1. science
2. physics

What is there to be explained
about
glasses and glassformers ?
Facts, questions, views
Gilles Tarjus
(LPTMC, CNRS/Univ. Paris 6)
1
III
Supercooled liquids and the glass
‶transition″
What is there to be explained ?
Salient properties
(1)
Dynamical facts
Part I
One of the most spectacular phenomena in
all of physics in terms of dynamical range
!"#$%&'&&( )$")$*+*%&%,%+-./%&01+$12134+%
53&+$34%1+1"4&01+$/#%/&/%+&6*2141/&)3$&7/&!".)"$+/./4+&6/&73&
01%!"%1+* ! 8*2141+1"4&3$91+$31$/&.31%&!".."6/&
Arrhenius plot of the viscosity: Log(viscosity/poise) versus 1/T.
/:)*$1./4+37/./4+;&
(Angell, 1995)
</)$*%/4+3+1"4&
Conventional (arbitrary)
A$$>/41#%B&
definition
of Tg:
Viscosity
= 103 Poise
7"C?+/.)%@&
2"4!+1"4&6/&'DE;
Similar results for relaxation time τ measured by various techniques.
<*%#7+3+%&%1.1731$/%&)"#$&6/%&+/.)%&6/&$/73:3+1"4&./%#$*%&)3$&&6=3#+$/%&.*+>"6/%
?+/.)%&6/&$/73:3+1"4&", #" @
!
One of the most spectacular phenomena in
all of physics in terms of dynamical range
FIG. 1. Frequency dependence of the dielectric constant in PC at various temperatures. The solid lines are fits with the CD function
performed simultaneously on , ! . The dotted line is a fit with the Fourier transform of the KWW function.
the KWW function is more widely used nowadays, in the
authors’ experience dielectric loss data in glass-forming materials are often described much better by the CD function
!22,26,27,47". The inset of Fig. 2 shows the temperature dependence of the frequency # $ !1/(2 % & $ ' ) which is virtually
identical to the peak frequency # p . Here & $ ' denotes
the mean relaxation time !48" calculated from & $ ' CD
! ( CD$ CD for the CD function and & $ ' KWW! $ KWW /
( KWW")(1/( KWW) () denoting the Gamma function* for
the KWW function. The results from the CD +circles* and the
KWW fits +pluses* agree perfectly well and are in accord
with previously published data !37,38,41–43,49". # $ (T)
ωpeak ∼
Frequency-dependent dielectric susceptibility (imaginary part) for
liquid propylene carbonate (Lunkenheimer et al., JCP 2001)
FIG. 2. Frequency dependence of the dielectric loss in propylene carbonate at various temperatures. The solid lines are fits with the CD
function, the dotted line is a fit with the Fourier transform of the KWW law, both performed simultaneously on , " . The dash-dotted line
indicates a linear increase. The FIR results have been connected by a dashed line to guide the eye. The inset shows # $ !1/(2 % & $ ' ) as
resulting from the CD +circles* and KWW fits +pluses* in an Arrhenius representation. The line is a fit using the VFT expression, Eq. +1*,
with T VF!132 K, D!6.6, and # 0 !3.2"1012 Hz.
1
τrelax
Fragility
Arrhenius plot with T scaled to Tg.
(Angell, 1993)
2
quid/Glass Physics
Tm
TA
sensitive to
Xtal structure
Tv
Landscape Collisional
ted
Transport
rt
dic condensed molecular phase
a dilute gas and a frozen glass.
rature, Tm the melting point.
e signalling the crossover to aclly but not always below Tm . Tg
ture which depends on the time
Tg the system is out of equilibuzmann temperature (see text).
which signals the quantization
FIG. 2: The viscosities of several supercooled liquids are plotted as functions of the inverse temperature. Substances with
almost-Arrhenius-like dependences are said to be strong liq-
Interlude: Explanations of slow dynamics
• Non-cooperative:
Arrhenius T-dependence for
chemical relaxation time
τ ∼ exp
!
E
T
τ
"
E
with a roughly constant
activation energy E.
• Cooperative:
Critical slowing down of relaxation (approaching a critical point at Tc)
✴
Diverging correlation length:
✴
Diverging relaxation time:
ξ ∼ |T − Tc |−ν
τ ∼ ξ z ∼ |T − Tc |−zν
practice unattainable, phase transition
would occur at a temperature T0 < Tg,
if the experiments were carried out
sufficiently slowly that local equilibrium
could be maintained5–8. Presumably, this
dynamically unattainable transition would
be a thermodynamic transition from
a supercooled liquid to a state referred
to as an ‘ideal glass’. It has also been
suggested9–11 that there is a well-defined
crossover temperature, T*, at which the
characteristic collective behaviour evinced
by the supercooled liquid begins. This
crossover could be thermodynamic10,
Interlude (contd.)
However...
correlation function, so if the puta
order is of a more subtle type, perh
could have eluded detection. Attem
to measure multipoint correlations
obviously of central importance, b
have not been successful so far.
Conversely, there are two obse
that are challenging for those theo
with no fundamental involvement
thermodynamics. The first is the fa
Kauzmann paradox15. The excess e
ΔS, which is defined as the differen
between the entropies of the super
liquid and the crystal, is a strongly
decreasing function of T from Tm t
and extrapolates to 0 at a temperat
which, for fragile glass-formers, is
20–30% below Tg. Even though the
is, as we argued above, not relevan
physics of the supercooled liquid,
sensible rationale for considering Δ
fragile glass-formers are molecular
in which a significant fraction of th
entropy is associated with intramo
motions. By subtracting the entrop
crystal, one hopes to eliminate mo
contributions from extraneous deg
freedom. A large change in the ent
something to be taken very seriou
The second observation is that
an empirical relation between ΔS a
slow dynamics5,16. Specifically, ther
to be a correlation between the dec
of ΔS(T) and the increase of Δ(T)
decreasing temperature.
• Viscous slowing down of relaxation seems of cooperative
(or collective) nature...
