~ 1 Pergamon Solid State Communications, Vol. 90, No. 8, pp. 527-532, 1994

Elsevier Science Ltd Printed in Great Britain

0038-1098/94 $7.00 + .00

0038-1098(94)E0145-2

FINITE-SIZE EFFECTS ON THE GLASS TRANSITION TEMPERATURE

Allen Hunt

Department of Soil & Environmental Sciences, University of California, Riverside, CA 92521, USA

(Received 5 November 1993 by A.A. Maradudin)

An expression for the activation energy of the viscosity in the per- colative transport regime demonstrates its proportionality to the peak in a distribution of barrier heights as well as to the width of the distribution. Such an expression implies that a "blocking" (slower than average) rate is responsible for the macroscopic relaxation time. This concept has recently been shown to account for a large number off phenomena related to the glass transition. Here it is shown that the average of a glass transition temperature over a large number of very small systems must correspond to an average barrier height; consequently the average glass temperature is reduced by confinement in pores because the average barrier height is smaller than the "blocking" barrier.

1. INTRODUCTION

THE CHARACTERIZATION of transport proper- ties of strongly disordered liquids has been a rather dif- ficult problem. At high temperatures, where the disorder is not so large, transport can be characterized as dif- fusive; in other words the various local environments present in the liquid (at any time) are seen by each particle over a time scale required to establish steady- state conditions. One could call such a liquid ergodic. The Mode-Coupling [1-4] theories suggest a transition to non-ergodicity at the Mode-Coupling temperature, To, a transition which may be alleviated through the existence and importance of hopping transport. On the other hand, it has been argued that below Tc transport properties are fundamentally percolative [5-7] in charac- ter, i.e. that transport is spatially inhomogeneous, but repeatable, rather than spatially homogeneous and non-repeatable (diffusive transport) without, how- ever, assuming that microscopic transitions differ above and below To. Such a description is similar to saying that Tc corresponds to a transition to non- ergodic behavior, but without claiming that the non- ergodicity is "relieved" by alternate hopping channels.

There are two fundamental propositions associ- ated with the viewpoint that Tc represents a cross- over from diffusive to percolative transport:

(1) Transport requires "classical" hopping [6-8]

(over a barrier) on both sides of Tc, for which each individual microscopic transition has a rate, wg = r~ I = Vph exp -Eo./kT (here vph is a vibrational frequency, roughly 1012Hz, and Eg denotes the height of the energy barrier connecting state i and state j).

(2) Different means to describe the macroscopic properties are required [6, 9-11] depending on the ratio of the width, ~, of relevant barrier heights, to kT. For tr/kT < 10 effective medium theories (or hydrodynamic theories) which employ identical equations for every particle, and assume thereby homogeneity, are more appropriate, while for a/kT > 10 percolative theories are required.

Such a cross-over in the description of transport need not be accompanied by a change in structure, although slow structural changes usually accompany super-cooling, and the effect of the change in the description of transport may be to produce an onset in structural changes at Tc, if cooling is relatively slow. Nevertheless, such structural change appears to be unnecessary to the general description of super- cooled liquids. Distributions of single particle transi- tions for hopping "over a barrier" arise from the spread of local environments present at any instant in a liquid. Although the time-average of local environ- ments may be homogeneous, at low T the time required for the system to appear homogeneous may

527

528 THE GLASS TRANSITION T E M P E R A T U R E Vol. 90, No. 8

be much larger than the time scale required for transport, indeed for crystallization. In the low temperature limit, one can associate one unique distribution of possible elementary transitions for each particle in the "glass"; however at the tem- peratures of interest one must reckon with a situation intermediate between homogeneity and complete distinguishability. Because a cross-over from time- temperature superposition to a T-dependence of primary relaxation peak widths occurs at T c, and various transport properties decouple at the same T, it seems clear that the best choice to invoke the cross- over is at the mode-coupling T c.

In view of the discussion in the previous paragraph, one is led to the conclusion that the appropriate distribution of microscopic transition rates is calculable from the distribution of local environments which would be visible if one could take a snapshot of the liquid at any arbitrary time. Although a number of individual transitions might (in principle) be accessible to each individual particle, the fastest one available would probably occur long before any other possible motions, because the exponential factors in rij = u;h I e x p E i J k T (for T < T¢) would be so much smaller, that any other transition would occur with negligible probability. As a consequence, whatever distribution of transition rates is appropriate f o r each particle for T > T c is adopted as the system-wide distribution for T < To. In other words, each particle is assigned a particular environment from the ensemble of particle environ- ments available; the distribution of available environments is not assumed to change drastically at T c, (or any other T), so the number of particles with any particular environment corresponds to the probability that that particular local environment is found. In order to obtain transport properties as functions of temperature which yield quantitative agreement with experiment, two general effects with reduction of T will have to be considered, however: (1) a systematic increase in the density brings particles into closer proximity, increasing heights of energy barriers (even in the absence of relaxation processes), (2) microscopic transitions (e.g. pair relaxations) tend to lower the system energy, also tending to raise barrier heights.

