Interpretation of the apparent activation energy of glass transition
IInterpretation of the apparent activation energy of glass transition
Koun Shirai
The Institute of Scientific and Industrial Research,Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
Abstract
The temperature dependence of the viscosity of glass is a major concern in the field of glassresearch. Strong deviations from the Arrhenius law make the interpretation of the activationenergy difficult. In the present study, a reasonable interpretation of the apparent activation energyis demonstrated along similar lines as those adopted in solid-state physics and chemistry. Incontrast to the widely held view that phase transition in glass occurs at the reference temperature T according to the Vogel–Fulcher–Tammann formula, in the present work the transition observedat the glass-transition temperature T g is regarded as a phase transition from the liquid to solidphases. A distinct feature of glass is that the energy barrier significantly changes in the transitionrange with width ∆ T g . This change in the energy barrier alters the manner in which the apparentactivation energy constitutes the Arrhenius form. Analysis of available experimental data showedthat the actual energy barrier is significantly smaller than the apparent activation energy, andimportantly, the values obtained were in the reasonable range of energy expected for chemicalbonds. The overestimation of the apparent activation energy depends on the ratio T g / ∆ T g , whichexplains the existence of two types of glasses strong and fragile glasses. The fragility can be re-interpreted as an indication of the degree of increase in the energy barrier when approaching T g from high temperatures. Since no divergence in viscosity was observed below T g , it is unlikely thata transition occurs at T . PACS numbers: a r X i v : . [ c ond - m a t . d i s - nn ] D ec . INTRODUCTION The core problem in glass research at present is what is the nature of the glass transition[1–4]. Grass transition is observed, for example, from specific heat ( C p ) versus temperature( T ) curves. The transition temperature T g is the temperature at which the specific heat ex-hibits a quick change. It can be experimentally observed, but its value varies to some extentdepending on experimental conditions such as the cooling rate. This gives rise to the believethat T g is not an intrinsic property of glass. Unlike the usual phase transitions, structuralrelaxation plays a crucial role in glass transition: the status of experiments on relaxation isreported in review papers [5–7]. The structural relaxation is reflected by viscosity η . Theviscosity of glass-forming liquid increases drastically by more than ten orders of magnitudewhen the liquid temperature approaches T g from a higher temperature (Fig. 1 (a)). The T dependence of viscosity is most commonly expressed by the Vogel–Fulcher–Tammann (VFT)formula [8–10], η ( T ) = C exp (cid:18) DT − T (cid:19) , (1)where C , D , and T are material-dependent constants. The reference temperature T is lessthan T g and is generally 0 . . T g [11]. While T is obtained by extrapolation, manyresearchers have attempted to identify its significance as the genuine transition temperature,which is an intrinsic property of a specific glass. The relation of T with the so-calledKauzmann temperature T K is being actively debated [12–18]. For normal liquids, the natureof viscosity is well understood as a thermally activated process, which obeys the Arrheniuslaw, η ( T ) = η exp (cid:18) Q a k B T (cid:19) , (2)where k B is Boltzmann’s constant, and η is a material-dependent constant. In this case, Q a is clearly the activation energy. This formula was derived from first principles [19]. Fornormal liquids, the Q a values are of the order of a few tenths of electron volts or less [20],which is a reasonable range considering the chemical energies of materials. If this standardformula, i.e., Eq. (2) is applied to the viscosity of glasses in a narrow range of temperature, Q a values are found to be surprisingly large more than 1 eV (in some cases, 10 eV) even fororganic glasses whose melting temperature T m is less than room temperature [21–23]. Thereis no convincing theory to interpret the large values of Q a . Furthermore, the pre-exponentialfactor A = 1 /η is extremely large: typically, the magnitude is of the order of e [23]. Thus2ar, no theory that accounts for this extraordinarily large value has been established.At present, the meaning of activation energy of glass is not being actively researched.Instead, by normalizing T with T g in the Arrhenius plot, which is now known as the Angellplot, researchers are attempting to discover something universal for glass transition [24–28].In the normalized form of the Angell plot, how largely the T dependence of η deviates fromthe Arrhenius law