Nonequilibrium Thermodynamics of the Kovacs Effect

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by  Eran Bouchbinder

Nonequilibrium Thermodynamics of the Kovacs Effect
Eran Bouchbinder
Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
J.S. Langer
Department of Physics, University of California, Santa Barbara, CA 93106-9530
(Dated: January 20, 2010)
WepresentathermodynamictheoryoftheKovacseffectbasedontheideathattheconfigurational
0 degrees of freedom of a glass-forming material are driven out of equilibrium with the heat bath by
1 irreversible thermal contraction and expansion. Weassume that the slowly varyingconfigurational
0 subsystem,i.e. thepartofthesystemthatisdescribedbyinherentstructures,ischaracterizedbyan
2 effectivetemperature,andcontainsavolume-related internalvariable. Weexaminemechanismsby
n whichirreversibledynamicsofthefast,kinetic-vibrationaldegreesoffreedomcancausetheentropy
a and the effectivetemperature of the configurational subsystem to increase during sufficiently rapid
J changes in the bath temperature. We then use this theory to interpret the numerical simulations
0 by Mossa and Sciortino (MS), who observe the Kovacs effect in more detail than is feasible in
2 laboratory experiments. Our analysis highlights two mechanisms for the equilibration of internal
variables. In one of these, an internal variable first relaxes toward a state of quasi-equilibrium

] determined by the effective temperature, and then approaches true thermodynamic equilibrium as
i theeffective temperature slowly relaxes toward thebath temperature. In the other mechanism, an
c
s internalvariabledirectlyequilibrateswiththebathtemperatureonintermediatetimescales,without
- equilibrating with the effective temperature at any stage. Both mechanisms appear to be essential
l
r for understandingtheMS results.
t
m
t. I. INTRODUCTION fects have been observed in many other glassy systems
a such as colloidal glasses [7, 8], ferroelectrics [9, 10],
m
gelatin gels [11], granular materials [12], superparamag-
The Kovacs effect reveals some of the most subtle and
- nets andsuperspinglasses[13]. It alsohas been the sub-
d important nonequilibrium features of glassy dynamics.
ject of various recent theoretical investigations [14–20].
n In particular, it provides detailed information about the
o ways in which glassy materials deform irreversibly and In this paper, we look at the Kovacs effect from the
c
remember their histories of deformation [1–4]. Here, we point of view of our recent attempts to develop a first-
[
develop a thermodynamic theory of the Kovacs effect, principles,statisticalformulationofnonequilibriumther-

1 motivated in large part by the molecular-dynamics sim- modynamics [21–23]. Generally speaking, our goal is to
v ulations of Mossa and Sciortino (MS)[5]. reinterpret the analysis of Kovacs et al. [2] in terms of
1
0 Ina Kovacsexperiment,the volume ofa glass-forming specific molecular processes. Our work differs from that
7 system is measured at fixed pressure as the temperature of Nieuwenhuizen, Leuzzi, and coworkers[15, 19, 20], for
3 is varied. A sample is first quenched from a high tem- example,inthatwestartfromafundamental, statistical
. perature T to a temperature T low enough – i.e. near statement of the second law of thermodynamics and use
1 h ℓ
0 enough to the glass temperature Tg – that some inter- ittoderiveequationsofmotionforrelevantinternalstate
0 nal degrees of freedom fall out of equilibrium with the variables as well as for an effective temperature. We dif-
1 heat bath. The system is then aged at T , for times fer also from Bertin et al [14], who have solved specific
ℓ
: insufficient to reach thermal equilibrium, and finally is modelsandhaveshownhowphenomenaanalogoustothe
v
i heated abruptly and held at a temperature Tf such that Kovacs effect emerge in interesting ways.
X
Tℓ < Tf < Th. The crucial observation is that, in this As in [22], our starting point is the assumption that
ar last stage of the Kovacs protocol, the volume does not a glass-forming material consists of two weakly inter-
increase monotonically as a function of time, but goes acting subsystems. The configurational (C) subsystem
throughamaximumbeforedecreasingslowlyto its equi- is specified by the set of mechanically stable molecular
libriumvalueatTf. Thefactthatthesystemcanexistin positions, that is, the inherent structures [24, 25]. The
two different states, on the upward and downward sides kinetic-vibrational (K) subsytem is specified by all the
of the Kovacs volume peak, at the same temperature, other degrees of freedom – the kinetic energies, the dis-
pressure and volume, indicates that these are not states placements of the molecules from their stable positions,
of thermal equilibrium. It is important to understand and,inthecaseofmorecomplexmoleculessuchasthose
how to characterize them. relevanttotheKovacseffect,theinternaldegreesoffree-
The Kovacseffect originally was observedin polyvinyl dom of these molecules. The physical rationale for this
acetate [1], but since then it has been observed in many separation is the distinction between the time scales for
other glassy polymers, see for example [6] for measure- dynamic processes in the two subsystems. In situations
ments in polystyrene. Qualitatively similar memory ef- where the system is driven by external forces, these two
2
subsystems can fall out of thermodynamic equilibrium potential. Theircrucialresultisthat,whenT isincreased
with eachother. The fast, K-subsystemremains in equi- from T to T , both the volume and the inherent struc-
ℓ f
librium with the heat bath at temperature T; the slower tureenergyincreaseandgothroughmaxima. Thisresult
C-subsystem, at least transiently, has an effective tem- tells us that the effective temperature and the vacancy
perature χ (in energy units) that is different from k T. population also increase and go through maxima, and
B
WeproposethatanymodeloftheKovacseffectshould that the thermal expansion driven by the change in the
include three essential ingredients. First, and most bathtemperatureispartiallyirreversible. Moreprecisely,
obviously, we need to specify the configurational (C- duringthermalexpansionorcontractionatnonzeropres-
subsystem) degrees of freedom that fall out of thermal sure, the system exchanges mechanical energy with its
equilibrium as the whole system is quenched into the surroundings. Someofthatenergyisdissipated,produc-
vicinity of its glass temperature. These degrees of free- ing configurational entropy.
