Table Of ContentBreakdown of Nonlinear Elasticity in Stress-Controlled Thermal Amorphous Solids
Vladimir Dailidonis,1 Valery Ilyin,2 Itamar Procaccia,2 and Carmel A.B.Z. Shor2
1Bogolyubov Institute for Theoretical Physics, 03680 Kiev, Ukraine
2Department of Chemical Physics, the Weizmann Institute of Science, Rehovot 76100, Israel
Inrecentworkitwasclarifiedthatamorphoussolidsunderstraincontroldonotpossessnonlinear
elastic theory in the sense that the shear modulus exists but nonlinear moduli exhibit sample to
sample fluctuations that grow without bound with the system size. More relevant however for
experiments are the conditions of stress control. In the present Communication we show that also
under stress control the shear modulus exists but higher order moduli show unbounded sample to
7 samplefluctuation. Theunavoidableconsequenceisthatthecharacterizationofstress-straincurves
1 in experiments should be done with a stress-dependent shear modulus rather than with nonlinear
0 expansions.
2
n
Amorphoussolidsarebornasaresultoftheglasstran- in future experimental measurements.
a
sition, by cooling sufficiently rapidly certainpure liquids Consider the standard non-linear elasticity theory for
J
or mixtures of liquids such as to avoid the equilibrium asolidunder simpleshearstrain,with γ =γ being the
4 xy
phase transition to a crystalline solid. In recent years onlynon-zerocomponentofthestraintensor. Thestress
] the theoretical understanding of amorphous solids is ad- iscomputedasaTaylorexpansionaroundzerostrain[9]:
ft vancing,showingthattheyhaveratherpeculiarmechan-
o icalproperties. Inparticularitwasclarifiedrecentlythat σ(γ)=B γ+ 1B γ2+ 1B γ3+... (1)
s generically amorphous solids do not have a regular non- 1 2! 2 3! 3
.
t linear elasticity theory [1–3]; describing the stress-strain
a where σ = σ is the only non-zero component of the
xy
m relation with as a power series in the strain for example
stress tensor and
is untenable. This was shown first for athermal amor-
-
d phous solids independently of the deformation protocol, dnσ
B ≡ . (2)
n andlaterforthermalamorphoussolidsunderstraincon- n dγn(cid:12)
o trol conditions. For a wide range of (low) temperatures (cid:12)(cid:12)γ=0
c (cid:12)
itisimpossibletodefinenon-linearelasticcoefficients,as B is the usual shearmodulus that is usually denoted as
[ 1
theydonotconvergetoaspecificvaluewiththeincrease µ, µ ≡ B . Inverting the power series (1) will give an
1
1 of the system size [3]. The physical reason for this lack equivalent expression:
v
of convergence is the anomalous statistics of stress fluc-
6 1 1
tuations in amorphous solids. It was also shown that in γ(σ)=γ +s σ+ s σ2+ s σ3...
6 0 1 2 3
straincontrolconditions nonlinear expansionof stress in 2! 3!
