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A gradient energy formalism for the nonexponential structural relaxation kinetics
250
Journal of Non-Crystalline Solids 102 (1988) 250-254 North-Holland, Amsterdam
A GRADIENT ENERGY F O R M A L I S M FOR T H E N O N E X P O N E N T I A L S T R U C T U R A L RELAXATION KINETICS Prabhat K. G U P T A Department of Ceramic Engineering, The Ohio State University, 177 Watts Hall, 2041 College Road, Columbus, OH 43210, USA
A phenomenological theory of structural relaxation is developed based on thermodynamics of systems which are spatially inhomogeneous with respect to structure. The theory yields for the kinetics an equation containing a diffusion-like term in addition to the usual relaxation term. This equation predicts two commonly observed features of structural relaxation: a broad distribution of relaxation times and a temperature independent shape of distribution function (i.e., thermorheological simplicity).
1. Introduction
Structural relaxation is the process of gradual approach to equifibrium of properties of a (viscous) liquid following a change in some external thermodynamic variable such as temperature. This phenomenon occurs in all liquids and in supercooled liquids. It forms the basis of the liquid to glass transition [1]. Structural relaxation has been studied extensively in a wide variety of glass forming systems (nitrate salts [2], oxides [1-3], halides [4], chalcogenides [2], organic systems [5,6] and metallic alloys [7]) and relaxations of many different properties (volume, refractive index, heat capacity, Young's modulus, etc.) have been investigated [1,2,7]. These studies show two features which appear to be common. These apparently universal features are: (1) A broad distribution of relaxation times g(~), which is skewed towards the short relaxation times. The relaxation function, M(t), which is related to g(~-) under isothermal isobaric conditions by the relation
M(t)
= f0~drg(~ ") exp( -
t/'r)
is approximately described by the KWW (or stretched-exponential) function [8,9]
M(t)=exp[--(t/~)b], where ~ and b are constant with respect to time and 0 < b < 1. 0022-3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2) The shape of g(~-) is a constant function of temperature, a phenomenon described by the term thermorheological simplicity (TRS). TRS implies that different relaxation times have the same activation enthalpies. In the language of KWW function, TRS implies that b is independent of temperature. The results show that TRS is obeyed only over a limited temperature range. The value of b decreases somewhat with decreasing temperatures [10,11]. Several attempts have been reported to describe mechanisms explaining some of these features [12-16]. While the microscopic approaches [16] are detailed, they are rather specific and frequently it is difficult to translate the ideas to other systems. For example the spin models of glass transition while applicable to spin glasses, fail to provide a clear meaning of spin in the case of non-spin glasses. On the other hand phenomenological treatments tend to be universal. Although they lack mechanistic details, they constitute a stepping stone for the more detailed microscopic models. This paper presents a phenomenological mode of relaxation. The motivation comes from the fact that this model directly leads to results which are consistent with some of the features mentioned earlier. The foundations of the gradient energy model lie in the internal parameter (or order parameter) formalism of glasses [17,18] which has been successful phenomenologically in explaining the thermodynamic and the kinetic behaviour of glasses
P.K. Gupta / Nonexponential structural relaxation kinetics
[2]. Because a frozen configuration is one of the fluctuation states of the liquid at the glass transition, structural variations are always present in a glass. These structural variations cause an increase in the free energy of the system whose magnitude depends on the structural gradients present in the system. It is useful to consider the structural variations in terms of the Fourier components. The steep gradients (i.e. short wavelength variations) are associated with high free energy and tend to decay rapidly. The more gradual variations (i.e. long wavelength variations) are associated with lower free energy and hence decay relatively slowly. This leads to a distribution of relaxation times which (as is shown later) has a cut-off at long relaxation times. Furthermore, the relaxation times for all wavelengths have the same temperature dependence (corresponding to the temperature dependence of a kinetic coefficient) because the wavelength dependent thermodynamic factor depends on temperature weakly. This picture of relaxation is consistent with the one advanced by others [11,14,19] in which relaxation in local regions (short wavelengths) proceeds faster compared to relaxation over larger regions (long wavelengths). In the following the gradient energy model is presented in its simplest (i.e. linear form) for the case of a one component system.
