J. Phys. Chem. 1994, 98, 662-669
662
Random-Walk Approach to Dynamic and Thermodynamic Properties of Supercooled Melts. 1. Viscosity and Average Relaxation Times in Strong and Fragile Liquids Vladimir I. Arkhipovf and Heinz BWler’ Fachbereich Physikalische Chemie und Zentrum f“urMaterialwissenschaften, Philipps- Universitiit, 0-35032 Marburg, Germany Received: July 6, 1993’
A random-walk concept of structural units in liquids is used to describe mechanical and thermodynamic properties of supercooled melts. Cooperative nonelastic excitations are considered as jumps of structural units in configurational space implying that the energetic distribution of metastable states that can be occupied by the structural units (density of possible metastable states (DPMS) function) is the principal characteristic feature of a particular liquid. Specific DPMS functions are proposed for both “strong” and “fragile” liquids. Overbarrier jumps of structural units involving bond breaking are assumed to be rate-limiting in the case of strong liquids while in weakly bonded fragile liquids jumps of structural units may occur without bond breaking. The model is able to explain (i) the non-Arrhenius and Arrhenius-like temperature dependences of viscosity for fragile and strong liquids, respectively, (ii) the dependence of the glass-transition temperature upon the cooling rate, and (iii) the temperature dependences of thermodynamic functions (free energy, entropy, etc.) for both strong and fragile liquids in a quantitative fashion.
Introduction Supercooled liquids are far from equilibrium and therefore are not in a thermodynamicallydefined state. Their properties depend upon the time scale of experiment which, in turn, is temperaturedependent. These facts indicate the existence of intrinsic relaxations associated with broad distributions of characteristic times and activation energies1-3 According to the temperature dependence of dynamical propertiessuch as viscous flow supercooledliquids can be classified as “strong” and “fragile”.4 Strong systems display an Arrheniustype temperaturedependence of the viscosity and relatively small changes in heat capacity at the glass-transition temperature T, while the fragile systems are characterized by pronounced deviationsfrom Arrhenius behavior and large ratios of liquid and glass heat capacities at Tg.2-uStrong liquids usually exist in the form of reinforcingtetrahedral network structures whereas fragile liquids do not possess directional bonds and often have ionic or aromatic character. Any theory of dynamic and thermodynamic properties of supercooled liquids must explain (i) the non-Arrhenius as well as Arrhenius-like temperature dependences of transport coefficients and averaged relaxation times for fragile and strong liquids, respectively; (ii) the dependence of dynamic properties of supercooled melts and glasses upon the thermal history of the sample, in particular, the dependence of the glass-transition temperature upon the cooling rate; and (iii) the occurrence of a broad distribution of relaxation times and characteristic frequencies for weakly bonded glasses and almost exponential relaxation functions for strongly bonded glasses. An often used framework for rationalizing the temperature dependence of transport coefficients in supercooled melts is the free volume concept.1J-9 It relates transport of constituting elements of the liquid to the exchange of free volume among the “cells” of the structure. Its central assumption is that if the free volume exceeds a critical value the excess free volume can be considered as free in the sense that its redistribution among the cells does not cost energy implying that the temperature dependence of properties like viscosity are solely determined by t On leave from the Moscow Engineering Physics Institute, Kashirskoye shosse 31. Moscow 115409, Russia. Abstract published in Aduance ACS Absrracrs. December 15, 1993.
