Athermal Shear-Transformation-Zone Theory of Amorphous Plastic Deformation II: Analysis of Simulated Amorphous Silicon

0.87 MB·English

by  Eran Bouchbinder

Athermal Shear-Transformation-Zone Theory of Amorphous Plastic Deformation II:
Analysis of Simulated Amorphous Silicon
Eran Bouchbinder1, J. S. Langer2, and Itamar Procaccia1
1 Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100 Israel.
2 Dept. of Physics, University of California, Santa Barbara, CA 93106-9530 USA
(Dated: February 6, 2008)
7
0 In the preceding paper, we developed an athermal shear-transformation-zone (STZ) theory of
0 amorphousplasticity. Hereweusethistheoryinananalysisofnumericalsimulationsofplasticityin
2 amorphoussiliconbyDemkowiczandArgon(DA).Inadditiontobulkmechanicalproperties,those
authorsobservedinternalfeaturesoftheirdeformingsystemthatchallengeourtheoryinimportant
n
ways. Weproposeaquasithermodynamicinterpretation oftheirobservations inwhich theeffective
a
J disordertemperature, generated bymechanical deformation well below theglass temperature, gov-
erns the behavior of other state variables that fall in and out of equilibrium with it. Our analysis
1
pointstoalimitationofeitherthestep-strainprocedureusedbyDAintheirsimulations,ortheSTZ

theory in its ability to describe rapid transients in stress-strain curves, or perhaps to both. Once
]
i weallowforthislimitation,weareabletobringourtheoreticalpredictionsintoaccurateagreement
c
with thesimulations.
s
-
l
r
t I. INTRODUCTION problematic. Intheirmoleculardynamics(MD)method,
m
DA used a different step-strain procedure in which each
t. In the preceding paper [1], we presented an athermal smallshearincrementwasfollowedby anMD relaxation
a STZ theory of plastic deformation in materials where at temperature T = 300K, with an average strain rate
m thermal activation of irreversible molecular rearrange- of order 108sec−1. In both procedures, an incremental
- ments is negligible or nonexistent. Here we use that the- shear was imposed only after the system was judged to
d
orytointerpretanextensivebodyofcomputationaldata havereachedastable, stationarystate followingthe pre-
n
o published recently by Demkowicz and Argon [2, 3, 4, 5], ceding step. We argue below that there are important
c hereafterreferredtooccasionallyas“DA.”Thoseauthors uncertainties associated with both of these step-strain
[ simulatedplasticdeformationinamorphoussiliconusing simulation methods.

2 asystemof4096atomsinteractingviaaStillinger-Weber The single most important feature of the DA simula-
v potential [6] in a cubic cell with periodic boundary con- tions is that, in addition to measuring the shear stress
6 ditions. They subjected this system to pure shear under (and keeping track of pressure and/or volume changes)
2 bothconstant-volumeandconstant-zero-pressure,plane- duringdeformation,DAalsoobservedlocalatomiccorre-
0
strain,conditions. Their reportsof these simulationsare lationswithintheirsystem. Heretheyweretakingadvan-
1
remarkably complete and detailed. They provide valu- tageoftheirnumericalmethodtoseeinsidetheirsystem
1
able and challenging information about: in a way that is seldom possible in laboratory experi-
6
0 mentsusingrealmaterials. Theyfoundthattheenviron-
• the relation between stress-strain response and
/ ments ofsome atomsweresolidlikeandothersliquidlike,
t sample preparation;
a andthatthe liquidlike regionsseemedto be, asthey say,
m the“plasticitycarriers.” Beforeanymechanicaldeforma-
• the theoretical description of nonequilibrium be-
- haviorinsystemssubjecttosteady-stateandtran- tion,theirfractionofliquidlikeregionsφwassmallwhen
d the system was annealed or cooled slowly, and was ap-
sient mechanical deformation;
n
proximately 0.5 when the system was quenched rapidly.
o
• thenatureoftheglasstransitioninsimulatedamor- Then, during constant-zero-pressuredeformations,φ ap-
c
: phous silicon; proached a value slightly less than 0.5, independent of
v
its initial value. Thus, φ behaved in a manner qualita-
i • the strengths and limitations of numerical simula-
X tively similar to the dimensionless effective temperature
tion techniques.
χ or, equivalently, the density of STZ’s Λ ∝ exp(−1/χ)
r
a Weaddresseachofthesetopicsinthebodyofthisreport. describedin[1]. The relationbetweenthese quantities is
Demkowicz and Argon used two different procedures one of the main topics to be addressed in this paper.
for simulating shear deformation. In their potential en- Amorphous silicon, like water, expands as it solidi-
ergy minimization method (PEM), each step in the pro- fies. Moreover,itspropertiesarehighlysensitivetosmall
cess consisted of a small, affine, shear displacement of changesindensity. Aslowlyquenched,morenearlyequi-
all the atomic positions, followed by a minimization of librated system is less dense than one that is rapidly
the potential energy during which the atoms relax to quenched,because the former containsa smallerpopula-
their nearby positions of mechanical equilibrium. Sup- tion of denser, liquidlike regions than the latter. When
posedly,PEMsimulationscorrespondtothelimitofzero strained, the more nearly equilibrated system initially
strainrateatzerotemperature,butthatinterpretationis responds elastically, and takes longer than a rapidly
2
quenched system to generate enough plasticity carriers turn, determines extensive quantities such as φ and ρ.
