Direct link between boson-peak modes and dielectric $\alpha$-relaxation in glasses

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by  B. Cui

Direct link between boson-peak modes and dielectric α-relaxation in glasses
Bingyu Cui1,2, Rico Milkus1, Alessio Zaccone1,3
1Statistical Physics Group, Department of Chemical Engineering and Biotechnology,
University of Cambridge, New Museums Site, CB2 3RA Cambridge, U.K.
2Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K. and
3Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, CB3 0HE Cambridge, U.K.
We compute the dielectric response of glasses starting from a microscopic system-bath Hamilto-
nianoftheZwanzig-Caldeira-Leggetttypeandusinganansatzfromkinetictheoryforthememory
function in the resulting Generalized Langevin Equation. The resulting framework requires the
7 knowledgeofthevibrationaldensityofstates(DOS)asinput,thatwetakefromnumericalevalua-
1 tionofamarginally-stableharmonicdisorderedlattice,featuringastrongbosonpeak(excessofsoft
0 modes over Debye ∼ωp2 law). The dielectric function calculated based on this ansatz is compared
2 with experimental data for the paradigmatic case of glycerol at T (cid:46)Tg. Good agreement is found
for both the reactive (real part) of the response and for the α-relaxation peak in the imaginary
b
part, with a significant improvement over earlier theoretical approaches, especially in the reactive
e
modulus. Onthelow-frequencysideoftheα-peak,thefittingsupportsthepresenceof∼ω4 modes
F p
at vanishing eigenfrequency as recently shown in [Phys. Rev. Lett. 117, 035501 (2016)]. α-wing

2 asymmetryandstretched-exponentialbehaviourarerecoveredbyourframework,whichshowsthat
these features are, to a large extent, caused by the soft boson-peak modes in the DOS.

]
t
f
o I. INTRODUCTION exponentialfunction∼exp[−(t/τ)]β providesagoodem-
s
. pirical fit of the relaxation function and of the dielectric
t
a Supercooled liquids that undergo a liquid-glass tran- loss [1, 2, 5, 15–17].
m
sition exhibit an abrupt and dramatic slowdown of the Mode-couplingtheory(MCT)developedbyW.Goetze
- atomic/molecular dynamics upon approaching the glass and co-workers, has provided a general interpretation of
d
transition temperature T [1–4]. The α-relaxation de- the α-peak in dielectric relaxation using a framework
n g
o scribes the slowest component of the time-relaxation (or where the many-body microscopic dynamics of charges
c autocorrelation function) of material response, including is treated statistically, in the same way as for an ensem-
[ mechanical relaxation, relaxation of density fluctuations ble of classically interacting spherical particles [4]. The

orofthedielectricpolarization[5]. Theα-relaxationphe- moststrikingsuccessofMCThasbeenthefirst-principles
2
v nomenon has always been associated with the collective derivationoftheKohlrauschstretched-exponentialrelax-
1 and strongly cooperative motion of a large number of ation for α-relaxation in the liquid phase.
4 atoms/molecules which rearrange in a long-range corre- While MCT has had tremendous success in describ-
0 lated way [1]. This process has also been interpreted, ing supercooled liquids at T > T , the situation is
g
5
within the energy landscape picture, as the transition of quite different at T (cid:39) T or in the glass at T < T .
0 g g
the system from one meta-basin to another, which in- Here, although MCT provides a theoretical foundation
.
1 volves the thermally activated jump over a large energy for Kohlrausch stretched-exponential behaviour, direct
0 barrier [6–8]. comparisons with experimental data have not been pos-
7
Moderntheoriesofdielectricresponseofmatter[9,10] sible due to the difficulty of calibrating various param-
1
: are based on the Lorentz model [11, 12], which approxi- eters in the theory. This scenario is the most striking
v mateselectronsasclassicalparticlesboundharmonically for the paradigmatic case of glycerol: this is the most
i
X to positive background charges. Upon assuming that all widely studied organic glass-former in the experimental
oscillators move at the same natural frequency, the re- literature,yetnomicroscopictheoryhasbeenusedtode-
r
a laxation function (cid:15)(t) is a simple-exponential increasing scribe the dielectric response of this material apart from
function of time, while the imaginary part (cid:15)(cid:48)(cid:48)(ω) of the empiricalmodels(e.g. Havriliak-Negami),whichhaveno
complex dielectric function (cid:15)∗(ω), features a resonance physics in them.
peak given by a Lorentzian function [11, 12]. Here we take a very different approach: instead of the
CorrectingtoaccountfortherotationalBrownianmo- liquid-stateapproachofMCT,wetaketheoppositepoint
tion in the case of strongly anisotropic molecules, as in of view, and describe the dynamics in analogy with a
the Debye dielectric-relaxation theory [9], does not alter disordered low-T lattice of particles which perform har-
the simple-exponential relaxation. While this may be a monic oscillations. Due to the disorder in the lattice
good approximation for gases and high-T liquids, it is (and in particular due to the absence of local inversion
not valid for glasses, as is well known since the time of symmetry[18]),thelow-frequencypartofthevibrational
Kohlrausch [13, 14]. For supercooled liquids in general, densityofstates(DOS)isdominatedbyanexcessofsoft
and for glasses in particular, the Kohlrausch stretched- modes over the Debye ω2 law valid for crystals.
p
2
This excess of soft modes in the DOS is universally ing dissipation, by keeping the analysis within the limits
known in the literature on glassy physics and disordered of linear theory and eventually leading to a Generalized
systems as the ”boson peak” [19, 20, 22, 23]. In the fol- Langevin Equation (GLE) where an ansatz in form of
lowingweusethisterminologyandwerefertothebroad stretched exponential is chosen for the memory kernel,
ensemble of all these excess soft modes over the Debye for the dynamics.
