Mechanical Yield in Amorphous Solids: a First-Order Phase Transition

0.29 MB·

by  Prabhat Jaiswal

Mechanical Yield in Amorphous Solids: a First-Order Phase Transition
Prabhat Jaiswal, Itamar Procaccia, Corrado Rainone and Murari Singh
Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
6
1
Amorphoussolidsyieldatacriticalvalueofthestrain(instraincontrolledexperiments);forlarger
0
strainstheaveragestresscannolongerincrease-thesystemdisplaysanelasto-plastic steadystate.
2
Alongstandingriddleinthematerialscommunityiswhatisthedifferencebetweenthemicroscopic
n states of the material before and after yield. Explanations in the literature are material specific,
a
but the universality of the phenomenon begs a universal answer. We argue here that there is no
J
fundamentaldifferenceinthestatesofmatterbeforeandafteryield,buttheyieldisabona-fidefirst
0 order phase transition between a highly restricted set of possible configurations residing in a small
1 region of phase space to a vastly rich set of configurations which include many marginally stable

ones. To show this we employ an order parameter of universal applicability, independent of the
]
t microscopicinteractions,thatissuccessfulinquantifyingthetransitioninanunambiguousmanner.
f
o
0.7
s
. A ubiquitous, and in fact universal, characteristic of 0.6
t
a the mechanical properties of amorphous solids is their
0.5
m stress vs. strain dependence [1]. Measured in countless
0.4
- quasi-static strain-controlled simulations (see for exam- σ
d 0.3
ple [2–8]) and experiments (see for example [9–11]), it
n
typically exhibits two distinct regions. In one region, at 0.2
o
c lowerstrain values, the stress σ increaseson the average 0.1
[ upon the increase of strain γ, although this increase is N=4000
0.0
punctuated by plastic events. A secondregion,athigher 0.00 0.04 0.08 0.12 0.16 0.20
1 γ
v values of the strain, displays a constant(on the average)
6 stresswhichcannotincreaseeventhoughthestrainkeeps
FIG.1: Atypicalstressvs. straincurveobtainedinastrain-
9 increasing. Ofcoursealsothis elasto-plasticsteady state controlledathermalquasistatic(AQS)shearingprotocolusing
1
branch is punctuated by plastic events. A typical such a Kob-Andersen 65-35% Lennard Jones Binary Mixture of
2
shearstressvs. shearstraincurve atzerotemperature is 4000 particles in 2d. Note the generic existence of a pre-
0
. shown in Fig. 1. The two regions are separated by what yield branch in which the stress is increasing on the average
1 whenthestrainincreases,andapostyieldsteadystatewhere
is referred to as “yield”. The actual shape of the stress
0 the average stress is constant. Both regions are punctuated
6 vs. straincurveneartheyieldpointdependsondetailsof by plastic events, with the stress drops being much larger in
1 thesystempreparation. Amorphoussolidspreparedbya the post-yield compared to the pre-yield branch. This kind
v: slow quench from the melt tend to display a stress peak of stress vs. strain curve is ubiquitous for a huge variety
i before yielding, whereasthose preparedby a fast quench of amorphous solids. Here and in the text we drop tensorial
X jointhesteadystatesmoothlywithoutastresspeak[12]. indicesfromthestressandthestrainfornotationalsimplicity.
r Ofcoursethesteadystatebranchitselfisindependentof
a
the preparation protocol; memory of the initial state is
lost in this regime. change of some sort must take place between the “elas-
Thephenomenonofthemechanicalyieldinamorphous tic” and “steady-state” parts of the stress-strain curve.
solids has been a subject of extensive study in recent As muchasthe qualitativepicture is evident, however,a
years. Many numerical studies on the subject have been lot of difficulty arises when trying to capture this quali-
performedusing athermal,quasi-staticshear(AQS) pro- tative change in a quantitative manner.