dom must describe structural features that change via One of the most remarkable results of Mossa and
slow molecular rearrangements and thus do not keep Sciortino[5]isshownintheirFig. 4. Therethey demon-
up with more rapid variations of the bath temperature. strate that, after the volume and the inherent structure
Since we are interested in volume changes, the simplest energy go through maxima, the system can be described
choiceofthisinternalvariableisapopulationofvacancy- by states of quasi-equilibrium fully characterized by the
like defects; but other internal degrees of freedom that effective temperature. During the earlier stages, how-
couple to the volume might serve our purposes equally ever,thisisclearlynotthecase. There,somethingelseis
well. happeningthatchallengesourunderstandingofnonequi-
Second,weneedtoensurethattheequationsofmotion librium thermodynamics. That “something else” is the
forthisout-of-equilibriumbutstatisticallysignificantde- central theme of the present investigation.
fectpopulationareconsistentwiththelawsofthermody- The scheme of this paper is as follows. In Sec. II,
namics. We have argued in [22] that the natural way to we describe our two-subsystem model and comment on
do this is to use the effective temperature ofthe configu- its physical ingredients. The thermodynamic equations
rational subsystem, in direct analogy with Gibbsian sta- of motion for this model are derived in Sec. III, where
tistical mechanics, to determine the states of maximum weshowhowthetwomechanismsofirreversiblethermo-
probability through which the configurational degrees of viscoelasticityemergefromournonequilibriumstatistical
freedom are moving. These configurational degrees of analysis. SectionIVisdevotedtoidentifyingappropriate
freedomhavewelldefinedenergiesU andentropies S ; dimensionlessvariablesandmakingfirstestimatesofthe
C C
thus they have an effective temperature χ=∂U /∂S , parameters that appear in the scaled equations.
C C
and their equations of motion must be based on their In Sec. V, we compare the predictions of our theory
effective thermodynamics. with the numerical simulations of Mossa and Sciortino
Third, and least obviously, we need mechanisms by [5], and confirm that each of our three ingredients of
which variations of the ordinary temperature of the K- a Kovacs model is, indeed, essential for understanding
subsystem can produce changes in the effective temper- theirdata. Duringthereheatingstage,asT risesquickly
ature of the C-subsystem. This means that we must from T to T , both the fast Kelvin-Voigt-type and the
ℓ f
understand how, and under what circumstances, ordi- somewhat slower Maxwell-type mechanisms are needed
nary thermal expansion and contraction become irre- as sources of configurational entropy. These sources in-
versible phenomena that can increase the entropy of crease the volume on short and intermediate timescales
the system as a whole. Our nonequilibrium thermo- anddriveanincreaseintheeffectivetemperatureχ. The
dynamic formulation suggests that there are two dis- volumecontinuestoriseasthevacancypopulationgrows
tinct thermo-viscoelastic mechanisms that can be rele- toward a quasi-equilibrium value determined by the in-
vant here. The first is a Kelvin-Voigt-type mechanism creased χ. After this quasi-equilibrium is established, at
in which the K-subsystem exhibits a bulk viscosity aris- aboutthe time that the volume reachesits Kovacspeak,
ingdirectlyfromfastmolecularinteractions. Thesecond thevacanciesremaininequilibriumwithχasthesystem
isasomewhatslowerMaxwell-typemechanisminvolving slowlyagestowarda finalstate withχ=k T . Inshort,
B f
volume-related internal degrees of freedom that equili- we recover the results of Mossa and Sciortino and fully
brate directly with the ordinary temperature T rather agree with their interpretation of them.