ω2 law as the boson peak. It is important to note that, This GLE is our microscopic equation of motion that
p
in the sub-field of dielectric spectroscopy of glasses, the we then use within the Lorentz model to obtain the di-
terminology ”boson peak” is used to designate an iso- electric function. Since this last step involves summing
lated peak in the THz frequency regime of the dielectric overallthemicroscopicdegreesoffreedomofthesystem,
loss modulus (cid:15)(cid:48)(cid:48). In our work we will never refer to or the vibrational density of states (DOS) is required for a
considerthisTHz-frequencypeakinthelossmodulus,so quantitative evaluation. Since the DOS cannot be de-
there is no ambiguity in our terminology and the term rived analytically from first-principles, we will make use
”bosonpeak”isusedexclusivelytodesignatetheensem- of the simplest meaningful DOS ρ(ω ), consistent with
p
ble of excess non-Debye modes in the low-ω part of the ourHamiltonian,torepresentaharmonicdisorderedlat-
p
vibrational DOS. tice, obtained by numerical diagonalization of a model
Famous physicists in the past have attempted to ex- randomlatticeofharmonically-boundsphericalparticles.
plainstretched-exponentialrelaxation(whichisthehall- ThislatticeisobtainedbydrivingadenseLennard-Jones
mark of the α-relaxation in glasses) in terms of the un- system into a metastable glassy energy minimum with
derlying cooperative coupling of vibrational degrees of a Monte-Carlo relaxation algorithm, and then replacing
freedom [15–17, 24, 25]. In spite of these efforts, the all the nearest-neighbour pairs with harmonic springs all
linkbetweenquasi-localizedsoftvibrationalmodesorbo- of the same length and spring constant [18]. Springs
son peak modes in the DOS, and the α-relaxation pro- are then cut at random in the lattice to generate disor-
cess,hassurprisinglyreceivedlessattention,withimpor- dered nearest-neighbour lattices with variable mean co-
tant exceptions like Ref. [26] and the macroscopic model ordination Z, from Z = 9 down to the isostatic limit
for viscoelasticity of Ref. [27]. This is despite the fact Z = 2d = 6. It is important to notice that this sim-
that both the boson peak in the vibrational DOS and plified model DOS is convenient for its simplicity and
the α-relaxation process display a strong T-dependence to single-out generic features of glassy behaviour, but in
near T (for the T-dependence of the boson peak, see ordertoobtainveryaccuratefittings, morerealisticsim-
g
e.g. [28, 29]). ulated DOS (e.g. from molecular simulations) will be
For the first time, we present a simple and explicit set employed in future work.
of relations between the dielectric relaxation functions
and the DOS of disordered lattices, based on the ansatz
that the microscopic Hamiltonian can be modelled using A. System-bath Hamiltonian for disordered
dissipative lattices
a system-bath coupling of the Zwanzig-Caldeira-Leggett
type. Within this framework, it is possible to show that
soft modes in the DOS in the boson-peak region are re- Caldeira-Leggett models [30] for the effective treat-
sponsible for the observed stretched-exponential relax- mentofanharmonicityleadingtotheemergenceofdissi-
ation in time and for the α-relaxation peak in the loss pative behaviour in condensed matter have been applied
modulus of glycerol at T (cid:46)T . mostly to low-temperature quantum physics problems,
g
particularly quantum tunnelling in superconductors and
inchemicalreactionratetheory. Wearenotawareofany
application of this approach to microscopic dynamics in
II. GENERALIZED LANGEVIN EQUATION
disordered solids such as glasses. We do this with the
FOR DIELECTRIC RESPONSE WITHIN THE
aimofderivingasuitableequationofmotionforatagged
LORENTZ MODEL
molecule in a low-T glass, which takes into account the
disorderedenvironmentaswellasthedissipation. Weuse
In the following, we work within the Lorentz dielectric
the Zwanzig-Caldeira-Leggett (ZCL) classical version of
model of disordered elastically bound classical charges
the model, which was actually proposed already in 1973
(basically on the same general coarse-graining level as
by Zwanzig [31], since it is particularly convenient in or-
in Ref. [3]). We will derive an average equation of mo-
der to build a connection with the DOS of a disordered
tionforataggedcharge/particlewhichisthemicroscopic
solid.
building block in our treatment. We start from an effec-
The general form of the Caldeira-Leggett Hamiltonian
tive Hamiltonian of the Zwanzig-Caldeira-Leggett type
fortheclassicaldynamicsofataggedparticlethatiscou-
which allows one to describe the motion of a tagged par-
pled to a large number of harmonic oscillators (which,
ticle which is coupled to a large number of harmonic os-
in our case, effectively represent the dynamics of all
cillators (the bath), in our case representing the other
other particles in the glassy system) is given by three
particles in the glass. As is known, the Caldeira-Leggett
terms [31]:
models [30] provide an effective way of accounting for
long-range anharmonic interactions, and for the result- H =H +H +H (1)
S B I
3
where H = p2/2m + V(q) is the Hamiltonian of fluctuation-dissipationtheoremisnotexpectedtoholdin
S
the tagged particle, HB = (cid:80)Nα=1(2pm2αα + 12mαωα2x2α) tlohwis-tfeomrmpe(rnaotrurine gbeenhearvailo)u.rSbinecloewweTa,rewientweriellstmedakine tthhee
is the Hamiltonian of the bath of harmonic oscilla- g
assumption that the noise is low, and set F = 0 above.
tors that are coupled to the tagged particle, H = p
I
(cid:80)NH (q,x ,ω ,q˙,q¨,...,x˙ ,x¨ ,...) is the coupling term Thisisconsistentwithfrozen-inmolecularpositions[32],
α α α α α α in the absence of an external driving force.
between the tagged atom and the bath. Note we also
assume that there is no interaction between bath oscil-
lators. Under these definitions, the equations of motion
C. The memory function for the microscopic
are:
friction coefficient
mq¨=−V(cid:48)(q)−(cid:88)∂Hα (2)
∂q The ZCL Hamiltonian does not provide any prescrip-
α
tion to the form of the memory function ν(t), which can
for the tagged particle, and takeanyformdependingonthevaluesofthecoefficients
c [31]. This is evident from looking at Eq.(6). Hence, a
α
∂H shortcoming of CL-type models, including ZCL, is that
m x¨ +m ω2x =− α. (3)
α α α α α x the functional form of ν(t) cannot be derived a priori for
α
a given system, because, while the DOS is certainly an
for an oscillator that belongs to the bath. easily accessible quantity from simulations of a physical
For the ZCL model, the Hamiltonian is given by system, the spectrum of coupling constants {c } is basi-
α
cally a phenomenological parameter.
