allow more flexibility in designing experiments since costly equipment would not be necessary. We have found that an excellent model system that may be used to address these issues is a triblock copolymer with glassy endblocks and a rubbery midblock that is swollen with a selective solvent for the center block. As discussed in a recent paper,15 these materials exhibit an elastic response at room temperature, show no tendency to flow over time periods appropriate for the adhesion tests, and undergo a rapid gelation when cooled from elevated temperatures (above the gel point) to room temperature. The gel is made from a parent triblock copolymer with poly (methyl methacrylate) endblocks and a poly (tert-butyl acrylate) midblock. The midblock is converted to n-butyl acrylate, and this converted polymer is swollen with 2-ethyl hexanol to form a low-modulus gel. Our work capitalizes on the soft, elastic nature of these triblock copolymer gels to investigate relatively weak adhesive interactions between polymer surfaces. By (21) Ko, M. B.; Kwon, I. H.; Jo, W. H. J. Polym. Sci. 1994, 32, 945951. (22) Tanaka, F.; Edwards, S. F. Macromolecules 1992, 25, 15161523. (23) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (24) Nijenhuis, K.t.; Winter, H. H. Macromolecules 1989, 22, 411414. (25) Dahmani, M.; Fazel, N.; J.-P., M.; Guenet, J.-M. Macromolecules 1997, 30, 1463-1468. (26) Raspaud, E.; Lairez, D.; Adam, M.; Carton, J.-P. Macromolecules 1994, 27, 2956-2964. (27) Balsara, N. P.; Tirrell, M.; Lodge, T. P. Macromolecules 1991, 24, 1975-1986. (28) ten Brinke, G.; Hadziioannou, G. Macromolecules 1987, 20, 486489. (29) Yu, J. M.; et al. Macromolecules 1996, 29, 5384-5391. (30) Hammer, D.; Tirrell, M. Annu. Rev. Mater. Sci. 1996, 26, 65191.
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forming a uniform gel layer with a thickness between 100 and 500 µm we have conducted a variety of experiments in order to develop a thorough understanding of the gels' adhesive and mechanical properties. In addition, we discuss the regimes in which friction between the indenter and gel layer plays a critical role in the amount of hysteresis observed between loading and unloading experiments. Last, we confirm through our studies that the adhesive nature of soft, compliant materials may be quantified without measuring the applied load, provided that the modulus, contact area, and displacement between the two bodies is known. We will begin with a brief review of the critical expressions describing the mechanics between a rigid indenter and a compliant film, followed by a brief description of the synthesis, modification, and characterization of the polymer gels. After describing the sample preparation and adhesion test apparatus, we will examine the adhesive response of a thin gel layer in contact with a rigid indenter. Finally, we confirm that the fracture mechanics expressions used to describe this geometry provide an accurate method for quantifying adhesion in these relatively thin films. Fracture Mechanics Analysis. The theory presented by Johnson, Kendall, and Roberts8 and generalized by Maugis and Barquins11 provides a convenient method for analyzing the adhesive interactions between compliant materials, such as two hemispherical caps or a cap and a flat substrate pressed together with a known force, P. By assuming that the adhesive interactions occur within the circular contact area of radius a, and that the lens is deformed in a linearly elastic manner, this analysis leads to a relationship between the contact radius and applied load which involves the energy release rate. This relationship can be written in either of the two following forms:
a3 )
R (P + 3πG R + {6πG RP + (3πG R)2}1/2) K G)
(Ka3/R - P)2
(1)
6πKa3R
where K is the effective modulus of the elastic medium and R is the radius of curvature of the indenter. At equilibrium, the energy release rate is equal to the thermodynamic work of adhesion, such that G ) W. If G is less than W, a driving force exists for the contact area to increase. For the case where G becomes larger than W, a driving force exists for the crack to advance, corresponding to a decrease in the contact area. For the geometry of a rigid indenter in contact with a soft, incompressible gel (Poisson’s ratio, ν, ) 0.5), the effective modulus of the gel, K, is related to Young’s modulus E by K ) 16E/9. In the absence of adhesive forces, when the work of adhesion is equal to zero, we recover the Hertzian expression for the applied load. In his analysis, Hertz did not account for the presence of adhesive interactions. The loads and displacements in this case are given by the following expressions:
Ph )
16Ea3 9R
δh )
a2 R
(2)
where Ph is the Hertzian load and δh is the Hertzian displacement. In the JKR theory, these expressions are modified to account for the fact that, at a given contact radius, the adhesive forces between the two bodies will cause the load and displacement to be less than the
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Hertzian values. The relevant equations can be derived in a more generalized form by describing G in terms of the compliance of the system:31
(P′ - P)2 dC (δ′ - δ)2 dC G))4πa da 4πaC2 da
(3)
