Athermal shear-transformation-zone theory of amorphous plastic deformation.II. Analysis of simulated amorphous silicon

Eran Bouchbinder,1 J. S. Langer,2 and Itamar Procaccia1

1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel2Department of Physics, University of California, Santa Barbara, California 93106-9530, USA

�Received 2 November 2006; revised manuscript received 1 January 2007; published 12 March 2007�

In the preceding paper, we developed an athermal shear-transformation-zone �STZ� theory of amorphousplasticity. Here we use this theory in an analysis of numerical simulations of plasticity in amorphous silicon byDemkowicz and Argon �DA�. In addition to bulk mechanical properties, those authors observed internalfeatures of their deforming system that challenge our theory in important ways. We propose a quasithermody-namic interpretation of their observations in which the effective disorder temperature, generated by mechanicaldeformation well below the glass temperature, governs the behavior of other state variables that fall in and outof equilibrium with it. Our analysis points to a limitation of either the step-strain procedure used by DA in theirsimulations, or the STZ theory in its ability to describe rapid transients in stress-strain curves, or perhaps toboth. Once we allow for this limitation, we are able to bring our theoretical predictions into accurate agreementwith the simulations.

DOI: 10.1103/PhysRevE.75.036108 PACS number�s�: 62.20.Fe, 46.35.�z, 83.60.�a

I. INTRODUCTION

In the preceding paper �1�, we presented an athermal STZtheory of plastic deformation in materials where thermal ac-tivation of irreversible molecular rearrangements is negli-gible or nonexistent. Here we use that theory to interpret anextensive body of computational data published recently byDemkowicz and Argon �2–5�, hereafter referred to occasion-ally as “DA.” Those authors simulated plastic deformation inamorphous silicon using a system of 4096 atoms interactingvia a Stillinger-Weber potential �6� in a cubic cell with peri-odic boundary conditions. They subjected this system to pureshear under both constant-volume and constant-zero-pressureplane-strain conditions. Their reports of these simulations areremarkably complete and detailed. They provide valuableand challenging information about the relation betweenstress-strain response and sample preparation, the theoreticaldescription of nonequilibrium behavior in systems subject tosteady-state and transient mechanical deformation, the natureof the glass transition in simulated amorphous silicon and thestrengths and limitations of numerical simulation techniques.We address each of these topics in the body of this report.

Demkowicz and Argon used two different procedures forsimulating shear deformation. In their potential energy mini-mization method �PEM�, each step in the process consistedof a small, affine, shear displacement of all the atomic posi-tions, followed by a minimization of the potential energyduring which the atoms relax to their nearby positions ofmechanical equilibrium. Supposedly, PEM simulations cor-respond to the limit of zero strain rate at zero temperature,but that interpretation is problematic. In their molecular dy-namics �MD� method, DA used a different step-strain proce-dure in which each small shear increment was followed byan MD relaxation at temperature T=300 K, with an averagestrain rate of order 108 s−1. In both procedures, an incremen-tal shear was imposed only after the system was judged tohave reached a stable, stationary state following the preced-ing step. We argue below that there are important uncertain-

ties associated with both of these step-strain simulationmethods.

The single most important feature of the DA simulationsis that, in addition to measuring the shear stress �and keepingtrack of pressure and/or volume changes� during deforma-tion, DA also observed local atomic correlations within theirsystem. Here they were taking advantage of their numericalmethod to see inside their system in a way that is seldompossible in laboratory experiments using real materials. Theyfound that the environments of some atoms were solidlikeand others liquidlike, and that the liquidlike regions seemedto be, as they say, the “plasticity carriers.” Before any me-chanical deformation, their fraction of liquidlike regions �was small when the system was annealed or cooled slowly,and was approximately 0.5 when the system was quenchedrapidly. Then, during constant-zero-pressure deformations, �approached a value slightly less than 0.5, independent of itsinitial value. Thus, � behaved in a manner qualitatively simi-lar to the dimensionless effective temperature � or, equiva-lently, the density of STZ’s ��exp�−1/�� described in Ref.�1�. The relation between these quantities is one of the maintopics to be addressed in this paper.

Amorphous silicon, similar to water, expands as it solidi-fies. Moreover, its properties are highly sensitive to smallchanges in density. A slowly quenched, more nearly equili-brated system is less dense than one that is rapidly quenched,because the former contains a smaller population of denser,liquidlike regions than the latter. When strained, the morenearly equilibrated system initially responds elastically, andtakes longer than a rapidly quenched system to generateenough plasticity carriers to enable plastic flow. Accordingly,the equilibrated system exhibits a more pronounced stresspeak of the kind illustrated in Fig. 1 �top curve, upper panel�of Ref. �1�. As the liquidlike fraction � increases, the systemcontracts if held at constant pressure; or else, if the system isheld at constant volume, the pressure decreases and mayeven become negative. It may be surprising to theorists butnevertheless is true that, in this material, the STZ’s must be

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associated with decreases rather than increases in free vol-ume.

