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Fragility of liquids using percolation-based transport theories
Journal of Non-Crystalline Solids 274 (2000) 93±101
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Section 7. Glass structure IV
Fragility of liquids using percolation-based transport theories Correlation between limiting slope of the viscosity and non-exponentiality of relaxation A.G. Hunt Atmospheric Sciences and Global Change Resources, Paci®c Northwest National Laboratory, Richland WA 99352, USA
Abstract Transport in super-cooled liquids becomes increasingly heterogeneous with reduction in temperature. This transport is assumed to occur by thermally activated hopping over barriers. At a temperature de®ned here to be Tc , transport becomes percolative rather than diusive, and, especially for mechanical relaxation, larger energy barriers must be surmounted in order to establish steady-state response. Use of the same distribution of energy barriers at high and low temperatures, but dierent theoretical approaches to calculate transport properties, relates expressions for the viscosity at high and low temperatures. Such an approach also allows calculation here of an expression relating the limiting slope, m, of the viscosity to the parameters of an assumed log-normal distribution of energy barriers. Careful calculation of the frequency dependence of transport properties in terms of this distribution of energy barriers also allows expression of the non-exponentiality parameter, b, in terms of the same parameters. Analytical calculations of both (within established approximation schemes) allows quantitative comparison between m and b. The analytical results are evaluated using typical frequencies and times from experiment. Agreement between experiment and calculations is fair. Nevertheless, sensitivity of the calculations to uncertainty in the values of the experimentally derived parameters makes it necessary to view the correspondence with caution. It is tentatively concluded that the major component of the curvature of the viscosity with decreasing temperature is indeed the change in transport type, and not changes in structure. Advances in theory as well as application of numerical methods are needed to help to reduce uncertainty in the conclusions. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The observed glass transition is a kinetic phenomenon [1,2]. The temperature-dependence of relaxation and transport phenomena in supercooled liquids, however, is quite intriguing, and, particularly in the case of `fragile' liquids, has been interpreted to imply the relevance of an underlying phase transition [3±5]. With diminishing tempera-
E-mail address: [email protected] (A.G. Hunt).
tures transport in fragile liquids becomes increasingly heterogeneous [6,7]. An increase in heterogeneity can lead to a cross-over in transport properties [7], and require (at this time, anyway) separate theoretical treatments on opposite sides of a temperature signi®cantly higher than the glass transition [7]. Changes associated with the transport cross-over include: maximum curvature of steady-state response times on Arrhenius plots (i.e., largest deviation from a simple, activated behavior), separation of bulk and mean relaxation times [8] (consistent with ®nite-size eects on the
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 0 6 - 4
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A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
glass transition temperature [9±13]), onset of nonergodicity with attendant loss of proportionality between resistivity and viscosity [14], increases in departure of relaxation from exponential, etc [1]. The temperature at which these changes set on has been called the mode-coupling temperature [15], but is interpreted here as signifying a change in the underlying transport mechanism from homogeneous to heterogeneous. If changes in transport produce temperature-dependence in the activation energy, then the relevance of temperature-dependent activation energy to structure is quite complex and inferences of an underlying phase transition [3±5] become problematic. Firstly, such an activation energy must be selected from a spectrum of molecular, ionic, or atomic activation energies [16±18]; secondly, the selection criterion changes with temperature [7]. A proposed theoretical treatment is based on a cross-over from eective-medium approaches at high temperatures to percolation type (critical rate) analysis at lower temperatures [7]. As speci®c justi®cation for such a proposal, I cite the observed proportionality of resistivity and viscosity at higher temperatures [14,19], and the loss of this proportionality [14] with diminishing T (interpreted in terms of a transport cross-over in [19]). Additional support is the relative success of percolation analysis in describing universal relaxation, pressure and concentration dependence of the glass transition temperature in ionic liquids [20,21], and ®nite-size eects on the glass transition [9±13,22]. The primary addition in this report concerns the distinction between strong and fragile liquids, and a possible explanation of the correlation between the fragility index and the non-exponentiality of relaxation [23]. The basis for changing theoretical approaches with reduction of T is actually simple [22], and does not require any change in structure, although structural changes certainly seem to accompany [24] the reduction in T in most systems. If the local environments would not change with reduction of temperature, allowing the individual particle energies, as well as barriers, E, for local rearrangements, to remain the same, the reduction of T would lead to signi®cant increase in the spread of values of E/kT. If transport occurs by hopping
over barriers [25] with relaxation times related to exp E=kT , the width of a distribution of relaxation times must increase greatly with reduction in T. Such an increase in heterogeneity of local transition rates leads to the following changes. Firstly, any theory of transport must be based on local heterogeneity. Secondly, when this heterogeneity becomes suciently large it introduces diering time scales of response and, portions of the system respond much more rapidly than others. Steady-state response may occur so rapidly along some `paths' that other regions of the system are unable to respond in the same time interval at all [22]. Thirdly, cross-over to percolation type descriptions of dynamics implies relevance of energy barriers larger than those from eective-medium theories, at least for the viscosity [7]. This increase in barrier height, occurring as it does with reduction in T, helps to account for the large increase in relaxation times with diminishing T. Fourthly, since dierent transport properties require dierent geometric adjustments (compare surfaces of viscous drag with paths of electrical charge transport) calculations of various transport properties must be accomplished separately, although the means to do so have not been universally established. These last two complications are related to the question of ergodicity, and the system becomes non-ergodic as the temperature is reduced [15]. While such liquids may be thought to be ergodic, in fact the lack of response of some regions to, e.g., an applied ®eld, is not consistent with the assumption that all portions of the system be equivalent. This lack of equivalence is a practical violation of ergodicity; while over an in®nite time period all positions in the liquid must be equivalent, over the time period establishing steady-state response equivalence is not observed. As a consequence, conventional transport/relaxation/ thermodynamic theories will fail. Fig. 1 is a schematic representation of the relationships between high and low temperature transport for two important transport properties, viscosity and resistivity. The cross-over typically occurs when transport speed has slowed about 10 orders of e 2:718, 4±5 orders of magnitude. An analogous cross-over is known in random impedance network simulations [26,27], and corresponds
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
Fig. 1. Schematic description of diusive and percolative transport regimes for the viscosity, g, and the resistivity, q. The region `Percolative with T-independent distribution' refers to classical hopping conduction in, e.g., ionic conducting glasses. The present paper focuses on the cross-over at Tc from high T diusive transport to low T percolative transport.
