Effective Temperature Thermodynamics and the Glass Transition: Connecting Time-Scales

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by  Ido Regev

Effective Temperature Thermodynamics and the Glass Transition:
Connecting Time-Scales
Ido Regev1,2, Xiangdong Ding1 and Turab Lookman1
1Theoretical Division Los Alamos National Laboratory, Los Alamos 87545, New Mexico
2Center for Non-Linear Studies, Los Alamos National Laboratory, Los Alamos 87545, New Mexico
(Dated: January 31, 2012)
Weproposeatheorybasedonsimplephysicalargumentsthatdescribesanonequilibriumsteady-
state by a temperature-like parameter (an “effective temperature”). We show how one can predict
the effective temperature as a function of the temperature of the environment for a specific case
2 of non-equilibrium behavior: radiation amorphization. The main idea that we present is that
1 the amorphization process is inherently connected to the dynamical arrest that a liquid undergoes
0 when it transforms into a glass. We suggest that similar arguments may hold also for the effective
2 temperature under plastic deformation.

n
a
The idea that the state of a system in a non- equation of state of the order parameter (assuming that
J
equilibrium steady-state obeys a Gibbs-like distribution the equation of state is still valid). In Bou´e et al. [11]
0
3 with a temperature-like parameter, called the “effective this approach was taken one step further by showing ex-
temperature” was studied extensively starting from the plicitly that several coarse-grained structural properties

] pioneering work of Edwards [1]. Edwards assumed that of at least two different models of amorphous solids un-
h
agranularmaterialunderexternalperturbationexplores derextremeshear(plasticdeformation)aredescribedby
c
e a set of equal probability states and therefore the micro- a Gibbsian measure controlled by an effective tempera-
m states are described by a stationary probability measure ture. Thevalueoftheeffectivetemperatureasafunction
- analogous to the Gibbs measure, but where a configura- ofthetemperaturewasmeasuredandwasfoundtoshow
t
a tional temperature (that he called “compactivity”) re- non-monotonous behavior. It was also shown that the
t places the thermal bath temperature. Cugliandolo et stress and the potential energy depend on the effective
s
. al.[2] showed, for mean field models, that such an ef- temperature. It is usually not possible to predict the ef-
t
a fective temperature has other properties desired for a fective temperature theoretically, which leads to the use
m
thermodynamicdescriptionofnon-equilibriumstates,for of fitting functions or scaling arguments.
- example it controls the direction of heat flow. Further- In this work we show, using molecular dynamics simu-
d
n more,itwasshownthatfluctuation-dissipationtheorems lations, that the steady-state structure of an amorphous
o are violated in a very specific way: at short time scales solid under irradiation can be described by a Gibbs-like
c the fluctuations are related to the susceptibilities in the distribution. More importantly, we show how one can
[
usual way, described by the fluctuation dissipation the- understandand predictthevalueofthe effectivetemper-

1 orem involving the thermal bath temperature. At large atureatthesteady-statebyconnectingthefastandslow
v time-scales the usual fluctuation dissipation theorem is dynamics in the system in a way that is deeply rooted
9 not obeyed but satisfies an analogous expression that in- in the nature of the glass transition. Our explanations
9
volves the effective temperature. In later work similar and theory are different than the ones given by Martin
0
6 behavior was observed for sheared glasses at tempera- in[12],whopredictedaninfiniteeffectivetemperatureat
1. tures below the glass transition (about 66%Tg) [3] and zero thermal bath temperature and also from the argu-
in sheared foam at zero temperature [4]. This suggests mentsgivenbyLundetal. in[13]whosuggestedthatthe
0
2 that a statistical mechanics description with a Gibbs- effectivetemperaturehasafinitevalueatzerobathtem-
1 like measure might apply for these systems. It was also peratureandfromBouchbinderetal. [14]whosuggested
: found that the response of amorphous materials to high athermodynamicexplanationforthenon-monotonicbe-
v
i shear-rates is related to their structure by an exponen- havior of the effective temperature as a function of the
X
tial function, which suggests that the structure might thermal bath temperature.