the energy required to generate the excess free volume. A statistical treatment leads to the well-known Vogel-Fulcher law or its equivalent the Williams-Landel-Ferry equation. The attractive feature about this formalism is its ability to explain the temperature dependence of the viscosity q over a large dynamic range and the possible physical significance of the divergence temperature.lO-I* On the other hand the phenomenological introduction of the temperature dependence of, say, r] via the thermal expansion coefficient above T, is unsatisfactory. A breakthrough in this field was the advent of mode-coupling theory for treating the dynamic properties of supercooled It rests on a descriptionof collective motions of particles and their interactionsand expressesthe viscosity in terms of density fluctuations whose decay is governed by the viscosity in a selfconsistent way yielding a scaling law for ?( T ) of the form v( T ) 0: (T- TJ-. Meanwhile it has been recognized, however, that thedivergencetemperature Tcisnot identicalwith thecalorimetric glass transition temperature T,. The temperature Tc marks a dynamical singularity at which the nonlinearly coupled density fluctuationsvanishwhile T,is the temperatureat which thesystem becomes nonergodic. The value of Tc is typically 1.2 T, and is difficult to determine experimentally. Nevertheless there is experimental evidence in favor of the existence of a singularity in the response of supercooled liquids at a temperature T 1.2 Tg.15-17 Although mode-coupling theory can make predictions concerning certain dynamic properties of supercooled melts at T < T,, such as the existence of &relaxation as a general phenomenon independent of the chemical constitution of the system elements,15 it cannot easily account for the occurrence of the calorimetric glass transition which is a kinetic phenomenon. So far only one attempt appears to have been made to solve the problemof the glass transition in term of mode-coupling theory.’* At the glass transition temperature displacive motion of the structural elements is arrested within a particular cell of phase space and remains locked at T < T,. The temperature at which this happens depends on the cooling rate of the system. This is a clear demonstration of the kinetic nature of the proce~s.~ Relaxation dynamics in an king spin system has been suggested by JIckle and Eisinger as a model for the more general case of hierarchically constrained dynamicsI9in supercooled melts. Other models are based upon the interactions of the liquid elements with their surroundings2Oand, more specifically,upon the random
QQ22-3654/94/2098-Q662504.50/0 0 1994 American Chemical Society
=
=
Properties of Supercooled Melts energy structure21 existing in the supercooled system at temperatures T, < T < T, at which collective density fluctuations are eliminated. It is straightforward to draw upon analogies between transport in viscous liquids whose elements occupy local minima in a multidimensionalhighly complexenergy landscapeand electronic transport in random solids in which an energy distribution of localized states exists. In fact, Dyre2Z was able to show that the adaptation of a multiple-trapping model with random energy distribution of the traps leads to a kinetic freezing effect upon cooling the system. Hunt arrived at a similar conclusion on the basis of percolation arg~ments.~3 BBsslerz4and later on Richert and BlsslerZSstarted from the idea that hopping transport of charge carriers or neutral excitations in organic glasses, characterized by a density of localized states with a Gaussian profile, may be considered as a reflection of the motion of the structural units within the energy landscape of a supercooled melt. They showed (i) that the observed temperature dependence of 1 is in accord with the prediction of that model which has been initially formulated to treat charge and energy transport and (ii) that the Occurrence of a freezing effect is a necessary consequence of the energy structure. The present work elaborates these ideas. Starting from a hopping approach an analytic theory will be developed that is able to account for the temperature dependence of the viscosity, the dependenceof the glass transition temperature on the cooling rate, and entropy changes near T,in a quantitative fashion, both for fragile and strong systems.
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 663
L 0)
a l
c
W
1 Configurational coordinate Figure 1. Schematicillustration of jumps of structural units on a potential surface. Structural disorder in configurational space is represented by the energetic distribution of metastable states and barrier heights, respectively. The density of states function is assumed to remain static on the time scale of a singlejump for a 'supercooled melt" and variable in case of a "real liquid".
The Model We assume that existence of two types of excitations in viscous liquids, namely, (i) elastic excitations (vibrations) which neither change the microscopic structure of the liquid nor contribute to viscous flow and loss phenomena and (ii) inelastic excitations, Density of states Le., displacements of structural units in configurational space which give rise to phenomena associated with energy-dissipating F i p e 2. Energetic distributions of metastable states for (a) weakly bonded (fragile) and (b) strongly bonded (strong) liquids. processes such as viscous flow. Due to strong interactionsbetween molecules in viscous liquidsone molecule cannot changeits position latter yet not in the former bonds have to be broken to allow without making its neighbors move as well. Thus, inelastic excitations in a viscous liquid must be highly c o o p e r a t i ~ e .A ~ ~ . ~ ~ motion of a structural unit other than oscillations within its potential well. transition from one configuration to another may be considered Let us first consider fragile systems starting with a normalized as a jump of a structural unit within a highly complex energy function landscape. The "structural unit" may either be a molecule or a subunit thereof embedded in its molecular environment or in a g(E) = ( a / 2 ~ 0 ) [ w d 1 - 1exP[-IE/EoI"l, 1 < a < (1) microscopic