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Physics of Amorphous Solids: their Creation and their Mechanical Properties
Itamar ProcacciaDepartment of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: December 1, 2009)
In this short review I summarize some progress achieved in my research group regarding the glasstransition and the mechanical properties of the resulting amorphous solids. Our main concerns wereon the one hand to understand the extreme slowing down over a narrow range of temperatures whichresults in a viscosity so high that the materials behave as solids under small mechanical strains. Onthe other hand we are interested in their mechanical yield to higher strains. After yielding thematerials can be in an elasto-plastic steady state which we want to understand and characterize.
First Part: The Glass Transition
I. INTRODUCTION
We study the glass transition by constructing variousmodels of point particles interacting via soft potentialswhich have the tendency to avoid crystallization uponcooling. Rather, they exhibit the so-called ‘glass transi-tion’ when the temperature is lowered. For typical glassformers with soft potentials the glass transition is accom-panied by a dramatic increase of relaxation times: theviscosity of the super-cooled liquid shoots up by fifteenorders of magnitude within a relatively short temperaturerange [1]. When the liquid is examined on microscopictime scales, the molecules seem jammed, trapped intomoving only within their cages. Of course, on the muchlonger viscous time scale these same systems do reachequilibrium by restructuring themselves through an er-godic exchange of particles between cages. As long asthe system is ergodic it lends itself, at least in principle[2], to a statistical mechanical treatment. Nevertheless,it is extremely hard to reach a useful statistical mechan-ical theory as long as one considers the glass former onthe level of its constituent particles because it is difficultto evaluate partition-function integrals in continuous co-ordinates. It is therefore tempting to find a reasonableup-scaling (coarse-graining) method that would define adiscrete statistical-mechanics with partition sums ratherthan integrals, with the sums running on a finite num-ber of quasi-species having well characterized degenera-cies and enthalpies. Indeed, in a number of examplesin 2-dimensions (2D) [3–7] and in one example in 3D[8] it was shown recently that such a discrete statistical-mechanics is possible and quite advantageous [9, 10] inproviding a successful description of the statistics and thedynamics of systems undergoing the glass transition. Inthis review we follow Ref. [11] and examine how far thepredictions of such a statistical mechanical theory canbe pushed. As the theory is developed on the level ofup-scaled quasi-species, one can ask whether the energy,entropy and other thermodynamic quantities appearingin the quasi-species language are identical numerically tothe corresponding quantities of the system itself when
computed on the level of constituting particles. One ofthe main results of the theory is an affirmative answer tothis question. This means that the statistical mechani-cal theory can be used to quantitatively predict physicalobservables of interest also beyond the range of measure-ments, hopefully providing a satisfactory understandingof the phenomenology of the glass formation.
To demonstrate the generality of the approach we ex-amine two extremely different models of glass formersto which we apply the same procedure of: i) identifyingan appropriate up-scaling ii) validating that the chosenup-scaling yields a consistent statistical mechanics andiii) demonstrating how the resulting theory provides agood understanding of the subtle geometric configura-tional changes as a function of temperature, of the ther-modynamic functions and of the relaxation time. Wewill stress the point that understanding the way that theconcentrations of the available up-scaled quasi-species de-pend on the temperature is tantamount to understandingthe scenario of the glass transition. To underline the gen-erality of the approach we deal below with two models,one model is studied in 3D in an NPT ensemble and theother in 2D in an NVT ensemble. The models were cho-sen, as explained below, to represent extremely differentmicroscopic characteristics; nevertheless once up-scaled,the statistical mechanics appears very similar, encourag-ing us to propose that the approach is rather general.
The structure of the first part of this review is as fol-lows: in Sect. II we introduce the two models studiedbelow, discuss the dynamics of their correlation func-tions as a function of temperature, and measure the re-laxation time τα that will be later connected to the sta-tistical mechanical theory. In Sect. III we present theupscaling for these two models. This upscaling definesquasi-species in terms of carefully chosen groups of par-ticles rather than individual particles. We then show howthe scenario of the glass formation can be characterizedby the temperature dependence of the concentrations ofthe various available quasi-species. Since this upscalingis not unique, we present a validation of the choice ofquasi-species. The validation is a demonstration thatthe choice of quasi-species provides a self-consistent sta-tistical mechanics. In other words, each quasi-species ischaracterized by its own enthalpy and its own degener-
2
acy. The statistical mechanics is then shown to be ableto accurately recapture the scenario of the glass forma-tion at temperatures where data is available, and to con-tinue to predict it for temperatures that are outside therange of numerical simulations. In Sect. IV we relatethe upscaled picture to the dynamics of the system. Weshow how a natural static length-scale is appearing, andhow this length scale characterizes the relaxation time τα.Finally, in Sect. IV D we demonstrate that the statisti-cal mechanics as defined on the upscaled quasi-species isable to provide the correct thermodynamic function forthe original system. In other words, we can compute theenergy, entropy, specific heat etc. directly from the up-scaled picture, in the range of temperatures accessible tosimulations as well as for temperatures that are inacces-sible to simulations. In Sect. V we discuss the degreeof generality of the present approach and stress what arethe remaining riddles for future research.
II. TWO MODELS OF GLASS FORMATION
A. Three-dimensional Binary Model in NPTensemble
The first model that we use here is a version of a muchstudied model [12–15], here of a 50:50 mixture of N point-particles in 3-dimensions (N = 4096 in our case), inter-acting via a binary potential. We refer to half the par-ticles as ‘small’ and half as ‘large’; they interact via thepotential U(rij):
U(rij) =
ε
[(σij
rij
)α
−(
σij
rij
)β
+ a0
], rij ≤ rc(i, j)
0 , rij > rc(i, j)(1)
Here, ε is the energy scale and σij = 1.0σ, 1.2σ or 1.4σfor small-small, small-large or large-large interactions, re-spectively. For the sake of numerical speed the potentialis cut-off smoothly at a distance, denoted as rc, whichis calculated by solving ∂U/∂rij |rij=rc = 0 which trans-
lates to rc = (α/β)1
α−β σij . The parameter a0 is cho-sen to guarantee the condition U(rc) = 0. Below weuse α = 8 and β = 6, resulting in rc =
√8/6 σij and
a0 = 0.10546875, see Fig. 1. Note that the potential ispurely repulsive, in a premeditated distinction from thesecond model discussed in Subsect. II B
The model dynamics were studied as a function of thetemperature keeping the pressure fixed at p = 10 (NPTensemble). The model was simulated by employing theVerlet integration scheme with the Berendsen thermostatand barostat [16]. The units of mass are the mass of theparticles m, energy is measured in units of ε and the fun-damental length scale is σ. In the figures below the timescale is in units of σ
√m/ε. All the simulations were run
for at least twice the τα relaxation time before startingto measure the decay of the correlation function. Theslowing down in the super-cooled regime is exemplified
0
0.5
1
1.5
2
2.5
r
U(r
)
σij rc(i, j)
FIG. 1: Color online: The pair-wise potential of the binarymodel
100
102
104
106
108
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fk(t
;T)
T = 1
T = 0.9
T = 0.8
T = 0.7
T = 0.6
T = 0.5
T = 0.4
T = 0.35
T = 0.3
T = 0.28
T = 0.26
1 2 3 4
100
102
104
1/T
τ α
FIG. 2: Color online: Time dependence of the correlationfunctions (2) for a range of temperatures (decreasing fromleft to right) as shown in the figure. The inset shows therelaxation time τα in a log-lin plot vs 1/T , compared to anArrhenius temperature dependence.
by measuring the self-part of the intermediate scatteringfunction [14] summed over the large particles only,
Fk(t;T ) ≡⟨
2N
N/2∑
i=1
exp {ik · [ri(t)− ri(0)]}⟩
. (2)
In Fig. 2 we show these correlation functions for k =5.1σ−1 and for a range of temperatures as indicated inthe figure. We see the usual rapid slowing down thatcan be measured by introducing the typical time scale τα
that is determined by the time where Fk(t = τα; T ) =Fk(0; T )/e ≡ 1/e. The relaxation times are shown in theinset of Fig. 2 as a function of 1/T in a log-lin plot tostress the non-Arrhenius dependence at lower T .
3
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.70
0.5
1
1.5
2
2.5
r
U(r
)
FIG. 3: The pair-wise potential for the hump model. In theinset we show a snapshot of the position of the point-particles(the circles represent the radius σ). Note that the two typicalscales rmin and r? appear as the typical distances betweenparticles. These scales will appear as two peaks in the 2-pointdensity static correlation function.
B. Two-dimensional ‘hump’ model in NVTensemble
The second model is constructed following [17] such asto have very different microscopic properties from the bi-nary model. We studied the present model in an NVTensemble to underline the fact that the method of up-scaling is not limited to one or the other ensemble. Wealso ran the present model in an NPT ensemble and madesure that the method works equally well. The interactionpotential is constructed as a piecewise function consistingof the repulsive part of a standard 12-6 Lennard–Jonespotential connected at r0 = 21/6σ to a polynomial inter-action P (x) =
∑i aix
i. The ai’s are tuned so that P (x)displays a peak at r = rhump and also such that there isa smooth continuity (up to second derivatives) with theLennard–Jones interaction at U(r0) = εh0 as well as withthe cut-off interaction range U(r?) = 0. The interactionpotential for the hump model reads:
U(rij) =
ε
[(σ
rij
)12
−(
σrij
)6
+ 14 + h0
], rij ≤ r0
εh0P(
rij−r0r?−r0
), r0 < rij ≤ r?
0 , rij > r?
(3)with the parameters as shown in table 1.
Note that the two typical distances that are definedby this potential, i.e. the distance at the minimum rmin
and the cutoff scale r?, appear explicitly in the amor-phous arrangement of the particles in the supercooledliquid, as shown in the inset in Fig. 3. The model hastwo crystalline ground states, one at high pressure with
h0 0.75
rhump 1.32
r? 1.65
a0 1.0
a1 0.0
a2 2.675405732987203
a3 46.593574934286437
a4 -212.021700143354
a5 308.7495934944721
a6 -188.1854149609638
a7 41.188540942572089
TABLE I: Parameters used in the potential of the hump model.
a hexagonal lattice and a lattice constant of the order ofrmin. At low pressure the ground state is a more openstructure in which the distance r? appears periodically.At intermediate pressures the system fails to crystallizeand forms a glass upon cooling. We have made sure thatthere exist no micro-cluster of either crystalline phase inour simulations.
The great difference between these two models will be-come even clearer after we define the up-scaled quasi-species below. On the face of it, the scenario for theglass transition appears similar as exemplified in Fig. 4in which the intermediate scattering function
Fk(t;T ) ≡⟨
1N
N∑
i=1
exp {ik · [ri(t)− ri(0)]}⟩
. (4)
is shown for k = 6.16σ−1. The hump model was sim-ulated using the same algorithms as the binary model,but without the Berendsen barostat that was unneededfor the NVT ensemble.
The relaxation time τα for the hump model is shownin Fig. 4 in which we stress again the difference be-tween the region in which there is Arrhenius behaviorand the region where the relaxation time grows fasterthan exp(E/T ).
III. CHOICE OF UP-SCALED VARIABLES ANDVALIDATION
Our aim is to provide a theory that captures, whenthe temperature goes down, the subtle changes in thestructural organization of the particles; this rearrange-ment is in fact directly responsible for the glass transi-tion. The first step in our approach is up-scaling, (orcoarse-graining) from particles to quasi-species that canbe characterized by their enthalpy and degeneracy. Up-scaling is an art, since it can be done in various ways andthere is no unique algorithm to select a-priori a ‘best’up-scaling. Therefore we need to validate the choice ofup-scaled variables using a criterion that was introduced
4
100
105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t
Fk(t
;T)
5 1010
0
105
1/T
τ α
T = 0.50
T = 0.25
T = 0.17
T = 0.13
T = 0.11
T = 0.10
T = 0.09
FIG. 4: Color online: Time dependence of the correlationfunction 4 in the hump model for a range of temperatures asshown in the figure. Inset: the temperature dependence ofthe relaxation time τα(T ).
.
in [8]. The up-scaling is typically different for differentmodels, and we describe it separately for the two modelsat hand.
A. Up-scaling of the binary model
The potential in the case of the binary model is purelyrepulsive and it has well defined range of interactionrc(i, j) for any given pair of particles. A natural up-scaling for this potential is provided by the particles andtheir nearest neighbors, where ‘neighbors’ are defined asall the particles j around a chosen central particle i thatare within the range of interaction rc(i, j). We refer tothe type of a central particle (small or large) in combi-nation with the amount of its nearest neighbors to definethe quasi-species. In the interesting range of tempera-tures we find 8 quasi-species with one ‘small’ central par-ticle and 3, 4 . . . 10 neighbors, and 9 quasi-species withone ‘large’ central particle with 6, 7 . . . 14 neighbors, allin all 17 quasi-species. Other combinations have negligi-ble concentration (< 0.5%) throughout the temperaturerange. We denote these quasi-species as Cs(n) and C`(n)with s and ` denoting the small or large central particle,while n denotes the number of neighbors. We measuredthe mole-fractions 〈Cs(n)〉(T ) and 〈C`(n)〉(T ) and theresults are shown in Fig. 5. We see that some concentra-tions decrease upon decreasing the temperature, othersincrease, and yet some first increase and further decrease.We submit to the reader that the subtle changes in con-figurational arrangement as seen in this plot encode thescenario of the glass transition in a way that we will at-
0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.05
0.1
0.15
0.2
T
〈Cs〉(
T)
Small Particles
n = 3
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
n = 10
0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.05
0.1
0.15
0.2
T
〈C`〉(
T)
Large Particles
n = 6
n = 7
n = 8
n = 9
n = 10
n = 11
n = 12
n = 13
n = 14
FIG. 5: Color online: Temperature dependence of the con-centrations of the quasi-species of the binary model. Symbolsare simulation data and the lines are a guide to the eye.
tempt to decode below.