than the effective temperature χ. (Our main reference
for models of thermo-viscoelasticityis Maugin’s book on
The Thermomechanics of Nonlinear Irreversible Behav-
iors, [26]). II. INGREDIENTS OF A TWO-SUBSYSTEM
MODEL
The work Mossa and Sciortino [5] offers a unique op-
portunity to test the ideas described above. These au-
thorsperformedmoleculardynamicssimulationsofaKo- Denotethetotal,extensive,internalenergyofthetwo-
vacs experiment using the Lewis and Wahnstro¨m model subsystem model by
[27]ofortho-terphenyl(OTP),inwhichthemoleculesare
rigid isosceles triangles interacting via a Lennard-Jones U =U (S ,V ,N )+U (S ,V ,N ). (2.1)
tot C C el v K K el a
3
Theconfigurational(C-subsystem)energy,U ,isafunc- units) is
C
tionofthe configurationalentropyS , anelasticvolume
C
∂U
Vel that is common to both subsystems, and an exten- χ= C . (2.4)
sive number of vacancy-like defects Nv whose energies (cid:18)∂SC(cid:19)Vel,Nv
andexcessvolumesaree andv respectively. Similarly,
v v Define the partial-pressure functions:
the kinetic-vibrational (K-subsystem) energy, U , is a
K
function of the kinetic-vibrational entropy SK, Vel, and p (χ,V )=− ∂FC ; (2.5)
the number N of what we call “misalignment” defects C el
a ∂V
with energies e and excess volumes v . (cid:18) el(cid:19)χ,Nv
a a
Assume that a fixed number of molecules, say N , oc- and
0
ecxucpeiesss tvhoeluemlaestoicf vthoeludmeefeVcetls,.wThhicehndtoheestnoottalinvcolluudmeethoef pK(θ,Vel)=− ∂FK ; (2.6)
∂V
the system is (cid:18) el(cid:19)θ,Na
where the free energies F are the Legendre transforms
V =V +N v +N v . (2.2) of the U’s. Note, however, that we do not immediately
tot el v v a a
identify p +p as the total applied pressure. We do,
C K
however,assume strictly linear elasticity by writing
Our picture of the vacancy-like defects in the C-
subsystem is a slight oversimplification but seems con- F (χ,V ) λ 2
ceptually simple. In contrast, the misalignment defects C V el = 2VC2 Vel−VC(χ) +fC(χ), (2.7)
require more discussion. The triangular geometry of an 0 0 h i
ortho-terphenyl molecule means that its volume and its where V0 is a reference volume, λC is a compression
energydependonitsorientationwithrespecttoitsneigh- modulus, VC(χ) is the relaxedC-subsystem volume,and
bors. Thus, even in the absence of the vacancy-like de- fC(χ) is a volume-independent free-energydensity. Sim-
fects that may characterize the slowly fluctuating con- ilarly,
figurational subsystem of OTP, there are local misalign-
F (θ,V ) λ 2
ment defects that couple to the volume and can partic- K el = K V −V (θ) +f (θ). (2.8)
V 2V2 el K K
ipate in energetically irreversible processes. If the for- 0 0
h i
mation energies of these defects are not too large, and if ThetemperaturedependentreferencevolumesV (χ)and
C
the energy barriers that resist their transitions from one V (θ) are different from each other. The elastic energy
K
orientation to another are small enough, then these de- oftheC-subsystemhasitsminimumatarelativelysmall
fects equilibrate quickly with the bath temperature and volume V (χ), because that system is cohesive at zero
C
canlegitimatelybeincludedinthekinetic-vibrationalK- pressure. In contrast, the energy of the K-subsystem
subsystem. Forsimplicity,wehaveassumedinEqs. (2.1) decreases as the volume increases, because the kinetic
and(2.2)thatthereis onlyonekindofmisalignmentde- energy is fixed and the vibrational modes become softer
fect. as the spacing between the molecules increases. Thus,
The assumption that the misalignment defects belong strictly speaking, V (θ) must actually be defined at a
K
in the fast K-subsystemis not trivial. Our model is sim- positive reference pressure;but, since we are considering
ilarto onestudied byIlg andBarrat[28], whoshowthat onlylinearelasticity,thereisnoneedtobespecificabout
the equilibration rate for a similar class of dynamical this definition. In any case, the partial pressures defined
inclusions in a driven glass former depends sensitively in Eqs. (2.5) and (2.6) are different from each other and
on the strengths of the thermal noise sources that acti- most likely have opposite signs, i.e. p <0<p . This is
C K
vate their transitions across internal barriers. We will indeed the case in the simulations of MS [5].
show, however, that the high-temperature assumption
works well for present purposes, and that these inter-
nal, orientational degrees of freedom produce a model III. THERMODYNAMIC EQUATIONS OF
of bulk thermo-viscoelasticitythat is consistentwith the MOTION
MS data.
The temperature of the K-subsystem is A. First and Second Laws
∂U The first law of thermodynamics is
K
θ =k T = . (2.3)
B
(cid:18)∂SK(cid:19)Vel,Na − pV˙tot =U˙tot, (3.1)
We assume that the K-subsystem, in addition to having where p is the applied pressure. With Eqs. (2.2), (2.3),
its own internal dynamics, plays the role of a thermal (2.4), (2.5) and (2.6), Eq.(3.1) becomes
reservoir,so that θ is the temperature that is being con-
∂U
trolledasafunctionoftimeduringaKovacsexperiment. χS˙ + pv + C N˙ + θS˙
C v v K
The effective temperature of the C-subsystem(in energy " (cid:18)∂Nv(cid:19)SC,Vel#
4
∂U TheinequalityinEq.(3.4)issatisfiedbywritinganequa-
+ pv + K N˙
" a (cid:18)∂Na(cid:19)SK,Vel# a tion of motion for Vel:
+ p−pC(χ,Vel)−pK(θ,Vel) V˙el =0. (3.2) τ0V˙el =−γ Vel−Veelq(χ,θ,p) , (3.8)
h i h i
The second law is
whereτ isa moleculartime scaleandτ /γ is abulk vis-
0 0
cosity. According to Maugin [26], this is a Kelvin-Voigt-
S˙ =S˙ +S˙ ≥0. (3.3)
tot C K type thermo-viscoelasticity. When rewritten in terms of
the pressures, Eq.(3.8) says that the driving force p is
Using Eq.(3.2) to eliminate S˙ , we write Eq.(3.3) in the
C equal to an elastic term, p +p , plus a viscous force
form: proportional to V˙ . C K
el
Toseeinmoredetailwhatishappeninghere,interpret
∂U
− pv + K N˙ (3.4) thefirst-lawinEq.(3.2)asanequationforχS˙ ,andnote
a ∂N a C
" (cid:18) a(cid:19)SK,Vel# that the contribution to the configurationalheating rate
from the term proportionalto V˙ is
∂U el
− pv + C N˙
0 v
∂N
" (cid:18) v(cid:19)SC,Vel# γλ¯ 2
− p−p (χ,V )−p (θ,V ) V˙ = V −Veq(χ,θ,p) .