to enable plastic flow. Accordingly, the equilibrated sys- Ourcentralhypothesisisthatthereexistquasithermo-
tem exhibits a more pronounced stress peak of the kind dynamic equations of state relating steady-state values
illustratedinFigure1(topcurve,upperpanel)of[1]. As of φ and ρ to χ (and also, in principle, to the pressure
the liquidlike fraction φ increases, the system contracts and shear stress). Once we have found these equations
if held at constant pressure;or else, if the system is held of state, we extend the quasithermodynamic model to
atconstantvolume,thepressuredecreasesandmayeven describe the way in which φ and ρ fall in and out of
become negative. It may be surprising to theorists but equilibrium with χ during transient responses to exter-
neverthelessistruethat,inthismaterial,theSTZ’smust naldrivingforces. Inthisway,wearriveataquantitative
beassociatedwithdecreasesratherthanincreasesinfree interpretation of the DA simulations.
volume. WebegininSec. IIwithabriefsummaryoftheather-
Fortheoreticalpurposes,itwillbesimplesttolookonly mal STZ theory developed in [1]. Section III contains
at constant-pressure simulations, because the changes a preliminary analysis of the DA stress-strain data. In
in pressure that occur at constant volume require addi- Sec. IV, we explain why we cannot accept the initially
tional,hard-to-controlapproximationsforbehaviorsthat goodagreementbetweentheSTZtheoryandthesimula-
are not intrinsic to the STZ hypotheses. System-specific tiondata. Theseargumentsmotivatethequasithermody-
parameters such as the yield stress s are likely to be namic hypothesis, introduced in Sec. V. We extend the
y
more sensitive to changes in pressure than in volume. quasithermodynamic ideas to nonequilibrium situations
For example, the Mohr-Coulomb effect implies that s in Sec. VI. Finally, in Sec. VII, we conclude with some
y
increaseswithpressure. DemkowiczandArgonpointout speculations concerning the validity of the quasithermo-
thatdirectevidence forpressuredependence canbe seen dynamic picture and its relation to the STZ theory in
in the graphs of the liquid fraction φ as a function of general.
strain γ in Figs. 10(c) and 15(c) of [3]. In the constant-
zero-pressuresimulations shown in their Fig. 15, φ equi-
librates to approximately the same value after deforma- II. STZ SUMMARY
tion for all four different initial conditions. That does
not happen in the constant-volume simulations in Fig. In order to make this paper reasonably self-contained,
10, implying that the internal states of these deformed westartby restatingEqs. (3.12)-(3.16)in[1]. Thefirst
systems differ from one another in nontrivial ways. of these equations is an expression for the total rate of
In addition to the questions pertaining to step-strain deformation tensor Dtot =γ˙/2 as the sum of elastic and
procedures, to be discussed in detail in Sec. IV, there plastic parts:
areotherlimitations ofthe DAsimulationsthatmustbe
s˜˙
taken into account. The DA simulation system is too Dtot = +Dpl(s˜,m,Λ), (2.1)
smalltomakeitlikelythatmorethanoneSTZ-triggered 2µ˜
eventistakingplacewithinacharacteristicplasticrelax-
where s˜ and µ˜ are the deviatoric stress and the shear
ationtime;anddataisreportedonlyforsinglenumerical
modulus measured in units of the yield stress s . The
experimentsperformedonindividualsamplesratherthan y
internal state variables m and Λ are, respectively, the
averages over multiple experiments. Thus, fluctuations
orientational bias and the scaled density of STZ’s. We
are large, and the results must be sensitive to statisti-
consideronlythecaseofpure,plane-strainshearinwhich
cal variations in initial conditions. In short, we must be
thematerialisstrainedatafixedrateγ˙,andthestressis
careful in our interpretations.
measured as a function of the strain γ. To describe such
On the positive side, Demkowicz and Argon’s remark-
experiments, we write Eq.(2.1) in the form
able combination of multiple simulation methods and
multiple observations allows us to use the athermal STZ
ds˜ 2ǫ Λ
0
theory to construct what we believe is an internally self- =µ˜ 1− q(s˜,m) , (2.2)
dγ (cid:18) γ˙ τ (cid:19)
consistent interpretation of their data. Going beyond 0
their stress-strain curves, and focusing on the behaviors
where ǫ is a number of order unity (the atomic density
0
oftheliquidlikefractionφandthedensityρ,wedevelopa
in atomic units multiplied by the incremental strain as-
quasithermodynamicpictureinwhichtheeffectivedisor-
sociatedwith anSTZ transition), τ is the characteristic
0
dertemperatureχplaysadominantrole. Inthispicture,
time scale for STZ transitions, and
the STZ’s, i.e. the active flow defects, are rare sites, out
in the wings of the disorder distribution, that are more s˜
q(s˜,m)≡C(s˜) −m . (2.3)
susceptible than their neighbors to stress-induced shear |s˜|
(cid:16) (cid:17)
transformations. They almost certainly lie in the liquid-
like regions of the system, but only a very small fraction The function C(s˜) describes the stress dependence of the
of the liquidlike sites are STZ’s. The energy dissipated STZtransitionrate;itisproportionalto|s˜−1|for|s˜|≫1
in the STZ-initiated transitions generates the effective andvanishessmoothlynears˜=0. AsdefinedbyEq.(3.6)
temperature χ, a systemwide intensive quantity that, in in [1], the smoothness of C(s˜) near s˜= 0 is determined
3
by a parameter ζ, which we take to be unity throughout
the following. 4
Similarly,werestateEqs. (3.9),(3.10)and(3.11)from
3.5
[1]:
3
dm 2 ms˜
= q(s˜,m) 1− e−1/χ ; (2.4)
dγ γ˙ τ0 (cid:16) Λ (cid:17) a)2.5
P
G 2
dΛ = 2 s˜q(s˜,m) e−1/χ−Λ ; (2.5) s (1.5
dγ γ˙ τ
0 (cid:0) (cid:1)
1
and
0.5
dχ 2ǫ
0
= Λs˜q(s˜,m)(χ∞−χ). (2.6)
dγ c γ˙ τ 0
0 0
0 0.1 0.2 0.3 γ 0.4 0.5 0.6
The constant c is a configurational specific heat per
0
atom in units k , which must be of order unity.
