Nonequilibrium dynamics and fluctuation-dissipation relation in a sheared fluid

17 Pages·0.76 MB·English

by  Ludovic Berthier

Nonequilibrium dynamics and fluctuation-dissipation relation in a sheared fluid
Ludovic Berthier1 and Jean-Louis Barrat2
1CECAM, ENS-Lyon, 46, All´ee d’Italie, 69007 Lyon, France
2D´epartement de Physique des Mat´eriaux, UCB Lyon 1 and CNRS, 69622 Villeurbanne, France
(February 1, 2008)
The nonequilibrium dynamics of a binary Lennard-Jones mixturein a simple shear flow is investi-
2 gatedbymeansofmoleculardynamicssimulations. TherangeoftemperatureT investigatedcovers
0 both the liquid, supercooled and glassy states, while the shear rate γ covers both the linear and
0 nonlinear regimes of rheology. The results can be interpreted in the context of a nonequilibrium,
2 schematic mode-coupling theory developed recently, which makes the theory applicable to a wide

n range of soft glassy materials. The behavior of the viscosity η(T,γ) is first investigated. In the
a nonlinear regime, strong shear-thinning is obtained, η ∼ γ−α(T), with α(T) ≃ 2/3 in the super-
J cooled regime. Scaling properties of the intermediate scattering functions are studied. Standard

4 ‘mode-couplingproperties’offactorization andtime-superposition holdinthisnonequilibriumsitu-
1 ation. Thefluctuation-dissipationrelation isviolatedintheshearflowinawayverysimilartothat
predictedtheoretically,allowingforthedefinitionofaneffectivetemperatureT fortheslowmodes
eff
]
of thefluid. TemperatureandshearratedependenciesofT arestudiedusingdensityfluctuations
t eff
f asanobservable. TheobservabledependenceofT isalsoinvestigated. Manydifferentobservables
o eff
arefoundtoleadtothesamevalueofT ,suggestingseveralexperimentalprocedurestoaccessT .
s eff eff
t. Itisproposedthatatracerparticleoflargemassmtrmayplaytheroleofan‘effectivethermometer’.
a WhentheEinstein frequencyof thetracersbecomessmaller thantheinverserelaxation timeofthe
m fluid,anonequilibrium equipartition theorem holdswith m v2 =k T , wherev isthevelocity
tr z B eff z
- in the direction transverse to the flow. This last result gives strong support to the thermodynamic
(cid:10) (cid:11)
d interpretation of T and makes it experimentally accessible in a very direct way.
eff
n
o PACS numbers: 64.70.Pf, 05.70.Ln, 83.60.Df
c
[ L’immense thermom`etre refl´etait modestement la temp´erature du milieu ambiant.
Vassili Axionov,L’ˆıle de Crim´ee.
2
v
2 I. INTRODUCTION ties with other, more standard, glassy systems.
1
By submitting the system to a homogeneous, steady
3
1 Glassesareusuallydefinedbythefactthattheirinter- shear flow, a different kind of nonequilibrium situation
1 nal relaxation time is larger than the experimental time is obtained. The steady shear flow, characterized by the
1 scale. In simple molecular systems, the associated glass shear rate γ, creates a nonequilibrium steady state, in
0 transition temperature corresponds to a very high vis- which time translation invariance is recovered12. The
t/ cosity, making it difficult to investigate experimentally shear flow can therefore be used as a probe of the glassy
a the rheological properties of glassy systems. In complex system, with the convenient feature of having the shear
m
fluids (e.g. colloids, emulsions) it is however possible to rate γ rather than the waiting time tw as a control pa-
- reach a glassy situation, in the sense of large relaxation rameter. Infact,theinverseshearrateintroducesatime
d
n times, with systems having viscosities or shear moduli scale in the problem, which plays a role similar to tw.
o that allow for a rheological investigation1. Such materi- Interesting phenomena are thus expected to show up as
c als have been described as ‘soft glassy materials’ in the soonasthis newtime scalecompeteswiththe relaxation
v: literature2. time of the fluid. As a result, it is also interesting to
i When it is quenched into its glassy state, a material study the ‘supercooled’ regime, which would correspond
X is by definition out of equilibrium. An important fea- to an equilibrated situation in the absence of a shear
r ture of this nonequilibrium situation is the absence of flow. Moreover, this way of probing the nonequilibrium
a
time translation invariance. Physical properties are a propertiesofglassysystemsispossiblymorerelevantex-
function of the time spent in the glassy phase, or wait- perimentallythantheagingapproach,atleastinthecase
ing time t . This is best seen through the measurement of soft glassy materials.
w
of time-dependent correlationswhich depend both on t Recently, a general scenario was proposed for glassy
w
and on the time difference. These dependencies on t systems subject to an external forcing13, based on the
w
are usually described as aging phenomena3. They have study of simple ‘mean-field’ models. The rationale of
been studied experimentally and theoretically in great this approach is that the equilibrium dynamics of these
detail4. Interestingly for the present work, the aging be- models is equivalent to the ‘schematic’ mode-coupling
havior of severalcomplex fluids has recently been exper- approach of slowing down in supercooled liquids14–16.
