Biman Bagchi

1
Relaxation in Glassy Liquids
Biman Bagchi
Indian Institute of Science, Bangalore, India.
Dr. Sarika Bhattacharyya Mr. Arnab Mukherjee Mr.
Dwaipayan Chakrabarti Dr. Rajesh Murarka
Adv. Chem. Phys. Vol. 116 (2001)
2
Plan of the talk

• Introduction (summary of some experimental
  results).
• Theoretical description (de Gennes narrowing,
  self-consistent mode coupling theory),
  Applications of MCT (Stokes-Einstein relation,
  Liquid Crystals)
• Some computer simulation results.
• More MCT (MCT of glass transition)

3
Basic Features

• Rapid rise of viscosity with lowering of T in
  the deeply supercooled liquid near the glass
  transition temperature. This rapid rise starts
  typically 30-50 deg C above Tg.
• This increase in viscosity can be described by
  Vogel-Fulcher expression. The same set of
  parameters can describe the rise for 4-5 orders
  of magnitude.

4
Basic Features (continued)

• Relaxation functions (that is, relevant time
  correlation functions) become markedly
  non-exponential in this temperature range.
  Stretched exponential (KWW) form provides good
  fit with a low value of the exponent (typically
  0.5).
• Computer simulation studies on binary mixtures
  have shown that at high pressure and low
  temperature, hopping becomes the effective mode
  of mass transport and orientational relaxation.
  The emergence of hopping is rather gradual, that
  is, it coexists with continuous (Brownian) mode
  of diffusion until some low temperature where the
  latter ceases to contribute to diffusion.

5
Stretched-exponential stress relaxation with
decreasing temperature P at constant 10.0
NPT Simulations of Binary Mixture AM BB, JCP
(2002)
ln Cs(t)
ln(t)
strong
Angel Plot
6
Anomalous observations on dynamics

• Structural relaxation (diffusion) decouples from
  mechanical relaxation(viscosity)- fragility of a
  liquid
• C. A. Angell, Science 1995

Orientational relaxation remains coupled to
viscosity Ediger, JCP 1996
Translational diffusion is decoupled from
orientational diffusion Sillescu, JCP 1996,
Ediger
7
Theories of Slow Relaxation

• Mode Coupling Theory
• Adam-Gibbs-DiMarzio Entropy Crisis Theory ---
  Concept of Cooperatively Rearranging Region
  (CRR).
• Energy Landscape Picture
• Random first order theory (RFOT)

8
How do molecules move in normal liquid?
Small displacements via structural relaxation
and transverse current
For many liquids D ? T/? ? Stokes-Einstein
relation
9
How to describe the continuos diffusion?

• Conventional descriptions
• (A) Kinetic theory extended by Enskog
• (B) Navier-Stokes Hydrodynamics Stick/Slip
  boundary condition

But it fails for small molecules which is due to
its failure to describe molecular length scale
processes
? Extended hydrodynamics and Remormalized
kinetic theory
? Mode Coupling
Theory (MCT)
10
Single Particle diffusion
Note that Ficks Law is phenomenological --- D is
obtained by Green -Kubo relation.
Exponential decay of wave- Number dependent
density fluctuation
11
IS THE INCOHERENT DYNAMIC STRUCTURE FACTOR
MEASURED BY NEUTRON SCATTERING EXPERIMENTS.
Coupling between Single particle and
collective variables missing in
hydrodynamic description.
12
Collective dynamics Variables are the
conserved Quantities density, Momentum and
energy --- but these are coupled To each
other. The simplest description Of coupled
equations is Given by Navier-Stokes Hydrodynamic
equations.
Note that stress-stress tcf gives viscosity
Linearization of Hydrodynamic equations
13
The hydrodynamic matrix -- note the decoupling
between the transverse and the longitudinal
current modes --the determinant (which determines
poles and hence the time constants of
relaxation) Becomes a prodcut of two.
14
The dynamic structure factor is a sum of three
peak Lorentzian known as Rayleigh-Brillouin
spectra which can be measured by Light
scattering experiments. The central peak is due
to density fluctuation and the corresponding hydro
dynamic mode is called the heat mode. The two
branches are due to the sound modes
Heat mode
Sound mode
15
De Gennes narrowing was the first indication
that dynamics at small length scales cannot be
described by conventional hydrodynamics.
16
The microscopic derivation of de- Gennes
narrowing is simple. It uses what is known as
Smoluchowski- Vlasov equation. This is an
equation of motion for singlet density with
a Mean-field force term.
The basic physics is that at intermediate to
large wavenumbers, both momentum and energy
relaxation is very fast. At such Small length
scales, momentum conservation is no longer a
constrain. Instead, local environment controls
density relaxation. Note that number density is
conserved at all length scales.

