## Abstract

We examine the influence of steady shear on structural relaxation in a simulated coarse-grained unentangled polymer melt over a wide range of temperature and shear rates. Shear is found to progressively suppress the α-relaxation process observed in the intermediate scattering function, leading ultimately to a purely inertially dominated β-relaxation at high shear rates, a trend similar to increasing temperature. On the basis of a scaling argument emphasizing dynamic heterogeneity in cooled liquids and its alteration under material deformation, we deduce and validate a parameter-free scaling relation for both the structural relaxation time τ_{α} from the intermediate scattering function and the “stretching exponent” β quantifying the extent of dynamic heterogeneity over the entire range of temperatures and shear rates that we can simulate.

## INTRODUCTION

Glass-forming liquids are often subjected to shear deformation and flow during their processing, leading to substantial changes in their relaxation dynamics in comparison to unsheared materials at equilibrium. Many measurements (*1*–*4*) and simulations (*5*–*7*) have shown that the structural relaxation time and effective viscosity of deformed complex fluids can decrease following material deformation, sometimes by orders of magnitude. Measurements have also shown that the relaxation dynamics of deformed glassy materials tend to recover to their equilibrium properties after long “aging” times (*8*), suggesting some sort of structural and dynamical recovery of the material. Despite the abundant experimental literature, there is limited understanding of these changes in dynamics from a theoretical perspective. This situation is perhaps not unexpected, since there is not even a consensus about the origin of the dynamics of glass-forming liquids under equilibrium conditions.

In the present work, we address the essential phenomenology of a model glass-forming polymer melt subjected to steady shear in the liquid regime at temperatures below the onset temperature *T*_{A}, where non-Arrhenius α-relaxation becomes prevalent (*9*). We restrict our attention to relatively short chains to avoid complications arising from chain entanglement. First, we characterize the slowing down of the α-relaxation through the study of the α-relaxation time *F*_{self}(*q*_{0}, *t*) at a scale *q*_{0} corresponding to the intermolecular distance, as determined by the primary peak of the static structure factor *S*(*q*). These equilibrium data are compared to the relaxation time *F*_{self}(*q*_{0}, *t*) becomes progressively suppressed with increasing *F*_{self}(*q*_{0}, *t*) at fixed *T* is insensitive to _{α} decreases with increasing _{α} follows a universal scaling relation as a function of a reduced stress, with apparently no adjustable parameters for both polymeric and small-molecule glass-forming liquids. The stretching exponent β of the α-relaxation in *F*_{self}(*q*_{0}, *t*) also shows a general transition from a diffusion-dominated regime to an inertially dominated ballistic regime and can be described by its own reduced variable description. In the context of the critical phenomena of fluids near their critical point for phase separation, it is well known that shear causes a progressive breakdown of transient particle clusters arising in connection with the incipient phase separation process, so that the lifetime of these transient clusters becomes independent of the average cluster size, as measured by the correlation length ξ for density and composition fluctuations in one and two component phase separation processes (*10*, *11*). This mechanism of shear thinning is probably very common in complex fluids exhibiting dynamic self-association, and we may likewise expect large changes in the stress relaxation time τ_{α} of glass-forming liquids under steady shear deformation to be accompanied by the breakdown of dynamical immobile particle clusters under steady shear that should occur generally in this broad class of fluids (*12*, *13*). Correspondingly, our analysis of shear thinning in our model polymeric glass-forming liquid is based on the working hypothesis that shear thinning in glass-forming liquids arises from the breakdown of immobile particle clusters under flow rather than from an alteration of the activation energy as in the conventional Eyring model for relaxation in deformed fluids (*3*). Previous work has shown that the lifetime of the immobile clusters in both polymeric and nonpolymeric liquids can be identified with the structural relaxation time extracted from the intermediate scattering function under equilibrium conditions (*12*, *13*), and the present work shows that this interpretation extends to our fluid under steady shear.