251
2.1. Thermodynamicsof a homogeneoussystem A general nonequilibrium state of the system is specified by T, P and Z. All properties such as the Gibb's free energy, G, are functions of (T, P, Z):
a= a(T, P, Z).
(1)
For specified values of external parameters (T, P ) the internal parameter Z assumes its equilibrium value, Z(e)(T, P), when the system is in equilibrium. Since an equilibrium state is associated with minimum in G (with respect to Z), one must have
( cl'e'
OZ ]r.e=O
(2)
and
]T,p>0
(3)
Eq. (2) can be solved for z(e)(T, P). Upon substitution of z(e)(T, P) in eq. (1), the equilibrium free energy G(e)(T, P) is obtained. A near-equilibrium expression for the free energy of glassy states is obtained by expanding G(T, P, Z ) about G(~)(T, P ) [201:
C(T, P, z) =G,e)(T, p) + ~B [ Z _ Z(e,(r, p ) ] 2 + .... (4)
2. The gradient energy model of structural relaxation Consider a one-component system which is homogeneous with respect to temperature T and pressure P. Equilibrium states of such a system are completely described by specifying the two external parameters T and P. Nonequilibrium states, for their description, require additional parameters which are called the internal (or order) parameters. Experimental evidence [1,2] indicates that at least two internal parameters are needed to describe the memory behaviour of glassy states. However in order to maintain simplicity the existence of a single internal parameter, Z, will be assumed. The treatment, however, can be readily extended to the case of multi-internal parameters.
where B = ( ~ 2 G / O Z 2 ) ( e ) is positive definite. The first order term vanishes because of eq. (2). By neglecting the higher order terms one obtains the near-equilibrium expression for the free energy of glass.
2.2. Thermodynamicsof structurally inhomogeneous states In structurally inhomogeneous systems the internal parameter is a nonuniform function of r where r is the position vector. Following the ideas of Cahn and Hilliard [21] for the thermodynamics of compositionally inhomogeneous systems, a local free energy density g(r) is defined such that the total free energy, G, of the system can be written as
G = fdrg(r). J~
(5)
P.K. Gupta/ Nonexponentialstructuralrelaxationkinetics
252
The local free energy density, g, is postulated to have the following functional form: R, K g(r) =g(e)(¥) q_ 2 [ Z ( r ) _ z(e)]2..}_ .~_(vZ) 2.
(6) The first two terms represent the local contribution. The first is the equilibrium free energy and the second is nonequilibrium contribution using eq. (4). The last term is a nonlocal contribution due to the gradient of Z. K, called the gradient energy coefficient, is positive definite since structural gradations (similar to diffuse interfaces) increase the free energy of the system. Addition of the last term (i.e. the gradient energy term) in eq. (6) represents a key concept in this formulation. Substitution of eq. (6) in eq. (5) gives the total free energy of a glass:
Clearly Z = Z (e) is a solution of eq. (11). Furthermore, eq. (7) shows that it has the lowest free energy and therefore represents the equilibrium state, as expected.
2.3. The relaxation equation By using the principles of the Linear Nonequilibrium Thermodynamics [22], a kinetic equation can be written as 3Z/3t = LA, (12) where the kinetic coefficient L depends on temperature and pressure. L may also depend on Z making eq. (12) nonlinear. That free energy must decrease monotonically requires that L be positive. L generally varies as the inverse of the volume viscosity. Substitution of eq. (10) in eq. (12) yields:
OZ
( Z - Z (e))
Ot
G(g)=G(e)+B fvdr[(g(r)-z(e))2
ro
..{-
R2 -
-
ro
~72Z,
(13)
where
q- R 2 ( ~7'Z)2],
(7)
where
R 2= K/B
(8)
It should be noted that Z (e~ does not depend on r because of the assumed uniformity of the system with respect to T and P. R has dimensions of length. R represents a correlation length over which the value of Z is positively correlated. R increases with increase in K because for larger values of K the system prefers to reduce the degree of inhomogeneity by increasing the size of the positively correlated regions. Parallel to the definition of affinity A for homogeneous systems where A = - (3G/OZ)r,e, one can define affinity for the free energy given in eq. (7) by the variational derivative [20]:
A(r) = -3G/3Z(r). (9) Using this definition it follows from eq. (7) that
TO= ( L B ) -1
(14)
Eq. (13) is the desired relaxation equation. It is a two parameter equation, the pertinent parameters being To and R. Eq. (13) reduces to Tool's equation [1,2] when R = 0. Therefore To is the relaxation time of Tool's equation. The effect of the gradient energy term appears as a diffusion-like term in eq. (13) and therefore RZ/'r0 may be thought of as a diffusion coefficient of the internal parameter. However it should be emphasized that the source of the diffusion term is the gradient energy in eq. (6). A similar equation was hypothesized by Montrose and Litovitz [23] following the ideas of Isakovich and Chaban [24] for relaxation dynamics of order parameter fluctuations in liquids. On the other hand, Mazumdar [25] has postulated an equation containing only the diffusion term.