B. Up-scaling of the hump model
The potential of the hump model has a distinct min-imum which suggests that quasi-species could consist ofeach particle and all its neighbors which are within theminimum. This type of up-scaling may have a limitedusefulness when the temperature is sufficiently high toallow easy escape over the maximum. The glass tran-sition occurs however in a temperature range where Tis lower than the maximum, and we expect the chosenup-scaling to be sensible throughout the interesting tem-perature range. In this range we find quasi-species with2, 3, 4, 5 and 6 neighbors within the minimum. In Fig. 6we show the temperature dependence of the quasi-speciesof this model which are denoted as Ci with i = 2, 3, . . . , 6being the number of neighbors within the minimum. Thereader should be sensitive to the distinction between thetwo models. The quasi-species in the binary model canchange upon every cage vibration; anytime the distancebetween two particles exceeds or goes below rc the def-inition of the quasi-speices changes. On the other handin the hump model the quasi-species are much more sta-ble, since an exchange of a particle calls for an escape orpenetration by going over the hump. We have chosen themodels to have these clear differences in order to test thegenerality of our approach.
5
0.1 0.2 0.3 0.4 0.5
0.1
0.2
0.3
0.4
0.5
0.6
T
〈Ci〉(
T)
i = 2
i = 3
i = 4
i = 5
i = 6
FIG. 6: Color online: Temperature dependence of the con-centrations of the quasi-species of the hump model. Symbolsare simulation data and the lines are a guide to the eye.
C. Validation of the up-scaling
Since there is no unique algorithm to choose the up-scaling, we need to have a criterion that validates or re-jects the choice of quasi-species. In ref. [8] such a crite-rion was introduced as explained next.
1. Validation of the up-scaling in the binary model
To decide whether the up-scaling provides a usefulstatistical mechanics for the binary model we now askwhether there exist free energies Fs(n; T ) and F`(n; T )such that
〈Cs(n)〉(T ) =e−Fs(n;T )/T
2∑10
n=3 e−Fs(n;T )/T,
〈C`(n)〉(T ) =e−F`(n;T )/T
2∑14
n=6 e−F`(n;T )/T. (5)
The free energies are found by inverting Eqs. (5) in termsof the measured concentrations. In doing so one can al-ways choose one of the concentration to have by defi-nition zero free energy. This is because the constraint∑
n Cs(n) +∑
n C`(n) = 1 makes the system of equa-tions over-determined. Then the free energies of all theother quasi-species are actually the difference from thischosen reference particle. We then plot these quantitiesas a function of the temperature, as demonstrated for thepresent case in Fig. 7. We say that our upscaling is vali-dated if Fs(n; T ) and F`(n;T ) can be well approximatedas linear in the temperature. Then we can interpret
Fs(n; T ) ≡ Hs(n)− T ln gs(n) ,
F`(n; T ) ≡ H`(n)− T ln g`(n) , (6)
0 0.2 0.4 0.6 0.8 16
8
10
12
14
T
Fs(T
)
Small Particles
n = 3
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
n = 10
0 0.2 0.4 0.6 0.8 1
13
14
15
16
17
18
19
T
F`(T
)
Large Particles
n = 6
n = 7
n = 8
n = 9
n = 10
n = 11
n = 12
n = 13
n = 14
FIG. 7: Color online: The approximate linear dependence ofthe free energies of the chosen quasi-species on the temper-ature. From the slope we read the degeneracy and from theintercept the enthalpies (up to normalization), cf. Eq. 6.Note that when the free energies are large we do not havedata: the concentrations become exponentially small and ina finite simulation box they disappear completely. Note thatpoints that are zero in Fig. 5 do not appear in this figure.The error bars in this figure are of the size of the symbols.
where now the degeneracies gs(n) and g`(n) (readfrom the slopes in Fig. 7) and enthalpies Hs(n) andH`(n) (read from the intercepts) are temperature-independent. This validates the choice of up-scaling.In other words, the approximate linearity of the invertedfree energies in the temperature means that we can writethe concentrations as
〈Cs(n)〉(T ) ≈ gs(n)e−Hs(n)/T
2∑10
n=3 gs(n)e−Hs(n)/T,
〈C`(n)〉(T ) ≈ g`(n)e−H`(n)/T
2∑14
n=6 g`(n)e−H`(n)/T. (7)
Then we can use these forms also as a prediction fortemperatures where the simulation time is too short toobserve the relaxation to equilibrium. The resulting de-generacies gs(n) and g`(n) can be easily modeled by aGaussian distribution. Denoting by nmp the most proba-ble number of nearest neighbors at T →∞ (for small andlarge particles respectively) we fit our data as follows:
gs(n) ∝ e−[(n−nsmp)2/2σ2
s ] , nsmp = 4.65, σ2
s = 1.55 ,
g`(n) ∝ e−[(n−n`mp)2/2σ2
` ] , n`mp = 7.50, σ2
` = 2.0 .(8)
The fit to the measured degeneracies is shown in Fig. 8,upper panel. The same figure shows in the middle panel
6
4 6 8 10
1
2
3
4
5
x 104 Small Particles
n
gs(n
)
datafit
6 8 10 12 14
1
2
3
x 104 Large Particles
n
g`(n
)
datafit
4 6 8 106
8
10
n
Hs(n
)
6 8 10 12 14
14
16
18
n
H`(n
)
0.4 0.6 0.8 10
0.05
0.1
0.15
T
〈Cs〉(
T)
0.4 0.6 0.8 10
0.05
0.1
0.15
T
〈C`〉(
T)
FIG. 8: Color online: Upper panel: The degeneracies gs(n)and g`(n) read from the slopes of Fig. 7 (in circles) and thedegeneracies according to the gaussian model Eq. (8). Mid-dle panel: the measured enthalpies. Lower panel: compari-son of the measured concentrations of quasi-species to thosecalculated from Eqs. (7) using the model degeneracies andmeasured enthalpies. Here symbols are data and lines aretheoretical predictions.
the enthalpies of the various quasi-species. One couldmodel the enthalpies as a linear function in n. These re-sults are easily interpreted; we have high enthalpies whenthere are large free volumes (few neighbors). The lowestenthalpies are found when there are many neighbors andthere is no much costly free volume. In other words, atthe present density and range of temperatures the pVterm dominates the energy in the enthalpy. Using thetheoretical degeneracies and the measured enthalpies wecompute the concentrations of all our quasi-species andcompare them with the measurement in the lowest panelof Fig. 8. The agreement that we have, especially con-sidering the number of quasi-species and the simplicityof the theory, is very satisfactory. Notice that the com-petition between degeneracy and enthalpy explains therather intricate temperature-dependence of the concen-trations of the quasi-species, sometimes declining whenthe temperature drops, sometime rising, and sometimehaving non-monotonic behavior.
2. Validation of the up-scaling in the hump model
Validating the up-scaling in the hump model followsthe same ideas as in the binary model. We choose thequasi-species n = 2 as the reference concentration withF2 ≡ 0 and then invert the data for 〈Ci〉(T ) to find the
0.1 0.2 0.3 0.4 0.5−2
−1.5
−1
−0.5
0
0.5
T
Fi(T
)
i = 2
i = 3
i = 4
i = 5
i = 6
FIG. 9: Color online: The approximate linear dependence ofthe free energies of the chosen quasi-species on the tempera-ture in the hump model. From the slope we read the degener-acy and from the intercept the energies(up to normalization),cf. Eq. 10.
H2 0. g2 1.
H3 -0.545053 g3 2.830972
H4 -0.564547 g4 7.289318
H5 -0.411079 g5 6.862693
H6 -0.217412 g6 1.552314
TABLE II: Temperature independent enthalpies and degen-eracies of the sub-species of the hump model. Note that theenthalpies in this case are not monotonic; this is a result of thecompetition between the energy and the free volume which isabsent in the binary model where the pV term dominates.
free energies via
〈Ci〉(T ) =e−Fi(T )/T
∑62 e−Fi(T )/T
. (9)
The observed linearity of Fi(T ) in the temperature meansthat we can write
Fi(T ) = Hi − T ln gi , (10)
where Hi is the enthalpy and gi the degeneracy of theith quasi-specie. The test of linearity from which we canread the energies and degeneracies is shown in Fig. 9.From the intercepts of the lines in Fig. 9 we read theenthalpies, and from the slopes we read the degeneracies.The resulting numerical values are shown in Table II.
Since the hump model is studied in the NVT ensemblethe reader may ask why we get enthalpies rather than en-ergies. The reason is that the upscaling using the indexi takes into account only the particles that reside inside
7
the hump. The energy of a quasi-species with i parti-cles within the hump is proportional to i, and thus forlow temperatures the energetically preferred subspecieswould be the one with no neighbors within the hump.Clearly this arrangement does not satisfy the constantvolume constraint which needs to be taken into account.Indeed, our enthalpies are non-monotonic in this modelbecause of the competition between energy that is in-creasing with i and free volume which is decreasing withi. This effect is absent in the binary model where the pVterm dominates.
D. Summary of the statistical mechanics
We can summarize the findings up to now by sayingthat at least in terms of capturing the scenario of thechanges in the spatial organization of the particles ourstatistical mechanics appears very suitable. Our quasi-species appear to have very well defined enthalpies anddegeneracies and we can therefore capture the temper-ature dependence of the concentrations of the quasi-particles with high accuracy. Our thesis is that this sce-nario is actually the scenario of the glass transition, andhidden in it is also the reason for the slowing down, asdemonstrated in the next section. Intuitively the pictureshould be clear already now. As the temperature re-duces the quasi-species with high free energy tend to dis-appear, leaving us with quasi-species of low free energy.Then the dynamics become more constrained, since it ismore costly to change objects of low free energy (creat-ing on the way objects of higher free energy) than at hightemperature when there are many objects with high freeenergy that can readily change to objects of lower freeenergy. What we need now is to make this observationmore quantitative.
IV. RELATION TO DYNAMICS
In this section we connect the structural theory to thedynamical slowing-down. To this aim we note that inboth models there are a number of quasi-species whoseconcentration goes down exponentially (or maybe faster)when the temperature decreases, and that the relaxationtime shoots up at the same temperature range. We re-fer to these quasi-species as the ‘liquid’ ones; These arethe quasi-species that are prevalent when the tempera-ture is high but are getting rare when the temperaturegoes down. We discuss now the connection between thisphenomenon and the slowing down.
A. The binary model
In this example the ‘liquid’ concentrations are thoseconsisting of small particles with 3-7 neighbors, and largeparticles with 6-11 neighbors, see Fig. 10 [18]. We sum
0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T
〈Cli
q〉(
T)
ns = 3
ns = 4
ns = 5
ns = 6
ns = 7
n` = 6
n` = 7
n` = 8
n` = 9
n` = 10
n` = 11sum
FIG. 10: Color online: The temperature dependence ofCliq(T ) is shown as the upper continuous line. The contribu-tions of the various ‘liquid’ sub-species are shown with sym-bols which are identified in the inset.
up these concentrations and denote the sum as 〈Cliq〉(T ).The dependence of 〈Cliq〉(T ) on the temperature is shownin Fig. 10. This concentration is used to define a typicalscale,
ξ(T ) ≡ [ρCliq(T )]−1/3 ; (11)
where ρ is the number density. This length scale hasthe physical interpretation of the average distance be-tween the ‘liquid’ quasi-species. It was argued before[2, 5, 6] that this length scale can be also interpreted asthe linear size of relaxation events which include O(ξ(T ))quasi-species. We can therefore estimate the growing freeenergy per relaxation event as ∆G = µξ(T ) where µ isthe typical chemical potential per involved quasi-species.This estimate, in turn, determines the relaxation time as
τα(T ) = τ0eµξ(T )/T , (12)
where τ0 is of the order of the cage microscopic time. Thequality of this prediction can be gleaned from Fig. 11,where we can see that the fit is excellent, with µ ≈ 0.3.The intercept in Fig. 11 is of the order of unity; this isvery reassuring, since this is what we expect when T →∞.
B. The hump model
In this model the quasi-species whose concentrationgoes to zero rapidly in the relevant temperature rangeare those with i = 2, 5 and 6. In Fig. 12 we see their tem-perature dependence and also their sum which is again
8
0 5 10 1510
−1
100
101
102
103
104
ξ(T )/T
τ α(T
)
FIG. 11: Color online: The relaxation time τα(T ) in termsof the typical scale ξ(T ) in the binary model. We show theexcellent fit to Eq. (12) with µ = 0.37. Note that the interceptat T →∞ is of the order of unity as it must be.