to the relevance to transport of percolation rather than eective-medium theories. A physical interpretation employs the concept of percolative, rather than diusive transport. The evidence shows that such heterogeneous transport and non-ergodicity become relevant at temperatures, T > Tg , the glass transition temperature; therefore theories for structural change near the glass transition cannot logically be based on equilibrium thermodynamics. Nor should even non-equilibrium thermodynamic predictions be compared with results of transport theories, which do not take heterogeneity into account. As a consequence, I argue that interpreting [3±5] Vogel± Tamman±Fulcher phenomenological descriptions of liquid dynamics in terms of a phase transition at the extrapolated divergence in relaxation times is meaningless. Correlations between high and low temperature transport (the reason Vogel±Fulcher phenomenology works reasonably well) have been explained using a cross-over in transport properties [28]. Further, as will be shown here, the relationship between fragility and non-exponentiality can also be understood with minimal impact from structural changes. As is also clear in this study, over the range of experimentally observed tem-
95
peratures, the large increase in characteristic relaxation times arises from an increase in activation energies typically of a factor 2 or so, and attributing such an increase to a hidden divergence is risky for many reasons. Since an important fraction at least, and perhaps nearly all of this increase, can be attributed to changes in transport, only a small fraction results from changes in structure. On the other hand, quantitative agreement between theoretical and experimental dependences of transport coecients in the vicinity of Tg cannot be generated if structural contributions to the temperature-dependence of the activation energy are overlooked, because some curvature in the viscosity continues all the way to the glass transition temperature [1]. Work based on percolation theory has not as yet directly addressed the issue of fragile vs. strong glass forming liquids. Using the description of the mixed-alkali eect as a model, I suggest that it will be possible to understand the distinction between fragile and strong glass-forming liquids by invoking statistical eects [29]. In the present case, the statistical eects are proposed to manifest themselves in a uniform additive constant to the activation energy of individual molecules (atoms, units). Thus the activation energy for relative motion of any particular atom or molecule is represented as, E E0 E0 , where E0 has a peaked distribution. E0 has been taken to be Gaussian distributed. Nevertheless, for large values of width to peak energy, a Gaussian form is not possible, and this question must be returned to. The contribution from E0 is assumed here to be statistical; consider a covalently bonded strong liquid, such as SiO2 . E0 then represents the bond breaking energy, which must be supplied in order to make a particular silicon tetrahedron mobile. In this case then, the width of the distribution of E0 , characterized by r, is very narrow, compared with E0 . The general type of motion described is analogous to trapping (re-formation of the bond). In the opposite limit, E0 vanishes, and the motion envisioned is more analogous to hopping in a disordered landscape. Allowing the relative magnitudes of E0 and r to vary permits description of a spectrum of liquids from strong to fragile, and transport to vary from trapping to hopping. It
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A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
should be noted that resulting transport calculations do not depend critically on the interpretation of E0 , only on the structure of the variable E0 and the magnitude of E0 . For calculation purposes it is dicult to distinguish between a statistical contribution, E0 , and E0 . In this case, E0 can simply be absorbed into the parameter, Em , describing the mean hopping energy. In this way the distinction between hopping and trapping is dropped. Such an ambiguity is consistent with an ambiguity in the pre-exponential of the transport coecient (because of the relevance of mobility, rather than an attempt frequency), but has no signi®cant eect on the exponential portion. Prior calculations, which typically assumed r=Em 1, were able to use the approximation that the distribution of activation energies was Gaussian. In some particular cases, such as random distributions of positive and negative charges, a Gaussian distribution of energy levels is justi®ed [30]. Here I also consider cases, where r=Em 1, and it becomes impossible to assume a Gaussian distribution. Under these conditions, the use of a log-normal distribution for activation energies might allow for an approximate description of the spectrum of strong to fragile liquids; when r=Em 1, fragile liquids are implied, but the opposite case, Em =r 1 is consistent with strong liquids. The form of the distribution assumed for r=Em 1 is proportional to expfÿ ln E=Em 2 = 2r2 g, so that Em and r are used analogously to the case of a normal distribution. Of course, here r2 is the variance of lnE, and is unitless. 2. Basic model equations A critically important aspect of percolative transport is its relationship with geometry. The selection, from a distribution, of a barrier height relevant for a steady-state response depends, in the percolative transport regime, on the property under investigation. This selection is geometrically motivated. As electrical conduction occurs along one-dimensional paths, dielectric relaxation times are de®ned using critical percolation [7]. On the other hand, shear occurs along surfaces, and the
connectivity of such surfaces requires a dierent criterion [7]. In dielectric relaxation individual particle transitions are organized into a path, which spans the system, thereby allowing steadystate response, which returns the system to its initial condition, but transports charge from one side to the other [7]. The analogous requirement for the viscosity requires that an entire surface be moved relative to another, but again with the stipulation that the system is returned to its initial state [7]. Construction of such a surface is accomplished from a superposition of one-dimensional paths such as provide for dielectric relaxation, but requires incorporation of many more transitions [7]. This perspective turns out to be a generalization of a proposal by Mott and coworkers [31]. Since dielectric relaxation preferentially utilizes the fastest individual transitions, the extra transitions required to construct an entire surface in relative motion make mechanical relaxation slower. The condition derived for the viscosity [7] implies that the rate-determining transition for the viscosity is much slower than the mean transition rate. This result has been explicitly veri®ed in experiments performed by McKenna [8], who found that at low temperatures the bulk mechanical relaxation times are slower than the average small system relaxation time. An important consequence of the faster mechanical relaxation is that the glass transition temperature is reduced in con®ned geometries, as derived [22]. While the experimental result was initially controversial, it has since been veri®ed by several groups [9±13]. An important corollary of the applicability of percolation theory is that the dielectric relaxation at the glass transition temperature is slowed in con®ned geometries, also veri®ed in experiments [10]. Finally, the difference between bulk and con®ned mechanical relaxation times also shows up when the system response has slowed about ®ve orders of magnitude [8], as predicted by a cross-over from diusive to percolative transport [7]. At temperatures higher than the regime of undercooling, the typical proportionality of resistivity and viscosity implies the relevance of eective-medium theories for the calculation of both [14]. This theoretical suggestion has been
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
subsequently veri®ed [32]. That a cross-over to percolative transport occurs with reduction in temperature is consistent with experiments [19]. Finally, one group of workers has succeeded using transport and magic angle spinning nuclear magnetic resonance (NMR) measurements in simultaneously observing the activation energy of the viscosity and that of breaking a single bond in silicate melts near the glass transition [33]. The energies were found to be the same [33]. If barriers to transport arise from the breaking of bonds of single molecules (or radicals), it is necessary to understand why the relevant bond energies tend to increase with diminishing T. So with the a posteriori justi®cation of the previous paragraph, the original predictions for the barriers relevant for the viscosity are reproduced, but adapted from a Gaussian distribution to a log-normal distribution of barrier heights (from Ref. [7]). In the present case, the viscosity near the glass transition is given by g g0 exp Eg =kT ;
1
where g0 is a pre-factor without exponential temperature-dependence, and, instead of Eg Em 0:75r, we ®nd in the same level of approximation Eg Em exp 0:75r:
2
In the high temperature regime, the high degree of cooperativity and the applicability of eectivemedium theories were assumed to make the viscosity an exponential function of the characteristic barrier height g g0 exp Em =kT :
3
The cross-over temperature is de®ned to be Tc .