B
We assume that the stress and strain tensors in these FIG.1: Theoreticalstress-straincurves(solidlines)compared
to the DA numerical simulation data [3]. The parameters
equationsremaindiagonalintwodimensionalpureshear
and plane strain, and that any correction in the third usedare: ǫ0 =1,sf =1.06GPa, µ=46GPa, sy =0.35 GPa,
c0 =0.18 and χ∞ =0.065. The initial effectivetemperatures
dimensionisnegligibleincomparisontootheruncertain-
are: χ0=0.0559, 0.0580, 0.0612, 0.0680(fromtoptobottom).
ties in this analysis. A basic premise of the STZ theory
The curvesare shifted by 0.5 GPa for clarity.
is that the zones are rare and do not interact with each
other;thusweassumefromthebeginningthatthequan-
tity ǫ0 exp(−1/χ) is small of order 10−3 (in fact, very 15(a) of [3], shown here in Fig. 1. These two equations
much less), so that the equations for m(γ) and Λ(γ) are involvetheparametersq (thedimensionlessstrainrate),
0
stiff compared to those for s˜(γ) and χ(γ). Then we can s (neededinordertoconvertfromdimensionlessstresses
y
safely replace Eqs. (2.4) and (2.5) by their stationary s˜to measuredstresses in units GPa), µ˜, c0, χ∞, and the
solutions: fourinitialvaluesoftheeffectivetemperatureχ(0)=χ .
0
We can obtain a certain amount of information about
s˜/|s˜| if |s˜|≤1
m=m (s˜)= (2.7) these parameters directly from features of the stress-
0 (cid:26)1/s˜ if |s˜|>1
strain curves before doing any computation. Note from
Eqs. (2.7) and (2.11) that no plastic deformation occurs
and
for s˜ < 1, no matter how large χ might be. Thus the
Λ=e−1/χ. (2.8) lowest observed value of a stress that marks a departure
fromelastic behavioris anupper boundfor s . The bot-
y
Eqs. (2.2) and (2.6) become: tomstress-straincurveinFig. 1,whichisthe curvewith
the largestinitialvalue ofφ, seemstohaveabreakpoint
ds˜ =µ˜ 1− 2ǫ0 e−1/χq(s˜) , (2.9) thatisaboutafactorofthreelessthantheflowstresssf,
dγ (cid:18) q (cid:19) which we see from the figure is about 1.06 GPa. There-
0
fore, s ∼= s /3 ∼= 0.35 GPa, and s˜ ∼= 3. We take the
y f f
and shearmodulusµ∼=46GPadirectlyfromtheslopeofthe
stress-strain curves in the elastic region; thus we choose
dχ 2ǫ
= 0 e−1/χs˜q(s˜)(χ∞−χ), (2.10) µ˜ ∼=130. Setting the right-hand side of Eq.(2.9) to zero,
dγ c q
0 0 we find that the flow stress s˜ satisfies
f
where
q =γ˙ τ =2ǫ e−1/χ∞q(s˜ ). (3.1)
0 0 0 f
s˜
q(s˜)=C(s˜) −m (s˜) , (2.11)
|s˜| 0 Using this relation with the DA estimate of γ˙ and the
(cid:16) (cid:17)
value of χ∞ given below, we confirm that τ0 is of order
and q =γ˙ τ . femtoseconds as implied by the Stillinger-Weber interac-
0 0
tions used in these simulations.
Once we have inserted Eq. (3.1) into Eqs. (2.9) and
III. FIRST FITS TO THE DA STRESS-STRAIN (2.10),wecansolvetheseequationsandvarytheremain-
CURVES ing parameters to fit the stress-strain curves in Fig. 15
(a)of[3]. Wefindthatthefitsaredisappointinglyinsen-
The firststepinourSTZ analysisofthe DAdatais to sitive to our choice of ǫ0 exp(−1/χ∞) so long as we stay
useEqs. (2.9)-(2.11)tofitthestress-straincurvesinFig. within our constraint that this product be small. How-
4
ever,ourlateranalysisoftheliquidlikefractionφreveals Note first the data for φ as a function of strain γ shown
thattheinternallyconsistentvalueofχ∞ is0.065,which by DA in Fig.15 of [3], also shown here in Fig.6. There
is slightly bigger than our a priori order-of-magnitude are four curves. Two of them have small initial values of
estimate basedonthe experimentaldata,asshownin[7] φ(γ =0)=φ andexhibitstresspeaks. Thecorrespond-
0
for example, but seems well within the accuracy of our ing values of φ(γ) rise monotonically to the steady-state
approximationforχ∞in[8]andtheuncertaintiesofthese value φ∞ ∼=0.46 as expected. Another curve starts with
small-scalesimulations. Thereforewehavechosenǫ0 =1 φ0 > φ∞. The corresponding stress-strain curve has no
andχ∞ =0.065. Agreementbetweentheoryandthenu- peak, and φ(γ) decreases to φ∞, again in accord with
merical simulations then can be obtained for all four of expectations. Inone case(the one withφ ∼=0.46),how-
0
the stress-strain curves by setting c0 = 0.18 and adjust- ever, φ0 is slightly above φ∞ but the stress still shows a
ingonlyχ foreachcurve. Ourbest-fitresultsareshown peak. Moreover, the corresponding density ρ decreases
0
in Fig. 1. The corresponding values of χ (from top to withγ,implying thatthe denserliquidlike fractionisde-
0
bottom in the figure) are 0.0559,0.0580,0.0612, 0.0680. creasing during the deformation.
