Probing relevant ingredients in mean-field approaches for the athermal rheology of yield stress materials

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by  Francesco Puosi

Probing relevant ingredients in mean-field approaches
for the athermal rheology of yield stress materials
5 FrancescoPuosi,∗a,b,c JulienOlivier,d andKirstenMartensb,c
1
0
2

g 1 Introduction
u
A Thetheoreticalunderstandingoftheyieldingtransitioninather-
mallydrivendisorderedsystemsisahighlychallengingproblem

6 andnoconsensushasbeenestablishedevenonthebasicingredi-
2
entsthatshouldunderliecoarsegraineddescriptionsofthenon-

] linearrheologicalresponseofyieldstressmaterials1–6.
t
f The only commonly accepted point of view is that disordered
o
materials,suchasglassesorsoftmatter,exhibitastronglyhetero-
s
. geneousdynamicswhendrivenbyanexternalshear. Fastparticle
t
a rearrangements, the so-called shear transformations, take place
m in small regions while the rest of the material deforms elasti- Fig. 1Schematicviewofthecoarsegrainedpicture:Theparticle
simulationboxisdevidedintosmallermesoscopicpartsthatcanhold
- cally7. These plastic events induce long-range elastic deforma-
d exactlyoneplasticevent.Theplasticactivityinthesurroundingofa
n tionsinthesystemleadingtocomplexcorrelationsoftheyielding mesoscopicregion(seegraphicalinterpretationontheright)leadsto
o regionsinformofplasticavalanches8–14. stressfluctuationsaroundthemeanvalueofthestress(dashedlinein
c One of the major concerns remains to know whether despite thegraphontheright),whichiscontrolledbytheexternalforcing.
[
this dynamical complexity it is sensible to describe the yielding
3 dynamics within mean-field descriptions1–3,15,16. We tackle this
v key question using particle based simulations, concentrating on In this work we show that the dynamically created noise in
4
flow responses in a regime that is relevant for many rheological shearedamorphoussystemscanbeencompassedthroughanor-
7
setups17?–26. We find that at high enough shear rates and/or mal diffusion in the local stresses33 (for a schematic view see
5
4 smallenoughsystemsizeswerecoveradynamicswelldescribed Fig. 1). In agreement with former works5,6, we consider that
0 bymean-fieldconsiderations27,similartonearmean-fieldcritical localstress diffusionis actingas noisein thetiltof thelocal po-
1. pointsinequilibriumphasetransitions28,29. tential energy landscape with an amplitude proportional to the
0 An important feature that found its way into several theo- plastic activity, an observable that can be measured experimen-
5 ries, e.g. the Soft Glassy Rheology (SGR) and the Shear Trans- tally21,24.
1
formation Zone Theory (STZ), is to describe yielding in a gen- We analyse within molecular dynamics simulations the cou-
:
v eralized thermodynamic description with an effective tempera- pling strength between the mechanical noise and the plastic ac-
Xi ture1,3.Buttheanalogywiththermallyactivatedeventsproposed tivity. Within our simulations we successfully relate this cou-
bytheSGRtheoryhasrecentlybeenquestionedforathermalrhe- pling strength, a dimensionless and density independent quan-
r
a ology5,6. Also it has been shown that the thermodynamic inter- tity, in a consistent manner to the flow response without any
pretationoftheeffectivetemperatureisproblematicintheather- further parameters. This finding leads us to reconsider a spec-
malregime30,31andthattheSTZtheoryisnotabletopredictthe ulative mean-field scenario put forward a long time ago in the
non-linearresponse32,expectedinthesmalldrivinglimit27. so-called Hébraud-Lequeux (HL) model2, that was at the basis
of important further developments15,22,34, providing so far one
of the best self-consistent mean-field description of mechanical
∗Correspondingauthor:[email protected] noiseinathermallysheareddisorderedsystems.
aLaboratoiredePhysiquedel’ÉcoleNormaleSupérieuredeLyon,UniversitédeLyon,
2 Mean-field approach
CNRS,46Alléed’Italie,69364Lyoncédex07,France.
bUniversitéGrenobleAlpes,LIPHY,F-38000Grenoble,France.
WithintheframeworkoftheHL-modelwecanestablishalinkbe-
cCNRS,LIPHY,F-38000Grenoble,France.
dAixMarseilleUniversité,CentreMathématiquesetInformatique,TechnôpoleChâteau tweenthedynamicalyieldstressandtheprefactorofaHerschel
Gombert,39RueFJoliotCurie,F-13453Marseille13,France. Bulkleytypepowerlawfitoftherheologicalcurvesofathermally
1–8 |1
sheared yield stress materials. This relation should, according
tothistheory, solelybedeterminedbythecouplingstrengthbe- 0.3 density
tweenmechanicalnoiseandtherateofactivity. Inthefollowing 1.149
wereviewbrieflythemainassumptionsandresultsofthemean- σxy1 11..128593
fieldapproach2.Themodeldescribesthestateofasoftglassyma- c 1.322
σ
terialviatheprobabilitydensityP oflocalshearstressesσ=σxy /y
inregionsofmesoscopicsizeW whilethematerialisshearedat σx 10-6 10-.4 10-2
γ
rateγ˙. ThetimeevolutionofP isgivenby
∂tP(σ,t)= −G0γ˙(t)∂σP−τ1θ(|σ|−σc)P 0.2
+Γ(t)δ(σ)+D (t)∂2P (1)
HL σ -4 -2
10 . 10
γ (G τ/σ )
whereθ(x)andδ(x)denoterespectivelytheusualHeavisideand 0 c
delta-distributions. The first term on the right hand side pro-
portional to the stress gradient of the probability density ∂σP Fig. 2Flowcurves:Rescaledmacroscopicshearstressσxy/σcasa
accounts for the linear elastic response. The following term de- functionoftherescaledappliedshearrateγ˙(G0τ0/σc)fordifferent
scribesthelossintheprobabilitydensityduetolocalyieldingof densities(seetext).ThedashedlineisafitwiththeHerschel-Bulkley
overstressed regions above a critical stress σc at a rate given by formσxy=σY+Aγ˙1/2.Theinsetshowsthesamedatabeforerescaling,
dashedlinesdisplayaguidetotheeyewiththesameHerschel-Bulkley
1/τ. Ithasbeenarguedthatthephenomenologicalparameterτ
fittingform.