17
Molecular hydrodynamic Equations of motion
Density functional Free energy
18
The eigen-values Of hydrodynamic Equations
undergo Sharp changes at Molecular length
scales Due to the presence of Intermolecular
correlations.
Heat mode. Note The near zero value

k
2
19
What is MCT?

• It is a self-consistent scheme which gets
• the short time behaviour (nearly) exactly
  correct (for two point correlation functions)
  because the lower order moments and satisfied.
• The long time behaviour is described by a
  correlator which is expanded in terms of the set
  of hydrodynamic eigenfunctions. The resulting
  equations are solved self-consistently.
• Thus, the long time diffusive limit is described
  fairly accurately.

20

• The mode coupling theory expressions can be
  derived in several different ways they all lead
  to the similar (if not the same) expression.
• One of the early elegant applications of the mode
  coupling theory expression for liquids was made
  by Gaskel and Miller who derived an expression
  for the velocity time correlation function of a
  tagged particle. They argued that a particle
  moves by coupling to the current mode. So, they
  projected the propagator on

21

• The resulting expression involved wave vector
  integration over the transverse and longitudinal
  current correlation functions.
• The longitudinal function decreases much faster
  than the transverse time correlation function.
  The latter decay as

When you combine all the factors you recover
Stokes-Einstein Relation, with 4?. This
calculation constitutes the first concrete
demonstration of MCT.
However, this early success of MCT was based on
assumptions which turn out to be untenable.
22

• The reason is that in dense liquids (and
  certainly in the supercooled state) it is the
  density relaxation of the surrounding solvent
  that is the slowest relaxation mode. The dramatic
  slowing down of density relaxation at wave
  numbers comparable to

This is of course de Gennes narrowing. In
principle, all the slow modes should be included.
23
Power laws in the orientational relaxation near
Isotropic-Nematic phase-transition (INPT)
24
The molecular dynamics simulation is run on a
system of 576 Gay-Berne ellipsoids in a
Micro-Canonical ensemble. The simulations were
run at temperatures T 1.0, 1.1, 1.2.
The variation of order parameter at different
temperatures along the density axis is shown here.
Phase diagram of Gay-Berne ellipsoids with aspect
ratio 3.
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New Experimental results (Fayer et al.2002)
optical Kerr effect data displaying the time
dependence of orientational dynamics of the
liquid crystal, 5-OCB at 347 K on a log plot.
M. Fayer (1996-2004)
Temperature dependent 5-OCB data sets displayed
on a log plot.
27
Time Scales involved

• Initial exponential decay occurs with a time
  constant in 1-5 ps range.
• The long time Landau-de Gennes exponential decay
  sets in after 100 ns or so, with a time constant
  few hundred ns.
• There is a big window between 10 ps to
• few hundred ns when decay is very slow.