tocols, wherein a glass is made by quenching a glass for- Devising a way to distinguish and study two differ-
mer downto zerotemperature,andthensubjecting it to ent states of matter, and the transition that connects
aquasi-static(γ˙ →0)shearprotocolwhereinthe system one with the other, means identifying an order parame-
is subject to small shear steps, and then allowed, after ter to act as a label for the states. This, however is a
eachstep, to finda new mechanicallystable minimumof challenging program in the present case: as much as the
its potential energy. This kind of protocol always gives “elastic”and “steady state” brancheslook different (one
rise to the same basic kind of phenomenology as seen in is able to increase the stress under a shear load, while
Fig. 1 independently of the detailed microscopic interac- the other cannot), a snapshot, say, of a particle configu-
tionbetweentheconstituents. Thisbasicphenomenology rations in both regimes is unable to detect any relevant
has been reported in very many publications, and there difference between the two. Since both states are any-
is a general consensus about the fact that a qualitative way amorphous, there is no trivial order parameter that
2
wouldallowustounambiguouslytellthemapart. Notice Imagine now that we begin to strain this glass. While
howthisdifficultyisnotpresentinthecaseofcrystalline the stress increases, there exist plastic events that begin
solids, which do exhibit evident structural peculiarities to cause irreversible displacements in the particle posi-
with respect to liquids, and whose mechanism of failure tions. Our order parameter Q will begin to respondto
12
iswellknownsincedecades. Ultimately,itallboilsdown these displacements and will begin to reduce from O(1)
to finding asuitable orderparameterforthe glassphase. to lower values. We will show now that all along the
The problem of finding an order parameter for a glass “elastic” branch hQ i will remain around unity, but as
12
is both practical and conceptual. First, as we said be- themechanicalyieldtakesplaceasharpphasetransition
fore, a snapshot of a typical glass configuration before occurs, whereupon sub-extensive plastic events [21–23]
and after yield does not show any difference: all glasses begintocausesubstantialdisplacements,allowingdiffer-
look the same to us, and all of them look like a liquid. ent regions of the configuration space to affect the order
Standard methods like structure functions, higher order parameter. In such a situation, the distribution P(Q )
12
correlationfunctions, Voronoitesselations,Delaunay tri- will have two peaks: one at high Q ≤ 1 corresponding
12
angulations etc. all failed to provide distinction between to configurations in the same patch and one for Q ≥0
12
typical configurations before and after yield. corresponding to configurations in different patches.
In this Letter we proposethat the difficulty in making To demonstrate this fundamental idea we can use any
a distinction between the pre- and post-yield configura- modelglass,since this orderparameterdescriptionis ex-
tions lies in the fact that there is really no distinction. pected to be universal. For concreteness we performed
The crux of the matter is not in the nature of configura- molecular dynamics simulations of a Kob-Andersen 65-
tions but in their number. The yield takesplace because 35% Lennard Jones Binary Mixture in 2d. We have two
of a sudden opening up of a vast number of configura- systemsizes,N =500andN =4000. WechoseQ with
12
tions that are not at the system’s disposal before yield. a = 0.3 in LJ units, but verified that changes in a leave
To establish this insight we employ an order parameter the emerging picture invariant. As a first step, we pre-
ofatypethatwasfoundusefulfirstinthecontextofspin paredaglassbyequilibratingthesystematT =0.4,and
glasses [13]. Following Refs. [13–20] we can define an or- thenquenchingit(therateis10−6)downtoT =1·10−6
der parameter with the idea of comparing two different intoaglassyconfiguration. Thesampleisthenheatedup
glassy configurations {r(1)}N and {r(2)}N , againto T =0.2,anda startingconfigurationofparticle
i i=1 i i=1
positions is chosen at this temperature. Note that while
N
1 at T = 0.4 equilibration is sufficiently fast, at T = 0.2
Q12 ≡ N Xθ(a−|ri(1)−ri(2)|) , (1) thecomputationtimeismuchshorterthantherelaxation
i
time. The configuration is then assigned a set of veloc-
whereinθ(x)isthe Heavisidestepfunction. Thevalueof ities randomly drawn from the Maxwell distribution at
theparameteraisfree,andisdeterminedbytrialander- T = 0.2, and these different samples are then quenched
ror. The quantity Q is called an “overlap”since it has downtoT =0atarateof0.1. Thisprocedurecanbere-
12
a value that goes from 0 (completely decorrelated con- peatedanynumberoftimes(say500times),anditallows
figurations) to 1 (perfect correlation). Its purpose is to ustogetasamplingoftheconfigurationsinsideonesingle
measure the degree of similarity between configurations. “patch”. Weverifythatthetypicaloverlapoftheensem-
Let us now consider a glass, made by quenching a bleofinherentstructuressoobtainedinonepatchisclose
super-cooled liquid with N particles down to a certain to hQ12i=1, signaling that indeed the ensemble is com-
temperature T ≥ 0 at a suitable rate. A glass is an pletely located in a single patch. Having one patch, we
amorphous solid wherein particles vibrate around an repeat the procedure starting from another equilibrated
amorphous structure. So, if we take two configurations configuration of the liquid to create another patch. The
{r(1)}N and{r(2)}N fromthisglass,theywillbemost results shown below for the system with 4000 particles
i i=1 i i=1
likely close to each other with Q of the order of unity. were obtained by having 100 different patches, each of
12
If one is able to obtain a good sampling of the typical whichcontaining500differentinherentstructuresdueto
configurations visited by the particles in the glass, one the velocity randomization. The results with 500 parti-
can measure the probability distribution of the overlap cles were based on 520 different patches each of which
P(Q ),whichwillbestronglypeakedaroundanaverage having 100 different inherent structures.