can be interpreted as the duration of a plastic event in the low
shear rate limit6. The corresponding gain term is given in the
thirdexpressionontherighthandside,wherethestressissetto lessshearrateandcouplingconstantα˜.
zeroafterayieldingeventwitharategivenbytheplasticactivity The rheological results in the small shear rate limit for this
rate model are well studied2,6,35–41. For small enough coupling
1(cid:90)
Γ(t)= P(σ,t)dσ. (2) strength α˜ <1/2 the HL model predicts a Herschel-Bulkley flow
τ |σ|>σc behaviour of exponent 1/2, (cid:104)σ(cid:105)≈σY+Aγ˙1/2, for the time aver-
Thelasttermencompassesthestresschangescreatedthrough aged macroscopic stress in the steady state (cid:104)σ(cid:105)=(cid:82)σP(σ)dσ,
otheryieldingeventsinamean-fieldmanner,assumingthatthis withthetwoconstantsσY (thedynamicalyieldstress)andA(the
mechanical noise can be approximated through a normal diffu- prefactor). These macroscopic constants of the model can only
sionofthemesoscopicstresses. Todescribethisnoiseinanself- depend on the last free control parameter, namely the coupling
consistent manner, the HL approach proposes that its amplitude constant α˜. This means that we will obtain a universal curve∗
√
shouldberelatedtotherateofplasticactivitythroughadimen- fortherescaledquantitiesA/ G0τσc versusσY/σc parametrized
sionlesscouplingconstantα˜ throughα˜ (seeFig.3).
Thephysicalinterpretationofthiscouplingconstantcouldbea
DHL(t)=α˜σc2Γ(t). (3) mechanicalfragility2thatwoulddependonthedetailsofthemi-
croscopicinteractionsandthusshouldbematerialdependent. In
Thislastrelationintroducesanon-linearityintoEq.(1), since
alaterworkbyBocquetetal.,theauthorsuseaspatialapproach
therateofplasticactivityitselfdependsonthedensityprobability
to the problem that allows for the derivation of an expression
of the mesoscopic stresses. It is this coupling that renders the
forthecouplingconstantα˜ ifoneassumesadecorrelationofthe
problem non-trivial and yields interesting results regarding the
plastic event dynamics. Within this approximation it is possible
behaviourofmacroscopicquantities.
toexpressα˜ asafunctionofthespecificformoftheelasticstress
This model is known to exhibit a unique stationary state for propagator15. This suggests that there should be classes of ma-
a finite shear rate in the large time limit, where the probability terialswithsimilarelasticresponsesthatnotonlysharethesame
density for the mesoscopic stresses becomes time independent. non-linearexponentintheflowcurve,butalsocomparablerela-
Todeterminethetimeaveragedmacroscopicstressinthesteady tionsbetweenthedynamicalyieldstressandtheprefactorinthe
stateoneaveragesoverthemesoscopicstressesweightedbythe flowcurve.
correspondingsteadystateprobabilitydensity In the following we aim at probing not only the above con-
(cid:90) stantrelationusingmicroscopicdynamicsimulations,butalsoto
(cid:104)σ(cid:105)= σP(σ)dσ. (4) test the the underlying assumptions, most importantly the self-
consistent description of the mechanical noise induced through
Usingappropriateunitswecanwritetheequationsindimension-
theplasticactivity. Totesttheabovetheoryweneedtomeasure
lessquantities,expressingstressrelatedvaluesinunitsofthelo-
allinvolvedparametersthatappearintheHLdescription.
calyieldstressσc,timequantitiesinunitsofτ,theshearratein Quantities and parameters like the local stress and the shear
units of σc/(G0τ) and the stress diffusion coefficient in units of
σ2/τ,leavingonlytwoindependentdimensionlessmodelparam-
c
eters that determine the flow behaviour, namely the dimension-
∗FortheanalyticalderivationofthisconstantrelationseetheAppendixA
2| 1–8
4 ingthevolumeV ofthesystem. Glassyconfigurationswerepre-
(a)
A (b) pared by quenching equilibrated configurations at T =1 to zero
4
temperaturewithafastcoolingrate. Simpleshearissetatarate
1/2) 3 d11e..11n48s99ity 23 γp˙obsyitidoenfso.rmTihnegqtuheenbcohxindgimpreontsoicoonlshaansdvriretmuaalplypinngo tehffeecptarstinicclee
c
σ 1.253 0 1 σ 2
τ 2 1.322 y we focus on the steady state shear (total imposed deformation
(G0 1/2σ)c00..43 (c) ∆γ>20%).