0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
T
〈Cliq〉(
T)
i = 2
i = 5
i = 6sum
FIG. 12: Color online: The quasi-species contributing toCliq(T ) in the hump model, and the temperature dependenceof the their sum, which is Cliq(T ).
denoted as Cliq(T ). Since in this model we are in twodimensions, the typical scale ξ(T ) is related to Cliq(T )according to
ξ(T ) ≡ [ρCliq(T )]−1/2 ; (13)
With this obvious change we expect Eq. (12) to be validalso here, and indeed in Fig. 13 we see that this expec-tation is wonderfully fulfilled.
0 10 20 30 40
100
102
104
106
ξ(T )/T
τ α(T
)
FIG. 13: Color online: The relaxation time τα(T ) in termsof the typical scale ξ(T ) in the hump model. We show theexcellent fit to Eq. (13) with µ = 0.34. Note that the interceptat T →∞ is of the order of unity as it must be.
C. Pertinent Remarks
A few points should be stressed. As we expect (cf. Ref[2]), in systems with point particles and soft potential,there is no reason to fit the relaxation time to a Vogel-Fulcher form [1] which predicts a singularity at finite tem-perature. In our approach we predict that ξ → ∞ onlywhen T → 0, and there is nothing singular on the way,only slower and slower relaxation. At some point thesimulation time will be too short for the system to re-lax to equilibrium, but we can use Eq. (12) to predictwhat should be the simulation time to allow the systemto reach equilibrium. It is important to bear in mindthat finite systems of point particles with soft poten-tial are different from finite granular media or systemsof hard spheres which can truly jam and lose ergodic-ity. Point particles with soft potential remain ergodic [2],and therefore should be amenable in their super-cooledregime to statistical mechanics. We argue that in or-der to construct simple, workable statistical mechanicsone needs to up-scale the system and find a collection ofquasi-species with well defined enthalpies and degenera-cies. In this paper we showed that the simple criterionto validate the choice of up-scaling which was suggestedin [8] can be applied to very different models once we se-lect the proper up-scaling. Once the structural theory isunder control, a natural length scale appears and can beused to determine the relaxation time, also for tempera-tures that cannot be simulated due to the fast growth ofthe relaxation time. The fact that the present approachworks equally well in two and three dimensions, in NPTand NVT ensembles and in very different models providesgood reason to believe that it has a substantial degree ofgenerality.
9
Another important remark is that in this paper weaddressed the relaxation time τα and not the full formof the correlation function, including the characteristicintermediate time plateau in the scattering function. Theunderstanding of this full form is possible using the sametheoretical tools, and this was demonstrated in Refs. [6,7].
D. The thermodynamics of glass forming systems
As a last issue we raise the following question: is thestatistical mechanical theory as defined on the up-scaledquasi-species determining also the thermodynamics ofthe original system on the particle level. The answerto this question is not obvious a-priori, but turns out tobe in the affirmative. As done throughout this paper, wedemonstrate this point separately for the two models athand.
1. Thermodynamics of the binary model
The binary model is studied in the NPT ensemble andtherefore the first question to ask is whether the averageenthalpy of the quasi-species represents the correct en-thalpy of the system. In other words, we have extractedthe enthalpies Hs(n, T ) and H`(n, T ). Is it then true thatthe enthalpy of the system, computed as the average po-tential energy summed over all the pairs of particles andsummed with pV should equal the enthalpy of the quasi-species:
10∑n=3
Cs(n)Hs(n, T ) +14∑
n=6
C`(n)H`(n, T )
=1
2N
N∑
i,j=1
〈U(rij)〉+ pV (T ) .(14)
In answering this question we remember that our en-thalpies are defined up to an arbitrary constant sincewe chose one of the quasi-species as a reference with zerofree energy. We can therefore add or subtract an arbi-trary constant from the LHS or the RHS of Eq. 14. InFig. 14 we show the ratio of the LHS of Eq. (14) to theRHS, with the conclusion that the enthalpy of the sys-tem is excellently reproduced by the up-scaled variables.Needless to say, we could therefore compute the specificheat at constant pressure, Cp, from either set of data,and from that we could get the entropy of the system,reassuring us that the up-scaling method, once validatedas above, provides also the correct thermodynamics forthe glass forming system. In the next subsection we showactual computations of such thermodynamic quantities.
0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
T
(∆H
th+
C)/
∆H
ex
p
FIG. 14: Color online: The ratio of the LHS of Eq. (14),denoted as ∆Hth + C (here, C = 2.8) over the RHS of Eq.(14), denoted ∆Hexp. We conclude that the enthalpy of thesystem is excellently reproduced by the up-scaled variables.
2. Thermodynamics of the hump model
The hump model was studied in the NVT ensemble,and therefore it is interesting to test whether one obtainsthe correct system’s energy. The potential energies of thequasi-species which were characterized by the number ofneighbors that are bound within the minimum are toan excellent approximation linear in that number, Ei =0.83i as measured with respect to the same zero pointas that of the potential (3). We therefore need to checkwhether
0.832
6∑
i=2
Ci(T )i + T =1
2N
N∑
i,j=1
〈U(rij)〉 . (15)
We have added T for the excitation around the groundstate of the quasi-species (two degrees of freedom in 2D).This test is shown in Fig. 15 where the predictions ofthe theory were extrapolated all the way to zero temper-ature. Obviously the data loses its predictability some-where around T = 0.05 where the specific heat becomesnegative. This artifact is seen even better in the plot ofthe specific heat which is offered in Fig. 16. Neverthelessthe plot of Cv vs. T indicates the expected typical spe-cific heat peak around T = 0.1 which could not be seenin the simulations due to the inherent limitations of com-puter time. Once the specific heat takes a dive, at somepoint we can no longer believe the extrapolation, andmore accurate data are necessary to be able to predictthe thermodynamic properties near zero temperatures.
V. SUMMARY AND THE ROAD AHEAD
In summary, we have shown how the scenario of theglass transition can be neatly characterized by the tem-perature dependence of the concentrations of the various
10
0.1 0.2 0.3 0.4 0.51.6
1.7
1.8
1.9
2
2.1
2.2
2.3
T
Ener
gy
per
part
icle
FIG. 15: Color online: Comparison of the LHS of Eq. (15) inred continuous line, to the RHS of the same equation, in sym-bols. We conclude that the energy of the system is excellentlyreproduced by the up-scaled variables down to temperaturesof about T ≈ 0.05 For lower temperatures the extrapolationobviously fails.
0 0.1 0.2 0.3 0.4 0.5−4
−3
−2
−1
0
1
2
T
Cv(T
)
FIG. 16: Color online: Symbols: the derivative with respectto T of the RHS in Eq. (15). Continuous line: derivative withrespect to T of the LHS in Eq. (15). The theory predicts theexpected specific heat peak and fails when the specific heatbecomes negative.
quasi-species that are obtained after up-scaling. Thosequasi-species that are readily changed because they arerich in free energy are of course those that deplete first.We are left with quasi-species that are low in free energy,and these would naturally be hard to change since theirchange necessarily increases the free energy. The pre-sumed ergodicity allows us to introduce statistical me-chanics which can beautifully encode the scenario, us-ing a small number of quasi-species together with theirenthalpies and degeneracies. The statistical mechanicsapplies also at temperatures that are beyond the rangeavailable to molecular simulations, and can be used topredict what is happening there. We have shown thatwhen the upscaling and the statistical mechanics workwell, the thermodynamics of the system can be under-stood very well using the up-scaled picture. We are nottoo concerned with the fact that at ultra low tempera-
tures the present accuracy is not sufficient; after all weare using a theory of almost non-interacting quasi-speciesto describe a system of N strongly interacting particles.It is quite amazing that we can do what is presented,and one cannot ask for too much in the highly complexregime T → 0.
There are still issues to understand. One of our mainconcerns is how to achieve a first-principles prediction ofthe parameter µ in Eq. (12). We have, so far, been un-able to bridge this gap, leaving the connection betweenthe dynamics and the upscaling theory still dependenton a phenomenological fit. A satisfactory prediction of µwould, in our opinion, close the loop and put to rest theriddle of the slowing down in the glass-forming systems.Also, the upscaling approach had been so far limited tocomputer model of glass formation, we propose that itwould be enlightening to try and apply it to real exper-imental systems. We hope that these and other issueswould be clarified in future research.
Part 2: Mechanical Properties of Amor-phous Solids
VI. INTRODUCTION
A. Motivation
Much of the theoretical analysis of deformation andplastic flow in amorphous solids and other non-crystallinematerials (structural glasses, metallic glasses, pastes,foams, gels etc.) is still influenced to a large degree by theunderstanding of crystalline materials. In the latter thedeformation and flow are governed by topological defectsknown as dislocations, whose dynamics are the basis ofthe theory of crystalline plasticity [19]. Indeed, followingthe pioneering work of Maeda and Takeuchi [20, 21] onmetallic glasses and those of Argon and coworkers [22, 23]on bubble rafts, workers in the field of amorphous elasto-plasticity [24, 25] proposed theoretical schemes based onthe notion that also in amorphous solids plastic eventsare carried by some sort of micro-structural defects, re-ferred to as “Shear Transformation Zones” (STZ). Whilethe precise nature of these STZ or how to measure themexperimentally or even simulationally has never beenfully clarified, their existence as the source of ‘quanta’of plastic relaxation carried by a small number of atomswas taken as a basis for developing mean-field models ofelasto-plasticity. Although these models are not uniqueand are sometimes even in disagreement with each other,careful attempts to apply them to a variety of phenom-ena, from shear banding [26] to fracture [27] and fromnecking instabilities [28] to grooving via Grinfeld instabil-ities [29] all showed considerable promise and a fair agree-ment with experiments or simulations. The fundamentalquestion of whether plasticity in amorphous solids is in-deed due to local events in which a microscopic numberof atoms (independent of the system size) are involved re-
11
mained unanswered. A second fundamental assumptionof the STZ theory is that there exists an order parameter,in the form of effective temperature, which dominates thedynamics. Also this assumption were never tested in acritical way.
A serious doubt on the first fundamental assertion wasrecently cast by Maloney and Lemaitre [30] with theirpresentation of a series of two-dimensional atomistic com-puter simulations of amorphous solids subject to simpleshear in the athermal, quasistatic limit. These authorsargued that the plastic events themselves were lines ofslip which span the length of the simulations cell. If itwere shown that these findings extend to physical exper-iments, this would put in question the very fundamen-tal assumptions underlying mean-field theories that wereput forward, irrespective of their relative success to pa-rameterize a number of interesting elasto-plastic phenom-ena. In a recent meeting at the Lorentz center in Leidenthe conclusions of Maloney and Lemaitre were criticized[31] on the basis of the algorithm used, in which aftereach step of strain the energy was minimized, irrespec-tive of the computer time needed for this minimization orwhether the trajectory followed is physical. This opensup the possibility that the spanning events seen in [30]would not be seen in a system with ‘natural’ dynamics inwhich the limit of zero strain rate is not to be confusedwith arbitrary waiting times between strain steps.
We begin our analysis by focussing on the fundamentalissue of the spatial extent of plastic deformations at zerotemperature and under quasi-static conditions, using avariety of algorithms and a number of simulational testsfor the locality (or rather non-locality) of these deforma-tions in two-dimensional amorphous solids. Among otherthings we provide evidence that the results found in [30]are generic. But we go further in analyzing the system-size dependence of various measures of elasto-plastic de-formation. Our conclusions are even more pessimisticthan those given by [30]; we conclude that it is quitedifficult to try to isolate the plastic contribution to anelasto-plastic event. The ‘plastic energy’ part of suchan event is ill-defined; every plastic deformation is ac-companied by an elastic deformation which is triviallylong ranged because elasticity is long ranged. The elasticcontribution to every energy change in an elasto-plasticevent can be much larger than the purely plastic energychange even if the latter were well defined. Measuring thelatter is almost like weighing the captain by weighing thecaptain with the ship and and ship without the captainand taking the difference. We therefore propose belowa number of new measures that are able to distinguishthe irreversible plastic contribution from the elastic affineand non-affine contributions. Our conclusion is that thegeneric plastic deformations are not localized, and thisis not because of the elastic response that accompaniesit. We will show that direct measures of the ‘size’ of the
purely plastic events scale with the size of the systemsimilarly to the scaling of the elasto-plastic events. Wereiterate that these results pertain to the athermal condi-tions and quasi-static strain. In Sect. XI we discuss thetremendous effects of finite temperature and finite strainrates. In Sect. XII we assess the second fundamentalassumption of the STZ theory, and find that also thisassumption is not supported by careful data analysis.
The structure of this discussion is as follows. In thesecond part of this introduction we describe the modelused in the majority of the paper below. In Sect. VIIwe describe an experiment of pulling one particle in aglassy system of varying sizes and measuring the maxi-mal force on this particle before the plastic deformation.We show that the average maximal force depends on thesize of the system as a power law, indicating that thesystem is never too large such that the walls do not mat-ter. The power law has a non-trivial exponent that wediscuss below. In Sect. VIII we consider systems of var-ious sizes under strain, and study the scaling propertiesof the distributions of stress and energy drops when thesystem reaches a steady-state plastic flow. We demon-strate that these distributions are characterized by sub-extensive scaling with non-trivial exponents, again indi-cating that the plastic events are not localized. We in-troduce a measure of the size of plastic events that filtersout the effect of non-affine elasticity, and show that thismeasure scales with essentially the same exponent as thetotality of the elasto-plastic energy drop. Finally, to re-move the last doubts, in Sect. IX we introduce a newmodel glass for which we can precisely measure the sizeand extent of a purely plastic event, filtering out com-pletely any possible elastic contribution. The conclusionis as above, that the size of plastic events scales in asub-extensive fashion with a non-trivial exponent. In thediscussion section X we provide a summary of the resultand discuss the apparent non-universality of the scalingexponents. It is stated that understanding the numericalvalues of the found exponents should be an importantstep in improving our understanding of the physics ofamorphous solids.
