Dynamical glassy behaviour on the basis of simple models described by master equations

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Journal of Non-Crystalline Solids 172-174 (1994) 371-377

Dynamical glassy behaviour on the basis of simple models described by master equations J.J. Brey*, A. Prados, M.J. Ruiz-Montero. Fisica Te6rica, Universidad de Sevilla, Apdo. Correos 1065, 41080 Seville, Spain

Abstract

Using a simple Ising model as illustration, general relaxation and evolution properties of systems described by master equations are investigated. The origin of the good agreement obtained by fitting the response function at low temperatures to a stretched exponential is discussed. The details of the freezing process under continuous cooling are analyzed by introducing appropriate timescales and the notion of demarcation mode. For heating processes, the relevance of the concept of normal curves shows up clearly. They explain the appearance of hysteresis cycles and can be useful to interpret experimental results.

1. Introduction

In the past years, a number of microscopic models have been proposed in order to understand some of the dynamical features of supercooled liquids and glasses. Some of these models have been reviewed in Ref. [1]. The interest has focused mainly on the non-exponential relaxation near the laboratory glass transition and the residual properties after a continuous cooling of the system. The models are quite different both in complexity and in the physical mechanisms of evolution they introduce. The simplest possibility seems to be considering two or more states separated by energy barriers, playing the role of activation energies. Examples

* Corresponding author. Tel: +34-95 462 6558. Telefax: + 34-95 461 2097. E-mail: [email protected].

are the two-level system [2] and Brawer's model [3]. A more basic starting point is adopted in the one-dimensional system of classical particles with competing and anharmonic interactions proposed by Schilling [4], where the dynamics follow from the Hamiltonian. At low temperatures, the evolution of the configuration of the system can be accurately described by means of an Ising chain with Glauber dynamics and extrinsic energy barriers [5]. Another model with energy barriers is the bond fluid model [6] where, in addition to the nearest neighbour interactions, two particles separated by a single hole may establish a bond between them. In all the above models, the kinetics of the system are fully governed by energetic considerations. There is also a number of models in which the dynamics are constrained by limiting the configurations from which the system can evolve. This is the case of the facilitated Ising models of Fredrickson

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J.J. Brevet al. / Journal o f Non-Crystalline Solids 172-174 (1994) 371-377

and Andersen [7] and the tiling models of Stillinger and Weber [8]. A general type of system with hierarchically constrained dynamics has been introduced and discussed in a rather heuristic way by Palmer et al. [9]. Models belonging to this category are the asymmetric facilitated Ising model and the lattice model with extended hard-core repulsion [10]. Another ingredient that has been considered is the existence of some type of disorder in the system. The disorder is introduced, for instance, by considering an ensemble of two-level systems with a distribution of excitation energies and barriers [11]. In spite of their quite different nature, most of the models lead to linear relaxation functions that can be accurately fitted by a KohlrauschWilliams-Watts (KWW) function, at least within limited time intervals and at low temperatures. However, usually, it appears only as a convenient numerical fitting or, in some cases, as an asymptotic behaviour valid in the limit of very long times, an exception being the diffusion model considered by Shore and Zwanzig [12]. Given the variety of models showing a K W W relaxation, it seems important to try to clarify whether there is a common mechanism that is responsible for that behaviour in all cases. We present here a general argument explaining why a K W W region will show up in a certain time window under very general conditions, and also a criterion to know if this window will lie in the relevant part of the relaxation. Regarding continuous cooling, the models show a freezing phenomenon that is similar to the laboratory glass transition. The dependence of the residual properties on the cooling process has been analyzed in some of the cases, mainly in the slow cooling limit. Very recently, a general result has been obtained for continuous heating processes in systems described by a master equation [13]. Given a law of variation of the temperature, it has been shown that there is a special solution that all the other solutions tend to approach, independently of the initial conditions. The existence of this 'normal solution' has a number of implications on the actual evolution of the system. In particular, it is responsible for the different behaviours shown by the system in cooling and heating processes, and,

therefore, for the hysteresis effects that are observed. In this paper, we show that the qualitative picture that one obtains in this way is in agreement with what is experimentally found in real glasses. Although many of the reasonings we present can be formulated in a general way, for the sake of concreteness we use an specific model, for which all the calculations can be carried out analytically. In any case, emphasis will be put in those results that are model independent.