where P′ is the nonadhesive load, δ′ is the nonadhesive displacement, and C is the compliance, defined as the slope of the displacement versus load curve at fixed contact area. The nonadhesive loads and displacements are defined as the values of these quantities obtained at a given contact area for the case where G ) 0. The constitutive expression of eq 3 is valid for a material that behaves in a linearly elastic manner, with the compliance written in the following form:
C)
δ′ - δ P′ - P
(4)
For thin films, it is necessary to include finite size corrections when calculating C, δ′, and P′. One of the basic assumptions of the JKR theory is that the deformed body can be regarded as an elastic half space. In other words, the thickness h of the compliant layer is assumed to be large relative to the contact radius. In some situations, the fracture mechanics equations need to be modified to account for the finite size effects of the thin layer.15,17,32 The origins of the following expressions have been described in detail elsewhere. 15,31 Briefly, comparisons to finite element data have been used to describe situations in which the ratio of the radius of contact area to the thickness of the soft, compliant layer becomes large, as is the case for the experiments described in this paper. Using these results, the alternative expressions for C and P′ are shown below:
Table 1. Properties of Synthesized Triblock Copolymers T1 (triblock1)
property MW of PMMA MW of PNBA PDI fraction diblock (w/w) PMMA-PNBA converted forms
TD1 (triblock-diblock 1)
23000 g/mol 92000 g/mol 1.11 0
15000 g/mol 137000 g/mol 1.12 0.35
carboxylated, methylated carboxylated
full-friction and frictionless forms of the nonadhesive displacement are both described by the following expression:
(
{
δ′ ) δh 0.4 + 0.6 exp -1.8
})
a h
(7)
These forms of the fracture mechanics equations allow us to describe situations of contact between a rigid body and a thin, elastic layer. By applying these expressions to our model gels as described in the following sections, we are able to quantify both the adhesive and elastic properties of these materials. Experimental Procedure
In these expressions, we have assumed that ν ) 0.5 since any deviations due to the compressibility of the finite gel layer are significant only for larger values of a/h than those encountered in our experiments.31 In eqs 5 and 6, β accounts for the friction between the elastic layer and the indenter. A full-friction boundary condition between the layer and substrate is assumed at all times. For the boundary condition of “full-friction” between the indenter and the elastic layer, β is equal to 0.33, and for the frictionless case β is equal to 0.15. These values for β, published previously, were obtained from a finite element analysis of a punch/adhesive interface in order to account for the inherent differences between a full-friction and frictionless boundary condition.31 In this paper, we compare our data with thin, gel layers to these previously derived expressions that account for both finite size corrections and frictional effects at the indenter/layer interface. The nonadhesive displacement is not affected by the boundary condition at the indenter/layer interface. The
Synthesis, Modification, and Characterization of Acrylic Polymer Gels. Physically cross-linked gels have been formed by diluting a poly(methyl methacrylate)poly(n-butyl acrylate)-poly(methyl methacrylate) triblock copolymer with 2-ethyl hexanol, a selective solvent for the midblock. Gel samples were prepared by forming solutions with polymer volume fractions Φp of 0.08, 0.14, and 0.25 at elevated temperatures of ∼60 °C and then cooling the solutions to room temperature. Two different polymers were studied: T1 polymer and TD1 polymer. These polymers were characterized using NMR and size exclusion chromatography, and the results are presented in Table 1. The parent copolymer to the T1(triblock 1) polymer, with a poly(tert-butyl acrylate) midblock, was anionically synthesized in a manner similar to the procedure outlined by Varshney et al. 33 The midblock of the copolymer was converted to poly(n-butyl acrylate) via an acid-catalyzed transesterification process. Details of the polymer synthesis and characterization are described in two recent publications.15,16 During the transesterification reaction a few mole percent of acrylic acid groups form to create a random copolymer of n-butyl acrylate and acrylic acid within the midblock. We refer to this polymer as the “carboxylated” version of T1. To determine whether these acrylic acid groups have a significant effect on the adhesive properties of the polymer gel, we methylated the acrylic acid groups in the carboxylated PNBA midblock with diazomethane, to form the “methylated” version.34,35 Methylation does not affect the molecular weight distributions and efficiently replaces the -OH functional groups with -OCH3 groups. The second polymer, TD1, was synthesized using the same techniques described earlier. Because some of the reactive sites of the difunctional initiator used to polymerize the triblock were terminated, a blend of diblock and triblock chains was formed. The fraction of diblock copolymer within the gel does not appear to significantly affect any of its mechanical or adhesive properties. The T1 and TD1 polymers are diluted with 2-ethyl hexanol in
(31) Shull, K. R.; Ahn, D.; Chen, W.-L.; Flanigan, C. M.; Crosby, A. J. Macromol. Chem. Phys. 1998, 199, 489-511. (32) Gent, A. N. Rubber Chem. Technol. 1994, 67, 549-558.