T-dependent effective
activation energy
30
15
τ ∼ exp
!
E(T )
T
5
"
25
m -Toluidine
∆(T )/T *
log10(!)
10
GeO2
20
15
Glycerol
10
0
o -Terphenyl
!
5
1
0.5
"#
$ !
molecular ?
"#
$
1.0
Tg/T
1.5
T */T
collective ?
Figure 1 Temperature-dependent effective activation
energy of several supercooled liquids (see equation
(1)) in units of the empirically determined crossover
temperature scale, T *. Ortho-terphenyl is one of the
formula
0 liquids, whereas GeO2
most ‘fragile’ glass-forming
is relatively strong. The original data are taken from
references listed in refs 3 and 4.
• ... But with an activated T-dependence:
e.g., empirical fit to VTF
τ ∼ τ exp
832
!
"
C
IMPORTANT DYNAMICAL FACTS
T − T0 The most important experimental
about fragile, supercooled liquids i
!
!
n
© 2008 Macmillan Publishers Limite
What is there to be explained ?
Salient properties
(2)
Thermodynamical and
structural facts
mum !FWHM"% and a Q range
!ii" 6.5 Å with 'E!25 # eV and
sample transmission was about
d was used for relative detector
quency and temperature range, all observables are expected
to have the same spectral distribution. This implies in particular that the neutron-scattering law S(Q, ) ) factorizes into
a Q- and a ) -dependent part:
No marked changes
in the (pair)
static
S ! Q, ) " !S ! Q " A G ! ) " .
!2"
structure
In Fig. 3 we show the rescaled intensities
roups of 3 He detectors was connumber of points by regrouping
ning time channels, keeping the
und of typically 10"2 . The data
constant Q with step 'Q!0.05
Q
S(Q, ) )/S(Q)A Q !G( ) ) for different wave numbers Q.
presentative experimental spectra
arbitrary but the same for all Q
emperatures at Q!1.45 Å "1 !a"
mbers at T!293 K !b" are shown
e recognizes a strong quasielastic
temperatures which vanishes for
es way to a distinct inelastic scatn peak". We notice that the specdifferent for different Q.
atic structure factor S(Q) as mea) max(Q)
d ) S(Q, ) )%. The double0
nd the temperature dependence of
well with the one measured on
0%. One recognizes particularly
ns around 0.85 Å "1 , a region not
FIG.
2. Static structure
factor of
S(Q)
of deuterated
for three
Static
structure
factor S(Q)
liquid
o-TP atOTP
several
temperatures
fraction experiment $20%.
different temperatures. The inset shows the temperature dependence
from
justatbelow
to justbyabove
the glass
of S(Q)
selectedmelting
values of (Tm=329K)
Q which are indicated
the arrows
LING REGIME
transition
(TgThe
=243K).
in the figure.
S(Q) have been scaled to their values at 320 K
kable predictions of MCT is the (Tölle
for clarity.
Note the strong temperature variations of S(Q) at
et al.,1997)
e + relaxation: In a certain fre-
Q!0.85 Å "1 and Q!1.95 Å "1 .
Rapid decrease of the entropy
Kauzmann ‶entropy paradox″
1.0
!S / !Sm
B2O3
0.5
Tg
n-propanol
salol
o-terphenyl
m-toluidine
Tg
TK
0.0
0.0
0.5
1.0
T / Tm
‶Configurational″ entropy, normalized by its value at
melting versus T/Tm.
∆S = Sliquid − Sxtal
What is there to be explained ?
Salient properties
(3)
Dynamical facts
Part II
./%&01+$12134+%
Nonexponential and multi-step
5$34%2"$.3+1"4&634%&7/&
relaxation
(1)
6".314/&6/%&2$*8#/4!/%
!"#$%&'&&( )$")$*+*%&%,%+-./%&01+$12134+%
!99:";<&)3$+1/&1.3=1431$/&6/&73&5>&6/&!:+;
Time versus frequency domain:
χ!! (ω) ∝ ω
!"
!
#99:";<&)3$+1/&1.3=1431$/&6/&73&
+∞
%#%!/)+1?171+* iωt
dte
−∞
φ(t)
#99:";&@&"A:BC5;&&!99:";
%&7/&
4!/%
/&6/&73&5>&6/&!:+;
/&6/&73&
"A:BC5;&&!99:";
Time-dependent response/correlation function
Frequency-dependent dynamic susceptibility
Nonexponential and multi-step
relaxation (2)
‶Stretched″ α-relaxation: φ(t) ! e
−( τt )β
(K.W.W.)
(χ0 − χ∞ )
χ(ω) = χ∞ +
(1 − iωτcd )βcd
a
0.9
b
0.8
2
""(!)
0.7
K.W.W.
C.D.
0.6
0.5
0.4
1
0.3
Debye
spectru
-1
0
0.1
0.0
-0.1
0
-2
0.2
Debye
spectru
autocorrelation function
1.0
1
log 10 (!! Hz)
2
3
4
10-5
10-4
10-3
10-2
10-1
100
101
102
103
t , sec
Stretched relaxation of liquid m-toluidine: left, dielectric spectrum (Debye and Cole-Davidson
fits); right, photon correlation spectroscopy (exponential and Kohlraush-Williams-Watt fits)
(C. Alba-Simionesco,2001)
Nonexponential and multi-step
relaxation (3)
W. KNAAK
et al.:
1 .o
0.8
v
0.6
v)
3
W.
!5.
v
v)
0.4
0.2
Neutron experiments
OBSERVATION O F SCALING BEHAVIOUR OF DYNAMIC CORRELATIONS
ETC.
533
Computer simulations
20
this case, the dynamic correlation function C(t) coincides with the incoherent intermediate scattering function Fs (q, t) [41], which is normally measured 363
in experiments. In systems other than
liquids, ϕk (t) can be any
-*-a-&
meaningful observable carrying a real space label (be it
particle or spin).