is the main information that can be obtained; fragility is a parameter usedto quantify the degree of the deviation. Depending on the fragility, glasses are classified asstrong or fragile. Although this manner of plot is useful for classifying glasses, it doesnot aid the chemical interpretation of the activation energy, which is the basic element forunderstanding properties of materials. Without a pertinent interpretation of the activationenergy, one cannot understand why the viscosity of glass changes so largely when approaching T g . It is necessary to appropriately interpret Q a obtained from the Arrhenius plot: this isthe aim of this study.A resolution of the problem of the activation energy of glass was achieved in a very dif-ferent context, namely, in terms of the state variables of glass. Today, the standard view isthat glass states are nonequilibrium states. Although the glass transition is a nonequilibriumphenomenon—transition itself represents nonequilibrium—unduly emphasizing the nonequi-librium character hides the thermodynamic nature of glass. A frequently-claimed reason forthe nonequilibrium character is that the state of glass is not determined by temperature andpressure ( p ) alone but is affected by the previous history of treatments. Thermodynamicsstates are to be specified solely by the current values of state variables; if the glass stateswere equilibrium, the properties must be completely determined solely by T and p , but thisis not true, from which the above conclusion is derived. However, the attribution of this dis-crepancy to nonequilibrium state is an easy escape from the difficulty. Careful observationreveals that this dependence of the past treatment that the solid underwent is a commonproperty of solids: any solid has, to a certain degree, the dependence of its properties on thepreparation conditions. The fundamental question then is that besides T and p which couldbe the other state variables of a solid: the term thermodynamic coordinate (TC) is used torefer to state variable. This question was at times considered by several authors to be aflaw of the theory [29–31] but was not solved until recently. The author examined this topicby reappraising the definition of equilibrium and TCs for solids, and defined in a consistentmanner. In short, the answer to the question of TCs for a solid is the time-averaged positions3 R j of all the constituent atoms of the solid: note that instantaneous positions are irrelevant,and only equilibrium positions have sense [32, 33]. This consideration can also be true forthe state during glass transition, provided some restrictions are imposed. The thermal re-sponse of solids is so fast—typically 10 ps—that the fast vibrational motion is adiabaticallydecoupled from slow changes in the structure. This behavior is expressed as the adiabaticapproximation of the second kind [33]. Then, the instantaneous state at timescales muchlonger than this response time can be regarded as an equilibrium state, i.e., temporal equi-librium [33]. The states in glass transition are states of temporal equilibrium. Even duringsuch a transition, all the averaged positions ¯ R j ( t ) as well as the time-dependent tempera-ture T ( t ) are well defined, and accordingly, the glass state is expressed by the instantaneousvalues of all TCs.A striking consequence of this conclusion is that energy barrier E b of the structuralrelaxation of glass during glass transition is determined solely by the present positions ofatoms, E b = E b ( (cid:8) ¯ R j (cid:9) ), irrespective of the previous history. Since during the transition,the structure is mainly determined by temperature, the structural dependence of E b ( { R j } )can be represented as the temperature dependence of energy barrier E b ( T ). The Arrheniusanalysis presumes that Q a is constant: however, this assumption does not hold for glasstransition. Hence, the apparent activation energies obtained from the Arrhenius plot resultsin extraordinarily large values in the usual sense of the term “chemical bond” [34].In this paper, the above conclusion regarding the temperature dependence of E b ( T ) isvalidated by analyzing experimental data on viscosity. Further, a pertinent interpretation ofthe apparent activation energy is provided. In a previous study [34], this interpretation wasderived from an analysis of the hysteresis in the C - T curve. Viscosity measurement is a moredirect method to obtain the activation energy. The rest of this paper is organized as follows:the theoretical basis for the present analyses is given in Sec. II. Then, Sec. III shows thefirst evidence for the proposed interpretation based on the relationship between the widthof glass transition and the apparent activation energy. The second evidence is obtainedby directly evaluating the temperature-dependent energy barrier: this evidence is describedin Sec. IV, which provides three examples of classes of glasses. An important implicationof the present results to glass transition is discussed in Sec. V. Section VI concludes thestudy. Throughout this paper, the numerical values of viscosity η are presented in termslog η (poise) with base of 10. 4 I. THEORETICAL BACKGROUNDSA. Glass state and glass transition