r be described by a Gibbsian-like measure, similar to the The structure and slow dynamics of a super-
a
ideas of Edwards [6, 7]. Two important theories of plas- cooledliquid: Inaseriesofrecentpapers(cf. [15,16])it
tic deformation in amorphous solids, the Shear Trans- wasshownthatthestructuralchangesthatasupercooled
formation Zone theory (STZ) [8] and the Soft Glassy liquid undergoes on cooling can be represented by a fi-
Rheology theory (SGR) [9] use the idea of an effective nite set of species which are indexed by 1,2, ,n. The
temperature as a control parameter. In STZ theory it is precisenatureofthesespeciesmaychangefrom···modelto
assumed that the distribution of soft areas in a material model,buttheyarealwaysformedbyparticlesandtheir
obeys a Gibbs-like measure. Bouchbinder et al.[10] have nearest neighbors. The main advantage of these species
extracted an effective temperature from non equilibrium is that they obey discrete statistical mechanics, in the
valuesofanorderparameterbyinvertingtheequilibrium sense that their temperature-dependent concentrations
2
C (T), are determined by a set of constant degenera- theexactnumericalvaluesoftheconstantsg and and
i i i
(cid:104) (cid:105) H
cies g and enthalpies such that: showed that equation 1 describes accurately the concen-
i i
H
trations of species measured from experiments, and that
C (T)= gie−Hi/T . (1) equation 3 describes the typical relaxation times of the
(cid:104) i(cid:105) (cid:80)ni=1gie−Hi/T autocorrelation function CR(t) [17, 18].
After applying this description we always find that some
speciesvanishattemperaturesthatareclosetotheglass
transitiontemperature. Thisqualitativeobservationwas
madequantitativebysummingthevaluesofoneormore
of the concentrations of the vanishing species to what is
calledthe‘liquid-like’concentration C (T). Theinverse
(cid:96)
(cid:104) (cid:105)
of this concentration provides a length-scale (the typical
distance between ‘fluid’ species):
ξ(T) [ C (T)]−1/d , ξ when T 0, (2)
(cid:96)
≡ (cid:104) (cid:105) →∞ →
where d is the space dimension. It was amply demon-
strated on a large variety of models that the relaxation
time τ (T) measured using correlation functions in the
α
super-cooledregimeisdeterminedbythisdivergingscale
FIG. 1: (Color online): Representative configuration of par-
according to
ticles in the ST model with examples for the three different
species. Particle with two favorable interactions are assigned
τα =τ0eµξ(T)/T , (3) as a “blue” species, particles with one favorable interaction
are labeled “green” and ones with no favorable interactions
where µ is a typical free energy per particle and τ0 at all are labeled “red”.
a typical vibration time. In the following we will use
the results of this analysis for a specific model: the
Simulations: The molecular dynamics protocol we
ST(Shintani-Tanaka)two-dimensionalliquidmodel[17].
implemented was the following: First, we carefully equi-
The ST model has N identical particles of mass m; each
librated a large number of independent configurations
oftheparticlescarriesaunitvectoru thatcanrotateon
i with N =1024 particles in the NVT ensemble using the
the unit circle. The particles interact via the potential
Berendsenthermostatoverawiderangeoftemperatures.
U(r ,θ ,θ )=U¯(r )+∆U(r ,θ ,θ ). HereU¯(r )isthe
ij i j ij ij i j ij After this, we turned our equilibrium super–cooled liq-
standardisotropicLennard-Jones12-6potential,whereas
uids into amorphous solids by minimizing their poten-
the anisotropic part ∆U(r ,θ ,θ ) is chosen such as to
ij i j tial energy using the conjugate gradient algorithm. This
favor local organization of the unit vectors of two in-
procedure can be thought of as quenching a liquid in-
teracting particles, such that they form 1260 with the
finitely fast into a disordered solid whose temperature
radius-vector between them. This kind of interaction
is formally T = 0. Then we irradiate at a constant rate
prefers structures with a five-fold symmetry, which frus-
whilemaintainingconstantthermalbathtemperatureT .