2. Model We consider a one-dimensional Ising model with Glauber kinetics [14] and extrinsic energy barriers. The latter do not affect the Hamiltonian of the system that is given by =

(1)

-- J E s i s i + l ,

i

where si = __+1and J ~>0. The time evolution of the system is described by a master equation where the transition rate of spin j is (2) In this expression,

7(T) = tanh (2J/kBT)

(3)

and a(T) = Ctoexp ( -

A/kBT).

(4)

The term 7(T) guarantees that detailed balance is verified and, therefore, the ergodicity of the system for T ¢:0. The effect of the energy barrier A is contained in ct(T), and it does not affect the equilibrium properties. We use the constant C~o to define the natural time scale and take it equal to 1. From a physical point of view, the origin of the barrier can be quite different. For instance, it may represent an activation energy required for hopping processes in configuration space, as is the case in Schillings's interpretation [4,5]. If the Ising model is used to represent a magnetic system, A may be associated to the activation energy needed to flip

J.J. Brey et al. / Journal of Non-Crystalline Solids 172-174 (1994) 371 377

the spin between the two possible values, showing the anisotropy of the system. We want to remark on the simplicity of the dynamics of this model. It does not present any type of asymmetry or disorder. Also, no dynamical constrains have been introduced. In fact, one of the objectives of this work is to show up that none of these elements is needed to get a glassy-like dynamical behaviour. To describe the evolution when the temperature of the heat bath is changing in time, we use the same master equation as in the case of constant temperature. At a qualitative level, it can be thought that this corresponds to assume that the temperature of the heat bath changes very slowly as compared with its equilibration time. Nevertheless, we want to point out that the modeling of heating and cooling processes is a non-trivial problem, and this applies also to computer simulation methods [16,17]. The quantity we focus on is the average energy per particle, e(t), and in all our considerations we restrict ourselves to homogeneous states.

3. Linear response

The linear response function in energy, ~b(t), to an homogeneous temperature perturbation has the form [15]

¢(t) = f dq g(q, T)e-t;(q'r}.

(5)

The explicit expression of the frequencies 2(q,T) and of the distribution 9(q,T) are given in Ref. [15]. Eq. (5) is exact, and contains no approximations. It can also be written in the form of an integral closed equation for ¢(t) with a rather complicated memory kernel. In the long time limit, one obtains

q~(t)

~{7(1 + ])!)1/2 1 e-;'l' \ 2~1 -- (1 -- 72)'/2 (2103/2,

(6)

while for short times it is qS(t) ~ e-;M',

(7)

where 21 and 2M are the minimum relaxation rate and the average relaxation rate, respectively. There-

373

I in A M .

/J /

¢

l

zo /

.//

[n~

Fig. 1. Sketch of the function ~(x) for a frequency distribution with a lower bound greater than zero. The two asymptotes corresponding to x ~ + oc and x---, - oo are also plotted. The K W W region is located around the inflexion point, xo.

fore, the behaviour of ¢(t) is essentially exponential both in the short and in the long time limits. It follows that there must be at least a time window where ~b(t) can be accurately approximated by a KWW function. Define x = In t and g' = In(In -~b). Then, ~b(x) ~ x + ln21 for x ~ + zc, and ~b(x) ~ x +ln2M for x ~ - oc. Consequently, ¢(x) must present an inflexion point at a given value Xo, as schematically illustrated in Fig. 1. Around this value we can approximate ~(x) ~_ OIXo) + q/(Xo)(X - Xo).

(8)

This expression is equivalent to the stretched exponential ~b(t) -~exp( - t/z) l~,

(9)

with fl and z defined by fl=q/(Xo) and In r =Xo -(@(Xo)/fl). The above argument is quite general and can be applied to any expression of the same form as Eq. (5), except when g(q,T)v ~ 0 or 9(q,T) ~ 2 p for 2 ~ 0. In the last case, ~b(t) has an algebraic long time tail. This generality is what explains why the K W W behaviour is observed in so many different systems, as discussed in the Introduction. Of course, the point now is to know whether the K W W time region will be in the relevant part of the response function, i.e., when it is not too small yet. We can estimate this by introducing the average relaxation time =

0

dt 4~(t).

