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Adhesion and Strength of Viscoelastic Solids. Is There a Relationship between Adhesion and Bulk Properties?† A. N. Gent Polymer Science and Engineering, The University of Akron, Akron, Ohio 44325-3909 Received October 17, 1995. In Final Form: February 7, 1996X Strength of adhesion depends upon the rheology of an adhesive as well as upon its interaction with a substrate. This is shown by studies using model joints prepared with different amounts of interlinking between two rubber sheets. The fracture energy G per unit of interfacial area appears to be a product of two terms: G ) Go [1 + f(R,T)], where Go is the intrinsic (chemical) strength of the interface and f(R,T), usually much larger than unity, reflects energy dissipated viscoelastically within the adherends at a crack speed R and temperature T. Values of Go range from virtually zero for nonbonded sheets up to the threshold tear strength of rubber, 50-80 J/m2, in proportion to the density of interfacial bonds. Values of G are as much as 1000× greater at high speeds and low temperatures. Like adhesion, the tear strength of a soft rubbery solid is also increased by internal energy losses and shows the same marked dependence upon rate of tearing. By comparing the rate R at which tear strength increases by a certain factor with the angular frequency ω at which the elastic modulus µ increases by the same factor the length δ of the dissipative zone at the crack tip can be estimated: δ ) R/ω. But values obtained in this way are only about 1 Å, too small to represent an actual dissipation zone in rubber. It seems likely instead that fracture takes place intermittently, the crack growing at high speed and then stopping. A possible mechanism of crack stopping is by splitting at the crack tip. Indeed, tip blunting can be inferred by comparing tear strength with resistance to cutting, when blunting is suppressed. Cutting resistance increases much more slowly with increasing rate and decreasing temperature. Thus, much of the observed tear strength of soft adhesives and soft solids is attributed to crack-tip blunting. Further research is needed to establish the exact nature of the blunting process and how it is related (as it clearly is) to viscoelastic and dissipative properties. Attention is drawn to other unresolved issues in adhesion science.
1. Introduction Strength of adhesion and cohesion of elastomers can be characterized by the amount of energy G required to advance an interfacial crack or, more generally, a fracture plane, by unit area. Values of G range from less than 1 J/m2 up to over 1 × 105 J/m2. In general, G appears to consist of two factors1
G ) Go [1 + f(R,T)]
(1)
where Go is a threshold value, below which no failure occurs. For tearing through a typical cross-linked elastomer, Go is calculated to be about 20 J/m2.2 Measured values of G at extremely low tear rates and high temperatures, when viscous effects are minimized, are of this order.3,4 The second term in eq 1, [1 + f(R,T)], is a function of rate R of separation and temperature T, reflecting energy expended in irreversible processes.1,5-8 For simple hydrocarbon elastomers, the effect of temperature can be completely accounted for by applying a simple multiplying factor, denoted aT, to the rate of separation R (see, for example, ref 8). Thus, f(R,T) ) f(RaT). Moreover, values of aT determined experimentally are equal to the ratio of the rate of internal molecular motion at a standard reference temperature Tg (the glass temperature) to that † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, held at Bar Harbor, ME, August 27 to September 1, 1995. X Abstract published in Advance ACS Abstracts, Sept. 15, 1996.
(1) Gent, A. N., and Schultz, J. J. Adhes. 1972, 3, 281. (2) Lake, G. J.; Thomas, A. G. Proc. R. Soc. London, 1967, A300, 108. (3) Muller, H. K.; Knauss, W. G. Trans. Soc. Rheol. 1971, 15:2, 217. (4) Ahagon, A.; Gent, A. N. J. Polym. Sci. Polym. Phys. 1975, 13, 1903. (5) Andrews, E. H.; Kinloch, A. J. Proc. R. Soc. London 1973, A332, 385, 401. (6) Ahagon, A.; Gent, A. N. J. Polym. Sci. (Phys.) 1975, 13, 1285. (7) Chang, R. J.; Gent, A. N. J. Polym. Sci. Polym. Phys. Ed. 1981, 19, 1619. (8) Gent, A. N.; Lai, S.-M. J. Polym. Sci. (Phys.) 1994, 32, 1543.