− p−pC(χ,Vel)−pK(θ,Vel) V˙el− (θ−χ)S˙K ≥0. C el K el el V0 el el
h i (cid:2) (3.i9)
h i
Ordinarily, this term is negligible. In the absence of a
Following the procedure described in [21, 22], we rec-
slow, internal, dissipative mechanism, the viscosity τ /γ
ognizethatEq.(3.4)consistsoffourseparateinequalities 0
is microscopically small. Suppose that some quantity on
associatedwiththeindependentlyvariablequantitiesN˙ ,
a the right-hand side of Eq.(3.8), perhaps p or θ, is var-
N˙ , V˙ , and S˙ , and therefore we must satisfy four sep-
v el K ied at an experimentally feasible rate, say ν/τ ≪γ/τ .
arate inequalities. In the next paragraphs, we look at 0 0
By dimensional analysis of Eq.(3.8), we find that ν/γ∼
these in reverse order of their appearance here. (V −Veq)/V . If we then integrate the right-hand side
el el 0
of Eq.(3.9) over a time of the order of ν−1, we find that
the change in the total heat energy is of the order of
B. Aging Rate λ¯V ν/γ, which vanishes when ν/γ →0. In this limit,
0
the system becomes thermodynamically reversible. The
The inequality proportionalto S˙ is satisfied by writ-
K solution of Eq.(3.8) is accurately p=p +p , which is
K C
ing the usual thermodynamic identity. In other words, we
haverecovereda specialexampleofthe generalrule that
χ
θS˙K =−A(χ,θ) 1− , (3.5) equilibrium thermodynamics is valid when systems are
θ
driven quasistatically.
(cid:16) (cid:17)
where A(χ,θ) is a non-negativethermaltransportcoeffi- Butthe Kovacseffectis anexceptionto this rule. The
cient. θS˙ is the rate at which heat is flowing from the original Kovacs observations were made with a glassy
K
C-subsystem to the K-subsystem. Since we assume that polymer, where the internal timescales τ0/γ are long, so
the coupling between the C and K-subsystems is weak, that it is possible to change temperatures and pressures
weexpectA(χ,θ)tobe small. Thisisthetermthatcon- relativelyrapidly. Itisalsoquiteeasytodothisinmolec-
trols the rate at which the system ages in the absence of ulardynamicssimulations,whichiswhathappens inthe
external driving. MScomputations. WewillseeinSec. VthattheKelvin-
Voigt-type dissipation is an important driving force for
the Kovacs effect.
C. Kelvin-Voigt-Type Thermo-Viscoelasticity
Next, consider the part of the inequality proportional D. Maxwell-Type Thermo-Viscoelasticity
toV˙ . UsethedefinitionsofthepartialpressuresinEqs.
el
(2.5)and(2.6),plustheelasticfreeenergiesinEqs. (2.7) We turn finally to the terms proportional to N˙ and
a
and (2.8), to write N˙ in Eq.(3.4). These terms produce a Maxwell-type
v
thermo-viscoelasticity, according Maugin [26]. To see
λ¯
p−p (χ,V )−p (θ,V )= V −Veq(χ,θ,p) , this, look at the time derivative of the expression for
C el K el V0 el el the totalvolume in Eq.(2.2). If there is no Kelvin-Voigt-
(cid:2) (i3.6) type viscous pressure proportional to V˙ , then the total
where λ¯ =λC +λK, and deformationrate V˙ is the sum of an eelalstic term V˙ =
tot el
1 −V0p˙/λ¯, and a viscoelastic term vaN˙a + vvN˙v. Our
Veelq(χ,θ,p)= λ¯ λKVK(θ)+λCVC(χ)−pV0 . (3.7) problemreducestofindinganexpressionforvaN˙a+vvN˙v
h i
5
thatwillserveasaconstitutiverelationfortheviscoelas- We assume that the entropy associated with N is the
v
tic (viscoplastic) part of the deformation rate. To solve same function S (N ) that we introduced in Eq.(3.12);
0 v
this problem, we follow steps outlined in [21]. therefore
Startwith the K-subsystem. Assume thatthe entropy
N
and energy of this system consist of separate, additive Neq(χ,p)= 0 , (3.21)
v ehv/χ+1
contributions,firstfromthedefectsand,second,fromall
the other degrees of freedom: where h = e + pv . We note that the idea that
v v v
an internal variable can be transiently driven out of
S (U ,V ,N )=S (N )+S (U ,V ); (3.10)
K K el a 0 a 1 1 el quasi-equilibrium with the effective temperature, as in
Eq.(3.20), was introduced earlier in [29], where it was
U (S ,V ,N )=N e +U (S ,V ); (3.11) usedtodescribetheinternaldynamicsofdeforming,sim-
K K el a a a 1 1 el
ulated, amorphous silicon.