imentally investigated5–11, showing interesting similari- The study of their nonequilibrium dynamics can thus be
1
seen as a nonequilibrium schematic mode-coupling ap- studiedunderconditionsthatinvolvedaconstantenergy
proach17. In this respect, it is interesting to note that input from an external driving force, analogous to the
the mode-coupling theory was recently extended to the effect of a shear flow on a fluid. The resulting station-
aging regime18 with results which were first predicted arynonequilibriumstatewasstudiedasafunctionofthe
from the study of mean-field models19,20. To our knowl- temperature T andexternaldrive,the intensity ofwhich
edge, the mode-coupling theory of supercooled fluid has will be, for convenience, denoted by σ.
not yet been extended to fluids under shear beyond the For σ =0, the model has an ideal glasstransition at a
linear response of the supercooled regime, i.e. at equi- finitetemperatureT wheretherelaxationtimediverges.
c
librium. However, in analogy to what was done for ag- For T > T , the correlation functions are described by
c
ing or supercooled systems, it seems sensible to bypass usualmode-couplingintegro-differentialequations15,and
thisaspectandtocarryoutadirectcomparisonbetween display the characteristic two-step relaxation predicted
the predictions from mean-field theories and experimen- by these equations, the α- and β-relaxations. Below T ,
c
tal or simulation results, with the hope that the general thesystemcannotequilibrate,anddisplaysagingbehav-
scenario is robust enough that the details of the system ior20.
under study are relatively unimportant. Under a finite external drive, σ 6= 0, the system is
Severalearlierstudieshavebeendevotedtosimulating stationary at all temperatures. For T <∼ Tc, and in the
fluid under shear. Onuki and Yamamoto21 concentrated asymptotic limit σ → 0, the correlation functions re-
their worksonthe understanding of the dynamics at the tain the characteristic two-step shape of the equilibrium
molecular level. Our study is focused on more global system, with an α-relaxation time that depends on σ.
aspects, with the aim of providing some experimentally For T < T , this relaxation time diverges as σ → 0,
c
testable predictions. Liu and Nagel22 also proposed to while for T > T it goes to its equilibrium value in this
c
use the shear rate as a relevant control parameter for limit. Although the driving force strongly influences the
jammingsystems. Liuandcoworkers23investigatedsome α-relaxation,itdoessobykeepingtheshapeofthedecay
aspects of the nonequilibrium dynamics of a model sim- of the correlationunchanged. This leads to the property
ilar to ours, with the difference of being athermal (zero that correlation functions may be collapsed by a simple
temperature) in the absence of shear. Some of their re- rescaling of the time. This scaling property is analogous
sults are closely related to ours, as will be discussed in tothetime-temperaturesuperpositionpropertyfoundat
Sections IV and VI. More phenomenological approaches equilibrium,andweshalldenotethispredictionby‘time-
may be found in Refs.2,10,24. shear superposition property’.
The aim of the present work is to check, on a realistic Based on a power dissipation argument, we were able
model of the fluid, some of the predictions that emerged for this simple system to define quantities equivalent
from the earlier study of driven glassy systems13. For to the viscosity and shear rate in a fluid under shear.
completeness,themainresultsofthisstudywillbebriefly This viscosity was found to have a characteristic shear-
recalled in section II. In section III, we describe our mi- thinning behavior, decreasing for increasing shear rates
croscopic model for the fluid under shear. Sections IV as a power law. The shear-thinning exponent is equal to
and V describe our results for the dynamic properties 2/3 for T > T , while for T < T , the viscosity diverges
c c
of the fluid, both at the macroscopic (rheological) and as σ → 0, and the shear-thinning exponent appears to
at the microscopic (wavevector dependent correlations) depend on temperature with a value between 2/3 and 1.
scales. Sections VI and VII are devoted to one of the The behavior of the effective temperature was also in-
most important predictions of the mean-field scenario, vestigated as a function of T and σ. We recall here that
namelythe mannerinwhichtheequilibriumfluctuation- the effective temperature T is defined, in a system in-
eff
dissipation theorem is violated in such nonequilibrium variant under time translation, by the relationship19
systems and the resulting notion of an effective temper-
ature. Section VIII proposes a numerical realization of 1 dC(t)
R(t)=− . (1)
a very simple experimental protocol to measure the ef- k T (C) dt
B eff
fective temperature through a nonequilibrium general-
ization of the equipartition theorem. We discuss exper- Here R(t) and C(t) are, respectively, a response func-
imental and theoretical consequences of our results in tion and the associated correlation function. In equilib-
Section IX. rium, the fluctuation-dissipation theoremcan be written
asT (C)=T. Inthe shearedsystem,the effectivetem-
eff
perature was found, for T < T , to be a discontinuous
c
II. SUMMARY OF THE NONEQUILIBRIUM function of C in the limit σ → 0. For C > q, one has
MODE-COUPLING RESULTS Teff = T while for C < q, one finds Teff(C) = Teff > T,
denoting an effective temperature larger than the bath
To make the paper self-contained, we briefly recall in temperature for long time scales19. The parameter q is
this Section the main results and predictions obtained the plateau value of the correlation function: it is called
within the mean-field approach of Ref.13. In this pa- theEdwards-Andersonparameterintheliteratureofdis-
per, a simple system, the p-spin mean-field model, was ordered systems, or nonergodicity parameter in the lan-
2
guage of mode-coupling theory. At finite shear, the ef- called SLLOD equations31
fective temperaturealsohasanalmostdiscontinuousbe-
dr p
havior as a function of C. For T >Tc, the discontinuity i = i +ri·∇v,
decreases and vanishes when σ →0, so that equilibrium dt mi
(3)
behaviour is recoveredin this limit. dpi =− ∂V(rij) −p ·∇v,
An important outcome of theoretical studies of the dt ∂r i
ij
concept of effective temperature for the slow modes of Xj6=i
a glassy system, is that the value of the effective tem-
where (p ,r ) are the momentum and position of parti-
i i
peratureisindependent,atthemean-fieldlevel,ofwhich
cle i, respectively. These equations are integrated by a
observableischosentocomputecorrelationandresponse standardleapfrog algorithm32, where time is discretized.