(10)
2.1. The relaxation equation for fietive temperature,
Eq. (10) represents a generalization of the concept of affinity to inhomogeneous system where nonlocal contributions to free-energy (because K v~ O) are important. A stationary state of the system is defined by
Using the definition of fictive temperature (Tf) as the temperature when a given value of Z corresponds to the equilibrium value, it is possible to write
A(r) =0.
Z-
A(r)=B[-(Z-Z(e))WR2172Z].
(11)
Z(°)(T) = O(~-- T),
(15)
253
P.K. Gapta / Nonexponential structural relaxation kinetics
3.2. Thermo-rheological simplicity
where
6) = (3z(e)/OT) will be treated as constant. Substitution of (15) in eq. (13) gives
at
--
To
q"
--
To
~72 Tf.
(16)
Eq. (16) is the same as eq. (13) except that the internal parameter Z has been cast in the form of fictive temperature.
Eq. (19) shows that all relaxation times have the temperature dependence of To because R 2 is only weakly dependent on T and thus thermorheological simplicity is automatically satisfied.
3.3. Distribution of relaxation times By defining relaxation function
M(t) as
fodz[Z(r, O)- Z(e'][Z*(r, t ) - Z *(e)] M(t) =
fvd¥[Z(r, 0)- z(e)][/*(r, 0)- Z *(e)]
3. Discussion
where * represents complex conjugation, eq. (21) gives
3.1. Range of relaxation times Clearly eq. (13) represents non-exponential relaxation. In order to obtain the range of allowed relaxation times, it is convenient to rewrite eq. (13) in the reciprocal space by defining a Fourier transform:
[Z(r)-- Z(e)l = fdkak ei*'" ,
ak
g(T)-
T0
(19)
1+ Rak 2 "
Eq. (19) shows that the maximum value of relaxation time is given by T0 (the same as in Tool's equation). This maximum value corresponds to k = 0 or the limit of large wavelengths (h >> R). The minimum value of Tk is zero for k = ~ (corresponding to small wavelengths, 2~<< R). Thus eq. (19) permits all relaxation times in the range 0 < T < TO. The solution of eq. (18) can be expressed as
ak(t ) = ak(0 ) e x p ( - - t / T k ) .
Z (e)] =
[1 - (T/T0)] 1/2 I ak(0) I 2
[2R3To~]
(T/To)5~ 2
,
(22)
0°~dk k 2 [ak(0 ) I 2 TO obtain information on (a2(0)), it is useful to substitute eq. (17) in eq. (7) to obtain an expression for the free energy density, G k, associated with an internal parameter fluctuation of wavevectot k.
Gk = B(1
(21)
R2k2)a 2.
kBT (a2)
fdk ak(O ) e x p ( - - t / T k ) e ik'r.
+
Using this equation it can be shown [26,27] that the average magnitude of fluctuations, ( a k2) , in the equilibrium state is given by
(20)
When substituted in eq. (17), eq. (20) gives
[Z(r, t ) -
1
where I2 is the normalizing factor given by
where Tk
Using this equation and eq. (19), it can be shown that
(18)
rk
3t
fo~dk k2 [ ak(0) 12
(17)
Substitution of (17) in eq. (15) gives
Oak
f 0 ~ d k k 2 l a k ( 0 ) l 2 exp(--t/Tk)
M(t)
B(1
+
RZk 2)
In a frozen state this equation becomes
kBTf (k) ( a 2 ( 0 ) ) = B(1 + R2k2) '
(23)
P.K. Gupta / Nonexponential structural relaxation kinetics
254
where T f ( k ) - the fictive t e m p e r a t u r e of a fluctuation of w a v e n u m b e r k - is the t e m p e r a t u r e w h e n the fluctuation freezes d u r i n g c o o l i n g of the equil i b r i u m system. A s s u m i n g the V o g e l - F u l c h e r temp e r a t u r e d e p e n d e n c e of T0.