Note that there is a gap in χ0 between the lowest three Thiskindofbehaviorappearselsewhereinthe DApa-
values,forwhichthestress-straincurvesarepeaked,and pers [2, 3, 4, 5]. It is clear in the constant-volume, MD
the largest, which shows no peak. simulations, where one of the stress-strain curves has a
If it were useful to do so, we could improve the agree- peak, but the corresponding φ(γ) remains nearly con-
ment between the simulated stress-straincurves and our stant,andthe pressureincreasesinsteadofdecreasingas
theorybymakingsmalladjustmentsofsy andµ˜ foreach it should if the liquidlike fraction were growing. And, as
curve, consistent with the likelihood that the four rela- we note below, all of the PEM simulations in [4] show
tively small computational systems are not exactly com- stresspeaks,eventhe onesinwhichφ is largeandcom-
0
parable to each other in their as-quenched states. In parable to φ∞.
the spirit of the discussion to follow, however, we have Itseemstousthatthemostlikelyexplanationforthese
chosen to assume that the four systems reach effectively discrepanciesisthattheubiquitous stresspeaksarearti-
identicalsteadystatesafterpersistentsheardeformation, factsofthe step-strainsimulations. Itisalsoquite possi-
and to attribute the small discrepancies to the statisti- ble, of course, that the STZ theory does not adequately
caluncertaintiesvisibleinthedata. Theonlysystematic account for the way in which stresses respond to rapid
feature of the stress-strain data that is not recovered by changes in the strain rate, and that the DA simulations
the theory is the slow decrease of the flow stress with are simply out of range of our STZ analysis. Perhaps
increasingstraininthe topcurve(the one withthe most both explanations are in part correct. Nevertheless, we
pronounced stress peak). We believe that this behavior prefer the first explanation for the following reasons.
is caused by the emergence of a nascent shear band as
Demkowicz and Argon performed only MD simula-
illustrated in Fig. 16(b) of [3].
tions, and not PEM,atconstant-pressure;but a number
of features of their PEM results may be relevant here.
Theyreportthatcascadesofeventswereevidentintheir
IV. SECOND THOUGHTS PEM simulations but were hard to detect in MD. Mal-
oney and Lemaitre [9, 10], who used only PEM, found
Fittingthestress-straincurves,however,isonlyapart thatcascadeswereprevalentandsometimessolargethat
of the challenge of interpreting the DA data [3]. We also they spannedtheir systems,which were only two dimen-
must understand how the liquidlike fraction φ and the sional but larger in numbers of atoms than those used
density ρ relate to our STZ variables, especially the ef- by DA. A related feature of the DA PEM simulations is
fective temperature χ. Therefore, it is essential to know that,evenforrapidlyquenchedsampleswithlargeinitial
whether we can trust the values of χ deduced from the disorder, the stress-strain curves exhibit marked stress
0
STZ analysisofthe stress-straincurves. We argueinthe peaksfollowedbystrainsoftening. Wesuspectthatthese
following paragraphs that these values are not quantita- behaviors may be generally characteristic of step-strain
tively reliable. processes, and correspondingly uncharacteristic of con-
One of the key tenets of plasticity theory is that tran- tinuous strainmechanismsthat are morecommonin the
sient peaks observed in the stress-strain curves for well laboratory.
annealed samples occur because there is an initial lack Notethat, betweeneachstrainincrementinPEM,the
of plasticity carriers in these systems, and new carriers system has a probability of dropping into a low energy
mustbegeneratedbydeformationbeforeplasticflowcan state from which it cannot escape without the applica-
begin and the stress can relax. Demkowicz and Argon tion of a large force. The large energy released when
use the correlation between stress peaks and the liquid- sucha trappedstate is destabilizedmay triggercascades
like fraction φ to argue that φ is a direct measure of ofsmallerevents. Thistrappingmechanismmaybeespe-
the population of plasticity carriers. The STZ theory, as ciallyimportantatthebeginningofasheardeformation,
presently constituted, predicts stress peaks when – and becausethentheenergyminimizationstartswithadisor-
only when – the initial STZ density is small. dered, as-quenched system. As a result, the first energy
This tenet is not confirmed by the DA simulations. drops may be large, and the initial stresses required to
5
set the system into motion may be anomalously high.
Thus we expect transient stress peaks in PEM, even for
0.6
initially disordered systems; and we expect large stress
fluctuations even in steady-state conditions. That is ex-
0.5
actly what is seen by Demkowicz and Argon.
The difference between step strains and continuous
0.4
shear has been demonstrated recently in bubble raft ex-
perimentsbyTwardosandDennin[11]. LikeDemkowicz 0
φ
0.3
and Argon, these authors subjected their strictly ather-
mal system to both steady, slow shear and to discrete
0.2
shear steps followed by relaxation periods long enough
for most motion to cease. They monitored the stress
0.1
during both procedures, with particular interest in the
distribution of stress drops associated with irreversible
0
plastic rearrangements. One oftheir mostinterestingre- 0.05 0.06 0.07 0.08
χ
sults is that the average size of stress events decreases
with decreasing shear rate for continuous strain, but in-

creases for step strains. That is, the stress relaxes via
a larger number of smaller drops for continuous defor- 1.08
mationthanfor step-strainmotion. Their interpretation
of this result seems to us to be roughly consistent with
ourdiscussionofPEMsimulationsintheprecedingpara- 1.06
graph;butneithertheynorweclaimafullunderstanding
c
ρ
of this phenomenon /0
ρ
In their MD simulations, Demkowicz and Argon use 1.04
step strains as opposed to a continuous strain rate. Ap-
parently, allowing the system to relax for a time at
T =300K between strainsteps removesthe effects of at 1.02
least some of whatever structural irregularities are pro-
ducingcascadesandstresspeaksinthePEMsimulations.