For theoretical purposes, it will be simplest to look only atconstant-pressure simulations, because the changes in pres-sure that occur at constant volume require additional, hard-to-control approximations for behaviors that are not intrinsicto the STZ hypotheses. System-specific parameters such asthe yield stress sy are likely to be more sensitive to changesin pressure than in volume. For example, the Mohr-Coulombeffect implies that sy increases with pressure. Demkowiczand Argon point out that direct evidence for pressure depen-dence can be seen in the graphs of the liquid fraction � as afunction of strain � in Figs. 10�c� and 15�c� of Ref. �3�. Inthe constant-zero-pressure simulations shown in their Fig.15, � equilibrates to approximately the same value after de-formation for all four different initial conditions. That doesnot happen in the constant-volume simulations in Fig. 10,implying that the internal states of these deformed systemsdiffer from one another in nontrivial ways.

In addition to the questions pertaining to step-strain pro-cedures, to be discussed in detail in Sec. IV, there are otherlimitations of the DA simulations that must be taken intoaccount. The DA simulation system is too small to make itlikely that more than one STZ-triggered event is taking placewithin a characteristic plastic relaxation time; and data isreported only for single numerical experiments performed onindividual samples rather than averages over multiple experi-ments. Thus, fluctuations are large, and the results must besensitive to statistical variations in initial conditions. Inshort, we must be careful in our interpretations.

On the positive side, Demkowicz and Argon’s remarkablecombination of multiple simulation methods and multipleobservations allows us to use the athermal STZ theory toconstruct what we believe is an internally self-consistent in-terpretation of their data. Going beyond their stress-straincurves, and focusing on the behaviors of the liquidlike frac-tion � and the density �, we develop a quasithermodynamic

picture in which the effective disorder temperature � plays adominant role. In this picture, the STZ’s, i.e., the active flowdefects, are rare sites, out in the wings of the disorder distri-bution, that are more susceptible than their neighbors tostress-induced shear transformations. They almost certainlylie in the liquidlike regions of the system, but only a verysmall fraction of the liquidlike sites are STZ’s. The energydissipated in the STZ-initiated transitions generates the ef-fective temperature �, a systemwide intensive quantity that,in turn, determines extensive quantities such as � and �.

Our central hypothesis is that there exist quasithermody-namic equations of state relating steady-state values of � and� to � �and also, in principle, to the pressure and shearstress�. Once we have found these equations of state, weextend the quasithermodynamic model to describe the way inwhich � and � fall in and out of equilibrium with � duringtransient responses to external driving forces. In this way, wearrive at a quantitative interpretation of the DA simulations.

We begin in Sec. II with a brief summary of the athermalSTZ theory developed in Ref. �1�. Section III contains a pre-liminary analysis of the DA stress-strain data. In Sec. IV, weexplain why we cannot accept the initially good agreementbetween the STZ theory and the simulation data. These ar-guments motivate the quasithermodynamic hypothesis, intro-duced in Sec. V. We extend the quasithermodynamic ideas tononequilibrium situations in Sec. VI. Finally, in Sec. VII, weconclude with some speculations concerning the validity ofthe quasithermodynamic picture and its relation to the STZtheory in general.

II. STZ SUMMARY

In order to make this paper reasonably self-contained, westart by restating Eqs. �3.12�–�3.16� in Ref. �1�. The first ofthese equations is an expression for the total rate of defor-mation tensor Dtot= � /2 as the sum of elastic and plasticparts

Dtot =s

2+ Dpl�s,m,�� , �2.1�

where s and are the deviatoric stress and the shear modu-lus measured in units of the yield stress sy. The internal statevariables m and � are, respectively, the orientational biasand the scaled density of STZ’s. We consider only the case ofpure, plane-strain shear in which the material is strained at afixed rate �, and the stress is measured as a function of thestrain �. To describe such experiments, we write Eq. �2.1� inthe form

ds

d�= �1 −

20�

��0

q�s,m�� , �2.2�

where 0 is a number of order unity �the atomic density inatomic units multiplied by the incremental strain associatedwith an STZ transition�, �0 is the characteristic time scale forSTZ transitions, and

FIG. 1. �Color online� Theoretical stress-strain curves �solidlines� compared to the DA numerical simulation data �3�. The pa-rameters used are 0=1, sf =1.06 GPa, =46 GPa, sy =0.35 GPa,c0=0.18, and ��=0.065. The initial effective temperatures are�0=0.0559,0.0580,0.0612,0.0680 �from top to bottom�. Thecurves are shifted by 0.5 GPa for clarity.

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q�s,m� � C�s�� s

�s�− m� . �2.3�

The function C�s� describes the stress dependence of the STZtransition rate; it is proportional to �s−1� for �s� 1 and van-ishes smoothly near s=0. As defined by Eq. �3.6� in Ref. �1�,the smoothness of C�s� near s=0 is determined by a param-eter �, which we take to be unity throughout the following.

Similarly, we restate Eqs. �3.9�, �3.10�, and �3.11� fromRef. �1�:

dm

d�=

2

��0

q�s,m��1 −ms

�e−1/�� , �2.4�

d�

d�=

2

��0

sq�s,m��e−1/� − �� , �2.5�

and

d�

d�=

20

c0��0

�sq�s,m���� − �� . �2.6�

The constant c0 is a configurational specific heat per atom inunits kB, which must be of order unity.