Our modeling of the reduction in τ_{α} under steady shear in terms of the breakdown of dynamic clusters of immobile particles accords with the mechanism of the reduction in the order parameter relaxation time under steady shear in critical fluids, which is one of the few complex fluids for which there is a rigorous, and extensively experimentally validated, theory of the nonlinear rheology of a real complex fluid (*10*, *11*). Nonetheless, our modeling of the shear thinning in glass-forming liquids is just a working model of a complex phenomenon for which there is no rigorous theory, even in the absence of shear deformation.

## RESULTS

The effective shear viscosity η, defined during the steady state as *T* dependence decreases up to the point where it is not appreciable for high shear rates.

Shear thinning is a common phenomenon in complex fluids (*14*) in which the structural organization of the fluid is broken down under shear. The shear thinning of solutions of self-assembled actin and tubulin molecules provides a protypical example of this phenomenon (*15*). As noted before, previous work has identified growing dynamical clusters of immobile particles forming in our model glass-forming liquid (*12*), so this interpretation of shear thinning in terms of “depolymerization” under flow seems plausible also for glass-forming liquids (we do not refer here to the breaking of chemical bonds, but rather to the transient associative bonds between molecules, as normally encountered in systems undergoing supramolecular assembly). We adopt this dynamic heterogeneity perspective below in our quantification of the shear-thinning effect that we observe.

To describe relaxation in our system under shear, we consider the self-part of the intermediate scattering function*q*_{0} ≃ 7.14 is the value at which the maximum of the static structure factor is observed. Following previous protocols (*7*), only *q* vectors in the *yz* plane orthogonal to the direction of the shear are considered.

Fig. 2 shows the intermediate scattering functions at the lowest temperature investigated for all shear rates. It is apparent that with increasing shear rate, the α-relaxation is suppressed, the α-relaxation time decreases, and, eventually, the α- and fast β-relaxations merge for high shear rates, similarly to the effect achieved by increasing *T*. Similar results have been observed before in two-dimensional (2D) (*6*) and 3D (*7*) model atomic liquids. To quantify these results, we fit our data to a two-step relaxation function, successfully used to describe *F*_{self} in thin films and bulk materials at equilibrium (*16*)

In qualitative accord with Eq. 2, the mode coupling theory likewise indicates (*17*) an additive contribution of a fast β-relaxation, with a *T*-independent β-relaxation time, and α-relaxation process, whose relaxation time is strongly dependent on *T* and whose amplitude is independent of *T* above the “glass transition” of this model, *T*_{g}. Correspondingly, we tentatively take τ_{f} and β_{f} to be fixed and determine α-relaxation time τ_{α}, *A*_{α}, and stretching exponent β as function of *T* and

We find that the values of the parameters τ_{f} and β_{f} of the initial “fast” relaxation process (see Fig. 2) can be fixed to τ_{f} = 0.3 and β_{f} = 1.3, respectively, defining the fast β-relaxation time as the exponent β_{f} quantifying the nonexponential nature of the fast relaxation process. The “mode amplitude” parameter *A*_{α} shows a relatively weak *T* and *A*_{α} to equal *A*_{α} = 0.7, except at the lowest temperature *T* = 0.43, where this parameter increases to a somewhat larger value, *A*_{α} = 0.73. The accuracy of the fit is not appreciably reduced in the *T* range investigated by fixing the three parameters (τ_{f}, β_{f}, and A_{α}). The lines in Fig. 2 are the fit of Eq. 2 to the data (circles) for all shear rates. From the fits of Eq. 2 to the simulation data, values of τ_{α} and β can be derived for all *T* and *F*_{self}(*q*_{0}, *t*) becomes increasingly “suppressed” with increasing *T* on *F*_{self}(*q*_{0}, *t*)]. In particular, we observe that τ_{α} becomes progressively reduced by *T* = *T _{A}*, where

*T*≈ 0.7 for this model) at a characteristic shear rate

_{A}_{β}as

*18*). For the present model, we find

*T*dependent and can be roughly estimated from the relaxation time data shown in the inset of Fig. 1. Below, we use these values of

_{f}, a time scale on the order of a picosecond in laboratory units.