28
Temperature dependent 3-CHBT data sets displayed
on a log plot.
Exponent 2/3
The short time portion of the 5-OCB data at 347 K
with the exponential contributions removed on a
log plot.
29
Comparison with relaxation in glassy liquids
The temperature dependence of the time
derivative of collective orientational time
correlation function is shown here (Cang et al.
J. Chem. Phys., 118, 9303 (2003)).
30
Mode coupling theory of orientational relaxation
near INPT
Origin of the slow down in relaxation can be
understood from a mean-field theory which gives
the following expression for ?LdG
where S20(k) is the wave number dependent
orientational structure factor in the
intermolecular frame. DR is the rotational
diffusion coefficient. Kerr experiments measure
the k 0 limit of the time derivative of the
collective orientational correlation function
C2m(k,t).
,
31
Zwanzig - Mori Continued fraction
Single Particle Rotation
Generalized Langevin equation
32
Mode coupling theory calculation of rotational
friction

• The baisc idea is that the torque tcf on a tagged
  ellipsoid slows down due to the marked slow down
  in orientational density relaxation.
• Unlike in supercooled liquid, this happens at
  small k.
• Expression for the torque can be obtained from
  the DFT free energy functional.

33

• MCT expression for rotational friction

C20(k) (20,20) component of the direct
correlation function F20(k,t) (20,20) component
of the dynamic structure factor
34
Isotropic-Nematic coexistence
Isotropic
Nematic
35
The molecular dynamics simulation is run on a
system of 576 Gay-Berne ellipsoids in a
Micro-Canonical ensemble. The simulations were
run at temperatures T 1.0, 1.1, 1.2.
The variation of order parameter at different
temperatures along the density axis is shown here.
Phase diagram of Gay-Berne ellipsoids with aspect
ratio 3.
36
These coefficients of expansion of angular pair
correlation functions can be calculated from
simulation using the expression
.
.
The coefficients of the spherical harmonic
expansion of pair correlation function tend to
diverge when isotropic nematic phase transitions
approached along the density axis.
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(JCP (2002))
39
Collective orientation
f20(k) 1/S20(k)
40
Thus, the leading term in the expansion varies as
t-1/2. The above analysis is valid only after the
initial short time decay, very close to the INPT.
Fayer et al JCP (2002,2003)
The time derivative of the theoretical
correlation function, C20(t). Also shown is a t
-0.63 power law (5-OCB). At short time,
the derivative of the theoretical correlation
function decays essentially as a power law.
41
Collective orientational correlation function
Slow down in the relaxation of collective
orientational time correlation function. The
regions where power law relaxation is dominant
are fitted to the function
at density3.1
42
The Log-log plot of derivative of the collective
orientational correlation clearly shows the power
law relaxation.
Experimental data in the power law region Top 4
curves are of liquid crystals and bottom 5 are of
supercooled liquids.
Data shown on left is for liquid crystal and
right is for supercooled liquid (Cang et al. J.
Chem. Phys., 118, 9303 (2003)).
43

• However, one should add that MCT is
  quantitatively accurate at normal liquids far
  far superior to the Brownian oscillator model
  (BSO). (ADDR, JLS, Rabani,Egorof .
• Please note that BSO has no diffusive behavior
  in the proper sense.

44
Basic MCT equations for friction
45
B. Bagchi and S. Bhattacharyya Adv Chem Phys,
116, 67 (2001)
46
Relationship with Stokes-Einstein

• It is important to realize that the
  Stokes-Einstein expression follows strictly from
  the current term, first derived by Gaskell
  Miller.

47
Self-consistent scheme

• Fs(k,t)exp(- q2lt?r2(t)gt/6)

lt?r2(t)gt 2 ?d? Cv(?)(t- ?)
Cv(z) kB T / m(z ?(z))
48
Power law and mass dependence of diffusion
D1 and D2 are self-diffusion coefficients of
solvent and solute with masses m and M
Straight line is the fitting
The slope of the line 0.099. MD simulation
studies predict the slope to be 0.1
S.Bhattacharyya and B. Bagchi, PRE, 61, 3850
(2000)
49
Microscopic analysis of Stokes-Einstein relation
Bhattacharyya and Bagchi, JCP(2001)
50
Projection Operator Formalism
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Basic ingredients of mode coupling theory

• Derivation of the Relaxation Equations for a set
  of dynamical variable A by using projection
  operator technique
• The resulting equation of motion for the
  correlation function is essentially a GLE

This equation is commonly written in the
frequency plane In the following form
55

• where M(q,z) is the memory function which is
  defined in terms of force-force time correlation
  function. The force F(q,t) must be orthogonal to
  the dynamical variable.
• In the next step, one usually writes down a
  continued fraction representation (equation) for
  M(q,z) because one often knows the t0 value of
  M(q,t0).