A/ τ In order to characterize the plastic activity of the system we
1 A/(G000..21 considertheD2minquantity3. Foragivenparticlei,D2minisdefined
astheminimumoverallpossiblelineardeformationtensorsε of:
0.2
σ/σ
00 0.1 0.2 0.3 y c 0.4 0.5 D2(i,t,δt)=∑(cid:2)rij(t+δt)−(I+ε)·rij(t)(cid:3)2 (5)
σ /σ j
y c
where the index j runs over all the neighbors of the reference
particle i and I is the identity matrix. We set the time lag to
Fig. 3HLpredictionsvs.MDsimulations:(a)Thepointsaretherescaled
δt=4. This value is a compromise between having a good sig-
HerschelBulkleyprefactorAasafunctionoftherescaleddynamical
yieldstressσY forfourdifferentdensities,errorbarsareestimatedfrom nal,i.e.largeirreversibledisplacements,andbeingabletoresolve
themeasurementsofthedifferentquantities.Thefulllinecorresponds individualplasticevents.
totheHLprediction.(b)Thisinsetshowsthedatafromthesimulation
fitsbeforetherescaling.(c)Thisinsetshowsazoomintothemain
panel,todisplaybettertheproximitytothetheoreticalcurveobtained 3.1 Macroscopicflowcurve
fromaverysimplifiedpicture. InFig.2weshowthedependenceofthemacroscopicshearstress
σxyontheappliedshearrateγ˙.Theflowcurvesarewelldescribed
bytheHerschel-Bulkley(HB)law,σxy=σY+Aγ˙n,withanexpo-
modulus G are well defined quantities in the microscopic sim-
0 nentn≈0.55(notshownhere)whichseemsnottodependonthe
ulations and rather easy to measure. Also, if stress fluctuations
density. Fixingtheexponentntothevalue0.5,theonepredicted
turn out to be diffusive, D has a well defined meaning. How-
HL by the HL model in the case of α˜ <1/2, gives indistinguishable
ever, other HL parameters like the local yield stress σc, the phe- results. Althoughotherworksonsheareddisorderedmaterialin
nomenologicalparameterτandtherateofplasticactivityΓ(t)are
twoandthreedimensionsreportsimilarvaluesfortheflowcurve
ratherdifficulttointerpretwithinthemicroscopicpicture. Inthe
exponent45,46,recentworksintheliteratureseemtosuggestthat
followingwedescribeourattempttorelatethedifferentparame-
a proper finite size scaling analysis close to the yielding transi-
terstomeasurablequantitiesinthemicroscopicdynamics.
tionrevealsdifferentcriticaldynamicalexponents13,14,27,44. We
3 Microscopic model wouldliketoinsisthereonthefact,thatourstudydoesnotaim
atmeasuringthecriticalexponentsofthetransition; insteadwe
We have investigated a generic two-dimensional (2D) model of
rathertesttheconsistencyoftheassumptionsmadeintheHLap-
a glass, consisting of a mixture of A and B particles, with NA=
proachinaparameterregimethatfitswellthemodelpredictions.
10400 and NB=5600, interacting via a Lennard-Jones potential
V (r)=4ε [(σ /r)12−(σ /r)6] with α,β =A,B and r be-
αβ αβ αβ αβ
ingthedistancebetweentwoparticles. TheparametersεAA,σAA 3.2 Sizeofanelementaryplasticregion
and mA define the units of energy, length and mass; the unit of Herewedescribetheprocedurewefollowedtoconvertthemicro-
(cid:112)
timeisgivenbyτ0=σAA (mA/εAA). WesetεAA=1.0,εAB=1.5, scopicsimulationsintoamesoscopicdescription.First,wedenote
εBB=0.5,σAA=1.0,σAB=0.8andσBB=0.88andmA=mB=1. aparticleiasactive,i.e.performingaplasticrearrangement,ata
Thischoiceisknowntopreventcrystallizationin2Datlowtem- giventimet ifthecorrespondingD2 (i,t)islargerthanathresh-
min
perature42. Thepotentialistruncatedatr=rc=2.5forcompu- oldvaluethatwefixequalto0.1. InFig.4(a)wecompareatypi-
tationalconvenienceandperiodicboundaryconditionsareused. calstress-straincurveandthecorrespondingevolutionofthetotal
The equations of motion are integrated using the velocity Ver- numberNpl ofactiveparticles. Thecorrelationbetweenthestress
let algorithm with a time step δt=0.005. The athermal limit is drops and the peaks in Npl suggests that the actual definition is
achieved by thermostating the system at zero temperature via a reasonable. Thenahigh-resolutiondiscretizationofthesystemis
Langevin thermostat43 with a damping coefficient ζ =1 which performedbydividingitintok×ksquareblocksoflengthw0=2.
corresponds to a strongly overdamped condition for the dynam- A small block is considered as active if it contains at least one
ics44. This model, which has been widely studied in different activeparticle.
versions3,10,44,45, is usually considered appropriate for colloidal The mean size, i.e. the mean linear extension, of a plastic
and other soft glasses. This choice is motived by the purpose of event (cid:104)l (cid:105) can be estimated by a cluster analysis of the spatial
pl
investigatingthegeneralaspectsoftherheologyofathermalyield arrangement of the active blocks in the configurations explored
stressmaterials. by the system. We employed a modified version of the Hoshen-
To investigate different athermal flow responses, we explore Kopelmanalgorithm? inordertoaccountforperiodicboundary
states with different number density ρ =(NA+NB)/V by chang- conditions. Ifweassume(cid:104)lpl(cid:105)=(cid:104)Apl(cid:105)1/2 whereApl istheareaof
1–8 |3
aplasticcluster,weobtain(cid:104)l (cid:105)≈6,withnorelevantdependence testthisideawedefinethecoarsemean-squarestressdifference
pl
onthedensityandshearrate(intherangeγ˙(cid:54)10−4). Thevalue as:
is in accord with previous works reporting plastic regions with (cid:104)(∆σxmy)2(t)(cid:105)=(cid:104)(σxmy(t0+t)−σxmy(t0)−G0γ˙t)2(cid:105) (7)
a size of a few particle diameters45,47. Furthermore, this agree-
where σm(t) is the stress in a given block at time t and the last
xy
mentjustifiesthecriterionweemployedtodefineactiveparticles.