We now show that the changes in *T* and *T*-dependent plateau followed by a decrease with increasing *T* dependence vanishes and the τ_{α} value is comparable with the fast β-relaxation time.

We draw upon basic scaling theory, symmetry arguments, and previous modeling of the shear rate dependence of the structural relaxation in sheared near-critical fluids to develop an approximant for the shear rate dependence of the structural relaxation time _{α} is invariant to the choice of the direction of shear so that *4*) provide some additional guidance regarding the form of this expansion, and the theory of shear thinning in fluids near their critical point gives us some physical and mathematical insights into how to model the shear thinning in glass-forming liquids (*10*). In particular, measurements on sheared near-critical Xe have indicated that the shear viscosity under steady shear is better described as a function of

The dynamical renormalization group (RG) theory of shear thinning in critical fluids indicates that the reduction in the “order parameter relaxation time” τ, the viscoelastic relaxation time describing the lifetime of dynamic particle clusters associated with incipient phase separation, can be understood from the progressive breakdown of these clusters under steady shear (*10*, *11*). In the absence of steady shear deformation, the shear viscosity and normal stresses of the near-critical fluid diverge as a power of the correlation length ξ describing the average size of the dynamic particle clusters, where the precise power relating the zero-shear viscosity *11*), so that the shear viscosity of critical fluids is completely independent of shear rate at high shear rates (*10*), i.e., the critical clusters are completely disintegrated at high shear rates. Below, we estimate the shear thinning exponent describing the dependence of *12*, *13*) are likewise progressively broken down with increasing shear rate, so that *12*, *13*) are the fundamental structural elements in glass-forming liquids responsible for the growth of the viscosity upon cooling.

The next step in our scaling argument requires a specification of the general functional form *19*)*a* is a small constant calculated from the RG theory, *20*), but our motivation for invoking this relation based on RG theory crossover scaling is different from conventional arguments for this equation.

An acceptable scaling expression for _{β} at high shear rates, _{β} is a temperature-insensitive time on the order of 1 ps in laboratory units. To account for this limiting behavior, we then revise our expression to the simple form

We find that our data best fit Eq. 5 by taking _{β} for our model glass-forming liquids and glass-forming liquids in general is on the order of 1 ps (*21*), then _{β}. More generally, this characteristic time probably has a weak temperature dependence (as in the case of τ_{β}), but this small variation is unessential in our data reduction below, as this only affects the data points at the extremely high shear rates. An analysis of the robustness of the data collapse as a function of

Many previous theoretical and experimental studies have fit shear relaxation data to the functional form of Eq. 4 or 5 or the more general Carreau-Yasuda functional form where ϕ is commonly taken to equal 2 and the constant of proportionality in the reduced shear rate is taken as an adjustable “nonuniversal” constant [see table 1.7 of Tanner’s *Engineering Rheology* (*22*)]. Correspondingly, we have found that we can fit our *23*, *24*) of _{c} signaling the onset of “elastic turbulence” where a sharp change in the “fluidity” of the melt and flow instabilities in polymer melt extrusion measurements are observed. The utilization of a physical condition for defining a reduced shear rate offers an attractive method for making our expression for