H = p2 +V(q)+1 (cid:88)N (cid:34) p2α +m ω2 (cid:18)x − Fα(q)(cid:19)2(cid:35). However, for a supercooled liquid, the time-dependent
2m 2 m α α α m ω2 friction, whichisdominatedbyslowcollectivedynamics,
α=1 α α α has been famously derived within kinetic theory (Boltz-
(4)
mann equation) using a mode-coupling type approxima-
tion by Sjoegren and Sjoelander [33] (see also Ref.[34]),
and is given by the following elegant expression:
B. Generalized Langevin Equation (GLE) of
molecular motion in low-T glasses ρk T (cid:90) ∞
ν(t)= B dkk4F (k,t)[c(k)]2F(k,t) (7)
6π2m s
IntheZCLHamiltonian,thecouplingfunctionistaken 0
to be linear Fα(q) = cαq, where cα is known as the where c(q) is the direct correlation function of liquid-
strength of coupling between the tagged atom and the state theory, F (q,t) is the self-part of the intermediate
s
α-th bath oscillator. The bilinear coupling assumption scattering function and F(k,t) is the intermediate scat-
can be related to a small-oscillation assumption in the tering function [33]. All of these quantities are func-
samespiritastheharmonicapproximation. Clearly, this tions of the wave-vector k. Clearly, the integral over
choice leads to a second-order inhomogeneous differen- k leaves a time-dependence of ν(t) which is controlled
tial equation for the position of the α-th oscillator of the by the product F (k,t)S(k,t). For a chemically homo-
s
bath. This solution can then be replaced into the equa- geneous system, F (k,t)S(k,t) ∼ F(k,t)2, especially in
s
tionofmotionforthetaggedparticle, whichleadstothe the long-time regime. From theory and simulations, we
following GLE know that in supercooled liquids F(k,t) ∼ exp(−t/τ)ξ,
with values of the stretching exponent that are typically
mq¨=−V(cid:48)(q)−(cid:90) t ν(t(cid:48))dqdt(cid:48)+F (t). (5) aroundξ =0.6[35]. Inturn,thisgivesν(t)∼exp(−t/τ)b
−∞ dt(cid:48) p with b ≈ 0.3. Hence, in our fitting of experimental data
we will take
where the non-Markovian friction or memory kernel ν(t)
is given by: ν(t)=ν e−(t/τ)b, (8)
0
(cid:88) c2 whereτ isacharacteristictime-scaleandν isaconstant
ν(t)= α cosω t. (6) 0
ω2 α pre-factor.
α α
The thermal noise term F is instead given by
p
the initial positions and momenta of the bath os- D. Final form of the GLE
cillators. For example, if the initial conditions
for the bath oscillators are taken to be Boltzmann- Further, by assuming no noise in the low-T limit, we
distributed ∼ exp(−H /k T), the noise satisfies a can generalize the formal GLE for the equation of mo-
B B
colouredfluctuation-dissipationtheorem,(cid:104)F (t)F (t(cid:48))(cid:105)= tion of an isolated tagged particle to the case of parti-
p p
k Tν(t−t(cid:48)). In our case of a glass far below T , the cles on a 3D nearest-neighbour lattice, where the tagged
B g
system is certainly not thermalized at t=0, so that the particle coordinate q is replaced by the position vector
4
r on the lattice, and the local conservative force-field 0.6
i
−V(cid:48)(q) is provided by the nearest-neighbours interac-
0.5
tions,takenheretobeharmonicasagoodapproximation
in the low-T limit, such that −V(cid:48)(q) ⇒ −H ˜r , where
ij j 0.4
summation over the repeated index j is implied. Here,
)
H = ∂V/∂r ∂r , is the Hessian matrix. Furthermore, ωp0.3
ij i j (
ρ
in the dielectric response problem, we also have an addi- 0.2
tional external force given by the electric field acting on
the particle charge (or partial charge) q . Thus we get 0.1
e
the following GLE-type equation of motion valid at low
0.0
T: 0.0 0.5 1.0 1.5 2.0 2.5 3.0
ω
p
(cid:90) t
m¨r + ν e−[(t−t(cid:48))/τ]br˙ dt(cid:48)+H r =q E. (9)
i 0 i ij j e
−∞
FIG. 1. Density of states (DOS) with respect to eigenfre-
quency ω at Z = 6.1 (solid line), i.e. close to the marginal
whereq isnowthepartialchargeonthetaggedparticle, p
e stability limit Z =6 that we identify here as the solid-liquid
and E is the externally applied electric field.
(glass) transition; plots of the DOS at Z = 7,Z = 8,Z = 9
This equation of motion describes the dynamics of
are also shown, and are marked as dashed, dot dashed and
an average particle harmonically bound locally to its
dotted lines, respectively.
nearest-neighbours, and non-locally coupled to many
other particles, due to the bilinear coupling term of the
ZCLHamiltonian. Thisnon-localcouplingrepresentsan
effective way of accounting for the effect of long-range parameteroftherelaxationprocesswhichinarealmolec-
anharmonicity in real materials, and is the root cause ular glass changes with T. Therefore, in order to use our
fordissipationandforthetime-dependentfrictioninthe numerical DOS data in the evaluation of the dielectric
memoryintegral. Equation(9)willbeusedbelowwithin function,weneedtofindaphysicallymeaningfulrelation
the Lorentz model of dielectrics to obtain the dielectric betweenZ andT attheglasstransition. Withinthispic-
function. ture, Z represents the effective number of intermolecular
contacts, which increases the number of positive charges
to which a negative charge is bound in the material.