term in Eq. (7) accounts for the stress increase due to the elas-
Indeed, if we improve the resolution in the analysis of the plas-
tic deformation of the system, being G the macroscopic shear
tic activity by decreasing the threshold on D2 by a factor 10, 0
min modulus. In Fig. 4(b), the short time behavior of (cid:104)(∆σm)2(t)(cid:105)
xy
while the number of active particles increase by a factor 4, the
is shown for a finite shear rate γ˙=10−6, approaching the qua-
meanextensionofaplasticevent(cid:104)l (cid:105)isreducedbyhalf,suggest-
pl sistatic limit. We observe that at short times, (cid:104)(∆σm)2(t)(cid:105) in-
xy
ingthatsingleparticlerearrangementsareerroneouslytakeninto
creases linearly with time. We define the stress diffusion coef-
account.
ficient as Dσ =(cid:104)(∆σxmy)2(t)(cid:105)/t. In the inset of Fig. 4(b) we show
Nextweimplementthecoarse-grainingofmicroscopicsimula-
thedependenceofD onthedensityofthesystem.
σ
tionsonthescaleofindividualplasticevents. Thesimulationbox
To define a duration of a plastic event we analyse the two-
is divided into M×M square blocks with M chosen in order to
time autocorrelation function of the D2 quantity for active
haveW ≈(cid:104)lpl(cid:105). Thelocalshearstressσxmy ofablockmisdefined particles. In Fig. 5(a) we show this cormrienlation function Cp =
as:
σxmy=−W12 ∑mvi,xvi,y+2W12 ∑ ∑N ∂V∂(rrijij)rxirjirjyij (6) (cid:104)sDer2mvien(tth0)aDt2mCinp(dt0e+caty)s(cid:105)/e(cid:104)x(Dpo2mnine(nt0t)ia)2ll(cid:105)yawsiathfuancchtiaornacotfertiismtiec.tiWmeeoτbp-,
i∈m i∈mj=1 that depends weakly on density (see the inset of Fig. 5(a)) and
where vi,x and vi,y are the x and y components of the velocity of thatisclosetothedampingtimeτd =ξ−1 oftheLangevinther-
theparticlei,rij isthedistancebetweentheparticlesiand jand mostat. Wechoosetointerpretthisdecorrelationtimeasthetyp-
the summation of i is performed over the particles in the block. icaldurationofaplasticevententeringtheHLmodeldescription.
The macroscopic stress tensor σxy is obtained by the summation
ofσm overalltheblocks.
xy 3.4 Localyieldstress
Inthissection,wepresentamethodwhichallowsustocalculate
(a) a local critical stress, i.e., the stress limit before a plastic rear-
300
G rangementoccurslocally. Forthispurpose,weadoptthe“frozen
0
y 0.5 200N matrix"method48–50. Thesystemisfrozenexceptforatargetre-
σx pl gion,i.e.,amesoscopicblock,andit’ssubjectedtoasimpleshear
100 deformation,withaquasi-staticprotocol. Thefrozenregioncan
onlydeformaffinelywhereasthetargetblockisallowedtorelax
0.45 non-affinely. For small deformations the target region behaves
0 0.002 0.004
∆γ elasticallyandthestressincreaseslinearly,withaslopecontrolled
100 bythelocalshearmodulus50. Asthestrainincreases,theelastic
(b)
~t behaviorgoesonuntilthelocalyieldstressisreachedandaplas-
>
t) 10-2 ticrearrangementtakesplaces. Thisisindicatedbyastressdrop
(
2 D
m)σxy10-4 density 8×10σ-5 iynietlhdesstrtreessssσ-smtrathinecvuarlvuee.oFfotrhaegloivceanlsbtlroecsks,awtethdeefifirnsetamsatxhiemluomca.l
<(∆ 10-6 111...112485993 4×10-5 In Fig. 5(bc) we show the distribution of σcm for the different
10-8 1.322 0 1.2 ρ 1.3 values of the density. First, the fact the local yield stress is dis-
101 102 103 104 tributed is in clear contrast with the HL assumption of a unique
t criticalstressσc. Theexposeddistributionsarewelldescribedby
asimpleGaussianforms. Themeanvalues(cid:104)σm(cid:105)andthevariance
c
Fig. 4Activityandstressdiffusion:(a)Thefulllinerepresentapartof (cid:104)δσcm(cid:105)ofthedistributionsareshownintheinsetofthefigure. As
themacroscopicstress-straincurveinthesteadystateregimefora thedensityincreases,(cid:104)σm(cid:105)increases,duetotheenhancementof
c
densityρ=1.149andashearrateγ˙=10−6.Theaverageslopeofthe therepulsiveinteractionsbetweenparticles.
elasticpartsonthecurveyieldsG0.Opensymbols:corresponding
With the frozen matrix method we estimated also the stress
evolutionofthenumberofactiveparticlesNpl inthesystem.(b)Coarse
mean-squarestressdifference(cid:104)(∆σm)2(t)(cid:105)asafunctionoftimefor releasefollowingalocalyieldingevent. Weobservethattheplas-
xy
differentvaluesofthedensity.Notethelinearbehaviouratshorttimes. ticeventonlypartiallyrelaxestheaccumulatedstressincontrast
Inset:stressdiffusioncoefficientDσ=(cid:104)(∆σxmy)2(t)(cid:105)/tasafunctionof with the assumption of the HL model of a complete relaxation.
density.