3. Calculations Application of the two conditions on the relaxation times (sTg is increased by 14 orders of magnitude, sTc by 5, with the subscripts Tg and Tc referring to the relaxation times at Tg and Tc ) yields the following two equations:
Em exp 0:75r=kTg 14 ln 10 32:2
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4
and Em =kTc 5 ln10 11:5:
5
Calculation of the limiting slope of the viscosity at the glass transition is, in the present context, equivalent to calculating the slope of the viscosity between Tc and Tg , where the viscosity is taken to be Arrhenius in form [7,15] m log10 sTg =sTc =1 ÿ Tg =Tc :
6
Using Eqs. (4) and (5), I rewrite Eq. (6) as m 9=1 ÿ 0:357 exp 0:75r:
7
In Eq. (7), I have used the log-normal form for the distribution of activation energies because a Gaussian distribution does not make sense in the limit of large fragility, r=Em 1. From Eq. (7) it can be seen that a width of barrier heights Em exp 1:5 4:5Em is sucient to make m arbitrarily large. It is also possible to ®nd b as a function of r by evaluating the electrical conductivity as a function of frequency [34]. In dielectric relaxation the critical frequency, xc , which de®nes the peak in the imaginary part of the dielectric constant, is de®ned using the critical percolation condition, exactly as for any dc conductivity due to the same mechanism. xc also represents the lower boundary of the frequency range, over which b is typically determined. A second frequency, x0 , typically about three decades higher in frequency corresponds to the upper end of this frequency range, and can also be used to de®ne the onset of cluster relaxation processes. The simplest method for calculating the electrical conductivity at these two frequencies is the pair approximation [35]. The result from [35] is a slight modi®cation of the pair approximation. This modi®ed procedure takes into account enhancement of the polarization at low frequencies of relaxation due to correlated hopping. Such correlations increase the conductivity at the critical frequency by a factor, f, and thereby increase b. The approximation for the electrical conductivity, r x, typically yields an approximate power law in
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A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
the frequency domain, and this power, called s, is found from the following relationship:
and dielectric relaxation, respectively, can be written down.
s ln r x0 =r xc = ln x0 =xc :
b Tg lnf =6:9 0:29=r2 :
8
The relationship b 1 ÿ s is then used to ®nd b. Using Eq. (8), and the fact that, in the pair approximation, the conductivity is always proportional to a product of the frequency, x, and the distribution of barrier heights, evaluated at kT ln mph =x s ln fx0 W kT ln mph =x0 = f xc W kT ln mph =xc g= ln fx0 =xc g:
9
Using a log-normal distribution for W leads to ÿ1
ÿ1
b ln x0 =xc lnf ÿ 2r2 ln f ln mph =x0 = ln mph =xc 2
ln kT =Em ln mph =xc ln mph =x0 g:
10
Speci®c values for x0 and xc must be used for comparison with experiment. xc appears to have the same temperature-dependence as the corresponding steady-state transport property [16,17]. Thus, percolation theoretical treatments predict explicit systematic dierences in xc for dierent transport properties. For mechanical (shear) relaxation, ln mph =xc 32:2. Since dielectric relaxation is normally much faster, ln mph =xc can be as small as 9.2±11.5 (or even smaller in highly conducting glasses, for which, however, the barrier heights seen by those particles responding to the electric ®eld are clearly much smaller on average than those which respond to mechanical driving forces). For a value intermediate between 9.2 and 32.2 (in steps of 2.3) I choose 20.7. x0 is much harder to evaluate analytically. But x0 represents the upper end of the frequency ranges over which b is customarily measured. Since departures from stretched exponential response occur at higher frequencies, x0 is actually experimentally constrained to lie 3±5 decades higher than xc , depending partly on the experimental quantity determined. I use three decades higher for dielectric relaxation. This is not really an adjustable parameter, per se, but uncertainty in the value chosen is admitted. Using these values for x0 and xc , expressions for b Tg for mechanical
11
Inverting to solve for r, and then substituting into Eq. (7) for m, leads to 1=2
m 0:9=1ÿ0:357exp 0:750:29= bÿ lnf =6:9
:
12 Remember that correlated hopping has been assumed to be relevant [34]. The modi®cation of the expression Eq. (11), for s involves f, proportional to [6] the natural logarithm of the one-third power of Eg /kTg . Such correlated hopping is very important for generating [36] universal scaling of the dielectric constant [37]. Solution of Eq. (12) for b m yields b 1=3 ln 32:2=6:9 0:17= ln2 2:8 ÿ 25:2=m: 13 Comparison of the calculated and observed values (from Ref. [23]) is shown in Fig. 2. Note that Fig. 2 contains bs obtained from many dierent relaxation phenomena. One might expect systematic dierences in b with dierent phenomena. No calculations of frequency dependent relaxation functions have been attempted with percolation
Fig. 2. Results of calculation deriving the non-exponentiality parameter for relaxation, b, as a function of the limiting slope of the viscosity, m. Theoretical prediction of b is from the pair approximation for the conductivity, using a log-normal distribution of barrier heights with arbitrary parameters. Calculation of an approximate frequency-dependent power law yields the power, s, which is then approximated as 1 ÿ b.
