An experimentalist's view of the physics of rubber elasticity

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Rubber elasticity is one of the most fascinating properties that stem from the nature of polymers. Crosslinked rubbers exhibit reversible high extensibility with a markedly low elastic modulus that originates from the entropic elasticity of the polymer chains—a property unparalleled in solid materials. Rubbery materials have been widely used in industrial products because of their unique properties. The research on rubber elasticity has a long history, and currently, it appears to have secured classical research status. However, despite having achieved this status, this field has many unsolved problems, and our current level of understanding is still far from complete. In this article, I draw attention to some important unsolved issues and future challenges with respect to the physics of rubber elasticity.

One of the most fundamental issues is the determination of the form of elastic free energy (Fel) governing the stress–strain relations, particularly on the basis of molecular understanding. The determination of Fel remains insufficient, even in the phenomenological form. If the form of Fel is known, we can obtain the stress–strain relation for any type of deformation in a straightforward manner. The high elasticity of crosslinked rubbers is dominantly entropic in origin, similar to gas pressure; therefore, the molecular approach for deriving Fel requires the consideration of how the number of conformations available to the constituent polymers changes under deformation. The classical theories derive Fel for Gaussian networks composed of phantom chains that can pass freely through their neighbors:1, 2

equation image(1)

where k is the Boltzmann constant, T is the absolute temperature, λi (i = 1, 2, or 3) is the principal ratio in the ith coordinate direction, and C is a constant related to the topological characteristics of networks such as the number of elastic chains and the number of crosslinks. The networks composed of phantom chains are qualitatively similar to an ideal gas with no intermolecular interaction, and the Gaussian network chains are infinitely extensible. In contrast, real rubbery networks have many chain entanglements because of the uncrossability of the network chains as well as limited extensibility because of a finite chain length. The deviation of the behavior of real networks from eq 1 was recognized long ago, and many efforts have been made to derive an alternative form of Fel for real networks. The investigation to determine the form of Fel for real networks is similar to seeking the equation of state for a real gas with intermolecular interactions and a finite molecular volume. Many molecular models have been proposed to model the entanglement effect in rubber elasticity.1, 2 All models assume purely entropic elasticity (FelkT), and the main difference among them lies in the form of the strain term in Fel. Many thermoelastic data show that the elasticity of real networks is not purely entropic, and a finite energetic contribution is present. More importantly, for most rubbers, the energetic contribution (usually <20%) is almost independent of the degree of deformation.2 The energetic contribution is closely related to the front factor with respect to T and is almost irrelevant to the strain term. The magnitude of the energetic contribution is well correlated with that of the T dependence of the chain dimension of the constituent polymers.2 Fortunately, as long as we focus on the stress–strain relation at constant T without appreciable volume change, we do not need to seriously consider the energetic contribution.

For a complete understanding of Fel, we require close cooperation between theories and experiments (and computer simulations as well). To characterize Fel experimentally, we examine the stress–strain relation of crosslinked rubbers under some types of deformations and compare the data with the theoretical predictions. So far, uniaxial deformation (stretching or compression) has often been employed in experiments because of its simplicity. It is noteworthy that uniaxial deformation is only a special case among all the physically possible deformations of elastomers, as shown in Figure 1. Because of the mechanical incompressibility of rubbers (λ1λ2λ3 = 1), the strains [or stresses (σ)] along two of the three principal directions are independently variable. In the case of uniaxial deformation (σ2 = σ3 = 0 and λ2 = λ3 = λ1−1/2), the information for obtaining the force is limited to only one direction. Consequently, uniaxial data are not sensitive enough to distinguish the differences between the theories; in fact, most of the theories can fit the data to some extent if we focus only on uniaxial deformation. The experiments relying only on uniaxial deformation provide quite dubious conclusions. A typical example is the overestimation of the familiar Mooney–Rivlin plot for uniaxial data [σ1/(λ1 − λ1−2) vs λ1−1]. This plot often yields a linear correlation with a finite slope that disagrees with the prediction of eq 1 (zero slope), and many attempts have been made to find the physical significance of the value of this slope. The free energy deduced from the Mooney–Rivlin plot [Fel = C112 + λ22 + λ32 − 3) + C212λ22 + λ22λ32 + λ32λ12 − 3)], however, yields a biaxial stress–strain relation that is far from the real behavior (e.g., see Fig. 8 in ref.3). This clearly means that a finite slope in the Mooney–Rivlin plot has no physical meaning other than the indication of the failure of the classical theories. A similar observation with respect to the limitation of the analysis using only uniaxial data has already been made in some earlier studies;1, 4, 5 however, this simple but important fact appears to be largely neglected. Such insufficient characterizations are partly responsible for the current incomplete understanding of Fel. In contrast, biaxial stretching with an independent variation in the two principal ratios covers all possible deformations,1, 6 as shown in Figure 1. The biaxial data (λ1, λ2, σ1, and σ2) provide a definite basis for determining the true form of Fel. Such biaxial experiments with λ1 ≠ λ2 are, however, quite few,1, 3, 7, 8 despite their significance.

