Thermodynamic Analysis of Polymer-Solid Adhesion: Sticker and Receptor Group Effects

Transcription

1 Thermodynamic Analysis of Polymer-Solid Adhesion: Sticker and Receptor Group Effects ILSOON LEE,* RICHARD P. WOOL Department of Chemical Engineering, University of Delaware, Newark, Delaware Received 9April 2001; revised 3May 2002; accepted 8July 2002 Published online 00 Month 2002 in Wiley InterScience ( DOI: /polb ABSTRACT: Adhesion of dense linear polymer chains containing asmall number of randomlydistributedstickergroups( X )toasolidsubstratecontainingreceptorgroups ( Y )has been analyzed by asingle-chain scaling approach. An entanglement sink probability (ESP) model motivated by vector percolation explains the nonmonotonic influences of sticker concentration ( X ),receptor concentration ( Y ),and their interaction strength ( ) on the adhesion strength G IC ofthe polymer-solid interface. The ESP model quantifies the degree of interdigitation between adsorbed and neighboring chains on the basis of the adsorbed chain domain with an extension of the scaling treatment of de Gennes. Here, the adsorbed chain domain changes thermodynamically with respect to the energy of interaction parameter, r X Y.This model considers the situation of ablend consisting of asmall volume fraction of adhesive molecules as acompatibilizerattheinterface,wherethesemoleculespromoteadhesionbyadsorbing tothesurfaceviasticker-receptorinteractions.thepercolationmodelscalessolelywith r X Y,and this parameter can be related to both the adhesive potential (G A )and thecohesivepotential(g C ).G A describesadhesivefailurebetweenadsorbedchainsand the solid surface and linearly behaves as G A r X Y.The cohesive strength between adsorbed and neighboring chains corresponds to G C r ( X Y ) When the fracture stresses for cohesive and adhesive failure are equal, the model predicts maximum adhesion strength at an optimal value of r* ( X Y )*. Thus, for agiven value, optimal values * X and * Y exist for the sticker and receptor groups, above or below which the fracture energy will not be optimized. Alternatively, if the X-Y interaction strength increases, then the number of sticker groupsrequiredtoachievetheoptimumstrengthdecreases.significantly,theoptimum strength is not obtained when the surface is completely covered with receptor groups ( Y 1) but is closer to 30%. For polybutadiene, the optimum value of r* was determinedexperimentally(lee,i.;wool,r.p.jadhesion2001,75,299),andtypically * X 1 3%, * Y 25 30% Wiley Periodicals, Inc. JPolym Sci Part B: Polym Phys 40: , 2002 Keywords: adhesion; polybutadiene; strength; structure-property relations; thermodynamics; fracture; functionalization of polymers; theory; thin films INTRODUCTION It is of both fundamental and practical interest to predict the adhesion potential of polymers in contactwithsolidsurfaces.theinterfacialproperties Correspondenceto:R.P.Wool( wool@ccm.udel.edu) *Present address: Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI Journal of Polymer Science: Part B: Polymer Physics, Vol. 40, (2002) 2002 Wiley Periodicals, Inc. of the polymer, for example, structure, strength, and interfacial tension, are closely related to each other and crucial in many industrial processes, such as coatings, reinforced composites, and microelectronic component packaging. 1 To gain a fundamental understanding of factors that control polymer-solid adhesion, previous work has focused on polymer areal density 2 ( ), chain length 3 (N), sticker group concentration 4 ( X ),receptor group concentration 5 ( Y ), interaction strength ( ) between sticker groups and receptor sites, 4,5 and bonding time. 4,5 (t). 2343

2 2344 LEE AND WOOL Table 1. Experimental Results of cpbd Adhesion: Sticker and Receptor Concentration Effects Case X Y (cal/mol) Optimum * X or * Y Optimum ( X Y )* (cal/mol) Maximum G* IC (J/m 2 ) 1. cpbd-al 3,000 7,000 * X 3 mol % (OCOOH vs OOH) At constant Y 100 mol % 2. cpbd-als 16,000 25,000 * X mol % (OCOOH vs ONH 2 ) At constant Y 100 mol % 3. cpbd-als 16,000 25,000 * Y 30 mol % (OCOOH vs ONH 2 ) At constant X 3 mol % cpbd, Al, and AlS are carboxylated polybutadiene, aluminum oxide surface, and mixed silane-modified aluminum oxide surface, respectively. The experimental details of cases 1 3 have been reported. 4,5,13 The adhesion potential has been related to surface energetics where the number and type of chemical group (e.g., via acid base interactions) are important in predicting the adhesion potential. 