The correlation function C(t) measures how quickly
correlations within the system decay in time. At high
temperatures we expect a very short-time ballistic regime
(for Newtonian dynamics), where particles move freely
with no mutual interactions, followed by a dissipative
regime, described by a normal exponential relaxation,
0
10-l~
10-
C(t) lo-”
= C0 exp(−t/τ
).
10-’0
t(s)
10-~
(56)
Fig. 3. - Normalized density
correlationthe
functions
calculatedtime
by Fourier
transformation
andparticresolution
In principle
relaxation
τ depends
on the
correction of TOF spectra
(solid
lines) and measured
by NSE (data
points)
experiments.
The solid
lines
ular
observable
ϕ.
However,
it
is
natural
to
expect
that
extended by the broken ones to long times were used in calculating the backtransformed solid lines in
at high T there is only one intrinsic time scale in the sysfig. 2.
tem, for example
the(normalized)
shear relaxation
time τR , and that
Time dependence
of the
dynamic
all the other time scales are a simple rescaling of it.
structure
S(Q,t)/S(Q)
for liquid
CKN
at various
The main
feature offactor
thealready
predicted
density
fluctuations
is the
existence
of two scaling
We
know
that
by lowering
the
temperature
the
regions governed
by
two
scaling
frequencies
temperatures
as obtained
fromvery
neutron
relaxation
time τR grows
sharply,time-of-flight
so that the decay
and neutron
experiments.
of C(t)spin-echo
is increasingly
slower approaching Tg . Therefore,
from a quantitative
point
of view, the dynamic corre(Knaak et al., 1988)
lation
functionof differs
significantly
from the
parameter
the transition
and D-1THz
is a structural
characteristic
where E is the control
correlation
function,
which shows
dramatic
tempermicroscopic frequency.
To first order
E is proportional
to the no
distance
from some
critical
temperature To. Theature
exponent
parameters close
a and bto
(0 c
0.5, 0 c b c 1)
are not
dependence
Tga.<However,
were
theuniversal,
sharp
increase of τ the only effect of lowering T , there would be
no qualitative signature of approaching glassiness. But
this is not what happens. Fig.11 shows the typical behaviour of the dynamic correlation function at various
temperatures, down to a temperature that is above, but
close to Tg . What the figure shows is that the qualitative
Time11:
dependence
of the self-intermediate
scattering
Figure
Two steps relaxation
The dynamic correlation
function
C(t)
in
a
Lennard-Jones
system.
In
this
case
C(t)
is
function Fs(Q,t) at various temperatures for a binary
the incoherent intermediate scattering function Fs (q, t), evalLennard-Jones model.
uated at the value of q where the static structure factor has the
1995)
main peak. At high(Kob-Andersen,
temperatures the decay
is exponential,
but when the temperature get close to Tg a plateau is formed
and relaxation proceeds in two steps. (Reprinted with permission from [52]; copyright of American Physical Society).
cal significance, after all, because something qualitatively
new shows up in the equilibrium properties of a liquid
close to Tg .
Let us describe more carefully the shape of the correlation function at low T (Fig.11). The time-scale over
which the correlation function arrives to the plateau does
Nonexponential and multi-step
relaxation (4)
FIG. 1. Frequency dependence of the dielectric constant in PC at various temperatures. The solid lines are fits with the CD function
performed simultaneously on , ! . The dotted line is a fit with the Fourier transform of the KWW function.
the KWW function is more widely used nowadays, in the
authors’ experience dielectric loss data in glass-forming materials are often described much better by the CD function
!22,26,27,47". The inset of Fig. 2 shows the temperature dependence of the frequency # $ !1/(2 % & $ ' ) which is virtually
identical to the peak frequency # p . Here & $ ' denotes
the mean relaxation time !48" calculated from & $ ' CD
! ( CD$ CD for the CD function and & $ ' KWW! $ KWW /
( KWW")(1/( KWW) () denoting the Gamma function* for
the KWW function. The results from the CD +circles* and the
KWW fits +pluses* agree perfectly well and are in accord
with previously published data !37,38,41–43,49". # $ (T)
Frequency-dependent dielectric susceptibility (imaginary part) for
liquid propylene carbonate (Lunkenheimer et al., JCP 2001)
FIG. 2. Frequency dependence of the dielectric loss in propylene carbonate at various temperatures. The solid lines are fits with the CD
function, the dotted line is a fit with the Fourier transform of the KWW law, both performed simultaneously on , " . The dash-dotted line
indicates a linear increase. The FIR results have been connected by a dashed line to guide the eye. The inset shows # $ !1/(2 % & $ ' ) as
resulting from the CD +circles* and KWW fits +pluses* in an Arrhenius representation. The line is a fit using the VFT expression, Eq. +1*,
with T VF!132 K, D!6.6, and # 0 !3.2"1012 Hz.
Nonexponential relaxation
Variation with temperature
of stretching exponent
!"#" $%&' ( )*+,-&. */ $*-01,234&..'-5 6*.'73 89: ;8<<<= 9>:?
M[
Stretching exponent β,
from
t
φ(t) ∼ exp[−( )β ]
τ
Temperature dependence of the stretching exponent beta
for various glass-forming liquids, from dielectric and light
scattering data. (Ngai, 2000)
X1". H. /&2@&%'#,%& 5&@&35&3(& $- #)& Y$)+%',0() &?@$3&3# # -$% 0$2& ()$0&3 "+'009-$%2&%0. K'N A/U8 0'+$+8 "+:(&%$+ '35 @%$@:+&3&
"+:($+ '0 ' -,3(#1$3 $- !! "! 8 '++ '%& -%$2 51&+&(#%1( %&+'?'#1$3 2&'0,%&2&3#0 -%$2 Z&-0. EMFPMO8GM8M[\I ]##&5 <: #)& Y$)+%',0()
-,3(#1$3 &?(&@# #)& ]++&5 (1%(+&08 ;)1() '%& -$% A/U -%$2 +1")# 0('##&%13" 2&'0,%&2&3#0 $- Z&-0. EH[PH^I. /)& &%%$%0 13 #)& *'+,&0 $- #
'%& #:@1('++: +\$\G. K<N # -%$2 #)& Y$)+%',0() -,3(#1$3 ,0&5 #$ ]# 510@&%01$3 $- 51&+&(#%1( %&+'?'#1$3 5'#' @+$##&5 '0 -,3(#1$3 $- !" "!