an expression for T f ( k ) can b e o b t a i n e d using eq. (19) a n d b y e q u a t i n g ~'k to an e x p e r i m e n t a l o b servation t i m e %xp: A
Tf(k)=
To+ [In(
"/'exp] + ln(l+
R2k2)] "
(24)
L \%) Substitution of eq. (24) in eq. (23) shows that (a~) is a rapidly decreasing function of (Rk): lim ( a ~ ) = ~ Rk ---~O
A To + - -
kBI
(25)
ln( %xp )'r~
and lira ( a ~ ) Rk--,~
kBT° B ( k 2 R 2) "
(26)
S u b s t i t u t i o n of eq. (23) in eq. (22) shows that the resulting d i s t r i b u t i o n , g('r), diverges in the r ~ 0 limit, a b e h a v i o u r in d i s a g r e e m e n t with the K W W form. In o r d e r to o b t a i n a K W W t y p e of d i s t r i b u t i o n f u n c t i o n a k ( 0 ) in eq. (23) has to b e m o r e singular at low values of k t h a n that given b y eq. (23). It is believed that i n c l u s i o n of higher o r d e r terms in eq. (4) - those which l e a d to n o n - l i n e a r terms in eqs. (10) a n d (13) - can give the desired f o r m of the d i s t r i b u t i o n function.
4. Summary and conclusions A general p h e n o m e n o l o g i c a l t h e o r y a p p l i c a b l e to all liquid a n d glass systems, b a s e d o n the i n t e r n a l p a r a m e t e r f o r m a l i s m of glasses, is presented b y t a k i n g i n t o a c c o u n t the n o n - l o c a l c o n t r i b u t i o n s to the free energy due to s p a t i a l v a r i a t i o n s in the i n t e r n a l p a r a m e t e r . It is s h o w n t h a t the r e l a x a t i o n e q u a t i o n c o n t a i n s a d i f f u s i o n t e r m in
a d d i t i o n to the usual r e l a x a t i o n t e r m as given b y T o o l ' s e q u a t i o n . This m o d e l of r e l a x a t i o n beh a v i o u r shows a d i s t r i b u t i o n of r e l a x a t i o n times with a cut-off at T0, the r e l a x a t i o n time in T o o l ' s equation. F u r t h e r m o r e all r e l a x a t i o n times exhibit i d e n t i c a l t e m p e r a t u r e d e p e n d e n c e . T h e r e f o r e therm o r h e o l o g i c a l s i m p l i c i t y follows directly f r o m the model.
References [1] G.W. Scherer, Relaxation in Glass and Composites (Wiley, New York, 1986). [2] C.T. Moynihan et al., Ann. NY Acad. Sci. 279 (1976) 15. [3] O.V. Mazurin, J. Non-Cryst. Solids 25 (1977) 130. [4] C.T. Moynihan et al., Polymer Eng. Sci. 24 (1984) 1117. [5] A.J. Kovacs. Ann. NY Acad. Sci. 371 (1981) 38. [6] I.M. Hodge and A.R. Berens, Macromolecules 15 (1982) 762. [7] H.S. Chen, in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworths, London, 1983) p. 169. [8] R.W. Douglas, J. Non-Cryst. Solids 14 (1972) 1. [9] G.W. Scherer, J. Am. Ceram. Soc. 69 (5) (1986) 374. [10] J. Tauke, T.A. Litovitz and P.B. Macedo, J. Am. Ceram. Soc. 57 (3) (1968) 158. [11] H.S. Chen and C.R. Kurkjian, J. Am. Ceram. Soc. 66 (9) (1983) 613. [12] G.S. Grest and M.H. Cohen, Adv. Chem. Phys. XLVIII (1981) 455. [13] A.K. Rajgopal, K.L. Ngai, R.W. Rendell and S. Teitler, J. Stat. Phys. 30 (2) (1983) 285. [141 S. Brawer, J. Chem. Phys. 81 (2) (1984) 954. [15] R.G. Palmer et al., Phys. Rev. Lett. 53 (10) (1984) 958. [16] G.H. Fredrickson and H.C. Anderson, Phys. Rev. Lett. 53 (13) (1984) 1244. [17] C.T. Moynihan and P.K. Gupta, J. Non-Cryst. Solids 29 (1978) 143. [18] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (10) (1976) 4136. [19] G. Adam and J.H. Gibbs, J. Chem. Phys. 43 (1) (1965) 139. [20] P.K. Gupta, J. Non-Cryst. Solids 71 (1985) 29. [21] J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28 (2) (1958) 258. [22] S.R. DeGroot and P. Mazur, Nonequilibrium Thermodynarnics (North-Holland, Amsterdam, 1963). [23] C.J. Montrose and T.A. Litovitz, J. Acoust. Soc. America 47 (5), Part 2 (1970) 1250. [24] M.A. Isakovich and I.A. Chaban, Soy. Phys. -Dokl. 10 (1966) 1055. [25] C.K. Mazumdar, Solid State Communications, 9 (1971) 1087. [26] R.A. Ferrel, Contemporary Physics, Vol. I (IAEA, Vienna, 1969). [27] D.J. Amit and M. Zanneti, J. Stat. Phys. 7 (1) (1973) 31.
Journal of Non-Crystalline Solids 102 (1988) 250-254 North-Holland, Amsterdam
A GRADIENT ENERGY F O R M A L I S M FOR T H E N O N E X P O N E N T I A L S T R U C T U R A L RELAXATION KINETICS Prabhat K. G U P T A Department of Ceramic Engineering, The Ohio State University, 177 Watts Hall, 2041 College Road, Columbus, OH 43210, USA