6 The general expectation is that as the number of X-Y interactions increases, the strength of the interface increases monotonically. One of the problems in predicting the adhesion potential from surface energetics is derived from macromolecules having the ability to restructure upon contact with a solid surface. As a result of the surface restructuring process, the chain connectivity near the interface differs from the bulk chain connectivity. Recent work on model polymer-substrate systems has shown that the fracture energy of polymer-solid interfaces is not a monotonic function of surface energetics. 5 Because the interface structure is key to the interface strength, simulations and theories have been used to investigate the chain conformation near solid surfaces. Recently, Brown and Russell 7 suggested that the entanglement concept of the bulk could be extended to the near-surface or interfacial regions of polymers. They proposed that the entanglement density of polymers decreases near a surface, although there is no supporting evidence. Terzis et al. 8,9 examined the entanglement network of a polypropylene brush and a melt interface near a solid surface using the self-consistent field model. They demonstrated that the surface density of entanglements between free and brush chains shows an optimum behavior with respect to the surface brush density. Although many simulations and theories have been used to investigate the interface structure, only recently has experimental work been done. However, ambiguity remains on the understanding of the true chain structure near a solid surface. For example, Jones et al. 10,11 reported that the chain dimensions parallel to the surfaces of polystyrene are not affected by the surfaces, as judged from neutron scattering experiments. Additionally, Brulet et al. 12 obtained different results after executing a similar experiment. The chain dimensions parallel to the surfaces for the case of strong adsorption are even less understood. Wool et al. 4,5,13 conducted experiments to investigate the influence of a small amount of sticker groups (OX) distributed randomly along the polymer chains, adhering with receptor groups (OY) distributed on the solid surface, and the X-Y interaction parameter, as summarized in Table 1. The problem was defined as the X-Y problem at the polymer-solid interface (see Fig. 1), 5 where one queried how the fracture energy G IC depended on the number of sticker X and receptor groups Y. In this polymer-solid interface study, G IC was explored as a function of interface structure, surface restructuring time and temperature, deformation rate in terms of polymer viscosity and non-newtonian rheology, Saffman Taylor meniscus instability phenomena and cavitation effects in the deformation zone, 1 microscopic deformation mechanisms involving disentanglement and bond rupture, and interrelationships between structure and strength of polymer-solid interfaces. Figure 1. Representation of the X-Y problem at a polymer-solid interface. X represents specific polymer sticker groups, and Y represents specific substrate receptor groups; is the molar percentage or the mole fraction of the groups.

3 POLYMER-SOLID ADHESION 2345 In the first experiment by Gong and Friend, 4 the influence of X on G IC at constant Y 1 was investigated. Carboxyl sticker groups were placed randomly on linear polybutadiene (PBD) chains, and the polymer melt was adhered to aluminum (Al) foil surfaces. Pure PBD chains adhere very weakly to Al surfaces. The X-Y interaction was determined by the hydrogen-bonding interaction between OCOOH and aluminum oxide. For this cpbd-al interface, with increasing sticker group concentration X, the G IC increased and then decreased as displayed in Figure 2. An optimal sticker group concentration, * X 3 mol %, gave a maximum fracture energy of about 300 J/m 2. The trend shown in Figure 2 seemed counterintuitive in terms of wisdom of adhesion but was reproduced several times using different equipment and synthetic techniques both at the University of Illinois by Friend and at the University of Delaware by Gong. There were significant effects of surface rearrangement times that varied with X such that the maximum fracture energy was obtained in the shortest time at the optimal sticker group concentration * X 3 mol %. Surface restructuring occurred at times several orders of magnitude longer than