(10)

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J.J. Brey et al. / Journal of Non-Crystalline Solids 172-174 (1994) 371-377

If ~ ,-,2~ 1, we can expect that the relaxation is essentially governed by the initial exponential, while for ~ >>2~ 1 the timescale over which the relaxation takes place is much longer than the one defined by the initial exponential, and the K W W region will show up clearly. For the model we are considering, the above criterion shows that the range described by the K W W function increases as the temperature is lowered. In fact, in the low temperature limit, three well defined different time regimes can be identified:

4~ (t)

=

exp[-- 2ot(2e)l/2t] for 20tt<< 1, exp[-- (32~et/n) 1/23 for 1 <<20~t<>e-1, (11) where e = 1 - ? . The details of the calculation are given in Ref. [,,153,where an expression valid for all times is also given. In Fig. 2, the exact ~b(t) is compared with the K W W function given above for the

intermediate time regime. It is seen that, at low enough temperatures, the K W W function is able to describe very accurately all the relaxation of tp(t). The relaxation of the spin time correlation function has been studied numerically by Anderson and also by Bozdemir [19], who found a similar behaviour but with fl = 0.63, instead of fl = 0.5. This is an example of how different properties of the same system can relax in a different way. An approximate description of the relaxation at low temperatures can be obtained by matching the initial exponential and the stretched exponential. This leads to a picture that is similar to the coupling model for glassy relaxation [20], that here appears in a natural way. It is interesting to realize that the inclusion of dynamical constrains in this model could imply that the low temperature region where the K W W function is dominant would become inaccessible and, therefore, the relaxation of the system could not be accurately fitted by that function.

4. C o n t i n u o u s c o o l i n g 1

tp

We have investigated the behaviour of the system when the temperature of the heat bath changes in time according to

0.5

dT/dt = --rcf(T), 0 -10

1 ~0

-6

i e= 1

-2 In c~t

0

-

' 4

2

I ~

6

'

)

0.5

0

-

'

-4

'

'

0

'

'

4

'

'%

8

in at Fig. 2. Energyresponse functionin the low temperature region for e = 10-2 and e = 10-4. The circles are the exact result, and the solid line is the KWW function appearing in Eq. (11).

(12)

where the cooling law f ( T ) is an adimensional function of the temperature, and rc is the cooling rate. From the analysis, it follows that the relevant timescale is I to

s=

dr' [,~(T')] - 1.

(13)

Here to is the extrapolated time for which the temperature vanishes and T' = T(t'). The time, s, is proportional to the number of'effective' relaxations that rest to the system before reaching T = 0. An estimation of the freezing temperature is obtained by making s = 1. Of course, the value of this temperature, T:, depends on the form of ~(T) and, therefore, on the property under consideration, the cooling law and the cooling rate. The exact analysis shows that this criterion leads to better results than

J.J. Brey et al. / Journal qf Non-Crvstalline Solids 172-174 (1994) 371 377

the simple picture that the system freezes when the rate of cooling equals some average transition rate. From the expression of Ts one can easily derive the functional form of the residual energy for a given cooling process [18]. To study in more detail the freezing of the system, we can define a timescale for each of the modes

0

375

'

-0.2 -0.4

-0.6 -0.8

as

-1 tO

s(q) =

dt' 2(q, T').

(14)

In this way, we can determine a freezing temperature for each q, and introduce a 'demarcation mode' qD(T) such that modes with q < qo are frozen, while modes with q > qD are still relaxing at that temperature. The departure of the system from equilibrium begins at T1 given by qo(T1)=O (slowest relaxation mode). On the other hand, the system will be fully frozen at T2 such that qD(T2) = n (fastest relaxation mode). The difference AT = T1 - 7"2 provides a measure of the width of the freezing. For a wide class of cooling programs, it is found that AT ~ (In re)- 1,

(15)

which agrees with the results obtained in real glasses [21].