A phenomenological theory of structural relaxation is developed based on thermodynamics of systems which are spatially inhomogeneous with respect to structure. The theory yields for the kinetics an equation containing a diffusion-like term in addition to the usual relaxation term. This equation predicts two commonly observed features of structural relaxation: a broad distribution of relaxation times and a temperature independent shape of distribution function (i.e., thermorheological simplicity).
1. Introduction
Structural relaxation is the process of gradual approach to equifibrium of properties of a (viscous) liquid following a change in some external thermodynamic variable such as temperature. This phenomenon occurs in all liquids and in supercooled liquids. It forms the basis of the liquid to glass transition [1]. Structural relaxation has been studied extensively in a wide variety of glass forming systems (nitrate salts [2], oxides [1-3], halides [4], chalcogenides [2], organic systems [5,6] and metallic alloys [7]) and relaxations of many different properties (volume, refractive index, heat capacity, Young's modulus, etc.) have been investigated [1,2,7]. These studies show two features which appear to be common. These apparently universal features are: (1) A broad distribution of relaxation times g(~), which is skewed towards the short relaxation times. The relaxation function, M(t), which is related to g(~-) under isothermal isobaric conditions by the relation
M(t)
= f0~drg(~ ") exp( -
t/'r)
is approximately described by the KWW (or stretched-exponential) function [8,9]
M(t)=exp[--(t/~)b], where ~ and b are constant with respect to time and 0 < b < 1. 0022-3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2) The shape of g(~-) is a constant function of temperature, a phenomenon described by the term thermorheological simplicity (TRS). TRS implies that different relaxation times have the same activation enthalpies. In the language of KWW function, TRS implies that b is independent of temperature. The results show that TRS is obeyed only over a limited temperature range. The value of b decreases somewhat with decreasing temperatures [10,11]. Several attempts have been reported to describe mechanisms explaining some of these features [12-16]. While the microscopic approaches [16] are detailed, they are rather specific and frequently it is difficult to translate the ideas to other systems. For example the spin models of glass transition while applicable to spin glasses, fail to provide a clear meaning of spin in the case of non-spin glasses. On the other hand phenomenological treatments tend to be universal. Although they lack mechanistic details, they constitute a stepping stone for the more detailed microscopic models. This paper presents a phenomenological mode of relaxation. The motivation comes from the fact that this model directly leads to results which are consistent with some of the features mentioned earlier. The foundations of the gradient energy model lie in the internal parameter (or order parameter) formalism of glasses [17,18] which has been successful phenomenologically in explaining the thermodynamic and the kinetic behaviour of glasses
P.K. Gupta / Nonexponential structural relaxation kinetics
[2]. Because a frozen configuration is one of the fluctuation states of the liquid at the glass transition, structural variations are always present in a glass. These structural variations cause an increase in the free energy of the system whose magnitude depends on the structural gradients present in the system. It is useful to consider the structural variations in terms of the Fourier components. The steep gradients (i.e. short wavelength variations) are associated with high free energy and tend to decay rapidly. The more gradual variations (i.e. long wavelength variations) are associated with lower free energy and hence decay relatively slowly. This leads to a distribution of relaxation times which (as is shown later) has a cut-off at long relaxation times. Furthermore, the relaxation times for all wavelengths have the same temperature dependence (corresponding to the temperature dependence of a kinetic coefficient) because the wavelength dependent thermodynamic factor depends on temperature weakly. This picture of relaxation is consistent with the one advanced by others [11,14,19] in which relaxation in local regions (short wavelengths) proceeds faster compared to relaxation over larger regions (long wavelengths). In the following the gradient energy model is presented in its simplest (i.e. linear form) for the case of a one component system.