We assume that the stress and strain tensors in these equa-tions remain diagonal in two dimensional pure shear andplane strain, and that any correction in the third dimension isnegligible in comparison to other uncertainties in this analy-sis. A basic premise of the STZ theory is that the zones arerare and do not interact with each other; thus we assumefrom the beginning that the quantity 0 exp�−1/�� is small oforder 10−3 �in fact, very much less�, so that the equations form��� and ���� are stiff compared to those for s��� and ����.Then we can safely replace Eqs. �2.4� and �2.5� by theirstationary solutions

m = m0�s� = s/�s� if �s� � 1,

1/s if �s� � 1, �2.7�

and

� = e−1/�. �2.8�

Eqs. �2.2� and �2.6� become

ds

d�= �1 −

20

q0e−1/�q�s�� �2.9�

and

d�

d�=

20

c0q0e−1/�sq�s���� − �� , �2.10�

where

q�s� = C�s�� s

�s�− m0�s�� �2.11�

and q0= ��0.

III. FIRST FITS TO THE DA STRESS-STRAIN CURVES

The first step in our STZ analysis of the DA data is to useEqs. �2.9�–�2.11� to fit the stress-strain curves in Fig. 15�a� of

Ref. �3�, shown here in Fig. 1. These two equations involvethe parameters q0 �the dimensionless strain rate�, sy �neededin order to convert from dimensionless stresses s to measuredstresses in units GPa�, , c0, ��, and the four initial values ofthe effective temperature ��0�=�0.

We can obtain a certain amount of information aboutthese parameters directly from features of the stress-straincurves before doing any computation. Note from Eqs. �2.7�and �2.11� that no plastic deformation occurs for s�1, nomatter how large � might be. Thus the lowest observed valueof a stress that marks a departure from elastic behavior is anupper bound for sy. The bottom stress-strain curve in Fig. 1,which is the curve with the largest initial value of �, seemsto have a break point that is about a factor of three less thanthe flow stress sf, which we see from the figure is about1.06 GPa. Therefore, sy �sf /3�0.35 GPa, and s f �3. Wetake the shear modulus �46 GPa directly from the slopeof the stress-strain curves in the elastic region; thus wechoose �130. Setting the right-hand side of Eq. �2.9� tozero, we find that the flow stress s f satisfies

q0 = ��0 = 20e−1/��q�s f� . �3.1�

Using this relation with the DA estimate of � and the valueof �� given below, we confirm that �0 is of order femtosec-onds as implied by the Stillinger-Weber interactions used inthese simulations.

Once we have inserted Eq. �3.1� into Eqs. �2.9� and�2.10�, we can solve these equations and vary the remainingparameters to fit the stress-strain curves in Fig. 15�a� of Ref.�3�. We find that the fits are disappointingly insensitive to ourchoice of 0 exp�−1/��� so long as we stay within our con-straint that this product be small. However, our later analysisof the liquidlike fraction � reveals that the internally consis-tent value of �� is 0.065, which is slightly bigger than our apriori order-of-magnitude estimate based on the experimen-tal data, as shown in Ref. �7� for example, but seems wellwithin the accuracy of our approximation for �� in Ref. �8�and the uncertainties of these small-scale simulations. There-fore we have chosen 0=1 and ��=0.065. Agreement be-tween theory and the numerical simulations then can be ob-tained for all four of the stress-strain curves by settingc0=0.18 and adjusting only �0 for each curve. Our best-fitresults are shown in Fig. 1. The corresponding values of �0�from top to bottom in the figure� are 0.0559, 0.0580, 0.0612,0.0680. Note that there is a gap in �0 between the lowestthree values, for which the stress-strain curves are peaked,and the largest, which shows no peak.

If it were useful to do so, we could improve the agreementbetween the simulated stress-strain curves and our theory bymaking small adjustments of sy and for each curve, con-sistent with the likelihood that the four relatively small com-putational systems are not exactly comparable to each otherin their as-quenched states. In the spirit of the discussion tofollow, however, we have chosen to assume that the foursystems reach effectively identical steady states after persis-tent shear deformation, and to attribute the small discrepan-cies to the statistical uncertainties visible in the data. Theonly systematic feature of the stress-strain data that is notrecovered by the theory is the slow decrease of the flow

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stress with increasing strain in the top curve �the one with themost pronounced stress peak�. We believe that this behavioris caused by the emergence of a nascent shear band as illus-trated in Fig. 16�b� of �3�.

IV. SECOND THOUGHTS

Fitting the stress-strain curves, however, is only a part ofthe challenge of interpreting the DA data �3�. We also mustunderstand how the liquidlike fraction � and the density �relate to our STZ variables, especially the effective tempera-ture �. Therefore, it is essential to know whether we can trustthe values of �0 deduced from the STZ analysis of the stress-strain curves. We argue in the following paragraphs thatthese values are not quantitatively reliable.