characteristic single chain-relaxation times, for example, the reptation time. Furthermore, non-newtonian viscoelastic effects dominated the deformation zone evolution and breakdown. A microscopic analysis of the mode of failure (atomic force microscopy, X-ray photoelectron spectroscopy, and scanning electron microscopy) indicated that cohesive failure occurred predominantly in a layer immediately adjacent to the metal surface. At low G IC and X values, simple adhesive or mixed adhesive cohesive failure occurred. Figure 2. Sticker group (X) effect on the fracture energy of cpbd-al interfaces by Gong et al. 4 The M w and M n of the polymer were 180 and 98k, respectively. The samples were annealed at room temperature for 1000 min. The peeling rate of the samples was 30 mm/min. Here Y (OOH) is constant and X is OCOOH. Role of X on G IC Role of on G IC In the second experiment by Gong and Wool, 13 the influence of was examined using aluminum oxide surfaces treated with amine-terminated silanes, creating ONH 2 substrate receptor groups with a much stronger X-Y interaction. Then, the sticker-concentration effect on the cpbd-als interface was investigated at constant Y 1. Similar trends to that shown in Figure 2 occurred, but the maximum G IC doubled to 600 J/m 2,and the optimal sticker concentration decreased to * X mol %. Again, the failure mode was cohesive at high G IC values and adhesive near X 0. Role of Y on G IC Finally, the third experiment by Lee and Wool 5 varied the coverage of active amine receptor groups Y in the range of 0 100% on the aluminum surface using mixed silanes (OCH 3 O nonadhering, and adhering ONH 2 terminated). A cpbd polymer with a constant sticker group concentration X 3 mol % (the same as * X in Fig. 2) was used in the peel experiments. The results of G IC versus Y are depicted in Figure 3 for surface-restructuring times of 10, 100, and 1000 min. Significantly, G IC reaches its maximum value of about 600 J/m 2 at an optimal partial coverage of * Y 30%. In the preceding experiments, the bonding dynamics were several orders of magnitude longer (up to 1000 min) than the characteristic relaxation time of the PBD bulk (reptation time 1 min). 5 Additionally, the adhesion dynamics were strongly dependent on the concentrations of the sticker and receptor groups. The trend with Y was similar to that of X and suggests that design rules exist for optimizing polymer-solid interface strength. The adhesion model that we present in this article considers the situation of a polymer blend

4 2346 LEE AND WOOL Figure 3. Receptor group (Y) effect on the fracture energy of cpbd-als interfaces by Lee and Wool. 5 The M w and M n of the polymer were 180 and 98k, respectively. The peeling rate of the samples was 30 min/min. The samples were annealed at room temperature for various times. On AlS, Y (ONH 2 ) Y (OCH 3 ) 1. Here X (OCOOH) was about 3 mol %. The data point at Y (ONH 2 ) 0 was based on the pure dispersive forces of PBD. consisting of a small volume fraction of chains with sticker groups as a compatibilizer at the interface. These molecules promote adhesion by adsorbing to the surface via sticker-receptor interactions where the stickers are randomly distributed along the chain, and the receptors are distributed on the surface. The first assumption is that the sticker sticker interaction in the polymer layer can be neglected in the case of a small amount of sticker groups X 1 mol % distributed along the backbone chain. The second assumption is that if we consider an isolated adsorbed chain on the solid surface, the single chain-scaling analysis as a function of X, Y, and can be considered as the averaged behavior of adsorbed chains. The latter assumption obviously precludes brushlike ordering where the sticker groups are located in blocks at the end of a chain or very high concentrations of sticker groups where the single-chain analogy becomes invalid. This model thermodynamically considers the adsorbed single-chain behavior as a function of a small amount of sticker and receptor groups. Leger et al. 14 explained the optimum adhesion behavior of a polymer brush and melt with a phenomenological definition of the interdigitation between the polymer brush and the melt. Motivated by their work, in this article we use the entanglement sink probability (ESP) as the degree of interdigitation of adsorbed and free chains. We essentially view the polymer-solid interface problem as a