56

• The calculation of the memory function for M(q,z)
  is often done by the mode coupling theory. The
  modes are usually the binary product of slow
  variables which are chosen to be orthogonal to
  the dynamical variable. In the language of
  projection operator technique, one projects the
  force term on the sub-space of binary product of
  slow variables.
• In the theory of glass transition, the primary
  dynamical variable in question is the wavevector
  dependent density. M(q,z) is then the
  longitudinal current tcf and the memory function
  of longitudinal current is longitudinal viscosity
  (vide hydrodynamics).

57
Start with the continued fraction representation
(1)
EXACT EXPRESSION
(2)
Replace the expression of the memory function
(3a)
58
Rewriting the longitudinal viscosity as
(3b)
In zeroth order, the longitudinal viscosity is
Use of this approximation in the continued
fraction gives two poles in the DSF
Use Of this DSF gives, to the first order,
59
The zero frequency value of the longitudinal
viscosity is
This value is greater than the zeroth order value
by
Each iteration increases the value of the
viscosity, leading to a divergence, leading to an
elastic peak in the. Dynamic structure factor.
This is the famous feed-back mechanism(Geszti,
1983). Therefore, we make the ansatz
(4)
Use of this expression gives the following
expression for the longitudinal viscosity
(5)
60
With the following value of the vibrational
contributing part of the of the viscosity
Use of the above expression in Eq.3a gives, after
comparison With Eq.4 gives the following
expression for strength f
(6)
The remaining part of the dynamic structure
factor is given by
(7)
61
Where
Eq.6 shows that the ansatz gives acceptable
solution Above the crtitical value ?1 . This
means the dynamic structure factor does not
decay to zero for ?gt1. Instead, they decay only
to f. Thus, the spectrum of density
fluctuation Will show a delta function peak at
??0. The value of f is wave-number dependent and
is called the order parameter of
the Transition. The transition is purely dynamic
in nature.
62
The vibrational part of the longitudinal
viscosity follows the following relation
where
Therefore, theory of dynamics in the glass phase
can be formulated fully in terms of the
vibrational component Of the dynamic structure
factor.
63
The Critical region Power Law Decay
The small frequency behaviour near the critical
point can be studied analytically. At small
frequencies, z/Dv(z) ltlt1. At the glass
transition point, one gets
At small z, Dv(z) exhibits a square root
singularity which means a power law decay for
the dynamics structure factor,
64

• is a solution of ??(1-2?)2?2 (1-?). This gives
  a value of ?0.395. That is, the square root
  dependence gets modified by a small extent by the
  frequency dependence of ?(z).
• The above equations imply a power law decay of
  the dynamic structure factor

This power law decay describes the decay of the
vibrational part to the plateau.
65
As the dynamic structure factor develops an
elastic peak, The longitudinal viscosity diverges
because the longitudinal Viscosity is time
integral of the square of DSF. The detailed
analysis shows that the longitudinal
viscosity Diverges as
Detailed numerical solutions have put the value
of the Exponent between 1.8 and 2.
66
Predictions of MCT

• Divergence of viscosity as a power law with
  exponent 2. This seems to fit the experimental
  and simulation data well at temperatures
  significantly above Tg.
• Theory predicts non-exponential relaxation
  functions. The theory also predicts power law
  decays at the end of beat relaxation and at the
  beginning of alpha relaxations which seem to be
  in good accord with experiments and simulations.
• The predicted critical point (Tc) is
  substantially above the glass transition
  temperature (by 30-50 K).