where, for simple defects without internal structure of
their own,
E. Equation of Motion for the Effective
S0(N)=N0 lnN0− N lnN −(N0−N) ln(N0− N), Temperature χ
(3.12)
and N , as defined earlier, is the total number of
0 Putting these pieces together, we rewrite Eq.(3.2) as
molecules. Then,
an expression for the heat flow into the C-subsystem:
UK(SK,Vel,Na)=Naea+U1 SK −S0(Na),Vel , χ S˙ =− h −χ∂S0(Nv) N˙ − h −θ ∂S0(Na) N˙
C v v a a
h (i3.13) (cid:20) ∂Nv (cid:21) (cid:20) ∂Na (cid:21)
so that γλ¯ 2 χ
+ V −Veq(χ,θ,p) +A(χ,θ) 1− . (3.22)
∂UK =e +θ dS0. (3.14) V0 el el θ
∂N a dN (cid:2) i (cid:16) (cid:17)
(cid:18) a(cid:19)SK,Vel a The first three of these terms are non-negative rates of
configurationalheat production; the last termis the (or-
TheinequalityassociatedwiththeN˙ terminEq.(3.4)
a dinarily negative) rate at which heat flows from the K-
hastheformofaClausius-Duhemrelation[26],enforcing
subsytem into the C-subsystem.
non-negative entropy production:
To convert this result into an equation of motion for
χ, write
∂U ∂G
− pv + K N˙ =− K N˙ ≥0,
" a (cid:18)∂Na(cid:19)SK,Vel# a (cid:18)∂Na (cid:19)θ,p a χS˙C = χ∂SC χ˙ +χ∂SC N˙v+χ∂SC V˙el
(3.15) ∂χ ∂Nv ∂Vel
where ∂S ∂p
= Ceffχ˙ +χ 0 N˙ +χ C V˙ . (3.23)
v e
G (θ,p,N )=h N −θS (N ); (3.16) ∂Nv ∂χ
K a a a 0 a
The second term exactly cancels the term proportional
and h =e +pv . We satisfy Eq.(3.15) by writing
a a a to ∂S /∂N on the right-hand side of Eq.(3.22), which
0 v
τ N˙ =Γ [Neq(θ,p)− N ], (3.17) becomes
0 a K a a
∂p ∂S (N )
where Γ is a dimensionless rate factor and τ is the Ceff χ˙ =−h N˙ − χ C V˙ − h −θ 0 a N˙
K 0 v v e a a
∂χ ∂N
samemoleculartimescalethatweintroducedinEq.(3.8). (cid:20) a (cid:21)
Naeq(θ,p) is the equilibrium value of Na determined by + γλ¯ V −Veq(χ,θ,p) 2+A(χ,θ) 1−χ . (3.24)
V el el θ
∂G 0
K =0 at N =Neq(θ,p); (3.18) (cid:2) i (cid:16) (cid:17)
∂N a a
(cid:18) a(cid:19)θ,p
IV. SCALING AND APPROXIMATIONS
which means that
Neq(θ,p)= N0 . (3.19) We have arrived at a complex set of equations with
a eha/θ+1 manyvariablesandparameters. Beforeusingtheseequa-
tions for data analysis, we rewrite them in terms of di-
The same analysis pertains to the vacancy-likedefects
mensionlessquantitiesand,wherepossible,identifyphys-
in the C-subsystem. The equation of motion for N is
v ically motivated estimates for some of the parameters.
the same as Eq.(3.17), but with v and e replacing v
v v a To start, rescale the time so that τ =1. Then define
0
and e , with χ instead of θ, and with a new rate factor
a the defect densities:
Γ :
C
N N
v a
τ0N˙v =ΓC [Nveq(χ,p)− Nv]. (3.20) nv = N0; na = N0; (4.1)
6
and the volume fractions: Lennard-Jones potential, we estimate the molecular vi-
brationperiodtobeabout1.5picoseconds. Therefore,it
V V Veq
φtot = Vto0t; φel = Ve0l; φeeql = Ve0l . (4.2) ipss.conSvimeniileanrlty,tofrcohmootsheeouchraurancitteroifsttiicmeenetorgybescτa0le=o1f
this potential, we estimate that h ∼ 0.1 eV, so that
We also need to express the defect volumes in units of v
T ∼1300K. The misalignment defects must have sub-
the volume per molecule: v
stantially smaller formation energies. If we guess that
N0 N0 the difference is roughlya factoroften, then Ta∼130K;
φv = vv; φa = va. (4.3) so thatthese defects are far frombeing frozenout atthe
V V
0 0
lowesttemperaturesusedbyMS,i.e. atT =150K. This
ℓ
Therefore same estimate of hv, combined with V0/N0 ∼ 0.4nm3,
and an estimate for the bulk modulus λ¯∼4 GPa, tells
φtot =φel+φana+φvnv; (4.4) us that b∼50, which means that the heating rate asso-
ciatedwiththe Kelvin-Voigt-typeterminEq.(4.10)may
and be substantial.
φ˙el =−γ(φel−φeeql). (4.5) These estimates have interesting implications for our
choices of the rate factors. Clearly, with T ∼130K, the
a
Measure χ in units of the vacancy enthalpy:
transitionrateforthe misalignmentdefects is notappre-
χ ciably limited by an activation barrier; so ΓK must be
χ˜= ; (4.6)
only moderatelyslowerthanthe molecularrate γ,which
h
v
by definitioncannotbe significantlydifferentfromunity.
thus, Our first guess is that Γ is in the range 10−2 to 10−1.
K
On the other hand, Γ should contain an effective ther-
C
n˙ =Γ 1 −n ≈Γ e−1/χ˜−n , (4.7) mal activation factor of the form exp(−∆˜C/χ˜), where
v C e1/χ˜+1 v C v ∆˜ is the excess barrier, in units of h , that the system
(cid:18) (cid:19) (cid:16) (cid:17) C v
must surmount in either creating or annihilating a va-
where the last approximation is valid in the low-density
cancy. If we assume that the system is fully equilibrated
limit,χ˜≪1. Forcomparisonwithexperimentaldata,itis
at T =T =400K, then the initial value of χ˜ is equal
h
convenient to express θ in units of absolute temperature to k T /h ∼0.33. Assuming that ∆˜ is not too much
B h v C
T, so that
smaller than unity, we conclude that Γ may be smaller
C
than Γ by a factor of ten or so.