12
value hQ i close to unity. The configurations visited by WethenapplytoeachinherentstructureanAQSpro-
12
the particles will then form a small connected “patch” tocol as described above. This will create for each value
in the configuration space of the system, selected by the of γ a strained ensemble of configurations in the patch
amorphous structure provided by the last configuration whoseP (Q )ismeasured. Theorderparameteriscom-
γ 12
that was visited by the liquid glass former before it fell puted by using all the unique pairs of configurationgen-
out of equilibrium while forming a glass. eratedin the strainedensemble atagivenγ. We present
3
1.0
0.6
0.8 Yield 0.5 2.0 γ=N0=.0480000
γ=0.084
〈〉Q12 00..46 QN1σ2=--γγ4000 00..34〈σ〉 〉Q)12 1.5 γγγ===000...000899827
0.2 P(γ 1.0
〈
0.2
0.1 0.5
0.0 0.0
0.00 0.05 0.10 0.15 0.20 0.0
γ 0.0 0.2 0.4 0.6 0.8 1.0
Q
12
FIG. 2: a superposition of a stress vs strain curve on the
dependence of hQ12i on γ. The stress vs. strain curve is FIG. 4: The probability distribution function hPγ(Q12)i in
obtained by averaging over 50,000 realizations of individual
thevicinity of thecritical point γ =0.088
such curves obtained from 100 initial patches, each of which Y
containing500inherentstructures. TheorderparameterQ12
was averaged on the same realizations to providehQ12i.
demonstrate the transition itself. In Fig. 4 we display
1.4 N=4000
γ=0.088 the change in hPγ(Q12)i in the vicinity of the critical
1.2 point γ as a function of γ. Within a very narrow range
Y
〉Q)12 01..80 olifkeγ,torafntshietioorndefrroomf ∆aγp≈df0w.0i1th7,dwoemoibnsaenrtvepaeafikrsatt-ohrdigehr
〈P(γ 0.6 values of Q12 to a dominant peak at low values of Q12.
0.4 We capture a very unambiguous and qualitative change
0.2 in behavior as the yielding point is reached.
0.0
0.0 0.2 0.4 0.6 0.8 1.0 Tosharpenthe understandingofwhatis happening in
Q12 thevicinityoftheyieldpointweexaminenexthow many
of our realizations loose the tight overlap with the ini-
FIG. 3: The probability distribution function Pγ(Q12) at tially prepared configuration and where the loss of over-
γ = 0.088 averaged over 100 initial configurations each of lap is taking place. To this aim we consider, for the
Y
which has 500 different realizations to obtain hPγ(Q12)i. At system of 4000 particles, all the 50,000 realizations that
thisvalueofthestrainthepdfhastwopeaksofequalheights. wehave. Theseareobtainedby100choicesofliquidreal-
Weidentifythisvalueofγasthepointofthephasetransition.