B. Model Description
Almost all the simulations below are performed in aglassy system consisting of poly-dispersed soft disks. Wework with point particles of equal mass m in two di-mensions with pair-wise interaction potentials. Eachparticle i is assigned a interaction parameter σi from anormal distribution with mean 1. The variance is gov-erned by the poly-dispersity parameter ∆ = 15% where∆2 = 〈(σi−〈σ〉)2〉
〈σ〉2 . With the definition σij = 12 (σi + σj),
the potential assumes the form (cf. Fig. 17)
12
U(rij) =
ε
[(σij
rij
)k
−k(k+2)8
(B0k
) k+4k+2
(rij
σij
)4
+ B0(k+4)4
(rij
σij
)2
− (k+2)(k+4)8
(B0n
) kk+2
], rij ≤ σij
(k
B0
) 1k+2
0 , rij > σij
(k
B0
) 1k+2
, (16)
1 1.2 1.4 1.6 1.8 2 2.2
−1
0
1
2
3
4
5
6
r/σ
U(r
)/ε
our model
LJ
ε(σ/r)12
FIG. 17: The potential Eq. (16) used in the present simula-tions in comparison to the more standard σ/r12 potential andto the Lennard-Jones potential.
The units employed here are as in the binary model abovebut with σ replaced by 〈σ〉. In the present simulationswe chose , B0 = 0.2. The choice of a quartic rather thana quadratic correction term is motivated by numericalspeed considerations, avoiding the calculation of squareroots. In all the simulations discussed below the numberdensity N/V = 1.176 and the boundary conditions areperiodic.
VII. SYSTEM-SIZE DEPENDENCE OF THEFORCE ON A SINGLE PARTICLE
To initiate the discussion of locality vs. non-localityissues we begin with a simulation of the lovely experi-ment presented in [32]. In this experiment one pulls asingle disk at a constant (low) velocity through a disor-dered array of disks whose diameters have two possiblevalues. We start by quenching a system of N = 16384particles from T = 1.0 to T = 0.05 at a cooling rateof 10−6 ε/(kBτ0). At this point we remove any residualheat by conjugate gradient minimization [33]. After thispreparation we choose one particle out of the N avail-able ones (referred to below as the ‘center’ particle), anddraw a circle of radius R around it. Next we freeze allthe particles outside this circle, leaving the particles in-side the circle to interact normally according to Newton’sequations. An example of the results of such a procedure
FIG. 18: Color online. The configuration of the pulling ex-periment. The center particle (in dark green) is pulled in arandom direction and the force exerted on it is measured, inaddition to the total energy of the system. The outer parti-cles (in red) are stationary and are not allowed to move. Weare interested in influence of the radius R on the measuredquantities.
is shown in Fig. 18. The procedure is then repeatedfor each and every one of the N particles for the sake ofstatistics.
The experiment performed consists of pulling the cen-ter particle at a velocity v0 = 0.005 〈σ〉/τ0. We reach thisvelocity with a smooth initiation as seen in Fig. 19 in or-der to minimize elastic shock waves. Obviously, whenthe particle begins to move it increases the elastic energyin the whole system. This increase continues until thefirst plastic event in which an irreversible re-organizationof the particle positions takes place. This event is irre-versible in the sense that until it happens one can reversethe motion and return to the initial condition, but afterthe event heat is released in the form of kinetic energyand reversibility is lost. An example of a typical run isavailable in a movie that can be found in [34], and theresults of this run are displayed in Fig. 19. One can seein the movie that the plastic event appears local to the
13
0 50 100 150 2000
10
20
30
40
time
forc
e on
par
ticle
0 0.2 0.4 0.6 0.80
10
20
30
40
displacement
0 50 100 150 2003795
3800
3805
3810
3815
time
pote
ntia
l ene
rgy
0 0.2 0.4 0.6 0.83795
3800
3805
3810
3815
displacement
FIG. 19: Color online. The force exerted on the center particleand the potential energy of the system as a function of time(left panels) and as a function of displacement (right panels)for a typical run with R = 18.0. Note the linearity of theforce vs. displacement before the plastic failure, this is theelastic branch which is linear up to the maximum. Sometimenon-affine elasticity destroys the linearity seen here. At zerotime the velocity increases smoothly from zero (up to secondderivative) in order to minimize shock waves.
eye. The main question of this paper is whether this isjust an eyeball impression or is it quite true. We studythis issue by changing the radius R and examining thedistribution of the maximal force exerted on the particlebefore the plastic failure. At each value of R we repeatthe experiment N times, once for each particle in thesystem being the center particle. In each trial the centerparticle is pulled in a random direction and we measurethe force exerted on it in the direction of the motion.
In Fig. 19 one can see the force as a function oftime and as a function of displacement for a typical run.Naively one could expect that if the plastic failure werea localized phenomenon, then the average (over N) max-imal force in the direction of pulling before that failureshould not depend crucially on the circle radius R, or if itdoes depend on R that dependence should fall off expo-nentially to an R-independent value as R increases. Totest this expectation we compute the average maximalforce for different values of R. In fact, when we increaseR we find that the maximal force falls off very slowly as afunction of R, as a power law, cf. Fig. 20. The distribu-tions of maximal force, p(fmax) moves systematically tosmaller values of fmax when R increases. For the presentsystem we find a power law decay in the average value ofthe maximal force, with the excellent fit exhibited in thelower panel of Fig. 20. The power law reads
〈fmax〉 = f∞ + BR−γ , (17)
0 10 20 30 40 50 60 70 800
0.02
0.04
0.06
0.08
fmax
p(f
max)
r = 06
r = 07
r = 09
r = 12
r = 16
r = 21
r = 27
r = 34
5 10 15 20 25 30 35
32
33
34
35
36
37
r
〈fm
ax〉(
r)
101
100
r
〈fm
ax〉(
r)−
f∞
FIG. 20: Upper panel: the distributions of maximal forcebefore plastic event in the pulling experiment, for differentradii of system sizes. The radius increases from right to left.Lower panel: the scaling of the mean maximal force as afunction of the system size, demonstrating the scaling lawEq. (17) in linear coordinates and in the inset as a log-logplot.
with B = 79.84, γ = 1.37 and f∞ = 30.67 being theasymptotic average maximal force for R → ∞. Thispower law is the first one found in this review with anon-trivial exponent, others will follow below. We havechecked that the exponent in the power law is not uni-versal, being dependent on the potential and other char-acteristics of the system. At this point we do not have asolid theory to predict the value of either this or the otherexponents discussed below, but we conjecture that thepresent exponent is determined by the fractal dimensionD of the force-chains created by the loading. Obviouslyit is highly desirable to develop such scaling relations inthe near future.
The reason of the slow decay in the average maximalforce is that in disordered systems one can always findregions that either respond via non-affine elasticity oryield plastically for smaller forces when the system in-creases in size and the availability of softer regions be-comes apparent. Nevertheless the convergence to a finiteaverage maximal force for R → ∞ shows that there ex-ists a material parameter, analogous to the yield stress,which determines the density of weak pathways per unitvolume when the system is sufficiently large. We cantherefore conclude that the plastic events incurred dur-ing the pulling of the center particle ‘know’ about thesize of the system, and the disorder does not screen theboundary from the local loading. One could argue thatthis is expected since the loading creates force chains that
14
must end at the boundaries, and that the long range ef-fects found here are nothing but a consequence of the factthat elasticity is long ranged. Indeed, it is very difficultto disentangle purely plastic from elastic contributions inany elasto-plastic material. Thus to drive our point fur-ther we turn next to the stress and energy drops duringplastic irreversible deformations under a different kind ofloading.
VIII. SYSTEM-SIZE DEPENDENCE OFELASTO-PLASTIC EVENTS
To explore the locality vs. non-locality issues of theplastic deformation we turn now to a direct explorationof the plastic failure when our system is subjected to asimple shear. In such simulations one expects to see anelastic branch where the stress increases linearly with thestrain until the yield-stress is achieved and plastic eventsbegin. There are a number of ways that such simulationscan be done. One way is by solving the so-called SLLODequations [16], another way is to introduce and move twowalls with no-slip boundary conditions [35], and the thirdis to impose small strain increments as described belowand then to minimize the energy of the resulting con-figuration under the constraints imposed by the strainincrement [30]. We opt for the third option in order tobe able to accurately measure the distribution of stressand energy drops in the plastic events which are well de-fined only in this method. In doing so we will be ablealso to validate the results of [30] and even to strengthenthem.
In detail, we prepare the system with N = 625, 1024,2500, 4096, 10000 and 20164, cooled at a rate of 10−3
ε/(kBτ0). Beginning from a quenched unstrained config-uration we impose a simple shear strain increment whichwill be denoted δε. This is achieved by applying the fol-lowing transformation on the particles coordinates:
rx → rx + ryδε ,
ry → ry , (18)
in addition to imposing Lees-Edwards boundary condi-tions. Typical stress-strain and potential energy-straincurves for a system with N = 4096 are shown in Fig. 21.One sees the main elastic branch after which the evolu-tion consists of small elastic branches ending with plasticdrops. We employ a basic strain increment of 10−4 forsystems smaller than N = 10000 and 5 × 10−5 for thelarger systems. To increase our precision in determiningthe stress and energy drops and to guarantee that wedo not overshoot and miss the next minimum we stopthe simulation after a drop is detected, backtrack to theconfiguration prior to the drop and half the strain in-crements. The way the detection of the plastic drop isdone is explained in detail Ref. [43] This procedure isrepeated until the strain increment is smaller than 10−6
for systems smaller than N = 10000 and 5×10−7 for thelarger systems. An example of a trajectory approaching
0 0.05 0.10
0.5
1
strain
stre
ss
0 0.05 0.1
1.28
1.29
x 104
strain
pote
ntia
l ene
rgy
0.0986 0.0987
0.8965
0.897
0.8975
0.898
0.8985
0.899
strain
stre
ss
FIG. 21: Typical stress-strain and potential energy-straincurves for a system with 4096 particles. The blow-up at theright panel demonstrates the procedure of reducing the incre-ment steps to increase numerical precision in determining thestress and energy drops.
the plastic drop is shown in the blown-up right panel ofFig. 21.
A. System size dependence of stress and energydrops
The statistics of the stress and energy drops is collectedfrom between 4500 and 9000 plastic drops for each sys-tem size, where all the considered drops are after steadystate plastic flow had been reached, with measurementscollected after about 40% strains.
The raw distributions of the energy and stress dropsare displayed in the upper panels of Fig. 22. The lowerpanel of the same figure display the system size depen-dence of the average energy drop and average stress droprespectively, in a log-log plot. The conclusion is thatthe mean drop are described to a very high precision bypower laws of the form
〈∆U〉 ∼ Nα, 〈∆σ〉 ∼ Nβ , (19)
with
α ≈ 0.37, β ≈ −0.63 . (20)
While we did not expect these scaling exponents, it isvery easy to understand that there exists a scaling rela-tion between them,
α− β = 1 . (21)
To see this, we note that in the athermal limit at very lowstrain rates the steady state plastic flow occurs around afixed value of the stress, which is the yield stress σY . Onthe average, for every elastic increase in the energy whichoccurs for a strain increment ∆ε we have a corresponding
15
0 50 100 150 20010
−4
10−3
10−2
10−1
∆U
P(∆
U)
62510242500409610K20K
0 0.5 1 1.510
−4
10−3
10−2
10−1
∆σ
P(∆
σ)
62510242500409610K20K
103
104
101
N
〈∆U〉
103
104
10−1
N
〈∆σ〉
FIG. 22: Color online. The raw distributions of energy drops(upper left) and stress drops (upper right). The lower pan-els exhibit the system-size dependence of the mean energydrop (lower left) and mean stress drop (lower right). Thissystem size dependence appears very accurately to conformwith power laws 〈∆U〉 ∼ Nα and 〈∆σ〉 ∼ Nβ with α ≈ 0.37and β ≈ −0.63. These scaling laws are used to rescale thedistributions as shown in Fig. 23
plastic drop ∆σ, and we can estimate the mean energydrop according to
σY × 〈∆ε〉 × V = σY × 〈∆σ〉µ
× V = 〈∆U〉 . (22)
Since in our systems of fixed density V ∼ N , the scalingrelation (21) follows immediately.
We can use the scaling laws Eq. (19) to re-scale the rawdistribution functions of Fig. 22. The resulting distribu-tions are presented in Fig. 23. We note that the datacollapse is superb for probabilities larger than about 1%.For lower probabilities the quality of the collapse deteri-orates; due to the paucity of data there we cannot deter-mine whether the deteriorating collapse is due to multi-scaling of higher moments or due to statistical errors. Ifwe attempt to rescale using a somewhat higher exponentswe can get the tails to collapse but then the high prob-abilities fail to collapse. The scaling Eq. (19) and thequality of the data collapse demonstrate that the plas-tic drops are not localized, but in fact are sub-extensive.This finding is in agreement with [30, 36], although thenumerical value of our scaling exponents α and β differsfrom theirs (both [30] and [36] report α = −β = 0.5).We return to the issue of non-universality of the scalingexponents in Sect. IX.