Figure 1.

(a) General biaxial deformations of incompressible elastomers (λ1λ2λ3 = 1) and (b) physically accessible deformations (hatched region) of incompressible elastomers as a function of I1 and I2. General biaxial strains with varying λ1 and λ2 values independently cover the entire hatched region. Conventional uniaxial stretching is only a particular case among all possible deformations.

It is also important to characterize structural parameters such as the number of elastically effective network strands and junctions to assess the molecular theories without ambiguity. Molecular theories involve these structural parameters in the expression of Fel. When no definite structural information about the network samples is available, these structural parameters are often employed as adjustable parameters in the data fit. This makes the assessment more ambiguous. In the case of the end linking of long precursor chains, we are able to evaluate the structural parameters of the resultant networks only from the reaction conditions, including the effect of the imperfect reaction.9 We carried out biaxial measurements for end-linked poly(dimethylsiloxane) networks with known structural parameters and compared the data with the predictions of the five existing entanglement models.10, 11 By this comparison, it was possible to unambiguously determine which model most successfully provided an account of the elastic behavior of real networks. Several new versions of tube models12, 13 were proposed after the study in ref.10. It is expected that similar tests of the theories will be performed with the biaxial data; however, the theoretical stress–strain expression for general biaxial deformations cannot be derived from their free energies in a straightforward manner because of mathematical difficulties.

What are the characteristics of a real network that satisfies the concepts of classical theories? Does an ideal network that corresponds to an ideal gas exist? Networks with no trapped entanglements would correspond to such networks (apart from the effect of the finite chain length). In principle, we will be able to prepare such networks by end-linking precursor chains in semidilute solutions in which the precursor concentrations are close to the overlapping concentration. Unfortunately, the stretching measurement of the networks will be very difficult because of their softness. With respect to this issue, the uniaxial data for highly swollen networks (originally formed in the melt or concentrated state) apparently obey the prediction of the classical theory (eq 1). In the Mooney–Rivlin plot, the slope becomes close to zero; that is, C2 = 0. This trend was recognized over 50 years ago14 and has been observed in many melt-crosslinked rubbers in the highly swollen state.1, 15 Does this result mean that melt-crosslinked rubbers in the highly swollen state correspond to ideal networks? The concentration of network strands is considerably reduced in the highly swollen state because of the presence of a considerable amount of the solvent. The trapped entanglements formed in the crosslinking stage are, however, not disentangled by swelling because the network topology is fixed upon crosslinking. In fact, several studies16, 17 have clearly revealed a significant contribution of the trapped entanglements to the small-strain modulus of fully swollen networks. On this basis, we simply expect the trapped entanglements to also contribute to a large-deformation behavior in the highly swollen state. Although we should again recall the ambiguity of the argument that relies only on uniaxial data, the evident suggestion here is that the forms of Fel before and after swelling differ significantly. To discuss this issue unambiguously, we need to perform a biaxial stretching experiment of fully swollen rubbers and observe how the high swelling affects Fel. However, these experiments may not be easy because of the fragility of highly swollen rubbers.

Slide ring gels,18 in which all the network strands are interconnected by mobile slip links along the network strands, are very attractive systems for studying rubber elasticity. Slide ring gels with no chemical crosslinks can be model systems used to elucidate the role of trapped entanglements in rubber elasticity. The details of the mechanical properties of the slide ring gels are yet unknown; however, they will be reported in the near future. It is expected that ideally the elastic behavior will be characterized in the bulk or preparation state of the slide ring networks so as to exclude a significant effect of the swelling, as mentioned previously. Unfortunately, the slide ring gels reported are not rubbery in the bulk state at room temperature, and some reported properties were obtained in a highly swollen state. Nevertheless, slide ring gels will provide a wealth of new and interesting aspects of the physics of rubber elasticity.19