(33) Varshney, S. K.; et al. Macromolecules 1991, 24, 4997-5000. (34) Ahn, D.; Shull, K. R. Langmuir 1998, 14, 3637-3645. (35) Aldrich Technical Bulletin. Number AL-180.
C)
3 (0.4 + {0.6 - 0.54(a/h)} exp[-1.8(a/h)]) (8Ea ) 1 + 2β(a/h) 3
P′ ) Ph(1 + β{a/h}|3)
(5) (6)
Properties of Thin Gel Layers
order to form the gel layers used in the adhesion experiments. The preparation of these thin gel layers and of the indenter itself are discussed in the following section. Gel Layer and Indenter Preparation. For the experiments described in this paper, we probe the surface of a thin, elastic layer with a rigid, optically smooth hemisphere, as shown in Figure 1. The displacement between the lens, defined by a radius of curvature R, and the gel layer, defined by a thickness, h, is controlled to produce a circular contact area. The diameter of this area is given as 2a. One advantage of the geometry that we employ during these adhesion tests is that the polymer volume fraction in the gel can readily be altered to create a higher or lower modulus material, while still enabling us to form a uniform thin layer. In contrast, for the geometry of the soft elastic lens in contact with a rigid substrate, it is very difficult to form a uniform spherical cap from a highly viscous liquid. The gel layers were formed by placing a small amount of the solid gel between two metal shims on a glass slide. The glass surface was heated to around 70 °C, to enable the gel to flow on the glass surface. A fluorinated glass slide, treated with a 10% solution of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (PCR, Inc.), was placed on top of the metal shims to spread the viscous polymer liquid evenly within the predetermined gap. After allowing the solution to cool to room temperature, the treated glass slide was removed, leaving a uniform gel film on the untreated glass surface. The thickness of the gel layer was confirmed through an optical microscope by imaging the profile of the layer. Hemispherical glass indenters with diameters of 3, 6, and 12 mm were obtained from International Scientific Products. To control the surface chemistry of the indenter, two of the 12 mm diameter hemispheres were coated with a polymer film of either PDMS or PMMA. A diblock copolymer of poly(styrene)-poly(dimethyl siloxane) (PSPDMS) was dissolved in toluene and cast onto the indenter to form a uniform coating on the upper surface of the lens. This PS-PDMS diblock copolymer has an overall molecular weight of 61000 g/mol and a polydispersity index of 1.3. The PMMA layer was formed by first dissolving PMMA from Polymer Laboratories, LTD (Mn)500,000 g/mol, Mw/ Mn ) 1.09) in toluene. A 1000 Å film was formed by spin coating the solution onto a glass slide and floating the layer off of the glass surface in water. The indenter was used to lift the film from the water and was annealed for 1 h at 60 °C to remove any excess water molecules. Optically, the coatings appeared to be very uniform in the area of interest on the indenter. Adhesion Test Apparatus. The adhesion test apparatus used for these experiments consists of an Inchworm stepping motor placed in parallel with a linear 50 g load transducer, as described in detail by Ahn et al. in a recent publication. 36 The motor is driven with a computer interface to ensure accurate control of the displacement. The crosshead velocity imposed by the motor can vary greatly, but during a typical experiment, the motor is moved at a constant rate of 2.2µm/s. Simultaneous measurements of the load (P), contact radius (a), and displacement (δ) are obtained as the glass indenter is pressed against the thin, gel layer and retracted. To verify that rates of loading and unloading did not have an effect on the acquired results, the gel layer was tested with a range of different cross-head velocities varying from 4 to 0.3 µm/s. An optical microscope is positioned directly above the two contacting bodies so that the circular contact area (36) Ahn, D.; Shull, K. R. Macromolecules 1996, 29, 4381-4390.