izations, each of which is velocity randomized 500 times
(chosen with Boltzmann probabilities). When the strain
γ is increased in our AQS algorithm, we keep comput-
the results for N = 4000 in Fig. 2. We can see how the
ingtheorderparameterQ wherethefirstconfiguration
initial ensemble for γ =0 shows a value of the order pa- 12
{r(1)}N inEq.(1)ischosenrandomlyfromalltheavail-
rameter Q12 = 1, signifying that our initial ensemble is i i=1
able configurations at that value of γ, and the second is
genuinely within one patch. As the ensemble is strained,
anyone ofthe other availableconfigurationsatthe same
the valueoftheorderparametergetslower,droppingto-
value of γ. We confirmed that changing the randomly
wards zerowhen the strainis increasedbeyondthe yield
strain. chosen{ri(1)}Ni=1 doesnotaffecttheresults. Next,choos-
To determine the yield strain γ accurately, we con- ing Q12 = 0.8 as a threshold value, we now count how
Y
sidertheprobabilitydistributionfunction(pdf)P (Q ). many of our observedconfigurations cross this threshold
γ 12
WedeterminePγ(Q12)foreachpatchofof500configura- and exhibit Q12 ≤ 0.8. The number of configurations
tions obtained as explained above, and then average the that do so as a function of the strain (superimposed on
result over the 100 available patches. We ask at which the stress.vs. strain curve) is shown in Fig 5. The con-
value of γ this averagedpdf has two equally high peaks, clusionofthistestisthatinthevicinityoftheyieldpoint
see Fig 3. The resulting hPγ(Q12)i determines the yield γY all the configurations lose their overlap with the ini-
point to occur at γ ≃ 0.088. Note that this criterion tialconfiguration,butnotbefore. Themechanicalyieldis
Y
implies a sharp definition of “yield” which seems absent tantamount to the opening up of a vast number of pos-
in the current literature. If accepted, it indicates that sible configurations, whereas before yield the system is
the mechanical yield occurs beyond the stress overshoot still constrainedto reside in the initial meta-basinof the
incorrespondencewiththemean-fieldresultsofRef.[24]. free energy landscape. This appears to be a first-order
phase transition [25].
Once we identify the phase transition point, we can To strengthen the propositionthat this is a first-order
4
1.0
Structures 4500000000 N=σ#4000 Yield 00..56 0.8 Yield 00..56
d Inherent 30000 4500000000 Yield 0.6 00..34〈σ〉 〈〉Q12 00..46 Q1σN2--=γγ500 000...234〈σ〉
mber of visite 1200000000 123000000000 0000 N=500 000...024 00..12 00..02 0.00 0.05 0.1γ0 0.15 0.20 00..01
Nu 0.00 0.05 0.10 0.15 0.20
0 0.0
0.00 0.05 0.10 0.15 0.20 1.4
γ
1.2
FIG. 5: The number of configurations which pass below the 〉Q)12 01..80
threshold value Q12 = 0.8 of the overlap order parameter as 〈P(γ 0.6
a function of the strain γ for N = 4000. In the onset we 0.4
show the same test for N = 500. The conclusion is that all 0.2 N=500
γ=0.122
the configurations lose overlap with the initial configuration 0.0
0.0 0.2 0.4 0.6 0.8 1.0
in thevicinity of theyield point γY Q12
.
2.0
1.5
ptihoansbeetcroamnseistisohnawrpeershwoiuthldindcermeaosnisntgrattheetshyasttetmhesitzrea.nAsit- 〉P(Q)γ12 1.0 N=500
presentwecannotproduceequallygooddataforsystems 〈 γ=0.102
0.5 γ=0.114
ofsizemuchlargerthanN =4000,butwehaveproduced γγ==00..112320
γ=0.140
equally extensive data for N = 500. The results for this 0.0
0.0 0.2 0.4 0.6 0.8 1.0
smaller system size are presented in Fig. 6. Indeed, the Q12
changein the values ofQ diminishes asseenin the up-
12
perpanel,theidentificationofthetransitionpointisless FIG. 6: Results similar to those shown in Figs.2-4 but for
sharp,andmostimportantly,therangeof∆γ overwhich system size N = 500. One observes a smearing out of the
a similar change in the peak structure is taking place is transition regionasisexpectedfrom afirst-orderphasetran-
sition.
now ∆γ ≈ 0.038. If we take just these two system sizes
as indicative we can roughly estimate the range of ∆γ
over which the transition is taking place to go to zero as
N−1/3 as N → ∞. Needless to say, at this point this grant STANPAS of the ERC and by the Minerva Foun-
should be taken as indicative only, and further accurate dation, Munich, Germany. We thank Ludovic Berthier
simulations should be conducted to solidify (excuse the for clarifying the role of “overlap” order parameters in
pun) this important issue. distinguishing amorphous configurations. Discussions
The upshot of these results is that we are able to put with Oleg Gendelman and Carmel Shor are gratefully
a finger on the essential feature that is responsible for acknowledged.