functions18,19,25,26. This prediction was recently chal-
The value ∆t = 0.01 was used throughout the simula-
lenged within a simple trap model27, with negative re- tions. Lees-Edwards boundary conditions32 are used in
sults. We will see in Section VI that this important pre-
a cubic simulation box of linear size L = 13.4442. With
diction is nicely verified in our realistic model.
these boundary conditions the flow is homogeneous and
The simple mean-field model studied in Ref.13 does
no instability, such as e.g. shear-banding, was observed
notallowanydiscussionofspatialdependency. However, in our simulations33,34. Note also that this is the shear
characteristic features of the spatial dependence usually
rate γ which is controlled in our simulations. The veloc-
expected from the solution of mode-coupling like equa-
itygradientisinthey direction,andthe fluidvelocityin
tions will be investigated in Section VC.
the x direction, i.e. v = γy e . Constant temperature
x
conditions are ensured by a simple velocity rescaling of
the z component of the velocities, at each time step35.
III. MODEL AND DETAILS OF THE
Theshearrateγ naturallyintroducesanewtimescale
SIMULATION γ−1 into the problem. Obviously, a simulation involv-
ing a steady shear state is possible only if the available
The system simulated in this work is a 80:20 mixture simulationtimeissignificantlylargerthanγ−1. Thislim-
of N = 2916 Lennard-Jones particles of types A and B, its our study to shear rates larger than typically 10−4τ ,
0
with interaction correspondingto 106 time steps. The temperature range
studiedhereisT ∈[0.15,0.6],whichcoverstheliquidand
12 6
V(r )=4ε σαβ − σαβ , (2) glassy phases, while the shear rates are γ ∈ [10−4,0.1]
αβ αβ"(cid:18)|rαβ|(cid:19) (cid:18)|rαβ|(cid:19) # which covers the linear and nonlinear response regimes
ofrheology. Allthedatapresentedinthe following,even
whereαandβ refertothetwospeciesAandB. Interac- those below T , are obtained in a stationary state. This
c
tion parameters are chosen to prevent crystallization28. is unusual for systems below the glass transition tem-
Inallthepaper,thelength,energyandtimeunitsarethe perature, which normally exhibit a non-stationary aging
standard Lennard-Jones units σAA (particle diameter), behavior.
ε (interaction energy), and τ = (m σ2 /ε )1/229,
AA 0 A AA AA
wherem isthe particlemassandthe subscriptArefers
A
tothemajorityspecies. Particleshaveequalmasses,and IV. MACROSCOPIC BEHAVIOR: NONLINEAR
the interaction parameters are ε = 1.5ε , ε = RHEOLOGY
AB AA BB
0.5ε ,σ =0.88σ ,σ =0.8σ . Withthesein-
AA BB AA AB AA
teraction parameters between species, equilibrium prop- In this Section, we study the rheological behavior of
erties of the system have been fully characterized28. At thesystem. Ourresultsareverysimilartothatobtained
thereduceddensityρ=1.2,wherealloursimulationsare in many different soft condensed matter systems, mak-
carried out, a ‘computer glass transition’ is found in the ing the Lennard-Jones mixture —which was originally
vicinity of Tc ∼0.435. The slowing down of the dynam- designed as a model for simple metallic glasses— a rea-
ics, T >∼ 0.45, is correctly described by mode-coupling sonable one for more complex systems.
theory, which breaks down when lowering further the Before discussing our results, let us briefly recall some
temperature, T <∼ 0.4528. The aging behavior of the possiblebehaviorsforthe flowcurves(stressσ asafunc-
system below this temperature has also been character- tion of shear rate) obtained in complex fluids. These
ized extensively, including the violation of the FDT in results are often represented in the phenomenological
the glassy phase30. form1,2
Inordertostudythesystemunderasteadyshearflow,
the classical Newton equations are replaced by the so- σ ≃σ +aγn. (4)
0
The case n = 1 corresponds to Bingham fluids, with a
yield stress σ , which vanishes for Newtonian systems.
0
For n < 1, Eq. (4) is often called the Hershel-Bulkeley
law1. For zero yield stress (σ = 0), Eq. (4) describes a
0
3
‘power-law fluid’, where the viscosity η decreases with For T > T , the behavior becomes Newtonian at low
c
increasing the shear rate as a power law, η ∼ γ−α, shear rates,
α≡1−n. Although these type of flow curves are found
in many different complex systems, they are not usually σ ≃η0(T)γ, (7)
supportedbyatheoreticalbasis,andshouldthereforebe
where
interpreted as a convenient representation of the results
on a limited range of shear rates. In our system, the η (T)= limη(γ,T >T ) (8)
0 c
microscopic stress σ is defined by the usual formula γ→0
xy
is the viscosity of the fluid in the linear regime. We
N N
1 pixpiy ∂V(rij) have checked that it is proportional to the equilibrium
σ ≡ − + r . (5)
xy L3  m ijx ∂r  relaxation time of the fluid21, and it is found to di-
i ijy
Xi=1 Xi=1Xj>i verge as the temperature is lowered as a power law
  η (T) ∼ (T −T )−2.45, in good agreement with equilib-
Theshearratedependentviscosityisdefinedbytheratio 0 c
rium simulations28. Figure 2 shows that in this temper-
σ ature regime, all the data can be rescaled onto a master
xy
η ≡ . (6)
γ curve of the form
η (T)
0
η(γ,T)= . (9)
2/3
1+γ/γ (T)
0
γ (T) is not a free param(cid:2)eter, but has(cid:3)the form γ (T)∼
(cid:11)=2=3 0 0
1 (T −T )−2.45×3/2, which ensures that the divergence of
c
the viscosity at T is a power law η(γ,T )∼γ−2/3.