As mentioned in the introduction, the work of [30] wascriticized on the basis of their algorithm; it was proposed
0 2 4 610
−4
10−3
10−2
10−1
P(
∆U
Nα
)
∆U
N α
62510242500409610K20K
0 50 100 15010
−4
10−3
10−2
10−1
P(
∆σ
Nβ
)
∆σ
Nβ
62510242500409610K20K
FIG. 23: Color online. The distributions of energy drops(left panel) and stress drops (right panel) shown in the upperpanels of Fig. 21 after rescaling by energy drops by Nα andthe stress drops by Nβ . Note that the data collapse is quiteperfect for probabilities larger than about 1%, and then beginto meander systematically; we cannot determine at this pointwhether this meandering is due to paucity of data or due tomultiscaling of the higher moments.
that the energy minimization procedure follows an un-physical trajectory in configuration space, and increasesartificially the amount of energy drop in a plastic event.Also, since the energy minimization does not correspondto real time units, the system has effectively infinite timeto reach the minimum energy, and this this is not a propersmall strain rate limit. We demonstrate now that thiscriticism is incorrect, and in fact molecular dynamics canoften lead to larger energy drops. Energy minimizationoften stops in the next available minimum, whereas truedynamics can often trigger subsequent energy drops andends up increasing the size of the average region involvedin the plastic event. This conclusion is demonstratedwith trajectories of the two methods in Fig. 24, and amovie of a multiple avalanche that is seen in true molec-ular dynamics is available in [34]. We conclude that theenergy minimization procedure produces a lower boundto the energy drops rather than an exaggerated result.One should understand that the true dynamics releaseskinetic energy after the first drop which is negligible onthe system scale, being the difference between a singlesaddle and a neighboring minimum. But before this neg-ligible energy is spread over the whole system it heats upconsiderably the local neighborhood and can easily trig-ger further energy drops which in turn heat up the localregion even further. Accordingly the ‘energy drop’ is welldefined only within the energy minimization procedure,where it is not in the eyes of the beholder.
One could argue that the reason that the energy dropsare sub-extensive is only because every change in anelasto-plastic medium involves also an elastic relaxation.This is of course true; since elasticity is long-ranged, onecould expect some kind of sub-extensive scaling. In fact,this underlines our belief that it is futile to talk about
16
7944 7945 7946
0.98
0.99
1
1.01
1.02
1.03
potential energy
stre
ss
minimizationreal−time dynamics
7900 7920 7940
0.4
0.5
0.6
0.7
0.8
0.9
potential energy
stre
ss
minimizationreal−time dynamics
FIG. 24: Color online. Left panel: an example for which theinitial and final energy and stress values are the same for theenergy minimization algorithm and the real-time dynamics.In dotted blue line is the latter trajectory and in continuousgreen line is the former. For the dynamics the green diamondrepresents the quenched configuration after the system hasreached a steady state without flow (with only thermal fluc-tuations around a minimum of the energy landscape). Rightpanel: an example for which the true real-time dynamics re-sults in larger energy and stress drops compared to the energyminimization procedure. Both panels are for systems withN = 2500 in the steady state plastic flow.
pure plastic energy changes, since the contribution of thepurely plastic part of an energy drop can be very smallcompared to the associated elastic contribution. Fur-thermore, any measure that employs the coordinates ofall the particles in the system, like various participationnumbers, will always collect some elastic relaxation con-tributions which can be quite large. Notwithstanding, wepropose here that the sub-extensive scaling of the energydrops is not only because of the elastic contributions,and that we can demonstrate this scaling also when wecarefully exclude the elastic contributions. This will bedone in the next subsection.
B. System-size dependence of purely plasticcontributions
To single out the purely plastic contributions to theelasto-plastic events we will track the neighbor lists dur-ing the straining simulations. Before every flow eventeach particle in the system is assigned a neighbor listconsisting of the particles residing within the range ofinteraction, cf. Eq. (16). After the elasto-plastic eventeach particle is checked against the original neighbor listand the number of neighbor changes is monitored. Itis important to state that upon loading, the neighborlist may change due to non-affine elastic effects when a
0 200 400 600 800
10−4
10−3
10−2
10−1
P(n
)
n
62510242500409610K20K
103
104
101.3
101.5
101.7
101.9
〈n〉
N
FIG. 25: The distributions of n for different size systems.Inset: the scaling of the average of n with the system size. Apower law is detected, see text.
neighboring particle leaves the range of interaction or anew particle enters that range. During an affine linearelastic loading the neighbor list does not change at all.However, during the loading we encounter also nonlin-ear non-affine elasticity. We have checked carefully anddetermined that during the latter there exist T1 pro-cesses in which the neighbor list does change. Such pro-cesses are reversible and can be traced back by unload-ing. In our simulations we find that in order to detectmore than one change in the neighbor list per particle inthe non-affine elastic processes we need to undergo verylarge strain intervals. We therefore conclude that duringthe plastic drops, where the displacement field changesonly little, it is unlikely that the non-affine elastic strainswould involve more than one change per particle in theneighbor list. We thus propose that by filtering thoseparticles whose neighbor list had undergone more thanone change during the elasto-plastic event we capture es-sentially only those that were a part of a purely plasticirreversible event. We formed a measure of the size n ofthe purely plastic event from summing up the numberof particles whose neighbor list changed by more thanunity. In Fig. 25 we show the raw distribution of n forsystems of varying sizes, with an inset exhibiting the av-erage of n. As before, we find that the raw distributiontend to higher and higher values of n when the systemsize increases, and as before we find that the average ofn scales nicely with the system size,
〈n〉 ∼ N ζ , (23)
with ζ ≈ 0.39. Note that with the present accuracy wecannot rule out that ζ = α.
17
0.9 1 1.1 1.2 1.3 1.4−1
−0.5
0
0.5
1
1.5
2
r/σ
U(r
)/ε
FIG. 26: The potential chosen for the model of section IX.
IX. YET ANOTHER MODEL TO REMOVE THELAST DOUBTS
The purpose of this section is to remove the last doubtsabout the sub-extensivity of the plastic events. To thisaim we introduce a new model which has a measure ofplastic deformation that is insensitive by construction tonon-affine elasticity. As above the system consists ofpoint particles in two-dimensions interacting via a pair-wise potential
U(r) =
ε[(
σr
)12 − (σr
)6 + 14 − h0
], r ≤ σx0
εh0P(
rσ−x0
xc
), σx0 < r ≤ σ(x0 + xc)
0 , r > σ(x0 + xc) ,(24)
which consists of a shifted repulsive part of the standardLennard-Jones potential, connected via a hump to a re-gion that is smoothed continuously to zero (up to secondderivatives), cf Fig. 26. The point x0 is the position atwhich the LJ potential is minimal, x0 ≡ 21/6, and theposition where the potential vanishes is σ(x0 + xc). Theparameter h0 determines the depth of the minimum. Thepolynomial P (x) is chosen as
P (x) =6∑
i=0
Aixi . (25)
with the coefficients given in table III. Note that withthese parameters the position of the shallow maximumin Fig. 26 is rb = 1.336 189 578 406 025.
We use the position of rb to define events that are plas-tic by definition and not elastic: whenever the distancebetween a pair of particles which were bonded exceeds rb
their energy drops irreversibly and spreads around. Sim-ilarly, whenever a pair of particles that were not bondedforms a new bond when their distance becomes smallerthan rb their energy drops irreversibly and is spreadaround. We can thus simply count the number of par-ticles that underwent a bond break or a bond creation
A0 -1.0
A1 0.
A2 0.642897426047121
A3 30.503000042685606
A4 -80.366384684339579
A5 72.652179536433835
A6 -22.431692320826986
TABLE III: The coefficients in Eq. (25)
0 100 200 300 400 500 60010
−4
10−3
10−2
10−1
100
P(n
b)
nb
10242500409610K
103
104
101.3
101.6
〈nb〉
N
FIG. 27: The distribution of the number of particles partic-ipating in a purely plastic event as a function of the systemsize. The number nb is defined such that elastic processescannot contribute to this measure by construction. In the in-set we show a log-log plot of the average 〈nb〉 as a function ofN ; the scaling law is Eq. (26).
during a stress drop, to obtain an unquestionable mea-sure of the size of the purely plastic event. To do this, werepeat the straining experiment in much the same wayas discussed above, but for systems with the present po-tential (24) for system sizes of N = 1024, 2500, 4096 and10 000, keeping the same value of the density ρ as before.In every stress drop event we count the number of par-ticles experiencing a bond change, which is denoted asnb. We stress that we carefully ascertained that there isabsolutely no change in this measure except during thedrops, even when the system undergoes very substantialnon-affine elastic displacements. Fig. 27 displays the dis-tribution of nb as a function of system size and also inthe inset the dependence of 〈nb〉 as a function of N in alog-log plot. The best fit reads
〈nb〉 ∼ Nχ , (26)
with χ ≈ 0.33. We believe that the difference between χand ζ is outside the error bars for either number, indi-cating that changing the potential may modify the valuesof the scaling exponents. To strengthen this conclusionswe measured also the scaling exponents α and β for thiscase. The measured numbers are consistent with χ, i.e.α ≈ χ and β ≈ χ− 1.
18
X. SUMMARY OF THE ATHERMALQUASI-STATIC SIMULATIONS
We have examined the issue of the non-locality of plas-tic deformation by a set of numerical experiments of in-creasing stringency in filtering out the elastic contribu-tions. First we examined the maximal force exerted on aparticle moving within a glassy medium of finite size, anddiscovered that the average maximal force before a plas-tic deformation depends on the system size as a powerlaw. Not being able to filter out the elastic from theplastic effects in this experiment, we turned to examin-ing the distribution of stress drops and energy drops in astraining experiment once the system exceeded the yieldstress and landed on the steady state plastic flow. Againwe found that both the stress and the energy drops haddistribution function that exhibited an interesting scalinglaw with the size of the system, strengthening the conclu-sion that the plastic events are not localized. The scalingexponents found were non trivial, but in agreement withthe presumably exact scaling relation Eq. (21). Hav-ing still difficulties in distinguishing elastic from plasticcontributions in these measurements, we turned to theneighbor lists whose large changes are most likely notdue to non-affine elasticity. Those changes scaled againwith the size of the system, and the scaling exponent ζwas numerically sufficiently close to α to indicate thatthey are the same exponents, and that the scaling of theenergy drops in the plastic events is the same as that ofthe purely plastic contribution. Finally, to remove thelast doubts we constructed a model in which the plas-tic events can be accurately separated from any elasticcontribution, and found that also there the size of theplastic events scales sub-extensively with the size of thesystem. The exponent χ was sufficiently different from ζto indicate a lack of universality in these exponents. In-deed, direct measurement of α in the last model resultedin α ≈ χ ≈ 0.33, with a different value from α ≈ 0.37 forthe previous potential. It is very likely that the scalingexponents seen here depend on the details of the models,on the nature of disorder and on parameters like pressure,density etc. Of course, the space dimensionality is alsoimportant, and there is evidence that in 3-dimensionsone get different exponents [38]. We propose that under-standing these exponents and finding a theoretical calcu-lation of them will shed important light on the physicsof amorphous solids.
XI. THE EFFECT OF FINITE TEMPERATUREAND FINITE STRAIN RATES
Having learnt that in quasi-static and athermal con-ditions the plastic events appear non-localized, the nextquestion is what are the modification, if they exist, thatare introduced by thermal effects and finite strain rates.Of particular relevance to the present discussion is Ref.[44] in which the authors studied the question for zero
temperature as a function of the strain rate. At low strainrates γ the plastic events were shown to be spatially cor-related with a system size dependence. At high strainrates (compared to elastic relaxation times) the correla-tion were cut-off proportional to γ−1/d where d is thespace dimension. Two crucial questions that remain are(i) what is the effect of temperature on this issue. Shouldtemperature fluctuations also cut-off the statistical cor-relations? and (ii) if temperature effects do cut off themagnitude of plastic flow events, which of the cut-offsdominates at a given temperature and strain rate?
The aim of this section is to address these two ques-tions. Following We will show that temperature effectsare as important, if not more important, in checking themagnitude of plastic events as the effect of a finite γ.We will present below some quantitative estimates of thevarious effects to compare their efficacy in bounding themagnitude of plastic flow events at a given temperatureand strain rate.
A. The effect of finite strain rate
As said already, Ref. [44] showed that finite strainrates may cut-off the magnitude of plastic flow events.To understand this effect we start by substituting Eq.(19) in Eq. (22) to obtain the scale s. With 〈λ〉 beingthe mean inter-particle distance we write:
s =εµ
σY 〈λ〉d =εµρ
σY m. (27)
Consider next the rate at which work is being done atthe system and balance it by the energy dissipation inthe steady state,
σY γV = 〈∆U〉/τpl , (28)
where τpl is the average time between plastic flow events.This time is estimated as the elastic rise time which is
τpl ∼ 〈∆σ〉µγ
∼ εNβ
σY 〈λ〉dγ . (29)
We increase our confidence in this estimate by substitut-ing it into Eq. (28) together with the other estimates, tofind perfect consistency.
