Chapter 7
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Entropic Approach to Relaxation Behavior in Glass-Forming Liquids Udayan Mohanty Eugene F. Merkert Chemistry Center, Boston College, Chestnut Hill, MA 02167
Adam-Gibbs picture of a "cooperative rearranging region" in glass-forming liquids is generalized. The extension leads to differential equations which are suggestive of the differential equations emerging in Wilson's theory of critical phenomena. A relation between the size of a "cooperative rearranging region" and the entropy emerges from solving these differential equations. The size of the region does not diverge near the glass-transition temperature. The line of metastable states of the supercooled liquid leads to a "twice unstable" fixed point This elusive fixed point is identified with the Kauzmann temperature. Polymorphism between fragile and strong liquids is accounted for in terms of a "discontinuity" critical "end point". The "discontinuity" fixed point leads to a non-critical phase via a first-order phase transition. The relaxation time of supercooled liquids is expressed in terms of the topography of the potential energy hypersurface in configuration space via a non– equilibrium generalization of the Adam-Gibbs model.
There has been renewed experimental and theoretical activity in trying to unravel and understand glass-forming liquids and the amorphous states of matter (1,2). Theoretical techniques to describe the supercooled and the glassy states are hampered by the fact that these states are far away from equilibrium (2,5). A theoretical framework is proposed that extends the seminal Adam-Gibbs picture (4) of a "cooperative rearranging region". The technique leads to differential © 1997 American Chemical Society
Fourkas et al.; Supercooled Liquids ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
95
96
SUPERCOOLED LIQUIDS
equations that appear in Wilson's renormalization group (5,6) approach to critical phenomena. The polymorphism of the glassy state as well as the Kauzmann temperature is controlled by fixed points of these differential equations. A nonequilibrium generalization of the Adam-Gibbs model is also described in terms of the inherent structures of the supercooled state (2,5,7). The notion of a cross-over temperature T below which motions are governed by "entropie" barriers emerges x
from the formulation. Adam-Gibbs Model. In elucidating the features of supercooled liquids, AdamGibbs (AG) introduced the idea of a "cooperatively rearranging region" or domains (2,4). It is defined to be part of the system, i.e. a "subsystem" that on "fluctuations in enthalpy" is capable of rearranging itself in a cooperative manner unconstrained of its surroundings (2,4).
In this model, the probability per unit time for
"cooperative rearrangements" is given by (2,4)
W( ',T).AC [^). Z
Here, k
(1)
W
is the Boltzmann's constant,
B
is the molar enthalpy, T is the absolute
temperature, A is a constant which is weakly temperature dependent, and z is the "minimum" number of molecules in a domain that leads to "cooperative rearrangements". In the Adam-Gibbs picture, the "minimum" size z is intimately related to the molar configurational entropy S (T) of the undercooled melt (2,4) C
* ,
where N
A
v
s
Na
is the Avogadro's number. The quantity s appearing in Eq. (2) is the
configuration entropy of "minimum" size domains (2,4). These are the smallest domains that allow cooperative relaxation. Since the specific heat difference AC
p
between the equilibrium melt and the glassy state at T is approximately constant, g
the configuration entropy S (T) is obtained via (2,4) C
T S (T)-f±^dT. c
Fourkas et al.; Supercooled Liquids ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
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7. MOHANTY
Relaxation Behavior in Glass-Forming Liquids
97
Here, T is a reference temperature usually identified with the Kauzmann 2
temperature. By definition, the configuration entropy vanishes at the Kauzmann temperature T . 2
A few comments are in order. First, Eq. (2) makes no assumption about the existence or the nature of the Kauzmann temperature. Second, according to Eq. (3), n
configurational entropy varies with temperature as AC
p
^|^"j
(2 4)> Third, the t
change of Gibbs-free energy A G - z Ap (2,4). On combining Eqs. (2) and (3) one observes that AG is singular as T - * T . 2