c c
(cid:27)
0.1 0:15
0:3
0:4 0:6
0:45 100 0:555
0:5 0:525
(cid:11)=0 0:525 0:5
0:555
0.01 0:6
0
(cid:17)
=
10(cid:0)4 10(cid:0)3 10(cid:0)2 10(cid:0)1 (cid:17)
10
(cid:13)
4
10
0:15
0:3
0:4 1
(cid:0)4 (cid:0)3 (cid:0)2 (cid:0)1 0 1
0:45 10 10 10 10 10 10
3 0:5 (cid:13)=(cid:13)0
10 0:525
0:555 FIG.2. Scalingbehavioroftheflowcurvesη(γ)forvarious
(cid:17) 0:6
temperatures aboveT . The full line is a fit to equation (9).
c
2
10
Equation (9) describes a crossover between a Newto-
nian behavior at small shear rates and a power law be-
havior with η ∼ γ−2/3 at higher shear rates. Remark-
1
10 ably, the same behavior, with the same exponent 2/3
(cid:0)4 (cid:0)3 (cid:0)2 (cid:0)1
10 10 10 10 were predicted by the mean-field theory of Ref.13. This
(cid:13)
coincidence, however, is not necessarily significant. A
FIG. 1. Top: Flow curves σ(γ) for various temperatures.
similar shear-thinning behavior, η ∼ γ−α, with expo-
FulllinesareNewtonianbehavior(α=0)andshear-thinning
nents in the range α = 0.5− 1.0 is obtained in many
behavior with α = 2/3. Bottom: Flow curves presented in
different soft systems, such as suspensions and concen-
thealternative form η(γ), for thesame temperatures.
trated polymer solutions1,36. Confined polymer layers
under constant normal load37 were also found to exhibit
Figure 1 presents the flow curves σ(γ,T) for various very similar shear-thinning behavior. In our case, the
temperatures. The same data are also represented in data could probably be fitted with a slightly different
the alternative form of viscosity versus shear rate. Two value for the exponent, and the choice of 2/3 is moti-
different regimes can be distinguished in these figures. vated by theoretical results13.
4
Althoughourresultsareobtainedon3decadesofshear
rates,we cannotreportadefinite functionalformfor the
1
(cid:12) flow curves. An alternative view of the low temperature
(cid:11)
flow curves is that the effective shear-thinning exponent
0.9 will saturate to the value α(T) = 1 in the (numerically
unreachable)verylowshearratelimit,indicatingtheex-
istence of a yield stress σ (T) ≡ lim σ(γ,T) for the
0 γ→0
0.8 system at finite temperatures. Hence, we also tried to
useEq.(4), inorderto extractapossible valueofσ (T).
0
Note that this is made possible by the fact that Eq. (4)
0.7
has more free parameters than a single power law. Fits
are thus satisfactory, and the values of the yield stress
0.6 obtainedatdifferenttemperaturesinthis mannerarere-
0 0.1 0.2 0.3 0.4 0.5 0.6 ported in Fig. 4. This ‘jamming phase diagram’ is very
T similartotheoneproposedbyLiuandNagel22,andstud-
FIG.3. Evolution of the effective shear-thinning exponent ied experimentally in Ref.38. It is also an interesting
α(T)andofthestretchingexponentβ(T). Thehorizontalar-
way of thinking of the glass transition, and as such was
rowshowsthevalueα=2/3obtainedaboveT . Thevertical
c theoretically investigatedin Ref.13. The ‘jammed phase’
dashed line is at T =0.435.
c would correspond here to the low-σ, low-T part of the
diagram.
V. MICROSCOPIC CORRELATION FUNCTIONS
1.5
Theshearratemaybeviewedasanewcontrolparam-
1.2 eter to access the glassy phase. As discussed earlier, one
mayexpectscalingpropertiesofthecorrelationfunctions
0.9 asafunctionofshearratetobesimilartothoseobserved
(cid:27)
when temperature is decreased. This basic remark was
0.6 alreadymadeinRef.21,but wenowhaveacleartheoret-
ical context in which these results may be understood13.
In this Section, we briefly consider the influence of the
0.3
shearrateonstaticcorrelations,andinvestigatenextthe
scalingpropertiesofthetime-dependentcorrelationfunc-
0
0.1 0.2 0.3 0.4 0.5 0.6 tions.
T
FIG. 4. The (σ,T) plane of the jamming phase diagram.
The dashed curves are the viscosity contour plots. η = 20, A. Static correlation functions
30, 50, 100, 200, 500, 1000 and 2000 (from top to bottom).
