Thermodynamic Interpretation of Soft Glassy Rheology Models

0.18 MB·English

by  Peter Sollich

Thermodynamic Interpretation of Soft Glassy Rheology Models
∗
Peter Sollich
King’s College London, Department of Mathematics, Strand, London WC2R 2LS, UK
Michael E. Cates
SUPA, School of Physics and Astronomy, The University of Edinburgh,
JCMB, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
(Dated: 15 Jan 2012)
Mesoscopic models play an important role in our understanding of the deformation and flow
of amorphous materials. One such description, based on the Shear Transformation Zone (STZ)
2
1 theory, has recently been re-formulated within a non-equilibrium thermodynamics framework, and
0 found to be consistent with it. We show here that a similar interpretation can be made for the
2 Soft Glassy Rheology (SGR) model. Conceptually this means that the “noise temperature” x,
proposedphenomenologically intheSGRmodeltocontrolthedynamicsofasetofslowmesoscopic
n
degrees of freedom, can consistently be interpreted as their actual thermodynamic temperature.
a
(Because suchmodes are slow toequilibrate, this generally doesnot coincide with thetemperature
J
of the fast degrees of freedom and/or heat bath.) If one chooses to make this interpretation, the
6
thermodynamic framework significantly constrains extensions of the SGR approach to models in
1
which x is a dynamical variable. We assess in this light some such extensions recently proposed in

thecontext of shear banding.
]
t
f
o
I. INTRODUCTION main benefits. Firstly, it gives an alternative way of
s
. looking at the effective temperature parameter x that
t
a Developingpredictivetheoriesforthedeformationand was previously proposed within SGR to set the noise
m
flow of amorphous materials remains an ongoing chal- level for stochastic mesoscopic dynamics. This param-
- lenge. The class of materials concerned is large, includ- eter can now, if desired, by interpreted as a genuine
d
n ing not only colloidal glasses, emulsions and foams but non-equilibrium thermodynamic temperature, governing
o also molecular glass formers, metallic glasses and possi- asubsetofdegreesoffreedomwhosedynamicscausesthe
c bly granularmaterials. The development of such predic- system to move among its various “inherent structures”
[ tive theorieswouldthereforebe ofgreatpracticalsignifi- or energy minima. Secondly, this interpretation signifi-

1 canceaswellasfundamentaltheoreticalinterest[1]. Two cantlyconstrainshowSGRcanbeextendedtoallowxto
v key challenges in understanding deformation and flow in evolve in time and space: it suggests a specific form for
5 amorphous materials are the absence of a reference con- the driving termin the dynamicalequationfor an evolv-
7 figurationrelativetowhichstraincanbedefined(incon- ing x. We find that this driving term is consistent, in
2 trast to ordered crystalline solids), and the difficulty of steady state, with one of two extensions of SGR postu-
3 encapsulating localflow events and/orlarger-scalestress latedrecentlytounderstandshearbandinginamorphous
.
1 redistributionsinacoarse-grainedvariableorcontinuum flows [15], but not with the other. Similar remarks ap-
0 field (analogous to the dislocation density for crystals). ply to two models for shear-thickening fluids developed
2 First principles approaches to glass rheology have previously [19].
1
madesignificantprogress[2,3],butentailmajorapproxi-
:
v mations(suchasmodecouplingtheory),usuallyfollowed
Xi by further simplification [4]. In the absence of a com-
prehensive microscopic theory, a number of models have
r
a beendevelopeddirectly ata mesoscopiclevelto describe
amorphous flow, see e.g. [5–10]. Two such approaches
We begin by reviewing briefly the SGR model, and
that have been investigated in some detail are the shear
then the non-equilibrium thermodynamic framework of
transformation zone (STZ) approach, reviewed recently
Bouchbinder and Langer [16–18] (Secs. II and III).
in[11],andthesoftglassyrheology(SGR)model[12–15].
Sec.IVhasthecoreofourargument,whichshowsthatin
One important feature of the STZ theory is that it can
this framework SGR can indeed be written as a thermo-
begivenanunambiguousthermodynamicinterpretation,
dynamically consistent model, and that this consistency
as discussedin detailin a seriesof recentpapers [16–18].
forces x to be the non-equilibrium thermodynamic tem-
Our purpose in this paper is to show that the SGR
perature of a set of slow degrees of freedom. In Sec. V
model can likewise be cast consistently within a non-
we discuss to what extent this argument can be made
equilibrium thermodynamic framework. This has two
to cover also generalizations of the SGR model. Sec. VI
has a summary of our results and discusses the implica-
tions for various extended SGR models that treat x as a
∗ [email protected] dynamical variable.