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at a test temperature T.9 This proves that the large effects of rate and temperature on adhesion and cohesion of these simple polymeric solids are due to viscoelastic processes. Owing to the second term in eq 1, values of G are as much as 1000× greater than Go at high speeds and low temperatures. Thus, strength properties are greatly enhanced by viscous resistance to internal motion or, more generally, by energy losses. But still-unresolved questions are: What is the exact relationship between strength and a loss function? What physical process does the function f represent? These are the main topics of the present essay. Some other outstanding issues in adhesion science are discussed at the end. Thomas10 proposed a relation between tear strength and the amount of energy Ub (J/m3) required to break a unit volume of material in tension at the tear tip
G ) dUb
(2)
where d is the diameter of the tear tip. An estimate of d is provided by the roughness of a torn surface, in the range 0.1-1 mm.11 On comparing eqs 1 and 2, the question arises: Do both the breaking energy density Ub and the tear tip diameter d vary greatly with rate and temperature, or is one parameter more sensitive than the other? Lake and Yeoh reduced the effective diameter of the crack at the tip of a tear to about 100 nm, by using a razor blade to cut rubber.12,13 Resistance to cutting was much less than the normal tear strength, as eq 2 would lead one to expect. Because variations in crack tip diameter are (9) Williams, M., L.; Landel, R. F.; Ferry, J. D. J. Am. Chem. Soc. 1955, 77, 3701. (10) Thomas, A. G. J. Polym. Sci. 1955, 18, 177. (11) Greensmith, H. W.; Thomas, A. G. J. Polym. Sci. 1955, 18, 189. (12) Lake, G. J.; Yeoh, O. H. Int. J. Fract. 1978, 14, 509. (13) Lake, G. J.; Yeoh, O. H. J. Polym. Sci. 1987, 25, 1157.
© 1996 American Chemical Society
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Figure 2. (a) Method of measuring tear energy Gc ()2F/t). (b) Method of measuring peel energy Ga ()2F/w).
Figure 1. Method of developing a known number of interfacial interlinking bonds.7,8
greatly reduced, it is interesting to look at the dependence of cutting resistance on rate and temperature. 2. Some Simple Experiments 2.1. Adhesive and Cohesive Strength.8 Three elastomers with quite different glass temperatures were used in the experiments: cis-polybutadiene (PB), Tg ) -96 °C; a styrene-butadiene copolymer containing 48% styrene by weight (SB), Tg ) -27 °C; and an ethylene-propylene copolymer (EPR), Tg ) -60 °C. A small amount of dicumyl peroxide was added in each case to cause chemical interlinking (“cross-linking”) of the polymer molecules and convert the original highly viscous liquid into a soft elastic solid. Bonded specimens were prepared by partially crosslinking two thin sheets for a time t1 and then pressing them together and heating them again for a further time t2 to complete the cross-linking reaction and simultaneously to interlink the sheets, Figure 1. The degree of cross-linking was characterized by the tensile modulus E, measured after various times of reaction. The degree of interlinking for two sheets cross-linked while in contact was inferred from the corresponding increase ∆E in modulus. In all cases the sheets were eventually crosslinked to the same extentsonly the density of interlinking bonds was changed. Fracture energy G was measured by peeling sheets apart (Ga) or by propagating a tear through a fully-cross-linked sheet (Gc), using a wide range of crack speeds and test temperatures. The experimental arrangements are sketched in Figure 2. In some cases samples were swollen with a light paraffin oil to minimize internal viscosity and approach threshold conditions. 2.2. Resistance to Cutting.14 Cutting experiments were carried out using the technique of Lake and Yeoh,12,13 shown schematically in Figure 3. Strips cut from crosslinked sheets of SB were pulled apart by weights W to minimize friction and cut along the centerline at a constant speed R by pushing a sharp razor blade into the cut tip. The cutting force F was monitored continuously. The total fracture energy G was computed from the sum of energy expended by the weights W and the force F as the fracture (14) Gent, A. N.; Lai, S.-M.; Nah, C.; Wang, C. Rubber Chem. Technol. 1994, 67, 610.
Figure 3. Cutting apparatus.12-14 Total fracture energy G is obtained by adding the contributions of tear forces Fw and cutting force F.