Thefractionoftherelaxedstressseemsnottodependondensity
being(cid:104)∆σm(cid:105)/(cid:104)σm(cid:105)≈0.2(seeinsetofFig.5).
c
In a former work50 it was shown that the mean value of the
3.3 Stressdiffusionanddurationofaplasticevent shearmodulusobtainedwiththefrozenmatrixmethoddepends
Theaboveintroducedmean-fieldmodelassumeslocalstressfluc- on the target region size W and that it converges, from higher
tuations, obeying a normal diffusion process in stress space. To values,tothemacroscopicmodulusasW increases. Thisisdueto
4| 1–8
1 tion of the mechanical noise, thus predict a diffusion coefficient
(a)
1.3 proportional to the square of this typical local stress jump ∆σHL
densities dividedbyatypicaltimescale,givenbytheinverseplasticityrate
t) 1.149 τ 1.2 Γ(t).
(p0.5 1.189 p
C 1.253 (a)Ourdataanalysisrevealsthatahomogeneousyieldstressis
1.322 1.1
ofcoursenotverifiedinadisorderedsystem,whereoneexpects
1.2ρ 1.3 adistributionofyieldstresses(seeFig.5b). However,ithasbeen
0 recentlyshownthattheexistenceofalocalyieldstressdistribu-
0 5 10 15
t tionintheHLdynamicsdoesnotstronglyalterthepredictionsfor
theflowbehaviour6.
0.6 (b) 10 2 > (b)Wetestedaswellthesecondassumptionofanormaldiffu-
m)c m><σc m><δσcm><∆σ mm>/<∆σσc soifosnhoefarthreatmesestohsactowpieccsotrnessisdeesr,,atnhdewmeeafisnudretmhaetnwtiothfitnhethaevrearnaggee
σ 0 0 <
( 0.3 1.2 ρ 1.4 1.1 1.2ρ1.3 mean-square stress differences indeed allows for the determina-
P
tionofawell-defineddiffusioncoefficient. TheHLcouplingcon-
stant α˜mHiLcro=Dσ/(σc2Γ) (see Eq. (3)) with Dσ from the results
0 in Fig. 4(b) and Γ from the cluster analysis turns out to be very
0 5 10 15
m small,withα˜HL oforder10−2forallconsidereddensities.
σ micro
c (c) However, our study suggests that the partial relaxation of
thelocalstressesafteryielding,(cid:104)∆σm(cid:105)/(cid:104)σm(cid:105)≈0.2(insteadofthe
c
Fig. 5Durationofeventsandlocalyieldstress:(a)D2 autocorrelation assumedtotalrelaxation)introducesanimportantcorrectivefac-
min
functionforactiveparticlesCp=(cid:104)D2min(t0)D2min(t0+t)(cid:105)/(cid:104)(D2min(t0))2(cid:105)asa torforthecouplingconstant. Sincethetypicalstressjumpinthe
functionoftime.Linesarefittingcurvesexp(−t/τp).Inset:dependence diffusion process is now reduced, we obtain for the stress diffu-
ofτponthedensity.(b)Distributionofthelocalyieldstressp(σYm)for sioncoefficientanalteredexpression
differentvaluesofthedensity.FulllinesareGaussiandistributionsfits.
Leftinset:Meanvalue(cid:104)σm(cid:105)andvariance(cid:104)δσm(cid:105)asafunctionofdensity.
Dashedlinesarepower-lacwguides.Rightinsect:Absolutelocalstress Dσ =α˜micro(∆σm)2Γ. (8)
drop(cid:104)∆σm(cid:105)andtherelativelocalstressdrop(cid:104)∆σm(cid:105)/(cid:104)σm(cid:105)asafunction
c
Wereportthevaluesobtainedforthisnewdefinitionofthecou-
ofdensity.Dashedlinesarepower-lawandconstantvalueguidestothe
eye.Symbolcodeisthesameasinpanel(a). plingconstantinTable1.Itturnsoutthatthesevaluesareingood
approximationdensityindependentandyieldanaveragevalueof
aboutα˜ ≈0.26. Thusthecouplingconstantdisplaysareasonable
the frozen environment which reduces the non-affine motion of valuesmallerthan0.5forwhichtheHLmodelpredicts6 awell-
theparticlesinthetargetregion. ForW ≈6thediscrepancyisup established Herschel Bulkley regime in the flow curve with an
to50%,asignificanterror.Weareawarethattheestimateof(cid:104)σm(cid:105) exponentof1/2.
c
maybeaffectedbythiseffect. Alsowehavetobecarefulinthe And indeed our flow curves from the microscopic simulations
interpretation of the result, because by using the frozen matrix can be well fitted with an Herschel Bulkley expression σxy =
method we measure the yield stress distribution, obtained from σy+Aγ˙1/2 (see inset of Fig. 2). In Fig. 2 we plot our data in
strainedconfigurations. Itwouldbeinterestingtotrytoinferthe dimensionless units, rescaled as suggested by the HL model, us-
inherent yield stress distribution from our measurements6, but ing dimensionless quantities σxy/σc and γ˙(G0τ/σc). The shear
thisisbeyondthescopeofthisstudyandleftforfurtherinvesti- modulus G0 can be easily accessed in simulations as the slope
gations. of the stress-strain curve in the elastic regime. We approximate
4 Robustness of the HL model the critical stress σc as the mean value (cid:104)σcm(cid:105) of the local yield
stress obtained with the frozen matrix method and we associate
InthefollowingweaimatprobingtherobustnessoftheHLmodel thephenomenologicalparameterτ withthedurationofaplastic
by comparing our data from the microscopic simulations to the eventinthelowshearratelimit6. Hence,wewillapproximateτ
various assumptions and predictions of the mean-field descrip- bythevalueofτpobtainedthroughthemeasurementoftheD2min
tion. two-time autocorrelation function. We observe a collapse of the
Letusfirstrecallsomeofthebasicassumptionsmadetowrite flow curves for different densities. This collapse is not perfect,
theevolutionEq.(1)fortheprobabilitydistributionofthemeso- butregardingthelargeerrorbarsintroducedbyourmethodsand
scopicstress:First(a)theyieldinghappensataconstanthomoge- estimationsitappearsstillconvincingandsuggestaverygeneric
neousyieldstressσc. Then(b)thelocalresponsetosurrounding flowbehaviour.