the mechanical yield. It is not that the configurations
visitedbythesystemafteryieldhavedifferentcharacter-
isticsfromtheconfigurationsbeforeyield. Rather,avery
constrained set of configurations available to the system
[1] S. Alexander, Phys.Rep.296, 65 (1998).
before yield is replaced upon yield with a vastly larger
[2] F.Varnik,L.Bocquet,and J.-L.Barrat, J.Chem.Phys.
set of available configurations. This much larger set is
120, 2788 (2004).
generic;itisnotselectedbyanycarefulcoolingprotocol, [3] C. E. Maloney and A. Lemaitre, Phys. Rev. Lett. 93,
andassuchitisexpectedtoincludemanymarginallyun- 016001 (2004).
stable configurations that will yield plastically with any [4] M. J. Demkowicz and A. S. Argon, Phys. Rev. B 72,
increase of strain [26, 27]. This is the fundamental rea- 245205 (2005).
sonfortheinabilityofthesystemtocontinuetoincrease [5] A.Tanguy,F.Leonforte, andJ.-L.Barrat, Eur.Phys.J.
E 20, 355 (2006).
its stress when strain is increased, leading to the steady
[6] C.E.MaloneyandA.Lemaitre,Phys.Rev.E74,016118
state branch. We propose this as a universal mechanism
(2006).
for the ubiquitous prevalence of stress vs strain curves [7] E. Lerner and I. Procaccia, , Phys, Rev. E, 80, 026128
that look so similar in a huge variety of glassy systems. (2009).
This work had been supported in part by an “ideas” [8] D.Rodney,A.TanguyandD.Vandembroucq,Modelling
5
Simul.Mater. Sci. Eng. 19, (2011) 08300. [19] L.BerthierandR.L.Jack,Phys.Rev.Lett.114,205701
[9] G. Subhash, Q. Liu and X-L Gao, Int. J. Impact Engi- (2015).
neering 32 1113 (2006). [20] A. Ninarello, L. Berthier and D. Coslovich, Molecular
[10] A.Kara, A.Tasdemirci and M. Guden,Materials & De- Physics, 113, 2707 (2015).
sign 49, 566 (2013). [21] E. Lerner and I. Procaccia, Phys. Rev. E 29, 066109
[11] A.L.Noradila, Z.Sajuri, J. Syarif, Y.Miyashita andY. (2009).
Mutoh, Materials Science and Engineering 46, 012031 [22] R. Dasgupta, H. G. E. Hentschel and I. Procaccia,
(2013). Phys.Rev. Lett., 109, 25502 (2012).
[12] Ashwin J., E. Bouchbinder and I. Procaccia Phys. Rev. [23] R.Dasgupta,O.Gendelman,P.Mishra,I.Procacciaand
E 87, 042310 (2013). C. A.B.Z. Shor, Phys. Rev.E., 88, 032401 (2013).
[13] S.FranzandG.Parisi,Phys.Rev.Lett.79,2486(1997). [24] C.Rainone,P.Urbani,H.YoshinoandF.ZamponiPhys.
[14] C. Cammarota, A. Cavagna, I. Giardina, G. Gradenigo, Rev. Lett.114, 015701 (2015).
T. S. Grigera, G. Parisi, and P. Verrocchio, Phys. Rev. [25] C. Rainone and P. Urbani,arXiv:1512.00341.
Lett.105, 055703 (2010). [26] S. Karmakar, E. Lerner and I.Procaccia, Phys.Rev. E,
[15] J.YeoandM.A.Moore,Phys.Rev.E86,052501(2012). 82, 055103(R) (2010).
[16] L. Berthier, Phys. Rev.E 88 , 022313 (2013). [27] H.G.E. Hentschel,P. K.Jaiswal, I.Procaccia and S.Sas-
[17] L. Berthier and D. Coslovich, Proc. Natl. Acad. Sci. try, “Stochastic Approach to Plasticity and Yield in
U.S.A.111 , 11668 (2014). Amorphous Solids”, ArXiv: 1509.04907
[18] G. Biroli, C. Cammarota, G. Tarjus, and M. Tarzia,
Phys.Rev.Lett. 112, 175701 (2014).