The full line represents the value of the yield stress σ (T),
0 Static correlations may be characterized through the
extrapolated from Eq. (4). Arrow marks the mode-coupling
structure factors of the fluid, defined as
temperature T .
c
N N
1
S(k)= exp ik· r (t )−r (t ) .
j 0 l 0
At low temperatures, the curves shown in Fig. 1 N
j=1l=1(cid:28) (cid:18) (cid:19)(cid:29)
presentacrosssoverbetweentwoshear-thinningregimes. XX (cid:2) (cid:3)
(10)
For relatively high shear rates, they can be described by
the same shear-thinning exponent as above Tc, α=2/3, We show first in Fig. 5 the structure factor for γ =
but at low shear rate, the shear-thinning behavior is 10−3, and T =0.3 in the three spatial directions. It can
more marked, α > 1. One can define an effective shear- be seen that the flow, somehow surprisingly, introduces
thinning exponent from the slope of the curves in Fig. 1 noobviousanisotropyinthis structuralquantity21. This
(α = dlnη/dlnγ)). It crosses over from α ∼ 2/3 at rel- canbeexplainedbythefactthatweareworkingatshear
atively high shear rate, to a larger value for the lowest rates which are small, γτ ≪ 1. A structural anisotropy
0
shear rate investigated here. The temperature depen- wouldprobablybepresentforlargershearrates,γτ0 >∼1.
dence of the latter is reported in Fig. 3, which shows WepresentnextinFig.6theevolutionofS(k )fordif-
z
that α(T) saturates to 1 when T → 0, which seems to ferent shear rates at constant temperature T < T . The
c
indicate that the system has a finite yield stress in this ‘glassy’ state is thus approached when the shear rate is
limit only. lowered. We shall see below that for these values of the
5
shear rate, the relaxation time increases by nearly two is the density fluctuation at point x and time t, ρ being
orders of magnitude. Only weak changes in the static the average density of the fluid. For simplicity, we will
structurefactorareobservedinthe sameshearrate win- limit ourselves to k vectors orthogonalto the flow direc-
dow. The situation is thus strongly reminiscent of the tions, for which the overall motion of the fluid does not
observation made at equilibrium that no structural sign affect the correlation function directly.
of the glass transition can be observed28,39.
1
kx
ky 0.8 T =0:5
3
kz
t) 0.6
(
) k
k 2 C
(
S 0.4
(cid:13)=0:05 (cid:13)=0
1 0.2
0
(cid:0)1 0 1 2 3
10 10 10 10 10
0
0 5 10 15 20 25 t
k
FIG. 5. Structure factor at fixed temperature, T = 0.3,
and shear rate, γ = 10−3, for the different orientations with 1
respect to theflow.
0.8 T =0:5
t) 0.6
(
k
C
0.4
(cid:13)=10(cid:0)3
(cid:13)=10(cid:0)2
3 (cid:13)=10(cid:0)1 0.2
) 0
k 2 (cid:0)4 (cid:0)3 (cid:0)2 (cid:0)1 0 1 2
( 10 10 10 10 10 10 10
S t=trel
FIG. 7. Top: Correlation functions for T = 0.5 > T and
c
1 differentvaluesof theshearrate, γ =0, 10−4, 5·10−4,10−3,
10−2,5·10−2 (fromrighttoleft). Bottom: Theslowdecayof
thecorrelationfunctioncanbecollapsedifthetimeisrescaled
by t (γ). The dashed line is a fit to a stretched exponential
0 rel
0 5 10 15 20 25 form, with an exponent β=0.77.
k
FIG.6. Structurefactor at fixed temperature T =0.3 and
different shear rates. The relaxation time increases by ∼ 2 To discuss the dynamic behavior of the system, we
orders of magnitudeon thesame shear rate interval. concentrateontheself-partoftheintermediatescattering
function, defined by
1 NA
B. α-relaxation and ‘time-shear superposition’ C (t)= exp ik· r (t+t )−r (t ) .
k j 0 j 0
N
A j=1(cid:28) (cid:18) (cid:19)(cid:29)
X (cid:2) (cid:3)
To investigate the dynamics of the fluid under (12)
shear, we focus on the number density fluctuations
hδρ(k,t)δρ(−k,t′)i where We have also computed this function for the minority
species,B, withverysimilarresults,thatwillnotbe dis-
N cussed here. These correlation functions are displayed
1
δρ(x,t)= δ x−r (t) −ρ (11) for two different temperatures and several shear rates in
j
N
* + Figs. 7 and 8. The wavector is k = 7.47e , which cor-
j=1 z
X (cid:0) (cid:1)
responds to the peak of the structure factor, see Fig. 5.