-$% G[_ <&3B:+ ()+$%15&`^[_ #$+,&3& K!N8 H!U K"N8 @%$@:+<&3B&3& K$N '35 L'+$+ K#N. L)$;3 '+0$ '%& # $- $%#)$#&%@)&3:+ KA/UN K,N '35
#$+,&3& K!N $<#'13&5 -%$2 5:3'21( +1")# 0('##&%13" 5'#'. /)& 3$%2'+1B&5 #&2@&%'#,%&0 !" "!! 1351('#&5 <: *&%#1('+ '%%$;0 '%& -$%
@%$@:+<&3B&3& KU!aN '35 H9<%$2$@&3#'3& KH!UN '35 #$+,&3&. /)& 3$%2'+1B&5 #&2@&%'#,%& !" "!4 10 3&'%+: #)& 0'2& -$% '++ +17,150 '35
10 1351('#&5 <: ' 013"+& *&%#1('+ '%%$;. /)& &%%$%0 13 #)& *'+,&0 $- # '%& #:@1('++: +\$\G.
!! ! ! ! !" #$ %&'() #)& *'+,& $- "! "!" # '# !" . /)&
0121+'% ()'3"&0 '(%$00 !4 '35 !! $- #)& 613&#1(
)1")&% #&2@&%'#,%&0 <: 1#0 4%%)&31,0 5&@&35&3(&
EMO8MMGPMMQI. /)& 3&'% ($13(15&3(& $- !! '35 !#
Heterogeneous dynamics (1)
When19approaching
J. Phys.: Condens. Matter
(2007) 113102
A
glass formation: fast and slow moving regions
Topical Review
B
Figure 5. From [99]. Reprinted with permission from AAAS. Three-dimensional rendering of
3-D
visualization
(confocal
of aparticles
concentrated
colloidal
colloidal
samples with
locations ofmicroscopy)
the fastest moving
(large spheres)
and other particles
!t . glass
The samples
(A) supercooled
with φ = 0.56 and
(smaller spheres),
over
a fixedand
time(B)
suspension
in (A)
liquid
states.areLarge
spheres:liquid
fast(er)
= 0.61.during
Clearly,τin ).the supercooled fluid, one can see large clusters of
(B) glassy
sample with
moving
particles
(0.5φ diam.
α
fast moving particles (there are 70 red particles clustered together), while these clusters are absent
(Weeks et al., 2000)
in the glassy sample.
!
Heterogeneous dynamics (2)
Meyers: Encyclopedia of Complexity and Systems Science — Entry 37 — 2008/4/18 — 17:06 — page 12 —
•Suggestive observations: nonexponential relaxation and translation/rotation decoupling
0
behavior
between periods of fast and slow motion
• Suggestive12observation: IntermittentGlasses
and Aging: A Statistical Mechanics Perspective
in a region of PVAc at 300K by AFM experiment (Vidal-Russel & Israeloff, 2000)
Glasses and Aging: A Statistical Mechanics Perspective, Figure 10
Time series of polarization in the AFM experiment performed by
Vidal Russell and Israeloff [161] on PVAc at T = 300 K. The signal intermittently switches between periods with fast or slow dynamics, suggesting that extended regions of space indeed transiently behave as fast and slow regions
P
orientational correlations, c(r; 0; t) / VN N
i; j
r i )Y(˝ i (0))Y(˝ j (t)), where ˝ i denotes the an
scribing the orientation of molecule i, r i (0) is t
tion of that molecule at time 0, and Y(˝) is some
priate rotation matrix element. Here, the ‘localit
probe comes from the fact that it is dominated by
term involving the same molecule at times 0 and
the contribution coming from neighboring molecu
dynamic susceptibility !4 (t) can thus be rewritten
Z
!4 (t) = " d3 rG4 (r; 0; t) ;
+ Exptal. evidence in molecular liquids and polymers
where
by selection of a subclass of slow(er) molecules
G4 (r; 0; t)etc...)
= hıc(0; 0; t)ıc(r; 0; t)i ;
(4D NMR, photobleaching, dielectric hole burning,
864
865
866
the spin glass susceptibility measuring the extent of amorphous long-range order in spin glasses. In this subsection,
we introduce these correlation functions and summarize
and translational invariance has been taken into
(" = N/V denotes the mean density). The above eq
show that !4 (t) measures the extent of spatial cor
between dynamical events between times 0 and t a
ent points of the system, i. e., the spatial extent of d
Heterogeneous dynamics (3):
Spatial correlations in the dynamics and
associated
length scale
!
Meyers: Encyclopedia of Complexity and Systems Science — Entry 37 — 2008/4
RK
ust 17, 2000
114
9:10
EDIGER
Need multi-point
space-time correlation/response functions...
AR109-04
Experimental challenge!!!!
Annual Reviews
Solid-state NMR in a polymer
Glasses and Aging: A Statistical Mechan
Computer simulation of Lennard-Jones model
where the multi-poin
by
ˇ
@F(t) ˇˇ
!T (t) =
@T ˇ N;
In this way, the expe
which quantifies th
functions F(t) to an i
be used in Eq. (7) to
over, detailed numer
TS2
ments [21,22] strong
Time dependence of the 4-point susceptibility.