competition between cohesive failure of the polymer polymer interface and adhesive failure of the adsorbed polymer-solid interfaces. The polymer polymer interface involves the interpenetration of the adsorbed and free chains in a layer adjacent to the surface, whereas the adsorbed polymer-solid interface considers the adhesive strength if the adsorbed chains in contact with the solid surface. Thus, as a chain becomes strongly adsorbed on the surface and its conformation flattens out on the surface, its adhesion potential increases, but its ability to interpenetrate with the free chains in the melt decreases. If the adhesive forces (not energy) required to debond the adhered chains exceeds the cohesive forces required to break the poorly interdigitated polymer polymer interface, then cohesive failure will be observed in the weak polymer polymer interface. If few X-Y interactions occur on the surface, then chains remain highly interdigitated, the adhesive forces are weaker than the cohesive forces, and a clean adhesive debond will be obtained. The failure prefers the weaker of the forces associated with either the adhesive potential (G A ) or the cohesive potential (G C ) that exist at the polymer-solid interface. Typically, G C is two to four orders of magnitude greater than G A, but the debond forces can be comparable. However, as discussed by Sharpe, 15 the mode of failure can be ambiguous. This ambiguity has been attributed to chemical reaction at the interface, 16 contamination before polymer attachment, 17 and geometric topography of a substrate surface combined with the irreversibility of fracture. From a practical point of view, partial modification of polymer and substrate surfaces is interesting for the design of polymer-solid interfaces. In these systems, the degree of modification is often a major factor in controlling the structure and strength of the polymer-solid interface, but its use may be limited because of either the processing cost or an undesired change of the polymer or substrate. In addition, it has frequently been shown that a high degree of modification could have no effect, or even a negative effect as compared with a small degree of modification. 18 The fracture energy of a polymer-solid interface is the energy required to separate the joined

5 POLYMER-SOLID ADHESION 2347 Figure 4. Representation of interfacial chain conformation (a) at low value (b) at critical value (c) at excessive value of X Y at equilibrium. X, the gray site, and the white site represent sticker groups (OCOOH), active receptor groups (ONH 2 ), and nonactive receptor groups (OCH 3 ). polymer-solid materials. In the case of a rubberlike polymer fracture, the observed fracture energy is dominated by the non-newtonian viscous disentanglement of polymer chains rather than the breaking of C C bonds. 1 Even in the case of glassy polymer fracture, because of the plastic deformation, the disentanglement process in the polymer-deformation zone is very important. 19 In this article, we use thermodynamics to express the conformation of the adsorbed layer via the radius of gyration R perpendicular to the surface. This gives R as a function of X, Y, and. The local entanglement density depends on R and is related to cohesive strength of the polymer polymer interface via the vector percolation approach described by Wool. 1 As the chain becomes more adsorbed on the surface, the cohesive strength decreases. However, the adhesive strength is determined by a relation similar to the nail solution 20 that increases monotonically with the number of X-Y interactions. Thus, the optimal behavior of G IC versus X or Y shown in Figures 2 and 3 will be dictated by the winner of the interplay of the adhesive and cohesive potentials for the two interfaces. A successful analysis of this scenario should produce a set of design rules for optimizing the strength and durability of polymer-solid interfaces. ESP V V 0 V V 0 (1) where V, V 0, and V are the pervaded volume of a bulk chain, a completely collapsed chain, and an adsorbed chain, respectively. Laub and Koberstein 22 similarly defined the entanglement prob- ESP The schematic representation of the volume shrinkage of a chain at the interface is illustrated in Figures 4 and 5. ESP is based on vector percolation theory 21 and is defined by Figure 5. Representation of chain dimension near the interface. R and R are the vertical chain dimension and parallel chain dimension.