1 K
n˙ =Γ −n , (4.8)
a K (cid:18)eTa/T +1 a(cid:19) The more interesting rate factor is ΓA, which, un-
like Γ in Eq.(4.7), is not the prefactor in a creation-
where k T ≡h . C
B a a rate formula that already contains an activation factor
For simplicity, assume that the relaxed volume of the
exp(−1/χ˜). In other circumstances, such as a calcula-
C-subsystem, V (χ), is independent of χ, and write
C tion of the α relaxation rate or the shear viscosity in
φeq(T,p)∼=φ +φ (T), (4.9) a glass forming material that is moving so slowly that
el 0 1 χ≈k T, the analog of Γ would be a super-Arrhenius
B A
where φ = 1−p/λ¯, and φ (T) describes the ordinary function of T. Here, although we are talking about the
0 1
thermalexpansionandcontractionthatdrivetheKovacs slowagingpartofaKovacsexperiment,wearelookingat
experiment. In general, φ (T) is a nonlinear function theearlytransientstagewhereχ˜isstillsomewhatbigger
1
over the range of temperature jumps used by MS, and thanT/TvinEq.(4.10). Accordingly,weassumethatthis
we will need to use their data to evaluate it. rate at which the two weakly coupled subsystems equili-
Theequationofmotionforχ˜,i.e. Eq.(3.24),nowreads brate with each other is limited by a substantial energy
barrier, say ∆˜ ≥1; and we write
A
T n
c˜eff χ˜˙ =−n˙ −β 1+ ln a n˙
v T 1−n a
(cid:20) a (cid:18) a(cid:19)(cid:21)
+ γb φel−φeeql(T,p) 2+ΓA(χ˜,T) TT −χ˜ .(4.10) ΓA(χ˜,T)∼=Γ(A0)(T)e−∆˜A/χ˜. (4.11)
h i (cid:18) v (cid:19)
Here, c˜eff = Ceff/N , k T = h , β = h /h = T /T ,
0 B v v a v a v
b=(λ¯V /h N ) and Γ (χ˜,T)=A(χ,θ)/(N k T). If the super-Arrhenius analogyis valid, Γ(0)(T) will be a
0 v 0 A 0 B A
rapidly varying function of T that becomes vanishingly
We can make rough estimates for some of these pa- smallbelowT . AtconstantT,whileχ˜>T/T ,Eq.(4.11)
g v
rametersusing knownproperties ofthe OTP model sim- implies that the aging rate slows exponentially as χ˜ de-
ulatedbyMS.Forexample,takingparametersfromtheir creases.
7
V. COMPARISONS WITH THE DATA OF
0.38
MOSSA AND SCIORTINO
0.37
0
N
In their molecular dynamics experiments, MS are able V/tot0.36
0.35
to resolve dynamic variations in their model of ortho-
terphenyl on time scales as small as tens of picoseconds. 0.340 0.5 1 1.5 2 2.5 3 3.5 4 4.5
log (t )
In addition to observing volume changes, they can ob- 10 e
serve the energy, the pressure, and the shape factor (a
0.4
measure of the width of the energy basins) of the inher- 0.35
ent structures throughout the Kovacs protocol. Thus, hv 0.3
χ/
theyprobetheKovacsphenomenatoadepththatseems 0.25
impossible for laboratory experiments. 0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Thereare,however,compromisesthatmustbemadein log (t )
10 e
suchaprocedure. MSsimulateasystemofonly343OTP
molecules. Althoughthey averagetheir resultsoverhun-
FIG. 1: Aging at T =280K after an instantaneous quench
dredsofinitialconfigurations,itishardtoruleouteffects ℓ
fromT =400K. Upperpanel: timeevolutionofthetheoret-
h
of numericalnoise, especially in the low-temperature ag-
ical volume per molecule V0/N0 (solid line) compared to the
ing calculations that must be dominated by very rare simulation data (open circles) extracted from MS Fig. 1(c)
events. [5]. The parameters used for the theoretical curve can be
Moreover, to control temperature and pressure, MS found in the text. Lower panel: the corresponding reduced
use a thermostat and a barostat with a time constant of effectivetemperature χ˜=χ/hv.
20 ps; and they state that their systems are too far out
of equilibrium for the data to be meaningful on shorter
time scales following the initial quench or the reheating
step. In spite of these uncertainties, we decided to try 0.42
to model the complete Kovacs data reported by MS. We
na 0.4
computedthevalues ofthe volume,the effectivetemper-
ature, and the defect densities at the end of the aging 0.38
stage, and used these values as the initial conditions for 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
log (t )
the reheatingstage. Notethatthe interestingfeaturesof 10 e
0.06
the Kovacs effect are very small; i.e. the fractional vol-
ume change associated with changes in χ near the Ko- 0.04
vacspeakisonlyoftheorderof10−3. Therefore,insome nv
0.02
places, we have adjusted the values of our parameters to
0
three or more significant figures in order to make quan- 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
log (t )
titative comparisons with the MS data. 10 e
Our data fitting procedure for the results presented 1
here started by choosing
φel0.95
φeq(T =400K)=φ =1−p/λ¯ =0.9947, (5.1)
el 0
0.9
which was based on the estimate p=16MPa and λ¯ = 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
log (t )
3GPa. We thenestimatedT =1300K,T =130K,v = 10 e
v a v
0.07nm3, and v =0.007nm3. Assuming full equilibrium
a
at T =400K, χ˜=T /T , we computed n and n at
thathtemperature. MShrepvortthattheirtotalvvolumeaper FIG. 2: na, nv and φel corresponding to Fig. 1.
moleculeat400K is0.378nm3. Thesenumbersuniquely
determine V /N =0.374nm3.