advanced. Samples were cut at various speeds and temperatures. 3. Results 3.1. Rate and Temperature Effects in Peeling Interlinked Sheets of PB. Peel results are shown in Figure 4 for samples with a density of interlinks of about one-half of the total cross-link density. Values of fracture energy Ga at different temperatures could be superposed along the rate axis by applying the WLF multiplying factors aT, calculated from the temperature difference between the test temperature and the glass temperature.9 The results then fell on a single curve versus effective rate of peeling at -20 °C, shown in Figure 5. Thus, the strength Ga of adhesion is governed by viscoelastic response of the adherends. Similar results were obtained with other levels of interlinking, as shown in Figure 5. Moreover, the curves appeared to be roughly parallel in a vertical sense, being displaced vertically by multiplying factors that are approximately equal to the ratios of threshold strengths Go. When reduced strengths G/Go were plotted against effective rate of crack propagation
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Figure 4. Fracture energy Ga vs peel rate for cross-linked sheets of polybutadiene, partially-interlinked together and then peeled apart at various rates and temperatures.8
Gent
Figure 6. Fracture energy G of cross-linked sheets of SB elastomer, interlinked to various degrees, vs effective peel rate at Tg: O, tear; 4, peel (t1 ) 30 min, t2 ) 70 min at 150 °C).8
Figure 7. Relative fracture energy vs effective tear rate R at Tg: 4, PB; 0, EPR; O, SB.8
Figure 5. Fracture energy G for cross-linked sheets of polybutadiene, interlinked to various degrees, vs effective peel rate at -20 °C; peel 1, cure times t1 ) 20 min, t2 ) 50 min at 150 °C (results from Figure 4); peel 2, cure times t1 ) 30 min, t2 ) 40 min.8 Symbols for test temperatures are as in Figure 4. Horizontal broken lines denote threshold fracture energies Go.
at Tg, a single broad curve was obtained over a considerable range of interlinking. 3.2. Threshold Strength. Threshold values were obtained for the strength Ga of adhesion at extremely low speeds and high temperatures, when viscoelastic effects are minimized. They are represented by horizontal broken lines in Figure 5. They varied linearly with interlinking density, measured by the increase in tensile modulus E, and the extrapolated value for a fully cured layer agreed reasonably well with the experimentally-measured tear strength. Thus, the threshold strength is roughly proportional to the density of tie molecules, as found earlier.7 3.3. Rate and Temperature Effects for Peel Strength of SBR. Results at different temperatures
again superposed satisfactorily using WLF multiplying factors for the rate R of peeling. Master curves for a partially-interlinked sheet and a fully-interlinked sheet (tearing) are shown in Figure 6. However, in this case they are parallel only over a limited range of rates of crack propagation, tending to converge at high rates. Thus, when measured strengths Ga and Gc are scaled with respect to corresponding threshold values Go, the results occupied a rather broad band. 3.4. Comparison between Strength and Dynamic Properties. For simplicity, only tear strength measurements are discussed heressimilar conclusions would be reached using adhesion measurements. Fracture energy G for three elastomers, scaled with respect to the corresponding threshold value Go, is plotted against effective rate of crack propagation at Tg in Figure 7. The same relation is obtained for all three materials, having widely different values of Tg. The fracture energy rose by over 3 orders of magnitude as the rate of crack propagation increased. Thus, not only is the strength of elastomers governed by their viscoelastic response but the relation
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Figure 8. Dynamic shear modulus µ′ vs effective frequency ωaT at Tg.8 Symbols as in Figure 7.
is the same for different polymers, as shown previously by Mullins.15 We now turn to a direct comparison. The variation of fracture energy with rate of crack propagation resembles the dependence of storage modulus µ′ upon frequency, as suggested by Knauss.16,17 Knauss assumed that energy is dissipated in a small region of length δ ahead of the crack tip. If δ remains constant, then the strength is expected to be proportional to modulus µ′. Values of µ′ are shown in Figure 8. They are similar for all three elastomers, when referred to their respective glass transition temperatures, increasing by about 3 orders of magnitude from the equilibrium value at low frequencies up to the glasslike value at high frequencies. In principle, the length δ of the dissipative region can be estimated from the ratio R/ω of rate R of tearing at which the strength increased by a certain factor, from Figure 7, and angular frequency ω of oscillation at which µ′ increased by a similar factor, from Figure 8. But values of δ obtained in this way are only about 1 Å, far too small to represent the actual size of a dissipation zone. A possible explanation of this anomalous result is that fracture takes place intermittently, the crack growing at high speed and then stopping. Energy losses associated with crack motion would then correspond to a high frequency and the inferred fracture zone size δ would be larger than that deduced from the average crack speed. A possible cause of crack stopping is splitting of the crack tip. As shown below, crack tip blunting can be inferred to occur in normal tearing by comparison with resistance to cutting by a sharp knife, when blunting is suppressed. 3.5. Fracture Energy G from Cutting Experiments.12 Fracture energies were determined for the SBR material by combined cutting and tearing, as shown in Figure 3. Results are plotted as a function of effective cutting speed at 25 °C in Figure 9. Cutting resistance at low speeds appeared to be rather independent of speed, although at higher speeds it increased markedly. Because the cut tip diameter d is presumably also constant, given by the blade tip diameter (about 0.1 µm), it follows from eq 2 that the intrinsic strength Ub is largely unchanged. The value deduced for Ub is about 1.5 GJ/m3, similar to (15) Mullins, L. Trans. Inst. Rubber Ind. 1959, 35, 213. (16) Knauss, W. G. In Deformation and Fracture of High Polymers; Kausch, H. H., Hassell, J. A., Jaffe, R. I., Eds.; Plenum Press: New York, 1974; pp 501-540. (17) Bowen, J. M.; Knauss, W. G. J. Adhes. 1992, 39, 43.