2
II. THE SOFT GLASSY RHEOLOGY MODEL III. NON-EQUILIBRIUM THERMODYNAMICS
FRAMEWORK
The SGR model [12–15] describes a sample of amor-
We next review the non-equilibrium thermodynamic
phous material as a collection of mesoscopic elements,
framework of Bouchbinder and Langer [16–18]. The ba-
chosen large enough that it makes sense to define local
sic premise is that the degrees of freedom of the ma-
strain and stress variables. The local strain l is defined
terial can be divided up, conceptually, into two weakly
relative to a local equilibrium state [20], and is assumed
interacting subsystems. The degrees of freedom in the
to change in time by following the change in the macro-
scopic shear strain γ, so that l˙ = γ˙. We focus here on configurationalsubsystem(C)encodewhichoftheexpo-
nentially many local energy minima, known in the glass
pure shear strain deformation but note that the model
communityasinherentstructures,the systemfinds itself
canalsobeextendedtogeneraldeformationtensors[21].
in; this subsystem is ‘slow’ because at low temperature
Thelocalstraincannotgrowindefinitely: oncethestored
any motion between inherent structures has to be ther-
elasticenergy,whichwewriteas(kv /2)l2intermsofthe
e mally activated or driven by external deformation. The
local shear modulus k and the element volume v , be-
e
kinetic-vibrationalsubsystem(K)gathersthe remaining
comes close to a local yield energy E, a yield event will
degreesoffreedomthatdescribefastmotionaroundthese
takeplace. Thisisassumedtoresetltozeroandcreatea
inherent structures. This subsystem is taken as strongly
newlocalequilibriumconfigurationwithcorrespondingly
coupled to a thermal reservoir(R) that sets the thermo-
anewyieldenergy,drawnfromadistributionρ(E). The
dynamic equilibrium temperature θ . The total internal
final ingredient is that this yield process is taken as ac- R
energy of the system is then written as
tivated by some effective temperature x. This is based
on the intuition that every yield event elsewhere in the
U =U (S ,Λ)+U (S )+U (S )
material causes a ‘kick’ locally via the associated stress tot C C K K R R
redistribution, and that many such kicks add up to an
where S , S and S are the entropies of the config-
C K R
effectively thermal noise.
urational and kinetic-vibrational subsystems and of the
The resulting equation of motion for the distribution thermal reservoir. In U , Λ denotes a set of internal
C
of strain and yield energy across elements, P(E,l,t), is state variables of the configurational subsystem. In the
STZ context these are taken as concentrations of shear
transformation zones in different orientations. For our
P˙(E,l,t)=−γ˙∂ P −Γ(E,l)P +Y(t)ρ(E,l) (1)
l application to SGR, we will take Λ as the distribution
P(E,l). Notethatbecausethisdistributionalreadycon-
Here Γ(E,l) is the yield rate from a given local config- tains information on all local elastic strains, we do not
uration, which represents the thermal activation in the includeseparatelyinUC astatevariableforstrainaswas
form Γ(E,l)=Γ exp[−(E−kv l2/2)/x]. (We will leave done in [17, 18].
0 e
Γ(E,l) general whenever possible in the following, to The thermodynamic analysis is based on the first and
assess the scope of our arguments.) We use k = 1 secondlaws. Thefirstlaw,i.e.energyconservation,takes
B
throughout, so that all temperatures are measured in the following form:
energy units. The time-dependent total yield rate is
U˙ +U˙ +U˙ =Vσγ˙ (3)
Y(t) = dEdlΓ(E,l)P(E,l,t). Finally, ρ(E,l) is the C K R
probabiliRty density of E and l for elements after a jump.
The r.h.s. is the external work rate that arises from
This is equal to ρ(E)δ(l) in the standard SGR model as
shearing a system of volume V at shear rate γ˙ against
described above. By choosing ρ(E) to have an exponen-
tial tail ∼e−E/E0 one finds that a glass transition arises a shear stress σ. In terms of the subsystem entropies
S , S and S , and the corresponding temperatures
at x/E = 1. For x/E > 1 the material is ergodic (al- C K R
0 0
χ= ∂U /∂S | andθ =∂U /∂S ,thefirstlawcanbe
beitwithinfinite viscosityfor1<x/E0 <2)whereasfor C C Λ K K
rewritten as
x/E < 1 the material is a nonergodic amorphous solid
0
with time-dependent (aging) material properties [14].
∂U
χS˙ +θS˙ +U˙ −Vσγ˙ +Λ˙ C =0 (4)
The SGR shear stress is an averageover elements, C K R ∂Λ (cid:12)
(cid:12)SC
(cid:12)
(cid:12)
WeonlywriteoneΛ-derivativetermhere,asashorthand
σ(t)= dEdlklP(E,l,t) (2) for a sum over the relevant derivatives w.r.t. the state
Z
variables gathered in Λ.
Thesecondlawstatesthattotalentropymustincrease,
with k the local shear modulus as defined above. Start- i.e.
ing from some initial condition P(E,l,0), this together
with the evolutionequationfor P(E,l,t)inprinciple de- S˙ +S˙ +S˙ ≥0 (5)
C K R
termines the stress σ(t) for any imposed shear history
γ(t). One can eliminate S˙ from this using the first law and
C
3
write S˙ =U˙ /θ to get One can now simplify using χ = ∂U /∂S | =
R R R C C Λ
∂U /∂S . Assuming further that the number of inter-
θ χ θ 1 1
− 1− U˙ − 1− θS˙ + U˙ +W ≥0 nal state variables in Λ is much smaller than the overall
R K R
(cid:18) θR(cid:19) (cid:16) θ(cid:17)(cid:18) θR (cid:19) numberofslowdegreesoffreedom[16],onehasS ≪S
Λ C
(6)
andχcanberegardedasafunctionofthisvastmajority
where
of configurational degrees only. The final expression for
∂U the dissipation rate is then
W =Vσγ˙ −Λ˙ C (7)
∂Λ (cid:12)
(cid:12)SC ∂U ∂S
(cid:12) W =Vσγ˙ −Λ˙ Λ −χ Λ (12)
BecausetimevariationsinU , S an(cid:12)dΛareinprinciple
R K (cid:18) ∂Λ ∂Λ (cid:19)
arbitrary,itisplausibletoargue[17]thatthethreeterms
in(6)shouldbe separatelynon-negative. Toalloweasier andthisshowsthatexternalworkthatisnotdissipatedis
comparison with [16–18], we follow this reasoning here. indeedstoredinthe freeenergyU −χS ofthe internal
Λ Λ
However, it turns out that this separate non-negativity state variables Λ.