region of a covalently bonded system. Since the structural unit is likely to encounter a new environment after for the self-energies of the structural units. Note that for a = having executed a jump in configurational space phase memory 2 eq 1 yields a Gaussian DPMS function. It is reasonable to is lost after every jump. This suggests mapping the sequence of assume that a unit in dynamical equilibrium occupying tail states elementary steps involved in inelastic processes, such as viscous of the DPMS can execute a jump after thermal excitation to the flow, by a random walk on a disordered network of hopping sites center of the DPMS located at E = 0. We shall in the following in configurational space characterized by a broad distribution of use the term "fluid state" to label states at or above E = 0 because energies. these states allow for transport in configurational space without further activation. Jumps that involve thermal activation over Since the intramolecular contributions to the energy of a the associated energy barrier will be designated as "over-barrier" structural unit depend upon a large number of configurational jumps (see Figure 1 top and Figure 2a) and a system that falls coordinates, each varying randomly, and since quadratic spectral into this category will be referred to as a "supercooled melt". In momentsof these contributionsdo not necessarilyexist, a stretched the present paper we assume that maxima of all barriers have the Gaussian function can be assumed to be realistic for the same energy E = 0 as the fluid level. distribution of possible ground-state energies.z8 Its variance In addition we have to consider direct jumps among the Eo-henceforth called the width of the density of possible metastable states of structural units. In this case, which, in fact, metastable states (DPMS)-and its amplitude are expected to was the situation dealt with in previous work,2s the activation be different for molecular ("fragile") and covalently bonded energy for a jump upward in energy is determinedby the difference ("strong") systems. While in van der Waals-type systems the in site energies. This situation can be realized if the energy mutual arrangement of the structural units is not constrained by landscape itself is fluctuating at a frequency that is large enough strong directional bonds, it is in network formers. As a to permit a unit to jump to an adjacent site in configurational consequence the density of available states will be much larger space once energy coincidence has been established thermally. in the former while Eo, which reflects the spread of the selfThis situation is realized in a real liquid (see Figure 1 bottom) energies of the units due to configurational disorder, may be and is analogous to adiabatic polaron hopping in a semiconduccomparable. Fragile and strong systems are expected to differ t0r.29 also with regard to the random walk of structural units. In the
664 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
Arkhipov and Blissler
For strong systems the energy required for bond breaking ( E d ) has to be added to the contribution resulting from the distribution of ground-state energies. Therefore an appropriate normalized distribution of energies required by a structural unit to reach the fluid state is
The average frequency of jumps cna be written as
g(E) = (a/2Eo)[r(l/a)l-’ exP[-I(E - Ed)/EOrl* 1 < OL < m, E d >> Eo (2) This is illustrated in Figure 2b. We start our consideration with the dynamics of structural units in supercooled melts assuming that an over-barrier jump leads to a totally new configuration of a structural unit implying that there is no correlation of energies of these two successive quasi-equilibrium states.20822 Under these conditions a master equation for the normalized energy distribution function f ( E , t ) of the structural units [JdE f(E,t) = 11 may be written as22-30
Substituting feq(E) given by eq 4 into eq 9 one obtains the equilibrium value of the total frequency of structural-unit jumps for the case of a real liquid
df(E,t)/dt = u&(E)
JOm
dE’f(E’,t) eXp(-E’/kBT) -
exp(-E/kBT)f(E,t) (3) where E is the energetic height of a barrier (as long as the peak heights of all barriers correspond to the energy of the fluid state located at E = 0, E represents the binding energy of a structural unit relative to the fluid state and, consequently, deeper states have larger energies), t is the time, YO the attempt-to-jump frequency, T the temperature, and kB the Boltzmann constant. The first term on the right-hand side of eq 3 describes jumps into the given state from any other states and the second term corresponds to jumps starting from a given state. Quasi-equilibrium of a supercooled melt corresponds to the equilibrium distribution of structural units over hopping states. This distribution is the stationary solution of eq 3, uO
&(E) = [ c d E g(E) exp(E/kBT)l-’g(E) exp(E/kBT) (4) Note that eq 4 actually presents the hopping-rate distribution of structural units. The hopping rate u(E) of a unit of energy E is given by
u(E) = Yo exp(-E/kJ) and the average frequency of unit jumps ( u ) is (’) = uOIJomdE
d E ) exp(E/kBT)l-’cdE
(5)
dE) =
vo[JomdE d E ) exp(E/kBT)l-’ (6) For the case of “real liquids” direct jumps of structural units between metastable states play a dominant role and a master equation takes the form
df(E,t)/dt = vo[JoEdE’f(E’,t) E)/k,T]lg(E)
+ JidE’f(E’,t)
- vo[fdE’g(E?
exp[-(E’-
exP[-(E - E?/kBT] -t S,’dE’ g ( E ?MEJ) (7)
Since in the regime of a real liquid, direct jumps of structural units occur over law barriers and do not require tunneling, eq 7 does not contain averaging over configurational coordinates and retains the condition that there be no correlation of energies of neighboring hopping sites. The equilibrium solution of eq 7 coincides with the stationary solution of eq 3 given by eq 4,but the frequency of a structural-unit jump from a state with energy E‘ into a state with energy E is now determined by u(E,E? = yo, E > E‘; u(E,E? = uo exp[-(E’E)/kBT], E
< E’ (8)
JidE’f(E’,t)
exp[-(E’-
E ) / k ~ q ]( 9 )
(v) =
Below we apply the above general results to describe relaxation processes in supercooled melts and real liquids.