Itseemslikely,however,thatsomeelementsoftheather- 0.05 0.06 χ 0.07 0.08
malPEM-like,step-strainbehaviorpersistintheMDre-
sults. Our hypothetical low-energy trapping states may
be less likely to be sampled in step-strain MD, in which FIG. 2: Upper panel: The equation of state φ = φ0(χ)
obtained by fitting the four data points (filled circles) plus
case the initial transients and steady-state stress fluctu-
φ∞ = φ0(χ∞) = 0.458 (filled square). The φ0’s are the
ations might be smoother, as indeed they are. But the
DA numerical simulation data [3]; the adjusted χ0’s are
step-strainMDprocedureisnotthesameasacontinuous
0.0559, 0.0589, 0.0663, and 0.0749, with χ∞ = 0.065. The
one in which the strain is incremented on the time scale solid line is our fit to these points in Eq. (5.1). The open
at which the molecular motions are resolved. In the DA circles are the observed φ0’s plotted against the unadjusted,
simulations,the strainisincrementedonlyonceinabout best-fit, χ0’s. The dashed line is intended only as a guide
ten or more atomic vibration periods. We see little rea- for the eye. Lower panel: The equation of state ρ = ρ0(χ)
son to believe that these two simulation procedures will obtained by a procedure similar to the one described above
producepreciselythesameresponsestorapidlychanging except that the point ρ∞ = 1.0335 (filled square) is not in-
cluded in the fit. We attribute its displacement from the
driving conditions.
equation of state to a stress-induced dilation. The density is
This situation forces us to conclude – reluctantly in
view of the quality of the apparent agreement between normalized by ρc, the density of crystalline silicon (diamond
cubic). The solid line is our smooth fit to the data. Again,
the simulations and STZ theory shown in Fig. 1 – that
theopen circles and and dashed line indicate the unadjusted
the values of the χ0’s stated above and in the caption to equation of state.
Fig. 1 cannot be trusted.
ertiesoftheconfigurationaldegreesoffreedombelowthe
V. QUASITHERMODYNAMIC HYPOTHESES glass transition are determined by the effective temper-
ature T = (E /k )χ, in close analogy to the way
eff STZ B
Our inability to deduce the χ ’s from the stress-strain in which they are determined by the bath temperature
0
data has led us to probe more deeply into the mean- T abovethattransition. In true thermodynamicequilib-
ing of the effective temperature in glass dynamics. We rium,extensivequantitiessuchasthedensityρ,theinter-
base our analysis on a set of quasithermodynamic hy- nal energy U, or the liquidlike fraction φ in the DA sim-
potheses in whichwe assume thatthe steady-stateprop- ulations are functions of T and the pressure P. In other
6
words, these quantities obey equations of state. Below
the glass transition, the configurational degrees of free- 4
dom–thatis,thepositionsoftheatomsintheirinherent
3.5
states – fall outof equilibriumwith T because thermally
activated rearrangementsare exceedingly slow or impos- 3
sible. The most probable configurations in such situa-
2.5
tions maximize an entropy, thus the statistical distribu- )
a
tion of these configurations is Gibbsian with T =T . P
eff G 2
Accordingly, our first hypothesis is that the high-T (

equilibriumequationsofstatearepreservedintheglassy s1.5
state as expressions for the configurational parts of the
1
extensivequantitiesρ,U,φ,etc. intermsoftheintensive
quantities Teff, P, and the shear stress s. In particular, 0.5
weproposethattheeffectivetemperaturesχ ,multiplied
0
by E /k ,are the bathtemperatures T atwhichthe 0
STZ B 0 0 0.1 0.2 0.3γ 0.4 0.5 0.6
DA simulation samples fell out of thermal equilibrium
during cooling, and that the configurational parts of ρ,
U,φ,etc. werefixedatthosetemperatures. Ourspecial-
FIG. 3: Modified stress-strain curves using χ0 =
izationtoconfigurationalpartsrecognizesthat,forexam-
0.0559, 0.0589, 0.0663, and 0.0749. All other parameters are
ple, ρ undergoes ordinary thermal expansion at small T thesame as those used in Fig. 1.
and that the total internal energyU includes the kinetic
energy. Both of those non-configurational parts are un-
interesting for present purposes. Since we consider only upper bound not too far above φ=1/2; thus, we expect
P = 0 situations and work only at fixed T, we omit ex- that φ (χ) saturates at some value φ = φ ≥ 0.55 for
0 0 lim
plicitdependencesonthosevariables. Sheardilationmay large χ.
be relevant, especially for determining ρ; but so long as
Yet another consideration is that, in principle, we
we are considering only pre-deformation properties with
ought to be able to deduce the T ’s directly from the
0
s = 0, we also omit explicit s dependence. We return graphsofdensityversustemperaturefordifferentquench
later to the question of shear dilation, and we also post-
rates shown in Fig. 1 of [3]. The slowest quench is the
pone a discussion of the potential energy. For the mo-
closest to full equilibrium, and the points at which the
ment, we write simply: φ=φ (χ), ρ=ρ (χ).
0 0 faster quenches fall away from it are the three higher
As a first test of this hypothesis, we show by the open T ’s. The lowest T must be near the inflection point on
0 0
circles and dashed lines in both panels of Fig. 2 the the slow-quench curve. Unfortunately, the best we can
functions φ0(χ) and ρ0(χ) obtained with the measured do with the data at hand is estimate the T0’s to about
valuesofφ0 andρ0 andourbest-fitvaluesofχ0fromFig. ±20K, and even then we must be cautious about sys-
1. We do, in fact, find qualitatively plausible behavior. tematic uncertainties in the simulations. Nevertheless,
With our uncertainties about the χ0’s, however, we are we can use this process as a rough check on the modifi-
obliged to look harder and bring other considerations to cations of the χ ’s that we propose below.