is sufficient but not necessaryfor (6): in Appendix A we
present a less restrictive but still sufficient set of condi-
tions for non-negative entropy production. Setting that IV. APPLICATION TO THE SGR MODEL
aside and assuming separate non-negativity of all three
terms, one obtains the conditions W ≥0 and
As mentionedearlier,inapplying the aboveformalism
θ to SGR our approach we take the distribution P(E,l)
U˙ =−B(θ −θ), θS˙ + U˙ =−A(θ−χ) (8)
R R K R as the set of internal state variables Λ. The arguments
θ
R
above rely on the number of degrees of freedom in Λ
where A and B are positive but can depend on the state
being much smaller than the overall number of degrees
ofthesystem,e.g.viathevarioustemperatures. Ourno-
of freedom in the configurational subsystem. This can
tationforAfollowsRef.[11],andwedefineBbyanalogy;
be ensured by initially considering, instead of P(E,l),
thecoefficientsin[16–18]differbyfactorsoftemperature.
thetotalprobabilitythisdistributionassignstoeachofa
Tosimplifyfurther,oneassumesstrongcouplingbetween
finitenumberofbinscoveringthe(E,l)plane. Thenum-
thethermalreservoirandthe fastsubsystem. Inthe cor-
ber of bins is then sent to infinity more slowly than the
respondinglimitoflargeB,onethenobtainsθ =θ ,but
R systemvolumeV inthethermodynamiclimitoflargeV.
this equality is approached in such a way that the prod-
We assume below that this limit has been taken appro-
uct B(θ −θ) stays finite. In fact it is only in this limit
R priatelyandwritedirectlytheversionofthecalculations
that(6)leadsto(8),asdiscussedfurtherinAppendixA.
in the large system limit, i.e. without binning.
One can now write the first law as
Clearly for SGR one should choose as the internal en-
χS˙ =−(θS˙ +U˙ )+W (9) ergy associated with Λ≡P(E,l),
C K R
Becauseoftheassumptionofstrongcouplingbetweenthe
U (Λ)=N dEdlP(E,l)u(E,l) (13)
reservoirand the fast subsystem, one can exploit θ =θ Λ e
R Z
to writethe firsttermonthe r.h.s.as−(θS˙ + θ U˙ )=
K θR R
A(θ−χ), so where Ne = V/ve is the number of SGR elements with
volume v each, and u(E,l) is some function of the lo-
χS˙ =W +A(θ−χ) (10) e
C cal yield energy and strain. The corresponding choice of
The upshot of this analysis [17, 18] is thus twofold. entropy is less obvious. A natural proposal is SΛ(Λ) =
−N dEdlP(E,l)[lnP(E,l)−1]. However,itturnsout
Firstly,oneobtainsanequationofmotionfor theconfig- e
urational entropy; this also then determines the dynam- that tRhis choice does not give a thermodynamically con-
sistent interpretation of the SGR model, essentially be-
ics of the non-equilibrium temperature χ as discussed in
Sec. VI. The second result is the constraint W ≥ 0. In causeeveryyieldeventresetsl=0andsolosesaninfinite
(10),W isthedissipationrate,i.e.thedifferencebetween amount of entropy. We argue instead that, because the
the externalwork(rate)andthe (rate of)increaseinthe l-dynamicsis deterministic apartfromjumps when rear-
free energy of the Λ-degrees of freedom. rangementsoccur,thesedegreesoffreedomare“slaved”.