One of the key tenets of plasticity theory is that transientpeaks observed in the stress-strain curves for well annealedsamples occur because there is an initial lack of plasticitycarriers in these systems, and new carriers must be generatedby deformation before plastic flow can begin and the stresscan relax. Demkowicz and Argon use the correlation be-tween stress peaks and the liquidlike fraction � to argue that� is a direct measure of the population of plasticity carriers.The STZ theory, as presently constituted, predicts stresspeaks when—and only when—the initial STZ density issmall.

This tenet is not confirmed by the DA simulations. Notefirst the data for � as a function of strain � shown by DA inFig. 15 of Ref. �3�, also shown here in Fig. 6. There are fourcurves. Two of them have small initial values of ���=0�=�0 and exhibit stress peaks. The corresponding values of���� rise monotonically to the steady-state value ���0.46as expected. Another curve starts with �0���. The corre-sponding stress-strain curve has no peak, and ���� decreasesto ��, again in accord with expectations. In one case �the onewith �0�0.46�, however, �0 is slightly above �� but thestress still shows a peak. Moreover, the corresponding den-sity � decreases with �, implying that the denser liquidlikefraction is decreasing during the deformation.

This kind of behavior appears elsewhere in the DA papers�2–5�. It is clear in the constant-volume, MD simulations,where one of the stress-strain curves has a peak, but thecorresponding ���� remains nearly constant, and the pres-sure increases instead of decreasing as it should if the liquid-like fraction were growing. And, as we note below, all of thePEM simulations in Ref. �4� show stress peaks, even theones in which �0 is large and comparable to ��.

It seems to us that the most likely explanation for thesediscrepancies is that the ubiquitous stress peaks are artifactsof the step-strain simulations. It is also quite possible, ofcourse, that the STZ theory does not adequately account forthe way in which stresses respond to rapid changes in thestrain rate, and that the DA simulations are simply out ofrange of our STZ analysis. Perhaps both explanations are inpart correct. Nevertheless, we prefer the first explanation forthe following reasons.

Demkowicz and Argon performed only MD simulations,and not PEM, at constant pressure; but a number of featuresof their PEM results may be relevant here. They report that

cascades of events were evident in their PEM simulations butwere hard to detect in MD. Maloney and Lemaitre �9,10�,who used only PEM, found that cascades were prevalent andsometimes so large that they spanned their systems, whichwere only two dimensional but larger in numbers of atomsthan those used by DA. A related feature of the DA PEMsimulations is that, even for rapidly quenched samples withlarge initial disorder, the stress-strain curves exhibit markedstress peaks followed by strain softening. We suspect thatthese behaviors may be generally characteristic of step-strainprocesses, and correspondingly uncharacteristic of continu-ous strain mechanisms that are more common in the labora-tory.

Note that, between each strain increment in PEM, the sys-tem has a probability of dropping into a low energy statefrom which it cannot escape without the application of alarge force. The large energy released when such a trappedstate is destabilized may trigger cascades of smaller events.This trapping mechanism may be especially important at thebeginning of a shear deformation, because then the energyminimization starts with a disordered, as-quenched system.As a result, the first energy drops may be large, and theinitial stresses required to set the system into motion may beanomalously high. Thus we expect transient stress peaks inPEM, even for initially disordered systems; and we expectlarge stress fluctuations even in steady-state conditions. Thatis exactly what is seen by Demkowicz and Argon.

The difference between step strains and continuous shearhas been demonstrated recently in bubble raft experimentsby Twardos and Dennin �11�. Similar to Demkowicz andArgon, these authors subjected their strictly athermal systemto both steady, slow shear and to discrete shear steps fol-lowed by relaxation periods long enough for most motion tocease. They monitored the stress during both procedures,with particular interest in the distribution of stress drops as-sociated with irreversible plastic rearrangements. One oftheir most interesting results is that the average size of stressevents decreases with decreasing shear rate for continuousstrain, but increases for step strains. That is, the stress relaxesvia a larger number of smaller drops for continuous defor-mation than for step-strain motion. Their interpretation ofthis result seems to us to be roughly consistent with ourdiscussion of PEM simulations in the preceding paragraph;but neither they nor we claim a full understanding of thisphenomenon

In their MD simulations, Demkowicz and Argon use stepstrains as opposed to a continuous strain rate. Apparently,allowing the system to relax for a time at T=300 K betweenstrain steps removes the effects of at least some of whateverstructural irregularities are producing cascades and stresspeaks in the PEM simulations. It seems likely, however, thatsome elements of the athermal PEM-like, step-strain behav-ior persist in the MD results. Our hypothetical low-energytrapping states may be less likely to be sampled in step-strainMD, in which case the initial transients and steady-statestress fluctuations might be smoother, as indeed they are. Butthe step-strain MD procedure is not the same as a continuousone in which the strain is incremented on the time scale atwhich the molecular motions are resolved. In the DA simu-lations, the strain is incremented only once in about ten or

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more atomic vibration periods. We see little reason to believethat these two simulation procedures will produce preciselythe same responses to rapidly changing driving conditions.This situation forces us to conclude—reluctantly in view ofthe quality of the apparent agreement between the simula-tions and STZ theory shown in Fig. 1—that the values of the�0’s stated above and in the caption to Fig. 1 cannot betrusted.