0 0
280Kandsubsequentagingasfunctionsofthetimeafter
Fromhere on, we chose parametersin accordwith our
quench t . (All times are stated in picoseconds.) The
rough estimates at the end of Sec. IV, and refined these e
estimates to improve the agreement with the data. In theoretical parameters were γ=10−1.13 and Γ(0)=101.8.
A
addition to those cited in the last paragraph, the fol- We need a Kelvin-Voigt-type viscosity with a small γ
lowing numbers were used throughout the calculations: becausetheelasticpartofthevolumeinitiallyrelaxeson
b = 30, ∆˜ = 3.5, Γ = 10−1.75, and Γ = 10−2.25. a time scale of the order of γ−1∼15ps. The resulting
A K C
Our best-fit values for the elastic volume fractions at dissipation drives a rapid increase in χ; and then both
T =150K and 280K were φeq(T =150K)=0.9037 and na and nv participate in the change of the total volume
el
φeq(T=280K)=0.9323. on time scales determined, respectively, by ΓK and ΓC.
el
In Figs. 1 and 2, we show theory and data for the MS Next consider the quench from T = 400K to T =
h ℓ
instantaneous quench from T =400K directly to T = 150K and subsequent aging at the latter temperature.
h f
8
0.38
N00.36 0.5
V/tot0.34 na0.4
0.3
0.32
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2
log (t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
10 e log (t)
10 e
0.35
0.06
hv 0.3 0.05
χ/ nv0.04
0.25
0.03
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.02
log (t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
10 e log (t)
10 e
1
FIG. 3: The same as Fig. 1, but for T = 150K, after a
ℓ
smooth quench from T =400K according to Eq.(5.2). See
h 0.95
text for the parameters used. The simulation data extracted φel
from MS Fig. 1(c) [5].
0.9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
log (t)
10 e
In this case, MS have told us that they used a smooth,
thermostatically controlled decrease in the temperature,
and started to measure the volume at about 30 ps after FIG. 4: na, nv and φel corresponding to Fig. 3.
thequenchwasstarted[30]. Accordingly,wehaveshifted
our time scale by 30 ps; and we have modeled the initial
temperature dependence by writing quickly, due to the combination of both Kelvin-Voigt-
type and Maxwell-type mechanisms. The density of K-
T(t )=T +(T − T ) exp − te , (5.2) subsystem defects, na, rises toward equilibrium with Tf
e ℓ h ℓ
(cid:18) τth(cid:19) at a rate ΓK; then the vacancy density nv rises toward
equilibrium with χ at a rate Γ ; and finally n , now at
C v
with τth=4ps. To use this equation for values of T be- its quasi-equilibrium value as a function of χ, decreases
φtweeqle(eTn)TbheatwndeeTnℓ,thweevhaalvueesmgaivdeenaalbinoevaerfionrteφreepqlo(lTahti)onanodf asFχigduercere7assehsowsloswthlyetKowovaardcskpBeaTkf.from Fig. 5 and the
φeeql(Tℓ). We used γ =1 and Γ0A =10−0.2. The results comparable peak for a shorter waiting time, te = 1ns,
are shown in Figs. 3 and 4, along with the MS data along with the MS data for both cases. The agreement
for the volume. With the more gradual quench and the seems excellent in view of the fact that we computed
largervalue of γ,the Kelvin-Voigt-typeeffect is less pro- the second curve only after having determined all of the
nounced but still present. The important feature here parameters from the preceding calculations.