In the case of dipoles in a viscous fluid, a distribution of barrier heights has been calculated [12]. The result was that the appropriate distribution should be a Gaussian. It may be [13, 14] that such a result is more generally appropriate, but even for various spatial distributions of electric (monopole) charges, a Gaussian distribution of electrostatic energies is not necessarily appropriate [15]. (What

the relationship is between a distribution of energies and a distribution of barriers in glasses is not known precisely anyway). But it is attractive to invoke a central-limit theorem type argument and expect a Gaussian distribution of barrier heights. In such a case, the parameters required to describe the distribution can be estimated by analyzing relation- ships between, e.g. the glass temperature and the melting temperature, Tm [16]. It seems a common condition that the width of the distribution, a, be roughly equal to the energy, E,,, of the barrier height at the distribution maximum. Strictly speaking, assumption of a Gaussian distribution with a ~ Em allows some (unphysical) negative energy barriers. However, it is known that these conditions lead to numerical results consistent with a number of experiments and they will be assumed to be approximately obeyed; they permit a calculation of numerical results for the finite-size dependence of Tg, which is also consistent with experiment. Still, the technique to actually calculate the appropriate distribution remains to be developed.

It should be noted that in usual effective-medium formulations of transport properties [17, 18], it is also necessary to provide distributions of barrier heights as an input function. But a basic problem is that even i ra distribution o f barrier heights is given, the state of percolation based theories for transport is (in the general case) far more primitive than effective medium theories, although the current sophistication is sufficient for the calculations given here. Never- theless, it has only recently become possible to make predictions for transport coefficients [6], while in the frequency domain, only the dielectric response has been treated [19, 20]. A percolation theory for the a.c. and d.c. conductivities (when the distribution of barrier heights can be independently obtained, i.e. from NSLR data) has been tested, and found to give good agreement with experiment in two ionic glassy systems [21, 22] (although a different value of the percolation probability was used). But in an inhomogeneous system, the relation of microscopic transitions to macroscopic quantities is not always unique; especially in the case of mechanical relaxation functions, ambiguity can arise if a probabilistic approach to time-dependent relaxation is taken. In dielectric relaxation, the motion of any charged particle a well-defined distance in a well-defined time contributes in a unique way to the macroscopic response (as a function of time, or frequency), but in mechanical relaxation, no such relationship exists, particularly when one is measuring the response of the system according to traditional oscillators, or in needle experiments. The motion of a particle between,

Vol. 90, No. 8

say, two particular sites, which are not in contact with the oscillator walls (or the needle) has a certain, in principle calculable, probability of affecting the motion of particles which are directly observed; but this probability depends on the type of motion which has occurred, as well as the environment in which it occurs (if not all environments are identical). The problem becomes a quite formidable exercise in percolation theory of hopping transport with a very high degree of correlation (but not high enough to employ hydrodynamic equations of motion).

However, it has been argued [6] that it is possible to obtain d.c. transport coefficients through judicious application off percolation conditions. In other words, conditions are deduced, which can define the appropriate barrier height for individual transport properties, and which therefore also define the associated time scales.

If such an inhomogeneous liquid contains charged particles, the d.c. resistivity has been argued [6] to be calculable from a condition defining the critical percolation of all transitions of charged particles with rates greater than or equal to vph exp - G . In this expression, t,ph is a fundamental (vibrational) fre- quency, while the random variable ~ is taken to be the ratio of a (random) barrier height E to kT. The value G is called the critical, or percolation, value. The d.c. resistivity is then [6]

Pat -- Po exp (c = P0 exp ~-~, (1)

where P0 is a prefactor (proportional to v;h ~) and

Eo

J dEn(E) 2.7 = T (2) 0

has been given as a condition to define the critical barrier height, Ep, in a distribution of barrier heights, n(E). Some uncertainty exists regarding the appro- priate fraction for percolation, some [17, 21, 22] prefer to use 1/4. But justification [21 ] of the value 1/4 based on agreement with experimental results for the a.c. conductivity may be inappropriate because the fit near the peak is affected by sequential [19] cor- relations, reducing the power of the conductivity, and making it appear as though a narrower distribution of barrier heights were involved. Such an impression could then lead to the choice of a cut-off (percolation) barrier which is too low, and also leave the physical causes of the higher frequency, smaller power regime [23, 24] unclear (a regime which corresponds to isolated pair transitions in [19]). While it is not the main pur- pose of this article to resolve such a debate, the ultimate conclusions will depend on the numbers obtained,

THE GLASS TRANSITION TEMPERATURE 529

and it must be made clear that the uncertainty exists.

