Open questions in the physics of deformation of polymer glasses


  • Edward J. Kramer

    Corresponding author
    1. Department of Materials and Department of Chemical Engineering, University of California, Santa Barbara, California 93106-5050
    • Department of Materials and Department of Chemical Engineering, University of California, Santa Barbara, California 93106-5050
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Polymer glasses have very appealing properties for a number of optical and other applications and their high resistance to plastic deformation, expressed as the ratio of low temperature yield stress to elastic modulus of order 0.05, is an additional bonus. Their normalized resistance to crack propagation, as measured by their ratio of fracture energy Gc to 2γ, twice the surface energy, is more than 5000 for polymer glasses whose molecular weight exceeds 10 Me, where Me is their entanglement molecular weight. These are potentially very tough materials, which nevertheless have an important flaw, their propensity to deform plastically by crazing, which competes with shear deformation. Because crazes are highly localized zones of plastic deformation that can break down to form cracks at very low overall strains, crazing is a source of brittleness.

Suppressing crazing and encouraging less localized shear deformation is thus an important molecular design goal. It has been known for some 20 years that two glassy polymer properties strongly influence whether crazing or shear deformation dominates, the physical aging of the polymer glass and its entangled strand density νe = ρNAv/Me, where ρ is the polymer strand mass density and NAv is Avogadro's number. Physical aging of the polymer glass increases the shear yield stress, as measured for example in compression, above the flow stress, the stress that is required to continue shear deformation at moderately larger strains. The flow stress has been shown to be independent of physical aging, giving rise to so-called strain softening, a decrease in (true) stress as the deformation continues past yield until the flow stress is reached. Recent experiments demonstrate unequivocally that a mild plastic predeformation by rolling erases the effects of physical aging (“rejuvenation”) so that if the polymer glass is deformed immediately after, even in tension, the yield stress and flow stress are the same. Shear deformation is favored over crazing even in polymers such as polystyrene (PS), which would normally craze in preference to deform by shear.1 Aging at room temperature for 48 h is sufficient to restore the normal yield and crazing behavior of PS. Obviously an understanding of the changes in structure of the glass during the initial stages of plastic deformation by shear is crucial to achieving a fundamental understanding of the competition between brittle (crazing) and ductile (shear) modes of deformation of polymer glasses, but to date very little is known except from molecular dynamics simulations of model Lennard-Jones nonpolymeric glasses under conditions of constant density at 0 K.2, 3 These simulations, viewed from the standpoint of energy landscape models, show that the rejuvenation by deformation is due to irreversible creation of structural states with shallower energy minima, that resemble, but are distinct from, those of the glass freshly quenched from an equilibrium melt, and that physical aging corresponds to the relaxation of these structural states toward those with deeper energy minima. This relaxation at constant density restores the yield point and strain softening behavior. Although previously this relaxation has been taken to be due to a decrease in free volume, the simulations show it occurs even in the absence of such a decrease. In addition, there is no experimental evidence of a volume increase during and immediately after moderate shear deformation that produces rejuvenation. In fact, a volume decrease has been reported.1 Recent positron annihilation experiments support this conclusion.4, 5 Simulations that treat the aging and rejuvenation of polymeric glasses, at constant pressure rather than constant density and at finite temperature, will certainly help us to clarify these important issues.

How increases in the entangled strand density act to suppress crazing is also not completely clear. One explanation is that νe influences the strain hardening of the polymer network and there is excellent evidence to support this view. But the mechanism for this strain hardening is mysterious. van Melick et al.,6 provide an excellent summary of the current puzzle, the prior literature, and their own recent results: Since the strain in a strain hardened glassy polymer can be completely recovered by heating for a short time above Tg, it is natural to imagine that the hardening is due to the entropic elasticity of Gaussian strands in the entangled network. Indeed σt(λ) the true stress in the strain hardening regime as a function of extension ratio λ closely obeys the following:

equation image(1)

in compression, and if instabilities leading to localized deformation such as necking or crazing can be avoided, in tension. Equation 1 is the constitutive relation for a Gaussian rubbery network and in fact GR, which van Melick et al., call the hardening modulus, scales linearly with the network strand density νe of either a homopolymer blend of PS and poly(phenylene oxide) (PPO) or of cross-linked PS, both of which were determined directly by van Melick et al. by measuring the rubbery plateau modulus of the blend Gmath image or the rubbery modulus Gx of the cross-linked PS above Tg. However, as realized by van Melick as well as previous workers, the values of the hardening modulus are much larger than those of the corresponding elastomer. Indeed, as shown by the replot of van Melick's data in Figure 1, the hardening modulus is larger than the corresponding rubbery modulus by 2 orders of magnitude. Moreover, instead of GR increasing linearly with T (K), the signature behavior of entropic elasticity, it decreases linearly with T, extrapolating to zero close to Tg. While deviations from the expected rubber elastic behavior of the plateau modulus are certainly present near Tg, the magnitude of GR implies that the forces to extend a strand in the entangled or cross-linked polymer are 100 times those of the relevant entropic spring. It is thus hard to see how the parameters of the entropic spring can be relevant to the hardening observed. For example, if one imagines a tube model with tube diameter dt ∼ √Me, to explain the magnitude of GR would require a tube diameter in the glass that is always a factor of 10 less than dt above Tg and one that would still scale as √Mx, where Mx is the strand molecular weight of the cross-linked networks. These tube diameters would approach or even lie below the statistical segment length of the polymer in question.

Figure 1.

Strain hardening modulus GR for PPO/PS blends and cross-linked PS as well as the rubbery plateau modulus Gmath image of PPO/PS and modulus Gx of rubbery cross-linked PS versus the strand density νe of the entanglement and cross-link density. Although both sets of moduli scale linearly with νe, the hardening modulus is two orders of magnitude larger than the true modulus of the entangled or cross-linked elastomer. (Data taken from Ref.6.)

Although clearly experiments that can monitor the changes in strand dimensions with λ can help us understand how to fundamentally explain strain hardening in polymer glasses and its dependence on νe, I think that simulations can perhaps play an even more important role. In this regard the work of Rottler, Robbins, and coworkers suggests that it should now be possible to arrive at a molecular level understanding of this phenomenon.7 I therefore issue this as a challenge to the simulation community and look forward to a much improved physical picture for this important property of polymer glasses.