251
2.1. Thermodynamicsof a homogeneoussystem A general nonequilibrium state of the system is specified by T, P and Z. All properties such as the Gibb's free energy, G, are functions of (T, P, Z):
a= a(T, P, Z).
(1)
For specified values of external parameters (T, P ) the internal parameter Z assumes its equilibrium value, Z(e)(T, P), when the system is in equilibrium. Since an equilibrium state is associated with minimum in G (with respect to Z), one must have
( cl'e'
OZ ]r.e=O
(2)
and
]T,p>0
(3)
Eq. (2) can be solved for z(e)(T, P). Upon substitution of z(e)(T, P) in eq. (1), the equilibrium free energy G(e)(T, P) is obtained. A near-equilibrium expression for the free energy of glassy states is obtained by expanding G(T, P, Z ) about G(~)(T, P ) [201:
C(T, P, z) =G,e)(T, p) + ~B [ Z _ Z(e,(r, p ) ] 2 + .... (4)
2. The gradient energy model of structural relaxation Consider a one-component system which is homogeneous with respect to temperature T and pressure P. Equilibrium states of such a system are completely described by specifying the two external parameters T and P. Nonequilibrium states, for their description, require additional parameters which are called the internal (or order) parameters. Experimental evidence [1,2] indicates that at least two internal parameters are needed to describe the memory behaviour of glassy states. However in order to maintain simplicity the existence of a single internal parameter, Z, will be assumed. The treatment, however, can be readily extended to the case of multi-internal parameters.
where B = ( ~ 2 G / O Z 2 ) ( e ) is positive definite. The first order term vanishes because of eq. (2). By neglecting the higher order terms one obtains the near-equilibrium expression for the free energy of glass.
2.2. Thermodynamicsof structurally inhomogeneous states In structurally inhomogeneous systems the internal parameter is a nonuniform function of r where r is the position vector. Following the ideas of Cahn and Hilliard [21] for the thermodynamics of compositionally inhomogeneous systems, a local free energy density g(r) is defined such that the total free energy, G, of the system can be written as
G = fdrg(r). J~
(5)
P.K. Gupta/ Nonexponentialstructuralrelaxationkinetics
252
The local free energy density, g, is postulated to have the following functional form: R, K g(r) =g(e)(¥) q_ 2 [ Z ( r ) _ z(e)]2..}_ .~_(vZ) 2.
(6) The first two terms represent the local contribution. The first is the equilibrium free energy and the second is nonequilibrium contribution using eq. (4). The last term is a nonlocal contribution due to the gradient of Z. K, called the gradient energy coefficient, is positive definite since structural gradations (similar to diffuse interfaces) increase the free energy of the system. Addition of the last term (i.e. the gradient energy term) in eq. (6) represents a key concept in this formulation. Substitution of eq. (6) in eq. (5) gives the total free energy of a glass:
Clearly Z = Z (e) is a solution of eq. (11). Furthermore, eq. (7) shows that it has the lowest free energy and therefore represents the equilibrium state, as expected.