67
(N P T) simulations in Binary Mixture
LJGB
L-J
Isolated Ellipsoid in a sea of spheres

• Mukherjee, S. Bhattacharyya, B. Bagchi, JCP
  (2002)
• S. Bhattacharyya, B. Bagchi, Phys. Rev. Lett.
  (2002)

68
Stretched-exponential stress relaxation with
decreasing temperature at constant P10.0
Dashed line is fitted to the equation below
Cs(t) Aexp(-t/?1) Bexp-(t/ ?2)?
Normalized stress correlation ln Cs(t) in binary
mixture
ln(t)
69
Stretched exponential relaxation with increasing
pressure at constant temperature T1.0
Dashed line is fitted to the equation below
Cs(t) Aexp(-t/?1) Bexp-(t/ ?2)?
Normalized stress autocorrelation function
ln Cs(t)
ln(t)
70
Non-Arrhenius increase in viscosity (?) with
temperature indicates that the studied model
liquid is weakly fragile
ln(?)
1/T
71
Pressure dependence of Viscosityat constant
temperature(T1.0)
Pressure dependence found to be exponential ---
though there is a distinct change in slope ----
signature of Landscape dominance?
Evidence of a dynamic transition?
ln(?)
Pressure(P)
72
Pressure dependence of viscosity in another
constant temperature (T0.5).
Nature of pressure dependence remains the same
as that of a higher T.
ln(?)
Pressure(P)
73
Computer Simulation Studies of Composition
Fluctuation in Binary Mixture

• We need to simulate a binary mixture where the
  two components interact strongly among
  themselves. Also, the interaction between the
  second (B) component should not be too strong.
• Thus, AB interaction should be stronger than BB
  interaction, while AA intercation can be
  comparable to AB interaction, preferably a bit
  weaker to avoid phase separation.

74
Local composition fluctuations in strongly
nonideal binary mixtures
Spontaneous local fluctuations ? rich and complex
behavior in many-body system
What is the probability of finding exactly n
particle centers within ?V(R) ?
R
?V(R)
In one component liquid ? local density
fluctuations are Gaussian
Binary mixtures that are highly nonideal, play an
important role in industry
75
N P T simulations of Nonideal Binary
Mixtures Study of Composition Fluctuations
Two models binary mixtures
Kob-Andersen model (glass-forming mixture) Equal
size model
xA 0.8 xB 0.2
mA mB m
76
Dynamical Correlations in Composition Fluctuation
Kob-Andersen Model
Stretched exponential fit
R 2.0?AA P 2.0
Slow Dynamics
R 2.0?AA P 4.0
Non-exponential decay Distribution of relaxation
times
77
A. Correlated Orientational and Translational
Hopping
Translational motion ?
Orientational motion ?
Orientational Correlation function at different
time intervals ?
P 10.0
78
Correlated Orientational and Translational Hopping
Translational motion ?
Orientational motion ?
Orientational Correlation function at different
time intervals ?
P 10.0
79
The Single Particle Potential Energy during the
time of Hopping.
Ellipsoid ?
Neighbour 1 ?
Neighbour 2 ?
80
B. Heterogeneous Rotational Dynamics of
the Ellipsoids
The orientational correlation function of the 4
ellipsoids in two different runs
? System in two different runs are in two glassy
minima
Bhattacharyya and Bagchi, Phys. Rev. Lett. (2002)
81
Decoupling between translation and rotational
motion
?R ? Orientational relaxation time
DT ? Translational diffusion
Normal liquids ? ?R? 10 -11 sec and DT ?10 10
sec
P 2.0 ? D ?R 0.144
P 5.0 ? D ?R 0.23
P 6.0 ? DT ?R gt 1
At higher pressures ? D ?R gtgt 1
82
Decoupling of diffusion of smaller particles from
solvent viscosity Test of ode coupling theory
DSE kBT/4??r
?r 0.5 long dashed line ?r 0.75 short dashed
line ?r 1.0 is solid line
?2 /?1 ?r
B. Bagchi and S. Bhattacharyya AC P, 116, 67
(2001)
D/DSE
MCT predicts weak decoupling -- at variance with
expts.
?
83
What went wrong with MCT?