Within the RG scaling scheme discussed above, it is natural to define a dimensionless shear stress by a quantity that delineates an approach to a strong coupling scaling regime, and we follow former experimental practice in how we identify our reduced stress variable to obtain a unified description of the shear dependence of *T* and *23*, *24*) have identified a critical shear stress based on the observation of a transition to unstable fluid flow, and this stress was used to define a reduced shear stress. Correspondingly, we identify the condition at which τ_{α} merges with τ_{β} as corresponding to this flow instability condition, because this condition implies that the relaxation of the fluid by momentum diffusion (shear viscosity) has been essentially suppressed, so that the fluid can be viewed as roughly equivalent to an inviscid fluid from a modeling standpoint. “Turbulent” or “chaotic” flow is a common phenomenon in the flow of fluids having a low viscosity, and even if there is no exact theory of this ubiquitous phenomenon, this general tendency can be qualitatively understood from recent computational investigations of particle motion in idealized inviscid fluids, where convincing numerical evidence is presented that the motion of a solid body through an incompressible, inviscid fluid, moving irrotationally and otherwise at rest, is inherently chaotic (*25*). On the basis of previous experimental practice and physical argument just stated for identifying the approach of τ_{α} to τ_{β} with the critical condition for unstable flow, we define our reduced stress σ_{R} by σ_{N}/σ_{c}.This critical stress can be estimated both by simulation and experiment, at least through extrapolation. The β-relaxation time of the unsheared material is on the order of 1 ps in the *F*_{self}(*q*_{0}, *t*) data discussed above [τ_{β} is on the general order of 1 ps for molecular liquids (*21*)] and in measurements (*26*), so *T* approaches the onset temperature *T _{A}* for non-Arrhenius relaxation (

*18*). If we define the critical shear rate

*27*). To further specify an appropriate stress variable that is defined in terms of experimentally and computationally measurable properties, we also need an estimate of the relation between

_{α}. For this estimate, we assume the validity of the Maxwell relation η

_{0}= Gτ for the fluid at equilibrium, where G is the experimentally measured shear modulus

*G*in a high frequency shear relaxation measurement, and τ is the shear stress relaxation time. In a glass-forming liquid at low temperatures,

_{g}*G*corresponds to the “glassy modulus” and τ to the relaxation time, but in the simple fluid regime, where appreciable chain diffusion occurs on the relaxation time of the fluid,

*G*corresponds to the Rouse or entanglement modulus and the relaxation time corresponds to the Rouse or “terminal” relaxation time of the polymer melt, respectively. Our focus on τ

_{α}in the present paper allows us to focus on the influence of shear on the segmental relaxation time, which is not sensitive to either molecular mass or large-scale polymer relaxation associated with large scale chain diffusion. We may then reasonably take

*G*in the Maxwell relation to correspond to the glassy modulus and the relaxation time τ to correspond to the segmental relaxation time, which is of the order of τ

_{α}. Recent work has shown that

*G*

_{g}of the glass-forming material should not be identified with the infinite frequency shear modulus

*G*

_{∞}, but rather with an intermediate time “plateau” in the shear relaxation function corresponding to the glassy relaxation process, in analogy with A

_{α}in the decay of the intermediate scattering function (see Eq. 2) (

*28*). Consistent with our proposed data reduction in our

*28*) high frequency modulus G

_{g}has previously been used for the effective data reduction in shear thinning measurements on the shear viscosity of soda-lime-silica glasses (

*29*). We note that the shear viscosity of polymeric glass-forming liquids inherently involves diffusion and relaxation processes at the scale of the polymer, so the viscosity of the polymer fluid should be much more sensitive to the polymer massthan τ

_{α}and it is not clear that the reduced shear rate deduced above for τ

_{α}should apply to the shear viscosity of polymer melts under steady shear. The α-relaxation time is a probe of dynamics at a segmental scale and is an inherently simpler quantity than the shear viscosity, diffusion coefficient, normal stresses, and other polymer melt transport properties.

Now, we have defined our reduced stress variable_{α} obtained from the intermediate scattering function (*6*, *12*, *24*), and the definition noted above of the critical stress σ_{c} in terms of the magnitude of the critical strain rate _{c}/*G*_{g}. Eq. 5, in conjunction with the approximate relation of Eq. 6, then provides a predictive relation for *G*_{g}(*T*) from *28*). We observe that Eq. 5 describes our *T*.