Temperature Dependence of Viscosity It seems reasonable to assume that jumps of structural units represent the only mechanism by which viscous flow occurs. Consequently the viscosity 9 has to be proportional to the inverse total frequency of jumps, i.e., 9 0: l/(u). Since the latter is determined by the DPMS function weakly and strongly bonded glases will behave differently. Viscosity of Weakly Bonded Glasses. As mentioned above two regimes of structural-unit jumps are possible for weakly bonded glasses, and consequently two characteristic regimes may appear in the temperature dependence of the viscosity of the weakly bonded melts. In the low-temperature regime (T < Tc) in which collective effects are unimportant over-barrier jumps will be rate-limiting. For this case eqs 1 and 6 yield 9 = 90[7r/2(OL - 1 ) p x
OL(2a-3)/2(a-1) [ r ( i / O L ) ] - 1 ( ~ o / ~exp[(a ( 2 ~ )/ 21)( xa - 1 ) ( T0/OLT)”/(“’) 1 (11) where 90 accounts for the contribution of structural-unit jumps toward viscosity and TO= Eo/kB. Equation 11 is valid for a sufficiently broad energetic distribution of states, i.e., (TO/T ) >> 1. Note that eq 11 may lead to large values of the apparent activation energy E, = kB[d In q/d(l/T)] = Eo(To/cYT)’/(~-’). In the high-temperature regime (T > TJ when the tunneling jumps of structural units become dominant eqs 1 and 10 yield
= Bo2(2a-1)/2(a-U( 1 / 4 r ( 1 / 4 ( T o / T ) exp[(OL- 1) x (1 - 24/(a-’))( To”LT)a/‘a-’’ 1 (12) Equation 12 is valid for TO/T > 1. The model thus predicts a temperature dependence of 9 of the type log q a PI(*’) (neglecting weak power-law terms) both in the low-Tand high-T regimes with the low-T slope being smaller by the factor (1 2-1/(a-l)). Note that for fragile liquids this factor is close to unity CY 1 and the two regimes are therefore practically indistinguishable as far as the Tdependence of the viscosity is concerned. Figure 3 shows a set of literature data on the temperature dependence of the viscosity for various glass formers (organic and inorganic) interpreted in terms of the Gaussian DPMS (a = 2). The straight lines observed in log 7 versus ( 1 / P ) plots indicate that the function 9( 7‘) can, in fact, be approximated by log 9 0: ( 1 / P ) within both low-T and high-T regimes with the ratios of slopes being close to the value 2 as predicted by the model. The temperature dependences of various relaxation frequencies for glycerol are collected in Figure 4. The agreement
-
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 665
Properties of Supercooled Melts
OX
0.8
0.6
1.0
Tg /T 0
20
100
40 60 80 1/T2 (1U6K-’)
Figure 3. Temperature dependences of the viscosity of poly~tyrene,~’ CH,(CHz)zOH?2 and B20333 in log q versus ( 1 / p ) representation. The ratios of s l o p of low- and high-temperature sections are 2.28,1.73, and 2.15, respectively.
01
, 10
14
18
22
26
1/T2 (106K-21 Figure 4. Temperature dependences of the characteristic relaxation frequencies in glycerol in log wml versus (l/’P) repre~entation.~~ The data are obtained by dielectric relaxation (u), ultrasonic attenuation (A),digital correlation spectroscopy (0),Brillouin scattering (M), and frequency-dependent specific heat (p.),34 The ratio of low- and hightemperature slopes is 1.80.