Figure 7 Solid-stateEstimate
NMR measurements
(66)of
of the
the size
of regions of heterogeneous
of the size
dynamic
Glasses
and
Aging:
A
Statistical
Mechanics
Perspective,
Figure
11
dynamics ξ het for polyvinylacetate
at Tg +for
10 K.
Data are shown
for experiments
using
(7) actually provides
heterogeneities
polyvinyl
acetate
at
The maximum shifts in time with τα and its
fluctuaTime dependence of !4 (t) quantifying the spontaneous
spin diffusion (one-filter experiment, used to obtain spin diffusion coefficient) and using
Tg+10K: size 2-4 nm (Tracht et al., 1999)
increases
withfunction
decreasing
T (Berthier- a lower bound.
tions ofamplitude
the intermediate
scattering
in a Lennard-Jones
both spin diffusion and dynamic selection (two-filter experiment). The two-filter experiment
Using this metho
Biroli,
2009)
supercooled
liquid.
For each temperature, !4 (t) has a maximum,
decays somewhat slower than the one-filter experiment, indicating that ξ het is larger than
13
able to obtain the ev
the average distance between C labels in the system. Fitting indicates that ξ het is inwhich
the shifts to larger times and has a larger value when T is derange of 2–4 nm. (Reprinted with permission from Reference 66.)
creased, revealing the increasing lengthscale of dynamic hetero- many different glass
geneity in supercooled liquids approaching the glass transition
regime. In Fig. 12 we
A selection of questions and
topics
about the viscous slowing
down and the glass transition
What is the important characteristic
temperature ?
No consensus!!!
14
(T-dependent viscosity of o-TP)
Experimental: Tb, Tm, Tg
Extrapolated: T0, TK
Crossover: TA, T*, Tc, TB, Tx,...
12
Tg
10
8
log10 ( ! / poise)
Illustration of the variety of possibly
important temperatures for
characterizing the viscous slowing
down and the glass transition.
6
Tm
4
2
Tb
0
TK
Tc T T
B x
-2
TA
T0
T*
-4
0.0020
0.0025
0.0030
0.0035
1 / T ( K-1)
0.0040
0.0045
0.0050
Is there a growing correlation length ?
More than one ?
What is its (their) charateristic size ?
What is its (their) nature ?
ure in
(τL);
shear
n this
quite
from
ant to
How universal is the alpha relaxation ?
J. Phys. Chem.,
Vol. 100, No. 31, 1996
13207
Comparison
of experimental
techniques
16
8
35
Glass
30
4
12
25
2
log 10 ( !!/ sec )
0
14
E(T) / T*
Tg
10
20
8
-2
T* / T
15
6
1.0
-4
4
-6
-8
supercooled
liquid
log 10 ("!/poise)
6
2
0
-10
-2
-12
Liquid
Tm
-14
-4
120 140 160 180 200 220 240 260 280 300 320 340 360 380
Figure 8. The log of the peak frequency νp as a function of inverse
temperature forDielectric
four samples:
glycerol
(]),
propylene
glycol
(open
symbols)
versus
specific
heat(4), salol
(0), and o-terphenyl
mixed with
o-phenylphenol (O). The open
spectroscopy
(filled33%
symbols):
symbols show Log
dielectric
relaxation
data,
andfortheo-TP,
corresponding
solid
of peak frequency vs 1/T
salol,
symbols show propylene
results from
specific
heat
spectroscopy.
glycol,
glycerol
(from
left to right) For these
samples, excellent
agreement
is
observed
between
these two techniques.
(Dixon et al., 1991)
Reproduced with permission from ref 81. Copyright 1991 NorthHolland.
can be answered using quasi-elastic neutron scattering. In
nonpolymeric liquids, “structure” at high temperatures exists
T, K
Calorimetry and dilatometry, photon correlation and
dielectric spectroscopies, depolarized light scattering,
neutron scattering, viscosity measurements:
Log of relaxation time or viscosity vs T for m-toluidine
(Alba-Simionesco,
2001)
Fig. 1
Universality... however,
Stretching exponent is usually technique dependent!
Structural relaxation in glass-forming liquids
559
558
Figure 4.
D. Prevosto et al.
Figure 3. a-relaxation time of the KWW function obtained from OKE measurements (^)
and dielectric spectra (&) ®t (for the procedure see the text). It should be noted that
OKE relaxation times rescaled for a constant factor (~) superimpose on the dielectric
values.
Stretching parameters of the KWW function obtained from OKE measurements
Comparison(*)
beween
dielectric
spectroscopy
and optical
and dielectric
spectra (&)
®t.
Kerr effect in liquid phenyl glycidyl ether.
Left: KWW stretching exponent
melting point of the system: Tm ˆ 276:6 K. This ®nding of an almost constant scald
o
Right: Relaxationingtime
factor (½KWW
k) in the high-temperatur e region (T
Tg ) seems to be a
=½KWW
general result. Indeed very similar results have been even found in previous studies
the interval explored; in fact, all the values are consistent although
the
spread
is
wide,
(Prevosto
et
al.,
2002)
o
comparing the relaxation processes of propylene carbonate (PC) (Schneider et al.
except that at 333:4 K. The mean value of these data results in !KWW ˆ 0:91. On the
d
other hand it can be seen that !KWW
shows a decreasing behaviour as the temperature is lowered, varying by about 13% in the whole temperature range. This diŒerence could be due to the larger temperature range investigated by dielectric
spectroscopy or to a diŒerent in¯uence of the temperature on the relaxation process
obtained by the two spectroscopies. However, concerning the former hypothesis it
can be noted that, even in the temperature region where data from both experiments
d
are available, !KWW
presents a more pronounced decreasing behaviour. At this
moment it is therefore di!cult to explain the reason for this diŒerence and a more
1999) and diglycidyl ether of bisphenol-A (DGEBA) (Comez et al. 1999) measured
by dielectric spectroscopy and light scattering experiments. However, it should be
noted that, according to Schneider et al. (1999), in a PC glass former the a processes
are characterized by faster relaxation times in the light scattering spectra with respect
to the dielectric spectra with a scaling factor of about three, while in the DGEBA
sample, according to (Comez et al. 1999), the a-relaxation time scales are longer
from optical spectroscopy than from the dielectric technique with a scaling factor of
about 0:13. In these comparisons all the tested glass formers (PC, DGEBA and
PGE) have very similar dynamic behaviours concerning the slowing down of the a
processes; however, the epoxy resins DGEBA and PGE show a scaling factor k < 1,
Y"
P1: FRK
August 17, 2000
hem. 2000.51:99-128. Downloaded from arjournals.annualreviews.org
US Jussieu - Multi-Site on 12/15/09. For personal use only.