It is also notable that the shear thinning that we observe, and its physical interpretation of the breakdown of self-assembled structures under flow, seem to be broadly consistent with observations on naturally occurring complex fluids exhibiting thermally reversible self-assembly (actin, tubulin, etc.) (*15*). All these fluids exhibit a similar “shear thinning” at high shear rates with a shear-thinning exponent near −1 in the limit of high shear rates, defining a hydrodynamic limit of flow instability, i.e., the flow shear stress becomes essentially indeterminant (*15*). We note that the shear thinning that we observe apparently has little to do with the polymeric nature of fluid but rather derives from the emergent dynamic heterogeneity of the cooled glass-forming liquid as these fluids are cooled. This is evidenced by the essential disappearance of the shear thinning upon raising the temperature toward the onset for non-Arrhenius relaxation *T _{A}* and by the observation of a very similar shear thinning in binary Lennard-Jones (LJ) fluids introduced to model metallic glass materials (see the Supplementary Materials). We view the polymeric nature of our glass-forming liquid as a convenient way to suppress any tendency toward crystallization so that dynamic heterogeneity occurs as an equilibrium phenomenon. The dynamics of crystallizing glass-forming materials, such as the binary LJ liquid, is complicated by equilibrium “aging” phenomena associated with the metastable nature of the supercooled state, yet this class of glass-forming liquids seems to exhibit a similar dynamics to our model polymeric glass-forming liquid.

By a similar reasoning to Eqs. 5 and 6, we may also deduce an approximate reduced variable description for the stretching exponent *30*) have shown that the β exponent for stress relaxation in solutions in which there are growing dynamic clusters can be exactly described by a sigmoidal variation between 1 at high temperatures to a value that saturates near 1/3 at lower temperatures, when the clusters have fully grown to large dimensions and persist for very long times. Since increasing shear has an effect on the immobile clusters that is similar to increasing temperature, we may generally anticipate a sigmoidal variation of β with shear rate as well. This expectation is again based on assumption that the decay of the intermediate scattering function and the structural relaxation time are physically closely related, a common, but unproven, assumption that explains why τ_{α} is designated the “structural relaxation time.” At any rate, simulations by Riggleman and co-workers (*3*, *31*), although for a different mode of deformation, have indicated a general tendency of β to increase under deformation, consistent with our cluster breakdown interpretation of the variation of β with applied shear. For our system, β increases monotonically with increasing ^{0} near 2/3 at ^{∞} that is larger than unity. β^{∞} apparently approaches a limiting value around 1.45 for large _{f} for the fast β-relaxation process. Here, we have further evidence that the α-relaxation process is being suppressed at high shear rates, where the overall relaxation progressively becomes more like the β-relaxation process seen only at short times in the quiescent fluid. A shortcoming of the equilibrium polymer model of stress relaxation for glass-forming liquids is that this model does not account for the fact that the immobile particle clusters form randomly branched polymers, which makes the low-temperature value of the β exponent somewhat larger in this type of dynamically heterogeneous fluid (*32*). Moreover, this model of a fluid exhibiting dynamic clustering corresponds to particles in two states at any given time: particles in polymeric clusters or undergoing Brownian motion in their “mobile” or “unassociated” particle state, so the short time dynamics in the equilibrium polymerization model is Brownian rather than inertial. Despite the quantitative disparities of the solution chain assembly model and the immobile clustering process observed in simulations of glass-forming liquids, the overall trend that we see in the shear dependence of β is qualitatively consistent with the expected behavior arising from immobile particle cluster breakdown under shear in glass-forming liquids, along with the contributing factor of the emergence of relaxation dominated by inertia at high shear rates. In particular, the transition from β^{0} to β^{∞} is consistent with a transition between diffusion-dominated relaxation to a predominantly inertially dominated relaxation regime where the α-relaxation is entirely suppressed. For each temperature, we then consider the characteristic shear *T*, going from 0.077 at *T* = 0.55 to 0.013 at *T* = 0.43. Introducing the reduced variable

Our scaling expression for ^{∞} should generally equal 1.5 for all liquids. Nonetheless, this cross-function should provide a good means of quantifying this crossover from a purely correlative perspective.