Next we note that τpl decreases when N increases. Onthe other hand there exists another crucial time scale inthe system, which is the elastic relaxation time τel ∼ L/c,where c is the speed of sound c =
√µ/ρ. Obviously
this time scale increases with N like N1/d. There willbe therefore a typical scale ξ1 such that for a system ofscale L = ξ1 these times cross. At that size the systemcannot equilibrate its elastic energy before another eventis triggered, and multiple avalanches must be occurringsimultaneously in different parts of the system, each ofwhich has a bounded magnitude. We estimate ξ1 fromτel ∼ τpl, finding
(ξ1/c) ∼ ε[N(ξ1)]β
σY 〈λ〉dγ ∼ ε[ξ/〈λ〉]dβ
σY 〈λ〉dγ . (30)
19
FIG. 28: A typical equilibrium configuration with 65,356 par-ticles. The particles are all point objects, and the ball aroundeach particle is of radius λi.
Using now the obvious fact that N(ξ1) ∼ (ξ1/〈λ〉)d wecompute
ξ1
〈λ〉 ∼[(
ε
σY 〈λ〉d) (
c
〈λ〉γ)]1/(1−βd)
. (31)
We observe the singularity for quasi-static strain whenγ → 0, where ξ1 tends to infinity, in agreement with theresults of quasi-static calculations. At low temperatures,before the thermal energy scale becomes important, thesize of plastic flow events can be huge indeed. Note thatthere exists a difference between our estimate of ξ1 andthat of Ref. [44]; this stems from our attempt to put amore stringent constraint, to see how big the system canbe while separating individual plastic drops. Ref. [44]was interested in avalanches which are not our main con-cern. We show next that thermal effects put a much morestringent bound on the magnitude of plastic flow events.
The effect of finite temperatures: During a plas-tic drop the energy released quickly spreads out in thesystem on the time scale of elastic waves. Thus everyparticle shares an energy of the order of εNα/N = εNβ .On the other hand the typical scale of thermal energyper particle is kBT where kB is Boltzmann’s constant.We thus expect thermal effects to start overwhelming thestatistics of athermal plastic events when
εNβ ∼ kBT . (32)
This equality will hold when the system size L = ξ2,where (ξ2/〈λ〉)d = N . Substituting the last equality inEq. (32) and then solving for ξ2 we find
ξ2/〈λ〉 = [kBT/ε]1/dβ. (33)
Recalling that β = α − 1 is negative, we again noticethe singularity at T → 0 in agreement with the athermalquasi-static simulations.
A Model Glass Example: To put some size esti-mates on these crucial length-scales, and to test theirconsequences, we need to choose a model glass. To thisaim we employ the same model that was described inSect. VI B. In the present simulations we chose k = 10,B0 = 0.2. In the 3D simulations below the mass densityρ ≡ mN/V = 1.3, whereas in 2D ρ = 0.85. In all casesthe boundary conditions are periodic and thermostatingis achieved with the Berendsen scheme [16]. In Fig. 28we present a typical 3D equilibrium configuration of thesystem with N = 65356. We measured for this 3D sys-tem the shear modulus µ = 15.7 and therefore the speedof sound is c ≈ 3.5. The value of σY at T = 0 is about0.7 and the typical value of σ∞ at higher temperaturesis of the order of 0.5.
The estimate of ξ1 depends of course on γ. In our3D simulations we have used γ = 5 × 10−5, and for thegiven values of the speed of sound and of σY we estimateξ1/〈λ〉 ∼ 2×105 which translates to about 1016 particles.Obviously this system size is hugely beyond the capabil-ities of molecular simulations. One could in principleincrease γ, but not beyond σy/(
√ρµL) [42]. It therefore
remains elusive to demonstrate the cross-over due to theelastic time-scale in numerical simulations. Neverthelessone should remember, in developing an athermal theoryof elasto-plasticity, that the plastic flow events are verylarge, a fact that cannot be disregarded with impunity.
The cross-over scale due to thermal energies is verywell within the range of system size available in numericalsimulations. Making the plausible estimate ε ≈ ε we seethat already at T = 10−3 ξ2/〈λ〉 is estimated (for β =−8/15 in 3 dimensions [38]) as ξ2 ≈ 102, which translatesto just 1 million particles. For T = 10−2 this estimatedrops down to about 1000 particles. Thus we expect avery rapid cross-over from correlated avalanches to un-correlated ones as the temperature rises above 10−3.
Demonstration of the Thermal Cross-over: Avery interesting and direct way of demonstrating thecross-over due to thermal effects is provided by mea-surements of the variance of the stress fluctuations asa function of the temperature and the system size. Thisvariance is defined by 〈δσ2〉 ≡ 〈(σ − σ∞)2〉 , where σ∞is the mean stress in the thermal steady state. In Fig.29 and 30 we display 2D and 3D measurements of thisquantity which is obtained by averaging the square of themicroscopic stress fluctuations in long stretches of elasto-plastic steady-states of the models described above at afixed γ = 2.5 × 10−5 in 2D and γ = 5 × 10−5 in 3D.It is evident that the variance of the stress fluctuationsdecreases as a function of N . Under quasi-static andathermal conditions the dependence is a power-law
〈δσ2〉 ∼ N2θ , (34)
where θ ≈ −0.4 both in 2D and 3D. One should noticethe difference between the exponent characterizing the Ndependence of
√〈δσ2〉 and of the athermal mean plastic
stress drop 〈∆σ〉, in the sense that θ 6= β. This differ-ence is due to very strong correlations between elastic
20
103
104
10−2
10−1
100
101
N
〈δσ
2〉
T = 0.0001
T = 0.001
T = 0.01
T = 0.05
T = 0.1
T = 0.15
AQS
FIG. 29: The variance of the stress fluctuations as a functionof the system size N for a 2D system and for various tempera-tures. The first power-law (data in squares) is obtained underathermal quasi-static conditions where we determine for thepresent model β = −0.61, θ = −0.40. The other plots goup in temperature as indicated. The plots are displaced by afixed amount for clarity. Note that the slope becomes morenegative as the temperature increases
103
104
105
10−3
10−2
10−1
100
N
〈δσ
2〉
T = 0.001
T = 0.01
T = 0.1
T = 0.3
T = 0.5
FIG. 30: The variance of the stress fluctuations as a func-tion of the system size N for a 3D system and for varioustemperatures. The plots are displaced by a fixed amount forclarity. Note that the slope becomes more negative as thetemperature increases
increases and plastic drops. At higher temperatures thedata in Figs. 29 and 30 indicate a clear cross-over to in-dependent stress fluctuations in which 〈δσ2〉 ∼ N−1 forhigh temperatures.
To capture the temperature and size dependence of thevariance, and to demonstrate unequivocally the thermalcross-over, we first need to separate the thermal from themechanical contributions to 〈δσ2〉. We write
〈δσ2〉 = 〈δσ2〉T + ˜〈δσ2〉 , (35)
where 〈δσ2〉T denotes the thermal contribution which canbe read from Eq. (10) of Ref. [48], i.e.
〈δσ2〉T ≈ µT/V . (36)
For the mechanical part we introduce a scaling functionwhich exhibits a cross-over according to Eq. (32). In
10−2
10−1
100
101
100
101
Nβ
T
〈δσ
2〉
N2
θ
T =0.001
T =0.01
T =0.1
T =0.3
T =0.5
10−2
100
102
100
101
Nβ
T
〈δσ
2〉
N2
θ
T = 0.0001
T = 0.001
T = 0.01
T = 0.05
T = 0.1
T = 0.15
FIG. 31: The scaling function g(x), cf. Eq. (38) for the 2Ddata (upper panel) and the 3d data (lower panel). Note thecross-over for x of the order of unity as predicted by Eq. (32).The power law decrease at low values of x are in agreementwith the prediction of ζ ≈ 0.33 in both cases. The two blacklines represent the theoretical prediction for the scaling func-tion g(x) for x ¿ 1 and for x À 1. Note that the full scalingfunction depends on γ, and the present one is an approximateversion for γ → 0. This is the reason for the curved data setsof the low temperature runs.
other words, we propose a scaling function g(x) to de-scribe the system-size and temperature dependence ofthe mechanical part of the variance:
˜〈δσ2〉(N,T ) = s2N2θg(εNβ/kBT ) . (37)
The dimensionless scaling function g(x) must satisfy
g(x) → g∞; for x →∞ ,
g(x) → g0xζ for x → 0 . (38)
The first of these requirements means that the fluctuationare in accordance with the athermal limit. The secondmeans that after the cross-over the fluctuations of thestress become intensive, requiring ζ = −(1 + 2θ)/β. Wecompute ζ ≈ 0.33 both in 2D and 3D.
We present tests of the scaling function for both our2D and 3D simulations in Fig. 31. Examining the scal-ing functions in Figs 31 we see that although the datacollapse is not perfect, the thermal cross-over is demon-strated very well where expected, i.e. at values of x of theorder of unity. The asymptotic behavior of the scalingfunctions agrees satisfactorily with the theoretical pre-diction for both the 2D and the 3D data.
21
0.5
1
1.5
〈σx
y〉
0 0.5 1 1.5 2 2.51.6
1.7
1.8
〈U〉/
N
γ
FIG. 32: Upper panel: Sress vs. strain curve in a 2-dimensional system of 6400 particles. Lower panel: the energyapproach to the steady state in the same system and the sameconditions as in the upper panel. Note that the elastic energyreaches its steady-state value together with the stress, butthen the configurational energy is slow to attain the steadystate.
We thus conclude this section by reiterating that thethermal cross-over appears much more aggressive thanthe shear-rate cross-over in cutting off the sub-extensivescaling of the shear fluctuations and mean drops. Formacroscopic systems it should be quite impossible toobserve plastic events that are correlated over the sys-tem size except for extremely low temperatures in thenano-Kelvin range. On the other hand nano particlesof amorphous solids may show at low temperatures andlow strain rates some rather spectacular correlated plas-tic events.
XII. TIME SCALES AND ORDERPARAMETERS IN ELASTO-PLASTIC FLOWS
Contrary to fluids for which the Navier-Stokes equa-tions provide an adequate description under a very widerange of conditions, we still do not have an acceptedtheory for amorphous solids that can describe their re-sponse to external loads from creep to steady elasto-plastic flows. The nonexistence of such an accepted the-ory is not for lack of trying. It is already half a centurysince the pioneering work of Cohen and Turnbull [54] andthree decades since the early work of Spaepen and Argon[24, 55], without an emerging theory that is accepted byone and all. In recent years the availability of more andmore powerful computers has allowed crucial progress inunderstanding the intricacies of the subject [30, 36, 39–41, 43–47]. In particular, numerical simulations can testall the basic assumptions that are made saliently or ex-plicitly in various theories of the mechanical response ofamorphous solids, allowing to weed out wrong assump-tions or to validate fertile concepts with the aim of finallyreaching a proper theory. In this Letter we contribute tothis path by focussing on the time-scales associated withthe various macro-variables that are used in current the-ories of elasto-plasticity.
To sharpen the issue consider for example a typical
stress-strain curve (for numerical details see below) whichis obtained in a 2-dimensional system of N = 6400 par-ticles with a strain rate of γ = 10−4, cf. Fig. 36 upperpanel. The stress reaches its steady state value, whichis known as the flow-stress, at values of γ of the orderof γ ≈ 0.4. On the other hand, the energy which isshown in the lower panel, reaches its steady state valueat much higher values of γ, values that in fact lie out-side the range of this graph. Since γ is constant in thisexperiment, a value of γ is also a time scale ∆t = γ/γ.The question is what determines these time scales, andwhat are the potential additional macro-variables thatcouple to the stress on the one hand and to the energyon the other hand such as to make these time-scale. Aclear proposition for an answer to this question exists forexample in the the ‘Shear transformation Zone’ theory ofLanger and coworkers [25, 26], which is certainly one ofthe more attractive theories of elasto-plasticity that hadbeen put forward in recent years. In this theory one as-serts that elastic relaxation occurs due to the yielding ofshear transformation zones which are relatively rare anduncoupled groups of molecules whose density is deter-mined by an ‘effective temperature’ which is a measure ofthe configurational disorder. The effective temperaturein this theory is the order parameter, and the time scaleof the stress reaching its steady state must be the same asthe time scale for the effective temperature, or any otherconfigurational variable, to reach its steady state. It isa fundamental assumption of this theory (and in fact ofany theory that assumes that an effective temperature isthe order parameter), that there cannot be a discrepancybetween the time scales of the stress and of the configu-rational variables in reaching the steady state. The datashown in Fig. 36 is in clear contradiction to these as-sumption. Since the elastic contribution to the energyreaches its steady state together with the stress, we seethat the configurational energy attains the steady stateon a much slower time scale then the stress. This phe-nomenon is generic rather than specific to this model oranother, as we show next.
Models and numerical simulations: To demon-strate the genericity of the phenomena discussed we em-ploy here four different models of glass formation, allcarefully studied before and all demonstrating the usualproperties of the glass transition and of the resultingamorphous solids. These models are
1. Binary mixture with repulsive potential. Here thepoint particles interact via a purely repulsive po-tential which diverges like (λij/rij)10 for r → 0 andwhich goes to zero smoothly with two derivativesat a cutoff length r = rc. There are three valuesof λij = 1, 1.18 and 1.4 for the interaction betweentwo ‘small’ particles, a ‘small’ and a ‘large’ particleand two ‘large’ particles respectively. Details of thepotential can be found in [43, 56, 57].
