Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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A9.2 Evaluation of integrals 481
where C
O
(ω
0
, t) is defined as
C
O
(ω
0
, t) = Re
t
0
dt
C
O
(t, t
)e
iω(t−t
)
=
1
2
[
C
O
(ω) +C
O
(−ω)
]
. (A9.1.14)
A9.2 Evaluation of integrals
We present here the evaluation of the various integrals used in the analysis of the asymp-
totic behavior of correlation and response functions in the FDT and aging regimes.
A9.2.1 Integral I
R
for the FDT solution
We compute here integral I
R
(say) in the last term on the RHS of (9.3.44). The integration
over t
extends from s to t, while that over s varies from t
to t (t > t
). Therefore both the
correlation and the response functions appearing in the integrand satisfy the FDT relation
(9.3.23) and the FDT relation (9.3.24) between and applies. Furthermore, in I
R
we
change the lower limit of the integral with respect to t
from s to t
. The extra contribution
arising from this change is a definite integral with t
lying between t
and s. This is equal to
zero since the integrand contains the response function R(t
, s), which vanishes for t
< s
due to causality. Hence the integral I
R
now reduces to the form
I
R
=
t
t
ds
t
t
dt
(t, t
)R(t
, s) =
t
t
dt
(t, t
)
t
t
ds R(t
, s)
=
t
t
dt
(t, t
)
t
t
ds R(t
, s). (A9.2.1)
Once again we have changed the upper limit of s from t to t
in the last integral since the
response function R(t
, s) is zero for s > t
. The expression for I
R
is further simplified
with a partial integration as follows:
I
R
=
t
t
dt
(t, t
)M(t
, t
) =
t
t
dt
∂
∂t
(t − t
)[1 − C(t
− t
)]
=[1 − C(t
− t
)](t − t
)
*
*
*
t
t
+
t
t
dt
(t − t
)
∂
∂t
C(t
− t
). (A9.2.2)
After some trivial rearrangements I
R
is obtained as
I
R
=
τ
0
dτ
(τ −τ
)
∂
∂τ
C(τ ) +[1 − C(τ )]
∞
p=2
a
p
, (A9.2.3)
where τ = t − t
.
482 Appendix to Chapter 9
A9.2.2 Integrals for the aging solution
Here we compute the following integrals appearing on the RHS of eqn. (9.3.49) for treating
eqn. (9.3.14) in the aging limit:
I
1
=
t
−δ
0
ds
A
(t, s)R
A
(t
, s), (A9.2.4)
I
2
=
t
t
−δ
ds
A
(t, s)R
F
(t
, s), (A9.2.5)
I
3
=
t−δ
0
ds
A
(t, s)C(s, t
), (A9.2.6)
I
4
=
t
t−δ
ds
F
(t, s)C
A
(s, t
). (A9.2.7)
In the aging regime the times t and t
are such that the ratio λ = t
/t < 1. Both t and
t
→∞, with (t − t
)/t
of O(1). The time scales δ and δ
are chosen such that they are
much smaller than t and t
, respectively. First we evaluate the integral I
1
with the use of
the scaling solution (9.3.29) in the aging regime, i.e.,
C
A
(t, t
) = qφ
h(t
)
h(t)
, R
A
(t, t
) =
h
(t
)
h(t)
R
h(t
)
h(t)
, (A9.2.8)
.
where h(t) is a monotonically ascending function of time and q is the nonergodicity
parameter. I
1
is obtained as
I
1
=
t
−δ
0
dt
∞
p=2
a
p
q
p−1
φ
p−1
(λλ
)
dλ
dt
R(λ
)
=
∞
p=2
a
p
q
p−1
1
0
dλ
φ(λλ
)R(λ
), (A9.2.9)
where we have defined λ
= h(t
)/ h(t
) and λ = h(t
)/ h(t). Since t
is very large com-
pared with δ
, the upper limit of the integral has been replaced as h(t
− δ
)/ h(t
) → 1.