is the very slow aging at long times, associated with Finally, in Fig. 8, we show the inherent-structure en-
the smaller value of Γ0A and the controlling effect of the ergy eIS for the reheating stage shown in Fig. 5. To fit
effective-temperature activation barrier ∆˜A. the MS data, we have used
The principal Kovacs effect occurs during and after
reheating from Tℓ=150K to Tf=280K, starting at the e ∼= λ¯V0 φ −φeq(T ) 2+n k T +n k T
end of the aging period t=te. Because MS increased T IS 2N0 el el ℓ a B a v B v
smoothly during reheating, we have used h ′ i
+ e¯ +e¯ k T χ˜ , (5.4)
IS IS B v
T(t)=Tf −(Tf −Tℓ) exp −t−te , (5.3) with e¯IS = −84.32 kJ/mol and e¯′IS = 0.08. Note
(cid:18) τth (cid:19) that our inherent-structure energy eIS contains not only
U (S ,V ,N ), but also a contribution from the mis-
C C el v
bageatwineewnitthheτltohw=er4apnsd, uapnpderwvitahlueasloinfeφaeerql(iTnt)e.rpFoolrattihoins UalKignfomretnhterdmefoedcytsnaNmaictrheaatsownes.eTarhlieercoanrvgeunetdionbaell,osntgattioc
stage,wehaveusedthesamevaluesofγ andΓ(0) thatwe definition of the inherent-structure energy requires this
A
usedforagingat280K. Theresultsforawaitingtimeof interpretation of e . We use the canonical variables T
IS
t =25ns(equivalenttothenominalMSvalueof25nson and χ in Eq.(5.4) instead of the entropies that we used
e
our shifted time scale) are shown in Figs. 5 and 6, along in the micro-canonical formulation in Eq.(2.1). Inter-
withtheMSdataforthevolume. Again,wefindthatwe estingly, the position of the peak in e (t) seems to be
IS
need allthree irreversiblemechanisms to understandthe determinedmoststronglyby the time dependence ofthe
observed behavior. The effective temperature increases configurational vacancy term, i.e. n k T , as opposed
v B v
9
0.356
0.355
0 0.355
/Not 0.35 0.354
Vt
0.345 0.353
0
N
0 0.5 1 1.5log (2t−t ) 2.5 3 3.5 V/tot0.352 log (t)=3 (MS data)
0.34 10 e 0.351 10 e
log (t)=3 (Theory)
10 e
0.35
log (t)=4.4 (MS data)
0.3 10 e
hv 0.349 log (t)=4.4 (Theory)
χ/ 10 e
0.26
0.348
0.5 1 1.5 2 2.5 3 3.5
0.22 log (t−t )
0 0.5 1 1.5 2 2.5 3 3.5 10 e
log (t−t )
10 e
FIG.5: ThesameasFig. 1,butforreheatingfromT =150K FIG. 7: The same as the upper panel of Fig. 5, but after an
ℓ
to Tf=280K according to Eq.(5.3), and after an aging time aging time log10(te)=3 (equivalent to the nominal MS value
log10(te)=4.4 (equivalent to the nominal MS value of 25ns of 1ns on our shifted time scale). The upper panel of Fig. 5
on our shifted time scale). See text for the parameters used. is copied here for comparison. Seelegend for more details.
The data in the upper panel are extracted from MS Fig. 2
[5]. TheKovacspeakappearstobesmallbecausethevertical
scale includes the initial thermal expansion. See Fig. 7 for a
configurations,their“agingdynamicspropagatesthesys-
closer look at thepeak.
tem through a sequence of configurationsnever explored
inequilibrium,anditbecomesimpossibletoassociatethe
aging system to a corresponding liquid configuration.”
They go on to ask whether “a thermodynamic descrip-
0.4
tioncanbe recovered[by]decomposingtheagingsystem
na0.3 ina collectionof substates,eachofthem associatedwith
a different fictive T ... or if the glass ... is trapped in
0.2 some highly stressed configuration which can never be
0 0.5 1 1.5 2 2.5 3 3.5 4
log (t−t ) associated with a liquid state.”
10 e
Weseemtobearrivingatarelatedbutdifferentinter-
0.04
pretation. Byfocusingoninternalstatevariables–inthis
nv0.03 case, the density of different kinds of defects in both the
0.02 configurational and kinetic-vibrational subsystems – in
0.01 addition to the effective temperature, we naturally gen-
0 0.5 1 1.5 2 2.5 3 3.5 4
log (t−t ) eratestates inwhichthe systemas awhole departs from
10 e
0.94 both ordinary thermal equilibrium with the bath tem-
perature and from quasi-equilibrium with the effective
φel0.92 temperature. Our technique for making this calculation
is the one we described in [21, 22]. We think that this
0.9 technique goes at least part of the way toward answer-
0 0.5 1 1.5 2 2.5 3 3.5 4
log (t−t ) ing the questions posed by Mossa and Sciortino, but we
10 e
recognize that it encounters conceptual problems that
eventually must be addressed.
FIG. 6: na, nv and φel corresponding to Fig. 5. The most obvious such problem, in our opinion, is the
onethatwefoundwhendecidingtoincludethemisalign-
ment defects among the fast degrees of freedom in the
kinetic-vibrationalsubsystem. TheresultsofIlgandBar-
to being determined predominantly by other configura-
rat [28] imply that some such internal variables may be
tional degrees of freedom and thus more directly related
neither completely fast nor completely slow but, rather,
to the time dependence of χ˜.
their dynamics might be activated by an intermediate
temperature or noise strength. This possibility might
be loosely related to the MS conjecture about “different
VI. CONCLUDING REMARKS
fictive [temperatures];” but the conjectures are intrinsi-
cally different from each other. Neither we nor Ilg and
In their own concluding remarks,Mossa and Sciortino Barrat are contemplating more than one effective tem-
[5] summarize their results by saying that, instead of perature. So far as we cantell, no complication ofeither
moving a system along a sequence of quasi-equilibrium kindisneededforunderstandingtheKovacseffectasob-
10
servedbyMS,nordoweseemtoneeditforshearflowin
amorphoussystems[23]orinpolycrystals[31]. Neverthe-
−82.9
less, we appear to be encountering some of the deepest
and most important open questions in nonequilibrium
−83
physics.
mol) −83.1
J/
k
(S−83.2
eI
Acknowledgments
−83.3
−83.4
We thank Stefano Mossa and Francesco Sciortino for
0 0.5 1 1.5 2 2.5 3 3.5 4 their remarkably incisive work, for sharing their data
log (t−t )
10 e with us, and for answering our questions about it. We
also thank Kenneth Kamrin for pointing out an error in
FIG. 8: The theoretical inherent structure energy eIS of our earlier thermodynamic analyses, which, had it not
Eq.(5.4) (solid line) compared to the simulation data (open been corrected, would have given us a wrong result in
circles) extracted from MS Fig. 3(b).
Eq.(3.24) in the present paper.
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