E. Vibrational DOS and its T-dependence In all experimental systems which measure the T-
dependent material response, the temperature is varied
at constant pressure, which implies that thermal expan-
As anticipated above, in the Lorentz dielectric model
sion is important. Following previous work, we thus
thedisplacementofallparticlesintheappliedoscillating
employ thermal expansion ideas [36] to relate Z and
electric field has to be evaluated to obtain the polariza-
T. Upon introducing the thermal expansion coefficient
tion. This require a sum over all degrees of freedom of
α = 1(∂V/∂T) and replace the total volume V of the
all particles, which can be done by using the vibrational T V
sample via the volume fraction φ = vN/V occupied by
densityofstatesandintegratingovertheeigenfrequency,
the molecules (v is the volume of one molecule), upon
as will shown later. The DOS that we will use is ob-
integration we obtain ln(1/φ) = α T +const. Approx-
tained from numerical diagonalization of the simulated T
network described above, and is expressed in terms of imating Z ∼ φ locally, we get Z = Z0e−αTT. Imposing
that Z = 12, as for FCC crystals at T = 0 in accor-
dimensionless eigenfrequencies ω . Generally, the eigen- 0
p
dance with Nernst principle, we finally get, for glycerol,
frequency ω obtained from numerics and the eigenfre-
p
quency ω(cid:48) of the real experimental systems are related Z ≈ 6.02 when T = 184K. This is very close to the
via ω(cid:48) ≈p(cid:112)κ/mω where m is the effective mass of the reported Tg for this material [37].
p p It is seen in Fig. 1 and in Fig. 2 that for the case
charged particle and κ the spring constant, under the
Z = 6.1, i.e. very close to the solid-liquid (glass) transi-
condition that (cid:82)0ωD(cid:48) ρ(cid:48)(ωp(cid:48))dωp(cid:48) = (cid:82)0ωDρ(ωp)dωp. We use tionthatoccursatZ =6,astrongandbroadbosonpeak
(cid:112)
the constant C = κ/m as a fitting parameter. κ and ispresentintheDOS.UponincreasingZ towardshigher
m are both equal to unity in the numerical simulation of values the boson peak is still present but its amplitude
the DOS, whereas their values are of course different for decreases markedly upon increasing Z. At Z = 6.1, the
different experimental systems (in the case of dielectric continuum Debye regime ∼ ω2 is not visible or absent,
p
response, m is to understood as an effective mass). whereas a very small gap between ω =0 and the lowest
p
Also, the DOS obtained from diagonalization of the eigenfrequency exists. Hence, under conditions close to
model random networks, depends on the average coor- theglasstransitionwherethesystemlosesitsshearrigid-
dination number Z. For example, the boson peak fre- ity,thevibrationalspectrumisdominatedbyalargeand
quencydriftstowardslowervaluesofω accordingtothe broad excess of soft modes with respect to Debye ∼ ω2
p p
scaling ωBP ∼ (Z −6). Hence, Z is the crucial control law at low frequency.
p
5
Upon multiplying through by the eigenvector vp, we
i
go back to a vector equation for the Fourier-transformed
displacement of particle i:
2ωp δ˜ri(ω)=−mω2−iων˜qe(ω)−mω2E(cid:101)(ω). (11)
/) p
p
ω
( Eachparticlecontributestothepolarizationamoment
D
p =q δr . Inordertoevaluatethemacroscopicpolariza-
i e i
tion, we need to add together the contributions from all
(cid:80)
microscopicdegreesoffreedominthesystem,P = p .
i i
In order to do this analytically, we use the standard pro-
cedureofreplacingthediscretesumoverthetotal3N de-
ωω
pp greesoffreedomofthesolidwiththecontinuousintegral
overtheeigenfrequenciesω ,(cid:80)3N...=(cid:80)3 (cid:80)N ...→
FIG. 2. The DOS normalized by Debye’s ωp2 law, for (from (cid:82) ρ(ω )...dω , which gives tphe fopllowing suαm=1rulei=in1 inte-
bottom to top): Z = 9,Z = 8,Z = 7,Z = 6, which gives p p
evidence of the boson peak at low ω . The eigenfrequency of gral form for the polarization in glasses
p
boson peak scales as ωBP ∼ (Z −6) as know from work for
p (cid:20)(cid:90) ωD ρ(ω )q2 (cid:21)
disorderedsystems withcental-forceinteractions [18,21, 22]. P(cid:101)(ω)=− mω2−iων˜p(ω)e−mω2dωp E(cid:101)(ω).
0 p
(12)
Here, ρ(ω )isthevibrationalDOS,andω isthecut-
p D
F. Dielectric response as a function of the
offDebyefrequencyarisingfromthenormalizationofthe
vibrational DOS
density of states. The complex dielectric permittivity (cid:15)∗
isdefinedas(cid:15)∗ =1+4πχ whereχ isthedielectricsus-
e e
In order to determine the dependence of the polariza- ceptibility which connects polarization and electric field
tion and of the dielectric function on the frequency of as [11]: P =χ E. Hence, we obtain the complex dielec-
e
the field, we have to describe the displacement r of each tric function expressed as a frequency integral as
moleculefromitsownequilibriumpositionundertheap-
pliedfieldE. Upontreatingthedynamicsclassically,the (cid:15)∗(ω)=1−(cid:90) ωD Aρ(ωp) dω (13)
equation of motion for a charge i under the forces com- ω2−i(ν˜(ω)/m)ω−C2ω2 p
0 p
ing from interactions with other charges and from the
(cid:112)
applied electric filed, is given by the GLE Eq.(9) derived where A is an arbitrary positive constant, C = κ/m,
above. and ω is the Debye cut-off frequency (i.e. the highest
D
The Hessian H = ∂U/∂r ∂r , where U is the total eigenfrequency in the vibrational DOS spectrum). As
ij i j
potential energy of the system, represents the restoring onecaneasilyverify,ifρ(ωp)weregivenbyaDiracdelta,
attractive interactions from oppositely-charged nearest- one would recover the standard simple-exponential (De-
neighbour charges, that tend to bring the charge i back bye) relaxation [11]. Note that this approach can be
to therest positionthat i had atzero-field. Tosolvethis extended to deal with molecules that have stronger in-
equation, the first step is to take the Fourier transform: ner polarizability by replacing the external field E with
r (t)→˜r (ω), resulting in the equation: the local electric field E , see e.g. Ref. [9]. The de-
i i loc
tailed derivation is provided in Appendix A. However,
−mω2˜r +iων˜(ω)˜r +H ˜r =q E˜, (10) we have checked that the qualitative predictions are the
i i ij j e
same with and without the Lorentz field correction.