Supercooled Liquids and Glasses
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286=
DYNAMICS
IN SUPERCOOLED
LIQUIDS
115
.*61$ 6 .*37-/68 ,-+2*3+-4/ 6/, .)* D-+;41+ K4<'
()* Chem.,
6228-;6:-8-.9
.)* No.
F41+*
J. Phys.
Vol.45100,
31,74,*8
1996-+ ;4/G/*,
13207
<-.)-/ .)* 086++L31::*3 .36/+-.-4/ A+45.*/-/0B 3*=
0-4/' ()* 286.*61$ .)* .*37-/68 ,-+2*3+-4/ 6/, .)*
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N)*.)*3 .)* 24897*3 ;)6-/+ 63* */.6/08*, 43 /4.$
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74,*+$ -/;81,-/0 .)* 84;68 +*07*/.68 74.-4/$ .)*
F41+* 74,*+ 6/, .)* .*37-/68 ,-+2*3+-4/ AD-+;4+=
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*>6;.89 .)* +67* 6741/.' ()-+ 2342*3.9 45 .)*
+2*;.317 -+ ;688*, .)*3743)*4840-;68 +-728-;-.9'
()-+ 6++172.-4/ .13/+ 41. .4 :* 568+* 6+ G3+.
+)4</ :9 P86Q*O 6/, ;4=<43O*3+ 4D*3 .)* 86+.
.)3** ,*;6,*+ H!"#$!"%$!#"L!#J& 6/, 3*2*6.*,89
;4/G37*, :9 4.)*3+ -/ 76/9 24897*3+
H!"#$!"%$!RI$!R#$!#ML!%%&' S43 6 3*D-*< +** F*5+'
H!"#$!"%&'
Universality... however,
9:10
Annual
!"#!# $%&' ()*+,-*./ 0+%&'1 +21.2&*+3 4)*5(+/6
Figure 8. The log of the peak frequency νp as a function of inverse
temperature for four samples:
glycerol (]),
propylene
salol
S43 */.6/08*,
24897*3+$
.)*glycol
84;68 (4),
+*07*/.68
(0), and o-terphenyl 3*86>6.-4/
mixed with.-7*
33%!"o-phenylphenol
The open
)6+ +-0/-G;6/.89(O).
+.34/0*3
.*7=
S-0' R!' X-T*3*/. .*72*36.13* ,*2*/,*/;*+ 45 .)* D-+;4+-.9 6/,
Figure 7. Variation of several relaxation times with temperature in
symbols show dielectric
relaxation
data, and
the.)*
corresponding
solid
2*36.13*
,*2*/,*/;*
.)6/
+)-5. 56;.43 45
.)*
84;68 +*07*/.68 3*86>6.-4/ .-7* 45 6.6.-; 2489234298*/* +)4</
Ca(NO3)2‚3KNO3: longitudinal relaxation time for phonons (τL);
symbols show results
from ,-+2*3+-4/
specific heat
spectroscopy.
For -/
these
:9 .)*-3 3*+2*;.-D* +)-5. 56;.43+ 6+ 51/;.-4/+ 45 .*72*36.13*' ()*
.*37-/68
43 .)*
D-+;4+-.9$ <)-;)
.13/
Figure
8
Translational
(D
)
and
rotational
(D
=
6/τ
)
diffusion
coefficients
in 5347
o- F*5+' H!#I$!%I$!%%& <*3* 4:.6-/*, -+4.)*376889
,6.6 .6O*/
t
c in
reorientation Decoupling
time (τreor); conductivity
relaxation time (τσ); shear
samples, excellent
agreement
is observed
between
these
two
techniques.
-+ +.34/0*3
.)6/r .)6.
45 .)*
+)-5.
56;.43
45 .)*
Terminal
versus segmental
Rotation-translation
decoupling
o-TP
between
:9 6 ;47:-/6.-4/ 45 ;3**2 ;4728-6/;*$ ,9/67-; +)*63 74,181+
time (τH). In
thisas a function
viscosity relaxation time (τs); enthalpy relaxationterphenyl
Reproduced
permission
from -/ref.)*
81.+45.*/-/0
Copyright
1991 North(12)
of with
viscosity
and74,*+
temperature.
Translation:
self-diffusion
(solid
F41+*
,-+2*3+-4/
H!"IL
6/, ,9/67-;
8-0). +;6..*3-/0 in
7*6+13*7*/.+'
()* .*72*36.13*
relaxation
a polymer
(Chang
et
al.,
1994)
conductivity
and
structural
multicomponent system, different relaxation processes show quite
Holland.
!"%&' ()*+*
,36+.-;6889
,-T*3*/.
,*2*/,*/;*+
45 .)* .<4 7*;)6/-+7+ 4D*3 6 ;4774/ .*72*36=
circles); tracer diffusion
for two probes
(open
box and
open.*72*36.13*
triangle).,*=Rotation:
selfdifferent temperature dependences. Reproduced with permission from
2*/,*/;*+ 63* +)4</ )*3* 543 4/* *>6728*$ .)*
.13* 36/0*
63*
;8*6389
,-T*3*/.'
()*
*3343+
-/
.)*
+)-5. 56;.43+ 63*
(Ngai,
2000)
relaxation
times in ionicdiffusion for deuterated o-terphenyl 67432)41+
( filledquasi-elastic
and open
Full A6PPB
lines
η−0.75
ref 8. Copyright
1991 North-Holland.
.92-;6889
<-.)-/ 6 56;.43 45 ! *-.)*3 <69'
6.6.-; diamonds).