trate crystallization. For full details of this model the b
We “irradiate” the material by giving a particle a very
reader is referred to Refs. [17–19]; here it suffices to
high momentum at random. This simulates the effect
know that with the parameters chosen in Ref. [17] the
of an energetic particle bombarding the material. The
model crystallizes upon cooling for ∆ < 0.6 whereas for
highmomentumcreatesacascadeeventwheretheparti-
larger values of ∆ the model exhibits all the standard
cle transfers its energy to the surrounding particles. The
features of the glass transition, including a spectacular
simulations were performed at a constant temperature
slowing down of the decay of the correlation functions of
with a constant rate (average time between hits) and in-
(cid:80)
the unit vectors C (t) (1/N) u (t) u (0) , which
R ≡ i(cid:104) i · i (cid:105) tensity (average momentum transfer) of radiation. The
isverywelldescribedbyEq. (3). In[18,19]and[11]the
rate of bombardment of the material was much slower
above statistical mechanics theory was constructed for
thantherateofthermalfluctuationsandwaschosensuch
the ST model. We chose to separate the particles into
that the material would not heat and melt.
three species, according to the possible number of neigh-
Effective temperature: In [11] we found that the
bors interacting in the most favorable way with a given
concentrations of species in steady-state plastic defor-
particle: particleswithtwofavorableinteractionsaredes-
mation C are described by the following Gibbsian
ignated as belonging to the “blue” species (n = 2), par- (cid:104) i(cid:105)ss
distribution:
ticles with one favorable interaction are labeled “green”
(n=1)andoneswithnofavorableinteractionsatallare C = gie−Hi/Teff , (4)
labeled “red” (n = 0). In previous work we have found (cid:104) i(cid:105)ss (cid:80)nj=1gje−Hj/Teff
3
where g and have the same values as in the equi- both cases the behavior is quite similar: At high tem-
i i
H
librium Gibbs distribution and the effective temperature peratures the dynamics is dominated by the fluctuations
T is a parameter that depends on the external forc- imposedbythethermalbathandtheirradiationhaslittle
eff
ing and thermal bath temperature in a non-trivial way. effect. In that regime the equilibrium Gibbs distribution
In figure 2 we show the concentrations C measured accuratelydescribesthestatistics. Asthetemperatureis
i
(cid:104) (cid:105)
from simulations in an initial amorphous configuration lowered, the rate of irradiation (strain-rate in the case of
(squares)andtheconcentrationsmeasuredatthesteady- plasticity) becomes comparable to the time scale associ-
state under irradiation (circles). The curves are the the- atedwiththermalfluctuations(eq. 3). Atthatpointthe
oretical concentrations given by the equilibrium Gibbs effective temperature starts to deviate from the thermal
measure for this system - equation 1. The concentra- bath temperature. What happens at lower temperatures
tions in the initial quenched state follow the Boltzmann iscompletelyun-intuitive: asthetemperatureislowered,
distribution. This is an indication that the structural theeffectivetemperaturedecreasesmoderatelyandthan
aspects that are reflected in the species are not sensi- increases again until it reaches a finite effective temper-
tive to the quench. Under irradiation the concentra- ature at zero temperature. But what is determining the
tions undergo a significant change (see inset) and reach values of the effective temperature? In the following we
new steady-state values. Remarkably, the new concen- will give a physical explanation of this behavior. The
trations, despite having changed significantly, still cor- basic idea underlying our analysis is that whenever an
respond to values of equations 1 with some “effective” energetic particle transfers its energy to one of the par-
temperature. For different radiation rates and thermal ticles in the material, it transfers some of its momentum
bath temperature the steady-state concentrations reach to the neighboring particles, which in-turn transfer some
different values but they all agree well with the equilib- of their momentum to their neighbors and so on, in a
riumGibbsdistributionwithsomeeffectivetemperature. “radiation cascade”. In many different materials, this is
equivalent to a local melting of the material around the
point of impact. Since the rest of the material is kept at
a much lower temperature, the kinetic energy dissipates
rapidlyandtheregioncoolsdown. Sincethecoolingpro-
cess is very fast it will result in a local glass transition.