where the tilde is used to denote Fourier-transformed It is important to emphasize that, in Eq.(13), low-
variables. Hence,ν˜(ω)istheFouriertransformofEq.(8). frequency soft modes which are present in ρ(ω ) neces-
p
We then implement normal-mode decomposition: sarilyplayanimportantrolealsoatlowapplied-fieldfre-
˜r (ω) = rˆ˜ (ω)vp, where the hat is used to denote the quencies ω, because of the ω2 term in the denominator.
i p i
coefficient of the projected quantity, and vp denotes an Aswewillseebelow,thisfactinourdescriptionimpliesa
i
eigenvector of the Hessian matrix. Thus the equation of directroleofthebosonpeakontheα-relaxationprocess.
motion is rewritten as
−mω2rˆ˜ +iων˜(ω)rˆ˜ +mω2rˆ˜ =q Eˆ˜, G. Finite-size effects and low-frequency limit in
p p p p e
the DOS
where ω denotes the p-th normal mode frequency. The
p
equation is solved by
Since we are using a DOS obtained numerically from
a system with a finite (∼ 4000) number of particles in
ˆ
rˆ˜ (ω)=− qeE(cid:101) . simulations,itisimportanttocorrectlytakecareoffinite
p mω2−iων˜(ω)−mω2 sizeeffectsinEq.(13). Innumericalsimulations,theDOS
p
6
ρ(ω ) is not a continuous function, but discrete and can III. APPLICATION TO DIELECTRIC
p
be conveniently represented as ρ(ω )∼ 1 (cid:80)3N δ(ω − RELAXATION DATA
p 3N p=1 p
ω ). Thus, we rewrite Eq.(13) as a sum over a discrete
p
distribution of ω :
p
Wenowpresentourtheoreticalfittingsofstate-of-the-
art experimental data [37, 38] on glycerol at T ≈ T
(cid:88) A g
(cid:15)∗(ω)=1− (14) using Eq.(15)-(16), also in comparison with the empir-
ω2−i(ν˜(ω)/m)ω−C2ω2
p p ical best-fitting Kohlrausch stretched-exponential relax-
ation fitting. In Fig. 3 we plotted the comparisons for
where A has absorbed the scaling constant. Since the (cid:15)(cid:48)(ω) at T = 184 K, i.e. slightly below Tg, obtained by
dielectric function is a complex quantity, we can split it implementing the numerical DOS of Fig.1 for Z = 6.1
into its real and imaginary parts, i.e. (cid:15)∗(ω) = (cid:15)(cid:48)(ω)− in Eq.(15). In this case, it is clear that our theoretical
i(cid:15)(cid:48)(cid:48)(ω): model performs significantly better than the Kohlrausch
best-fitting (that is optimized for the joint the fitting
(cid:15)(cid:48)(ω)=(cid:15)(cid:48)(∞)+ (15) of dielectric loss below). This suggests that excess soft
modes are important for the fitting of the dielectric re-
(cid:88) A1(C2ωp2−ω2+ν˜2(ω)ω/m) , sponse at the glass transition.
(C2ω2−ω2+ων˜ (ω)/m)2+(ων˜ (ω)/m)2
p p 2 1 InFig. 4wepresentfittingsofthedielectricloss,(cid:15)(cid:48)(cid:48)(ω)
(cid:15)(cid:48)(cid:48)(ω)=(cid:88) A2(ων˜1(ω)/m) . fortheMarkoviancaseν˜=ν˜1 =constinEq.(16). Inthis
(C2ω2−ω2+ων˜ (ω)/m)2+(ων˜ (ω)/m)2 case,itisseenthatourframeworkeveninitsMarkovian-
p p 2 1
frictionversion, providesareasonablygoodfittingofthe
(16)
α-peak on both the left-hand and the right-hand side of
the peak, and the overall quality of the fitting is com-
The Markovian friction case is retrieved by simply
parable to the one of the Kohlrausch best empirical fit-
setting ν˜ = ν˜ = const in the above expressions.
1 ting. Our theoretical model provides the crucial connec-
A ,A ,(cid:15)(cid:48)(∞),m are re-scaling constants that need to
1 2 tionbetweenthesalientfeaturesoftheDOSnearT and
be calibrated in the comparison with experimental g
the corresponding features of the response. Of course,
data. It is important to note that the experimental
at the higher-frequency end of the α-wing, other effects
data of dielectric permittivity and dielectric loss are
mayaswellbeimportantwhicharenotdescribedbyour
not necessarily given in the same units and there is, in
model: inparticular,theexistenceofJohari-Goldsteinβ-
general, no coherence between the offsets in the plots of
relaxation-type contributions to the loss modulus in this
the (cid:15)(cid:48) and (cid:15)(cid:48)(cid:48) curves. For this reason, the values of A
1 regime has been shown for a variety of systems [40–44].
and A do not necessarily coincide. ν˜ and ν˜ are real
2 1 2
and (minus) imaginary parts of ν˜(ω) in Fourier space,
On the left-hand ascending side of the peak, the scal-
ν˜(ω)=ν˜1(ω)−iν˜2(ω). ingρ(ω )∼ω4 leadstothelinearbehaviour∼ω1,asde-
p p
rived in Appendix C, for the ascending part of the peak.