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Figure 2. Load and contact radius for a 360 µm thick carboxylated polymer gel of Φp ) 0.14 in contact with a 12 mm diameter glass indenter: (a) load relaxation after compression to 0.9 mN (b) loads (0), and contact radii (b) during repeated loading/unloading in compression and in tension.
can be monitored over the course of the experiments. The microscope is connected to a video cassette recorder, which provides the capability of capturing images every 0.03 s. By using an image analysis program, Image Pro Plus, the changing circular contact area can be measured over the course of the experiment. The base of the apparatus rests on a vibration isolation table to aid in dampening disturbances in the signal from the load transducer. Results and Discussion Adhesive Properties of Thin, Gel Layers in Contact with a Glass Indenter. While we have discussed similar adhesive experiments between a soft, elastic gel and another polymer substrate in a recent paper,15 the experiments presented in this paper are unique for two reasons. First, use of this geometry allows us to increase the modulus of the gel so that the applied loads are large enough to be measured accurately with our load cell. Second, we are able to extend our results to larger a/h ratios, enabling us to probe the effects of confinement on the properties of the gel layers. For this experiment, we investigate the adhesive and frictional responses of a thin gel layer probed with an untreated 12 mm diameter glass hemisphere. Figure 2 illustrates the behavior of a carboxylated T1 gel with Φp ) 0.14 and h ) 360 µm, during a stress relaxation experiment. In part a, the glass indenter is pushed into the layer at a rate of 2.2 µm/s until a compressive load of 0.9 mN is reached. At this point, the displacement between the indenter and gel layer is fixed as the voltage signal from the transducer is recorded for a time period of 450 s. The contact area remains constant during the course of the experiment, indicating that there was no bulk flow of the polymer. One possible explanation for the gradual decrease in load at a fixed contact area is
4970 Langmuir, Vol. 15, No. 15, 1999
Figure 3. Adhesive properties of a 12 mm diameter glass indenter in contact with a 360µm thick carboxylated polymer gel with Φp ) 0.14: (a) Load-displacement relationship showing significant hysteresis between the loading and unloading data. The dashed line represents the load/displacement relationship in the absence of adhesive forces (P′ vs δ′), as obtained from eqs 2, 6, and 7, with β ) 0.33. (b) Contact area vs displacement for the same experiment, showing virtually no hysteresis between the loading and unloading data. The solid line represents the relationship between contact radius and the nonadhesive displacement (δ′) as obtained from by eqs 2 and 7.
that shear stresses at the gel/indenter interface are gradually relaxing. This hypothesis will be discussed in greater detail later. A similar experiment was performed where the displacement between the lens and gel layer was fixed for 200 s at a tensile load during the pulloff portion of the test. Figure 2b shows both the applied load (squares) and contact radius (circles) over the course of this test. At a time of about 30 s, there is a dramatic decrease in the load from the baseline value. This decrease can be attributed to the adhesive interactions between the glass probe and the gel layer, as initial contact between the two bodies is established. The load increases to approximately 7.5 mN as the indenter is pressed into the gel and decreases to a value of -2 mN as the motor retracts the gel layer from the glass hemisphere. At a time of 80 s into the test, the displacement between the two bodies is held constant as the load and contact area are recorded. Again, we see some relaxation in the load yet no deviation in the contact area measurements during the experiment. Beginning at a time of 280 s, the loading/unloading/holding cycle is repeated two additional times. The reproducibility of the experiment further confirms our belief that the material is behaving elastically. The relationships between the load, contact area and displacement allow us to directly determine the energy release rate G. Figure 3 illustrates the behavior of the TD1 polymer gel with Φp ) 0.14 and h ) 360 µm, in contact with a glass indenter with R ) 6 mm. As seen in part a of this figure, the signature drop in load indicates that intimate contact between the glass lens and the gel layer
Flanigan and Shull
has been established. The large decrease in the load is less pronounced for smaller lenses, yet it still provides us with an excellent means for synchronization of the contact area, which is obtained from the videotape, with the load and displacement information, which is recorded on the computer. By our convention δ ) 0 when initial contact is made between the two bodies. Positive values of P and δ correspond to compressive loads and displacement, i.e., these quantities increase during the loading portion of the test. After establishing contact, the gel layer is pressed against the glass probe at a rate of 1 µm/s until a load of 5 mN is reached, at which point the motor is retracted at the same cross-head velocity. Because of the adhesive interaction between the gel and glass, the gel remains in contact with the indenter up to a negative displacement of about 25 µm. For the experiment presented in Figure 3, there is a significant amount of hysteresis between the loading and unloading cycles in the load-displacement data. The area of the hysteresis loop gives the hysteresis energy, which when normalized by the maximum contact area gives