6
Theoverallshapeisverysimilartowhatisusuallyfound be seen that such a scaling is indeed nicely obeyed for
in supercooled systems, namely a first ‘microscopic’ re- theα-relaxationofthecorrelationfunctions. Thescaling
laxationwhich is independent of the controlparameters, functioniscorrectlydescribedbyastretchedexponential,
followed by a slow approach to a ‘plateau’ which we
will describe as ‘β-relaxation’in analogy with the mode- F (x)≃exp −xβ(T) , (14)
k
coupling terminology. The decay beyond the plateau, or
(cid:16) (cid:17)
terminal relaxation, will be described as ‘α-relaxation’.
withthestretchingexponent0<β(T)<1. Thetemper-
Obviously,theα-relaxationtime is a stronglydecreasing
atureevolutionofβ(T)isshowninFig.3,whichdemon-
function of shear rate40.
strates that the decay becomes less stretched when the
temperature is lowered. The stretching exponent seems
to saturate to the value 1 as T decreases to 0. In this
1 figure, a close relationshipbetween this stretching of the
correlation function and the shear-thinning exponent is
0.8 T =0:3 alsoclearly apparent. This is physically plausible: β =1
corresponds to a pure exponential form, i.e. a single
t) 0.6 relaxation time; β < 1, on the other hand, is associ-
(k ated with a broader distribution of relaxation times. If
C
β = 1 = α, there is a single relevant time scale, and the
0.4
scaling η ∼ γ−1 ∼ t shows that this time scale is sim-
rel
ply given by the inverse of the shear rate, as would be
0.2
obtainedby a naive dimensional analysis. This suggests,
as an interesting corollary, that a shear-thinning expo-
0 nent α < 1 implies the existence of a broad distribution
(cid:0)1 0 1 2 3 4
10 10 10 10 10 10
of time scales.
t
1 C. Spatial dependence and β-relaxation
0.8 T =0:3 The wavevector dependence of the number-number
correlation function is displayed in Fig. 9. As usual28,
t) 0.6 theoverallshapeisqualitativelythesameforallwavevec-
(
k tors,theplateauvalueandthestretchingexponentbeing
C
simple functions of the wavevector.
0.4
0.2
1
0
(cid:0)4 (cid:0)3 (cid:0)2 (cid:0)1 0 1 2
10 10 10 10 10 10 10
0.8
t=trel
FIG. 8. Top: Correlation functions for T = 0.3 < T and
different values of the shear rate, γ = 10−4, 3·10−4, c10−3, (t) 0.6
3·10−3, 10−2, 3·10−2, 10−1 (from right to left). Bottom: Ck
Theslowdecayofthecorrelation functioncanbecollapsed if 0.4
the time is rescaled by t (γ). The dashed line is a fit to a
rel
stretched exponential form, with an exponent 0.95.
0.2
0
Itisatheoreticalpredictionthatthe slowdecayofthe (cid:0)2 (cid:0)1 0 1 2 3
10 10 10 10 10 10
correlationfunctionmay be collapsedbya simple rescal- t
ingofthetime,the‘time-shearsuperpositionproperty’13. FIG. 9. Correlation function for T = 0.3, γ = 10−3, and
As in equilibrium28, the relaxation time is defined from different values of the wavevector regularly spaced between
the relation Ck(trel)≡e−1. The scaling form k=0.93 and k=24.76 (from top to bottom).
t
C (t)=F , (13)
k k t (γ) Inmode-coupling theories16,18, aninterestingandeas-
(cid:18) rel (cid:19)
ily testable property of the correlation function is the
where the function Fk(x) is a master function that may so-called‘factorizationproperty’which characterizesthe
dependontemperature,is testedinFigs.7and8. Itcan
7
spatial dependence in the β-relaxation region. This fac- defined by the ratio of the response to correlation func-
torisation property can be written, for the correlation tion in the following way. Consider two physical observ-
function under study, in the form Ck(t) = qk +hkf(t), ables O(t) and O′(t). The correlation function COO′(t)
whereq isthewavevector-dependentEdwards-Anderson is defined as
k
parameter, and f(t) a universal function, i.e. indepen-
dent of the chosen correlation function. More generally, COO′(t)=hO(t+t0)O′(t0)i−hO(t0)ihO′(t0)i, (16)
it implies that in the β-regime the ratio
where t is an arbitrary initial time and h···i indicates
0
ϕ(t)−ϕ(t′′) anaverageoverdifferentinitialtimest . Theconjugated
R (t)= , (15) 0
ϕ ϕ(t′)−ϕ(t′′) response function is defined by
where ϕ(t) is any slow observable [e.g. Ck(t)], and t,t′ δhO(t+t0)i
and t′′ are in the β-relaxationwindow, is independent of ROO′(t)= , (17)
δhO′(t0)
the observable28,30.