As remarked above and as observed in all numerical On the high-ω side of the peak, where the dynamics is
calculations of the DOS in the vicinity of the mechanical dominated by the soft boson-peak modes and the DOS
stability point of disordered solids, there exists a lowest is approximately flat as a function of ω in Fig.1, our
p
non-zero eigenfrequency ωp,min, and a vanishingly small model, reproduces, remarkably, the asymmetric α-wing
gap between ωp =0 and ωp,min =0.019. A recent study behaviour still in good agreement with the experimental
has pointed out that a scaling ∼ ω4, possibly related data.
p
to soft anharmonic modes, exists even below the lowest
Goldstone modes [39]. Even though we cannot settle In Fig. 5, we present the same fitting, but now with
this issue here, since it reaches far beyond the scope and a non-Markovian friction given by ν(t) = ν e−4tb used
0
interest of this work, we have performed an asymptotic in Eq.(16), with b = 0.3 suggested by previous studies
analysis of the limiting behaviour of ρ(ωp) at ωp → 0 on glassy dynamics. Overall, the non-Markovian friction
in the context of the dielectric response. The analysis provides a better fitting, which suggests that memory
is reported in Appendix C, and clearly shows that only effects in the atomic dynamics are non-negligible. How-
asymptotic scalings ρ(ωp) ∼ ωpn, with n > 3 can lead ever, the memory kernel due to non-Markovian friction
to meaningful behaviour of (cid:15)(cid:48)(cid:48)(ω). This finding lends doesnotappeartobeessentialtogenerateandreproduce
further support to this form of the DOS in the zero the α-wing asymmetry.
frequency limit, and we therefore use this scaling in the
finite-size gap between ω = 0 and ω which results This comparative analysis therefore demonstrates
p p,min
in a linear behaviour of the left flank of the α-peak in quite clearly that while memory effects are important,
(cid:15)(cid:48)(cid:48)(ω) in perfect agreement with experimental data as the main cause for the α-wing asymmetry is the excess
discussed below. of soft vibrational modes in the DOS, which is a very
important outcome of our study.
7
100
50 10
nits] nits]
u u
b. b.
[ar [ar 1
ω) 10 ω)
ϵ'( ϵ''(
5
0.1
10-4 0.1 100 105 10-6 10-5 10-4 10-3 10-2 0.1 1
ω [Hz] ω [Hz]
FIG. 3. Real part of the dielectric function as a function of FIG.4. Dielectriclossmodulusasafunctionofthefrequency
the frequency of the applied field. Symbols are experimen- of the applied field. Symbols are experimental data of the
tal data of the real part of the dielectric function of glyc- imaginarypartofdielectricfunctionofglycerolatT =184K
erol at T = 184 K from Ref. [38]. The solid line is our fromRef.[38]. ThesolidlineisourcalculationfortheMarko-
theoretical calculation for the Markovian friction case, i.e. vian friction case, i.e. ν =const in Eq.(16). The dot-dashed
ν = const in Eq.(15). The dot-dashed line is the real part line is the imaginary part of the Fourier transform when
of the Fourier transform when we consider the best-fitting weconsiderthebest-fitting(empirical)stretchedexponential
(empirical)stretched-exponentialfunctionwithβ =0.65. We (Kohlrausch) function with β = 0.65. In our calculation, we
havetakenC =10,ν/m=1620andA1 =0.039. Fortheem- have taken C = 10,ν/m = 1620 and A2 = 0.0437. For the
pirical fitting with β = 0.65, τ = 6555. Rescaling constants empirical fitting β =0.65, τ =6555. Rescaling constants are
are used to adjust the height of the curves. used to adjust the height of the curves.
IV. DIELECTRIC RELAXATION IN THE TIME
DOMAIN
We are also interested in the dielectric response in the
10
time domain. In order to keep the derivation amenable s]
to analytical treatment, we focus on the case of Marko- nit
u
vian friction, ν = const. The time dependent dielectric b.
function (cid:15)(t) and complex dielectric function (cid:15)∗(ω) are ar 1
[
related as: ω)
(
d(cid:15)(t) 1 (cid:90) ∞ ϵ''
= ((cid:15)∗(ω)−(cid:15)(ω =∞))eiωtdω (17) 0.1
dt 2π
−∞
(cid:90) ∞ d(cid:15)(t) 10-5 10-4 10-3 10-2 0.1 1
(cid:15)∗(ω)=(cid:15)(ω =∞)− e−iωtdt (18) ω [Hz]
dt
0
.
By using Eq.(13) for Markovian friction, we can write FIG. 5. Dielectric loss modulus as a function of the fre-
theanalyticalformof(cid:15)(t)asfollows(seeAppendixBfor quency of the applied field. Symbols are experimental data
the details of the derivation): of the imaginary part of dielectric function of glycerol at
T = 184K from Ref. [38]. The solid line is the theoreti-
cal description presented in this work for the non-Markovian
(cid:15)(t)=B+
friction case, i.e. ν˜(ω) in Eq.(16) is the Fourier transform
(cid:90) ωD Aρ(ω )(cid:18)e(K−ν/2m)t e−(K+ν/2m)t(cid:19) of Eq.(8). The dot-dashed line is the imaginary part of the
p + dω , (19)
2K K−ν/2m K+ν/2m p Fourier transform when we consider the best-fitting (empir-
0 ical) Kohlrausch relaxation function (cid:15)(t) ∼ exp(−t/τ)β with
. (cid:113) β = 0.65. In our calculation, we have taken C = 555,
where K = (Cωp)2− 4νm22, while B is a re-scaling con- A2 = 120, ν0 = 6×106, and b = 0.3(for the physical justifi-
stant. This equation is a key result: it provides a direct cation of the latter value, see Section IIC). For the empirical
andquantitativerelationbetweenthemacroscopicrelax- Kohlrausch fitting β = 0.65, τ = 6555, as before. Rescaling
ation function of the material and the DOS. As we show constants are used to adjust the height of the curves.
below, the presence of a boson peak in ρ(ω ) directly
p
causes stretched-exponential decay in (cid:15)(t) via Eq.(19).