5. Heating and cycles Let us now consider that, after the continuous cooling to low temperatures described by Eq. (12), the system is reheated with a law, g(T), and rate, rh. During the heating process, the energy is given as a function of the temperature as the sum of two components [18]:

e(T) = eN(T) + ep(T),

(16)

where the contribution ep(T) contains all the information from the previous cooling, while eN(T), which will be referred as the normal component of the energy, only depends on the heating law and the heating rate. Both components are positive, and the normal one lies below the equilibrium curve e(°)(T),

al . . . . . . .

-6

~

,

-4

I

-2

,

0

ln~ Fig. 3. Hysteresis cycle when the system is cooled (O) to very low temperatures and afterwards heated ( o ) with the same law (( = exp( - 4J/kBT)). In the case shown, the cooling rate was 100 times smaller than the heating rate. Also plotted are the equilibrium curve em)(T) ( ) and the normal solution eN(T)

(--,--.--).

coinciding with it for T--- 0 and T ~ @. Besides, for large T, ep(T) tends to zero, meaning that the system forgets the initial conditions. The physical picture that follows is that, when the system is being heated, it tends to approach the normal curve, and not the equilibrium one, although both coincide at high enough temperatures. Since the system deviates from equilibrium in the cooling process having larger energies than at equilibrium, it has to cross the equilibrium curve in the heating processes in order to approach the normal curve. This will imply the appearance of hysteresis cycles when the system is cooled and reheated. An example is given in Fig. 3. We want to remark that the above analysis is not restricted to heating processes following a previous cooling of the system. Starting from an arbitrary initial condition and an arbitrary history, the energy of the system will always tend to approach the normal value. Even more, there is a curve of normal states for each continuous heating process of the system, from which the normal curves corresponding to different properties can be derived. The existence of a curve of normal states has been proved under very general conditions for models formulated in terms of master equations [13].

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J.J. Brey et al. / Journal of Non-Crystalline Solids 172-174 (1994) 371-377

6. Heat capacity

'

Given a law of variation of the temperature, we can define an apparent specific heat per particle as (17)

which in general will be different from the true specific heat c(T), computed along the equilibrium curve. For the model we are considering, one can easily prove that for continuous cooling processes it is c,(T) > 0. This agrees with the results found in real glasses., Another experimental relation [21] is c(T) >>.ca(T), which for linear and faster than linear coolings is also verified in our model. Nevertheless, there are cooling schedules, slower than the linear one and leading to residual energy, that violate the above inequality. We do not know whether this is a peculiarity of the model or the same behaviour is to be expected on a more general basis. If the system is reheated after the cooling, it follows from Eq. (16) that C'a(T) = c~(T) + cp(T),

(18)

where the first term on the rhs is the contribution of the normal curve to the specific heat, and the second one contains the contributions coming from the previous history. We use the prime to indicate that they refer to heating processes. From the explicit expressions, one obtains that c~(T) is positive, vanishing at T = 0, while C'p(T)< 0 for all T. It follows that C'a(T) ~< c~(T),

,

i

i

i

12

r

4J c, kB

c.(T) = de(T)/dT,

f

a

1

-8

~6

- - 4

In Fig. 4. Apparent specific heat in two continuous heating processes (( = exp( - 4J/kBT)). The heating program was the same in both cases, but the rate of the previous cooling was 1000 times smaller in one experiment (,) than in the other (o). Also plotted are the equilibrium curve ( ) and the specific heat along the normal curve ( , , ~ ) .

decreases. Besides, it moves towards lower temperatures. Both facts are nothing else but the manifestation of the apparent specific heat approach to the normal one. More specifically, it can be proved that lim C'p(T) --- 0, rc~o

d/drclc'p(T)[ > 0.

(20)

With respect to the width of the initial negative region, it is seen that it increases with r¢ and decreases with rh.