9
0 powersof strainarenot necessary,since the stress-strain =γ + ∂γ(σ) σ+ 1 ∂2γ(σ) σ2+...(3)
0 curves can be fully characterized using a theoretically 0 ∂σ (cid:12)σ=0 2! ∂σ2 (cid:12)σ=0
. meaningful (local) γ-dependent shear modulus µ(γ) [2]. (cid:12) (cid:12)
1 The relations between(cid:12) the coefficients (cid:12)in Eq. (1) and
0 Eq. (3) are given by [10]
Additional interest in this phenomenon has emerged
7
1 due to its relation to the putative Gardner transition [4] 1
B =
: which is believed to exist in generic glass formers [5–7]. 1 s
v 1
One ofthe mainconsequencesofthe Gardnertransition, s
Xi if it exists, is precisely the breakdown of nonlinear elas- B2 =−s32
1
r ticity inthe sense discussedabove[8]. This increasesthe
a motivation to identify the breakdown of nonlinear elas- B = 3s22−s1s3
ticity in experimental system. Experiments are however 3 s51
rarelydoneinstraincontrolledconditionssinceitismuch 10s s s −s2s −15s3
B = 1 2 3 1 4 2 ···
easiertocontrolthe stressonapieceofmaterialthanits 4 s7
1
strain. It is therefore necessary to examine the elasticity
theoryofamorphoussolidsinstresscontrolcondition. In To compute the derivatives appearing in Eq. (3)
this Communication we examine the nonlinear elasticity in terms of strain fluctuations we need a statistical-
theory of stress controlled simulations, where the strain mechanical expression for the mean strain. Consider a
isthefluctuatingquantity. Thecentralconclusionisthat system with N particles in positions R ≡ {ri}Ni=1 in a
also in stress-controlled condition the nonlinear elastic- volume V. When the external stress σext is fixed, the
ity theory breaks down, and should not be used in the ensemble average is provided by (see, e.g., [11])
context of amorphous solids. Using a stress-controlled
simulationtakesusonestepclosertosystemsusedinex- γe(−U(γ,R)+Vσextγ)/TdγdR
γ(σext)= , (4)
periments, and perhaps similar results can be obtained R e(−U(γ,R)+Vσextγ)/TdγdR
R
2
whereT isthetemperatureinunitswheretheBoltzmann In this equation the α component of the displacement
constant is unity, U(γ,R) is the energy of the system. vector of a particle is given by
The computation of the derivatives in Eq. (3) is now
straightforward[12], yielding δrˆα =∆rˆmax(2ξα−1), (9)
V where∆rˆ isthemaximumdisplacementandξα isan
s = (hγ2i −hγi2) max
1 T 0 0 independent random number uniformly distributed be-
V 2 tween0and1. After thesestandardsweepsthe transfor-
s2 =(cid:18)T (cid:19) hγ3i0−3hγi0hγ2i0+2hγi30 mation h changes according to
(cid:0) (cid:1)
V 3 hnew =hold+δh, (10)
s = hγ4i −4hγ3i hγi −3hγ2i2
3 (cid:18)T (cid:19) 0 0 0 0
(cid:0) where elements of the random symmetric matrix δh are
+12hγ2i0hγi20−6hγi40 defined by
... (cid:1) (5)
δh =∆h (2ξ −1), i≤j. (11)
ij max ij
where the moments of the strain distribution at zero ex-
ternal shear stress are defined by Here ∆hmax is the maximum allowed change of a ma-
trix element and ξ is an independent random number
ij
γne−U(γ,R)/TdγdR uniformly distributed between 0 and 1. In this particu-
hγni0 = R e−U(γ,R)/TdγdR . (6) lar case the h matrix was chosen to induce simple shear
conditions
R
Themainquestionnowiswhatisthestatisticsofstrain
1 γ 1 δh
fluctuation that determine these moments. In an amor- h=L =L , (12)
phous solid the system is confined to a compact set in a (cid:18)0 1 (cid:19) (cid:18)0 1 (cid:19)
restricted domainoftheconfigurationspace;accordingly
whereListhe lengthofthe squaresimulationboxandγ
for each realization of the amorphous solid the integral
is the simple shearstrain,with the volume ofthe system
(6) is computed over this set of configurations which are
V=L2 being conserved. The value of ∆h and the
visited by the glass particles, confined around a given max
maximum displacement of particle positions ∆rˆ are
amorphous structure [2, 13]. This fact is the origin of max
selected to obtain a desired acceptance rate of 30%. For
sample-to-samplefluctuations;thesefluctuationsmayei-
eachkindofmovethetrialconfigurationisacceptedwith
therdecreaseorincreasewiththesystemsize,depending
probability
on the moment involved. This is precisely what we need
to learn using the numerical simulations described next. ∆G
Having the expressions (5) for the coefficients sn we Ptr =min(cid:20)1,exp(cid:18)− T (cid:19)(cid:21). (13)
will measure them below in numerical simulations for
different realizations of the same glass. We will then Here ∆G is the enthalpy change due to the move. Since
use Eqs. (4) to determine the moduli Bn. Needless to in this case we chose volume conserving transformations
say, for each realization of the glass with N particles we withoutanyexternalstresswasappliedthe expressionis
will find a number for Bn. As explained, the interesting simply
questionisthesample-to-samplefluctuation,dotheyde-
crease or increase with the system size. In other words, ∆G=U(γ+δγ,rnew)−U(γ,rold) , (14)
i i
wewillmeasurethedistributionsoftheseobjectsandthe
moments of sample-to-sample variances where γ is the system’s strain. In the current study no
external stress is implemented, and thus the energy dif-
(δB )2 ≡(B −B )2 , (7) ference is only due to the affine deformation itself. Thus
n n n
the strainγ is free to fluctuate aroundsome meanvalue.