an energy per unit area of 19 mJ/m2. This particular value of the hysteresis energy is low compared with a typical value of 30 mJ/m2. The dotted line in Figure 3a shows the relationship between load and displacement in the absence of adhesive interactions, i.e., P′ vs δ′ as given by elimination of a from eqs 6 and 7, using β ) 0.33. Figure 3b shows the relationship between the actual displacement and the contact radius. At zero displacement, defined where the lens and layer initially come into contact, there is a large contact area increase, due to the adhesion energy G. The balance between G and the elastic deformation energy of the gel layer determines the measured contact area for each value of the displacement. The circular points in this figure show that there is very little hysteresis in this system. This behavior suggests that dissipative processes do not play an important role and that G can be equated with W, the thermodynamic work of adhesion. We will expand on this idea in a following section. We also point out that the ratio of contact radius to gel thickness ranges between 1 and 2 during the experiment. The magnitude of a/h in these experiments requires the use of finite size corrections to correctly describe the adhesive contact between the two surfaces. The difference in hysteresis observed between the loaddisplacement and contact radius-displacement relationships is quite remarkable. To explore this unique feature, we tested a range of materials including the carboxylated and methylated forms of the T1 and TD1 polymer gels with polymer volume fractions ranging from 0.14 to 0.25. In addition, we varied the maximum load from 1 to 15 mN in order to rule out possible contributions to the hysteresis associated with bulk yielding of the gels. In all cases the materials behaved in a reproducible elastic manner, with no evidence of yielding. In addition, the observed hysteresis was always much larger for the load-displacement data than for the contact radius-displacement data. We also find that the acrylic acid groups in the carboxylated T1 gel do not alter the adhesive behavior of the swollen polymer network. No difference in adhesive behavior between the methylated and carboxylated T1 gels was observed. Furthermore, changes in motor rate from 0.3 to 4 µm/s did not affect these results. Adhesive Test of Gel Layer Probed with Coated Indenter. To explore the discrepancy between the amount of hysteresis observed in Figure 3a and b, we extended our studies to probe the effects of a thin gel layer in contact with coated indenters with well-defined surface chemistries. Since the surface chemistry of a glass interface is
Properties of Thin Gel Layers
Figure 4. Adhesive characteristics of a 12 mm diameter glass indenter, coated with PDMS, in contact with a 360 µm thick carboxylated polymer gel of Φp ) 0.14. (a) Load-displacement relationship indicates that the PDMS coating reduces the observed hysteresis in comparison to the uncoated indenter. (b) Illustration of no hysteresis in the contact radius-displacement relationship between loading and unloading cycles. The dashed and solid lines in parts a and b have the same meaning as in Figure 3, with β ) 0.15.
not necessarily uniform, we coated the hemispherical indenter with a ∼100 nm layer of PMMA and repeated the axisymmetric adhesion test described earlier. We found that these data were extremely reproducible and that they corresponded to the behavior observed with the noncoated lenses. PMMA is a glassy polymer with properties similar to inorganic glasses in some respects. To modify the surface properties of the lenses more substantially, we also used poly(dimethyl siloxane) (PDMS) surface coatings, which are well-known for their ability to reduce friction and adhesion.37,38 The PDMS coatings we have used are actually grafted PDMS layers obtained from block copolymers as described above. Figure 4 shows the behavior of a PDMS coated glass lens with R ) 6 mm in contact with a 360 µm gel layer with Φp ) 0.14. These results can be directly compared to the results for the bare glass indenter, which are shown in Figure 3. Two differences are immediately apparent. First, as shown in Figure 4b, very low values of the contact radius are maintained to remarkably large negative displacements. The shape of the contact area-displacement curve near the pulloff is fundamentally different from the shape of this curve for the bare glass and PMMAcoated indenters. While we currently do not have a detailed explanation for this effect, we have found that it is quite reproducible. A potentially important factor is that while the polymer molecules in the gel probably absorb preferentially to the PMMA and glass surfaces, it is possible that the 2-ethylhexanol exists preferentially at the PDMS surface. Our focus here is on the second notable effect of the PDMS coating, which is the significant reduction of the
Langmuir, Vol. 15, No. 15, 1999 4971
Figure 5. Effect of a PDMS coating on the adhesion between a 12 mm diameter glass indenter and a 360 µm carboxylated polymer gel with Φp ) 0.14. (a) Relationship between a3 and P for the PDMS coated indenter and (b) relationship between a3 and P for the glass hemispherical indenter. The arrows represent the directions of loading and unloading as the indenter is pressed into the gel and retracted. The solid line shows the relationship between P′and a3 taking into account finite size corrections in the load calculation, as given by eq 6. The dashed line illustrates the significance of these finite size effects by showing the relationship between Ph and a3 as given by eq 2. The square of the distance between P and P′ is directly proportional to the energy release rate.