0
bear on this analysis. We come now to our second and more speculative
We know some qualitative features of these equations hypothesis – that the point (χ∞, φ∞) ought to lie on
of state. First, we are dealing with a glass transition, the equilibrium equation-of-state curve; that is, φ∞ =
and therefore we expect that φ0(χ) should extrapolate φ0(χ∞). Here we are assuming that persistent shear de-
to some fixed value, possibly zero, at a nonzero Kauz- formation in an athermal system rearranges the atomic
mann temperature. In the following discussion, we de- configurations in a way that is statistically equivalent to
fine χK to be the effective temperature at which the ex- thermallydrivenrearrangement,exceptthattherelevant
trapolated φ0(χ) vanishes; but we make no assumption temperatureistheeffectivedisordertemperatureTeff in-
aboutwhetherthisisactuallyaKauzmanntemperature, stead of the bath temperature T. We also are assuming
orwhetheranidealglasstransitionactuallyoccursatthis that φ∞, being a ratio of two populations, is insensitive
or any other point. to changes in the volume of the systemthat might occur
Another feature is suggested by the binary, solid- during constant-zero-pressure simulations. In contrast,
like/liquidlike, nature of the atomic structure described ρ∞ woulddecreaseif, forexample,the systemundergoes
by Demkowicz and Argon. If this were just a simple a shear-induced dilation.
binary mixture, and if the energy of the mixture were With these considerations in mind, our next step is
proportionaljust to the density ofthe higherenergy,liq- to refine the estimates of the T ’s by requiring that
0
uidlike component, then the entropy would be a maxi- theyproducesmoothequationsofstatewiththequalita-
mum at φ = 1/2 and the temperature would diverge at tive features hypothesized above, and that the observed
that point. Obviously, amorphous silicon is not so sim- φ∞ ∼= 0.458 be equal to φ0(χ∞). Accordingly, we have
ple. Nevertheless, the observed φ does seem to have an computed φ = φ (χ) and ρ = ρ (χ) by adjusting the
0 0
7
χ ’s so as to optimize agreement with all the previously stress. Thus their result anticipates and fits accurately
0
discussedconstraints. Ourresultsareshownbythe solid into our quasithermodynamic picture.
lines in Fig. 2. As expected, ρ∞ <ρ0(χ∞), because the
flow stress s˜ at that point seems easily large enough to
f
cause dilation. (Throughout this paper, values of ρ are VI. DEPARTURES FROM
in units of ρ =2323.8kg/m3, the density of crystalline, QUASITHERMODYNAMIC EQUILIBRIUM
c
diamond-cubic, silicon.) The effective Kauzmann tem-
perature is χ = 0.051, and the upper bound for φ is Thenextquestioniswhethertheequilibriumequations
K
φ =φ(χ→∞)∼=0.6. The smooth curve that we have ofstateshowninFig. 2areobeyedundernonequilibrium
lim
fit to the equation of state for φ, shown in the upper conditionsduringdeformation. Apparentlytheyarenot.
panel of Fig. 2, is To see this, in Fig. 4, we have added to our equation-of-
state graphs four data sets showing the observed values
φ (χ)=φ 1−e−a(χ−χK) , (5.1) of φ and ρ as functions of our calculated values of χ for
0 lim
(cid:16) (cid:17) the four different deformation histories. In all cases, the
observed values of φ and ρ initially fall off the equilib-
where a = 100. Other important parameters are the
rium curves and do not come together again until they
four T ’s: 1100K,1160K,1305K, 1475K, which are
0 approachχ∞. Thisnonequilibriumbehaviorcanbe seen
consistent with our crude direct estimates from Fig.
more directly by comparing the three panels of Fig. 15
1 in [3]. Note that the large gap between the low-
in [3]. It is clear there that the rate at which the stress
est three unadjusted T ’s and the upper one has disap-
0 relaxestotheflowstressisfasterthantheratesatwhich
peared, and that the adjusted points fit on a smoother
either φ or ρ relax to their steady-state values.
curve than the unadjusted ones. We also find that
To account for the transient, nonequilibrium behavior
T∞ = 1280K. Then, with χ∞ = 0.065, we obtain
ofφ, we proposeanequationof motionanalogousto Eq.
E /k ∼= 21,000K, or about 1.3 ev. The associated
STZ B (17) in [5], but with a form similar to our Eq. (2.10):
values of χ are 0.0559, 0.0589,0.0663,and 0.0749.
0
Using these adjusted values of the χ ’s, we have re- dφ 2ǫ
0 = 0 e−1/χs˜q(s˜) (φ (χ)−φ), (6.1)
computed and plotted the stress-strain curves in Fig. 3, dγ c q 0
1 0
shown again in comparison with the simulation data.
As expected, the non-STZ peak has disappeared, and wherec1 isaconstantsimilartoc0. Theideahereisthat
there is more rounding in the bottom, most rapidly the disorderdescribedby χ may rise atits ownrate, but
quenched case, but the stress peaks for the two most the liquidlike-solidlike reorganization associated with φ
deeply quenched systems remain almost unchanged. may not catch up instantaneously. We assume that the
underlyingmechanismthatdeterminestheseratesisstill
Finally, within the context of quasithermodynamic
therateofenergydissipation. Thereisnoothercompara-
equilibrium, we note that a relation between the poten-
bly simple andbasic couplingbetween mechanicaldefor-
tial energy and χ has been proposed and confirmed by
Shietal..[12]Theseauthorsdescribemolecular-dynamics mation and the internaldegrees of freedomthat satisfies
the requirement that it be a non-negative scalar. Note
simulations of shear banding in two-dimensional, non-
that Eq.(6.1) is supplementary to the STZ equations of
crystalline, Lennard-Jones mixtures. Their analysis of
motion, Eqs.(2.9) and (2.10); it assumes that the inten-
these simulations goes beyond our own in at least one
sivequantitiess˜(γ)andχ(γ)areunchangedfromtheSTZ
important way. They note that the plastic strain rate
predictions and that they continue to controlthe behav-
predicted by the STZ theory, as well as by other flow-
ior of φ even away from equilibrium. This is a strong
defect theories, has the form exp(−1/χ) multiplied by a
assumption, especially near the initial stress transients
function of the shear stress, essentially our factor q(s˜)
where we already know that there is some mismatch be-
defined in Eq.(2.11). Force balance requires that s˜be a
tween the STZ theory and the DA simulations.