between data obtained by various experimental techniques such as dielectric and ultrasonic relaxation, photon correlation spectroscopy, Brillouin and specific-heat spectroscopy demonstrates the universality of the thermal behavior and indicates that these dynamic phenomena have a common microscopic origin. All data can be fitted by two straight lines in a log wml versus (1/ P) plot with a slope ratio of 1.80. Viscosity of Strongly Bonded Glasses. To obtain the temperature dependence of viscosity for strongly bonded glasses we have to substitute the DPMS function (2) into eq 6. The result is
Figure 5. Arrhenius representation of liquid viscosities. The curves 1-6 are calculated from eq 13 for the following set of parameters: a = 2; (qg/q0)=2.15X 10i4;(T0/Td)1 ~ , 2 ~ ) . 1 , 3 - 0 . 2 , 4 - 0 . 5 , 5 - 1 . 0 , 6 - = . T h e curves 7-9 are plotted according to eq 1 1 for ( q g / q 0 ) = 2.15 X lo1‘; a = 7-1.5, 8-1.25, 9-1.125.
example, for pure fused quartz (Si02) and germanium (Ge02) the energies are Ed = 7.5 eV and Ed = 3.3 eV, respectively.* Consequently, before measuring the viscosity a sample of strongly bonded glass has to be “stabilized” during a sufficient length of time at the given temperature in order to establish quasiequilibrium. As mentioned above viscous flow in strongly bonded glasses must occur via the breaking of bonds. Comparing the activation energy of the viscosity (7.5 eV for Si02) with the energy of a covalent Si-0 bond (4 eV)35indicates that two broken bonds are necessary for a structural unit to make a jump via “fluid” states in fused quartz. Besides, in comparison with weakly bonded glasses strongly bonded glasses have anomalously small values of the prefactor viscosity 70, for instance, 70 = 3 X 10-14 N s m-2 for Si02 and 70= 1.4 X It7 N s m-2 for GeOz. Several arguments may be put forward to explain this fact. (i) Structural-unitjumps in strongly bonded glasses are much less cooperative than those in weakly bonded glasses. Consequently, the effective mass of a structural unit is small and the attempt-to-jump frequency is large in strongly bonded as compared to weakly bonded glasses. (ii) The energy of an excited state is very high in strongly bonded glasses, and it may be difficult to dissipate the energy liberated in the course of the recombination of broken bonds. It seems possible that this energy or part of it can be used for breaking another bond in the neighborhood. Thus each thermal excitation can give rise to a series of “flow” events and consequently lead to the anomalously large contribution of structural-unit jumps into the viscous flow. (iii) Because of the enthalpy-entropy relation a large activation energy always relates to a large transition entropy.36
Glass-Transition Temperature where Td = &/&Be This equation describes the temperature behavior of viscosity which can be referred to as “intermediately fragile”. Under the condition Td >> TOthat is typical for strongly bonded glasses the temperature dependence of viscosity given by eq 13 becomes Arrhenius-like especially a t high temperatures when T becomes comparable to TO.The predicted temperature dependence of the viscosity of both fragile and strong liquids for various parameter sets is presented in Figure 5 . The agreement with experiment is striking. The residual temperature dependence of the apparent activation energy for strongly bonded glasses arises from the shift of the equilibrium occupational density of states of the structural units within a relatively narrow distribution of ground states. It requires bond breaking to be established. Strongly bonded glasses are usually characterized by rather large values of energy Ed. For
The distribution of structural units within the rough potential energy surface under conditions of dynamic equilibrium depends on temperature. If the temperature is lowered the mean of the distribution moves to deeper states. Attainment of a new equilibrium distribution upon lowering the temperature will not, however, be completed before the relaxation time has elapsed, which strongly increases with decreasing temperature. For a given cooling rate the relaxation time will sooner or later exceed the equilibration time and the distribution of structural units will be “frozen-in”. This is obviously the glass transition. From the above arguments it is clear that the glass-transition temperature TBmust depend upon the cooling rate j3. Now we embark on a calculation of this dependence. The glass transition occurs within the low-temperature regime when over-barrierjumps dominate. Our consideration, therefore, has to be based on a model of thermally activated structural-unit
666 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 jumps via fluid states and the dependence of T, vs @canbe found from the solution of eq 3. Assuming a linear decrease of temperature with time starting at an initial temperature Ti,, T = Tin - @t,and introducing the new variables T = Tin - @t and f*(E,T) =flE,(qn-T)/j3] in eq 3 we obtain an equation for the distribution function f*(E,T) as a function of temperature,
Presuming that at T = Tin dynamic equilibrium is established,
Arkhipov and B i d e r equilibrium form of the function +(e$) 4(c,O) = A(8) r(4 exp(e/b)