Previous simulation studies by Starr *et al.* (*12*) have shown that fractal dynamic particle clusters of relatively immobile polymer segments exist in our coarse-grained polymer model and that the lifetime of those clusters is proportional to *33*, *34*). Riggleman and co-workers have further shown that immobile clusters exist in the glassy state, where they persist for extremely long time scales (*35*). Evidence for immobile particle clustering can also be inferred from nanoscale imaging of the interface of polymeric and metallic glass-forming materials (*36*). We note that our interpretation of the reduction in the relaxation time through the breakdown of immobile particle clusters is consistent with the recent comprehensive experimental studies by Ediger and co-workers (*2*), where it is emphasized that a wide range of measurements indicate that the application of stress accelerates mobility in immobile regions in the material much more than the mobile regions, as evidenced by the increase in β under deformation sufficient to cause flow. Correspondingly, we next check our hypothesis that the variation of

Starr *et al.* (*12*) define an autocorrelation function defining the persistence of the immobile particles in their “caged” state, and this method was also used in a recent work on metallic glass-forming liquids (*13*), where similar results were obtained. We follow these previous works in our definition of the immobile particles and then determine how the application of shear alters the persistence time of the immobile particle clusters and how this time relates to *et al.* (*12*) for a detailed discussion. In the present work, we confine our attention to the *yz* plane to be consistent with our analysis of *F*_{self}(*q*_{0}, *t*) to consider how *C*(*t*) at the relatively low temperature *T* = 0.43.

Consistent with the equilibrium analysis of the lifetime of the immobile clusters under variable temperature conditions (*12*), the decay time is comparable to the structural relaxation time τ_{α} derived from the intermediate scattering function, becoming shorter with increasing *12*)] in a time *r*_{c} = 1.46) have a different color. It is apparent that while the equilibrium system shows larger immobile particle clusters, they are indeed progressively broken down into smaller and more spatially heterogeneous clusters with increasing *T* at equilibrium.

Of course, the mobile particles also play an important role in glass-forming liquids at equilibrium (*12*) as these particle determine the rate at which particle exchange between their mobile and immobile states, and string-like motion has been observed in experimental systems under shear (*15*, *14*). A more detailed analysis of the string-like motion associated with the most mobile particles under shear will be presented in future works (see Discussion).

## DISCUSSION

Given the practical importance of material deformation and flow on the processing and performance of glass-forming polymer materials, there have naturally been numerous experimental and computational studies of how relaxation and transport processes in glass-forming materials are influenced by molecular and thermodynamic properties and flow conditions. There is an extended scientific literature on this topic, recently reviewed by Hebert *et al.* (*37*), who summarize the findings of numerous experimental and computational studies. The measurements summarized in this work indicate, under constant strain rate conditions and for temperatures 10 K to 20 K below *T*_{g}, that the relaxation time in the postyield regime decreases with an apparent power law behavior, with an exponent close, but slightly below, −1. A corresponding reduction in the relaxation exponent β with strain rate was also observed. For example, Hebert and co-workers indicate a shear thinning exponent of about −0.8 to −0.9 for cross-linked polymers slightly below *T*_{g}, which compares well with our simulations at a much higher shear rate and at a temperature above *T*_{g}. For example, a log-log plot of our raw data of τ_{α} as a function of *T* = 0.43) indicates a power law dependence with a slope of −0.79 ± 0.03 (see the Supplementary Materials for graphical illustration of this finding). The dynamic mean field theory of Schweizer and co-workers (*38*) also indicates an effective power law having an exponent with similar magnitude (≈0.86) for shear thinning flow when the results are analyzed in a similar fashion. We note that the model of Schweizer and co-workers (*38*) and its antecedent mean-field model by Eyring, emphasizing changes on the activation energy for relaxation under material deformation, cannot address changes in β since this class of models assumes that glassy materials are perfectly homogeneous.

