Microscopic origin of strain hardening in polymer glasses: the Gaussian-form scaling of the dissipative stress, full-regime energetic–dissipative partition, and a microscopic theory beyond entropic network models

Molecular simulations by Hoy and Robbins (Phys. Rev. Lett. 99, 117801, 2007) showed that the strain hardening of polymer glasses, traditionally modeled with entropic rubber-elasticity network theories, is in fact dominated by an energetic term that rises when chain segments are pulled taut between entanglements and a dissipative term tied to the rate of local plastic rearrangements — yet the dissipative stress follows approximately the same linear dependence on chain stretch $g(\lambda)=\lambda^2-\lambda^{-1}$ that Gaussian-chain network theories predict for the entropic stress. The problem, following the open agenda the source itself leaves, has three strands: (i) give an analytic explanation of why the plasticity-driven dissipative stress tracks the Gaussian entropic functional form, i.e., derive the coupling between the loss of accessible chain conformations and the rate of plastic rearrangements; (ii) obtain an experimental energetic-versus-dissipative decomposition of the hardening stress over the entire strain-hardening regime — by differentiating deformation-calorimetry measurements of mechanical work and heat into $\sigma_U$ and $\sigma_Q$ — and compare it with the simulated trends, including their entanglement-density dependence; and (iii) construct the microscopic theory of strain hardening that the source argues should be based on glassy-state physics rather than rubber elasticity, in place of entropic network fits whose parameters lack direct physical significance. Each strand is a distinct open task bearing on the same underlying question: what carries and what dissipates the hardening stress in polymer glasses. To date no accepted derivation of the Gaussian-form coincidence, no calorimetric partition beyond the plastic-flow regime, and no glassy-physics microscopic theory that reproduces the simulated stress and conformation changes and has been tested against such a partition are known: the problem remains open.

Background

When a polymer glass — an amorphous polymer cooled below its glass-transition temperature $T_g$ — is deformed past yield, the stress required to continue deforming it rises with strain instead of remaining at the plastic-flow plateau. This strain hardening stabilizes thermoplastics against necking, crazing (cavitated fibrillar deformation zones), and fracture, and it has been modeled for decades with entropic network theories borrowed from rubber elasticity: the glass is treated as a temporary network in which entanglements (topological constraints where mutually wound chains cannot pass through each other, spaced by a mean number of monomers $N_e$ between entanglements) act like crosslinks, and the hardening stress is attributed to the entropy lost as Gaussian chains between constraints stretch. For incompressible uniaxial deformation with stretch $\lambda$, such theories predict the hardening stress to grow linearly with the strain measure $g(\lambda) = \lambda^2 - \lambda^{-1}$, with a slope (hardening modulus $G_R$) fixed by the entanglement density $\rho_e$; the neo-Hookean model (a network of Gaussian chains whose strands deform affinely, i.e., following the macroscopic deformation field) and the eight-chain model (which averages the stretch over eight chains oriented along the diagonals of a cube, improving fits at large stretch) are the standard versions.

Two long-standing anomalies undermined this picture: fitting experimental hardening curves yields effective $N_e$ values systematically smaller than the intrinsic entanglement spacing measured from the melt plateau modulus (the rubbery plateau of the shear modulus in the melt, set by the entanglement network) or primitive-path analysis (a topological construction that contracts each chain onto its taut path between entanglements), and the hardening moduli of glasses are orders of magnitude larger than rubbery moduli at comparable network density, where entropic models predict $G_R=\rho_e k_B T$. Hoy and Robbins (PRL 2007; companion paper PRE 77, 031801, 2008) addressed this with coarse-grained molecular-dynamics simulations (bead–spring chains with Lennard-Jones intermonomer interactions) of glasses under uniaxial and plane-strain compression. Writing the first law for isothermal deformation, they split the stress into an energetic part $\sigma_U = \partial U/\partial\epsilon$ (internal-energy density change per unit strain, dominated by rising tension in covalent bonds as chain segments of length $\sim N_e$ are pulled taut, which produces the strong upward curvature at large strain) and a thermal part $\sigma_Q$ (heat flow out per unit strain). They found $\sigma_Q$ to be mostly irreversible and directly proportional to the rate of plastic rearrangements $P$ (measured by counting large non-affine — i.e., departing from the imposed homogeneous deformation — changes of interchain 'bonds' per unit strain), with a proportionality constant that is nearly independent of $N_e$ and chain length, so that the $P$ and $\sigma_Q$ curves collapse for all $N_e$ up to a single vertical scale factor, fluctuations included. Two distinct entanglement dependences must be kept apart here: what is $N_e$-independent is the $\sigma_Q$-to-$P$ proportionality constant, whereas the thermal hardening modulus $G_{\mathrm{therm}}$ obtained from linear fits $\sigma_Q = \sigma_0 + G_{\mathrm{therm}}\,g$ decreases rapidly with increasing $N_e$, apparently as $N_e^{-1}$ — the same density scaling that Gaussian rubber elasticity predicts for $G_R$ — while remaining orders of magnitude larger than the entropic estimate (PRE Sec. III B and Table II). Stresses for all chain lengths collapse when plotted against a microscopic measure of chain stretch rather than macroscopic strain, and unentangled short chains also harden. Recovery tests show almost none of the $\sigma_Q$ work is recovered on unloading (about 5% of that work, and about 10% of the strain, in the simulations), so it cannot be entropic in origin. Yet the total hardening stress can still be fit well by the Gaussian network form, and — the central puzzle — the dissipative, plasticity-driven $\sigma_Q$ itself rises approximately linearly in $g$ over the hardening regime, mimicking the entropic prediction whose physical basis the same simulations refute, and doing so even in the $T\to 0$ limit where any entropic stress must vanish. Hoy and Robbins conjectured that a decrease in the number of accessible chain conformations necessitates larger-scale plastic rearrangements, so the plasticity rate inherits the conformational-entropy scaling — a picture that also accounts for the observed correlation between plastic flow stress and hardening modulus and for the collapse of data for different chain lengths when plotted against the microscopic effective stretch $\lambda_{\mathrm{eff}}$. They called for analytic investigations of this scenario, for deformation-calorimetry experiments (simultaneous measurement of mechanical work $W$ and evolved heat $Q$ during deformation, differentiated to give $\sigma_U$ and $\sigma_Q$) extended beyond the plastic-flow regime into the full hardening regime where existing studies stopped, and — noting that the fit parameters of network models, including those of their own modified four-parameter eight-chain fit, do not appear to have physical significance — observed that the correct microscopic theory of strain hardening should be based on glassy-state physics rather than rubber elasticity. Deformation calorimetry is the experimental technique that measures stored energy $\Delta U = W - Q$ directly, and its strain coverage is the bottleneck for testing the simulated $\sigma_U/\sigma_Q$ trends.