2. Repulsive potential with multi-dispersed length pa-rameters. Here again point particles interact via
22
the same potential as in the binary case, butthe length parameter 2λij = λi + λj where eachλi is taken from a Gaussian distribution with〈(λi−〈λ〉)2〉
〈λ〉2 = 15%. Details can be found in[43, 56, 57].
3. The Shintani-Tanaka model. This model is a glass-forming system whose constituents interact via ananisotropic potential depending on the angle of aunit vector carried by each particle. Details of thepotential can be found in [5, 7, 58].
4. The hump model. In this model the identical par-ticles are interacting via a pair-wise potential thathas a minimum, then a hump, and then it goessmoothly to zero at a finite distance rc with twoderivatives. Details of the model can be found forexample in [11, 43]
One important difference between models 1 and 2 onthe one hand and models 3 and 4 on the other is that thelatter have a minimum in the potential where nearest-neighbor particles can reside for a long time. The formermodels are purely repulsive and can exchange nearest-neighbors on a vibrational time-scale. For all these mod-els we have measured the stress and the energy ap-proach to the steady state in two-dimensions (2D) andin three-dimensions (3D). Results are presented in Fig.33. Examining the results we conclude that (i) the phe-nomenon discussed is generic, and (ii) it appears to bemore pronounced in 2D than in 3D. We will argue nextthat this phenomenon can be ascribed in part to thestress being relatively insensitive to the configurationaldegrees of freedom, whereas the energy, after reachingits elastic steady state, is evolving further precisely onthe time scale of the configurational degrees of freedom.
The energy times scale and the configurationaldegrees of freedom: To demonstrate the relation be-tween the time scales of the configurational degrees offreedom and the energy in attaining the steady state weuse two approaches. In the first we simply recognize thatchanging the configurational degrees of freedom resultsin a temporal dependence of the mean number of nearestneighbors. In every model ‘nearest neighbors’ may meansomething slightly different; thus in the multi-dispersedmodel a ‘neighbor’ is any particle that resides within theinteraction range, before the cut-off. In the hump modela ‘neighbor’ is counted as such if its distance is smallerthan the position of the hump. Irrespective of these slightdifferences, one can see in Fig. 34 that the dynamics ofthe configurational energy follows verbatim the dynam-ics of the average number of neighbors. In all cases theelastic energy reaches its asymptotic value rapidly, onthe time scale of the stress, and then the configurationalenergy follows the configurational change. Note the rel-atively higher fluctuations in the number of neighborsin the multi-dispersed model 2 compared to the humpmodel 3; this follows from the definition of neighborsthat can switch from one particle to the other on the
0 0.4 0.8 1.2 1.6 20
0.2
0.4
0.6
0.8
12D
Poly
disper
se
〈σx
y〉
0 0.4 0.8 1.2 1.6 20
0.012
0.024
0.036
0.048
0.06
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
13D
0 0.2 0.4 0.6 0.8 10
0.0095
0.019
0.0285
0.038
0.0475
〈δU〉/N
0 0.4 0.8 1.2 1.6 20
0.1
0.2
0.3
0.4
0.5
Bin
ary
〈σx
y〉
0 0.4 0.8 1.2 1.6 20
0.004
0.008
0.012
0.016
0.02
0 0.4 0.8 1.2 1.6 20
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 20
0.023
0.046
0.069
0.092
0.115
〈δU〉/N
0 0.4 0.8 1.2 1.6 20
0.3
0.6
0.9
1.2
1.5
Hum
p
〈σx
y〉
0 0.4 0.8 1.2 1.6 20
0.035
0.07
0.105
0.14
0.175
0 0.8 1.6 2.4 3.2 40
0.3
0.6
0.9
1.2
1.5
γ
0 0.8 1.6 2.4 3.2 40
0.05
0.1
0.15
0.2
0.25
〈δU〉/N
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Tanaka
〈σx
y〉
γ0 1 2 3 4 5
0
0.07
0.14
0.21
0.28
0.35
FIG. 33: Stress vs. strain curve and energy vs. strain as com-puted for models 2,1,3 and 4 respectively in 2D (left panels)and in 3D (right panels). The Tanaka model 4 is only definedin 2D. Note that the slow attainment of the configurationalenergy to the steady state is very clear in 2D, but it exists in3D as well, as seen for example in the hump model 3. The ap-parent identity of time scales in 3D between stress and energyin models 2 and 1 is accidental.
time scale of a single vibration. For the hump modelthe switch is thermally activated and therefore much lessreadily made at low temperatures. The other two modelsexhibit similar results, allowing us to conclude that for allmodels and all dimensions the energy appears to followthe configurational change, and if its time scale appearsin some simulations to be the same as that of the stress,this is accidental.
To solidify these conclusions we use yet a second mea-sure of the configurational change. To this aim we denotethe nearest neighbors of the ith particles as n.n.(i) anddefine the local non-affine deformation measure qi via[25]:
qi(γ, γ′) ≡∑
j∈n.n.(i)
∑α
(rijα (γ)−ΨανΥ−1
νβ rijβ (γ′)
)2
,
(39)
23
0 0.4 0.8 1.2 1.6 2
5.715
5.72
5.725
5.73
5.735
5.74
2D
Poly
disper
se
〈i〉
0 0.4 0.8 1.2 1.6 20
0.009
0.018
0.027
0.036
0.045
0 0.2 0.4 0.6 0.8 14.4
4.42
4.44
4.46
4.48
4.53D
0 0.2 0.4 0.6 0.8 10
0.0075
0.015
0.0225
0.03
0.0375
〈δU〉/N
0 0.4 0.8 1.2 1.6 23.76
3.83
3.9
3.97
4.04
4.11
Hum
p
〈i〉
γ0 0.4 0.8 1.2 1.6 2
0
0.032
0.064
0.096
0.128
0.16
0 0.4 0.8 1.2 1.6 26.43
6.464
6.498
6.532
6.566
6.6
γ0 0.4 0.8 1.2 1.6 2
0
0.015
0.03
0.045
0.06
0.075
〈δU〉/N
FIG. 34: Upper panels: Comparison between the attainmentof steady state by the energy and by the mean number ofneighbors. In 2D we employed a system with N = 20164, in3D with N = 16384. Lower panels: the same comparison forthe hump model, in 2D with a system of N = 6400 and in 3Dwith N = 32768.
where rijα ≡ rj
α − riα, and
Ψαβ =∑
j∈n.n.(i)
rijα (γ)rij
β (γ′) ,
Υαβ =∑
j∈n.n.(i)
rijα (γ′)rij
β (γ′) .(40)
Clearly, qi ≥ 0 for any γ, γ′, and is non-zero if the vicin-ity of particle i undergoes either a plastic deformation ornon-affine elasticity. The measure of non-affine deforma-tion is than taken as Q(γ):
Q(γ, γ′) ≡∑
i
qi(γ, γ′) . (41)
In order to demonstrate the correlation between theenergy equilibration and the non-affine deformation pro-cesses, we performed an athermal quasi-static strainingexperiment, during which the energy change and the de-formation measure Q were computed. At each strainstate the energy change ∆U ≡ U(γ)−U(γ− δγ) and themeasure of deformation Q(γ, γ − δγ) are evaluated withrespect to the previous recorded strained state in the tra-jectory. In Fig. 35 we show the correlation plots betweenthe magnitude of the energy changes and the changes inthe degree of mixing. We see the almost perfect correla-tions which are a general feature spanning across modelsand dimensions.
In summary, we showed that an explicit measure of theevolution of the configurations space of the shearing ma-terial (the average number of neighbors) and a measure ofthe character of collective particle movements (the non-affinity measure) are both strongly coupled to the evo-lution of the energy. We conclude that it is the energy,not the stress, that provides an accurate measure of theshearing system’s progression towards its steady state.
The mechanism of the rapid attainment of the stressto its own steady state is still under investigation and
−150 −100 −50 00
500
10002D
Poly
disper
seQ
(γ,γ
−δγ
)
−200 −100 00
2000
4000
6000
3D
−40 −20 00
20
40
60
∆U
Hum
pQ
(γ,γ
−δγ
)
−100 −50 00
100
200
300
400
∆U
FIG. 35: Correlations between the energy drops and the in-creases in Q(γ)
is not entirely understood. It is worthwhile to specu-late on the basis of the findings reported here that thestress is much less sensitive to the presence of dynamicalheterogeneities than the configurational energy. Indeed,imagine that the system has regions that are reluctant torelax, be them soft modes of the Hessian of the systemor maybe even crystalline clusters that are very slow todisappear and mix. Such dynamical heterogeneities willobviously affect the time scale of the attainment of con-figurational energy to the steady state. All parts of thesystem contribute to the energy. In this sense energy is atrue reflection of the state of the entire system. Stress, incontrast, is determined by some subset of the system thatmay even live on a fractal. In any instantaneous config-uration you will have low energy (hard) and high energy(soft) regions. The stress is supported by the network ofhard regions. The smaller the cross-sectional area of thisnet, the higher the stress for a given strain. This meansthat a relatively small increase in the amount of highenergy (soft) regions can lead to a significant decrease inthe cross-sectional area of the hard backbone and hence alarge increase in stress. The evolution of stress is fast, inother words, because it reflects not only the time depen-dence of the amount of soft regions (i.e. the energy) butthe more tenuous distribution of the remaining hard do-mains. Evidently these speculations must be supportedby observations and further theory which at this pointhas to await further research.
XIII. THE BAUSCHINGER EFFECT AND NEWORDER PARAMETERS FOR VISCO-PLASTIC
FLOWS
In this last section we only hint towards the roadahead, which includes finding observables in strainedamorphous solids that can be used to describe the com-plex elasto-plastic behavior. Here we describe an observ-able that can explain the Bauschinger effect which wedescribe next.
24
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
|γ|
|σxy|
0 0.05 0.1 0.15 0.20
0.1
0.2
0.3
0.4
0.5
0.6
|γ − γ0|
|σxy|
positive strainnegative strain
γ0
FIG. 36: Color online: Stress-strain curves. Left panel: start-ing the experiment from a freshly prepared sample results ina symmetric trajectory for γ → −γ. Right panel: starting theexperiment from the zero-stress state with γ = γ0 results inan asymmetric trajectory, see text for details. Data was av-eraged over 500 independent stress-strain curves at T = 0.01.
An amorphous solid which is freshly produced by cool-ing a glass-forming system from high to low temperatureis isotropic up to small statistical fluctuations. Put un-der an external strain, its stress vs. strain curve shouldexhibit symmetry for positive or negative strains. Thisis not the case for the same amorphous solid after ithad been already strained such that its stress exceededits yield-stress where plastic deformations become nu-merous, resulting in an elasto-plastic flow state. Thephenomenon is clearly exhibited in Fig. 36. A typicalstress-strain curve for a 2-dimensional model amorphoussolid (see below for numerical details) starting from afreshly prepared homogenous state is shown in the leftpanel, with a symmetric trajectory for positive or nega-tive shear strain. Once in the steady flow state, the sys-tem is brought back to a zero-stress state, which servesas the starting point for a second experiment in whicha positive and negative strain is put on the system asshown in the right panel of Fig. 36. Even though theinitial state is prepared to have zero mean stress, the tra-jectory now is asymmetric, with positive strain exhibit-ing ‘strain hardening’ [49], but reaching the same levelof steady state flow-stress, whereas, the negative strainresults in a ‘strain softening’ and a faster yield with even-tually reaching the same value of steady-state flow-stress(in absolute value). This simple phenomenon, sometimereferred to as the Bauschinger effect [50], shows that thestarting point for the second experiment retains a mem-ory of the loading history, some form of anisotropy, whichis the subject of this Letter.
How to identify the order parameter which is responsi-ble for the anisotropy underlying the Bauschinger effectis a question that hovers in the elasto-plastic communityfor some while [22–24]. One obvious concept, i.e. of ‘back
stress’ [51] or ‘remnant stress’ for explaining the asym-metry seen in the second experiment in Fig. 36 can beruled out simply by verifying that the initial point haszero mean stress. A more sophisticated proposition is em-bodied in the ‘shear-transformation zone’ theory (STZ)in which it is conjectured the plasticity occurs in local-ized regions whose densities differ for positive and nega-tive strains, denoted n+ and n− [25, 52]. The normalizeddifference between these, denoted as m, is a function ofthe loading history and can, in principle, characterize theanisotropy that we are seeking. Unfortunately the pre-cise nature of the STZ’s was never clarified, and it isunknown how to measure either n+, n− or m, makingit quite impossible to put this proposition under a di-rect test. More recently it was proposed that the soughtafter anisotropy can be characterized in granular mat-ter by the fabric tensor F = 〈nn〉 which captures themean orientation of the contact normals, n, through thespatial average of their diadic product [53]. This orderparameter was generalized for silica glass where n waschosen as a unit vector in the direction of the vector dis-tance between Si atoms, disregarding the oxygens. At-tempting to test this proposition in the context of thebest-studied model of glass-forming, i.e. a binary mix-tures of point particles with two interaction lengths, orin the case of multi-dispersed point particles (see belowfor details), did not reveal any systematic signature ofanisotropy. We thus conclude that this order parameteris not sufficiently general to be of universal use in the de-velopment of the theory of elasto-plasticity, and that thequestion of identifying a missing order parameter remainsopen.
|γ − γ0|
|σxy|
positive strainnegative strain
FIG. 37: Color Online: Model stress-strain curves as obtainedfrom σ ∼ σ∞ tanh(µ|γ − γ0|/σ∞) + β|γ − γ0|2e−|γ−γ0|2 withβ positive.