2489234298*/*
can be answered using
neutron
scattering.
In show
glassformer CKN
and η−1 dependences. Translational diffusion has a weaker dependence on viscosity be-
nonpolymeric liquids, “structure” at high temperatures exists
The question of one or many R processes is most relevant to
only Translational
on time scales and
shortrotational
compared diffusion
to molecular
reorientalow
290
K
than
above.
have
different tempeature
(Angell,
1991)
a fundamental understanding of supercooled liquids in the case
tion.82,83
Unfortunately,
present there
are used
no experiments
dependences
below 290
K. (Adapted
from at
Reference
12 and
with permission.)
of single-component systems. Figure 3 shows that
four different
which directly probe structural relaxation on time scales much
experimental techniques give very similar results for the R
longer than a nanosecond (on the relevant length scale of 0.5process in o-terphenyl. Very nearly this same temperature
10 nm). The appropriate time scales can be probed on longer
dependence is also observed for probe molecule reorientation66
84 dramatically
of
the
same
molecule
arescales
compared
here,
datacorrelation
emphasizes
that the
length
with visible
lightthis
photon
spectroscopy.
4
and viscosity over 14 decades in each of these quantities. For
X-ray
photon
correlation
spectroscopy
probes
structural
relax79
between
this well-studied system and others, the connection
different
temperature dependences for rotation and translation in Figure 8 cannot
ation on the relevant length scales but for the present lacks
the viscosity and various measures of molecular
reorientation
be ascribed to the sensitivity.
fact that85rotation of one molecule (OTP) was being compared
seems well established, at least to a good approximation. In
with
translation
of
another
(a probe).indicate
[Over that
the some
rangerelaxation
where rotation
and transRecent experiments
times
some cases, small differences between the temperature dependassociated
structure
may be
considerably
longer than
ences of these variables have been interpretedlation
as evidence
for
measurements
can bewith
made
on neat
OTP,
these results
arethe
in good accord
time scale for molecular rotation near Tg. This implies that
a diverging length scale as Tg is approached.80 Others have
with the probe results shown in Figure 9 (46).]
Respective roles of density and temperature
✴
Density driven
✴ Temperature
driven
congestion, jamming
thermal activation
Use additional info from pressure measurements
At constant P, variation of T implies both change of T at constant rho
and change of rho at constant T
τ (P, T ) → τ (ρ(P, T ), T )
Respective roles of density and temperature:
Empirical rescaling of density effect in
When no high-T data are available ,
glassforming
liquids
polymers
extension to polymers
withand
e(!) $ E%(!)
How well established ?
4
!
4
PVAc
2
0
2
!#=0.62
PVME
log
x=2.7
Patm
50MPa
100MPa
150MPa
200MPa
250MPa
300MPa
-6
-8
-10
0.0020
0.0025
e(rho)/T
PVME
0
0.0030
0.0035
!x/T
!x/ T
0.0040
0.0045
%$ &) /log (
(!,
-4
x= 1.4
323 K
333 K
343 K
353 K
363 K
373 K
383 K
393 K
403 K
413 K
sec)
#
!( " (
/sec),T)
-2
"
e(!) '
x
e(ρ)
(
)
log("# (!, T ))=log(τ
F *( (ρ,
where
e
!
=
!
or e(! )=! $!*
%
T
))
=
F
T
α
)
&
T
-2
-4
-6
PVAc
-8
!#=0.29
-10
0.0008
e(rho)/T
0.0012
0.0016
(!"!# ) /T
!"!#/ T
0.0020
0.0024
e(rho)/T
31
Scaling L-J
of themixtures
relaxation time with e(rho)/T:
(a) Binaryliquid
Lennard-Jones
Molecular(PVME)
liquid (o-TP);
Binary
molecular
(oTP) model; (b)polymers
(c) Polymer (PVME) (Alba-Simionesco et al., 2002-2005)
However... not observed in soft ‶jamming″
systems
Figs. 2a, 2b where we plot
nd WCA for all densities beon of the inverse of the scaled
construction all curves coperature above some ‘onset’,
the viscous regime roughly
imple Arrhenius fit becomes
t temperature, we find that
se onto a master curve (with
he lowest density of 1.1), as
or the three highest densities
dence between LJ and WCA
hose densities, and the curves
s the density to reach values
rcooled liquids, i.e. ρ = 1.2
The isochoric ‘fragility’ of
density-dependent, which is
found in dense fluids of har. This is clearly at variance
choric fragility of the LJ.
play in Fig. 2c the density
τα (ρ, T )/τ∞
106
(a)
ρ = 1.1
1.2
1.4
1.6
1.8
104
102
Scaling !
Standard binary Lennard-Jones model
LJ
exp(E∞ /T )
100
10
τα (ρ, T )/τ∞
e data at high T with a good
effective activation energy
n then be used to compare
esence and in the absence of
temperature scale T /E∞ (ρ)
ove hypothesis. No emphasis
g of this Arrhenius fit, which
d nonsingular representation
Example from computer simulations: binary Lennard-Jones model versus
3
purely repulsive
WCA model
0
2
4
E∞(ρ)/T
6
8
(b)
ρ = 1.1
1.2
1.4
1.6
1.8
104
102
No scaling !