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
theory other than dielectric, so no systematic variability in calculations of b is known. Not shown in Fig. 2 is the rather high sensitivity of the relationship between m and b to changes in x0 and xc . While the values chosen are within a range constrained by experiment, they are not the only choices possible. Most other choices yield poorer agreement, some yield better agreement. Thus the results are not robust, although they have not been optimized either. 4. Discussion It should be mentioned ®rst that choice of a Gaussian distribution of activation energies will not, in the present hypothesized relationship between percolation and diusion, generate results for b m compatible with experiment. In general, bs are much too small. This result is independent of the means chosen to calculate b, whether numerical [38], or the pair approximation. In either case, incorporation of sequential correlations in the hopping, while lessening the discrepancy between experiment and theory, is quite insucient to bring about compatibility (results can be derived analogously to the approach shown here). On the other hand, for the log-normal distribution, compatibility can be attained. The problem with the Gaussian distribution occurs for relatively large r, for which b is too small, as might be expected from the implication that under such conditions negative energy barriers would be common. In such cases, the appropriate distribution cannot be Gaussian. In the above calculations, it was tacitly assumed that individual molecules, atoms, ions, etc. were the hopping entities. On the other hand, there is nothing to prevent identi®cation of the transitions with clusters of molecules with dierent relaxation times. In other contexts, such as the calculation [20] of the pressure dependence of Tg in CKN [21], the relevant barrier heights were clearly demonstrated to be associated with hard-sphere repulsion under the eects of the Coulomb potential between individual ions, and the objects undergoing transport were single ions. Other investigators have found the same results in silicate glasses [33]. Further, the sizes of any correlated regions are
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restricted by the heterogeneous response in con®ned geometries to be less than 10 intermolecular separations [6,9±13,22]. Such a size restriction is consistent with assumption of relaxation and transport by individual molecules or ions, as well as a correlated region of linear dimension Eg =kTg 1=3 , in units of particle size [6]. The compatibility of the calculated and observed relationships between the limiting slope of the viscosity at the glass transition temperature and the non-exponentiality of relaxation, suggests that the majority of the change in activation energy with reduction in T is related to transport changes. Nevertheless, the lack of a quantitative agreement between percolation theory and experiment, together with the high degree of sensitivity to value of x0 chosen, makes it clear that either the theory is still inadequate, or that structural changes in¯uence the activation energy with reduction in T. This assertion can be understood as follows. Any tendency to increase the system order with reduction in T would increase the average strength of the bonds between particles. Such an increase in bond strength must increase the typical hopping energy. But the accompanying increase in order must reduce the width of the distribution of hopping energies. The viscosity is a bilinear function of the width and the typical barrier height [7] (for a Gaussian distribution), but increases faster with r than with Em in the log-normal case, as shown here. In either case, the viscosity does not change greatly due to a small increase in order, and could actually diminish. But the non-exponentiality of the relaxation, which is independent of the typical excitation energy, is signi®cantly reduced by an increase in order. Thus the fact that the derived b is still typically too small in a fairly wide range of m is suggestive of this diculty, and evidence for additional change in the activation energy due to changes in structure. 5. Conclusions Application of percolation theoretical methods to the electrical conductivity (and by approximate analogy, to other responses) together with the concept of a cross-over from diusive transport at
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high T to percolative transport at low T in supercooled liquids has been shown to be compatible with the correlation between the limiting slope of the viscosity and the non-exponentiality of relaxation. It appears to be productive to interpret the range of strong vs. fragile liquids in terms of the parameters of a distribution for individual particle excitation energies. When the width of the distribution is much larger than the peak value, the liquid is fragile. When the width is much smaller than the peak value, the liquid is strong. Previous work showing correlations between, e.g., the glass and melting temperatures used the ansatz [39] (which can be justi®ed in at least some individual cases [20]) that the width and peak values are frequently approximately equal, meaning that the classi®cations strong and fragile are endpoints in a spectrum whose most important classi®cation scheme is disorder, referred to an energy landscape. In the low temperature regime, changes in structure increase mean heights of barriers to transport but reduce the spread in barrier heights. Since, in this regime, both an increased spread in barrier heights and an increased mean, slow transport, the two eects oppose each other. Therefore, structural changes have a smaller eect on transport than otherwise expected, accentuating the role of transport accordingly. So much of the curvature in Arrhenius plots of the viscosity is due to the cross-over in transport types, from diusive to percolative. I think it is important that I attempt to answer a question posed by the referee, who doubts my assertion of the relevance of percolative transport to calculation of relaxation properties generally. The referee notes that Vogel±Fulcher forms for characteristic relaxation times in NMR, dielectric, and light scattering experiments are observed, although these properties do not measure transport, and thus interpretation in terms of changes of the mechanism of transport are not obvious. First I note that changes in activation energy with temperature are largest for properties, which involve transport, such as the viscosity. Moreover, in the theoretical description I have given here, dielectric relaxation is clearly related to long-range motion, or transport in many systems. Even in systems,
where there is no intrinsic dc conductivity, such as in dipolar liquids [37], the dielectric relaxation has been interpreted in terms of percolation theory [36]. Nevertheless, I admit that for such properties as dielectric relaxation of dipolar liquids, NMR, and light-scattering, interpretation in terms of percolation theory is never as straightforward as for properties clearly measuring long-range motion, and I appreciate the refereeÕs concerns. Finally, the choice of parameter values was not made for optimization of agreement with experiment, which clearly could be better. Nevertheless, values which lead to worse agreement, could also be used. The frequencies involved should ultimately be numerically determined on the basis of percolation arguments.