There exists a large gap in comprehension of the glass state between glass physics andother areas of solid-state physics. For non-experts in the field of glass research, glass is almostevidently a solid. The glass transition that occurs at T g is a phase transition between thesolid and liquid phases. However, glass researchers do not think so. Many of glass researchersconsider glass as a special type of liquid, specifically, kinetically frozen liquid . Hence, it isnecessary at the outset to state the present view and to explain the differences from theviewpoint of glass literature.The signature of the glass transition is observed in a quick change in the properties ofglass, such as specific heat and thermal expansion, in a narrow range of temperature [35].Here, the range is specified by T g, and T g, with its width ∆ T g = T g, − T g, , as shown inFig. 1(b). In this paper, the term the glass state is restricted to refer to the phase of aglass substance below T g, , and is regarded as a solid phase. Above T g, and below T m , thestate of the glass substance is a supercooled-liquid state; there C p obeys almost the classicallimit of Dulong-Petitte law, 3 R ( R is the gas constant) [36]. The state between T g, and T g, is referred to as the transition state . The C p - T curve often exhibits complicated shapesdepending on the preparation conditions, which makes it difficult to identify the width ∆ T g .Usually, an averaging value is used for T g , unless special interest is paid for the width. Inspite of the preparation dependence, it is meaningful to determine the unique value for T g within a reasonable range, provided suitable conditions are used [37, 38]. In the relationshipbetween the viscosity η and T , in contrast, there is no characteristic temperature otherthan T when η ( T ) is recasted to the VFT form. However, it will be shown later thatthis is not true: see Sec. IV A. Operationally, the T g can be defined as the temperature atwhich log η = 13. Many of glass researchers do not think that there is a sharp boundarybetween the supercooled-liquid state and the glass state. Even in the glass state, atoms aremigrating with extremely slow velocities. From this view, T g cannot be the temperatureof a phase transition. Instead, T in Eq. (1) deserves the genuine transition temperature,below which the so-called ideal glass appears. The reason that the ideal glass has not beenobserved is attributed to its extremely slow motions. From this, that the observed glass is5hermodynamically a nonequilibrium state is concluded. TC T g,1 T g,2 C l C g glass transition region liquid TLog η T g
13 T ? S T (a)(b)(c) T m ( l )( c )( g ) (s l )T K FIG. 1: Glass transition represented in terms of viscosity η (a) and specific heat C p (b). Evolutionof the energy barrier for atomic movement is shown in (c). The causes behind the above contrasting views are deeply rooted in the fundamentalproblems of thermodynamics: the definition of equilibrium and TC. The rigorous definitionof equilibrium has been only recently established [39], and it enables us to define TC in aconsistent manner. The author applied this principle to solids and found that the equilibriumpositions of all the atoms (cid:8) ¯ R j (cid:9) are TCs for solids [32]. The present viewpoints are basedon this quite general conclusion [33]. The important points relevant to this study are listedbelow, with contrasting the general views in glass research [2–4, 40–42]: however, some ofthe views are not necessarily unique even in the glass research.1. Glass is a solid : it is viewed as a frozen liquid in glass literature. An often-claimed6eason for the lack of distinction between the liquid and glass states is that atomicpositions are random in both states. However, this is only true when the instantaneouspositions of atoms R j ( t ) are compared. Thermodynamically, equilibrium positions ¯ R j ,which are obtained by time averaging, have sense. They are well defined for solids,but not for liquids.2. Glass is an equilibrium state : it is viewed as a nonequilibrium state in glass literature.This is the natural consequence of the preceding conclusion. Although the changingstate in the course of time during the transition is a nonequilibrium state, the valuesof TCs can be specified for each moment. When the change in the external conditionsis halted and keep it constant for a time longer than the relaxation time, the transitionstate finally reaches the equilibrium state even in the transition region of temperature.3.