hysteresis energy obtained from the load-displacement data. The hysteresis energy from Figure 4a is approximately 10 mJ/m2, in contrast to the 30 mJ/m2 typically seen in the glass-gel layer experiments. This difference can also be seen by plotting the relationship between a3 and P. This comparison is made in Figure 5,which also demonstrates the importance of including corrections to the JKR expressions when using thin, elastic layers. The connected dots in Figure 5a and b indicate the relationship between the cube of the contact area and the load as the materials are loaded together and pulled apart. The solid lines show the value of the nonadhesive load P′, taking into account the finite size effects in the thin layers. The Hertzian loads (P′ ) Ph) are also plotted in Figure 5, as dashed lines. At low values of a/h the P′ values converge. However, for most of the a/h values, finite size corrections to Ph are significant. Corrections to the calculated G values are even more significant, as discussed in more detail below. We assert here that shear forces imposed by the uniaxial loading of a confined material can significantly affect the observed adhesion hysteresis. These effects only become important for thin layers, where h is relatively small compared to the lateral dimensions of the confined material. For a thin layer of an incompressible material, an applied normal displacement will necessarily result in
4972 Langmuir, Vol. 15, No. 15, 1999
Figure 6. Relationship between the nonadhesive loads and displacements obtained from eqs 6 and 7 for a 12 mm diameter lens in contact with a 360 µm gel layer. The two curves are a comparison between the full-friction boundary condition (β ) 0.33) and a frictionless boundary condition (β ) 0.15) for the interface between the gel and the lens. The area between these two curves corresponds to an energy per unit area of 100 mJ/ m2. This quantity represents the extra stored elastic energy available to be dissipated by frictional sliding along the indenter/ gel interface.
a lateral displacement, giving rise to shear forces at the confining interfaces. The full-friction boundary condition corresponds to the case where there is no relaxation of these shear forces, whereas the frictionless boundary condition corresponds to the case where these interfacial shear forces relax instantaneously, with no energy dissipation. We believe that our experiments correspond to the intermediate case, where the interfacial shear forces relax over time due to frictional sliding, dissipating energy while not affecting the actual contact area. Indeed, the magnitude of the energy dissipation one would expect from this sort of process is consistent with our experimental observations. This point is illustrated in Figure 6, which compares the nonadhesive load-displacement relationships for the full-friction and frictionless boundary conditions for an indenter (R ) 6 mm) in contact with a gel layer with h ) 300 µm and E ) 1.5 × 104 Pa. The two curves are obtained by eliminating a from eqs 6 and 7 and comparing the results for β ) 0.15 (frictionless case) and β ) 0.33 (full-friction case). The stored elastic energy in each case is obtained from the integral of the loaddisplacement curve. The difference in stored elastic energies for the two curves represents the energy available to be dissipated by frictional sliding. After normalization by the maximum contact area, this energy difference is approximately equal to 100 mJ/m2. Thus, good agreement was obtained with the results obtained from the PMMAcoated and bare glass indenters if approximately 30% of the available energy is dissipated by frictional sliding. The PDMS-coated indenter provides a nearly frictionless boundary condition, so the adhesion hysteresis is quite low in this case. The predicted contact areas for these two boundary conditions are identical,15 therefore, the relationship between the contact area and displacement is not affected by the relaxation of the shear forces. Admittedly, this argument is qualitative, since we have neglected the effects of adhesive forces on the actual load-displacement relationship, i.e., we have made this comparison for G ) 0. Very similar results are obtained, however, if we use more realistic values of G. We believe that this picture
Flanigan and Shull
Figure 7. Comparison of experimental load-displacement data for the PDMS coated indenter in contact with the (filled circles) gel layer to the (squares) calculated load/displacement relationship. The calculated load-displacement relationship is obtained by using eq 8 to obtain predicted displacements from the measured loads, for E ) 1.5 × 104 Pa. The experimental data are identical to those shown in Figure 4a.