We can describe this process using two rate equations
(based on Fourier’s cooling law):
i
Ci
h 1
T˙ = [T T ], (5)
K −τ K − b
th
1
T˙ = [T T ]. (6)
eff −τ (T ) eff − K
α K
Equilibrium theory
Initial h C i i from simulations The first equation describes the relaxation of the kinetic
Final h C i i from simulations temperature T of the molten region. After long enough
K
time, all of the kinetic energy will dissipate and the ki-
netic temperature will equilibrate with the thermal bath
FIG. 2: (Color online): Concentrations of species with two temperature T . The rate of relaxation is controlled by
b
strong bonds’ (blue), one strong bond (green) and no bonds the thermal conductivity of the material and is mate-
at all(red) as measured from simulations against the equilib-
rial dependent. The second equation describes an aging
riumtheory(smoothlines,colorsthesameasconcentrations).
process. At times longer than the relaxation time τ (T)
Squaresrepresenttheconcentrationataninitialrandomcon- α
the structure of the molten area - described by the ef-
figurations quenched from a temperature T = 0.29. Circles
representtheconcentrationsofthespeciesatnon-equilibrium fective temperature Teff - reaches thermal equilibrium
steady-state under radiation. Inset: concentrations as func- (note that one can use instead of equation 3, the Vogel
tion of time starting from an amorphous solid. Fultcher expression to a similar effect). The solution to
the first equation is:
Theoretical description of the steady-state ef- T (t,T )=[T (t=0) T ]e−t/τth +T , (7)
fective temperature: In figure 3 we show the effective K b K − b b
temperaturemeasuredatthesteady-stateforagivenra- and to the second equation:
diation rate as a function of the thermal bath tempera-
tsmutaortedee(lbvualulnuedeecsrirocpflleasts)ht.eicIendffeetfhcotreimvienastteioetnmwpaeetrcatawtnuorseedeioffftehrteehnestt,esacadomnye-- Teff(t,Tb)= (cid:82)0te(cid:82)0t(cid:48) τα[TKdt((cid:48)t(cid:48)(cid:48)(cid:48),Teb)(cid:82)]0tτατTα[TK[TK(Ktd((cid:48)tt(,(cid:48)(cid:48)tT,(cid:48)Tb,T)bb))]]dt(cid:48)+Teff(t=0).
stant strain rates (see [11]). One can observe that in (8)
4
By choosing a cutoff time τ we get an expression for uesofthesteady-stateeffectivetemperatureofamaterial
cf
the effective temperature as a function of the thermal under irradiation. This result is based on a deep con-
bath temperature T (T ) = T (τ ,T ). This equa- nection between the effective temperature and the glass
eff b eff cf b
tion is analytically solvable if we approximate and as- transition which was proposed in the past [20] and is ex-
sume that T = T . However, for small temperatures plainedhere. Duetotheobvioussimilaritytothebehav-
K b
this assumption fails and the approximation is not accu- ior of the effective temperature in plastic deformation,
rate. The non-Arrhenius increase of the relaxation time we suggest that a modified theory will be able to predict
atlowtemperaturesmeansthatthemoltenareawillnot the effective temperature in plastic deformation as well.
reach the equilibrium structure but will undergo a glass Future work will focus on applying the same formalism
transition and get “stuck” in some metastable state. In to more realistic simulations and experiments and also
order to find the effective temperature that corresponds on relating structural properties to the effective temper-
to this state, we numerically integrate the above equa- ature, for example in irradiation induced flow (see for
tions with a high initial kinetic temperature which we example [21]).
guess to be: T (t = 0) = T (t = 0) = 0.5, repre-
K eff Ido Regev would like to thank Laurent Bou´e for very
senting the temperature of the molten region just after
useful discussions. This work was supported by the US
a cascade event. We also chose τ = 200 and introduce
th DepartmentofEnergy’sLosAlamosNationalLaboratory
a time-cutoff τ = 400 after which we assume that the
cf through contract DE-AC52-06NA25396.
structure does not evolve. The time scale τ is related
cf
to the frequency of cascade events since this frequency
controls the typical time between re-melting of a specific
region. In figure 3 we observe the agreement between
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