2.3. The relaxation equation By using the principles of the Linear Nonequilibrium Thermodynamics [22], a kinetic equation can be written as 3Z/3t = LA, (12) where the kinetic coefficient L depends on temperature and pressure. L may also depend on Z making eq. (12) nonlinear. That free energy must decrease monotonically requires that L be positive. L generally varies as the inverse of the volume viscosity. Substitution of eq. (10) in eq. (12) yields:
OZ
( Z - Z (e))
Ot
G(g)=G(e)+B fvdr[(g(r)-z(e))2
ro
..{-
R2 -
-
ro
~72Z,
(13)
where
q- R 2 ( ~7'Z)2],
(7)
where
R 2= K/B
(8)
It should be noted that Z (e~ does not depend on r because of the assumed uniformity of the system with respect to T and P. R has dimensions of length. R represents a correlation length over which the value of Z is positively correlated. R increases with increase in K because for larger values of K the system prefers to reduce the degree of inhomogeneity by increasing the size of the positively correlated regions. Parallel to the definition of affinity A for homogeneous systems where A = - (3G/OZ)r,e, one can define affinity for the free energy given in eq. (7) by the variational derivative [20]:
A(r) = -3G/3Z(r). (9) Using this definition it follows from eq. (7) that
TO= ( L B ) -1
(14)
Eq. (13) is the desired relaxation equation. It is a two parameter equation, the pertinent parameters being To and R. Eq. (13) reduces to Tool's equation [1,2] when R = 0. Therefore To is the relaxation time of Tool's equation. The effect of the gradient energy term appears as a diffusion-like term in eq. (13) and therefore RZ/'r0 may be thought of as a diffusion coefficient of the internal parameter. However it should be emphasized that the source of the diffusion term is the gradient energy in eq. (6). A similar equation was hypothesized by Montrose and Litovitz [23] following the ideas of Isakovich and Chaban [24] for relaxation dynamics of order parameter fluctuations in liquids. On the other hand, Mazumdar [25] has postulated an equation containing only the diffusion term.
(10)
2.1. The relaxation equation for fietive temperature,
Eq. (10) represents a generalization of the concept of affinity to inhomogeneous system where nonlocal contributions to free-energy (because K v~ O) are important. A stationary state of the system is defined by
Using the definition of fictive temperature (Tf) as the temperature when a given value of Z corresponds to the equilibrium value, it is possible to write
A(r) =0.
Z-
A(r)=B[-(Z-Z(e))WR2172Z].
(11)
Z(°)(T) = O(~-- T),
(15)
253
P.K. Gapta / Nonexponential structural relaxation kinetics
3.2. Thermo-rheological simplicity
where
6) = (3z(e)/OT) will be treated as constant. Substitution of (15) in eq. (13) gives
at
--
To
q"
--
To
~72 Tf.
(16)
Eq. (16) is the same as eq. (13) except that the internal parameter Z has been cast in the form of fictive temperature.
Eq. (19) shows that all relaxation times have the temperature dependence of To because R 2 is only weakly dependent on T and thus thermorheological simplicity is automatically satisfied.
3.3. Distribution of relaxation times By defining relaxation function
M(t) as
fodz[Z(r, O)- Z(e'][Z*(r, t ) - Z *(e)] M(t) =
fvd¥[Z(r, 0)- z(e)][/*(r, 0)- Z *(e)]
3. Discussion
where * represents complex conjugation, eq. (21) gives
3.1. Range of relaxation times Clearly eq. (13) represents non-exponential relaxation. In order to obtain the range of allowed relaxation times, it is convenient to rewrite eq. (13) in the reciprocal space by defining a Fourier transform:
[Z(r)-- Z(e)l = fdkak ei*'" ,
ak
g(T)-
T0
(19)
1+ Rak 2 "
Eq. (19) shows that the maximum value of relaxation time is given by T0 (the same as in Tool's equation). This maximum value corresponds to k = 0 or the limit of large wavelengths (h >> R). The minimum value of Tk is zero for k = ~ (corresponding to small wavelengths, 2~<< R). Thus eq. (19) permits all relaxation times in the range 0 < T < TO. The solution of eq. (18) can be expressed as
ak(t ) = ak(0 ) e x p ( - - t / T k ) .