The integral I
2
is evaluated as
I
2
=
t
t
−δ
ds
A
(t, s)R
F
(t
, s) ≈
A
(t, t
)
t
t
−δ
ds R
F
(t
, s)
=
∞
p=2
a
p
q
p−1
φ
p−1
(λ)
t
t
−δ
ds
∂C
F
(t
, s)
∂s
. (A9.2.10)
A9.2 Evaluation of integrals 483
For the range of integration of s from (t
− δ
) to t
we have treated (t, s) as a constant
(t, t
),
I
2
=
∞
p=2
a
p
q
p−1
φ
p−1
(λ)
t
t
−δ
ds
∂
∂s
C
F
(t
− s),
=
∞
p=2
a
p
q
p−1
φ
p−1
(λ)[C
F
(0) − C
F
(δ)]
=
∞
p=2
a
p
q
p−1
φ
p−1
(λ)(1 − q). (A9.2.11)
Next we calculate the integral I
3
by computing I
31
, I
32
, and I
33
. The integral I
31
is obtained
from (9.3.51) as
I
31
=
t
−δ
0
dt
A
(t, t
)C
A
(t
, t
)
=
∞
p=2
( p − 1)a
p
t
−δ
0
dt
C
p−2
A
(t, t
)R(t, t
)C
A
(t
, t
)
=
∞
p=2
( p − 1)a
p
q
p−2
h(t
−δ)
h(t)
0
dt
φ
p−2
(λ
)
d(λ
)
dt
R(λ
)qφ(λ
/λ)
=
∞
p=2
( p − 1)a
p
q
p−1
λ
0
dλ
φ
p−2
(λ
)R(λ
)φ(λ
/λ), (A9.2.12)
where we have defined λ
= h(t
)/ h(t) and ignored terms of O(δ) in the limit δ t
,
such that h(t
− δ)/h(t) →λ. The integrals I
31
and I
33
are very similar, differing only in
the limits. So the latter is evaluated in exactly the same way as the former, giving
I
33
=
t−δ
t
+δ
dt
A
(t, t
)C
A
(t
, t
),
=
∞
p=2
( p − 1)a
p
q
p−1
1
λ
dλ
φ
p−2
(λ
)R(λ
)φ(λ/λ
), (A9.2.13)
where we have used the fact that over the range of the integration t
> t
. On adding eqns.
(A9.2.12) and (A9.2.13) we obtain
I
31
+ I
33
=
∞
p=2
( p − 1)a
p
q
p−1
1
0
dλ
φ
p−2
(λ
)R(λ
)φ((λ
/λ)
sg(λ−λ
)
). (A9.2.14)
484 Appendix to Chapter 9
Next we compute the integral I
32
,
I
32
=
t
+δ
t
−δ
ds
A
(t, s)C
F
(s, t
)
=
A
(t, t
)
t
+δ
t
−δ
ds C
F
(s −t
). (A9.2.15)
Since t − t
∼ O(t
), and t and t
are very large, we take
A
(t, s) ≈
A
(t, t
) over the
range of the integral. The contribution from integral I
32
is therefore calculated as
I
32
=
∞
p=2
( p − 1)a
p
q
p−1
φ
p−1
(λ)R(λ)
dλ
dt
(δ
) ≈ 0 (A9.2.16)
to leading orders. Now we consider the final integral I
4
:
I
4
=
t
t−δ
ds
F
(t, s)C
A
(s, t
) ≈ C
A
(t, t
)
t
t−δ
ds
∂
∂s
F
(t, s)
= qφ(λ)
∞
p=2
a
p
C
p−1
(t, s)
*
*
*
t
t−δ
=
∞
p=2
a
p
q
1 − q
p−1
φ(λ). (A9.2.17)
Next we compute the integrals appearing on the RHS of eqn. (9.3.55), which follows
from eqn. (9.3.15) in the aging regime. For J
1
we obtain
J
1
=
t
+δ
t
ds
A
(t, s)R
F
(s, t
). (A9.2.18)
Over the range t
to t
+ δ
for variation of s, we approximate
A
(t, s) by
A
(t, t
).For
the response function in the FDT regime, since s > t
we use the corresponding relation
R
F
(s, t
) =−∂
s
C
F
(s −t
) to write the integral J
1
as
J
1
≈−
A
(t, t
)
t
+δ
t
ds
∂
∂s
C
F
(s −t
)
=−
A
(t, t
)[C(δ
)) − C(0)]
= R(λ)
dλ
dt
∞
p=2
a
p
( p − 1)[1 −q]q
p−2
φ
p−2
(λ), (A9.2.19)
A9.2 Evaluation of integrals 485
where λ = h(t
)/ h(t).ForJ
2
we obtain
J
2
=
t−δ
t
+δ
dt
A
(t, t
)R
A
(t
, t
)
=
∞
p=2
a
p
( p − 1)q
p−2
1
h(t
)
h(t)
dt
φ
p−2
(λ
)
dλ
dt
R(λ
)R(λ/λ
)
1
λ
dλ
dt
=
dλ
dt
∞
p=2
a
p
( p − 1)q
p−2
1
λ
dλ
λ
φ
p−2
(λ
)R(λ
)R(λ/λ
). (A9.2.20)
For J
3
we obtain
J
3
=
t
t−δ
ds
F
(t, s)R
A
(s, t
)
≈ R
A
(t, t
)
t
t−δ
ds
∞
p=2
a
p
( p − 1)C
p−2
F
(t − s)R
F
(t − s)
= R(λ)
dλ
dt
∞
p=2
a
p
t
t−δ
ds
∂
∂s
C
p−1
F
(t − s)
= R(λ)
dλ
dt
∞
p=2
a
p
[1 − q
p−1
]. (A9.2.21)
10
The thermodynamic transition scenario
In Chapter 4 we introduced the Kauzmann temperature T
K
as a possible limiting temper-
ature for the existence of the supercooled liquid phase. The original hypothesis due to
Kauzmann proposes eventual crystallization in the supercooled liquid at very low tem-
peratures as a possible way out of the paradoxical situation in which the entropy of the
disordered state becomes less than that of the crystal. Another possible explanation of the
Kauzmann paradox could be that the simple extrapolation of the high-temperature result