Hence we should only consider the entropy of the distri-
To go beyond the formal expression for W above re-
quires further assumptions on the relation between the bution across yield energies. We also allow for a prior
distribution (or density of states) R(E) in the entropy,
state variables Λ and the other configurational degrees
offreedom[16–18]. Theslowsubsystemasawholeisde- and take:
scribedvia U (S ,Λ)=U (Λ)+U (S −S (Λ)) where
C C Λ 1 C Λ P(E)
U (S ) captures all the slow degrees of freedom other S (Λ)=−N dEdlP(E,l) ln −1 (14)
1 1 Λ e
Z (cid:20) R(E) (cid:21)
than Λ. Then
∂UC ∂UΛ ∂U1∂SΛ where P(E) = dlP(E,l). The argument that the en-
= − (11)
∂Λ (cid:12)(cid:12)SC ∂Λ ∂S1 ∂Λ tropyshouldbeRindependentofthel-degreesoffreedomis
(cid:12)
(cid:12)
4
somewhatanalogoustotheobservationmadeinSTZthat We nowspecializetothe conventionalSGRmodelwhere
in orderto derive the standardSTZ equations of motion ρ(E,l) = ρ(E)δ(l), Γ(E,l) = Γ eu/χ, and the obvious
0
one has to assume that the shear transformation zone priortouseinthedefinitionofS isR(E)=ρ(E). Also,
Λ
orientation makes no contribution to the entropy [11]. the quantity
To ensure that SGR fits into the thermodynamic
π(E,l)≡Γ(E,l)P(E,l)/Y (21)
framework described above, we need to show that with
the above choices for U and S , one has W ≥ 0 al- isanormalizeddistributionbydefinitionofY. Sowecan
Λ Λ
ways, which means that the rate at which external work express W as
is performed is never less than the rate at which free en-
W Γ(E,l)P(E)
ergy is stored reversibly in the internal state variables = dEdlπ(E,l)ln
N χY Z Γ ρ(E)
Λ≡P(E,l). e 0
The inequality W ≥0 has to hold at alltimes t, so we e−E/χP(E)
− dEρ(E)ln (22)
drop all time arguments in the following. If we unpack Z ρ(E)
our shorthand notation, the contribution to W in (12)
To show that this is non-negative we now use a cross-
from the change of P(E,l) becomes
entropy inequality of the form
∂(U −χS ) δ(U −χS )
Λ˙ Λ Λ ≡ dEdlP˙(E,l) Λ Λ (15) dzP(z)ln[P(z)/Q(z)]≥−ln dzQ(z) (23)
∂Λ Z δP(E,l) Z Z
This is valid if P is a probability distribution and Q is a
where the functional derivative on the right hand side is
measure(i.e.a probabilitydistributionbut notnecessar-
δ(U −χS ) P(E) ilynormalizedto 1). The secondterminW is alreadyof
Λ Λ
=N u(E,l)+χln (16)
δP(E,l) e(cid:20) R(E)(cid:21) theform(23). Inthe firstterm,wecangetπ(E,l)inside
the log by writing P(E)P(l|E)=P(E,l); here P(l|E) is
Incombinationwiththe externalwork,this givesfor the the conditional distribution of l given E. This yields
dissipation rate
W Yπ(E,l)
= dEdlπ(E,l)ln
P(E) N χY Z Γ P(l|E)ρ(E)
W =Vσγ˙ −N dEdlP˙(E,l) u(E,l)+χln e 0
e
Z (cid:20) R(E)(cid:21) ρ(E)
+ dEρ(E)ln (24)
(17) Z e−E/χP(E)
WenowwanttocheckthatW ≥0,fortheSGRequation
of motion (1). Inserting the latter into the expression ≥−ln(Γ /Y)−ln dEe−E/χP(E) (25)
0
Z
above and also expressing σ as an average over P(E,l)
from (2) shows dEdle(kvel2/2−E)/χP(E,l)
=ln ≥0 (26)
R dEdle−E/χP(E,l)
W P(E)
N =Z dEdl (cid:26)γ˙kvelP +γ˙∂lP (cid:20)u+χlnR(E)(cid:21) This shows that the RSGR model is thermodynamically
e
consistent in the sense that its equation of motion en-
P(E)
+(ΓP −Yρ) u+χln (18) sures that the dissipation rate W is non-negative as it
(cid:20) R(E)(cid:21)(cid:27)
shouldbe. Notethatthis argumentworkedonlybecause
in writing down the yield rate Γ(E,l) above we had al-
where P, ρ, u are short for P(E,l), ρ(E,l), u(E,l) re-
ready identified the SGR effective temperature x with
spectively. Now integrate by parts over l:
thethermodynamictemperatureχofthe slowdegreesof
W freedom. This is an important conclusion, which we will
= dEdl γ˙P (kv l−∂u)
N Z (cid:26) e l develop in more detail in the next Section.
e
eu/χP(E) We remark finally that the three inequalities used
+χ(ΓP −Yρ)ln (19) above to prove W ≥ 0 (two cross-entropy ones, and the
R(E) (cid:27)
lastonethatthe averageofexp[kv l2/(2χ)]overanydis-
e
Since the sign of γ˙ is arbitrary, and so is the shape of tribution is ≥ 1) all become equalities if (and only if)
P(E,l), the first term must vanish pointwise, i.e. ∂ u = P(E,l)∝ρ(E)eE/χδ(l). As is physically sensible, this is
l
kv l. This ofcoursemakes sense: the stresscontribution the (Boltzmann) steady state ofthe SGRmodel without
e
ofeachelementisjustthestrainderivativeofitsinternal flow, where W = 0. In all other situations, W will be
energy density. So u(E,l)−kv l2/2 is a function of E positive.
e
only,andfromthemeaningoftheyieldenergyweexpect
this function to be −E itself so that u(E,l)=kv l2/2−
e
E. V. CONSTRAINTS ON MORE GENERAL
SGR-LIKE MODELS
With the flow term thereby eliminated,
W ΓP eu/χP(E) The natural question arising from the results of the
= dEdl −ρ ln (20)
N χY Z (cid:18) Y (cid:19) R(E) previous Section is whether more general versions of the
e
5
SGR equation of motion are thermodynamically consis- Γ(E,0)P(E)δ(l). Choose a P(E) such that this is
tent in the same sense. ρ(E)[1+ηr(E)]δ(l)withsomesmallηandasofarunspec-
ified function r(E) satisfying dEρ(E)r(E) = 0. Then
the first contribution to W isRO(η2). The second and
A. Deviations from local linear elasticity third combine to
e−E/χ
Considerfirstdeviationsfromlocallinearelasticity,as −η dEρ(E)r(E)ln (31)
Z Γ(E,0)
contemplated e.g. in the tensorial version of SGR [21].