7. Conclusions

(19)

being, in particular, c'(0) < 0. Therefore, the apparent specific heat measured in a given continuous heating experiment is bounded by the specific heat measured along the corresponding normal curve. Besides, it presents an initial negative region. In Fig. 4 we show the apparent specific heat obtained in two heating experiments following a previous cooling of the system. In both cases, the system was heated in the same way, but the previous cooling rates were different. Also plotted are the equilibrium specific heat and the normal one. The apparent specific heat shows a maximum that tends to increase as the previous cooling rate

We have shown that very simple models exhibit many of the dynamical features of supercooled fluids and glasses. In particular, the K W W relaxation and the hysteresis effects appear as general properties of systems whose dynamics can be described by a master equation. To explain them one does not have to introduce complex relaxation mechanisms. For heating processes, the theory leads in a natural way to the introduction of the concept of normal curves, which we believe can be useful tools to analyze and classify experimental data. They provide information about the inherent dynamics of the system, with independence of the initial conditions.

J.J. Brey et al. / Journal q/" Non-Crystalline Solids' 172-174 (1994) 371 377

The generality of the results reported here indicates that this can be a fruitful way of approaching the study of relaxation in complex systems. On the other hand, one has to keep in mind that simple models like the one considered here cannot reflect all the complexity of real glasses and, in this sense, they are limited in their scope. We want to finish by noting that if, as Goldstein said [22], not every kinetic freezing out process can properly be called a glass transition, the crucial question, not answered yet, is which are the key features of glasses that a model must explain [23]. Partial support from the Direcci6n General de Investigaci6n Cientifica y T6cnica (Spain) through Grant No. PB92-0683 is gratefully acknowledged.

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[5] W. Kob and R. Schilling, Phys. Rev. A42 (1990) 2191. [-6] J.J. Brey and M.J. Ruiz-Montero, Phys. Rev. B44 (1991) 9259. [7] G.H. Fredrickson and H.C. Andersen, Phys. Rev. Lett. 53 (1984} 1244; G.H. Fredrickson and S.A. Brawer, J. Chem. Phys. 84 (1986) 3351. [8] T.A. Weber, G.H. Fredrickson and F.H. Stillinger, Phys. Rev. B34 (1986) 7641; T.A. Weber and F.H. Stillinger, Phys. Rev. B36 (1987) 7043. [9] R.G. Palmer, D.L. Stein, E. Abrahams and P.W. Anderson, Phys. Rev. Len. 53 (1984} 958. [-10] J. Jfickle and S. Eisinger, Z. Phys. B84 ( 1991 ) 115; J. Jfickle, these Proceedings, p. 104. [11] D.A. Huse and D.S. Fisher, Phys. Rev. Lett. 57 (1986) 2203. [12] J.E. Shore and R. Zwanzig, J. Chem. Phys. 63 (1975) 5445. [13] J.J. Brey and A. Prados, Phys. Rev. E47 (1993) 1541. [14] R.J. Glauber, J. Math. Phys. 4 (1963) 294. [15] J.J.Brey and A. Prados, Physica A197 (1993) 569. [16] J.J. Brey and J. Casado, J. Stat. Phys. 61 (1990) 713. [17] J.W. Dufty, A. Santos, J.J. Brey and R.F. Rodriguez, Phys. Rev. A33 (1986) 459. [18] J.J. Brey and A. Prados, Phys. Rev. B49 (1994) 984. [19] J.E. Anderson, J. Chem. Phys. 52 (1970) 2821: S. Bozdemir, Phys. Status Solidi B104 11980) 37. [20] K.L. Ngai, Commun. Solid State Phys. 9 (1979} 127; K.L. Ngai, R.W. Rendell, A.K. Rajagopal and S. Teitler, Ann, NY Acad. Sci. 484 (1986) 150. [21] G.W. Scherer, Relaxation in Glass and Composites (Wiley, New York, 1986). [22] M. Goldstein, in: Complex Systems, ed. K.E. Ngai and G.B. Wright (Government Printing Office, available from National Technical Information Service, 5825 Port Royal Road, Springfield, VA 22161, USA, 1985) p. 13. [23] PK. Gupta, Rev. Solid State Sci. 3 t1989) 221.