• Relaxation functions are decomposed as
• G(r,r,t) Ga (r,r,t) G? (r,r,t)
• MCT includes only two point correlations.
• Thus decay due to many body fluctuations is
  neglected!!
• However, decay can indeed happen due to many
  body fluctuations!! In fact, in highly viscous
  liquids, these many body fluctuations are the
  important ones.

84
Idea of Entropy Crisis Adam-Gibbs picture
A cooperatively rearranging region (CRR) is
defined as the region which contains at least two
distinct configurational states.
As the entropy of the systems decreases, this z
must grow in size because by definition, z
contains at least two states. It is shown that
z 1/Sc, where Sc is the configurational entropy
of the system. Relaxation time scales as
exp(z), so as entropy of the system goes to zero
(as in Gibbs-DiMarzio theory), the relaxation
time diverges.
85
Difficulties with Adam-Gibbs Scenario

• Experimental and computer simulations have failed
  to find to any diverging length scale. The
  heterogeneous domains are found to be typically
  2-3 nm long.
• No prediction on non-exponentiality.
• The failure may be due to neglect of interactions
  among CRR. It is a mean-field theory description.

86
Random First Order Theory (RFOT) (Wolynes et al.)

• This theory considers relaxation and fluctuation
  to occur via formation of an entropy droplet. The
  nucleation free energy is obtained by standard
  method except that here surface tension is a
  function of the size of the droplet (nucleus) and
  decreases as 1/vr , where r is the radius of the
  droplet.
• An interesting prediction of the theory is that
  the activation free energy is inversely
  proportional to the configuration entropy

87
A Mescoscopic Model of Glassy Dynamics
Motivation

• Jump motion is the dominant mode of diffusion
  near Tg.
• Hopping is a highly collective phenomenon with
  strong nearest neighbor correlations.
• Growth of spatially heterogeneous domains that
  span 2-3 nm near Tg.

88
Correlated Orientational and Translational
Hopping in a Ring Like Tunnel
Translational motion ?
Orientational motion ?
Orientational correlation at different time
intervals ?
89
Mesoscopic Model of glassy dynamics

• An a process is promoted by coherent excitation
  of a certain minimum number of ß processes within
  a cooperatively rearranging region (CRR).
• A ß process is envisaged as a transition in a
  two-level system (TLS).
• As the temperature is lowered towards the glass
  transition temperature, the number of ? processes
  which need to be excited to attain a relaxation
  increases.

90
The model

• Each of the two wells, labeled 1 and 2, comprises
  a collection of Ni (i 1, 2) identical,
  non-interacting TLSs of such kind.
• For a collection of Ni TLSs, a variable ?ji(t) (j
  1, 2 , , Ni) is defined, which takes on a
  value 0 if at the given instant of time t the
  level 0 is occupied and 1 if otherwise.
• We define

• An a process occurs only when
• all the ß processes in a well are
• simultaneously excited, i.e.,
• Qi Ni.

91
The model

• A CRR is characterized by an Nß number of
  identical non-interacting TLSs.
• A collective variable Q(t) is defined
• ?j(t) 0 or 1 is the occupation variable
  (j 1, 2 , , Nß).
• The waiting time before a transition can occur
  from the level i ( 0, 1) is drawn from a
  Poissonian probability density function
• where ti is the average time of stay in the
  level i.

92
The model

• Detailed balance gives
• where e is the separation between the two
  levels.
• From the TST results
• The rate of a relaxation depends crucially on
  Q. For simplicity, we assume that the relaxation
  occurs with unit probability at the instant Q
  reaches for the first time Nc, an integer greater
  than the most probable value of Q.