Glass transition that occurs at T g is a genuine phase transition : it is viewed as anextrinsic change in glass literature. Although the transition has dependence on thepreparation conditions, this dependence, small or large, is a common feature of allphase transitions. In molecular-beam-epitaxy method, the growth temperature varieswith the kinetics of source gases.4. The transition state is a mixture of liquid and solid phases : it is viewed as asupercooled-liquid state after a sufficiently long time in glass literature. Althoughthe final states according to the two viewpoints are different, both share the commonthought that the transition region is a crossover region between different phases. Theterm dynamic heterogeneity in glass literature is also compatible with the present viewof the mixed state.A natural consequence of the above-described views is that the energy barrier E b foratomic movement depends on the structure, that is, E b = E b ( { R j } ), and hence it depends ontemperature [34]. It follows that E b varies strongly in the transition region from the energybarrier of the solid state ( E b,g ) to that of the liquid state ( E b,l ), as shown schematicallyin Fig. 1(c). The temperature dependence of the activation energy becomes clearer inmolecular-dynamic simulations [43, 44]. The idea of the T -dependent energy barrier wasat times proposed by observing the deviations from the Arrhenius law [45–48]. However,this idea did not gain wide acceptance. The difficulty in interpreting the T dependence of7he viscosity—even if the structural dependence of energy barrier is accepted—is that thevalues of the activation energy at T g are too large on account of the magnitude of the energiesof chemical bonds, as mentioned in Introduction. Further, the pre-exponential factor is toolarge: the absolute values of log η are greater than 100 for organic glasses [23]. B. Theory of viscosity
Many theories have been proposed for explaining the viscosity of glass since Eyring derivedthe quantum-mechanical formula for viscosity of liquids [19, 49]. These theories have beenreviewed in textbooks by Nemilov [24] and Rao [50]. The current trend of study has shiftedto microscopic dynamics, focusing on topics such as the mode-coupling theory, dynamicfacilitation, and first-order random transition [1–4, 40, 42, 51, 52]. The approach employedin this study is the traditional one based on the reaction rate theory, with the aim offacilitating the chemical interpretation of the viscosity of glass.For normal liquids, formula Eq. (2), including the pre-exponential factor η , can be de-duced from the microscopic theory, without assuming any model [19, 49]. Eyring’s theoryexplains well the viscosity of normal liquids [20]. For an isotropic media, the pre-exponentialfactor η in Eq. (2) is expressed as η = (cid:16) a λ (cid:17) ha Z n Z a , (3)where a is the mean interatomic distance, and λ is the average distance between the equilib-rium positions of the original and slipped states. The product ( λ/a ) a may be interpretedas the activation volume V a , which is the volume swept by the slipped atoms. Planck’sconstant h is a universal constant but is equal to k B T /ν m at thermodynamic equilibrium,where ν m is the mean frequency of the frequency spectrum at T and is interpreted as theattempt frequency. Z n and Z a are the partition functions of the normal and the activatedstates of the slipped atoms, respectively. Although it is difficult to calculate the partitionfunction Z , the values of Z in Eq. (3) are obtained only as the ratio Z n /Z a , which is ofthe order of unity in most cases, and hence, we can ignore this term. By assuming λ ≈ a , η = h/V gives log η = − . V = 10 ˚A . This result is well in agreement with theexperimental results: in many cases, log η = − − Q a is ofthe order of 0.1 eV or less [20, 53]. 