V. QUASITHERMODYNAMIC HYPOTHESES

Our inability to deduce the �0’s from the stress-strain datahas led us to probe more deeply into the meaning of theeffective temperature in glass dynamics. We base our analy-sis on a set of quasithermodynamic hypotheses in which weassume that the steady-state properties of the configurationaldegrees of freedom below the glass transition are determinedby the effective temperature Teff= �ESTZ/kB��, in close anal-ogy to the way in which they are determined by the bathtemperature T above that transition. In true thermodynamicequilibrium, extensive quantities such as the density �, theinternal energy U, or the liquidlike fraction � in the DAsimulations are functions of T and the pressure P. In otherwords, these quantities obey equations of state. Below theglass transition, the configurational degrees of freedom—thatis, the positions of the atoms in their inherent states—fall outof equilibrium with T because thermally activated rearrange-ments are exceedingly slow or impossible. The most prob-able configurations in such situations maximize an entropy,thus the statistical distribution of these configurations isGibbsian with T=Teff.

Accordingly, our first hypothesis is that the high-T equi-librium equations of state are preserved in the glassy state asexpressions for the configurational parts of the extensivequantities �, U, �, etc., in terms of the intensive quantitiesTeff, P, and the shear stress s. In particular, we propose thatthe effective temperatures �0, multiplied by ESTZ/kB, are thebath temperatures T0 at which the DA simulation samplesfell out of thermal equilibrium during cooling, and that theconfigurational parts of �, U, �, etc., were fixed at thosetemperatures. Our specialization to configurational parts rec-ognizes that, for example, � undergoes ordinary thermal ex-pansion at small T and that the total internal energy U in-cludes the kinetic energy. Both of those nonconfigurationalparts are uninteresting for present purposes. Since we con-sider only P=0 situations and work only at fixed T, we omitexplicit dependences on those variables. Shear dilation maybe relevant, especially for determining �; but so long as weare considering only predeformation properties with s=0, wealso omit explicit s dependence. We return later to the ques-tion of shear dilation, and we also postpone a discussion ofthe potential energy. For the moment, we write simply�=�0���, �=�0���.

As a first test of this hypothesis, we show by the opencircles and dashed lines in both panels of Fig. 2 the functions�0��� and �0��� obtained with the measured values of �0 and�0 and our best-fit values of �0 from Fig. 1. We do, in fact,find qualitatively plausible behavior. With our uncertaintiesabout the �0’s, however, we are obliged to look harder andbring other considerations to bear on this analysis.

We know some qualitative features of these equations ofstate. First, we are dealing with a glass transition, and there-fore we expect that �0��� should extrapolate to some fixedvalue, possibly zero, at a nonzero Kauzmann temperature. Inthe following discussion, we define �K to be the effectivetemperature at which the extrapolated �0��� vanishes; but wemake no assumption about whether this is actually a Kauz-mann temperature, or whether an ideal glass transition actu-ally occurs at this or any other point.

Another feature is suggested by the binary, solidlike/liquidlike, nature of the atomic structure described byDemkowicz and Argon. If this were just a simple binarymixture, and if the energy of the mixture were proportional

FIG. 2. �Color online� Upper panel: The equation of state�=�0��� obtained by fitting the four data points �filled circles� plus��=�0����=0.458 �filled square�. The �0’s are the DA numericalsimulation data �3�; the adjusted �0’s are 0.0559, 0.0589, 0.0663,and 0.0749, with ��=0.065. The solid line is our fit to these pointsin Eq. �5.1�. The open circles are the observed �0’s plotted againstthe unadjusted, best-fit, �0’s. The dashed line is intended only as aguide for the eye. Lower panel: The equation of state �=�0���obtained by a procedure similar to the one described above exceptthat the point ��=1.0335 �filled square� is not included in the fit.We attribute its displacement from the equation of state to a stress-induced dilation. The density is normalized by �c, the density ofcrystalline silicon �diamond cubic�. The solid line is our smooth fitto the data. Again, the open circles and and dashed line indicate theunadjusted equation of state.

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just to the density of the higher energy, liquidlike compo-nent, then the entropy would be a maximum at �=1/2 andthe temperature would diverge at that point. Obviously,amorphous silicon is not so simple. Nevertheless, the ob-served � does seem to have an upper bound not too farabove �=1/2; thus, we expect that �0��� saturates at somevalue �0=�lim�0.55 for large �.

Yet another consideration is that, in principle, we ought tobe able to deduce the T0’s directly from the graphs of densityversus temperature for different quench rates shown in Fig. 1of Ref. �3�. The slowest quench is the closest to full equilib-rium, and the points at which the faster quenches fall awayfrom it are the three higher T0’s. The lowest T0 must be nearthe inflection point on the slow-quench curve. Unfortunately,the best we can do with the data at hand is estimate the T0’sto about ±20 K, and even then we must be cautious aboutsystematic uncertainties in the simulations. Nevertheless, wecan use this process as a rough check on the modifications ofthe �0’s that we propose below.

