## INTRODUCTION

Mechanical deformation of polymer glasses has been studied for many decades, and the basic features of the stress–strain curves are well known.1 At very small strains the response is elastic. At slightly larger strains, yielding occurs when intermolecular barriers to segmental rearrangements are overcome. Following yield, the material may exhibit strain softening, a reduction in stress to a level corresponding to plastic flow. At higher strains, the stress increases again as the chain molecules orient, in a process known as “strain hardening.” The balance of strain softening and strain hardening is critical in determining material properties such as toughness. Polymers that exhibit greater strain hardening, such as polycarbonate, are tougher and tend to undergo ductile rather than brittle deformation, because strain localization is suppressed.

The complex stress–strain behavior of polymer glasses has often been modeled using rubber elasticity theory.2 Glasses are assumed3 to behave like a crosslinked rubber, with the number of monomers between crosslinks equal to the entanglement length *N _{e}*. The contribution of strain hardening to the stress is then associated with changes in the entropy of the entanglement network under macroscopic deformation. For uniaxial stress with a longitudinal stretch λ, this contribution is given by

where σ(λ) is the true stress, *s* is the entropy per unit volume, and *T* is the temperature. In the simplest case, known as “Gaussian” hardening, eq 1 yields σ(λ) = *G _{R}*(λ

^{2}− 1/λ), with

*G*the “strain hardening” modulus.

_{R}*G*is predicted to be linearly proportional to both

_{R}*T*and the entanglement density ρ

_{e}:

*G*= ρ

_{R}_{e}

*k*

_{B}*T*.

It is not clear why an entropic argument should apply in the glassy state where chains cannot move freely to sample the configurational entropy,4 but the network model of strain hardening has had much success in describing experimental results on polymer glasses. For example, Gaussian hardening has been observed in many uncrosslinked glasses.5, 6 More recently, van Melick et al. performed experiments7 that showed *G _{R}* has the predicted linear dependence on ρ

_{e}. However, in contrast with the entropic prediction,

*G*was not proportional to

_{R}*T*but instead decreased linearly with increasing

*T*. Also,

*G*was found7 to be about 100 times larger than ρ

_{R}_{e}

*k*

_{B}*T*, even near the glass transition temperature

*T*. The higher modulus can be attributed8 to “frictional forces” or to the greater energy necessary to plastically deform a material below

_{g}*T*, but quantitatively little is known. Other open questions about glassy strain hardening remain as well, as summarized recently by Kramer.4

_{g}In this paper we examine the effect of entanglement density, temperature, chain length, and strain rate on the strain hardening behavior of model polymer glasses. Several previous simulation studies have considered strain hardening,9–14 but none have examined the factors controlling *G _{R}* over a wide parameter space. This is desirable to understand the results of van Melick et al. and other recent experiments.7, 15, 16 To examine chemistry-independent factors controlling

*G*, we use a generic coarse-grained bead-spring model.17 The lower computational cost of this model allows us to simulate a wide variety of relatively large systems, allowing for good statistics and precise measurements of

_{R}*G*.

_{R}We find that the functional form of the stress–strain curves at fixed temperature and strain rate is consistent with entropic elasticity as defined by eq 1. Both Gaussian hardening and the more dramatic “Langevin” hardening2 are observed. Moreover, the transition between these two forms is consistent with rubber-elastic predictions.2 In addition, we reproduce the key result of van Melick et al., *G _{R}* ∝ ρ

_{e}, over a comparable range of entanglement densities.

Other simulation results reveal dramatic inconsistencies with the entropic network model. As in the experiments of van Melick et al.,7*G _{R}* drops linearly with increasing

*T*. This drop extends to the

*T*→ 0 limit, which is clearly inconsistent with eq 1. The ratio of

*G*to ρ

_{R}_{e}

*k*

_{B}*T*is also comparable to experiment, remaining of order 100 even near

*T*. Our results for the variation of

_{g}*G*with

_{R}*T*, intermolecular interactions, and the rate of deformation can be understood if

*G*scales with the plastic flow stress σ

_{R}_{flow}rather than a network entropy. Indeed entire stress–strain curves at different strain rates and interaction strengths collapse onto a universal curve when scaled by σ

_{flow}.

It is known that *G _{R}* decreases with decreasing molecular weight, and this has been attributed to greater relaxation of the entanglement network.15 We study the entire range of molecular weights from the

*N*≪

*N*to

_{e}*N*≫

*N*limits, with

_{e}*N*the degree of polymerization. We find significant strain hardening even in unentangled systems. At small strains the chains deform affinely, and their increased length and alignment leads to strain hardening that is very similar to that of entangled chains. Only at large strains do the alignment and strain hardening begin to drop below those in entangled systems. The chain length dependence combined with the rate dependence discussed above suggests that strain hardening can be expressed as a product of the flow stress and a factor that represents the amount of local plastic deformation required to maintain connectivity of the chains.

Our simulations also allow us to examine microscopic quantities that are not easily accessible in experiments. Entangled chains deform affinely at large scales, as expected if entanglements act like crosslinks, and there is little entanglement loss through slippage at chain ends. The underlying entanglement structure is studied using primitive path analysis.18 A primitive path is the shortest path a chain fixed at its ends can take without crossing any other chains.19 The scaling of primitive path lengths with increasing strain is well described by a model assuming affine stretching of paths, and also by the nonaffine tube model of Rubinstein and Panyukov.20 The degree of plastic deformation is studied by examining the nonaffine component of deformation at low temperatures. Results for different entanglement densities fall on a universal curve for low strains, but increase more rapidly for higher entanglement densities at large strains. Strain hardening is related to microscopic plastic events, which are required to maintain chain connectivity. As the stress rises with increasing strain, both the number of events and the energy dissipated per event increase.

In the following section we describe the polymer model used in our simulations, and the protocols used to strain the system and identify primitive paths and entanglement lengths.18 Next we describe the effect of entanglement density, temperature, interaction strength, strain rate, and the microscopic rearrangements of monomers, chains, and primitive paths. The final section contains conclusions.