6 2348 LEE AND WOOL ability function and estimated the adhesion strength of polymer brushes and melts. For special limiting cases, such as polymer brushes and completely collapsed chains to the solid surfaces, V approaches V 0, leading to ESP 0. If the adsorbed chain dimensions, vertical chain dimension (R ), and parallel chain dimension (R ) are known, ESP can be defined through the relations V R R 2 V 0 br 2 V R 3 (2) where b is the monomer unit length. Hence, ESP is defined by layer is uniform), the free energy, F, can be written as 23 F N N kt R chain 2 (5) R 2 where N/R represents the entropy loss per chain, S/k chain, because of the confinement of N segments within a thickness of R, and N/R represents the enthalpic energy gain per chain, H/kT chain, as a result of the contact of N/R monomers within the surface. The parameter is the X-Y interaction parameter representing the normalized adsorption potential, and k represents Boltzmann s constant. For example, when the potential is short ranged, it can be roughly described by 24 ESP R R 2 2 br R 3 2 br (3) z V z kt (6) Equation 3 describes a relative measure of interpenetration of the adsorbed chains with the free chains. When the chain is fully adsorbed, R b and ESP goes to zero; when R R, ESP 1, and we obtain the maximum interpenetration. For brushlike ordering, V approaches V 0, and the entanglement density decreases to zero at the polymer polymer interface. In most cases, with high-molecular-weight polymers, b can be negligible as compared with R and R, and ESP effectively reduces to ESP R R (4) Therefore, we have the important structural relation of the adsorbed polymer chain illustrated in Figure 5. Quantification of the ESP or percolation parameter p depends on how we relate the dimensions of an adsorbed chain, R and R, to the sticker and receptor group concentrations. ESP FROM SINGLE-CHAIN SCALING ANALYSIS Consider a chain of length N monomers adsorbed to a planar surface (Fig. 5) with the following well-known scaling argument. 23 Assuming that a Gaussian chain is adsorbed to the planar surface with thickness R (perpendicular to the solid surface and the monomer density in the adsorbed where V(z) is the interaction potential energy between a polymer segment at distance z from the surface, and (z) is a delta function. The optimum value of R that minimizes the free energy in eq 5 is given by R 1 (7) This suggests that as the adsorption potential increases, the vertical chain dimension of an adsorbed chain decreases by losing its entropy. In other words, the adhesive potential energy, G A, between an adsorbed and a solid surface increases with increasing, but simultaneously the cohesive potential energy, G C, between adsorbed and neighboring chains decreases. Sticker-Concentration Effect on ESP In the limit of small sticker group concentration X, a useful picture of the random A-X copolymer proposed by Marques and Joanny 24 is a grouping of monomers into blobs, each of them containing one sticker X. The average number of monomers per blob is g 1/ x, and the number of blobs per chain is N b N/g N x, where N is the degree of polymerization of the chain. Under the assumptions that the blob size distribution is small and that the A-solid surface and A-X interaction can be ignored as compared with a strong X surface interaction, this is a rather good approximation. The conforma-

7 POLYMER-SOLID ADHESION 2349 tion of the adsorbed chain is then similar to that of an adsorbed homopolymer but with renormalized monomers, namely, the blobs. The vertical chain dimension, R, is also estimated as before. We have an expression similar to eq 5 F N kt R chain 2 N (8) g R where the sticker fraction, g 1/ X, was introduced to consider the adsorption energy of a blob. Minimizing eq 8 with respect to R leads to R g X 1 (9) which reduces to eq 7 when X 1. In the limit of strong adsorption ( ), all the stickers are on the surface, that is, all the blobs touch the interface, and each of them still follows Gaussian statistics. The chain is confined at the interface over a distance of order g 1/2 so R is given by The scaling form of the free energy of a chain confined in a two-dimensional plane with impurities, in the limit of vn 3 where the height of impurity is greater than R of an adsorbed chain, is F N R* 2 Nvn R* (12) where is the Kuhn length. Combining this free energy with the adsorption free energy for a layer of thickness R, the total free energy is given by Baumgartner and Muthukumar 25 F N kt R chain 2 N N 2 N vn R R R R R (13) where F is now written for N/R segments, which are in the parallel plane. Here, R is the parallel chain extension on the surface. Minimizing F with respect to both R and R, the optimum values of the layer of thickness perpendicular to the surface and chain extension parallel to the surface were obtained 25 lim R g 1/2 X 1/2 (10) 3 R 1 v (14) Receptor-Concentration Effect To quantify the receptor concentration ( Y ) effect on polymer adhesion, we considered two different approaches. The first is the direct consideration of the correspondence of with Y using an effective potential, Y. In the case of a high-molecularweight polymer, the chain domain size is much greater than that of a receptor group; this can be a good approximation if we do not consider the local chain behavior. The ESP model considers only the overall chain domain, using a single-chain scaling analysis. The second approach considers impurities randomly distributed on a high energetic solid surface that decreases the receptor group density. For this case, Baumgärtner and Muthukumar 25 considered a highly energetic planar surface containing n impurities randomly distributed and adjusted the scaling argument. For two dimensions the rootmean-square end-to-end distance (R* ) for the chain depends on the