The viscosity in the percolative transport regime is determined by a different condition. Requiring [6] that the cross-over to d.c. transport (non-local relaxation) occur when two-dimensional sheets, or cylindrical surfaces, can be translated relative to one another means that a much larger number of microscopic processes must be included to construct the relaxation surface, than in the case of dielectric relaxation, where a one-dimensional path suffices. The d.c. viscosity was found [6] to be

E~ r/d c = r/0 exp~---~, (3)

where

G 4.3

I dEn(E) = T ' (4) 0

Here ~ is a prefactor, whose value [25] is rather uncertain. In the calculations which follow, the glass temperature, Tg, is essentially found by setting the relaxation time for the viscosity equal to the experimental time. As a consequence, T s cx E,, and to the inverse of In rt0, and even large uncertainties in r/0 are not of great significance. The particular fraction, 4.3/6, was arrived at by requiring the separation of paths contributing to the macroscopic mechanical relaxation to be approximately the length of the individual hopping transitions on the paths.

Tg, was then found by setting % = v#expE~/ k T = texp, with v# roughly 1012Hz and texp the experimental time, taken arbitrarily [25] to be 100s. The result [26] was

G G (5) Tg ~ k In (texpV#) - 32k'

For example, a Gaussian distribution of barrier heights, n(E) cx exp [E - Em]2/2a 2, with Em ~ a the result,

1.75tr Tg = 3 2 k ' (6)

is obtained [l 7, 26] from equation (4). The approximate proportionality between the width of the Gaussian distribution and the value of the energy at its maximum value was chosen to limit negative energy barriers [16], and was found to be consistent with various tendencies in experiment (the tendency to correlate the glass temperature with the melting temperature [16], and with the extrapolated divergence [26] of the viscosity in Vogel-Fulcher phenomenology, etc.).

The above results should not be considered as having no (numerical) uncertainty. But for any

530 THE GLASS TRANSITION T E M P E R A T U R E Vol. 90, No. 8

numerical fraction larger than 3/6 = 0.5, the macro- scopic viscosity is determined by a "blocking" transition. And, in addition to the results mentioned, it has been shown [6] possible with the present numerical values to understand the decoupling of the viscosity and the resistivity at To the slowing of the bulk modulus relaxation in comparison with the shear modulus [27], the correlation of the logarithm of the decoupling index with the non- exponentiality of the dielectric relaxation [28], and the correlation of the limiting slope of the viscosity with the non-exponentiality of relaxation [29]. Here it will be shown that the finite-size dependence [30, 31] of the glass transition temperature is also interpret- able using the same results.

2. CALCULATION OF THE FINITE-SIZE EFFECT ON Tg

The origin of the largest curvature of lnr/vs l I T near T c, involves a cross-over from a typical rate to a blocking rate, slower than the mean value. The mean Tg (for a given cooling rate) of disordered liquids in pore spaces of size L is found from a mean relevant barrier height (the "blocking" tran- sition in individual pores is subject to a stochastic variation), and is reduced in comparison with Tg in bulk. In the limit L ~ r 0, a molecular size calculation of (Ts) must be given in terms of a mean barrier height, since in this (unattainable) limit, a single barrier is relevant to each pore. The calculation of (Tg) as a function of L is a difficult problem in stochastic processes; without precise, numerical knowledge of the macroscopic condition for Eo, it seems pointless at present to try to calculate (Tg(L)) exactly. But if the limits (Tg(ro)) and (Tg(oo)) are known, and if the basic physics of the confinement to pore spaces can be deduced, then a reasonable solution may be obtained. The confinement in pore spaces will clearly reduce the role of "blocking" transitions; any which would be found on the boundary layers will no longer prevent equilibration within an individual pore. In order to address this question, the statistics of the individual transitions is further discussed.