This prediction is tested in Fig. 10, which shows that where hO′ is the field thermodynamically conjugated to
the ratios (15) obtained for different wavevectorsare in- O′(t). In a system at thermodynamic equilibrium, the
deed independent of the wavevector. Again, these data FDT reads
areverysimilartowhatisobservedatequilibrium28,and
in the aging regime30. 1 dCOO′(t)
ROO′(t)=− (18)
T dt
The more easily accessible physical quantity is the sus-
5 ceptibility
t
χOO′(t)= dt′ROO′(t′) (19)
0 Z0
)
(tk t0 00 whichis obtainedbyapplyingasmall,constantfieldhO′
R t in the time interval [0,t]. In the linear response regime,
-5 hO′ →0, one gets
hO(t)−O(0)i
χOO′(t)≃ (20)
hO′
-10
10(cid:0)2 10(cid:0)1 100 101 102 103 The equilibrium FDT thus implies a linear relation be-
t tween the susceptibility and the correlation, namely
FIG. 10. The correlation functions of Fig. 9 are rescaled
intheβ-regimeusingthemode-coupling-called ‘factorization 1
property’, Eq. (15). Times are t′ = 0.57 and t′′ = 14, as χOO′(t)= T COO′(0)−COO′(t) . (21)
indicated bythe arrows. (cid:0) (cid:1)
In a system out of equilibrium, the FDT is not ex-
pected to hold. However, in a system invariant under
time translation, a modified version of Eq. (18) can be
VI. EFFECTIVE TEMPERATURES used to define the functions TeOffO′(COO′) through19
A. Existence of an effective temperature 1 dCOO′(t)
ROO′(t)=−TeOffO′(COO′) dt . (22)
Nonequilibrium mean-field or mode-coupling theories
have shown that whereas correlation functions were the The TOO′(x) are a priori arbitrary functions of their ar-
eff
only relevant quantities to be studied at equilibrium, it gument,andmay depend onwhichobservablesO(t) and
is of primary interest to separatly study susceptibilities O′(t) are under study. However, they can be measured
in out of equilibrium situations13,18–20. At equilibrium, byfollowingthesamelinearresponseprocedureasinthe
these two families of dynamic functions are related by equilibrium case, so that
fluctuation-dissipation theorems (FDT). In nonequilib-
rnairouewmreissliectvulaeaantrti4ot.nhsRa,teFctDheneTtsittshuaedoyprrioeifosrtoihfnenoctoonsraereqticustfiiiloeibndrs.iutHomotwhsyeevsFteerDm,Tist χOO′(t)=Z0tdt′(cid:18)−TeOffO′(1COO′)dCOdOt′′(t′)(cid:19) (23)
COO′(0) dx
have indeed introduced the notion of effective tempera- =
TOO′(x)
ture associated with the modification of the fluctuation- ZCOO′(t) eff
dissipation theorem19,26. This effective temperature is
8
The existence ofaneffective temperature is thus demon- times,andviolatedatlargertimes. Fromthesefunctions,
strated if a straight line is obtained in a susceptibility- aparametricplotisbuilt,asshownintheinsetofFig.11.
correlation plot parameterized by the time. As was It can be seen that the parametric plot can be described
shown in several cases26, the same system may exhibit to an excellent approximation by two straight lines, of
different temperatures on different time scales. If this is slopes−1/T (largevalueofthecorrelationcorresponding
the case, the parametric plot will consist in several dif- toshorttimes)and−1/T (smallvalueofthecorrelation
eff
ferent straight lines, the slope of each line determining corresponding to large times).
an effective temperature. Cases are also known41,42 in This observationis thus in perfect agreementwith the
which the parametric plot is ill-defined at short times behavior predicted theoretically13, and was the main re-
(e.g. becauseofanoscillatorybehaviorofthecorrelation sult in Ref.43.
function) but yields a well-defined effective temperature
on longer time scales. It is therefore generally incorrect
to define an effective temperature through the approxi- 2
FDT
mate expressionχOO′(t→∞)/COO′(0), as is sometimes 3
done23. This corresponds to taking the average slope of 1.5
the parametric plot as an effective temperature, there- (cid:31)
1
fore losing all information on the time scale dependence
2
of the effective temperature.
0.5
Obviously, the introduction of an effective tempera-
ture is of ‘thermodynamic’ interest only if this quantity 0
0 0.2 0.4 0.6 0.8 1
isactuallyindependentoftheobservablesO(t)andO(t′) 1 C
under consideration, TOO′ ≡ T . This is true at the
mean-field level18,19,25,eaffnd thiseqffuestion will be investi- (cid:13) =10(cid:0)3, T =0:3
gated in detail below.
0
Asafirstinvestigationoftheexistenceandbehaviorof 10(cid:0)2 10(cid:0)1 100 101 102 103
the effective temperature and fluctuation-dissipation re- FIG.11. Susceptibility χ (tt) (squares) and [1−C ](t)]/T
lation,weconsiderthecaseofsingleparticledensityfluc- k k
tuations, for which the corresponding correlation func- (circles) versus time, for k=7.47ez, T =0.3, and γ =10−3.
Thetwo curvesaresuperposed when theequilibrium FDTis
tion has been studied in detail above. Corresponding
satisfied. Inset: Parametric susceptibility versus correlation
observables are in this case
plot for the same data. The dashed line is a linear fit to the
1 NA small-C partofthedata,withTeff =0.65. Thefulllineisthe
O(t)= ε exp ik·r (t) , (24) equilibrium FDT.
j j
N
A
j=1
X (cid:0) (cid:1)
and
B. Effective temperature in the plane (γ,T)
NA
O′(t)=2 ε cos k·r (t) , (25)
j j The effective temperature is defined as (minus) the
j=1
X (cid:0) (cid:1) inverse slope of the parametric susceptibility-correlation
where ε =±1 is a bimodal random variable of mean 0. plot for long times. This effective temperature will, gen-
j
The relation erally speaking, depend on how strongly the system is
externally driven. One may ‘reasonably’ expect that
Ck(t)=COO′(t), (26) a stronger drive implies a higher effective temperature.
[Seethe discussionofthe word‘reasonably’atthe endof
where the horizontal line means an average over the re- the paper.] That this is indeed the case is demonstrated
alizationsof{εj},easilyfollows. Tocomputethe suscep- in Fig. 12, where parametric susceptibility-correlation
tibility, a force plots are displayed for several different shear rates.
Two different situations are considered in this figure.