Truncated, purely repulsive, Lennard-Jones
model (WCA): model for soft jamming
systems
WCA
exp(E∞/T )
100
0
2
20
(c)
10
5
(Berthier-Tarjus, 2009)
6
E∞ ∝ ρ5
4
E∞(ρ)/T
6
8
Empirical correlations
al. 2003
l. 2000
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Buchenau et al. 2004
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! #, #""! #,fragility
Correlation
between
'$,%'$( &$%+ $D 1 \"4$+$&! 9$)%N"/+( %+$ ()/=& 9E+
Böhmer
etexponent
al. 1993
and stretching
(Bohmer et al., 1993)
_"D1 W1
`/++$-%!"/&
9$!=$$&
$&)%&,$'
'"4.("/&
Correlation
between
translation-rotation
-/D '"! #, #""! #, #>A!\>A ( %! $D %&' !)$ K/)-+%.(,) $F*/&$&! /0 !)$
decoupling and stretching at Tg for liquids
*+/9$ +/!%!"/&%- ,/++$-%!"/& 0.&,!"/& %! $D "& 0/.+ #%!+",$(3
2;<3 and
;TM3polymers
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Correlation fragility & nonlinear relaxation
Universality of alpha relaxation
+ Density/temperature scaling
+ etc...
ature effects on correlations between fragility and glassy properties – p.2/18
Correlations between slow dynamics
and thermodynamics
7
7
40
Data
2
R =0.52
Data
2
R =0.52
!Cp (kb per particle)
30
!Cp (kb per particle)
30
40
20
10
20
0
0
50
100
150
m
Correlation between heat capacity
jump at Tg and fragility for
FIG. 3: Left: Illustration of the Adam-Gibbs correlation: plot of log 10 τα /τ0 as a function of the inverse configurational entropy,
molecular liquids
revealing a reasonably linear region between
10−7 sec. (corresponding to T = T ∗ ) and 10 sec. (T ≈ Tg ). From [18]. Right:
0
2005)
correlation between the fragility parameter
m and the jump
specific heat
a collection of 20 molecular glasses
0
50 of (Stevenson-Wolynes,
100 at Tg , ∆Cp , for
150
10
for which we were able to cross-check the results from different m
sources. The correlation is significant, albeit far from perfect
(R2 = 0.52). The regression gives ∆Cp ≈ 0.15m. A better correlation, with much less scatter, is obtained when ∆Cp is counted
“bead” the
[37]. inverse of
Log(τ/τ0per
) versus
+ Correlation between dynamic and thermodynamic
T times the configurational
fragilities (Ito et al.,1999)
entropybetween
Sc for liquid
the two2-MTF
is deemed fortuitous. We will come back to this point in sections VI A,VI B.
FIG. 3: Left: Illustration of the Adam-Gibbs
correlation:
plot of log 10 τα /τ0 as a function of the inverse configurational entropy,
Finally, 1998)
as discovered
more recently,
dynamics is glasses
is spatially heterogeneous
and temporally
intermittent [40,
+ Correlation
(Richert-Angell,
revealing a reasonably linear
region between
10−7 sec. (corresponding
to T = T ∗ )between
and 10 sec. VTF
(T ≈ T
Tg0).and
FromKauzmann
[18]. Right: TK
41]. A number of experimental, numerical and theoretical papers have established that the slowdown of supercooled
correlation between the fragility parameter
m and theisjump
of specific
heat atbyTthe
of 20 molecular
g , ∆C
p , for
etc...
liquids as temperature
reduced
is+accompanied
growth
of aa collection
purely dynamical
correlation glasses
length ξd , with no
for which we were able to cross-check
the
results
from
different
sources.
The
correlation
is
significant,
albeit
far from perfect
static counterpart. This length can be for example defined by measuring how a local perturbation
of the system (for
(R2 = 0.52). The regression givesexample
∆Cp ≈a0.15m.
A better
correlation,
much less
when ∆Cp isξdcounted
local density
change)
influences with
the dynamics
at ascatter,
distanceisr obtained
from the perturbation,
is the decay length
of this response function (see [42, 43]).
per “bead” [37].
The fact that dynamics in glasses is heterogeneous is thought to have observable consequences on macroscopic
β
quantities [40, 41]. For example, the stretched exponential nature e−t of the relaxation is often interpreted in terms
of a mixture of exponential functions with different local relaxation time, reflecting the coexistence of slow and fast
Correlations between slow dynamics
and fast dynamics (in liquid or glass)
Boson Peak and
mP
Example:
Large intensity of boson peak small fragility
Sokolov et al. 1993
gDB (ω)
g(ω)
(T = Tg )
R = IIquasi
BP
Novikov et al. 2005
Correlations
between slow dynamics and fast
Correlations
dynamics (in liquid or glass)
Correlations
Correlations between fragility and: Poisson ratio of elastic moduli, relative
amplitude of Boson peak, mean-square displacement, ergodicity parameter
at Tg, etc...
Correlations
Sokolov et al. 1993
Sokolov et al. 1997
Buchenau et al. 2004
Sokolov et al. 1993
Sokolov et al. 1993
Scopigno et al. 2003
et al. 1997
Sokolov etSokolov
al. 1997
Yannopoulos et al. 2000
Buchenau et al. 2004
Scopigno et al. 2003
Yannopoulos et al. 2000
Yannopoulos et
al. 2000 et al. 2000
Yannopoulos
Scopigno et al. 2003
Novikov et al. 2004
Novikov et al. 2004
Yannopoulos et al. 2000
Buchenau et al. 2004
+ Correlation
Huang
et al. 2000 between alpha relaxation
time and shear modulus
oral.
meanBöhmer et
1993
square displacement (Dyre), etc...
Density versus temperature effects on correlations between fragility and glassy properties – p.2/18
Huang et al. 2000
Böhmer et al. 1993
Density versus temperature effects on correlations between fragility and glassy properties – p.2/18
Caution!!!
•Correlation does not mean causality
•Large body of data and systems are required to ascertain
correlation
•Large error bars
•Correlations may not be robust when studying effect of
additional control parameters (pressure, molecular weight,...)
cf. work by Alba-Simionesco, Niss et al.
Scopigno et al. 2003
Assessment of correlation between ‶kinetic″ and
‶thermodynamic″ fragilities for large data set
2004
Correlation between dynamic (from relaxation time)
and thermodynamic (from heat capacity) fragilities
(Huang-McKenna, 2000)
Huang et al. 2000
Böhm
Diversity of views on glasses, glass
formation and the glass ‶transition″
• Variety of questions...
• Variety in the scope of systems under study...