impurity density, v, according to Edwards and coworker 26,27 R* N1/2 vn 3 0 v 1/2 vn 3 (11) where v is the impurity density on the energetic surface. Also, R v 1/2 as the derivation started in the limiting case of vn 3, where the height of the impurities is greater than R. On the other hand, when the height of the impurities is less than R, then a mushroomlike conformation is expected. Here R is the height of the mushroom obeying eq 14. The core of the mushroom obeys R v 1/2, whereas the width of the cap of the mushroom must be the standard result of the radius of gyration of a Gaussian chain in two dimensions, R N 1/2. The numerical prefactor of the linear term in v in eq 14 is unknown, and v 0 means that the surface is a pure energetic surface ( Y 1). When v 1, the surface is not favorable to the polymer ( Y 0). Hence, eq 14 is expressed by a linear interpolation as R 1 Y (15) Here, R of the entire chain remains constant, scaling as N 1/2. As discussed in the introduction, the experimental study of R for a chain near a surface is not well understood. The most up-to-date research with polystyrene using neutron scattering by

8 2350 LEE AND WOOL Jones et al. 10,11 and Brulet et al. 12 is not in agreement. Further, R for the case of strong adsorption as in our system is even less understood. Thus, the basic premise of R is not clearly resolved, and following the aforementioned derivation we assume that R N 1/2. THERMODYNAMIC ANALYSIS OF POLYMER-SOLID ADHESION Now consider the chain dimension, R and R,of an adsorbed chain on a planar surface (Fig. 5) influenced by, X, and Y. The free energy, F, can be written by where the parameter r F N kt R chain 2 r N (16) R r X Y (17) is the equilibrium energy of interaction at the interface. Again, N/R 2 represents the entropy loss per chain, S/k chain. The second term, rn/r, corresponds to the enthalpic energy-gain term per chain, H/kT chain, as a result of the sticker-receptor interactions, where the contacts between the monomer units containing stickers and the surface receptor units are dominant among all other contacts. Minimizing eq 16 with respect to R gives R 1 r 1 X Y (18) and R N 1/2 remains constant. Therefore, the interpenetration ESP parameter based on the conformational volumes of an adsorbed chain on a solid surface is a function of the molecular parameters, X, and Y as ESP 1 r (19) From the enthalpic energy gain term in eq 16, the adhesive energy gain per chain due to stickerreceptor interactions within the surface, G A1 ( H/kT chain ) is expressed by G A1 r N R (20) Figure 6. Adhesive potential per surface area, G A, between adsorbed chains, and a solid surface monotonically increases with X Y. With the consideration of the relation (R ) 1 r, G A1, is given by G A1 r 2 (21) The total adhesive potential per surface area, G A, can be obtained by multiplying G A1 by the number of chains per surface area, R /N. G A G A1 r (22) As shown in Figure 6, G A linearly corresponds to the enthalpic energy term, r, at the interface (as classically expected by many people). For example, Fowkes and Mostafa 6 linearly related the number of acid base interactions, n ab, at the interface to the thermodynamic work of adhesion, W A. A similar behavior can also be seen in the nail-solution adhesive fracture by Wool et al. 20 They suggested an analytical form of an interface strength model that is linear in areal density ( ) and compared well with experimental data. 1 The cohesive failure energy G C is proportional to ESP, which is similar to the entanglement probability function defined by Laub and Koberstein 22 and the vector percolation entanglement fraction by Wool. 1 The entanglement probability function quantifies the degree of interdigitation between

9 POLYMER-SOLID ADHESION 2351 G IC 2 (24) Because G A r, the stress to adhesively debond the interface is obtained as A r (25) where is a proportionality constant. Using the vector entanglement percolation fracture theory, 1 the cohesive stress is determined by C p p C (26) where p is the normalized entanglement density and is essentially identical to ESP, and p c is the percolation threshold. Because p 1/r, C is derived as C 1 r 1 r C (27) Figure 7. Cohesive potential, G C, between adsorbed chains and its neighboring chains monotonically decrease with X Y. polymer brushes and melts. Phenomenologically, the polymer brushes and strongly adsorbed polymer chains on solid surfaces are the same as described by Guiselin. 28 Because G C ESP, the G C near a surface is given by G C 1 r (23) Figure 7 determines that the G C decreases as r increases. As a result of this, the polymer chains near the interface become less entangled with the bulk, causing a decrease in the cohesive strength. The G A and G C potentials alone would fail to capture abnormal adhesion behavior. However, when used together as in the ESP model, their results accurately predict the adhesion behavior given in Figures 2 and 3. where is a constant. Equating ESP to the vector percolation entanglement fraction, p, the adhesive failure stress at C near the interface can be expressed as a function of A as C (28) A Ac in which 1/r C is replaced by 2 / 2 Ac. This relation predicts that C (near interface) decreases as A (at the interface) increases. Figure 8 determines the behavior of both A and C with increasing r values. As the chains adhere more strongly with increasing r, the cohesive stress decreases until eventually the stresses ADHESIVE VERSUS COHESIVE FRACTURE STRESS In the mode-i crack propagating at a polymersolid interface, the strain energy release rate, G IC, can be related to the failure stress in the polymer-deformation zone by Figure 8. Crossover point of adhesive versus cohesive failure stress at the critical value of equilibrium energy of interaction at interface r* ( X Y )*.