Thesearesimpletoincorporate: onecouldhaveu(E,l)= Unless the argument of the logarithm is a constant, one
u(l)−E,withthestressthentheaverageofu′(l)/v over can arrange the sign of r(E) so as to make the integral
e
all elements. As long as u(l) has its global minimum positive, and thus W negative to leading order in η. To
u(0)=0atl =0,the argumentabovegoesthrough,and avoid this contradiction to W ≥0, Γ(E,0) must be pro-
the model is thermodynamically consistent. portional to e−E/χ, say Γ(E,0) = Γ0e−E/χ. This ar-
gument shows that for thermodynamic consistency, the
effectivetemperaturexofSGRmustbeequaltothenon-
B. More general yield rates Γ(E,l) equilibriumthermodynamictemperatureχoftheconfig-
urationalsubsystem,aspromisedinthepreviousSection.
Turning next to l-dependence of Γ(E,l), we can write
One can also generalize the SGR model by allowing a
this as Γ(E,l) = Γ e−E/χg(E,l) with g(E,0) = 1, and
moregeneraldependenceoftheyieldratesΓ(E,l)onthe 0
hence
state of the local element, i.e. E and l. (This was done,
for instance, in [19] in the context of shear thickening W π(E,l)
= dEdl[π(E,l)−P(l|E)ρ(E)]ln
models; wediscussthe implicationsforthatworkinSec- N χY Z P(l|E)ρ(E)
e
tionVI.)Tounderstandthe behaviourofW inthis case, ekvel2/(2χ)
we start from the still general expression (20), choosing + dEdlπ(E,l)ln
Z g(E,l)
again R(E)=ρ(E) as the prior in the entropy:
W eu/χP(E) + dEdlP(l|E)ρ(E)lng(E,l) (32)
Z
= dEdlπ(E,l)ln
N χY Z ρ(E)
e The first term here is of cross-entropy form, while the
e−E/χP(E) secondandthirdtermare(pointwise)non-negativeif1≤
−Z dEρ(E)ln ρ(E) (27) g(E,l) ≤ ekvel2/(2χ). In summary, a sufficient condition
for W ≥0 is
As before, one can now write P(E) = P(E,l)/P(l|E)=
Yπ(E,l)/[Γ(E,l)P(l|E)] so that the first term becomes Γ(E,l)=Γ0e−E/χg(E,l) with 1≤g(E,l)≤ekvel2/(2χ)
(33)
eu/χP(E) Thefirstpartisalsonecessaryasshownabove. Whether
dEdlπ(E,l)ln =
Z ρ(E) thesecondpartislikewisenecessaryisnotclear,i.e.there
Yπ(E,l)eu/χ may be other g(E,l) that also keep W ≥ 0. But physi-
= dEdlπ(E,l)ln (28) cally the sufficient condition given here is broad enough:
Z Γ(E,l)P(l|E)ρ(E)
the yieldrateshouldincreasewithl2,i.e.it shouldnever
Similarly, the second term is go below its l = 0 value for given E; on the other hand
theyieldrateshouldnotgoupmorequicklywithl2 than
ρ(E)
prescribed by the reduction in activation energy. If any-
dEρ(E)ln
Z e−E/χP(E) thing, one might expect the rate to level off when the
Γ(E,l)P(l|E)ρ(E) barrier −u=E−kvel2 becomes small or even negative,
=Z dEdlP(l|E)ρ(E)ln e−E/χYπ(E,l) (29) for example
Γ(E,l)=Γ min(eu/χ,1) (34)
Putting these two together and isolating the cross- 0
entropy terms gives or
W π(E,l) Γ(E,l)=Γ /(1+e−u/χ) (35)
= dEdl[π(E,l)−P(l|E)ρ(E)]ln 0
N χY Z P(l|E)ρ(E)
e It is clear from the above arguments that one could also
eu/χ allowashear-ratedependenceoftheattemptrateΓ ,e.g.
+ dEdlπ(E,l)ln 0
Z Γ(E,l) replacing Γ0 by (Γ20+γ˙2)1/2 in the spirit of recent pro-
posals [22] on how the relaxation time varies with shear
Γ(E,l)
+ dEdlP(l|E)ρ(E)ln (30) rate. These more general forms can of course be com-
Z e−E/χ
bined with deviations from local linear elasticity as in
To say something about necessary conditions on Γ(E,0) the previous subsection, with the upper limit on g(E,l)
first, consider P(l|E) = δ(l), so that π(E,l) ∝ then eu(l)/χ instead of ekvel2/(2χ).