8or solids, the definition of viscosity is not so simple because of its nonlinearity. Never-theless, the rate theory of Eyring was widely applied to solid dynamics beyond the originalobject of viscosity. Plastic deformation of crystalline solids is explained by the rate theory,in which the motion is mediated by dislocations [54]. For the case of metallic glass, themotion is mediated by free volumes [55] or by shear transformation zones [56]. The physicsinvolved in the processes is different from those of liquids. Despite this, the formal expressionfor the rate of deformation is almost the same as Eq. (3), if the quantities in this equationare suitably interpreted. Hence, there is no reason not to apply this formula for glasses ingeneral. However, the problem in this case is that the material-dependent “constants” arenot constant during the transition, because the structure changes during the transition.Now, let us ask the question why the T dependence of the viscosity does not deviate fromthe Arrhenius law for strong glass. From the present viewpoint, the energy barrier stronglyvaries with temperature, and hence a large deviation from the Arrhenius law is expectedfor all glasses. The apparent energy barrier Q ∗ a is obtained from the derivative of ln η withrespect to 1 /T , Q ∗ a = ∂ ln η∂ (1 /T ) . (4)We are so accustomed to this formula to obtain energy barriers in numerous applicationsthat we seldom consider how the Arrhenius form is altered when the energy barrier has T dependence. Let us consider a simple case of the linear dependence of the energy barrier ontemperature, E b ( T ) = E b,g − b ( T − T g, ) , (5)where b = ( E b,g − E b,l ) / ∆ T g is a constant. In the Arrhenius form, the linear term in T iscanceled by the denominator in the exponent of Eq. (2). Thus, Eq. (2) becomes η ( T ) = η e − b/k B exp (cid:18) Q ∗ a k B T (cid:19) , (6)where Q ∗ a = E b,g + bT g, . Hence, T dependence is not seen in Q ∗ a . Instead, there exists alarge separation between Q ∗ a and E b . Since ∆ T g is small, Q ∗ a is approximated by Q ∗ a ≈ T g ∆ T g ( E b,g − E b,l ) . (7)The factor k = T g / ∆ T g acts as a magnification factor for the barrier height. Althoughthe linear term disappears from Eq. (7), its effect appears in the pre-exponential factor as9 (cid:48) = η e − b/k B . Sometimes, the entropy of atomic migration contributes to the transportcoefficients through the T -independent term e b/k B in a manner similar to Eq. (6). However,the above derivation shows that it is not appropriate to interpret the e b/k B term in termsof entropy. In fact, b/k B often becomes as large as 100. It is impossible to explain thismagnitude by entropy. Even for vaporization, the increase in entropy is at most of the orderof 10 times k B . To obtain the genuine energy barrier E b , the term e − b/k B should be retainedin the Arrhenius analysis. Thus, E b can be obtained by E b ( T ) = k B T ln (cid:18) η ( T ) η (cid:19) , (8)provided that η is independent of T . Unfortunately, η may not be expected to remainconstant during phase transition because the structure changes and does the size λ of themoving unit. In addition, the ratio Z n /Z a may not be ignored because the moving unitsbecome more collective motions for solids. This problem has to be solved. III. WIDTH OF GLASS TRANSITION
The present theory predicts that the apparent activation energy Q ∗ a is magnified by thefactor k = T g / ∆ T g . Let us examine this magnification in Q ∗ a by analyzing the experimentaldata. The data for this purpose are taken from measurements of the specific heat. Whilenumerous data have been accumulated for T g , data for ∆ T g are very rare. The width issensitive to the shape of the C - T curve in the transition region, which is largely affectedby the conditions of sample preparation and measurement [38]. It is, therefore, importantto use the references in which the experimental conditions are well documented. Moynihancompiled data of ∆ T g for 17 inorganic glasses with T g well above room temperature [57].These glasses include chalcogenides, heavy-metal fluorides, and network oxide glasses. Hetook special care when collecting experimental data: i.e., only the data obtained undercommon conditions, such as within an acceptable range of heating rate, were considered.He found that there is a good correlation between Q ∗ a in viscosity measurement and ∆ T g inheat-capacity measurement Q ∗ a k B ∆ (cid:18) T g (cid:19) = 4 . ± . . (9)The values of Q ∗ a are obtained in the range 11 ≤ log η ( T g ) ≤