There has apparently been no previous attempt to obtain a universal reduced variable description of *4*) and polymeric liquids (*23*, *24*) have introduced reduced variable descriptions of the shear viscosity that are close in spirit to our simulation findings, however. Our *1*, *2*). The small variability in the apparent inverse power law scaling with shear rate probably derives from the fact that power law scaling in Eq. 5 is only approached asymptotically at extremely large strain rates. We then suggest that the somewhat smaller exponents, sometimes reported in glass-forming liquids and also seen in measurements on sheared self-assembled protein solutions (*15*), are probably a consequence of this type of crossover effect (see the Supplementary Materials).

We briefly comment on modeling changes of τ_{α} based on the assumption of high shear limit corresponding to a quasi-static deformation of the material in the limit of zero temperature. Our simulations indicate that the dynamics in the high shear limit approaches a state that is more similar to a gas than a zero-temperature solid. For our highest shear rates, the structural relaxation time is reduced by steady shear to a point that the material can no longer support any shear stresses at any frequency and particle motions become nearly ballistic rather than caged by surrounding particles (see the Supplementary Materials where the near ballistic nature of molecular displacements are indicated at high shear rates). At extremely high shear rates, but lower than the “critical” value at which *A*_{α} (see the Supplementary Materials). Moreover, the potential energy landscape becomes fundamentally altered when shear is imposed so that drawing an equivalence of the material under steady shear to a material at equilibrium cannot be possible beyond the weak deformation case.

This lack of equivalence between strain rate and temperature has also been demonstrated in the four-point density correlation analysis where anisotropy in transport is indicated in the steadily strained material (*6*). The two-point density correlation as well as the static and dynamic structure factors are insensitive to this shear-induced anisotropy, and we view this as roughly supporting the idea of a shear rate–dependent effective temperature in a mean field sense. At any rate, relaxation in sheared glass-forming liquids remains an activated process, and increasing the shear rate progressively suppresses α-relaxation processes mediated by diffusion in favor of inertial relaxation processes and the same general trends arise when we raise the temperature of the fluid toward its liquid-vapor critical point.

In summary, we find a predictive relation for the structural relaxation time under shear *G*_{g} of the liquid at equilibrium. The theoretical framework adopted is consistent with the idea that the clusters of immobile particles relevant for the dynamics of supercooled liquids at low temperatures are broken down under shear (*37*). It should be appreciated that clusters of highly immobile particles that also arise in cooled liquids are implicated in understanding the rate of molecular diffusion and relaxation processes, and those clusters have been emphasized by Onuki and co-workers (*39*) in understanding shear thinning of model glass-forming liquids.

Mobile particles involving collective particle exchange have been shown to be highly correlated with changes of the activation free energy of diffusion and relaxation so that these clusters must also influence τ_{α} through their effect on the rate at which the immobile particle exchange with the surrounding fluid particles. It is this type of change in the activation free energy under deformation that is envisioned in the Eyring model, and evidence indicates that this type of model has merits at low degrees of deformation before the material exhibits “yield” and a transition to material flow. It is anticipated that this distinct type of dynamic heterogeneity (*12*) plays an important role in understanding shear banding, plastic instabilities, and other instabilities, brought about by the progressive conversion of immobile particles into mobile particles through the application of shear. This idea is supported by experimental reports of increases in mobility by many orders of magnitude in shear bands (*40*) and modeling indicating that shear banding occurs from the deformation-induced emergence of regions of transiently high mobility in the glass-forming liquid (*41*). Simulation observations of large-scale rearrangements of relatively mobile particles undergoing cooperative rearrangements in metallic glass materials undergoing plastic deformation (*42*), and an observed quantitative relation in molecular dynamics simulations between the extent of string-like collective motion and the activation energy of Johari-Goldstein β-relaxation process in metallic glasses, a relaxation process highly correlated with the occurrence of plastic deformation and brittleness of metallic glasses (*43*), further support this possibility. The emergence and significance of the mobile particle clusters under strong shear conditions have been emphasized by Tanaka and co-workers (*6*), on the basis of a four-point density correlation analysis in which the corresponding cooperative motions are found to be more prevalent than for *F*_{self}(*q*_{0}, *t*). There have also been experimental (*14*) and simulation (*44*) reports of the appearance of string-like organization of particles in colloids and molecular fluids at high rates of *18*). For

As a final point, we note that care must be taken in inferring change of the shear viscosity based on estimates of the relaxation time *G*_{g} can also be expected to depend on *N*_{1} and *N*_{2}) under shear deformation, in a separate publication.