constant across the shear band, thus a measurement of
Our results for the four functions φ(χ), determined
the position-dependent strain rate is a measure of the
position dependence of χ. Shi et al. then postulate using Eq.(6.1), are shown in Fig.5. The corresponding
functions φ(γ) are shown in Fig.6, here in comparison
that the potential energy depends linearly on χ. They
with the DA data. The only adjustable parameter is c ,
computetheposition-dependentpotentialenergydirectly 1
which we choose to be 0.42. The agreement seems to be
from their simulation data and find that it maps accu-
within the uncertainties of the data.
ratelyontoχ aspredictedbythe spatiallyvaryingstrain
We continue our development of the nonequilibrium
rate. The importanceofthis observationisthattheyare
quasithermodynamictheorybywritinganequationanal-
lookingataweaklynonequilibriumsituationinwhichthe
ogous to Eq. (6.1) for the function ρ(γ):
potentialenergyandχarevaryingcontinuouslyinspace,
but only very slowly in time, along the postulated linear
dρ 2ǫ
equation of state. Their system remains in quasithermo- = 0 e−1/χs˜q(s˜) (ρ0(χ)−ρ−δρ0), (6.2)
dγ c q
dynamicequilibriumduringthisvariationbecause,unlike 1 0
theDAsimulationsdiscussedhere,itisnotundergoinga Here,δρ ∼=0.012isthedilation-inducedshiftoftheequi-
0
fasttransientresponsetoasuddenchangeintheapplied librium density shown by the displacement of ρ∞ from
8
0.5 0.5
0.4
0.4
φ
φ
0.3
0.3
0.2
0.2
0.05 0.055 0.06 0.065 0.07 0.075 0.08
χ 0.05 0.055 0.06 0.065 0.07 0.075 0.08
χ
1.08
1.08
1.06
1.06
c
ρ 1.04 c
ρ/ ρ 1.04
/
ρ
1.02
1.02

0.05 0.055 0.06 0.χ065 0.07 0.075 0.08 0.0 5 0.055 0.06 0.065 0.07 0.075 0.08
χ
FIG. 4: Upperpanel: The DA numerical simulation data for
FIG. 5: Upper panel:φ as a function of χ using the nonequi-
φversusthetheoreticalχforthefourcasesconsideredinRef.
[3]. The equation of state φ=φ0(χ) was added to stress the libriumEq. (6.1)withc1 =0.42. Theequilibriumequationof
state was added for illustration. The curves should be com-
nonequilibrium nature of the deformation-induced dynamics
pared to the DA data shown in the upper panel of Fig. 4.
of φ. Lower panel: The corresponding plot for ρ/ρc. The
Lower panel: ρ as a function of χ using the nonequilibrium
points (χ∞, φ∞) and (χ∞, ρ∞) are marked by solid squares.
Eq. (6.2) with c1 = 0.42. These curves should be compared
totheDAdatashowninthelowerpanelofFig. 4. Thepoints
(χ∞,φ∞) and (χ∞, ρ∞) are marked by solid squares.
the equilibrium curve ρ (χ) in the lower panel of Fig.2.
0
To integrate Eq.(6.2), we have approximated ρ (χ) by
0
a smooth polynomial. The factor c1 = 0.42 needed to ationofφ(γ) andρ(γ)ascomparedto thatofs˜(γ). This
fit the density data is the same as in Eq. (6.1). The level of success seems to be as much as we can expect
corresponding graphs of ρ(γ), along with the DA data, from the theory at this stage in its development.
are shown in Fig. 7. Again, the agreement seems to
be within the uncertainties; but here we are pushing the
theorytoofarforcomfort. Ourquasithermodynamichy-
potheses imply that ρ (χ) ought to be a function of the VII. SUMMARY AND CONCLUDING
0
REMARKS
shear stress s˜as well as χ, and that the dilation approx-
imated here by δρ should be part of that generalized
0
equation of state. In particular, δρ should be the dila- The unique aspectof the DemkowiczandArgonsimu-
0
tional change in the density when s˜is equal to the flow lationsistheirmeasurementofextensivequantities–the
stresss˜ . We haveinfacttriedδρ ∝s˜2 (as innonlinear mass density ρ and the liquidlike fraction φ – in parallel
f 0
elasticity),buttheresultsaredistinctlyunsatisfactoryat with conventional stress-strain curves. They argue con-
small γ where neither Eq.(6.2) nor the STZ theory itself vincinglyandimportantlythatφiscloselyrelatedtothe
may accurately describe the fast transient. Equations density of plasticity carriers. However, their remarkably
(6.1) and (6.2) do account for the relatively slow relax- complete report includes some results that make it seem
9
a theoretical interpretation of the DA simulations that
seemsphysicallysatisfying,internallyself-consistent,but
interestingly incomplete. If confirmed by further tests,
0.55
this quasithermodynamic theory could become a useful
0.5 toolforpredictingthenonequilibriummechanicalbehav-
ior of amorphous solids.
0.45
There are many open issues. Perhaps the most ur-
0.4 gent of these is the interpretation of the DA step-strain
φ
simulation technique. The DA data do show that a well
0.35
annealedsamplewithaninitiallysmallφexhibitsatran-
sient peak in its stress-strain curve. However, the con-
0.3
verseseemsnotnecessarilytobetrue. Stresspeakssome-
0.25 times appear in the DA results in cases where the initial
φ is large and does not increase during plastic deforma-
0.2
tion, a behavior that is inconsistent with the presumed
0 0.1 0.2 0.3 0.4 0.5 0.6
γ relation between φ and the density of plasticity carriers.