(22)
if the condition exp(c/t9) < 9 (23) is satisfied for the energy c. A(9) is a normalizing factor. It seems plausible to define the glass-transition temperature T, as that temperature at which the structural units occupying the center states of the DPMS distribution at E = E(m) can no longer equilibrate, This definition is equivalent to &e calorimetric definition of the glass-transition temperatudrecorded at a given relaxation time (usually 102 s). On the other hand the glasstransition temperature is often determined as a temperature at which the viscosity reaches some large value, usually 1012Pa s.3*6 The latter procedure yields values that differ from calorimetric glass transition temperaturesS6 This accounts for the fact that various melts have different values of the viscosity at the calorimetric glass-transition temperature ranging from 109 Pa s for weakly bonded to 10l2Pa s for strongly bonded glasses.6 Substituting eq 16 into eq 23 we obtain the following transcendental equation for the glass-transition temperature
voTgexp[-Eg) (Tg)/kTg] = B For a particular DPMS function the dependenceE(m)(T,) can be found using the equilibriumenergetic distribution o?structural units given by eq 4. In the following we consider the cases of weakly bonded and strongly bonded glasses separately. Glass-Transition Temperature for Weakly Bonded Glasses. Using eq 1 one can easily obtain a simple formula for the energy E:) at the temperature T = To:
r d c y(c) = JomdE'g(E) = 1, Jomdc 4(t,t9) = C d E f.(E,T) = 1 (19)
-
and the equilibrium distribution of the form given by eq 18 represents a solution of eq 17 for t9 a. As long as the purpose of our calculations is to determine the dependence of the glass-transition temperature on cooling rate it is appropriate to consider a solution of eq 17 for T > TBwhen the energetic distribution of structural units is in equilibrium. We then have to find the temperature at which the system falls out ofequilibrium at a given cooling rate, Le., becomes nonergodic. This temperature is the calorimetric glass-transition temperature TB. The quasi-equilibrium solution of eqs 17 and 18 reads
y(c)SJJ"dtY[Kdc ~ ( e )exp(c/b')]-' exp[-J;dtP"
X
exp(-c/d/')] (20) The first term on the right-hand side of eq 20 accounts for the contribution of those structural units which did not execute a jump since the beginning of the cooling process. However, in order to render an experiment aimed at determining TBmeaningful the starting temperature Tinmust be sufficiently in excess of TB implying that the energetic distribution of structural units has completely been reordered once the temperature has reached Tg and, consequently, that the first term on the right-hand side of eq 20 is negligible. We now have to determine the lowest temperatureat which thelatter term iscompatiblewithattainment of equilibrium. Taking into account the condition t >> 9 which is valid for the range of energies of practical importance eq 20 reduces to
exp[-(s/*/c)[exp(-c/rs') - exp(-t/8)]] (21) An analysis of the integral in eq 21 shows that it gives an
E:) = E ~ ( T ~ / ~ T ~ ) ~ / ( * ~ ) Substituting eq 25 into eq 24 yields
(25)
voTgexp[-a( To/aTg)"/("')] = (3 To a first-order approximation the solution of eq 26 is
(26)
Tg =
To[ln(voTo/(3)]-'+~'~")
Glass-Transition Temperature for Strongly Bonded Glasses. Equation 2 combined with eq 4 gives
E(m) #I = Ed
+ Eo(T"(rT,)*/'*''
(28)
For strongly bonded glasses having a narrow distributionof ground states the first term on the right-hand side of eq 28 is usually much larger than the second one. Substituting eq 28 into eq 24 yields an equation for TB, voTBexp[-(Td/Tg) - C~(T~/CYT~)"/'*''] = fl (29) Under the assumption Td > To(To/cYT~)~/(*~) an approximate solution of eq 29 takes the form TB= Td[ln(v,Td/@)I-' (30) Equation 30 shows that for strongly bonded glasses the glasstransition temperature is more sensitive to the cooling rate than for weakly bonded glasses. Equations 27 and 30 reveal almost linear correlations between the glass-transition temperature T, and the characteristic energy of the DPMS function EOfor weakly bonded glasses or E d for strongly bonded glasses with proportionality factors being smoothly decreasing functions of To or Td, respectively. The correlation between the glass-transition temperature and the characteristic energies of the DPMS function is illustrated in Figure 6 for both weakly and strongly bonded glasses. In Figure 7a,b experimental data for the dependence of the glass-transition temperature on cooling rate are plotted on 1/T,
Properties of Supercooled Melts
Figure 6. Correlation between the glass transition temperature TIand the characteristic temperatures TO (weakly bonded glasses) and Td (stronglybonded glasses),respectively,for a seriesof organicglass formers (B) and the inorganic glasses Se, BzO3, borosilicate glass, As2Se3 ( O ) , SiO2, Ge203, Be% P205, Na20.2Si02 (A). Values of TOand Td are obtained by fitting of corresponding viscosity data. Theoretical curves are calculated from q 27 for weakly bonded glasses (@= 10-1 K s-I, YO = 1Olo s-I) and from eq 30 for strongly bonded glasses (6 = 10-2 K s-I, YO = 1013 s-1).