Problem Statement

The problem is to replace the entropic-network description of strain hardening in polymer glasses with a microscopic account grounded in glassy-state physics, along the three open directions identified by Hoy and Robbins (2007, 2008); each is a distinct task, and the source states them as separate suggestions rather than as one conjunctive deliverable, but all three concern the same physical question — what carries and what dissipates the hardening stress in polymer glasses. (i) Analytic explanation of the dissipative-stress scaling — derive, within a microscopic model of glassy-polymer dynamics, why the thermal (dissipative) stress component $\sigma_Q$, which in the source's coarse-grained simulations is directly proportional to the rate of local plastic rearrangements $P$ and is largely irreversible on unloading, grows with chain stretch $g$ in approximately the same functional form (near-linear in $g(\lambda)=\lambda^2-\lambda^{-1}$, and persisting in the $T\to 0$ limit where any entropic stress must vanish) that Gaussian-chain entropic network theories predict for the entropic stress; in particular, establish analytically the relation the source conjectures between the decrease in the number of accessible chain conformations and the rate of plastic rearrangements — that the growing conformational constraint necessitates larger-scale plastic rearrangements, so that the plasticity rate inherits the conformational-entropy scaling — a picture that would also account for the correlation between plastic flow stress and hardening modulus and for the collapse of stresses for different chain lengths when plotted against the microscopic effective stretch $\lambda_{\mathrm{eff}}$. The explanandum is this near-linear functional dependence on $g$; an explanation must in addition be consistent with the accompanying simulation results, namely that the $\sigma_Q$-to-$P$ proportionality constant is nearly independent of $N_e$ and chain length, while the thermal hardening modulus $G_{\mathrm{therm}}$ fitted from $\sigma_Q = \sigma_0 + G_{\mathrm{therm}}\,g$ decreases roughly as $N_e^{-1}$ yet remains orders of magnitude larger than the entropic prediction $G_R=\rho_e k_B T$. (ii) Experimental energetic–dissipative partition over the full hardening regime — obtain, by deformation calorimetry (simultaneous measurement of mechanical work and heat) or an equivalent experiment on ductile glassy polymers deformed isothermally through the entire strain-hardening regime well beyond the plastic-flow plateau, the separated stress components $\sigma_U(\epsilon)$ and $\sigma_Q(\epsilon)$, and compare their trends — including the large-strain energetic upturn associated with chain segments of length $\sim N_e$ being pulled taut and its dependence on entanglement density, which the source singles out as particularly useful — with the simulation predictions. (iii) Microscopic theory beyond network models — construct a theory of strain hardening based on glassy-state physics rather than rubber elasticity that captures the changes in stress and chain conformation observed in the simulations (the energetic upturn once taut segments reach length $\sim N_e$, a dissipative stress proportional to the plasticity rate, hardening of unentangled chains, and the collapse of stresses against $\lambda_{\mathrm{eff}}$), and that resolves why fits of network models to the total stress yield effective entanglement spacings smaller than intrinsic ones and different values for different strain states, i.e., why those fit parameters lack direct physical significance. The premise of direction (i) is established within the source's coarse-grained simulations, but its counterpart in real materials is contested: experiments on cold-drawn polycarbonate and polystyrene attribute the stress to intrachain retractive tension of a strained chain network rather than to dissipative intersegmental interactions (refs 8, 9; see also ref 10). Direction (i) therefore asks for an analytic account of the dissipative-stress scaling reported in those simulations, while direction (ii) is the measurement that would test which molecular picture holds in real ductile glasses. Because no calorimetric partition extending beyond the plastic-flow regime is known, direction (ii) is to obtain the separated trends $\sigma_U(\epsilon)$ and $\sigma_Q(\epsilon)$ over the full strain-hardening regime, including their entanglement-density dependence.

References

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