Figure 9. Fracture energy G for SB at various cutting speeds and temperatures, plotted against effective cutting speed at 25 °C.14 Open squares represent tear energy from Figure 7.
the theoretical value for C-C bond rupture, about 5 GJ/ m3.2 Thus, the threshold cutting resistance appears to be in accord with eq 2 and is approximately independent of effective cutting speed. In contrast, the tear strength increased dramatically as the temperature was lowered or the tear rate increased, by over 2 orders of magnitude, Figure 9 (results taken from Figure 7). We infer that the diameter of the tear tip increased simultaneously, by a similar factor. Thus, as Lake and Yeoh concluded, the anomalously high strength of elastomers and adhesives is mainly governed by blunting of the crack tip. Further research is needed to establish the precise nature of the blunting process and how it is related (as it clearly is) to viscoelastic and dissipative properties. 4. Conclusions 1. Tear energy Gc is approximately the same for simple elastomers when the tear rate is adjusted for differences in glass temperature.15 Energy Ga required to peel apart interlinked sheets depends on rate and temperature in the same way.8 Both Gc and Ga increase by about 3 orders of magnitude as the rate of crack propagation increases, to reach a maximum value of about 105 J/m2 at a tear rate of about 1 m/s at Tg. These large effects of rate and temperature are in good accord with the rate and temperature dependence of simple viscoelastic processes.9 2. At low rates and high temperatures a threshold strength Go, of 1-50 J/m2, is observed. Go appears to be directly proportional to the density of interfacial bonds and increases with their length.2,7 3. Fracture energy G at other test conditions is approximately proportional to Go. However, at high degrees of interlinking the strength increases somewhat less rapidly with increasing peel rate, so that the results tend to converge at high rates. 4. The increase in fracture energy with peel rate R resembles the increase in dynamic modulus with frequency ω. A scaling distance δ is obtained from the ratio R/ω but
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the values are too small, only about 1 Å,8 to represent a physically-realistic length of the fracture zone. 5. Cutting resistance is substantially unchanged over a wide range of effective cutting speeds, in contrast to tear energy.14 Thus, the intrinsic strength of rubber is relatively unaffected by crack speed and temperature. We conclude that the volume of material actually ruptured in tearing or detachment must be a strong function of speed and temperature, probably as a result of branching of the crack tip.13 Moreover, crack branching is inferred to be governed by viscoelastic processes, although the mechanism remains to be elucidated. 6. It is inferred that crack tip blunting is the main cause of enhanced strength and adhesion at high speeds and low temperatures.
actual number of bonds that are fractured and whether it increases in the same way as the strength does. Similarly, although the fracture energy increases with increasing concentration of interlinking bonds under all test conditions, we have no evidence that these are the only bonds that break. Quantitative analysis is needed of the number of interfacial bonds that break during adhesive or cohesive rupture. 3. Enhanced adhesion or tear strength has been tentatively attributed to blunting or splitting of the crack tip and intermittent fracture, but there is no direct evidence for these processes. Examination of a propagating tear or debond by high-resolution microscopy would shed light on what is at present merely a hypothesis.
5. Unresolved Issues
Acknowledgment. An earlier draft of this essay was presented at the International Adhesion Symposium held in Yokohama, Japan, November 1994. The research on which this essay is based was carried out in cooperation with several colleagues whose names are given in the references. Support of the research was provided at various times by the Office of Naval Research, the Edison Polymer Innovation Corporation of Ohio, Lord Corporation, 3M Company, Kumho Tire and Rubber Company, and Westvaco and is gratefully acknowledged.
1. Interfacial bonds can be created, at least in principle, with different dissociation energies, lengths, and concentrations. A systematic study of the effect of these features on the strength of adhesion under various test conditions would provide an important test of the ideas about the connection between interfacial chemistry, rheology, and adhesion that have been outlined in this essay. 2. Although adhesive and cohesive strength is greatly enhanced as the rate of fracture increases or as the test temperature is lowered, we have no knowledge of the
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