to very low temperature is not correct and the entropy difference between supercooled liq-
uid and crystal remains finite down to very low temperature (Donev et al., 2006; Langer,
2006a, 2006b, 2007), finally going to zero only near T =0. Either of these resolutions,
however, leaves us with no understanding of the dramatic slowing down and associated
phenomenology of the supercooled region above T
g
. The difference of the entropy of the
supercooled liquid from that of the solid having only vibrational motion around a frozen
structure represents the entropy due to large-scale motion and is identified with the config-
urational entropy S
c
of the liquid. The rapid disappearance of the configurational entropy
of the disordered liquid or the so-called “entropy crisis” poses an important question that
is essential for our understanding of the physics of the glass-transition phenomena and the
divergence of the relaxation time at T
g
. Apart from having a characteristic large viscosity,
the supercooled liquid shows a discontinuity in specific heat c
p
at T
g
due to freezing of
the translational degrees of freedom in the liquid. The features described above are almost
universally observed in all liquids.
10.1 The entropy crisis
We will now explore some ideas proposed in recent times to understand the entropy-crisis
characteristics of supercooled liquids. We discuss the physics of the deeply supercooled
state on the basis of a plausible scenario for an underlying phase transition. Our focus is
primarily on structural glasses interacting through short-range potentials. The ENE tran-
sition of the MCT at temperature T
c
discussed in the earlier chapters marks a crossover
in the dynamics of the supercooled liquid in a temperature range higher than T
g
. Beyond
T
c
, with increasing supercooling the local motion of the molecules comes mainly to rep-
resent vibrations around frozen configurations and it is widely separated from large-scale
486
10.1 The entropy crisis 487
diffusive motion of the particles. In thermodynamic terminology the system remains trapped
in a local minimum of the free energy. The entropy related to the large-scale motion is the
configurational entropy (which is also called complexity in spin-glass terminology) S
c
.Itis
related to the number of free-energy minima N of the N -particle system through the basic
relation of statistical mechanics,
N = exp[NS
c
(T )]. (10.1.1)
10.1.1 The Adam–Gibbs theory
In 1956 J. Gibbs proposed a resolution of the Kauzmann paradox with the hypothesis
that the vanishing of the configurational entropy signifies (Gibbs, 1956) a second-order
phase transition at T =T
K
. The configurational entropy comes down sharply to zero at
T
K
and remains zero below T
K
. The entropy vs. temperature curve thus has a kink at this
transition temperature. Let us consider the supercooled state in which crystallization is
avoided until close to T
K
. Indeed, there are amorphous states in which crystallization is
very unlikely. Even before reaching T
K
, it is clear that with increasing supercooling the
entropy of the liquid rapidly approaches that of the crystal. The Gibbs–Di Marzio (Gibbs
and Di Marzio, 1958) and subsequently Adam–Gibbs theory (Adam and Gibbs, 1965) is
based on this hypothesis of a phase transition at T
K
. The model estimates the entropy of
the given state using the relation (10.1.1). The Adam–Gibbs theory is based on the concept
of cooperatively rearranging regions (CRRs). The term “cooperatively rearranging” here
implies that by definition this is the smallest size of the system which cannot be further
subdivided, and that each such region can be independently rearranged.