of frictional sliding provides the best explanation for the observed relationships between the load, displacement, and contact radius. Validity of Fracture Mechanics Analysis for Thin, Elastic Layers: Calculations of Load, Displacement, and Energy Release Rate. To verify that our expressions for describing the adhesive and elastic properties of thin layers are valid, we calculate displacement values from the experimentally determined values of a and P, and we compare these calculated values to the actual displacements. Calculated displacements are obtained from the following rearrangement of eq 4
δcalc ) δ′ + C(P - P′)
(8)
where eqs 5-7 are used for δ′, C, and P′. Because we are comparing to experimental data obtained with the PDMScoated indenter, we use β ) 0.15, corresponding to a frictionless boundary condition for the gel/indenter interface. In Figure 7, we show a typical load-displacement curve for a rigid PDMS-coated indenter against a 360 µm gel layer with Φp ) 0.14. We chose this particular lens surface chemistry because of the low hysteresis which is observed in this case. For each measured contact area, the measured load is plotted as a function of the calculated displacement, assuming E ) 1.5 × 104 Pa. Excellent agreement is obtained between the experimental and calculated values of the displacement for intermediate displacements between -10 and 40 µm. The slight deviation between the experimental and calculated values at the extreme values of displacement may be attributed to the fact that there is a degree of uncertainty in both the thickness h and Young’s modulus E of the lens. Also, for very large deformations, the gels can no longer be assumed to be linearly elastic. We are also able to compare the experimental loads to calculated loads obtained from the following expression:
Pcalc )
δ - δ′ + P′ C
(9)
where δ′, C, and P′ are obtained from eqs 5-7 as before.
Properties of Thin Gel Layers
Langmuir, Vol. 15, No. 15, 1999 4973
Figure 8. Comparison between calculated (squares) and experimental (filled circles) load/displacement relationships. The experimental data are identical to those shown in Figure 7, but here eq 9 is to used to obtain the calculated load from the measured displacement, assuming E ) 1.5 × 104 Pa.
Figure 8 is a comparison of the experimental and calculated loads, plotted against the measured displacement. The main significance of the excellent agreement between the actual and calculated loads is that we are able to accurately determine the load in situations where it is either inconvenient or impossible to measure. If we know the thickness of the layer and its elastic modulus, the load (and hence the stresses) can be obtained directly from the measured displacements and contact radii. Alternatively, in situations where P, δ, and a are all measured, the elastic modulus can be used as an adjustable parameter to force agreement between calculated and measured displacements or between calculated and measured loads. Once the elastic modulus is known, the energy release rate G can be calculated from the relationship between the contact radius and the displacement, or from the relationship between the contact radius and the load. Analytic expressions for G are obtained by substituting the compliance expression of eq 5 into the generalized form of the energy release rate given by eq 3. Expressions for Gfriction, assuming a full-friction (β ) 0.33) boundary condition between the indenter and thin, elastic layer, are shown in eqs 10 and 11.
Gfriction (P) )
[
]
Figure 9. Correction factors for (a) the compliance and (b) the derivative of compliance with respect to contact radius to account for a frictionless boundary condition between the indenter and elastic layer. Solid lines represent empirical fits used in eqs 12 and 13. Symbols represent values obtained from eq 5, with β ) 0.33 for the full-friction case and β ) 0.15 for the frictionless case.
tions 12 and 13 describe the modifications to eqs 10 and 11, respectively.
Gfrictionless(P) ) Gfriction(P)
(0.75 + a/h + {a/h}3)2
(10) Gfriction (δ) )
2E(δ′ - δ)2 [1 + 2.67(a/h) + 5.33(a/h)3] 3πa (11)
The prefactor in each of these expressions is the JKR form for the energy release rate, valid for a/h ) 0. The factors in brackets represent the a/h corrections, obtained from an algebraically simpler compliance expression which is essentially equivalent to the full-friction case (β ) 0.33).31 To describe situations where the interface between the rigid indenter and thin layer is assumed to be frictionless, eqs 10 and 11 are modified to account for corrections to the compliance (C) and derivative of the compliance with respect to contact radius (dC/da), as shown below. Equa-
)
dCfrictionless/da ) dCfriction/da
(
a ( h) (P) a 1 + 0.13( ) h
Gfriction
Gfrictionless(δ) ) Gfriction(δ)
(
)
Cfriction Cfrictionless
2
(
(
3
)
(12)
dCfrictionless/da × dCfriction/da
a ( h) (δ) a 1 + 0.13( ) h 1 + 0.29
) Gfriction
)
3
1 + 0.29
3(P′ - P)2 0.75(0.75 + 2(a/h) + 4(a/h)3) 32πEa3
(
)(
3
3
(ha) a 1 + 0.66( ) h 1 + 0.3
3
)
2
3
(13)
These corrections, plotted in Figure 9, are obtained from empirical fits to the ratios of the compliance expressions (eq 5) derived from finite element data. Cfriction is obtained by setting β ) 0.33 in eq 5, and Cfrictionless is obtained by setting β ) 0.15. Because the analytic solutions obtained in this manner are quite complicated, we have used the approximate solutions represented by eqs 12 and 13 instead. Gfriction gives the driving force for crack extension for the case where shear forces at the gel/indenter interface are not allowed to relax. The values plotted in Figure 9 give the ratios of Cfriction to Cfrictionless and dCfriction/da to
4974 Langmuir, Vol. 15, No. 15, 1999
Flanigan and Shull
eq 10 is equal to 0.4, and the bracketed correction factor in eq 11 is equal to 23. Our primary conclusion from the results presented in this section is that the mechanics expressions do indeed allow the elastic and adhesive properties of thin layers to be accurately determined from a simple experimental geometry. Summary
Figure 10. Energy release rate as a function of displacement, over the range where the gel exhibits a linearly elastic response. The experimental data is obtained from the trial described in Figure 4a. (circles) Equations 12 and (squares) 13 are used to determine the value of G, assuming E ) 1.5 × 104 Pa.