We come now to our second and more speculativehypothesis—that the point ���, ��� ought to lie on the equi-librium equation-of-state curve; that is, ��=�0����. Here weare assuming that persistent shear deformation in an athermalsystem rearranges the atomic configurations in a way that isstatistically equivalent to thermally driven rearrangement,except that the relevant temperature is the effective disordertemperature Teff instead of the bath temperature T. We alsoare assuming that ��, being a ratio of two populations, isinsensitive to changes in the volume of the system that mightoccur during constant-zero-pressure simulations. In contrast,�� would decrease if, for example, the system undergoes ashear-induced dilation.

With these considerations in mind, our next step is torefine the estimates of the T0’s by requiring that they producesmooth equations of state with the qualitative features hy-pothesized above, and that the observed ���0.458 be equalto �0����. Accordingly, we have computed �=�0��� and�=�0��� by adjusting the �0’s so as to optimize agreementwith all the previously discussed constraints. Our results areshown by the solid lines in Fig. 2. As expected,����0����, because the flow stress s f at that point seemseasily large enough to cause dilation. �Throughout this paper,values of � are in units of �c=2323.8 kg/m3, the density ofcrystalline, diamond-cubic, silicon.� The effective Kauzmanntemperature is �K=0.051, and the upper bound for � is�lim=���→���0.6. The smooth curve that we have fit tothe equation of state for �, shown in the upper panel ofFig. 2, is

�0��� = �lim�1 − e−a��−�K�� , �5.1�

where a=100. Other important parameters are the fourT0’s: 1100, 1160, 1305, 1475 K, which are consistent withour crude direct estimates from Fig. 1 in Ref. �3�. Note thatthe large gap between the lowest three unadjusted T0’sand the upper one has disappeared, and that the adjustedpoints fit on a smoother curve than the unadjusted ones. Wealso find that T�=1280 K. Then, with ��=0.065, we obtainESTZ/kB�21 000 K, or about 1.3 eV. The associated valuesof �0 are 0.0559, 0.0589, 0.0663, and 0.0749.

Using these adjusted values of the �0’s, we have recom-puted and plotted the stress-strain curves in Fig. 3, shownagain in comparison with the simulation data. As expected,the non-STZ peak has disappeared, and there is more round-ing in the bottom, most rapidly quenched case, but the stresspeaks for the two most deeply quenched systems remain al-most unchanged.

Finally, within the context of quasithermodynamic equi-librium, we note that a relation between the potential energyand � has been proposed and confirmed by Shi et al. �12�These authors describe molecular-dynamics simulations ofshear banding in two-dimensional, noncrystalline, Lennard-Jones mixtures. Their analysis of these simulations goes be-yond our own in at least one important way. They note thatthe plastic strain rate predicted by the STZ theory, as well asby other flow-defect theories, has the form exp�−1/�� mul-tiplied by a function of the shear stress, essentially our factorq�s� defined in Eq. �2.11�. Force balance requires that s be aconstant across the shear band, thus a measurement of theposition-dependent strain rate is a measure of the positiondependence of �. Shi et al. then postulate that the potentialenergy depends linearly on �. They compute the position-dependent potential energy directly from their simulationdata and find that it maps accurately onto � as predicted bythe spatially varying strain rate. The importance of this ob-servation is that they are looking at a weakly nonequilibriumsituation in which the potential energy and � are varyingcontinuously in space, but only very slowly in time, alongthe postulated linear equation of state. Their system remainsin quasithermodynamic equilibrium during this variation be-cause, unlike the DA simulations discussed here, it is notundergoing a fast transient response to a sudden change inthe applied stress. Thus their result anticipates and fits accu-rately into our quasithermodynamic picture.

VI. DEPARTURES FROM QUASITHERMODYNAMICEQUILIBRIUM

The next question is whether the equilibrium equations ofstate shown in Fig. 2 are obeyed under nonequilibrium con-

FIG. 3. �Color online� Modified stress-strain curves using�0=0.0559, 0.0589, 0.0663, and 0.0749 �from top to bottom�. Allother parameters are the same as those used in Fig. 1.

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ditions during deformation. Apparently they are not. To seethis, in Fig. 4, we have added to our equation-of-state graphsfour data sets showing the observed values of � and � asfunctions of our calculated values of � for the four differentdeformation histories. In all cases, the observed values of �and � initially fall off the equilibrium curves and do notcome together again until they approach ��. This nonequi-librium behavior can be seen more directly by comparing thethree panels of Fig. 15 in Ref. �3�. It is clear there that therate at which the stress relaxes to the flow stress is faster thanthe rates at which either � or � relax to their steady-statevalues.