plastic events is encompassed through local diffusive stress fluc- One of the strongest prediction of the HL model is that both
tuationswithawelldefineddiffusioncoefficientDHL(t). Andthe the prefactor A as well as the dynamical yield stress σy are de-
thirdassumption(c)concernstherelaxationofthelocalstressto terminedsolelybythespecificvalueofthecouplingconstantbe-
zero,onceasiteyields,leadingtoatypicalstressjump∆σHL≈σc. tween diffusion and plastic activity. In Fig. 3 we compare the
The HL model assumptions, aiming for a self-consistent descrip- dimensionless yield stress σy/(cid:104)σcm(cid:105) and prefactor A/(G0σcτ)1/2,
1–8 |5
density α˜micro/10−1 α˜flow/10−1
1.149 2.81±0.21  3.21±0.12 
1.189 2.47±0.27  3.30±0.19 
2.57±0.26 3.28±0.16
1.253 2.27±0.24  3.28±0.18 
1.322 2.74±0.30 3.34±0.18
Table1Comparisonofdifferentcouplingconstantmeasurementsasexplainedinthetext.
obtainedfromaHerschelBulkleyfitoftheflowcurves,withthe originalversionofHLmodelequationssuchasthepartialrelax-
analytically obtained parametric curve. We observe that all the ationofthelocalstressafterayieldingeventandtheintroduction
data belonging to different density values collapse as expected ofayieldstressdistributionobtainedfromthemicroscopicsimu-
by the previous collapse of the data roughly onto a single point lations. Inafutureworkweplantotestthecoherenceofsucha
close to the theoretical curve, that lies within the estimated er- modifiedmodelwithourmicroscopicapproach. Furtheritwould
ror. This result points to a universal determination of the flow behighlydesirabletotestthedegreeofgeneralityoftheresults
curves,aprioristronglysensitivetothedensity(notethedensity onothermodelsystems46andexperimentalsetups,thathaveac-
dependenceofAandσyintheinsetofFig.3),throughonesingle cesstothemeasureoflocalplasticactivity21,23–26.
densityindependentparameter,namelythecouplingconstantα˜. Acknowledgments KM acknowledges financial support of the
AssuggestedbytheKEPmodel15,weexpectthisparametertobe French Agence Nationale de la Recherche (ANR), under grant
onlydependentonthespecificformoftheelasticpropagator9. ANR-14-CE32-0005 (project FAPRES) and FP acknowledges fi-
If we assume the HL parametric relation between prefactor nancial support from ERC grant ADG20110209. Most of the
A(α˜) and dynamical yield stress σy(α˜) to hold, it is possible to computations were performed using the Froggy platform of the
estimate the coupling constant α˜ in an alternative way through CIMENT infrastructure supported by the Rhône-Alpes region
the macroscopic measure of the flow curve, in the following re- (GRANT CPER07-13 CIRA) and the Equip@Meso project (refer-
ferredtoasα˜flow.Despitealltheroughapproximationswehadto enceANR-10-EQPX-29-01). FurtherwewouldliketothankElis-
make,thattendtointroducelargeerrorbarsonthedata,wefind abethAgoritsas,EricBertinandJean-LouisBarratforfruitfuldis-
thecomparisonbetweenthecouplingconstantα˜microwiththeal- cussionsandvaluablerevisionsofourmanuscript.Alsowewould
ternativemeasurementusingtheexpressionfortherescaledyield liketothankJörgRottlerforassistanceintheclusteranalysisof
stress† σy/σc(α˜) rather convincing (see table 1). Both measure- plasticevents.
ments suggest a density independent result with a quite small
Appendix A
relativeerrorofapproximately20%.
Altogether our data suggests that revisiting the rule of setting
The parametric relation between the yield stress and the HB
thestresstozeroafterayieldeventintheHLmodelequations,by
prefactor
changingthegaintermintheevolutionEq.(1)togetherwiththe
In this section, we compute the relation between the dynamic
introductionofamorerealisticyieldstressdistribution,seemsto
beapromisingroutetoreachamorerealisticmean-fieldmodel- yieldstressσY andtheprefactorAintherelation(cid:104)σ(cid:105)=σY+Aγ˙1/2
derived from the HL model (as given in Fig. 3). We use the
ingofathermallyshearedamorphoussystems.
methoddevelopedin38: theprobabilitydensityfunctionsolving
5 Conclusion the(stationary)HLequation(2)ofthepaperisexpandedinthe
followingway
Inthisstudyweaimedattestingsomeofthemostbasicassump-
tionsandpredictionsofmean-fieldmodelingfortherheologyof P(σ)=Q0(σ)+γ˙1/2Q1(σ)+···, forσ in[−σc,σc] (9)
athermallyshearedamorphoussystems. Inconclusionweobtain
anocisoensiinsttehnet rpeigctimureeoofflharogwe teonomuoghdedlrcivoirnrgecrtlaytetsh(efamrefcrohmanitchael P(σ)=γ˙1/2R1±(cid:18)|σγ˙|1−/2σc(cid:19)+···, for±σ in[σc,+∞[ (10)
truecriticalpoint27). Wefindthatwecanincorporatethenoise
This ansatz has been proved to be correct in the case where
into a normal diffusion of local stresses with a noise amplitude
solelygovernedbytherateofplasticactivityasproposedbythe α <σc2/2 (again see38). Moreover, term-by-term integration is
Hébraud-Lequeux (HL) model2. We not only confirm this phys- allowedwhichmeansthat
ical picture using molecular dynamics simulations, but we also
showthatthecouplingstrengthbetweendiffusionandtherateof (cid:90) (cid:18)(cid:90) (cid:19)
(cid:104)σ(cid:105)= σP(σ)dσ= σQ0(σ)dσ
plasticactivity,adimensionlessanddensityindependentquantity,
(cid:124) (cid:123)(cid:122) (cid:125)
seemstodeterminethespecificformoftherheologicalresponse.