12, where the T dependence ofviscosity obeys the Arrhenius law. The relationship between T g / ∆ T g and Q ∗ a /k B T g which is10aken from Moynihan’s data is plotted in Fig. 2. The acronyms and the chemical composi-tions are explained in his paper. There is a good correlation between T g / ∆ T g and Q ∗ a /k B T g . T / ∆ T g g Q * / R T ga SNWDGNBS711NBS710NaKSBSCKS NaS BIZnYbTCLAP ZBLAN20ZBLAHBLAN20ZBLAN10As Se
B O
Ge As Se
23 23 54
Ge As Se
33 12 55
FIG. 2: Plot of Moynihan’s data showing the relationship between T g / ∆ T g and Q ∗ a /k B T g [57]. Thesources of the original data are quoted in Ref. [57]. From the approximation of Eq. (7), Eq. (9) is rewritten as E b,g − E b,l k B T g = 4 . . (10)This ratio seems to be reasonable when compared with the range of energy barrier E b ofimpurity diffusion. In silica glass, the energy barriers E d of impurity diffusions are reportedto range from 0.3 to 0.8 eV [58]. Since for silica glass T g = 1450 K, the ratio E d /k B T g rangesfrom 2 to 6. Although diffusion and viscose motions are different modes of motions, theenergy barriers should not be so different [59, 60]. Thus, the relationship shown in Eq. (10)must give reasonable values for E b . This infers that the previous values for the activationenergy for glass transition were significant overestimations.Among various methods available, one definition of fragility is m = ( ∂ ln η/∂ (1 /T )) /k B T g ,meaning the activation energy normalized with the glass-transition temperature [61]. Hencethe relationship Eq. (9) is also rewritten as follows: m = 4 . k. (11)11lthough the factor was determined as 4.8 from a certain group of glasses, this may notlargely change among various glasses: Ito et al. showed that this is indeed the case [13].Thus, one can immediately see that fragility represents the magnification factor k . Therefore,the greater fragility, the more magnified Q ∗ a is compared to the true value E b . Therefore,fragile glasses exhibit large Q ∗ a , in spite of their low melting temperatures. This naturallyleads to the following interpretation of fragility m : that is, it represents how rapidly theenergy barrier varies with varying T . This interpretation is more appealing because theenergy barrier is the standard terminology in solid state physics and chemistry. A similarinterpretation for the m was inferred by Dyre and Olsen [62]. The present study gives anadditional meaning of m as the magnification factor for the activation energy Researchersattempted to find correlations between fragility with other properties of glasses. Fujimori andOguni found the correlation of m with a special index representing the difference between α and β relaxations [63]. Scopigno at al. found another correlation of m with the decorrelationof the density fluctuations [64]. These correlations can be understood better by using afamiliar term, namely, energy barrier. IV. ANALYSIS OF THE TEMPERATURE DEPENDENCE OF VISCOSITYA. Silicate glass
Let us examine the full-scale temperature dependence of viscosity for three classes ofglasses. The first class of glasses comprises silicate glasses, which are typically strong glasses.It is said that the T dependence of η for strong glasses obeys the Arrhenius law, but it is trueonly in a relative sense compared with fragile glasses. If the T dependence of η is examinedover a wide range of T , one sees a large variation in the activation energy: for example,the Q ∗ a of silica glass increases from 4.0 eV above T = 2000 ◦ C to 8.1 eV at T = 1400 ◦ C astemperature declines [25, 65–67].Figure 3 shows the apparent activation energies Q ∗ a obtained using Eq. (4) for soda-lime-silicate (SLS) glass, as reported by several authors. The chemical composition of SLS glassis formally SiO : Na O : CaO = 75 : 15 : 10, and the transition temperature T g is 530 ◦ C.The numerical data of η ( T ) were retrieved from the original figures by using a digitizer. Theretrieved data were then processed by fitting them with smoothing functions, and Q ∗ a was12 T (°C)