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Section 7. Glass structure IV
Fragility of liquids using percolation-based transport theories Correlation between limiting slope of the viscosity and non-exponentiality of relaxation A.G. Hunt Atmospheric Sciences and Global Change Resources, Paci®c Northwest National Laboratory, Richland WA 99352, USA
Abstract Transport in super-cooled liquids becomes increasingly heterogeneous with reduction in temperature. This transport is assumed to occur by thermally activated hopping over barriers. At a temperature de®ned here to be Tc , transport becomes percolative rather than diusive, and, especially for mechanical relaxation, larger energy barriers must be surmounted in order to establish steady-state response. Use of the same distribution of energy barriers at high and low temperatures, but dierent theoretical approaches to calculate transport properties, relates expressions for the viscosity at high and low temperatures. Such an approach also allows calculation here of an expression relating the limiting slope, m, of the viscosity to the parameters of an assumed log-normal distribution of energy barriers. Careful calculation of the frequency dependence of transport properties in terms of this distribution of energy barriers also allows expression of the non-exponentiality parameter, b, in terms of the same parameters. Analytical calculations of both (within established approximation schemes) allows quantitative comparison between m and b. The analytical results are evaluated using typical frequencies and times from experiment. Agreement between experiment and calculations is fair. Nevertheless, sensitivity of the calculations to uncertainty in the values of the experimentally derived parameters makes it necessary to view the correspondence with caution. It is tentatively concluded that the major component of the curvature of the viscosity with decreasing temperature is indeed the change in transport type, and not changes in structure. Advances in theory as well as application of numerical methods are needed to help to reduce uncertainty in the conclusions. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The observed glass transition is a kinetic phenomenon [1,2]. The temperature-dependence of relaxation and transport phenomena in supercooled liquids, however, is quite intriguing, and, particularly in the case of `fragile' liquids, has been interpreted to imply the relevance of an underlying phase transition [3±5]. With diminishing tempera-
E-mail address: [email protected] (A.G. Hunt).
tures transport in fragile liquids becomes increasingly heterogeneous [6,7]. An increase in heterogeneity can lead to a cross-over in transport properties [7], and require (at this time, anyway) separate theoretical treatments on opposite sides of a temperature signi®cantly higher than the glass transition [7]. Changes associated with the transport cross-over include: maximum curvature of steady-state response times on Arrhenius plots (i.e., largest deviation from a simple, activated behavior), separation of bulk and mean relaxation times [8] (consistent with ®nite-size eects on the
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 0 6 - 4
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glass transition temperature [9±13]), onset of nonergodicity with attendant loss of proportionality between resistivity and viscosity [14], increases in departure of relaxation from exponential, etc [1]. The temperature at which these changes set on has been called the mode-coupling temperature [15], but is interpreted here as signifying a change in the underlying transport mechanism from homogeneous to heterogeneous. If changes in transport produce temperature-dependence in the activation energy, then the relevance of temperature-dependent activation energy to structure is quite complex and inferences of an underlying phase transition [3±5] become problematic. Firstly, such an activation energy must be selected from a spectrum of molecular, ionic, or atomic activation energies [16±18]; secondly, the selection criterion changes with temperature [7]. A proposed theoretical treatment is based on a cross-over from eective-medium approaches at high temperatures to percolation type (critical rate) analysis at lower temperatures [7]. As speci®c justi®cation for such a proposal, I cite the observed proportionality of resistivity and viscosity at higher temperatures [14,19], and the loss of this proportionality [14] with diminishing T (interpreted in terms of a transport cross-over in [19]). Additional support is the relative success of percolation analysis in describing universal relaxation, pressure and concentration dependence of the glass transition temperature in ionic liquids [20,21], and ®nite-size eects on the glass transition [9±13,22]. The primary addition in this report concerns the distinction between strong and fragile liquids, and a possible explanation of the correlation between the fragility index and the non-exponentiality of relaxation [23]. The basis for changing theoretical approaches with reduction of T is actually simple [22], and does not require any change in structure, although structural changes certainly seem to accompany [24] the reduction in T in most systems. If the local environments would not change with reduction of temperature, allowing the individual particle energies, as well as barriers, E, for local rearrangements, to remain the same, the reduction of T would lead to signi®cant increase in the spread of values of E/kT. If transport occurs by hopping
over barriers [25] with relaxation times related to exp E=kT , the width of a distribution of relaxation times must increase greatly with reduction in T. Such an increase in heterogeneity of local transition rates leads to the following changes. Firstly, any theory of transport must be based on local heterogeneity. Secondly, when this heterogeneity becomes suciently large it introduces diering time scales of response and, portions of the system respond much more rapidly than others. Steady-state response may occur so rapidly along some `paths' that other regions of the system are unable to respond in the same time interval at all [22]. Thirdly, cross-over to percolation type descriptions of dynamics implies relevance of energy barriers larger than those from eective-medium theories, at least for the viscosity [7]. This increase in barrier height, occurring as it does with reduction in T, helps to account for the large increase in relaxation times with diminishing T. Fourthly, since dierent transport properties require dierent geometric adjustments (compare surfaces of viscous drag with paths of electrical charge transport) calculations of various transport properties must be accomplished separately, although the means to do so have not been universally established. These last two complications are related to the question of ergodicity, and the system becomes non-ergodic as the temperature is reduced [15]. While such liquids may be thought to be ergodic, in fact the lack of response of some regions to, e.g., an applied ®eld, is not consistent with the assumption that all portions of the system be equivalent. This lack of equivalence is a practical violation of ergodicity; while over an in®nite time period all positions in the liquid must be equivalent, over the time period establishing steady-state response equivalence is not observed. As a consequence, conventional transport/relaxation/ thermodynamic theories will fail. Fig. 1 is a schematic representation of the relationships between high and low temperature transport for two important transport properties, viscosity and resistivity. The cross-over typically occurs when transport speed has slowed about 10 orders of e 2:718, 4±5 orders of magnitude. An analogous cross-over is known in random impedance network simulations [26,27], and corresponds
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
Fig. 1. Schematic description of diusive and percolative transport regimes for the viscosity, g, and the resistivity, q. The region `Percolative with T-independent distribution' refers to classical hopping conduction in, e.g., ionic conducting glasses. The present paper focuses on the cross-over at Tc from high T diusive transport to low T percolative transport.