8
fitting is able to reproduce the α-wing asymmetry and
although memory effects in the atomic-scale dissipative
60 dynamics are non-negligible, the most important factor
s] that causes the asymmetry is represented by the excess
nit of soft modes in the DOS (boson peak). For the loss
u40
b. modulus (cid:15)(cid:48)(cid:48)(ω), on the low-frequency side of the α-peak,
r
a we show that the response is controlled by the scaling
[
ϵ(t)20 ∼ ωp4 as ωp → 0 in the DOS. In the time-domain re-
sponse, remarkably, our framework recovers a stretched-
exponentialrelaxationwithβ =0.56,overtheentiretime
0
domain. These unprecedented results show, for the first
0.1 10 1000 105 107
t[s] time, that stretched-exponential relaxation in glasses is
directly caused by the quasi-localized boson-peak excess
modes contribution to the relaxation spectrum. These
FIG. 6. Time-dependent dielectric response. The solid line resultsopenupnewopportunitiestounderstandthecru-
is calculated using our Eq.(19) with physical parameters cal- ciallinkbetweenα-relaxation,bosonpeakanddynamical
ibratedinthefittingofFig.3. Thedashedlinerepresentsthe heterogeneity [48, 49] in glasses.
stretched-exponential Kohlrausch function that more closely
approximatesourprediction,calculatedusingtheparameters
β = 0.56 and τ = 5655. Rescaling constants are used to
ACKNOWLEDGMENTS
adjust the height of the curves.
Many useful discussions with R. Richert, W. Goetze,
and with E. M. Terentjev are gratefully acknowledged.
In Fig. 6, we plot predictions of Eq.(19) with the
parameters calibrated in the glycerol data fitting for
the case ν = const, along with the Kohlrausch func- Appendix A: Lorentz local field effect on the total
polarization
tion [45, 46], for the relaxation in the time domain.
It is seen that our description based on soft modes
is able to perfectly recover stretched-exponential relax- In condensed matter systems, the electric field that
ation, with stretching-exponent β = 0.56, over many effectively acts on a molecule locally is equal to the ex-
decades in frequency. Without the boson-peak modes ternalfieldonlyinthelimitofvanishingpolarizabilityof
in the DOS, we have checked that stretched-exponential themolecule[9]. Thisisawell-knowneffectwherebythe
relaxation cannot be recovered, and the decay is simple- field in the medium is affected (diminished) by the local
exponential. Hence, our Eq.(19) provides the long- alignmentofthepolarizedmolecules. ThesimpleLorentz
sought cause-effect relationship between soft modes and cavity model works well in materials where the build-
stretched-exponentialrelaxation,evenforthesimplecase ing blocks are not pathologically shaped or anisotropic,
of Markovian friction where the response is clearly dom- and is applicable to random isotropic distribution of the
inated by the density of states. building blocks. Without loss of generality, we present
an analysis for the case of Markovian friction ν =const.
The derivation of the local field or Lorentz field can be
found in many textbooks, e.g. in Refs. [9, 50]
V. CONCLUSIONS
4π
E =E+ P. (A1)
We have re-considered the nature of the dielectric α- loc 3
relaxation of simple glass-formers from the standpoint
of soft modes, dissipation and lattice dynamics. We
started from the same presumption of Ref. [3] that di-
With E replaced by E , we now write equation of
loc
electric α-relaxation emerges from many-body dynamics
motion as
in a statistical way, transcending the details of charge
4π
dynamics. This should especially be true for glycerol, m¨r +νr˙ +H r =q (E+ P). (A2)
a paradigmatic glass-forming molecule. Starting from i i ij j e 3
a microscopic system-bath Hamiltonian, we are able to As a consequence, we instead have
reproduce the dielectric response of glycerol in reason-
q 4π
able agreement with state-of-the-art experimental data. δ˜r (ω)= (E˜(ω)+ P˜(ω)). (A3)
Ourphysically-informedtheoreticalfittingcompareswell i mω2−iων−ωp2 3
with empirical Kohlrausch fittings. For both the re-
The total polarization is
active modulus (cid:15)(cid:48)(ω) and the loss modulus (cid:15)(cid:48)(cid:48)(ω), our
modelprovidesasignificantlybetterfittingthanthebest P =((cid:88)qδr +αE ), (A4)
i loc
Kohlrausch fitting. For the loss modulus, our model
i
9
whereαisthemicroscopicelectronicpolarizability. Com- f(x) = 1[f(x−) + f(x+)] for all x. Then f(x) =
2
biningtheaboverelationstogetherandsummingoverall lim 1 (cid:82) e−iξxe−(cid:15)2ξ2/2fˆ(ξ)dξ, x ∈ R. Moreover, if fˆ ∈
contributions from all the building blocks, we obtain (cid:15)→02π
L1, then f is continuous and f(x) = 1 (cid:82) e−iξxfˆ(ξ)dξ,
2π
χ(ω) x∈R.
(cid:15)(ω)=1+4π ,
1− 4πχ(ω)
3 The uniqueness of the inverse Fourier transform is
χ(ω)=q2(cid:90) ωD ρ(ωp) dω +α (A5) guaranteed by this theorem. If we can find a function of
e mω2−mω2+iων p time, whose Fourier transformation gives back the com-
0 p
plex dielectric function (cid:15)∗(ω), then this function would
whereweusedD =(cid:15)E =E+4πP. Wehavecheckedthat be the time derivative of the dielectric relaxation (cid:15)(t).
accountingfortheLorentzfieldandusingEq.(A5)forthe We use the following ansatz
fittingproducesverysimilarresultsanddoesnotalterthe
fitting of the dielectric relaxation data qualitatively. e−γtsin(Kt)
H(t) (B4)
K
(cid:113)
Appendix B: Derivation of Eq.(19) for the relaxation where γ = ν and K = − ν2 +(Cω )2 and H(t) is a
function in the time-domain 2m 4m2 p
Heaviside step function, whose Fourier transformation is
expressed as 1 .