The number of particles n involved in the CRRs changes with temperature. It is assumed
that the number ν
c
of independent configurations that a given CRR can assume is indepen-
dent of its size and of the temperature. This simplifying assumption allows us to estimate
the configurational entropy. Indeed, ν
c
under general conditions should increase with n or
as the temperature rises, but we ignore this dependence in the simplest form of the theory.
With a total number of N particles, the number of CRRs each having n particles is equal
to N/n. Hence the thermodynamic probability of a given state is obtained as
W
N
={ν
c
}
N/n
. (10.1.2)
Therefore the configurational entropy is estimated using the basic relation (10.1.1) con-
necting thermodynamics with statistical mechanics,
S
c
(T ) =
1
N
ln W
N
=
ln ν
c
n(T )
. (10.1.3)
The configurational entropy and the number of particles involved in the CRRs are therefore
inversely related. With the lowering of the temperature the number n of particles moving
in a cooperative manner, i.e., the size of the CRRs, increases and hence S
c
falls. From
this purely thermodynamics-based argument it follows that the vanishing of S
c
is directly
related to a diverging size of the CRR.
488 The thermodynamic transition scenario
The configurational entropy S
c
is related to the jump in the specific heat of the liquid at
the transition. To demonstrate this, we need to extend the formalism of standard equilib-
rium thermodynamics to metastable states. Since S
c
is defined as the difference between
the entropies of the liquid and the crystalline states, denoted by S
LQ
and S
CR
, respectively,
its derivative with respect to temperature is therefore
dS
c
dT
=
1
T
T
dS
LQ
dT
− T
dS
CR
dT
=
c
p
T
. (10.1.4)
c
p
is the difference of specific heats between the supercooled metastable liquid and the
crystal. Upon integrating the above relation we readily obtain
S
c
(T ) − S
c
(T
K
) =
T
T
K
d
¯
T
¯
T
c
p
. (10.1.5)
Assuming c
p
to be weakly dependent on temperature and S
c
(T
K
) =0 by definition, the
above relation reduces to
S
c
(T ) = c
p
ln
T
T
K
. (10.1.6)
For low temperatures T ∼ T
K
the RHS of eqn. (10.1.6) reduces to the linear form
S
c
(T ) = c
p
T − T
K
T
K
. (10.1.7)
It is important to note here that the free energy of the supercooled liquid remains continuous
through the transition at T
K
. The specific heat c
p
shows a jump but there is no latent heat
absorbed. From the thermodynamic point of view, the transition envisaged at T
K
is thus
a continuous second-order transition. However, in this transition we can also identify an
order parameter that shows a finite jump at the transition. The inverse of the radius of
the cage seen by each particle can be treated as an order parameter like this. In the liquid
phase this is zero and jumps discontinuously to a finite value in the glassy phase, which is
contradictory to the general nature of second-order transitions.
10.1.2 Dynamics near T
K
The theory of the deeply supercooled liquid state presented so far primarily deals with
its thermodynamics properties, e.g., the calculation of the configurational entropy of the
metastable state. An important aspect of the glassy behavior that has been left out of these
discussions so far is the slowing down of the dynamics. With increasing supercooling the
relaxation time for the liquid sharply increases. Relaxation in the present context implies
that for a typical fluctuation around the disordered liquid state at a temperature T < T
m
.
The nature of the single-particle dynamics in the deeply supercooled liquid changes over
from liquid-like continuous motions to activated hopping. In a structural glass this occurs
10.1 The entropy crisis 489
even without any external agent coming into play, i.e., the slow dynamics is self-generated.
The growth of the relaxation in the Adam–Gibbs theory is considered in a simple way in
the following.