where (•) denotes the averageoverdifferent realizations. The reader should note that for any given realization of
To simulate a system in which the strain is free to theglassthemeanvalueofγisnotnecessarilyzero. Only
fluctuate we employ the “variable shape Monte-Carlo” whenthesemeansareaveragedovermanyglassyrealiza-
technique, cf. Ref. [14, 15], explained in more details in tioninthe senseofthe overlineaverageEq.(7)the result
Ref. [11]. This method starts by defining a square box should vanish.
of unit area where the particles are at positions rˆ. Next To collect data we used the same glass former used
i
onedefinesalineartransformationh,takingtheparticles in Ref. [3], i.e. a two-dimensional Kob-Andersen 65:35
to positions r via r = h·rˆ. The actual area of the binary Lennard-Jones mixture [16, 17], slowly quenched
system becomies the ideterminaint V =| h |. The Monte formaliquidatT =0.4downtoT =10−6usingmolecu-
Carloprocedureemploystwokindsofmoves. Firstlyone lardynamics withquenchrate T˙ =10−6. Here tempera-
performs n standard Monte Carlo moves ture and all other quantities are measured in dimension-
lessunits. Thenumberdensityisρ=1.162. Eachsample
rˆnew =rˆold+δrˆ, 1≤i≤N. (8) wasthanrelaxedinthetargettemperaturefori=5×105
i i
3
0.4 N=200 0.010
N=500 20 0.050
0.35 N=1000 0.100
N=2000
0.3
10
0.25 2
B()1 0.2 Bδ)1 5
P (
0.15
2.5
0.1
0.05
05 10 B15 20 25 200 500N 1000 2000
1
3.5x 10−3 N=200 0.010
N=500 0.050
3 N=1000 106 0.100
N=2000
2.5
B)2 2 B2)2105
( δ
P 1.5 (
1 104
0.5
−100 00 −500 B0 500 1000 200 500N 1000 2000
2
x 10−6
N=200
12 N=500 0.010
N=1000 0.050
N=2000 1012 0.100
10
B()3 68 B2)31011
P δ
(1010
4
109
2
0
− 4 −2 B03 2 x 1045 200 500N 1000 2000
FIG.1. Thedistributionofvaluesofthefirstthreecoefficients FIG. 2. The system size dependence of the variances of the
in the expansion Eq (1) over the realizations, for different firstthreecoefficientsintheexpansionEq(1). Herethetem-
systems size. The temperature here is T =0.05. peratures are T=0.01, 0.05 and 0.1.
the strain aroundtwo (or more) mean values were elimi-
MC swaps,in whichthe parameters∆h ,∆rˆ were
max max
natedfromtheensembleandwerenotconsideredforthe
calibrated. Than each realization was ran with zero ex-
ternal stress σext = 0 for τ = 5×106 MC swaps, from calculation of the sample-to-sample variances.
which the moments of the strain fluctuations hγni were In Fig. 1 we show the distributions of B ,B , and B
1 2 3
calculated. The number of steps τ was chosen to be suf- over the different sets of realizations as explained above.