6
C. Frustration scopic degrees of freedom do not equilibrate sufficiently
evenamongthemselvesfortheirnon-equilibriumtemper-
ature to ever be defined. In that case, however, it re-
We had previously considered [13] the possibility that
mains useful to know that the empirical choice made for
thelocalstrainlmightnotrelaxfullytozeroafterayield
the noise-driven dynamics in the standard formulation
event because disordered stresses from the environment
of SGR is exactly equivalent to having such a temper-
prevent this, and termed this effect ‘frustration’. Math-
ature. Anyone convinced that such a temperature does
ematically, it would correspond to ρ(E,l) = ρ(E)r(l|E)
notexistwouldprobablywishtoaddtermstobreakthis
with r(l|E) 6= δ(l). For an SGR model extended in this
way it is doubtful that an argument can still be made equivalence – which amounts to an implicit assumption
of detailed balance among slow modes in the quiescent
that l shouldbe left out of the entropy S as l no longer
Λ
has deterministic dynamics slaved to the last yield time state.
We have further shown that the thermodynamic con-
of each element. If one nevertheless leaves S as before,
Λ
then one can show that the extended model does not sistency extends from the original SGR model to more
general versions that allow for nonlinear local elasticity.
guaranteeW ≥0,i.e. there arechoicesforP(E,l)which
lead to W < 0. It is possible that the model would still Likewise, a fairly broad class of yield rates Γ(E,l) is al-
lowed. This includes most physically plausible choices,
be thermodynamically consistent because such distribu-
tions might not be accessible by the dynamics starting but always maintaining the thermally (χ-)activated de-
pendence on yield energy.
from reasonable initial conditions, but we have not been
able to establish this. Beyond checking consistency, the thermodynamic
framework gives an equation of motion (10) for the en-
tropy S of the configurational subsystem; see the Ap-
C
pendix for a generalization of this. As discussed in [11],
VI. SUMMARY AND DISCUSSION
one can convert Eq. (10) to an equation of motion for
the effective temperature (a more involved scenario is
With plausible expressions for the entropy and inter-
discussed in [24]). To see this, recall the decomposi-
nal energy of the P(E,l)-degrees of freedom, we have
tion we assumed for the energy of the slow subsystem,
been able to show that the SGR model admits an in-
U (S ,Λ) = U (Λ) + U (S − S (Λ)). The effective
terpretationconsistentwiththenon-equilibriumthermo- C C Λ 1 C Λ
temperature χ = ∂U /∂S | = ∂U /∂S depends then
dynamics framework of Bouchbinder and Langer. This C C Λ 1 1
on S = S −S . Assuming as before that S ≪ S
interpretation requires that the effective temperature x, 1 C Λ Λ C
because Λ only contains a vanishing fraction of the slow
postulated phenomenologically within the SGR model,
degrees of freedom, χ becomes a function of S ≈ S
is in fact the thermodynamic temperature χ of a con- 1 C
only. Conversely, χ then fully determines S , so that
figurational subsystem describing mesoscopic degrees of S˙ =(∂S /∂χ)χ˙ and Eq. (10) becomes C
freedom. (These degrees of freedom have slow energy C C
transfer to the thermal bath of ‘fast modes’ and hence Ceffχ˙ =W +A(θ−χ) (36)
V
their temperature is a nonequilibrium one.)
Webelievethisrepresentssignificantprogressinunder- Here CV (χ) is an effective heat capacity at constant
eff
standing the conceptual foundation of the SGR model. volume, and A(χ,θ) is as before positive but otherwise
The original empirical motivation for introducing x unspecified; in principle, A could also depend on other
was (and remains) intuitively reasonable: an effective quantities like γ˙ and possibly even P(E,l). What is in-
noise temperature arising ultimately from yield events teresting, however, is that the form of the driving term
throughout the sample. However a more solid formal W is not negotiable; this is from (20)
grounding is now provided by the link to a thermody-
namically defined temperature that governs the explo- W eu/χP(E)
= dEdlΓ(E,l)P(E,l)ln
ration of the system’s inherent structures, through the N χ Z ρ(E)
e
dynamics of its mesoscopic degrees of freedom. These e−E/χP(E)
degrees of freedom are taken to be in thermal equilib- −Y dEρ(E)ln (37)
Z ρ(E)
rium with each other, but not with the rest of the sys-
tem. If one accepts that, the thermodynamic interpre- While not obvious from this expression, it is clear from
tation gives new insights by showing, for example, that thedefinition(7)thatinsteadystate,whereΛ≡P(E,l)
x should be viewed as intrinsic to the state of the sys- doesnotchange,W =Vσγ˙ isjusttheexternalworkrate.
tem, and thereby to its history, as opposed to being set This is the form that was assumed recently in exploring
directly by the current rate of external driving. (The the extension of the SGR approach to understand shear
latter would make x-mediated effects second order, and banding: it was called ‘Model 1’ in [15]. Steady state
hence negligible, within linear response theory [23].) flow curves of shear stress σ versus γ˙ as predicted from
Of course, the consistency of the thermodynamic in- (36)wouldthereforelookexactlylikethosefoundin[15].
terpretation of x does not make this the only possible On the other hand, ‘Model 2’ from that paper, where
interpretation of the SGR model. It would be perfectly the driving term for χ is taken as proportional to the
defensible to believe that, in many soft glasses, meso- yield rate Y, is not thermodynamically consistent. We
7
note that even with model 1, Eq. (37) predicts different glass transition at χ = E . It is clear that allowing χ
0
transient behaviour, because the driving term W gener- to evolve dynamically according to (36) in line with the
ally differs from the external work rate Vσγ˙. It would thermodynamic picture will contribute additional aging
be interesting to explore in future work what effects this effects. However, what quantitative form these take can
thermodynamicallymotivatedformforW hasonthepre- only be fully assessed once physically motivated param-
dicted shear banding dynamics. eter dependences have been established for the effective
We can similarly assess the thermodynamic consis- heat capacity Ceff and the coefficient A determining the
V
tency of two extensions of the SGR model that were rate of heat transfer between the slow and fast subsys-
proposed some time ago to model shear thickening ef- tems.