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 667 of measurements.ls~40This renders an experimental investigation of the inelastic contribution to the specific heat difficult. There exists, nevertheless, a characteristic value of the inelastic specific heat which can be determined unambiguously. It is the change of specific heat at the glass-transition temperature ACE. This value also depends upon the relaxation time and, concomitantly, upon the cooling rate j3, but this dependence is taken into account by measuring AC, at a given T, depending on j3. As long as an equilibrium distribution of structural units (at T > T,) is described by a similar function for both weakly and strongly bonded glasses, the averaged inelastic energy (6)per one mole may easily be related to the energy E, = EO(To/cu~)~/(*~) of the maximum of the energy distribution of structural units. The results are:
(6) = goNMEO( TO/anl'(a-l)
for weakly bonded glasses (3 1a)
and
( 6 ) = 6O-NM[Ed + Eo(T0/a7)1/(a-1)] for strongly bonded glasses (31b) where 60is the inelastic energy of the fluid state and NM the number of structural units in 1 mol of a liquid. Note that eq 3 la is valid for both the regimes of real liquid and supercooled melt implying that there is no singularityin the temperature dependence of specificheat at T = Tc. When a melt is cooled to a temperature below T, the system of structural units preserves the state it had at T = T, and consequently the energy (6)remains constant for T < T,, Le.,
(6)= 60-NMEO(TO/aTg) T < T,
1&-1)
9
for weakly bonded glasses (32a)
and
( 6 ) 6O-NM[Ed + EO(TO/aTg)l/(a'-l)], T < T, for strongly bonded glasses (32b) Figure 7. Dependenceof glass transition temperature on cooling rate for strong (a, top) and fragile (b, bottom) liquids. Experimental data for P20537(0),borosilicateglass3*(.),and A s ~ S e (B) 3 ~ ~are fitted by straight linesaccordingt o q 30 for strong and q 27 for fragileliquids,respectively, with the following parameter sets: (a) P205, Ed = 2.07 eV, vo = 6.7 x 10" s-1; bo rosilicate glass, Ed = 3.06 eV, YO = 1.2 x 1022 s-l; AszSe3, Ed 3.31 ev, Yo 6.0 X S-I. (b) P205, Eo 0.34 ev, YO 4.2 X 104 s-I; borosilicate glass, EO = 0.36 eV, vo = 0.8 X lo8 s-I; As2Se3, EO = 0.35 eV, vo = 1 X s-I. Although both date analyses yield reasonable values for the characteristic energies, the fit values of the attempt-to-jumpfrequencyindicatethat P 2 0 is~ the strong liquid whereas borosilicate glass and AszSe3 are fragile ones.
versus log j3 and 1/ pversus log j3 scales, respectively. The latter corresponds to fragile glasses with CY = 2. Experimental points may be fitted by straight lines in either plot. But only interpretation of the experimental data in terms of either eq 27 or 30 gives reasonable values of parameters for As2Se3 and borosilicate glass as examples of 'fragile" liquids and for PzOs as a "strong" liquid, respectively. It should be emphasized that the obtained values of attempt-to-jump frequency YO strongly depend upon which DPMS function is used for the interpretation of cooling-rate dependence of the glass-transition temperature.
Temperature Dependences of Thermodynamic Functions We start our consideration from the temperature dependence of the specific heat C(7). Both elastic and inelastic excitations contribute to the value of C with both contributions being temperature dependent. Moreover, the inelastic part of the specific heat depends upon the relaxation time during the course
Comparing eqs 3 la,b and 32a,b we see that the change of specific heat at the glass-transition temperature is determined by the derivative d ( C ) / d T at T = T, yielding Acg
= [k,NM/(a-
l)1 ln(voTO/b> for weakly bonded glasses (33a)
and
AC, = [ak,NM/(a - l)](TO/aTd)a/'a-l'[MYoTd/b) 1 +-? for strongly bonded glasses (33b) Equation 33b predicts relatively small changes in the specific heat at T = T, for strongly bonded glasses, characterized by To/T,j Tg the system comes into equilibrium and the specific heat decreases with increasing temperature. Thus, on heating at a sufficiently high rate the specific heat has a peak at a temperature close to T,.