10 2352 LEE AND WOOL are equal at the crossover point r*. At r r*, the applied stress A C *, so r* is determined from the preceding equations as r* r C r C 2 1 (29) When r C 2 / 2 (1/r C is negligible), the critical value of r* reduces to r* cohesive stress coefficient adhesive stress coefficient (30) When r r*, A C and adhesive failure dominates at the adsorbed polymer-solid interface such that one expects G 1C r. When r r*, A C, and cohesive failure dominates in the weaker polymer polymer interface such that G IC decreases as G 1C 1/r. Overall, the applied macroscopic stress follows the weakest link and controls the optimization trends given in Figures 2 and 3 where the maximum fracture energy corresponds to r*. The critical value of r* was experimentally determined to be approximately 150 cal/ mol for PBD. The parameter r* ( X Y )* suggests that when any of the parameters are changed, the other critical parameters must shift to keep the same product value of r*. DISCUSSION AND CONCLUSIONS To apply the aforementioned analysis to real polymer-solid interfaces, we consider first the most common polymer-solid interface in which Y 1 (receptor groups totally cover the solid surface), and the polymer contains a variable amount of sticker groups X, as depicted in Figure 2. Thus, for this interface we have r* ( X )*. Because is constant, r X and when X * X, the fracture energy G 1C r (eq 22) where adhesive failure dominates, and we obtain G 1C G* 1C X / * X r r* (31) in which G* 1C is the maximum fracture energy. In Figure 2, G* 1C 300 J/m 2 at * X 3%, and when X 1%, we predict a fracture of G 1C 100 J/m 2, as noted. However, when r r* or when X * X, cohesive failure dominates (eq 23) and G 1C 1/r such that G 1C G* 1C * X / X r r* (32) For example, in Figure 2 at X 6%, one expects that G 1C 1 2 G* 1C or about 150 J/m 2 as observed. If the surface becomes contaminated such that Y decreases from 100 to 80% coverage, then * X needs to be increased from 3 to 3.75% to obtain the same fracture energy. Therefore, with regard to sticker group variation, we see that the optimization hill shape is quite asymmetric as a result of the nature of adhesive (G 1C r) versus cohesive failure (G 1C 1/r). Thus, when designing interfaces for optimal adhesion in which and Y are constant, values of X on the low side of * X can weaken the interface more than on the high side of * X. In another important polymer-solid interface example, consider the case as depicted in Figure 3 where the sticker group concentration X and bond strength are both constant, but the receptor group concentration Y can be varied from 0 to 100% coverage. In Figure 3, the bonding factor was increased considerably by ONH 2 Y groups to bond with the OCOOH X groups as compared with the native oxides used on the aluminum surfaces (Fig. 2). Thus, when Y 1, the optimal sticker group concentration X for this interface is determined by * X1 * X2 2 / 1 (33) For example, when * X2 3% for the oxide and 2 for the NH 2 OY groups is much greater than the oxide bond strength, we expect * X2 (NH 2 )tobe much less, as noted by Gong, who obtained values of * X % at maximum strength. In Figure 3, we used X 3%, which was the same carboxylated polymer as in Figure 2 at optimal strength. Therefore, to optimize the interface found in Figure 3 with X 3.0% [because r* ( X Y )*], it follows that the optimal receptor group concentration * Y1 at constant is given by * Y1 * Y2 X2 / * X1 (34) Because * Y2 1at * X2 1.0%, when X1 3.0%, we expect the optimal receptor group concentration to be about 30%, as observed. It is quite remarkable that surfaces with partial coverage of receptor groups can be stronger than those that are completely covered. This suggests that surfaces containing impurities (in the preceding case, up to 66%) can form stronger interfaces than the pristine clean surfaces, depending on the polymer that is adhering. For plasma modification of surfaces to promote adhesion by form-