∂
Fj(k,t)=−∂rj(cid:18)−hO′O′(t)(cid:19) (27) Aretduhcigehs ttoemthpeerbaatuthretse,mTp>erTatcu,rtehewehffeencttihveesthemeaprerraatteuries
decreased. Aspredictedtheoretically13,wefindherethat
is exerted on each A-particle. The response function
the effective temperature of the slow modes is equal to
χ (t) [see Eq. (20)] is computed by averaging the re-
k thebathtemperatureaslongasthesystemisinitsNew-
sponse obtained in different realizations of the {ε }.
j tonian regime, η ∼ const. This is physically reasonable,
The time-dependence of the dynamic functions χ (t)
k since the linear regime of rheology corresponds roughly
and [1−C (t)]/T is displayed in Fig. 11. In that case,
k toshearratessuchthatγt <1,forwhichthedynamics
200 realizations of the {ε } have been considered. This rel
j is weakly affected13,21.
figureshowsthattheequilibriumFDTisobeyedatshort
9
In the ‘glassy’ low temperature region, on the other VII. OBSERVABLE INDEPENDENCE OF Teff
hand, the effective temperature also decreases with de-
creasingshearrate,butsaturatesforthelowershearrates As we already mentioned, the notion of an effective
that canbe investigatedin the simulation at a value dif- temperature derived from the fluctuation-dissipation re-
ferent from the bath temperature, as theoretically pre- lation is much more relevant if it can be shown not to
dicted13. The existence of such a limiting value is phys- dependonthe observableunder consideration. ThatT
eff
ically interpreted by the fact that at low temperature, a is observable independent is indeed one of the crucial
Newtonian (η ∼ const) or near-equilibrium (Teff ∼ T) predictions of mean-field approaches18,19,25. In this Sec-
regime defined as above by the condition γtrel < 1 does tion, we explore this important issue by computing the
not exist, since trel ∼∞. susceptibility-correlationparametricplotsforseveraldif-
ThelimitingvalueTeff(γ →0)correspondstotheeffec- ferent observables. Since the computation of response
tive temperaturethatwasobservedinthe corresponding functions is numerically verydemanding,the method we
aging studies of the same system at this temperature30. usedconsistedincomputingveryaccuratelytheeffective
InRef.30,thelimitingvalueTeff ∼0.62isreported(how- temperaturefromthescatteringfunctionfork=7.47ez.
ever with much less accuracy than in the present pa- ¿From the data in Fig. 11, one has the estimation T ≃
eff
per), while we find Teff ∼ 0.65 at the same temperature 0.65 for T = 0.3 and γ = 10−3. We can then compute
T = 0.3. This is again expected from the theory13, and for the same parameters (T, γ) the response associated
confirms the interpretation of Teff as a signature of the with other observables, and check whether these data
geometry of phase space explored by the system on the are compatible with this value of T . Since we have ex-
eff
simulation time scale, independently of the exact proce- plained in some detail the generic numerical procedure
dure which is followed13,19,42. to measure T in the previous Section, less details will
eff
be given here. Instead, we carefully discuss the possi-
ble experimentalrealizationsof ourmeasurements. Note
2 that in experiments, a simultaneous measurement of the
(cid:13)=10(cid:0)4
(cid:13)=10(cid:0)3 response and of the correlation is very difficult, and has
(cid:13)=10(cid:0)2 beenachievedonlyinveryfewcases44–47. Whileusualdi-
1.5 FDT electric, magnetic or mechanical methods yield response
functions, scattering experiments probe only correlation
)
t
( functions5,9.
(cid:31)k 1
0.5 A. Position correlations for different wavevectors
T =0:3
As a first check, we have investigated the wavevector
0 dependenceoftheparametricplotfortheself-partofthe
0 0.2 0.4 0.6 0.8 1
intermediate scattering function. The results, displayed
Ck(t)
inFig.13,areindeedperfectlyconsistentwithawavevec-
tor independent effective temperature. Parametric plots
2 for different wavevectors differ only through the value
(cid:13)=0
(cid:13)=5(cid:1)10(cid:0)5 of the correlation function at which the crossover from
(cid:13)=10(cid:0)3
1.5 (cid:13)=(cid:13)5=(cid:1)1100(cid:0)(cid:0)22 bmaatrhkstotheeffeccrtoisvseovteermbpeetrwateuerne ‘tsahkoerst’,pleaqcue.ilibTrahtisedvaalnude
FDT
) ‘long’,outofequilibriumtime scales. Italsocorresponds
t
(
(cid:31)k 1 to the plateau value qk in the correlation functions dis-
playedin Fig. 7. Note that the wavevectorsstudied here
cover a range of length scales roughly between 0.5σ
AA
0.5 and 4σ . That these different length scales have the
AA
same off-equilibrium behavior, as observed through the
T =0:5
effective temperature, is not a priori obvious. It is in
0
opposition, for instance, to the simple view of the sys-
0 0.2 0.4 0.6 0.8 1
Ck(t) temrapidlyequilibratingitssmalllengthscales,whileits
FIG. 12. Parametric plots for T = 0.3 (top) and T = 0.5 largelengthscalesneedmuchmoretimetoreachequilib-
(bottom), and various shear rates. In both figures, the full rium, a generic scheme which is put forward in systems
lineis theFDT, and hasaslope −1/T. Thedashed lines are such as spin glasses48.
linear fits to the small-C part of the data, for γ > 0. The The‘chemical’dependenceoftheeffectivetemperature
wavevector is k=7.47ez for all thecurvesof this figure. can also be investigated by computing the correlation
function C (t) and the corresponding response, associ-
k
10