93
Theoretical Treatment

• Q(t) is a stochastic variable in the discrete
  integer space
• 0, N? (single flip assumption)
• The mean first passage time t(n, T), which is the
  mean time elapsed before Q starting from its
  initial value n reaches Nc for the first time at
  temperature T, is given by
• where F is the hypergeometric function.

94
Theoretical Treatment

• We assume that an a process corresponds to a
  large scale change in configuration within a CRR.
• The relevant time correlation function is the
  survival probability that a CRR refrains from
  undergoing an a process
• The average relaxation time tCRR(T) of a CRR,
  characterized by a set of values for Nß, Nc, and
  e, is

95
The Model (Continued)

• In a heterogeneous environment within a bulk
  sample, a fluid-like region can be envisaged as
  having at any instant of time, on the average, a
  larger number of ß processes in the excited state
  than what is in the solid-like region.
• A Gaussian distribution of e among CRRs gt
  Heterogeneous domains.
• The model assumes Nc to grow as the reduced
  inverse temperature Tg/T increases until Nc
  reaches Nß at Tg.

96
Results

• The present calculation takes the following set
  of values for the relevant parameters
• lt e gt kBTm
• se 0.05 kBTm
• Tg (2 / 3) Tm
• e1 4kBTm
• Time is scaled by t1 at Tg.

The long time behavior of S(t) fits well to the
stretched exponential function. Inset Monotonic
decrease of the Stretching exponent ßKWW as Tg
is approached from above.
97
Results

• ? Nc / Nß
• O gt ? varies linearly from 0.6 to 1.0 as Tg / T
  increases from 0.68 to 1.0 with Nß held fixed at
  20.
• gt ? grows linearly with Tg / T, both rising
  from 0.7 to 1.0, with Nc and Nß varied together,
  the latter from 10 to 20.
• T0 0.689 Tg (solid line)
• T0 0.789 Tg (dashed line)

Arrhenious plot showing the scaled
characteristic relaxation time as a function of
the reduced inverse temperature.
98
Anomalous Temperature Dependence of Heat Capacity
During a Cooling-Heating Cycle in Glassy systems
Motivation

• A sharp rise in the measured heat capacity during
  heating
• A shift to higher values of the limiting fictive
  temperature TfL obtained upon cooling and the
  glass transition temperature Tg for faster rates.
• For qc qh q,
• An identical dependence of TfL on qc.

99
The Model

• The model, based on the framework of ß organized
  a process, envisages a ß process as a transition
  in a two-level system (TLS).
• An a process is conceived as a cooperative
  transition from one well to another in a
  double-well, subject to the establishment of a
  certain condition.

100
Theoretical Treatment

• Qi is a stochastic variable in the discrete
  integer space
• 0, Ni.
• where the and - signs in the indices
  of the Kronecker delta are for i 1 and 2,
  respectively.
• The total energy of the system at time t is

101
Theoretical Treatment

• The system, when subjected to cooling or heating
  at a constant rate, can be envisaged to undergo a
  series of instantaneous temperature changes, each
  in discrete step of ?T, in the limit ?T ? 0, at
  time intervals of length ?t, whence q ?T / ?t.
• We calculate the heat capacity from the following
  equation
• Here tobs ?t.

102
Results

• N1 6
• N2 10
• ?T 0.0015 in reduced units of kBT / e
• e1 8 e
• Time is expressed in reduced units, being scaled
  by t1(Th)
• k-1 0.50 in reduced time units
• For e / kB 600 K, the cooling and heating rates
  explored here range from 0.0085 to 0.35 K s-1.

q 7.5 x 10-5 in reduced units
103
Results
q 3.0 x 10-4, 7.5 x 10-5, 2.0 x 10-5, 7.5 x
10-6 in reduced units from top to bottom.
The fictive temperature is calculated in terms of
energy.

• q 3 X 10-4 (solid line), 7.5 x 10-5 (dashed
  line), both in reduced units.
• Inset q 7.5 x 10-5
• The fictive temperature evolves in an
  identical fashion as the energy.