to the relevance to transport of percolation rather than eective-medium theories. A physical interpretation employs the concept of percolative, rather than diusive transport. The evidence shows that such heterogeneous transport and non-ergodicity become relevant at temperatures, T > Tg , the glass transition temperature; therefore theories for structural change near the glass transition cannot logically be based on equilibrium thermodynamics. Nor should even non-equilibrium thermodynamic predictions be compared with results of transport theories, which do not take heterogeneity into account. As a consequence, I argue that interpreting [3±5] Vogel± Tamman±Fulcher phenomenological descriptions of liquid dynamics in terms of a phase transition at the extrapolated divergence in relaxation times is meaningless. Correlations between high and low temperature transport (the reason Vogel±Fulcher phenomenology works reasonably well) have been explained using a cross-over in transport properties [28]. Further, as will be shown here, the relationship between fragility and non-exponentiality can also be understood with minimal impact from structural changes. As is also clear in this study, over the range of experimentally observed tem-
95
peratures, the large increase in characteristic relaxation times arises from an increase in activation energies typically of a factor 2 or so, and attributing such an increase to a hidden divergence is risky for many reasons. Since an important fraction at least, and perhaps nearly all of this increase, can be attributed to changes in transport, only a small fraction results from changes in structure. On the other hand, quantitative agreement between theoretical and experimental dependences of transport coecients in the vicinity of Tg cannot be generated if structural contributions to the temperature-dependence of the activation energy are overlooked, because some curvature in the viscosity continues all the way to the glass transition temperature [1]. Work based on percolation theory has not as yet directly addressed the issue of fragile vs. strong glass forming liquids. Using the description of the mixed-alkali eect as a model, I suggest that it will be possible to understand the distinction between fragile and strong glass-forming liquids by invoking statistical eects [29]. In the present case, the statistical eects are proposed to manifest themselves in a uniform additive constant to the activation energy of individual molecules (atoms, units). Thus the activation energy for relative motion of any particular atom or molecule is represented as, E E0 E0 , where E0 has a peaked distribution. E0 has been taken to be Gaussian distributed. Nevertheless, for large values of width to peak energy, a Gaussian form is not possible, and this question must be returned to. The contribution from E0 is assumed here to be statistical; consider a covalently bonded strong liquid, such as SiO2 . E0 then represents the bond breaking energy, which must be supplied in order to make a particular silicon tetrahedron mobile. In this case then, the width of the distribution of E0 , characterized by r, is very narrow, compared with E0 . The general type of motion described is analogous to trapping (re-formation of the bond). In the opposite limit, E0 vanishes, and the motion envisioned is more analogous to hopping in a disordered landscape. Allowing the relative magnitudes of E0 and r to vary permits description of a spectrum of liquids from strong to fragile, and transport to vary from trapping to hopping. It
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should be noted that resulting transport calculations do not depend critically on the interpretation of E0 , only on the structure of the variable E0 and the magnitude of E0 . For calculation purposes it is dicult to distinguish between a statistical contribution, E0 , and E0 . In this case, E0 can simply be absorbed into the parameter, Em , describing the mean hopping energy. In this way the distinction between hopping and trapping is dropped. Such an ambiguity is consistent with an ambiguity in the pre-exponential of the transport coecient (because of the relevance of mobility, rather than an attempt frequency), but has no signi®cant eect on the exponential portion. Prior calculations, which typically assumed r=Em 1, were able to use the approximation that the distribution of activation energies was Gaussian. In some particular cases, such as random distributions of positive and negative charges, a Gaussian distribution of energy levels is justi®ed [30]. Here I also consider cases, where r=Em 1, and it becomes impossible to assume a Gaussian distribution. Under these conditions, the use of a log-normal distribution for activation energies might allow for an approximate description of the spectrum of strong to fragile liquids; when r=Em 1, fragile liquids are implied, but the opposite case, Em =r 1 is consistent with strong liquids. The form of the distribution assumed for r=Em 1 is proportional to expfÿ ln E=Em 2 = 2r2 g, so that Em and r are used analogously to the case of a normal distribution. Of course, here r2 is the variance of lnE, and is unitless. 2. Basic model equations A critically important aspect of percolative transport is its relationship with geometry. The selection, from a distribution, of a barrier height relevant for a steady-state response depends, in the percolative transport regime, on the property under investigation. This selection is geometrically motivated. As electrical conduction occurs along one-dimensional paths, dielectric relaxation times are de®ned using critical percolation [7]. On the other hand, shear occurs along surfaces, and the
connectivity of such surfaces requires a dierent criterion [7]. In dielectric relaxation individual particle transitions are organized into a path, which spans the system, thereby allowing steadystate response, which returns the system to its initial condition, but transports charge from one side to the other [7]. The analogous requirement for the viscosity requires that an entire surface be moved relative to another, but again with the stipulation that the system is returned to its initial state [7]. Construction of such a surface is accomplished from a superposition of one-dimensional paths such as provide for dielectric relaxation, but requires incorporation of many more transitions [7]. This perspective turns out to be a generalization of a proposal by Mott and coworkers [31]. Since dielectric relaxation preferentially utilizes the fastest individual transitions, the extra transitions required to construct an entire surface in relative motion make mechanical relaxation slower. The condition derived for the viscosity [7] implies that the rate-determining transition for the viscosity is much slower than the mean transition rate. This result has been explicitly veri®ed in experiments performed by McKenna [8], who found that at low temperatures the bulk mechanical relaxation times are slower than the average small system relaxation time. An important consequence of the faster mechanical relaxation is that the glass transition temperature is reduced in con®ned geometries, as derived [22]. While the experimental result was initially controversial, it has since been veri®ed by several groups [9±13]. An important corollary of the applicability of percolation theory is that the dielectric relaxation at the glass transition temperature is slowed in con®ned geometries, also veri®ed in experiments [10]. Finally, the difference between bulk and con®ned mechanical relaxation times also shows up when the system response has slowed about ®ve orders of magnitude [8], as predicted by a cross-over from diusive to percolative transport [7]. At temperatures higher than the regime of undercooling, the typical proportionality of resistivity and viscosity implies the relevance of eective-medium theories for the calculation of both [14]. This theoretical suggestion has been
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
subsequently veri®ed [32]. That a cross-over to percolative transport occurs with reduction in temperature is consistent with experiments [19]. Finally, one group of workers has succeeded using transport and magic angle spinning nuclear magnetic resonance (NMR) measurements in simultaneously observing the activation energy of the viscosity and that of breaking a single bond in silicate melts near the glass transition [33]. The energies were found to be the same [33]. If barriers to transport arise from the breaking of bonds of single molecules (or radicals), it is necessary to understand why the relevant bond energies tend to increase with diminishing T. So with the a posteriori justi®cation of the previous paragraph, the original predictions for the barriers relevant for the viscosity are reproduced, but adapted from a Gaussian distribution to a log-normal distribution of barrier heights (from Ref. [7]). In the present case, the viscosity near the glass transition is given by g g0 exp Eg =kT ;
1
where g0 is a pre-factor without exponential temperature-dependence, and, instead of Eg Em 0:75r, we ®nd in the same level of approximation Eg Em exp 0:75r:
2
In the high temperature regime, the high degree of cooperativity and the applicability of eectivemedium theories were assumed to make the viscosity an exponential function of the characteristic barrier height g g0 exp Em =kT :
3
The cross-over temperature is de®ned to be Tc .