11 POLYMER-SOLID ADHESION 2353 ing functional groups that act as receptor groups, it is clear that an optimal amount of modification is required, which typically corresponds to an optimal plasma exposure time. In the aforementioned example, we again note that the optimization effect is asymmetric with respect to receptor groups Y. By the same arguments, it follows that when adhesive failure dominates that G 1C G* 1C Y / * Y Y * Y (35) When cohesive failure dominates G 1C G* 1C * Y / Y Y * Y (36) From Figure 3, we can deduce from eq 35 that when Y 10%, * Y 30% and G* 1C 600 J/m 2, one should obtain G 1C 200 J/m 2. When Y 50%, we predict (eq 36) that G 1C 360 J/m 2, which is close to that observed. Given the complexity of these interfaces and the simplicity of our assumptions in this analysis, strength predictions using values far from the optimal values are considered dubious. Overall, the qualitative and semiquantitative predictions, using the parameter r* ( X Y )*, agree with a range of experimental observations and suggest that we are garnering the essential features of the interface strength development, namely, the competition and interplay between adhesive and cohesive failure mechanisms as a function of X, Y, and. The authors are grateful to Hercules Inc. (Wilmington, DE), the National Science Foundation (NSF DMR ), and the Center for Composite Materials at the University of Delaware for their financial support of this work. REFERENCES AND NOTES 1. Wool, R. P. Polymer Interface: Structure and Strength; Hanser/Gardner: New York, Creton, C.; Kramer, E. J.; Hui, C. Y.; Brown, H. R. Macromolecules 1992, 25, Duchet, J.; Chapel, J. P.; Chabert, B.; Gerard, J. F. Macromolecules 1998, 31, Gong, L. Z.; Friend, A. D.; Wool, R. P. Macromolecules 1998, 31, Lee, I.; Wool, R. P. Macromolecules 2000, 33, Fowkes, F. M.; Mostafa, M. A. Ind Eng Chem Prod Res Dev 1978, 17, Brown, H. R.; Russell, T. P. Macromolecules 1996, 29, Terzis, A. F.; Theodorou, D. N.; Stroeks, A. Macromolecules 2000, 33, Terzis, A. F.; Theodorou, D. N.; Stroeks, A. Macromolecules 2000, 33, Jones, R. L.; Kumar, S. K.; Ho, D. L.; Briber, R. M.; Russell, T. P. Nature 1999, 400, Jones, R. L.; Kumar, S. K.; Ho, D. L.; Briber, R. M.; Russell, T. P. Macromolecules 2001, 34, Brulet, A.; Boue, F.; Menelle, A.; Cotton, J. P. Macromolecules 2000, 33, Gong, L. Z.; Wool, R. P. J Adhes 1999, 71, Leger, L.; Raphael, E.; Hervet, H. Adv Polym Sci 1999, 138, Sharpe, L. H. J Adhes 1998, 67, Gallez, D.; DeWit, A.; Kaufman, M. J Colloid Interface Sci 1996, 180, Eilbeck, J.; Rowley, G.; Carter, P. A.; Fletcher, E. J. Int J Pharm 2000, 195, Inagaki, N.; Tasaka, S.; Narushinma, K.; Mochizuki, K. Macromolecules 1999, 32, Sha, Y.; Hui, C. Y.; Kramer, E. J. J Mater Sci 1999, 34, Wool, R. P.; Bailey, D. M.; Friend, A. D. J Adhes Sci Technol 1996, 10, Sahimi, M. Application of Percolation Theory; Taylor & Francis: Bristol, PA, Laub, C. F.; Koberstein, J. T. Macromolecules 1994, 27, de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, Marques, C. M.; Joanny, J. F. Macromolecules 1990, 23, Baumgartner, A.; Muthukumar, M. J Chem Phys 1991, 94, Edwards, S. F.; Muthukumar, M. J Chem Phys 1988, 89, Muthukumar, M. J Chem Phys 1989, 40, Guiselin, O. Europhys Lett 1992, 17, 225.