σY
Our data analysis suggest some important modifications in the
(cid:18)(cid:90) (cid:19)
+ σQ1(σ)dσ γ˙1/2+··· (11)
(cid:124) (cid:123)(cid:122) (cid:125)
†giveninAppendixA A
6| 1–8
wherethedotsaretermsofhigherorderthanγ˙1/2. Nowallthat In this system d˜is an unknown coefficient to be simultaneously
is left to do is to identify Q0 and Q1. By plugging (9) in (1) we computedwithQ1.ThecomputationofQ1istediousbutstraight-
obtainthefollowingequationonQ0: forward. The expression of Q1 are quite lengthy but can be
 checkedoutonasymboliccomputationprogram: if−σc≤σ≤0
−d∂σ2Q0+τG0∂σQ0= αdδ0 then
Q(cid:82)−σ0σc(c±Qσ0c()σ=)d0σ=1 (12) Q1(σ)= σc√d(cid:16)1τG−0e−G0dτσc(cid:17)eGd0τσ
Note that in this equation d is an unknown coefficient used to
ednifffourscioenthceoeinffitecgieranltcboynτdDiti∼ondoγ˙n. Q0;itisphysicallyrelatedtothe −d2σc(cid:16)1d˜G−0eτ−G0dτσc(cid:17)(σ+σc)eGd0τσ
ThissystemcaneasilybeintegratedandwefindQ0tobe
1 (20)
+ ×
Q0(σ)= σc(1−1e−σcdG0τ)(cid:16)1e−σeGd0(τσ−−σdc)eG−0τσcdτG0(cid:17) −0≤σcσ≤≤σσ≤c0 (13) (cid:32)eGσd0τcσGdc0−τe(cid:32)−Gdd0dστ˜σcc−√σc2d(cid:33)+ddσ˜c(cid:16)1+e−Gd0τσc(cid:17)(cid:33)×
anddisselectedsothatthefollowingequationholdstrue:
α = d tanh(cid:18)σcG0τ(cid:19) (14) (cid:16)eGd0τ(σ+σc)−1(cid:17)
σc2 σcG0τ 2d andif0≤σ≤σc
Nonoltyeitfha0t<thαis/σeqc2u<ati1o/n2.(inHdow) hevaesr,aiunnviqieuwe sooflu(t1io3n),di(tαis) iefaasinedr Q1(σ)= σc√d(cid:16)1τ−G0e−G0dτσc(cid:17)eGd0τ(σ−σc)
to express everything in terms of the parameter d instead of α:
tehqeunivtahleenltimtoitdα→/σ0c2.→1/2isequivalenttod→+∞andα→0is +d2σc(cid:16)1d˜−G0eτ−G0dτσc(cid:17)(σ−σc)eGd0τ(σ−σc)
NowusingtheformulaforcomputingσY weobtainbyintegra-
tion,
σσYc = 12coth(cid:18)σc2Gd0τ(cid:19)−σcGd0τ (15) +eGd0τσc−1e−Gd0τσc×
tatTeostohbetacionmtphuetpatrieofnacotofrR1±A.wUesimngusthtecocmonptuinteuiQty1owf∂hσicPhnaetc±esσsci-, (cid:32)−σcGd0τ(cid:32)ddσ˜c−√σc2d(cid:33)+ddσ˜c(cid:16)1+eGd0τσc(cid:17)(cid:33)×
onecanfindthatR1(z)arefunctionssatisfying
±
(cid:16)1−eGd0τ(σ−σc)(cid:17)
−d∂2R1 +R1 =0
∂zR1+z(0±)=∂σ±Q0(σc)=− τG−0σcτG0 (16) (21)
dσc(1−e d ) Theparameterd˜isselectedtoenforcethevanishingintegralcon-
−∂zR1−(0)=∂σQ0(−σc)= dστcG(10−ee−−σcσdτcGτd0G0) ditionwhichamountstotaking
Again,itiseasytosolvethissystem: d˜=−d3/2eτGd0σc −e−τGd0σc +2τGd0σc (22)
R1+(z)= √dσc(1τ−G0e−σcdτG0)e√−dz (17) σc eτGd0σc −e−τGd0σc −2τGd0σc
R1−(z)= √dστcG(10e−−σecd−τGσ0cτdG0)e√−dz (18) Finally we can compute the prefactor A which is equal to
NowwecanwritedowntheequationssatisfiedbyQ1usingthe (cid:82)σQ1(σ)dσ. Allinall,weobtain(letusnoteu=(τG0σc)/d):
continuityofP atσ=±σc: σY 1 1
= coth(u)− (23)
 σc 2 u
Q−1d(∂σσ2cQ)1=+RG1+0(τ0)∂σ=Q√1d=σcd(˜1(τ−αG1e0δ−−0σσccd+ττGG00∂)σ2Q0) (19) √σcAτG0 = 2√1ueu(ucosuh((s2iun)h+(22u)si−nhu()2u)+u) (24)
Q(cid:82)σ1c(−Qσ1c()σ=)dRσ1−=(00)= √dτσGc(01e−e−dσcτdG0) −2√1uu(6sui2nhco(2thu()2−u)u)
−σc
1–8 |7
References 25 A.LeBouil,A.Amon,S.McNamaraandJ.Crassous,Physical
ReviewLetters,2014,112,246001.