After examining other possibilities and looking at other
examples of step-strainprocedures (see Sec.IV), we have
FIG. 6: Theoretical predictions of φ(γ)for all four quenches,
concluded that the stress peaks observed in cases with
based on Eq. (6.1) with c1 = 0.42, compared with the DA
large initial φ are most likely artifacts of the numerical
simulation data.
step-strain procedure. Once we allowed ourselves that
flexibility and found alternative ways of estimating the
initial effective temperatures for those cases, we found
1.08
that our quasithermodynamic picture fits together quite
1.07 well.
In our opinion, one of the most important next steps
1.06
in determining the limits of validity of the quasithermo-
dynamic STZ theory would be to redo the DA analysis,
1.05
including the measurement of the liquidlike fraction φ,
c
ρ 1.04 with continuous strain MD. So far as we know, nobody
/
ρ
beforeDemkowiczandArgonhaseverdoneanythinglike
1.03
this – making independent, simultaneous measurements
of the density of plasticity carriers and the stress-strain
1.02
response for differently quenched systems. Discovering
1.01 the relation between continuous and step-strain simula-
tions in this context seems likely to be extremely useful
1 at least for understanding the numerical simulations. It
0 0.1 0.2 0.3 0.4 0.5 0.6
γ
could also be a big step forward in understanding amor-
phous plasticity.
FIG. 7: Theoretical predictions of ρ(γ) for all four quenches, AnotherobviouslyopenissueisthevalidityoftheSTZ
based on Eq. (6.2) with c1 = 0.42, compared with the DA theory in situations where the system is responding to
simulation data. rapidchangesinloadingconditions,asintheDAnumeri-
calsimulationswherethestrainrateislargeandisturned
on instantaneously both at the beginning of the process
that this relationship may be neither simple nor direct. and ateach strainstep. The presentversionof STZ the-
The main purpose of our investigation has been to learn ory [1] is a mean-field approximation in which interac-
more about that relationship. tions between zones are included only on average, and
Our proposed interpretation of the DA simulations is fluctuations are neglected. It is possible that neither the
based on the athermal STZ theory [1], which we believe STZtheoryaspresentlyformulated,northeconventional
captures the central features of amorphous plasticity – explanation of stress peaks in terms of plasticity carri-
at least for processes that are not too rapidly varying as ers, can fully account for stress transients during rapid
functions of space or time. In order to discuss quanti- changes in loading conditions, even in continuous-strain
ties like φ, however, we have had to go beyond the STZ MD simulations. From this point of view, a continuous-
theory. We have hypothesized that, in low-temperature, strainversionoftheDAsimulationsseemsdoublyimpor-
steady-state,nonequilibriumconditions,ρandφandpre- tant. If new simulations showed no qualitative discrep-
sumablyothersuchquantitiesarerelatedtotheeffective ancy between continuous and step-strain procedures, we
disorder temperature χ via quasithermodynamic equa- wouldknowthattheSTZtheoryismissingsomeessential
tions of state. With this hypothesis, we have developed ingredients. We might then try to include correlations
10
and fluctuations by developing a more detailed statisti- ditions, where strain rates are very much smaller than
caltheoryofSTZ’swithvaryingthresholds,perhapswith those in MD simulations? Or does it generally become
noisy interactions betweenthem as proposedrecently by inaccurate near stress peaks? Can it be used to pre-
Lemaitre and Caroli [13]. dict plastic deformation near the tip of an advancing
Inasimilarvein,wemustaskaboutthelimitsofvalid- crack? Moregenerally: Isthe effectivetemperatureχre-
ity ofourquasithermodynamichypotheses. Itwill be es- allysuchadominantstatevariable? Whatotherinternal
pecially important to look harder at Eqs.(6.1) and (6.2), variables might become relevant for describing fast pro-
whichdeterminetheratesatwhichφandρrelaxtotheir cesses? What real or computational experiments might
quasithermodynamic equilibrium values. Our quasither- help to answer such questions?
modynamic picture is based on the assumption that the
effective temperature χ is the intensive variable – the
thermodynamic force – that drives the extensive quanti-
ties φ, ρ, and the STZ variables Λ and m. According to
Acknowledgments
ourequationsofmotion,Λandmaretightlyslavedtoχ.
Wehavepostulatedequationsofstaterelatingφandρto
χunderequilibriumorsteady-stateconditions,andhave We thank M.J. Demkowicz and A.S. Argon for pro-
further proposedinEqs.(6.1)and(6.2), onwhatseemto vidingthe data onwhichthis researchwasbasedandfor
us to be general grounds, that φ and ρ are more loosely permissiontouseithere. J.S.L.alsothanksC.Caroli,M.
slavedtoχthanareΛormduringexcursionsfromequi- Falk, A. Lemaitre, and C. Maloney for important ideas
librium. Here, as in the questions regarding the stress and information. E. Bouchbinder was supported by a
response, we need to learn whether the equations of mo- doctoralfellowshipfromthe HorowitzComplexityFoun-
tionforφandρareaccuratewhenthoseexcursionsfrom dation. J.S. Langer was supported by U.S. Department
equilibrium are faster,say,than the relaxationrate ofχ. ofEnergyGrantNo. DE-FG03-99ER45762. I.Procaccia
In short, we are asking how far this theory can be acknowledges the partial financial support of the Israeli
pushed. Can it, for example, always be used to predict Science Foundation, the Minerva Foundation, Munich,
stress-strain transients under realistic experimental con- Germany, and the German-IsraeliFoundation.
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