To understand the development of very long relaxation times in the deeply supercooled
state, we need to introduce the idea that energy barriers build up to resist molecular rear-
rangement in the jammed state. Relaxation under such conditions occurs through thermally
assisted hopping over the barrier (E
B
, say). The probability of such a jump will be con-
trolled by the Boltzmann factor exp[−E
B
/(k
B
T )]. Thus estimation of the relaxation time
is closely linked to that of the energy barrier E
B
which must be overcome so that a local
fluctuation can relax. In a many-particle system with short-range forces, relaxation of a
fluctuation takes place locally. To understand this, we consider the following simple pic-
ture. Every particle is affected by a set of n of its neighbors forming the CRR. Local
relaxation of the fluctuations will involve rearranging n particles cooperatively. Therefore
the barrier E
B
for this process is proportional to n.Usingeqn. (10.1.3) we obtain a simple
relation between E
B
and S
c
,
E
B
=
c
S
c
(T )
, (10.1.8)
where
c
is a constant. Note that the energy barrier E
B
is now temperature-dependent
through that of the configurational entropy S
c
. For temperature-independent barriers E
B
the relaxation time (e
E
B
/(k
B
T )
) will grow with temperature in accord with a simple Arrhe-
nius law as is the case with the so-called strong liquids. However, the fragile glass-forming
liquids show a much sharper change with temperature and the effective barrier height
increases with lowering of the temperature. The present analysis is therefore more applica-
ble to relaxation in fragile systems. The relaxation time τ
α
(T ) corresponding to the barrier
(10.1.8) is therefore obtained using the Arrhenius relation
τ
α
= τ
0
exp
β
c
S
c
(T )
. (10.1.9)
Equation (10.1.9) represents the crucial link between the configurational entropy and the
relaxation time in the deeply supercooled liquid state. For temperatures near the Kauzmann
temperature the relaxation time τ
α
(T ) is obtained as
τ
α
= τ
0
exp
E
VF
T − T
K
, (10.1.10)
where E
VF
=
c
/(k
B
c
p
). This is the Vogel–Fulcher form for the relaxation time with
the corresponding divergence temperature T
0
≡ T
K
.
While the analysis above provides a natural resolution of the Kauzmann paradox and
a basis for the empirical Vogel–Fulcher fitting form, the validity of the whole scenario is
uncertain due to the rather simplified assumptions. The equality of the two temperatures T
0
and T
K
has been tested in a wide number of glassy systems. As we have seen above, due
to the intervention of the calorimetric glass transition at T
g
, i.e., the liquid becoming too
sluggish to equilibrate on laboratory time scales, it is impossible to locate precisely both
490 The thermodynamic transition scenario
of the temperatures T
0
and T
K
. The temperature T
0
is obtained by extrapolating a highly
nonlinear fit of the data well above T
g
to a different temperature regime below T
g
. T
K
is also
inferred through extrapolation of the entropy data below T
g
. The above equality, however,
suggests a link of deeper significance on considering the fact that the physics differs greatly
for the two temperatures T
0
and T
K
. T
0
is the temperature at which the relaxation time for
the supercooled liquid diverges and basically relates to the dynamics. On the other hand,
the Kauzmann temperature T
K
is related to the vanishing of the thermodynamic property
of configurational entropy of the metastable liquid. Hence the equality of T
0
and T
K
,or,
more appropriately, the linking of the sharp increase of relaxation time to the entropy crisis,
actually represents effects of structure on the dynamics.
It is plausible to assume that an increase of the number of particles n in a CRR implies
growth of the size of the latter. This leads to the idea of a growing (static) correlation
length in the supercooled liquid ξ
s
, which is related to the number of particles n in the
cluster: n ∼ ξ
d
s
for a d-dimensional system. Since the barrier E
B
for the relaxation process
also grows with the number of particles n, we obtain that E
B
∼ ξ
d
s
.Fromeqn. (10.1.9) we
therefore obtain
τ
α
= τ
0
exp
B
0
ξ
d
s
k
B
T
, (10.1.11)
where B
0
is a constant. It is important to compare this with the case of divergence of length
and time scales in standard critical phenomena. It is well known that the relaxation time
increases sharply on approaching the second-order phase-transition point and cooperativity
over domains of the size of the correlation length requires that the relaxation time τ
R
∼ ξ
z
s
,
where z is a dynamic exponent. Since the correlation length ξ
s
in this case diverges with a
power law on approaching the critical point, the relaxation time τ
R
also follows a power-
law divergence on approaching T
c
. In a glass-forming liquid, on the other hand, using eqn.
(10.1.11), it follows that growth of the static length scale by only an order of magnitude
produces the required growth in the time scale near T
g
. In the case of structural glasses,
evidence of such a growing static length scale has been hard to find (see below for further
discussion). For example, the magnitude of the structure factor of the liquid or the density
does not acquire any special feature on supercooling (Dyre, 2006).
10.2 First-order transitions
The freezing transition T
m
of the disordered liquid to the crystalline state is a first-order
phase transition with finite latent heat absorbed. The glass transition, however, is not asso-
ciated with any latent-heat absorption. At T
g
, as the system freezes into an amorphous
solid-like structure, there is a drop in the specific heat due to the removal of the transla-
tional degrees of freedom. The free energy of the supercooled liquid, however, does not
show any discontinuous change. At deep supercooling there is a large number of available
metastable structures into which the liquid can freeze and hence a considerable entropic
drive is present for the process. There have been theories for the vitrification process that