The mean separation of blocking transitions is a property only of the distribution chosen, and the result, equation (4), defining the blocking transition rate. Whether a large spread of such separations exists is also relevant to a stochastic description of relaxation in small systems, and relates to the maxi- mum spatial scale on which the system may be regarded as homogeneous. In traditional, non-interacting percolation theory, microscopic transitions are

assumed to be associated completely at random, with no correlations. The construction of a network of such random processes acquires fractal dimension near the percolation threshold [32]. Thus, ifa glass has a completely random distribution of microscopic transition rates, one can choose a value of the barrier height energy as a parameter; when that value is near the percolation threshold, the set of all the transitions faster than, or equal to the chosen transition rate form a fractal network. Of course, a small change in the chosen parameter changes the resulting fractal form, and can move the considered subset of the glass completely through the percolation transition. So, all structures, below, at and above critical percolation are present simultaneously in a random glass and macroscopic properties which involve the statistics and association of processes with rates near the critical (percolating) rate may relate (indirectly) to fractal structures. But statistics of microscopic transition rates of interacting (traditional) glasses are not random. There exist theoretical agruments as well as experimental evidence for homogeneity on length scales greater than 3-5 molecular separations. The experimental evidence consists of the uniformity of relaxation spectra derived from local probes which "average" over distances smaller than 3-5 "shells" [33], and the apparent explanation of anomalous light scattering in terms of fluctuations in density over similar length scales [34] (also approximately equal to the separation of blocking transitions calculated here, as will be seen). On the other hand, in ionic glasses, e.g. temperature-dependent screening of Gaussian charge fluctuations should introduce homogeneity over distances larger than a screening length; separ- ations of "blocking" transitions will not exceed maximum values (in a completely random glass, such separations have wide spreads about their mean values), and cut-offs [14] in distributions of barrier heights could result. The fraction of slow "rate- determining" transitions near T 8 calculated consistently with equation (4) is a ratio of the number of rate- determining transitions, % = v~h t exp E~/kT to all the transitions faster than (or equal to) %,

E~ + O.Sk r,

J dEexp [ - (E - Em)2/2a 2]

f = ~.,-0.Skr, • ~ kT J2 .5 o . E,

J dEexp [ - (E - E,,)2/20 ~1

0 (7)

The limits on the upper integral define f in terms of processes within one order of e = 2.718.. . of the rate-limiting transitions. Using equation (6) for T s in

Vol. 90, No. 8

terms of o one finds that the distance between such transitions is roughly,

[2.5a] i/3 X =f-'/3ro = C~--~'g ] r0 ~ 3.6r0, (8)

with r 0 the typical hopping length, or approximately the typical separation of molecules, or atoms constituting the disordered liquid. This value is also consistent with a recent observations that medium-range order extends to [35] approximately 3-5 inter-molecular distances.

So, when L is on the order of 2X (or possibly 3X), finite size effects should set on. In the limit L ~ r0

N

(Tg(t ~ r0) ) =/imoo E Elk In (tvph)]-' I = l

= [kln(tuph)] -1 i E n ( E ) d E , (9)

0

with N the number of pores and (Tg) is, as stated, proportional to a mean barrier height. Using the Gaussian form for n(E)

(T,) ~ Em+ 0.25a 1.25o (10) k In (tr,vh) ~ 32~

and (Tg)cx (E). A heuristic argument (noting the relevance of surface effects to results [30, 31] which scale as ro/L) for the dependence on L assumes that (otherwise) rate-limiting transitions located on pore boundaries are unnecessary for thermal equilibration of adjoining pores; different pores simply do not communicate. The fraction of rate-limiting processes on boundaries is equal to the fraction of all processes on the boundary, i.e.

47rL2r0 g - (4/3)7rL 3 oc ro/L. (l l)

Assume then that the fraction of pores with internal rate limiting microscopic transitions is proportional to the probability that a rate limiting process is in the interior. This assumption, reasonable for very small pores, is likely to be inaccurate for larger ones, but there the finite-size effect is rather small. Then

re(L) = re(r°) + k In (tllph)

with E found from Tg(ro) = [Era + 0.25~]/[k In (tVph)] and the result for T~r(oc), so

0.5tr r 0 (13) Tg - Tg(L) -- kin (wvh) L"

For any T, percolation and effective medium calculations of p yield rather similar values. This

THE GLASS TRANSITION TEMPERATURE 531

similarity is a consequence of the chosen fraction, 2.7/ 6; any treatment based on a typical relaxation time will not give a very different value (depending on the asymmetry of the distribution, an effective medium theory, including interactions, might yield a slower, or possibly even a faster relaxation). Thus the rate of dielectric relaxation is not in general enhanced in pore spaces. But dielectric relaxation at Tg is significantly slowed, since Tg is reduced.

The onset of the finite size dependence, its magnitude, its functional form, a tendency to slow dielectric relaxation at Tg, and the fact that the size dependence is stronger in more fragile systems (with larger a) are all verified experimentally [30, 31].