dCfrictionless/da in order to calculate Gfrictionless, the “frictionless” crack driving force for the case where the interface is not able to support a shear stress. Note that the correction to Gfriction associated with the “frictionless” boundary condition is relatively small, reaching asymptopic values close to 2.2 for P-based calculations (eq 12) and 0.45 for δ-based calculations (eq 13.) In Figure 10 we plot values of G obtained from measured values of P and a, using the frictionless expressions for the energy release rate given by (circles) eq 12 and (squares) eq 13. For this trial, we show the displacements for which good agreement between measured and calculated load-displacement profiles is obtained. Because we use the value of E obtained by fitting the load-displacement curve, we have not had to use any adjustable parameters. Thus, by simultaneously measuring the load, displacement and contact radius, we are able to determine both the elastic modulus of the gel layer and the energy release rate. Because the adhesion hysteresis is quite low, we can equate G with the thermodynamic work of adhesion, which is in turn given by γgel + γpdms - γgel/pdms.39 Here γgel and γpdms are the respective surface energies of the gel and of the PDMS coating, and γgel/pdms is the gel/PDMS interfacial free energy. From estimated values for these quantities, we obtain an expected value of approximately 40 mJ/m2 for G, which is in reasonably close agreement with the predictions of eqs 12 and 13. This degree of agreement is especially gratifying, given the magnitude of the a/h correction, i.e., the bracketed factors in eqs 10 and 11. For δ ) 60 µm, (a/h ) 1.5) the bracketed correction factor in (37) Newby, B.-M. Z.; Chaudhury, M. K.; Brown, H. R. Science 1995, 269, 1407-1409. (38) Newby, B.-M. Z.; Chaudhury, M. Langmuir 1997, 13, 18051809. (39) Cherry, B. W. Polymer Surfaces; Cambridge University: Cambridge, 1981.
We have shown in this paper that the elastic and adhesive properties of a thin gel layer can be accurately probed by monitoring the load, displacement, and contact area of the gel against a hemispherical indenter. In comparison to adhesion tests performed with soft, elastic caps, the experimental technique of forming thin elastic layers provides increased flexibility for varying both the polymer volume fraction within the gel and the thickness of the gel layer. The following conclusions have arisen from our application of a fracture mechanics methodology to studies of model gel layers: 1. The fracture mechanics expressions accurately describe the relationships between the applied load, normal displacement, and contact area. 2. Interfacial shear forces become important in these experiments when a/h, the ratio of the contact radius to the thickness of the elastic layer, is appreciable. In situations where this ratio is between 1 and 3, we have shown that discrepancies between different types of hysteresis observed between loading and unloading can be attributed to the response of the system to these shear forces. For a typical experiment, an energy per unit area of ∼30 mJ/m2 is observed in the hysteretic loop between loading and unloading for the load-displacement data. Our interpretation is that this energy is dissipated by frictional sliding at the gel/indenter interface. The relationship between the contact radius and the displacement is not affected by this energy dissipation, so there is very little hysteresis in the relationship between these two quantities. 3. Accurate estimates of G, the energy release rate, may be obtained from measurements of contact area and displacement, provided that the modulus is known. The simple, analytical expressions for G described in this paper account for both finite size corrections and frictional boundary conditions between the rigid indenter and thin, elastic layer. Acknowledgment. We thank A. J. Crosby for many fruitful discussions regarding the adhesion test and fracture mechanics analyses. We are very grateful for the donation of the triblock copolymer PMMA-PNBA-PMMA by D. Ahn and the diblock copolymer PS-PDMS by J. Hooker. Acknowledgment is also to N. Furgiuele and M. Woo for assistance with GPC measurements. The authors also appreciate the financial support of the National Science Foundation under Grant DMR-9457923. LA9810556