ficiently large to reach saturation of the measured quan- The data presented here are for T =0.050, similar plots
tities and independence of the averaging window. To were made for T = 0.010,0.100. For each of the coeffi-
calculatethevariancesacrosstheensembles,foreachsys- cients, and for each system size, the values of B across
n
tem size N = {200,500,1000,2000} between 300 to 800 the whole ensemble (N = 300−800 realizations) were
different realizations were prepared and allowed to fluc- binnedandnormalizedtoproducethe PDF.As inprevi-
tuate independently. In order to keep the data clean, ous reports ([1, 3]), one can see that the distribution of
realizationsthat were detected exhibiting fluctuations of the shear modulus µ = B converge to a delta-function
1
4
at a temperature dependent sharp value B =15.9, 14.9, Nevertheless it is interesting that the shear modulus is
1
14.1attemperaturesT=0.01,0.05,0.1respectively. The expected to exist, and the variances of B with k ≥ 3
k
second modulus B does not converge with the system are expected to diverge with the system size, in agree-
2
size, and the variance of the third modulus B diverges ment with the predictions of Ref. [1] and the findings of
3
with the systems size. It is therefore interesting to ex- the present paper. In Ref. [8] it is also predicted that
amine the variance of the distributions as a function of the phenomenon should disappear when the system is
system size, as shown in Fig. 2. The ensemble variances heated above the (protocol dependent) Gardner temper-
(δB )2 is presented in a log-log plot for different system ature. Whether this is occurring before the shear modu-
k
sizes,andfitted to a linearcurve. Fromthis data we can lus itself vanishes ornot depends onthe speed ofquench
extract the following scaling laws: from high to low temperatures. A search of a Gardner
temperature would require repeating our analysis on ex-
(δB1)2 ∼Nα1 ,(δB2)2 ∼Nα2 ,(δB3)2 ∼Nα3 , (15) tremely slowly quenched glasses as a way to provide a
good separation of the Gardner point and the point of
To determine the exponents we estimate the variance disappearanceoftheshearmodulus[8]. Suchananalysis
by fitting a Gaussian to the distribution. In our experi- is beyond the scope of the present Communication but
ence this method is superior to measuring the variance appears to be a worthwhileendeavorfor future research.
fromtherawdatathatmayhaveoutliersthatreducethe
Finally we should discuss the implications of our find-
accuracyofthe estimate. Wemeasureα =−0.95±0.04
1 ing to stress-controlled experiments. Clearly, one can
with only slight temperature dependence. For α we
3 repeat the procedure described above at a given stress
measure the values α = 1.6± 0.24. The distribution
3 value insteadof zero stress. For any givenstress one can
of B seems marginal, and indeed the measured value
2 determine the mean strain, plot a stress vs. strain curve
of the exponent α is close to zero, except at the lowest
2 and determined the local slope. The prediction that is
temperaturewherethereisanapparentweakdivergence.
implied in the present Communication is that this local
This is in line with the results in the athermal case [1],
slope will be equivalent to the value of the shear modu-
but differs from the thermal strain-controlled ensemble
lus µ = B that can be measured from the moments of
1
[3],whereadivergencewasfoundalsoforvarianceofB .
2 the strain fluctuations. There is no point in attempting
Itisinterestingtorelatethesefindingstoarecentthe-
to fit a nonlinear stress vs. strain curve since the non-
oretical work [8] predicting a so-called Gardner transi-
linear moduli have no theoretical value. This prediction
tion [4] in thermal glass forming liquids [7, 8]. The phe-
warrants a careful experimental verification.
nomenon seen in this transition is that at some temper-
ature, lowerthan the glasstransition temperature, there
appearsaqualitativechangeinthefree-energylandscape,
ACKNOWLEDGMENTS
generating a rough surface with arbitrarily small barri-
ersbetweenlocalminima. Theconnectiontothepresent
workis thatthis is accompaniedby a breakdownofnon- This work has been supported in part by the Min-
linear elasticity in much the same way reported here. erva foundation with funding from the Federal German
The available theory pertains to a mean field treatment Ministry for Education and Research, and by the Israel
and comparisonof exponents is probably not warranted. Science Foundation (Israel Singapore Program).
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