To account for the transient, nonequilibrium behavior of�, we propose an equation of motion analogous to Eq. �17�in Ref. �5�, but with a form similar to our Eq. �2.10�:

d�

d�=

20

c1q0e−1/�sq�s���0��� − �� , �6.1�

where c1 is a constant similar to c0. The idea here is that thedisorder described by � may rise at its own rate, but theliquidlike-solidlike reorganization associated with � may notcatch up instantaneously. We assume that the underlyingmechanism that determines these rates is still the rate of en-ergy dissipation. There is no other comparably simple andbasic coupling between mechanical deformation and the in-ternal degrees of freedom that satisfies the requirement that itbe a non-negative scalar. Note that Eq. �6.1� is supplemen-tary to the STZ equations of motion, Eqs. �2.9� and �2.10�; itassumes that the intensive quantities s��� and ���� are un-changed from the STZ predictions and that they continue tocontrol the behavior of � even away from equilibrium. Thisis a strong assumption, especially near the initial stress tran-sients where we already know that there is some mismatchbetween the STZ theory and the DA simulations.

Our results for the four functions ����, determined usingEq. �6.1�, are shown in Fig. 5. The corresponding functions���� are shown in Fig. 6, here in comparison with the DAdata. The only adjustable parameter is c1, which we chooseto be 0.42. The agreement seems to be within the uncertain-ties of the data.

We continue our development of the nonequilibrium qua-sithermodynamic theory by writing an equation analogous toEq. �6.1� for the function ����:

d�

d�=

20

c1q0e−1/�sq�s���0��� − � − ��0� . �6.2�

Here, ��0�0.012 is the dilation-induced shift of the equilib-rium density shown by the displacement of �� from the equi-librium curve �0��� in the lower panel of Fig. 2. To integrateEq. �6.2�, we have approximated �0��� by a smooth polyno-mial. The factor c1=0.42 needed to fit the density data is thesame as in Eq. �6.1�. The corresponding graphs of ����,along with the DA data, are shown in Fig. 7. Again, theagreement seems to be within the uncertainties; but here weare pushing the theory too far for comfort. Our quasithermo-dynamic hypotheses imply that �0��� ought to be a functionof the shear stress s as well as �, and that the dilation ap-proximated here by ��0 should be part of that generalizedequation of state. In particular, ��0 should be the dilationalchange in the density when s is equal to the flow stress s f.We have in fact tried ��0� s2 �as in nonlinear elasticity�, butthe results are distinctly unsatisfactory at small � where nei-ther Eq. �6.2� nor the STZ theory itself may accurately de-scribe the fast transient. Equations �6.1� and �6.2� do accountfor the relatively slow relaxation of ���� and ���� as com-pared to that of s���. This level of success seems to be asmuch as we can expect from the theory at this stage in itsdevelopment.

VII. SUMMARY AND CONCLUDING REMARKS

The unique aspect of the Demkowicz and Argon simula-tions is their measurement of extensive quantities—the mass

FIG. 4. �Color online� Upper panel: The DA numerical simula-tion data for � versus the theoretical � for the four cases consideredin Ref. �3�. The discrete markers sets, from top to bottom, corre-spond to the largest initial � �fastest quench� down to the smallestinitial � �slowest quench�, respectively. The equation of state�=�0��� �solid line� was added to stress the nonequilibrium natureof the deformation-induced dynamics of �. Lower panel: The cor-responding plot for � /�c. The discrete markers sets, from top tobottom, correspond to the largest initial � �fastest quench� down tothe smallest initial � �slowest quench�, respectively. The points��� ,��� and ��� ,��� are marked by solid squares.

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density � and the liquidlike fraction �—in parallel with con-ventional stress-strain curves. They argue convincingly andimportantly that � is closely related to the density of plastic-ity carriers. However, their remarkably complete report in-cludes some results that make it seem that this relationshipmay be neither simple nor direct. The main purpose of ourinvestigation has been to learn more about that relationship.

Our proposed interpretation of the DA simulations isbased on the athermal STZ theory �1�, which we believecaptures the central features of amorphous plasticity—atleast for processes that are not too rapidly varying as func-tions of space or time. In order to discuss quantities such as�, however, we have had to go beyond the STZ theory. Wehave hypothesized that, in low-temperature, steady-state,nonequilibrium conditions � and � and presumably other

such quantities are related to the effective disorder tempera-ture � via quasithermodynamic equations of state. With thishypothesis, we have developed a theoretical interpretation ofthe DA simulations that seems physically satisfying, inter-nally self-consistent, but interestingly incomplete. If con-firmed by further tests, this quasithermodynamic theorycould become a useful tool for predicting the nonequilibriummechanical behavior of amorphous solids.

There are many open issues. Perhaps the most urgent ofthese is the interpretation of the DA step-strain simulationtechnique. The DA data do show that a well annealed samplewith an initially small � exhibits a transient peak in itsstress-strain curve. However, the converse seems not neces-sarily to be true. Stress peaks sometimes appear in the DA

FIG. 5. �Color online� Upper panel: � as a function of � usingthe nonequilibrium Eq. �6.1� with c1=0.42. The equilibrium equa-tion of state was added for illustration. The curves, from top tobottom, correspond to the largest initial � �fastest quench� down tothe smallest initial � �slowest quench�, respectively. These curvesshould be compared to the DA data shown in the upper panel of Fig.4. Lower panel: � as a function of � using the nonequilibrium Eq.�6.2� with c1=0.42. The curves, from top to bottom, correspond tothe largest initial � �fastest quench� down to the smallest initial ��slowest quench�, respectively. These curves should be compared tothe DA data shown in the lower panel of Fig. 4. The points ��� ,���and ��� ,��� are marked by solid squares.