## MATERIALS AND METHODS

We model polymers as fully flexible linear chains of beads linked by harmonic springs. Each chain has *M* = 20 monomer, far below the entaglement regime. *N*_{c} = 500 chains are simulated for each system, for a total of *N* = 10,000 monomers. Nonbonded monomers belonging to the same or different chains interact with a truncated LJ potential*r*_{cut} = 2.5σ, σ* = 2^{1/6}σ is the position of the potential minimum with depth ε. The value of the constant *U*_{cut} is chosen to ensure *U*^{LJ}(*r*) = 0 at *r* ≥ *r*_{c} = 2.5 σ. Bonded monomers interact with a harmonic potential *U*^{b}(*r*) = *k*(*r* − *r*_{0})^{2} with *k* = 555.5 ε/σ^{2} and *r*_{0} = 0.97 σ. Henceforth, all quantities are expressed in terms of reduced LJ units, i.e., ε = 1, σ = 1, with unit monomer mass and Boltzmann constant. The reduced units can be mapped onto physical units relevant to generic nonequilibrium fluids, by taking molecular dynamics (MD) time, length, and energy units as corresponding roughly to about 2 ps, 0.5 nm, and 3.7 kJ/mol, respectively. MD simulations were carried out with the LAMMPS (large-scale atomic/molecular massively parallel simulator) code (http://lammps.sandia.gov). The systems were initially equilibrated in the NPT ensemble (constant number of particles N, pressure P and temperature T) using a Nose-Hoover thermostat and barostat with 〈*P*〉 = 0 to allow full correlation loss of the end-to-end vector of the polymer chains. After equilibration, shear deformation was applied to the simulation box at fixed shear rate in the *xy* plane. Periodic boundary conditions are used during the NPT equilibration. A range of temperatures from *T* = 0.43 to *T* = 0.55 and a range of shear rates from

### Deformation protocol

Simple shear is performed in the *xy* plane at fixed rate * _{xy}* is considered as a function of the total strain

*F*and

_{xkl}*r*are the

_{xkl}*x*components of the force between the

*k*th and the

*l*th monomer and their separation, respectively, and

*v*is the average per-monomer volume, i.e.,

*v*=

*L*

^{3}/

*N*, where

*L*is the box size.

The main panel of Fig. 5 reports the typical stress loading curve for our systems during shear. The stress along the plane of shearing σ* _{xy}* initially increases with increasing strain for small deformations. After reaching a maximum value (the yield strength), it then drops to a lower value (strain softening) and remains constant. The properties of the stress overshoot are widely studied and present a rich phenomenology (

*38*), but this interesting transient effect is not the focus of the present paper. We focus on the steady-state flow regime that develops after the overshoot. Every system under study is initially deformed up to total strain γ = 10

^{2}, beyond the overshoot that is observed around γ ∼

*O*(1) for all temperatures and shear rates studied. The inset of Fig. 5 shows the steady-state stress value for all the

*T*and

## SUPPLEMENTARY MATERIALS

Supplementary materials for this article is available at http://advances.sciencemag.org/cgi/content/full/6/17/eaaz0777/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**

**Funding:**This work was supported in part by National Institute of Standards and Technology (NIST) Award 70NANB15H282. A grant of computing time from IT Center, University of Pisa and Dell EMC Italia is also acknowledged.

**Author contributions:**A.G., J.F.D., F.P., and D.L. conceived the research. A.G. and J.F.D. designed the simulations and analyses. A.G. and J.F.D. conducted the analyses. All authors contributed to the writing of the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).