In contrast to a large deformation, the small-deformation behavior of elastomers is expected to be very simple because, in principle, it must obey the infinitesimal linear elasticity theory. In the usual expression of the stress–strain relation, problems with respect to the small-deformation behavior will be imperceptible. An unusual behavior at small strains is recognizable when the data are viewed in the form of the derivatives of Fel with respect to the first and second invariants of the deformation tensor (I1 = λ12 + λ22 + λ32 and I2 = λ12λ22 + λ22λ32 + λ32λ12, respectively). When Fel is expressed with I1 and I2 as the variables, the derivatives ∂Fel/∂I1 and ∂Fel/∂I2 as a function of I1 and I2 are obtained from the biaxial stress–strain data.1 It has been observed in the case of many elastomers that in the small-deformation limit (I1 → 3 and I2 → 3), ∂Fel/∂I1 and ∂Fel/∂I2 exhibit a steep upswing and downswing, respectively.1, 3, 7 Although the derivatives at the small deformation are sensitive to the experimental error, the trend of the swing behavior is evident. In particular, ∂Fel/∂I2 even becomes negative in the small-strain limit. What is the physical origin of such behavior of the derivatives? None of the existing molecular or phenomenological theories account for this unusual behavior of the derivatives at small strains.5, 10 An attempt20 was made to explain this as asymptotic behavior toward the limiting values on the basis of the linear elasticity theory for a system with a finite compressibility in which Poisson's ratio is not exactly equal to 1/2. This approach is qualitatively successful in explaining the small-deformation behavior for some elastomers; however, the applicability of the theory is limited to the small-strain region. Some efforts have been made to establish a phenomenological form of Fel for describing the large-deformation behavior as well as the swing behavior of the derivatives at small strains;21 however, presently these efforts have been unsuccessful.

The size and shape variations of the constituent network chains in response to the imposed macroscopic strains have also been an important matter of investigation.2 The single-chain form factor of the labeled constituent chains in polymer networks has been investigated as a function of the uniaxial stretching strain by the small-angle neutron scattering technique2, 22 and dynamic molecular simulations.23 The applicability of the affine deformation at the microscopic level is an issue that has been debated for a long time. Some recent studies have revealed that the scale of affinity is strongly dependent on the scale of interest of the chain length.22, 23 The crosslinks in rubbery networks are not frozen; in fact, they fluctuate with time around the average positions as a direct result of the micro-Brownian motion of the network chains. The spatial range of the dynamic fluctuations of the crosslinks revealed by neutron spin-echo measurements is smaller than that predicted by the phantom network theory.24 This deviation suggests that the entanglements reduce the thermal fluctuations of crosslinks below the expectations of the phantom network theory. The molecular models differ in the constraint effect of the entanglements on the thermal fluctuation of the crosslinks. The length-scale dependence of the microscopic deformation of the network chains and the constraint effect on the thermal fluctuation of the crosslinks must be understood together with the macroscopic stress–strain relations on the same molecular basis.

We can achieve a marked improvement in the mechanical properties of elastomers by controlling the topological characteristics of networks without relying on the filler effect or chemical modifications. A bimodal length distribution of network chains yields a steep upturn of the stress at a high elongation, similar to a reinforcing effect by fillers.2 A finite chain-length effect of the short chains in bimodal networks leads to an enhancement in the stress at break. The effect of other types of chain length distributions (trimodal2 or an extremely broad, pseudo-unimodal distribution25) is also an interesting matter of investigation. The corresponding experiment of the latter system, however, has not yet been performed. A deswollen network that is made by the removal of the solvent from the gel prepared by the end linking of long precursor chains at a low concentration exhibits a marked strain at break greater than 3000%.26 This high extensibility results from two characteristics of the network topology: a small amount of trapped entanglement stemming from a low overlapping degree of precursor chains upon crosslinking and a compact conformation of network strands due to a large volume reduction by deswelling.26, 27 Some theoretical efforts are continuing to describe theoretically an unusually weak strain dependence of stress as well as the preparation concentration dependence of the modulus for the deswollen networks.28 Irregular networks having many dangling chains show a high damping that originates from the slow viscoelastic relaxation of a branched structure with a broad size distribution.29 It is noteworthy that this damping is insensitive to temperature because of the broad relaxation spectrum, unlike the conventional damping elastomers with the glass transition for energy dissipation. A noticeable advantage of these approaches based on network topology is that the availability is independent of the chemical structures of the network constituents. Network topology seems to still have the potential for the emergence of excellent mechanical properties. This is an interesting challenge from the viewpoint of not only industrial applications but also the physics of rubber elasticity.

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