Z (e)] =
[1 - (T/T0)] 1/2 I ak(0) I 2
[2R3To~]
(T/To)5~ 2
,
(22)
0°~dk k 2 [ak(0 ) I 2 TO obtain information on (a2(0)), it is useful to substitute eq. (17) in eq. (7) to obtain an expression for the free energy density, G k, associated with an internal parameter fluctuation of wavevectot k.
Gk = B(1
(21)
R2k2)a 2.
kBT (a2)
fdk ak(O ) e x p ( - - t / T k ) e ik'r.
+
Using this equation it can be shown [26,27] that the average magnitude of fluctuations, ( a k2) , in the equilibrium state is given by
(20)
When substituted in eq. (17), eq. (20) gives
[Z(r, t ) -
1
where I2 is the normalizing factor given by
where Tk
Using this equation and eq. (19), it can be shown that
(18)
rk
3t
fo~dk k2 [ ak(0) 12
(17)
Substitution of (17) in eq. (15) gives
Oak
f 0 ~ d k k 2 l a k ( 0 ) l 2 exp(--t/Tk)
M(t)
B(1
+
RZk 2)
In a frozen state this equation becomes
kBTf (k) ( a 2 ( 0 ) ) = B(1 + R2k2) '
(23)
P.K. Gupta / Nonexponential structural relaxation kinetics
254
where T f ( k ) - the fictive t e m p e r a t u r e of a fluctuation of w a v e n u m b e r k - is the t e m p e r a t u r e w h e n the fluctuation freezes d u r i n g c o o l i n g of the equil i b r i u m system. A s s u m i n g the V o g e l - F u l c h e r temp e r a t u r e d e p e n d e n c e of T0.
an expression for T f ( k ) can b e o b t a i n e d using eq. (19) a n d b y e q u a t i n g ~'k to an e x p e r i m e n t a l o b servation t i m e %xp: A
Tf(k)=
To+ [In(
"/'exp] + ln(l+
R2k2)] "
(24)
L \%) Substitution of eq. (24) in eq. (23) shows that (a~) is a rapidly decreasing function of (Rk): lim ( a ~ ) = ~ Rk ---~O
A To + - -
kBI
(25)
ln( %xp )'r~
and lira ( a ~ ) Rk--,~
kBT° B ( k 2 R 2) "
(26)
S u b s t i t u t i o n of eq. (23) in eq. (22) shows that the resulting d i s t r i b u t i o n , g('r), diverges in the r ~ 0 limit, a b e h a v i o u r in d i s a g r e e m e n t with the K W W form. In o r d e r to o b t a i n a K W W t y p e of d i s t r i b u t i o n f u n c t i o n a k ( 0 ) in eq. (23) has to b e m o r e singular at low values of k t h a n that given b y eq. (23). It is believed that i n c l u s i o n of higher o r d e r terms in eq. (4) - those which l e a d to n o n - l i n e a r terms in eqs. (10) a n d (13) - can give the desired f o r m of the d i s t r i b u t i o n function.
4. Summary and conclusions A general p h e n o m e n o l o g i c a l t h e o r y a p p l i c a b l e to all liquid a n d glass systems, b a s e d o n the i n t e r n a l p a r a m e t e r f o r m a l i s m of glasses, is presented b y t a k i n g i n t o a c c o u n t the n o n - l o c a l c o n t r i b u t i o n s to the free energy due to s p a t i a l v a r i a t i o n s in the i n t e r n a l p a r a m e t e r . It is s h o w n t h a t the r e l a x a t i o n e q u a t i o n c o n t a i n s a d i f f u s i o n t e r m in
a d d i t i o n to the usual r e l a x a t i o n t e r m as given b y T o o l ' s e q u a t i o n . This m o d e l of r e l a x a t i o n beh a v i o u r shows a d i s t r i b u t i o n of r e l a x a t i o n times with a cut-off at T0, the r e l a x a t i o n time in T o o l ' s equation. F u r t h e r m o r e all r e l a x a t i o n times exhibit i d e n t i c a l t e m p e r a t u r e d e p e n d e n c e . T h e r e f o r e therm o r h e o l o g i c a l s i m p l i c i t y follows directly f r o m the model.
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