• Activated dynamics for intra-well transition
• Trapped into non-equilibrium glassy state
• Delayed energy relaxation on subsequent
• heating

104
Mode Coupling Theory of frequency dependent
specific heat

• Total energy-energy time correlation function can
  be expressed by using the mode coupling theory in
  terms of dynamic structure factor.
• C(t) is given in terms of wave number (q)
  integral over F(q,t)/S(q)2, where F(q,t) is the
  intermediate scattering function and S(q) the
  structure factor of the liquid.
• Thus, MCT predicts a large vibrational peak,
  well-separated from the alpha peak, in the
  frequency dependence of the specific heat.

105
Conclusions
Even a modest growth in the number of coupled
beta transition is sufficient to provide a severe
entropic bottle-neck to make relaxation slow.
The fragility of the system is related to the
growth of beta states. But this growth is
limited. No divergence of length scale is
involved.
Non-exponentiality enters naturally into the
dynamics.
106
Acknowledgement

• Sarika Bhattacharyya
• Arnab Mukherjee
• Rajesh Murarka
• Dwaipayan Chakrabarti

107
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108
VFT fitting signifying the super-Arrhenius nature
of the temperature dependence of
Viscosity (?)
? ? exp -
C/(T-T0)
ln(?)
1/(T-T0)
109
MCT prediction for temperature dependence of
viscosity(?)? ? (T - Tc)-?

• ln(?)

ln(T-Tc)
110
Correlated Orientational and Translational
Hopping in a Ring Like Tunnel
Translational motion ?
Orientational motion ?
Orientational correlation at different time
intervals ?
111
The Single Particle Potential Energy during the
time of Hopping.
Ellipsoid ?
Neighbor 1 ?
Neighbor 2 ?
112
Heterogeneous rotational dynamics and decoupling
of orientational and translational motion are
also observed.
113
Results
? Nc / Nß
e kBT, Nß 10, Nc 6, 7, 8
Single exponential dependence of the average
relaxation time within a CRR on Nß at fixed
?. Inset slope m ??, ? 3.4
Single exponential decay of SCRR
114
Conclusion
One of the possible scenario of decoupling
between rotation and translational diffusion is
that molecule can jump retaining their
orientation
Decoupling between diffusion and viscosity might
occur due to the presence of the hopping mode.
Hopping found to persist even when stress
autocorrelation function ceases to decay
Translational motion of the ellipsoid in ring
like tunnels is observed. Both orientational and
translational hopping are gated process where the
free energy barrier is entropic in nature.
115
Anomalous observations in Supercooled Liquids
Statics

• The overshoot of the heat capacity
• during heating is taken to be a
• signature of a glass to liquid
• transition.

• Is there any underlying
• thermodynamic phase
• transition ?

116
Potential Energy landscape Picture

• Goldstein originally proposed that below certain
  temperature, activated processes become important
  in the dynamics of glassy liquids. Inherent
  structure concept of Stillinger and Weber
  substantiated this view.
• Angell has proposed a powerful classification of
  glass forming liquids in terms of strong and
  fragile liquids which can be explained in terms
  of topology of the potential energy surface
  (density of local minima, barrier heights)

117
Potential Energy landscape Picture(continued)

• Difficulty is that the role of spatial
  correlations is left unclear. Strong liquids are
  expected to have significant static pair and
  higher order correlations even in less viscous
  liquids.
• Lack of correlation of fragility with
  molecular/microscopic properties.
• Correlation with free energy landscape has been
  proposed.

118
Probability Distributions of Composition
Fluctuation
Kob-Andersen Model
R 2.0?AA
T 1.0 P 2.0
?NA? 27.3 ?A 1.995
Gaussian distribution
? NB ? 6.74 ?B 1.995
Both A and B fluctuations are large
System is indeed locally heterogeneous
119
Joint Probability Distribution Function
Kob-Andersen Model
R 2.0?AA
Nearly Gaussian
Corr?NA , ?NB - 0.203 ? Fluctuations in A and
B are anticorrelated