3. Calculations Application of the two conditions on the relaxation times (sTg is increased by 14 orders of magnitude, sTc by 5, with the subscripts Tg and Tc referring to the relaxation times at Tg and Tc ) yields the following two equations:
Em exp 0:75r=kTg 14 ln 10 32:2
97
4
and Em =kTc 5 ln10 11:5:
5
Calculation of the limiting slope of the viscosity at the glass transition is, in the present context, equivalent to calculating the slope of the viscosity between Tc and Tg , where the viscosity is taken to be Arrhenius in form [7,15] m log10 sTg =sTc =1 ÿ Tg =Tc :
6
Using Eqs. (4) and (5), I rewrite Eq. (6) as m 9=1 ÿ 0:357 exp 0:75r:
7
In Eq. (7), I have used the log-normal form for the distribution of activation energies because a Gaussian distribution does not make sense in the limit of large fragility, r=Em 1. From Eq. (7) it can be seen that a width of barrier heights Em exp 1:5 4:5Em is sucient to make m arbitrarily large. It is also possible to ®nd b as a function of r by evaluating the electrical conductivity as a function of frequency [34]. In dielectric relaxation the critical frequency, xc , which de®nes the peak in the imaginary part of the dielectric constant, is de®ned using the critical percolation condition, exactly as for any dc conductivity due to the same mechanism. xc also represents the lower boundary of the frequency range, over which b is typically determined. A second frequency, x0 , typically about three decades higher in frequency corresponds to the upper end of this frequency range, and can also be used to de®ne the onset of cluster relaxation processes. The simplest method for calculating the electrical conductivity at these two frequencies is the pair approximation [35]. The result from [35] is a slight modi®cation of the pair approximation. This modi®ed procedure takes into account enhancement of the polarization at low frequencies of relaxation due to correlated hopping. Such correlations increase the conductivity at the critical frequency by a factor, f, and thereby increase b. The approximation for the electrical conductivity, r x, typically yields an approximate power law in
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the frequency domain, and this power, called s, is found from the following relationship:
and dielectric relaxation, respectively, can be written down.
s ln r x0 =r xc = ln x0 =xc :
b Tg lnf =6:9 0:29=r2 :
8
The relationship b 1 ÿ s is then used to ®nd b. Using Eq. (8), and the fact that, in the pair approximation, the conductivity is always proportional to a product of the frequency, x, and the distribution of barrier heights, evaluated at kT ln mph =x s ln fx0 W kT ln mph =x0 = f xc W kT ln mph =xc g= ln fx0 =xc g:
9
Using a log-normal distribution for W leads to ÿ1
ÿ1
b ln x0 =xc lnf ÿ 2r2 ln f ln mph =x0 = ln mph =xc 2
ln kT =Em ln mph =xc ln mph =x0 g:
10
Speci®c values for x0 and xc must be used for comparison with experiment. xc appears to have the same temperature-dependence as the corresponding steady-state transport property [16,17]. Thus, percolation theoretical treatments predict explicit systematic dierences in xc for dierent transport properties. For mechanical (shear) relaxation, ln mph =xc 32:2. Since dielectric relaxation is normally much faster, ln mph =xc can be as small as 9.2±11.5 (or even smaller in highly conducting glasses, for which, however, the barrier heights seen by those particles responding to the electric ®eld are clearly much smaller on average than those which respond to mechanical driving forces). For a value intermediate between 9.2 and 32.2 (in steps of 2.3) I choose 20.7. x0 is much harder to evaluate analytically. But x0 represents the upper end of the frequency ranges over which b is customarily measured. Since departures from stretched exponential response occur at higher frequencies, x0 is actually experimentally constrained to lie 3±5 decades higher than xc , depending partly on the experimental quantity determined. I use three decades higher for dielectric relaxation. This is not really an adjustable parameter, per se, but uncertainty in the value chosen is admitted. Using these values for x0 and xc , expressions for b Tg for mechanical
11
Inverting to solve for r, and then substituting into Eq. (7) for m, leads to 1=2
m 0:9=1ÿ0:357exp 0:750:29= bÿ lnf =6:9
:
12 Remember that correlated hopping has been assumed to be relevant [34]. The modi®cation of the expression Eq. (11), for s involves f, proportional to [6] the natural logarithm of the one-third power of Eg /kTg . Such correlated hopping is very important for generating [36] universal scaling of the dielectric constant [37]. Solution of Eq. (12) for b m yields b 1=3 ln 32:2=6:9 0:17= ln2 2:8 ÿ 25:2=m: 13 Comparison of the calculated and observed values (from Ref. [23]) is shown in Fig. 2. Note that Fig. 2 contains bs obtained from many dierent relaxation phenomena. One might expect systematic dierences in b with dierent phenomena. No calculations of frequency dependent relaxation functions have been attempted with percolation
Fig. 2. Results of calculation deriving the non-exponentiality parameter for relaxation, b, as a function of the limiting slope of the viscosity, m. Theoretical prediction of b is from the pair approximation for the conductivity, using a log-normal distribution of barrier heights with arbitrary parameters. Calculation of an approximate frequency-dependent power law yields the power, s, which is then approximated as 1 ÿ b.