The proposed order parameter: To see what mayserve as a general order parameter we examine first thesituation with the isotropic amorphous solid which is ob-tained after a quench without any loading history. De-noting the shear stress by σ and the shear strain by γ,we observe that isotropy dictates that all the even deriva-tives d2nσ/dγ2n must vanish by symmetry. For examplea function that can model the stress-strain curve with
25
this constraint in mind may be σ ∼ σ∞ tanh(µγ/σ∞)where σ∞ is the flow stress (the mean value of the stressin the elasto-plastic steady state), and µ is the shearmodulus. For γ → −γ this function is perfectly anti-symmetric as required for an isotropic system. Imaginenow that we add even derivatives to this function, sayσ ∼ σ∞ tanh(µγ/σ∞) + βγ2e−γ2
with β having the di-mension of stress. The effect will be to change the stress-strain curve as seen in Fig. 37, which is quite reminiscentof the Bauschinger effect. We therefore propose that it isadvantageous to focus on the even derivatives of σ vs. γ,with the most important one being the second derivative.
Statistical Mechanics: Under external loads the dis-placement field u describes how a material point movedfrom its equilibrium position. The strain field is defined(to second order) as
εαβ ≡ 12
(∂uα
∂xβ+
∂uβ
∂xα+
∂uν
∂xα
∂uν
∂xβ
), (42)
where here and below repeated Greek indices are summedupon. We expand the free energy density F/V up to aconstant in terms of the strain tensor
FV
= Cαβ1 εαβ + 1
2Cαβνη2 εαβενη + 1
6Cαβνηκχ3 εαβενηεκχ .
(43)The mean stress is defined as σαβ ≡ 1
V∂F
∂εαβ, and
σαβ = Cαβ1 + Cαβνη
2 ενη + 12Cαβνηκχ
3 ενηεκχ . (44)
In our simulations we apply a simple shear deformationusing the transformation of coordinates according to
xi → xi + δγyi ,
yi → yi , (zi → zi in 3 dimensions)(45)
where δγ = γ − γ0 is a small strain increment from somereference strain γ0. The explicit 2D strain tensor follow-ing Eq. (42) is
ε =12
(0 δγ
δγ δγ2
), (46)
with an obvious generalization in 3D. Since εxx = 0, themean shear stress reduces to the form (equally valid in2D and 3D)
σxy = Cxy1 +Cxyxy
2 δγ+ 12 (Cxyyy
2 +Cxyxyxy3 )δγ2+O(δγ3) .
(47)As discussed above, in isotropic systems where σxy isantisymmetric in δγ, Cxy
1 = 0 and the sum Cxyyy2 +
Cxyxyxy3 = 0. Our proposition is to use this sum as the
characterization of the anisotropy that we seek.Models and numerical procedures: Below we em-
ploy a model system with point particles of equal massm and positions ri in two and three-dimensions, inter-acting via a pair-wise potential in which the interactionlength-scale between any two particles i and j is λij
as defined in [43, 46]. In the three-dimensional simu-lations each particle i is assigned an interaction param-eter λi from a normal distribution with mean 〈λ〉 andλij = 1
2 (λi + λj). The variance is governed by the poly-
dispersity parameter ∆ = 15% where ∆2 = 〈(λi−〈λ〉)2〉〈λ〉2 .
In the two dimensional simulations we use the same po-tential but choose a binary mixture model with ‘large’and ‘small’ particles such that λLL = 1.4, λLS = 1.18and λSS = 1.00. Below the units of length are λ = λSS
in 2D and λ = 〈λ〉 in 3D. The units of energy (ε), mass(m) and temperature (ε/kB) are as defined in [46? ].The unit of time is τ =
√mλ2/ε. In the 3D simulations
below the mass density ρ ≡ mN/V = 1.3, whereas in 2Dρ = 0.85. In all cases the boundary conditions are peri-odic and thermostating is achieved with the Berendsenscheme [16]. We employ the sllod equations of motionfor imposing deformations, and integrate them using astandard leap-frog algorithm [16]. The strain rate is cho-sen to be γ = 10−4τ−1 for all simulations described be-low. Initial configurations were prepared by equilibrat-ing at least 1000 independent systems in the supercooledtemperature regime, followed by quenching to the targettemperature at a rate of 10−4 ε
kBτ . If not stated oth-erwise all the simulation below were obtained with sys-tems of N = 20164 in 2-dimensions and N = 16384 in3-dimensions.
We choose to measure the sum
B2(γ0) ≡ limT→0
[Cxyyy2 + Cxyxyxy
3 ]= limT→0
∂2σxy
∂γ2
∣∣∣∣γ=γ0
,
(48)using an athermal, quasi-static scheme [43]. This schemeconsists of imposing the affine transformation (45) toeach particle of a configuration, followed by a poten-tial energy minimization under Lees-Edwards boundarycondition [16]. We choose the stopping criterion for theminimizations to be |∇iU | < 10−9 ε
λ for every coordi-nate xi. Within this method one can obtain purely elas-tic trajectories of stress vs strain [43]. Given a config-uration of our simulation, we first cool the system toT = 10−3 using molecular dynamics during a time inter-val of 50τ . This initial treatment brings our system toan elastically stable state. We then apply the athermalquasi-static scheme to measure the finite differences ap-proximation to ∂2σxy
∂γ2 ≈ σxy(δγ)+σxy(−δγ)−2σxy(0)δγ2 by sam-
pling a small elastic trajectory, using strain incrementsof δγ = 2.5 × 10−6. We have checked that stricter stop-ping criteria for the minimizations or smaller strain in-crements do not significantly alter our results. We em-phasize that although we measure B2 in the athermallimit, the configurations on which we perform this mea-surement are sampled from various finite temperatures,see below. This athermal measurement is motivated bythe requirement to probe the purely elastic response, ex-cluding plasticity from the discussion. Using this methodwe can compute B2 at any point of the trajectory. Wereiterate that B2 is not the second derivative of the av-eraged stress-strain curve, but the second derivative of
26
0
0.2
0.4
σx
y
0 0.1 0.2 0.3 0.4 0.5
−150
−100
−50
0
50
γ
B2
T = 0.01
T = 0.1
T = 0.2
T = 0.3
FIG. 38: Color Online: Upper panel: Trajectories of stressvs. strain for four different temperature at the same strainrate γ = 10−4. Lower panel: the corresponding values of B2
as a function of strain. Data was averaged over 1000 indepen-dent stress-strain curves at each temperature. Note that B2
is negative even when the averaged stress-strain curve has apositive curvature, see text for discussion.
the purely elastic branch of each individual stress trajec-tory. These saw-tooth individual stress-strain curves aresmeared out by the averaging.
Results and discussion: The average trajectoriesof both stress vs strain (upper panel) and B2 vs strain(lower panel) are shown for four different bath tem-peratures in Fig. 38; the system was strained untilγ = 1/2 and then strain was reversed until the meanstress dropped to zero. The strain value was then γ0
from which the experiment in Fig. 1 right panel wasstarted with positive and negative straining with respectto γ0. The resulting trajectories of stress vs. strain areshown in Fig. 39 for the 2D system at the same fourvalues of the temperature as in Fig. 38. We observe thatthe value of B2 at the point of zero stress γ0 reduceswhen the temperature increases, and in accordance withthat the magnitude of the Bauschinger effect goes downas seen in Fig. 39.
We can draw the conclusion that the magnitude of B2
is correlated with the amplitude of the Bauschinger ef-fect (measured as the area of difference between the pos-itive and negative stress-strain curves, see inset in Fig.39). But even more detailed information which is highlyrelevant to the elasto-plastic behavior can be gleanedfrom the probability distributions functions (pdf’s) of B2.These pdf’s have rich dynamics along the stress-straincurves, as can be seen in Fig. 40. When measured in theisotropic zero-stress systems that are freshly quenchedthe distribution is symmetric as expected, with zeromean. In the elasto-plastic steady state the distributionmoved to have a negative mean, in accordance with thelow panel of Fig. 38, and it sends a tail to −∞ to accom-modate the sharp changes in first derivative (the shear
0 0.05 0.1 0.15 0.2
0
0.2
0.4
0.6
0.8
1
|γ − γ0|
|σxy|
T = 0.01
T = 0.1
T = 0.2
T = 0.3 20 40 60 800.003
0.005
0.007
B2
area
FIG. 39: Color Online: Upper panel: The Bauschinger effectfor the four temperatures shown in Fig. 38, increasing fromtop to bottom. The trajectories are displaced by fixed amount(∆|σxy| = 0.15) for clarity. Note the reduction of the effectwith increasing temperature. Data was averaged over 500independent stress-strain curves at each temperature. Inset:the shaded area of difference between the stress-strain curveswith positive and negative strain as a function of B2. Themagnitude of the Bauschinger effect saturates for T → 0.
modulus) during the plastic drops [30]. In a system largeenough there is always a location that is about to makea plastic drop, and this is reflected by the tail of this pdf.At the Bauschinger point γ0 the mean stress is zero, butthe pdf of B2 gains a positive asymmetry, sending a tailtowards +∞. In the inset of Fig. 40 we exhibit the sizedependence of the pdf at the Bauschinger point, to showthat the asymmetry and the general shape of the pdf isquite independent of the number of particles N , alwayshaving long tails, indicating that near the Bauschingerpoint there are close-by lurking plastic instabilities thatare heralded by the tail of our pdf.
To confirm that the qualitative findings reported aboveremain unchanged in 3-dimensions we repeated similarsimulation for the model described above. In Fig. 41 wepresent a representative averaged stress-strained curve inthe upper panel and the corresponding trajectory of B2,both at T = 0.01.
It is not obvious at this point in time whether a theoryof elasto-plasticity should take into account the full pdfof B2, or whether it would be sufficient to take the meanvalue of B2 into account. We propose however that thisobject and its pdf are a tempting analogue of the objectm of the STZ theory as discussed above, with the obvi-ous advantage that it can be easily measured. In fact,in a follow up paper we will show that this object canbe expressed as a sum over the particles in the system,and therefore the measurements of the pdf can be donenaturally and rapidly, making them highly accessible forfurther research. We stress that the value of B2 whichhas been defined as the limit T → 0 in Eq. (48) can be
27
−1000 −800 −600 −400 −200 0 200 400 600 800 10000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
B2
P(B
2)
zero−stress, isotropicsteady−statezero−stress, anisotropic
−1000 0 1000 20000
0.02
0.04
0.06
0.08
B2
P(B
2)
N = 1024
N = 2116
N = 4096
N = 10000
N = 20164
FIG. 40: Color Online: Probability distribution function ofB2 at various points on the stress-strain trajectory. Data wascollected from 3000 independent stress-strain trajectories atT = 0.01. In red (continuous) line we draw the symmetricpdf of the freshly prepared samples with γ = 0. In green(dashed) line we show the pdf in the steady state, where itgains a negative asymmetry. In blue (dashed-dotted) linewe see the pdf at the Bauschinger point γ = γ0 where itgained a positive asymmetry. The dynamics of these pdf’sand their means are correlated with the shapes of the stress-strain curves and are proposed to be a crucial ingredient inany theory of elasto-plasticity. Inset: the N dependence ofthe pdf at the Bauschinger point γ = γ0. Data was averagedover 1000 independent samples for each system size.
measured experimentally at sufficiently low temperatureswhere the Bauschinger effect is expected to be saturated.It appears worthwhile to measure this quantity in suchlow-temperature experiments and to correlate the valuewith the amplitude of the Bauschinger effect.
XIV. FINAL REMARKS
In this review I have pointed out some interesting is-sues in the context of amorphous solids, in their creation
via the glass transition and their unusual properties atlow temperatures. It is obvious that there is a lot moreto discover and to develop. At this time it seems to methat the issue of the stress relaxation on the presumablytenuous structure of ‘hard’ regions, be it fractal or not,is the most crucial question that needs to be figured outbefore a proper theory of elasto-plasticity is developed.
Acknowledgments
The research reviewed above was carried out by anumber of collaborators without whose work and inge-nuity nothing could be achieved. These include Lau-rent Boue, George Hentschel, Valery Ilyin, Smarajit Kar-makar, Edan Lerner, Ido Regev and Jacques Zylberg.
0
0.5
1
σxy
0 0.1 0.2 0.3 0.4 0.5−100
−50
0
γ
B2
FIG. 41: A representative averaged stress-strain curve(averaged over 600 independent trajectories—) for the 3-dimensional model (upper panel) and the corresponding tra-jectory of B2 in the lower panel for T = 0.01.
Over time I profitted from discussions, both e-mail andface-to-face, with Eran Bouchbinder, Peter Harrowell,James Langer and Anael Lemaitre. This work had beensupported in part by the Israel Science Foundation, theGerman Israeli Foundation and the Minerva Foundation,Munich, Germany.
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