fects [19]. That paper considered two models. In one, Acknowledgements: We thank Eran Bouchbinder,
the effective temperature x ≡ χ is taken as a function Mike Falk, Suzanne Fielding, Lisa Manning, and partic-
x(σ) of the overall stress. The second model, in a more ularly Jim Langer, for illuminating discussions. We also
radical departure from the standard SGR picture, takes thankKITPSantaBarbara,wherethisresearchwassup-
x(l)asafunctionofthelocalstrainineachelement. The portedinpartbytheNationalScienceFoundationunder
firstcase canfeasibly be accommodatedwithin the ther- Grant No. NSF PHY05-51164. MEC is funded by the
modynamic framework,at least if we consider situations Royal Society.
where the effective temperature x, in its time evolution
according to (36), has reached a steady state. Assuming
that θ ≪ x for simplicity, so that the effective tempera- Appendix A: More general conditions for second law
tureismuchlargerthanthebathtemperature,thesteady
state condition is x = W/A = Vσγ˙/A. Given that A is We have in the main part of the paper adopted the
only restricted to be positive but allowed to depend on argumentof[17]thatthethreetermsin(6)mustbe sep-
e.g. γ˙ and σ, there are certainly choices for this depen- arately non-negative. This is a sufficient condition for
dence that would give a steady state x depending (only) the overallsum to be non-negative andis also physically
on σ. plausible. However it is not mathematically necessary.
The second model in [19], where x=x(l), is different. We discuss here a somewhat more general set of condi-
Having a temperature that differs from one element to tionsthatarealsosufficient. Theseadmitthethermody-
the next is a significant departure from the idea of a namicconsistencyofawiderclassofmodelsthanallowed
single effective temperature for the slow configurational by the conditions of [17] as used above.
subsystem. Consistent with this, one can check that a Ifwewritethefirstlawasin(4)andinsertU˙R =θRS˙R,
generically chosen x(l) will violate the conditions (33). we obtain
ThisdoesnotofitselfproveW <0,butwehavechecked
that there are indeed conditions under which negative χS˙C +θS˙K +θRS˙R =W (A1)
W does arise, e.g. for the specific (stepwise decreasing)
This needs to be satisfied together with the second law
form of x(l) considered in [19]. Accordingly this second
model of [19] is not consistent with a thermodynamic S˙ +S˙ +S˙ ≥0 (A2)
C K R
interpretation of the effective temperature as described
here. With the reversible degrees of freedom (Λ) of the slow
Finally,weshouldnoteagainthatourthermodynamic system already treatedseparately in W, these two equa-
viewpoint for SGR was developed above by closely fol- tionsarecompletelysymmetricintherolesplayedbythe
lowingthe route firsttakenfor STZ by Bouchbinder and slow, fast and reservoir subsystems. A more general set
Langer[16–18]. Theseauthorsusedthesecondlawtoin- of conditions for the second law to hold then suggests
fer conditions on the dynamics of macroscopic variables. itself as W ≥0 and
However these conditions, while both plausible and suf-
χS˙ =α W +A(θ−χ)+C(θ −χ) (A3)
ficient,arenotinfactnecessary: inthe Appendix we de- C C R
scribe a less restrictive, but still sufficient, set. The true θS˙ =α W +A(χ−θ)+B(θ −θ) (A4)
K K R
value of thermodynamic consistency as a constraint on θ S˙ =α W +B(θ−θ )+C(χ−θ ) (A5)
R R R R R
mesoscopic rheological models will not be known with-
out a set of necessary, as well as sufficient, conditions where A, B, C, α , α , α are all non-negative and
C K R
for compliance of such models with the second law. It α +α +α =1. Thelastconditionisenoughtoensure
C K R
is possible that an explicitly model-independent formal- that the first law is satisfied, as the heat flow terms all
ism, such as that developed in [25], could proveuseful in cancel in pairs. For the total rate of entropy change one
identifying such conditions. gets
A further interesting direction for future work will be
α α α (χ−θ)2
toexplorethermodynamiceffectsontheagingbehaviour S˙ +S˙ +S˙ =W C + K + R +A
C K R
(cid:18) χ θ θ (cid:19) χθ
predicted by the SGR model. An important feature of R
the model is that aging effects are present [13, 14] even (θ−θ )2 (χ−θ )2
R R
+B +C (A6)
when χ is constant, as long as its value lies below the θθ χθ
R R
8
which is evidently non-negative. The more restrictive Λdegreesoffreedom. Insoftglasses,therefore,α could
R
condition of [17] corresponds essentially to α = α = besubstantial,andcorrespondinglyα <1. Analterna-
K R C
C = 0. One has then U˙ = θ S˙ = −B(θ −θ) as in tive strategy might be to exclude this contribution from
R R R R
(8), and theworkinput,byseparatingoffaglobalNewtoniansol-
ventcontribution. However,as the example of thin films
θS˙ + θ U˙ =A(χ−θ)+ B (θ −θ)2 (A7) betweenemulsiondropletsshows,themesoscopicandsol-
K θ R θ R vent contributions cannot be treated as simply additive:
R R
in principle the droplet organization controls both the
This generalizes the second equality of (8) to the case thicknesses of films and the shear rates within them. It
where θ 6= θR. In the limit where B becomes large at thereforeseemsadvisablenottoassumethatkindofsep-
constant B(θR −θ), the term on the r.h.s. proportional aration within a thermodynamic description, at least in
to B goes to zero so (8) is retrieved. That a term pro- the case of soft glasses.