1 P. Sollich, F. Lequeux, P. Hebraud and M. E. Cates, Physical
26 K. E. Jensen, D. A. Weitz and F. Spaepen, Physical Review E,
ReviewLetters,1997,78,2020–2023.
2014,90,042305.
2 P.HébraudandF.Lequeux,PhysicalReviewLetters,1998,81,
27 C. Liu, E. Ferrero, F. Puosi, J. Barrat and K. Martens,
2934–2937.
arXiv:1506.08161,2015.
3 M.L.FalkandJ.S.Langer,Phys.Rev.E,1998,57,7192–7205.
28 M. Kac, G. E. Uhlenbeck and P. C. Hemmer, J. Math. Phys.,
4 D.Rodney,A.TanguyandD.Vandembroucq,ModellingSimul.
1963,4,216.
Mater.Sci.Eng.,2011,19,083001.
29 W.Klein,H.Gould,N.Gulbahce,J.B.RundleandK.Tiampo,
5 A.Nicolas,K.MartensandJ.L.Barrat,Epl,2014,107,6. PhysicalReviewE,2007,75,031114.
6 E.Agoritsas,E.Bertin,K.MartensandJ.-L.Barrat,TheEuro- 30 N. Xu and C. O’Hern, Physical Review Letters, 2014, 94,
peanPhysicalJournalE,2015,38,71. 055701.
7 A.S.Argon,ActaMetallurgica,1979,27,47–58. 31 J.PuckettandK.Daniels,PhysicalReviewLetters,2013,110,
8 J.D.Eshelby,Proc.R.Soc.LondonA,1957,241,376. 058001.
9 F. Puosi, J. Rottler and J. L. Barrat, Physical Review E, 2014, 32 J.Langer,arXiv:1501.07228,2015.
89,042302. 33 S. Karmakar, E. Lerner, I. Procaccia and J. Zylberg, Physical
10 A.LemaîtreandC.Caroli,Phys.Rev.Lett.,2009,103,065501. ReviewE,2010,82,031301.
11 K.Martens,L.BocquetandJ.-L.Barrat,Phys.Rev.Lett.,2011, 34 V.Mansard,A.Colin,P.ChauduriandL.Bocquet,SoftMatter,
106,156001. 2011,7,5524–5527.
12 K.M.SalernoandM.O.Robbins,PhysicalReviewE,2013,88, 35 Y.Gati,PhDthesis,EcoledespontsParisTech,2004.
062206. 36 E.Cances,I.CattoandY.Gati,SIAMJ.Math.Anal.,2006,37,
13 Z. Budrikis and S. Zapperi, Physical Review E, 2013, 88, 60.
062403. 37 E. Cances, I. Catto, Y. Gati and C. L. Bris, Multiscalce Model.
14 J. Lin, E. Lerner, A. Rosso and M. Wyart, Proceedings of the Simul.,2006,4,1041.
National Academy of Sciences of the United States of America, 38 J.Olivier,PhDthesis,UniversitédeGrenoble,2011.
2014,111,14382–14387.
39 J.Olivier,Z.Angew.Math.Phys.,2010,61,445.
15 L. Bocquet, A. Colin and A. Ajdari, Physical Review Letters,
40 J. Olivier and M. Renardy, SIAM J. Appl. Math., 2011, 71,
2009,103,036001.
1144.
16 K.A.Dahmen,Y.Ben-ZionandJ.T.Uhl,NaturePhysics,2011,
41 J.Olivier,ScienceChina-Mathematics,2012,55,435–452.
7,554–557.
42 R.Brüning,D.A.St-Onge,S.PattersonandW.Kob,J.Phys.:
17 I.CantatandO.Pitois,JournalofPhysics: CondensedMatter,
Condens.Matter,2009,21,035117.
2005,17,S3455.
43 T.SchneiderandE.Stoll,Phys.Rev.B,1978,17,1302–1322.
18 L. Bécu, S. Manneville and A. Colin, Physical review letters,
44 K. Salerno, C. Maloney and M. Robbins, Physical Review Let-
2006,96,138302.
ters,2012,109,105703.
19 J.Goyon,A.Colin,G.Overlez,A.AjdariandL.Bocquet,Na-
45 A.Nicolas,J.RottlerandJ.-L.Barrat,Eur.Phys.J.ESoftMat-
ture,2008,454,84–87.
ter,2014,37,37–50.
20 J.Goyon,A.ColinandL.Bocquet,SoftMatter,2010,6,2668–
46 C.Fusco,T.AlbaretandA.Tanguy,EuropeanPhysicalJournal
2678.
E,2014,37,43.
21 A. Amon, V. B. Nguyen, A. Bruand, J. Crassous and
47 A.Tanguy,F.LeonforteandJ.-L.Barrat,Eur.Phys.J.E,2006,
E.Clement,PhysicalReviewLetters,2012,108,135502.
20,355–364.
22 K. Kamrin and G. Koval, Physical Review Letters, 2012, 108,
48 P.SollichandA.Barra,Inpreparation.
178301.
49 P. Sollich, School on Glass Formers and Glasses, Bangalore,
23 K.W.Desmond,P.J.Young,D.D.ChenandE.R.Weeks,Soft
India,2010.
Matter,2013,9,3424–3436.
50 H.Mizuno,S.MossaandJ.-L.Barrat,Phys.Rev.E,2013,87,
24 E.D.Knowlton,D.J.PineandL.Cipelletti,SoftMatter,2014,
042306.
36,6931–6940.
8| 1–8