A.G. Hunt / Journal of Non-Crystalline Solids 274 (2000) 93±101
theory other than dielectric, so no systematic variability in calculations of b is known. Not shown in Fig. 2 is the rather high sensitivity of the relationship between m and b to changes in x0 and xc . While the values chosen are within a range constrained by experiment, they are not the only choices possible. Most other choices yield poorer agreement, some yield better agreement. Thus the results are not robust, although they have not been optimized either. 4. Discussion It should be mentioned ®rst that choice of a Gaussian distribution of activation energies will not, in the present hypothesized relationship between percolation and diusion, generate results for b m compatible with experiment. In general, bs are much too small. This result is independent of the means chosen to calculate b, whether numerical [38], or the pair approximation. In either case, incorporation of sequential correlations in the hopping, while lessening the discrepancy between experiment and theory, is quite insucient to bring about compatibility (results can be derived analogously to the approach shown here). On the other hand, for the log-normal distribution, compatibility can be attained. The problem with the Gaussian distribution occurs for relatively large r, for which b is too small, as might be expected from the implication that under such conditions negative energy barriers would be common. In such cases, the appropriate distribution cannot be Gaussian. In the above calculations, it was tacitly assumed that individual molecules, atoms, ions, etc. were the hopping entities. On the other hand, there is nothing to prevent identi®cation of the transitions with clusters of molecules with dierent relaxation times. In other contexts, such as the calculation [20] of the pressure dependence of Tg in CKN [21], the relevant barrier heights were clearly demonstrated to be associated with hard-sphere repulsion under the eects of the Coulomb potential between individual ions, and the objects undergoing transport were single ions. Other investigators have found the same results in silicate glasses [33]. Further, the sizes of any correlated regions are
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restricted by the heterogeneous response in con®ned geometries to be less than 10 intermolecular separations [6,9±13,22]. Such a size restriction is consistent with assumption of relaxation and transport by individual molecules or ions, as well as a correlated region of linear dimension Eg =kTg 1=3 , in units of particle size [6]. The compatibility of the calculated and observed relationships between the limiting slope of the viscosity at the glass transition temperature and the non-exponentiality of relaxation, suggests that the majority of the change in activation energy with reduction in T is related to transport changes. Nevertheless, the lack of a quantitative agreement between percolation theory and experiment, together with the high degree of sensitivity to value of x0 chosen, makes it clear that either the theory is still inadequate, or that structural changes in¯uence the activation energy with reduction in T. This assertion can be understood as follows. Any tendency to increase the system order with reduction in T would increase the average strength of the bonds between particles. Such an increase in bond strength must increase the typical hopping energy. But the accompanying increase in order must reduce the width of the distribution of hopping energies. The viscosity is a bilinear function of the width and the typical barrier height [7] (for a Gaussian distribution), but increases faster with r than with Em in the log-normal case, as shown here. In either case, the viscosity does not change greatly due to a small increase in order, and could actually diminish. But the non-exponentiality of the relaxation, which is independent of the typical excitation energy, is signi®cantly reduced by an increase in order. Thus the fact that the derived b is still typically too small in a fairly wide range of m is suggestive of this diculty, and evidence for additional change in the activation energy due to changes in structure. 5. Conclusions Application of percolation theoretical methods to the electrical conductivity (and by approximate analogy, to other responses) together with the concept of a cross-over from diusive transport at
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high T to percolative transport at low T in supercooled liquids has been shown to be compatible with the correlation between the limiting slope of the viscosity and the non-exponentiality of relaxation. It appears to be productive to interpret the range of strong vs. fragile liquids in terms of the parameters of a distribution for individual particle excitation energies. When the width of the distribution is much larger than the peak value, the liquid is fragile. When the width is much smaller than the peak value, the liquid is strong. Previous work showing correlations between, e.g., the glass and melting temperatures used the ansatz [39] (which can be justi®ed in at least some individual cases [20]) that the width and peak values are frequently approximately equal, meaning that the classi®cations strong and fragile are endpoints in a spectrum whose most important classi®cation scheme is disorder, referred to an energy landscape. In the low temperature regime, changes in structure increase mean heights of barriers to transport but reduce the spread in barrier heights. Since, in this regime, both an increased spread in barrier heights and an increased mean, slow transport, the two eects oppose each other. Therefore, structural changes have a smaller eect on transport than otherwise expected, accentuating the role of transport accordingly. So much of the curvature in Arrhenius plots of the viscosity is due to the cross-over in transport types, from diusive to percolative. I think it is important that I attempt to answer a question posed by the referee, who doubts my assertion of the relevance of percolative transport to calculation of relaxation properties generally. The referee notes that Vogel±Fulcher forms for characteristic relaxation times in NMR, dielectric, and light scattering experiments are observed, although these properties do not measure transport, and thus interpretation in terms of changes of the mechanism of transport are not obvious. First I note that changes in activation energy with temperature are largest for properties, which involve transport, such as the viscosity. Moreover, in the theoretical description I have given here, dielectric relaxation is clearly related to long-range motion, or transport in many systems. Even in systems,
where there is no intrinsic dc conductivity, such as in dipolar liquids [37], the dielectric relaxation has been interpreted in terms of percolation theory [36]. Nevertheless, I admit that for such properties as dielectric relaxation of dipolar liquids, NMR, and light-scattering, interpretation in terms of percolation theory is never as straightforward as for properties clearly measuring long-range motion, and I appreciate the refereeÕs concerns. Finally, the choice of parameter values was not made for optimization of agreement with experiment, which clearly could be better. Nevertheless, values which lead to worse agreement, could also be used. The frequencies involved should ultimately be numerically determined on the basis of percolation arguments.
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