668 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
E
e
Lo
..
T (io3K) Ngure 8. Temperature dependenceof the normalized excess entropy for a wries of strong and fragile liquids.6 Full curvh~are calculated from q 34 for a = 2 without any other adjustable parameter.
The difference in characteristic widths of the DPMS functions for weakly and strongly bonded glasses permits one to explain the different temperature dependences of excess entropy S observed for strong and fragile melts, respectively. It seems reasonable to assume that the deeper states are associated with smaller entropy implying that relaxation of a structural unit into a deeper state is accompaniedby a loss of entropy ISin proportion to the energy difference between initial and final states, Le., A S = (E/T,), where 1/T, is a proportionality factor of dimension of [temperature]-I. This yieldsS(7') = [ ( C ) ( T )- ( & ) , ] / T , ,whereSis the excess entropy per mole and (C),the inelastic energy per mole at the glass-transition temperature (see eqs 32a and b). Using eqs 31a,b and 32a,b yields
for both strongly and weakly bondedglasses and without predicting singularity at T = Tc in the latter case. The factor (Eo/T,) in eq 34 accounts for different rates of the variation of the entropy with temperature for weakly and strongly bonded glasses observed in experiment. Figure 8 compares experimental data normalized to the value of the entropy at the melting point, which eliminates the factor NM(Eo/T,)from eq 34, with model predictions. The agreement is excellent. Note that there is no adjustable parameters.
Conclusions In the present paper a simple phenomenological model is proposed for describing dynamic and thermodynamic properties of supercooled liquids. It is based on the random-walk concept assuming some energetic distribution of metastable states for structural units in configurational space. Different behavior of socalled "strong" and "fragile" liquids is associated with different energetic distributions of metastable states. A broad energetic distribution of "ground" metastable states with high total density has been associated with weakly bonded glasses ('fragile" liquids) that permits jumps of structural units between metastable states or jumps via some "fluid" state without involving bond breaking. Strong systems, on the other hand, are also characterized by a distribution of ground-state energies resulting from local deformations of the structural network but a constant energy term resulting from bond breaking has to be taken into account when calculating the activation energy for the jump of a structural unit in configurational space. The primary intention of this work was to provide a framework for describing properties of liquids below the temperature T, at which collective effects are eliminated. The results nevertheless indicate that the phenomenological changes observed in the temperature dependence of the viscosity of fragile systems above Tc can be recovered on the premises that (i) being determined by intramolecular forces, the densities of available states for the structural units are constant throughout the temperature range T, C T C Tmand (ii) that above Tc structural fluctuations are fast enough to eliminate energy barriers separating adjacent local
Arkhipov and BlIssler energy minima, i.e., render thermally activated jumps adiabatic. Apparently the latter assumption is a simple, yet useful, zeroorder approach to account for the collective effects that become important above TBand are usually described in a more rigorous way by mode-coupling theory. Structural-unit jumps in strongly bonded glasses ("strong" liquids) requires bond breaking. Consequently,only over-barrier jumps of structural units are possible with the fluid state being associated with the energy of broken bonds. Therefore, temperature dependences of both dynamic and thermodynamic characteristics of "strong" fluids are described by universal functions within the entire temperature interval T, C T C T,,,. A wide energy gap between ground and excited states results in almost constant value of activation energy for jumps of structural units. Only slight deviations from Arrhenius behavior are noted that result from the temperature-dependent shift of the occupational distribution of structural units within the manifold of ground states. The establishment of dynamic equilibrium at temperatures not far above T, requires over-barrier jumps. Since a characteristic energy of the distribution and, consequently,the activation energy of structural-unit jumps increase with decreasing temperature, the rate of structural readjustment will sooner or later be insufficient for the establishment of quasi-equilibrium at a given cooling rate. This means that a system of structural units must fall out of thermodynamic equilibrium at a certain temperature identified as the calorimetric glass-transition temperature. This approach yields simple analytic formulas for the dependenceof T, on the cooling rate for both "strong" and "fragile" melts consistent with the results of Hunt.23 The random-walk model can also be used for an analysis of susceptibility spectra of supercooled melts which reflects both the temperature dependence of characteristic average times and the distribution of microscopic relaxation times. A moredetailed description of relaxations in complex liquids requires more sophisticatedconsiderationof structural-unit dynamics accounting for some additional phenomena that are not very important for temperature dependencesof the viscosity and average relaxation times, though. An elaboration of the random-walk approach for beating susceptibility spectra will be presented in a following paper.
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