WerecallthattheFouriertransformofafunctionf(x), ω2−iνω/m−(Cωp)2
However, we need to put care in taking ν (cid:29) 2mCω ,
in this article, is defined as: D
which amounts to restricting our analysis to the high-
(cid:90) ∞ friction overdamped dynamical regime. In this way, we
fˆ(ω)= f(x)e−iωxdx (B1) finally obtain (for t>0)
−∞
(cid:113)
while the inverfs(exF)o=uri1er(cid:90)tra∞nsffo(ωrm)eiiωsxddeωfi.ned as (B2) d(cid:15)d(tt) =e−2νmt (cid:90)0ωD Aρ(ωp)(cid:113)sin4hνm2(2 −4ν(m2C2ω−p)(2Cωp)2t)dωp.
2π
−∞ (B5)
Upon further integrating over t, we recover Eq.(19) in
From Eq.(17) in the main article, we firstly need to find
the main article.
the time derivative of (cid:15)(t):
d(cid:15)(t) 1 (cid:90) ∞(cid:90) ωD Aρ(ω )eiωt
=− p dω dω.
dt 2π 0 0 ω2−(Cωp)2−iων/m p Appendix C: Behavior of (cid:15)(cid:48)(cid:48) when ω→0
(B3)
We can change the order of integration, which gives:
Withoutlossofgenerality,wespecializeontheMarko-
(cid:90) ωD (cid:90) ∞ 1 eiωt vian case ν =const and take the limit ω →0 in Eq.(13):
Aρ(ω ) − dωdω .
0 p 0 2πω2−(Cωp)2−iων/m p lim (cid:15)(ω)∗ = lim (cid:18)1−(cid:90) ωD Aρ(ωp) dω (cid:19)
Note that, for the inner integration, i.e., ω→0 ω→0 0 ω2−i(ν/m)ω−C2ωp2 p
(cid:82)a0l∞yt−ic21cπoωn2t−in(uCaωetpii)ωo2tn−iωoνf/ωmdtωo,thweecocmopulledx mplaaknee aannd uanse- =1−Aωli→m0(cid:90)0ωD ω2−i(νρ/m(ω)pω)−C2ωp2dωp.
contour integration to evaluate the complex integral. (C1)
However, we can achieve the same result via a simpler
route just using the Fourier inversion theorem [52] that We Taylor-expand ρ(ωp) around ωp =0:
we recall below.
ρ(cid:48)(cid:48)(0) ρ(3)(0)
Theorem (The Fourier Inversion Theorem). Suppose ρ(ωp)=ρ(0)+ρ(cid:48)(0)ωp+ 2 ωp2+ 6 ωp3+... (C2)
f is integrable and piecewise continuous on R, de-
fined at its points of discontinuity so as to satisfy Thus, after substituting Eq.(C2) into Eq.(C1), we have
10
lim (cid:15)(ω)∗ =1−A lim (cid:90) ωD ρ(0)+ρ(cid:48)(0)ωp+ ρ(cid:48)(cid:48)2(0)ωp2+ ρ(36)(0)ωp3+...dω
ω→0 ω→0 0 ω2−i(ν/m)ω−C2ωp2 p
=1−A lim (cid:90) ωD [ρ(0)+ρ(cid:48)(0)ωp+ ρ(cid:48)(cid:48)2(0)ωp2+ ρ(36)(0)ωp3+...](ω2+i(ν/m)ω−C2ωp2)dω
ω→0 0 ω4−2C2ω2ωp2+C4ωp4+ω2ν2/m2 p
=1−A lim (cid:90) ωD W0(ω)+W1(ω)ωp+W2(ω)ωp2+W3(ω)ωp3+...dω (C3)
ω→0 0 ω4−2C2ω2ωp2+C4ωp4+ω2ν2/m2 p
where W (ω) = ρ(0)(ω2 + i(ν/m)ω),W (ω) = flank of the α-peak, we can set ρ(ω ) ∼ ω4, then from
0 1 p p
ρ(0)(cid:48)(ω2 + i(ν/m)ω),W (ω) = −ρ(0)C2 + ρ(0)(cid:48)(cid:48)(ω2 + Eq. (C3), we want to study the behavior of
2 2
i(ν/m)ω),W (ω)= (ω2+i(ν/m)ω)ρ(3)(0)−C2ρ(cid:48)(0).Inorder
3 6
tolettheintegrandbecontinuousforbothrealandimag- (cid:90) ωD ω4
lim p dω . (C4)
inarypart,(ω,ωp)∈R+∪{0}×R+∪{0}(itmakessense ω→0 0 ω2−iν/mω−C2ωp2 p
to change the order of integration/limit at (0,0)), we
must have ρ(ω ) ∼ 0,ρ(cid:48)(ω ) ∼ 0,ρ(cid:48)(cid:48)(ω ) ∼ 0,ρ(3)(ω ) ∼
p p p p
0asω →0. Thereisnorestrictionforρ(4)(ω )orhigher Without loss of generality for the asymptotic analysis
p p
order as ω ∼ 0. Hence, we must have ρ(ω ) ∼ ωn for we can take ω = 1, and the integral can be evaluated
p p D
n > 3. In order to study the scaling of (cid:15)(cid:48)(cid:48) on the left analytically to the following expression
(cid:90) 1 ωD(ω )
(cid:15)(cid:48)(cid:48)(ω)= p dω (C5)
(ω2−ω2)2+ω2 p
0 p
(cid:18) (cid:20) (cid:21) (cid:20) (cid:21)(cid:19)
1 i i
=ω+ (ω2+iω)3/2arctan √ −(ω2−iω)3/2arctan √ (C6)
2 ω2+iω ω2−iω
from which we obtain
π
(cid:15)(cid:48)(cid:48)(ω)≈ω+ ((ω2+iω)3/2−(ω2−iω)3/2) (C7)
4
and hence (cid:15)(cid:48)(cid:48)(ω) ∼ ω in the limit of small frequency, in
agreement with the experimental data.
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