portionaltoB(θ −θ)2 shouldbepresentingeneralhere
R Despite the above generalizations the condition that
is clear because θS˙ +(θ/θ )U˙ =θ(S˙ +S˙ ) and this
K R R K R W ≥0 remainsinplace. So farwe havenotbeen ableto
rate of entropy change should have a non-zero contribu-
showmathematicallythatW ≥0isnecessary(orindeed
tion when there is heat transfer at a finite rate between
that there are not broader conditions on the heat flows
the reservoir and the kinetic-vibrational subsystem.
that would satisfy the second law), although physically
Returning to our more general conditions above, the
it is certainly plausible that the dissipation rate should
effectofthecoefficientC isprobablyunimportantsolong
be non-negative. The main possible generalization aris-
as we are indeed interested only in the limit θ = θ
R ing from the arguments above is then in the equation of
because of strong thermal coupling to the reservoir: one
motion for χ, which would become (replacing (36))
can then just combine A and C.
The effect of the α coefficients is more subtle. In soft
Ceffχ˙ =α W +A(θ−χ) (A8)
materials, where the reservoirshould include the solvent V C
degrees of freedom, it seems plausible that a substantial
amountofthe workperformedonthe system(minus the Here one now has an additional undetermined factor
amount stored reversibly in Λ, which gives W) could be 0 ≤ α ≤ 1 in front of the driving term W. One
C
dissipateddirectlyinthesolvent. Forinstance,inadense couldspeculatewhetherthereisadifferencebetweene.g.
emulsion,workcouldbedissipateddirectlyinthesolvent metallic and soft glasses in the size of α , which could
C
films between droplets without ever passing through the be of order unity for the former and small for the latter.
[1] For a recent overview, see e.g. the Topical Issue on the [14] S. M. Fielding, P. Sollich, and M. E. Cates, J. Rheol.,
Physics of Glasses, Eur. Phys. J. E, 43(9), 2011. 44, 323 (2000).
[2] J.M.Brader,T.Voigtmann,M.E.Cates, andM.Fuchs, [15] S.M.Fielding,M.E.Cates, andP.Sollich,SoftMatter,
Phys.Rev.Lett., 98, 058301 (2007). 5, 2378 (2009).
[3] J. M. Brader, M. E. Cates, and M. Fuchs, Phys. Rev. [16] E. Bouchbinder and J. S. Langer, Phys. Rev. E, 80,
Lett., 101, 138301 (2008). 031131 (2009).
[4] J. M. Brader, T. Voigtmann, M. Fuchs, R. G. Larson, [17] E. Bouchbinder and J. S. Langer, Phys. Rev. E, 80,
and M. E. Cates, PNAS,106, 15186 (2009). 031132 (2009).
[5] P. H´ebraud and F. Lequeux, Phys. Rev. Lett., 81, 2934 [18] E. Bouchbinder and J. S. Langer, Phys. Rev. E, 80,
(1998). 031133 (2009).
[6] G.Picard,A.Ajdari,L.Bocquet, andF.Lequeux,Phys. [19] D. A. Head, A. Ajdari, and M. E. Cates, Phys. Rev. E,
Rev.E, 66, 051501 (2002). 64, 061509 (2001).
[7] L. Bocquet, A. Colin, and A. Ajdari, Phys. Rev. Lett., [20] M. Tsamados, A.Barra, and P. Sollich, unpublished.
103, 036001 (2009). [21] M. E. Cates and P. Sollich, J. Rheol., 48, 193 (2004).
[8] O.PouliquenandY.Forterre,Phil.Trans.R.Soc.Lond. [22] H. M. Wyss, K. Miyazaki, J. Mattsson, Z. B. Hu, D. R.
A,367, 5091 (2009). Reichman, andD.A.Weitz,Phys.Rev.Lett.,98,238303
[9] M.Talamali,V.Petaja,D.Vandembroucq, andS.Roux, (2007).
Phys.Rev.E, 84, 016115 (2011). [23] E.BouchbinderandJ.S.Langer,Phys.Rev.Lett.,106,
[10] V. Mansard, A. Colin, P. Chauduri, and L. Bocquet, 148301 (2011).
Soft Matter, 7, 5524 (2011). [24] E. Bouchbinder and J. S. Langer, Soft Matter, 6, 3065
[11] M. L. Falk and J. S. Langer, Ann. Rev. Cond. Matt. (2010).
Phys., Vol2, 2, 353 (2011). [25] H. C. O¨ttinger, Beyond equilibrium thermodynamics
[12] P. Sollich, F. Lequeux, P. H´ebraud, and M. E. Cates, (Wiley, New Jersey, 2005).
Phys.Rev.Lett., 78, 2020 (1997).
[13] P.Sollich, Phys. Rev.E, 58, 738 (1998).