3. CONCLUSIONS

The following conclusions can be made:

(1) The same model which is consistent with experimental results for a wide range of relaxation data, from a.c. conductivity, to the pressure depen- dence of Tg, to the correlation of the limiting slope of the viscosity with the non-exponentiality of relax- ation, is consistent with the known features of the finite-size dependence of Tg.

(2) This model is based on the selection of differ- ent microscopic transition rates (from a distribution of rates) to describe various macroscopic transport properties. The selections are made on the basis of topo- logical conditions relating the motion of the glassy material to the boundary conditions of experiment.

(3) In general, different topological descriptions of the onset of steady-state response arise from the different experimental conditions accompanying measurement of various properties; in strongly inhomogeneous liquids, these different topologies may select different relevant barrier heights from a distribution of barriers, leading to greatly different macroscopic responses.

REFERENCES

1. W. Goetze & L. Sjoegren, Rep. Prog. Phys. 55, 241 (1992).

2. E. Leutheusser, Phys. Rev. A29, 2765 (1984). 3. U. Bengtzelius, W. Goetze & A. Sj61ander, J.

Phys. C17, 5915 (1984). 4. S. Dis, G.F. Mazenko, S. Ramaswamy & J.J.

Toner, Phys. Rev. Lett. 54, 118 (1985). 5. A. Hunt, Phil. &lag. !164, 563 (1991). 6. A. Hunt, Solid State Commun. 84, 701 (1992). 7. A. Hunt, Int. Conf. Hopping and Related

Phenomena, World Scientific (1993) (in press). 8. J. Dyre, J. Appl. Phys. 64, 2456 (1988). 9. S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).

532

10.

11.

THE GLASS TRANSITION TEMPERATURE Vol. 90, No. 8

C.H. Seager & G.E. Pike, Phys. Rev. BI0, 1435 (1974). M. Pollak, in Disordered Semiconductors (edited by M. Pollak), Chap 5b. CRC Press, Boca Raton (1987).

12. E.R. Grannan, M. Randeria & J.P. Sethina, Phys. Rev. 1141, 1492 (1991).

13. J.C. Dyre, Phys. Rev. Lett. 58, 792 (1987). 14. J.R. MacDonald, J. Appl. Phys. 62, R51 (1987). 15. A.L. Efros & B.I. Shklovskii, Electronic Pro-

cesses in Doped Semiconductors, Springer, Berlin (1984).

16. A. Hunt, J. Phys. CM4, L429 (1992). 17. J. Dyre, Phys. Rev. B48, 12511 (1993). 18. M. Wagener & W. Schirmacher, Ber. Bunsen-

ges. Phys. Chem. 95, 983 (1991). 19. A. Hunt, J. Phys. CM:3, 7831 (1991); J. Phys.

CM:4, 5371 (1992). 20. A. Hunt, J. Non-Cryst. Solids 160, R183

(1991). 21. I. Svare, F. Borsa, D.R. Torgeson & S.W.

Martin, Phys. Rev. 1348, 9336 (1993). 22. I. Svare, F. Borsa, D.R. Torgeson & S.W.

Martin, International Discussion Meeting on Relaxation in Complex Systems (Proceedings; in review) [as in refs. 24 & 27].

23. K.L. Ngai, U. Strom & O. Kanert, Phys. Chem. Glasses 33, 109 (1992).

24. A.S. Nowick, International Discussion Meeting on Relaxations in Complex Systems (Proceed- ings; in review).

25. W. Goetze, in Liquids, Freezing, and the Glass Transition (Edited by J.P. Hansen, D. Levesque & J. Zinn-Justin, North Holland, Amsterdam (1972).

26. A. Hunt, Solid State Commun. 88, 377 (1993). 27. T. Christenson & N.B. Olsen, Int. Discuss.

Meet. Relax. Complex Systems. 28. A. Hunt, J. Non-Cryst. Solids (1994) (in

press). 29. A. Hunt, J. Non-Cryst. Solids (1993) (in

review). 30. J. Zhang, G. Liu & J. Jonas, J. Phys. Chem. 96,

3478 (1992). 31. C.L. Jackson & G.B. McKenna, J. Non-Cryst.

Solids. 32. D. Stauffer, Phys. Rep. 54, 1 (1979). 33. R. Richert, Chem. Phys. Lett. 216, 223 (1993). 34. C.T. Moynihan & J. Schroeder, J. Non-Cryst.

Solids 160, 52 (1993). 35. S.R. Elliott, Journal de Physique IV (Colloque)

2, 279 (1992).