New Picture of "Cooperative Rearranging Region". A d a m - G i b b s considered an isothermal-isobaric ensemble of "cooperatively rearranging regions". Any given member of the ensemble has the same number of molecules (2,4). The partition function of the supercooled liquid is constructed so that the phase space associated with crystal-like packing configurations has been excluded. The partition function A(7\P,z) of the ensemble is (2,4) A ( z , P , F ) . ^V z)cKil(-fiE-pPV) 9
(4)
9
EM
where /3 - —— and co(£,V,z) is the density of states at energy E and V. Since kT B
the Gibbs-free energy is defined as /3G(z, P, T) - - In A(z, F, P ) , the ratio of the subsystems that allows transitions, to the total number of subsystems in the ensemble, is proportional to exp(/3AG) (2,4). Each term in the sum over E and V in Eq. (4) is analytic in 7 \ P , and z. But, near the temperature T , A G find a "singular" behavior of the partition function 2
(2,4). Various approximations to the partition function then lead to the expression for transition probability
r j for "cooperative rearrangements" as given by
Eq. (1). However, w( -,t) is not an analytic function as T -* T . z
2
T o deduce such "singular" behavior in thermodynamic functions, such as the free-energy, by explicitly evaluating the partition function is not a simple task for several reasons. First, evaluation of the "singular" part of the free-energy requires precision that is hard to accomplish by conventional techniques and approximations (2,5,9). Second, "singular" feature of the partition function can appear only in the so-called thermodynamic limit (2,5,9).
Fourkas et al.; Supercooled Liquids ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
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SUPERCOOLED LIQUIDS
It is suggested that generalization of the Adam-Gibbs picture in terms of differential equations of the renormalization group may provide insights into the low temperature relaxation dynamics of glass-forming liquids (2,9,10). Since it is easier for singularities to arise by solving differential equations, the hope is that these differential equations would lead to non-analytic behavior (if any) in the freeenergy. 3
The total volume, L , of die system is partitioned into cubic "blocks" or "cells", each of length L (9,2). Each block has z number of molecules. As the temperature L
of the system is lowered, the configurations available to a subsystem for "cooperative rearrangements" decrease (9,2). What this suggests is the relevance as well as the importance of accounting for correlations between the "Mocks" (9,2). Consequendy, one introduces the "correlation length" f to denote die size over which the blocks are correlated. As the temperature is lowered, the "blocks" grow in size compared with typical molecular scale length Long wavelength fluctuations govern the relaxation behavior of supercooled liquids in the A G model. T o put it differentiy, the short wavelength fluctuations are integrated out form the partition function under renormalization group transformations
(5,6). The exact nature o f the
renormalization group transformation is not necessary, except to note that the partition function is die same before and after such a transformation (5,6). Consequendy, the Gibbs free-energy of a "block" and the Gibbs free-energy per unit volume of the subsystem obey the relation (2,9,5) a
g{z.a)^C t{z a ). L9
(5)
L
Further, the correlation length f ( z , a ) of a "block" is linked to die correlation L
L
length f (z, a) of the subsystem (2,9,5) %{z,a)-I4{z ,a ) . L
(6)
L
In the framework of renormalization group, die a's include the "relevant" as well as the "irrelevant" variables (5). Example of a "relevant" variable is the number of flex bonds introduced by Gibbs-DiMarzio in their theory of amorphous polymers (2,8). Differentiating Eq. (5) with respect to temperature leads to the relation (9,2)
3
s(z,a)- L" s(z ,a ) + ( p ) ^ g ( z , a ) , L
L
L
L
Fourkas et al.; Supercooled Liquids ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
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7. MOHANTY
99
Relaxation Behavior in Glass-Forming Liquids
3
3
where j ( z , a ) is the entropy of a "block". The ratio of the volumes L / L is a L
L
measure of the number of "blocks". If one takes a mole of the substance, then there N are also — ^ "blocks". This observation together with Eq. (7) provides the link between the size of a "block" and the entropy of the subsystem (9,2) „
N
s
A (ZL>