FIG. 6. �Color online� Theoretical predictions of ���� for allfour quenches, based on Eq. �6.1� with c1=0.42, compared with theDA simulation data. The discrete markers sets and the solid lines,from top to bottom, correspond to the largest initial � �fastestquench� down to the smallest initial � �slowest quench�,respectively.

FIG. 7. �Color online� Theoretical predictions of ���� for allfour quenches, based on Eq. �6.2� with c1=0.42, compared with theDA simulation data. The discrete markers sets and the solid lines,from top to bottom, correspond to the largest initial � �fastestquench� down to the smallest initial � �slowest quench�,respectively.

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results in cases where the initial � is large and does notincrease during plastic deformation, a behavior that is incon-sistent with the presumed relation between � and the densityof plasticity carriers. After examining other possibilities andlooking at other examples of step-strain procedures �see Sec.IV�, we have concluded that the stress peaks observed incases with large initial � are most likely artifacts of thenumerical step-strain procedure. Once we allowed ourselvesthat flexibility and found alternative ways of estimating theinitial effective temperatures for those cases, we found thatour quasithermodynamic picture fits together quite well.

In our opinion, one of the most important next steps indetermining the limits of validity of the quasithermodynamicSTZ theory would be to redo the DA analysis, including themeasurement of the liquidlike fraction �, with continuousstrain MD. So far as we know, nobody before Demkowiczand Argon has ever done anything similar to this—makingindependent, simultaneous measurements of the density ofplasticity carriers and the stress-strain response for differ-ently quenched systems. Discovering the relation betweencontinuous and step-strain simulations in this context seemslikely to be extremely useful at least for understanding thenumerical simulations. It could also be a big step forward inunderstanding amorphous plasticity.

Another obviously open issue is the validity of the STZtheory in situations where the system is responding to rapidchanges in loading conditions, as in the DA numerical simu-lations where the strain rate is large and is turned on instan-taneously both at the beginning of the process and at eachstrain step. The present version of STZ theory �1� is a mean-field approximation in which interactions between zones areincluded only on average, and fluctuations are neglected. It ispossible that neither the STZ theory as presently formulated,nor the conventional explanation of stress peaks in terms ofplasticity carriers, can fully account for stress transients dur-ing rapid changes in loading conditions, even in continuous-strain MD simulations. From this point of view, acontinuous-strain version of the DA simulations seems dou-bly important. If new simulations showed no qualitative dis-crepancy between continuous and step-strain procedures, wewould know that the STZ theory is missing some essentialingredients. We might then try to include correlations andfluctuations by developing a more detailed statistical theoryof STZ’s with varying thresholds, perhaps with noisy inter-

actions between them as proposed recently by Lemaitre andCaroli �13�.

In a similar vein, we must ask about the limits of validityof our quasithermodynamic hypotheses. It will be especiallyimportant to look harder at Eqs. �6.1� and �6.2�, which deter-mine the rates at which � and � relax to their quasithermo-dynamic equilibrium values. Our quasithermodynamic pic-ture is based on the assumption that the effective temperature� is the intensive variable—the thermodynamic force—thatdrives the extensive quantities �, �, and the STZ variables �and m. According to our equations of motion, � and m aretightly slaved to �. We have postulated equations of staterelating � and � to � under equilibrium or steady-state con-ditions, and have further proposed in Eqs. �6.1� and �6.2�, onwhat seem to us to be general grounds, that � and � are moreloosely slaved to � than are � or m during excursions fromequilibrium. Here, as in the questions regarding the stressresponse, we need to learn whether the equations of motionfor � and � are accurate when those excursions from equi-librium are faster, say, than the relaxation rate of �.

In short, we are asking how far this theory can be pushed.Can it, for example, always be used to predict stress-straintransients under realistic experimental conditions, wherestrain rates are very much smaller than those in MD simula-tions? Or does it generally become inaccurate near stresspeaks? Can it be used to predict plastic deformation near thetip of an advancing crack? More generally, is the effectivetemperature � really such a dominant state variable? Whatother internal variables might become relevant for describingfast processes? What real or computational experimentsmight help to answer such questions?

ACKNOWLEDGMENTS

We thank M.J. Demkowicz and A.S. Argon for providingthe data on which this research was based and for permissionto use it here. J.S.L. also thanks C. Caroli, M. Falk, A.Lemaitre, and C. Maloney for important ideas and informa-tion. E. Bouchbinder was supported by the Horowitz Com-plexity Foundation. J.S. Langer was supported by U.S. De-partment of Energy Grant No. DE-FG03-99ER45762. I.Procaccia acknowledges the partial financial support of theIsraeli Science Foundation, the Minerva Foundation, Mu-nich, Germany, and the German-Israeli Foundation.

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