LillieNapolitanoJonesKawamura200 400 600 800 1000 1200 Q ( e V ) * a T g Mazurin FIG. 3: Apparent activation energy Q ∗ a of silicate glasses obtained by the conventional method.Data are for soda-lime-silicate (SLS) glass samples, except for Kawamura’s data. Kawamura’ssamples are high-level radioactive waste (HLW) glass. The sources of data are as follows: Lillie[68], Jones [69], Napolitano [70], Mazurin [71], and Kawamura [72]. obtained by taking the derivatives of the fitting function according to Eq. (4). All the originaldata and the subsequent process of analysis are provided in Supplemental material. As seenin Fig. 3, the apparent activation energy Q ∗ a increases as temperature decreases to T g . Asthe temperature approaches T g , Q ∗ a reaches about 6.8 eV and then quickly decreases. Themaximum value is more than twice the value at T = 1200 ◦ C. The linearity in the Arrheniusplot holds only for the range of 11 ≤ log η ≤
14, which corresponds to a temperature rangeof 600 ◦ C ≥ T ≥ ◦ C.Let us calculate the corrected value E b according to Eq. (8). The constant η is notknown. The lowest value of the measured viscosity cited above is about log η = 2, whichwas reported by Lullie [68]. The pre-exponential factor η is less than this value but is largerthan a typical value of normal liquids, i.e., − .
5. Since silicate glasses are among the mostviscous materials, tentatively, log η = 0 was used. The calculated values are plotted inFig. 4. The maximum value of E b reduces significantly to 2.2 eV from the correspondingvalue Q ∗ a of 6.8 eV. This E b value of 2.2 eV is already smaller than the Si-O bond energyof SiO (4.6 eV). If an increase in η with decreasing T is taken into account, the value E b is further reduced. The E b value can also be estimated by using Eq. (7). For the transition13 T (°C) E ( e V ) b
200 400 600 800 1000 1200 T g LillieNapolitanoJonesKawamuraMazurin FIG. 4: Energy barrier E b of SLS glass. The same symbols as those used in Fig. 3 are also usedhere. All data are calculated by assuming log η = 0. Under the assumption that log η = 10 . width ∆ T g of the SLS glass, the value of NBS710 ( T g / ∆ T g = 17, as shown in Fig. 2) isadopted. Equation (7) leads to E b,g − E b,l = 0 .
37 eV. The model shown in Eq. (5) assumesthat the change in E b occurs only within ∆ T g . However, as seen in Fig. 4, the change in E b begins from far higher temperatures and is more gradual against T . In this respect, Eq. (7)may be an overcorrection. The true value E b lies between 0.37 and 2.2, which covers thediffusion barrier E d of silica glass (0 . ∼ . E b below T g . It is noteasy to measure viscosity values larger than log η = 13. Despite this difficulty, viscositybelow T g were measured even in early studies [69, 73, 74]. For SLS glass, Jones reporteda significant reduction in Q ∗ a from 2.3 eV above T g to 0.25 eV at T = 350 ◦ C, at which η increases to about log η = 18 [69, 74]. The values reported by Jones are shown in Fig. 3:these values are obtained by the present analysis employing a smoothing functions, and hencethe values are slightly different from the values reported by Jones. A similar reduction wasreported by Shen et al [75]: a reduction from Q ∗ a =5.2 eV above T g to 1.2 eV at T = 450 ◦ Cwas observed for SLS glass. Surprisingly, on applying the correct form of Eq. (8), thesequick reductions in Q ∗ a below T g are drastically modified so that they are nearly constant, asshown in Fig. 4. This is consistent with the present model shown in Fig. 1(c). This behavior14s reasonable because the structure does not change once the glass substance is frozen.There are some literatures describing the activation energy as being saturated around T g [24, 76], which means a recovery of the Arrhenius law below T g . Recently, Kawamura etal. performed viscosity measurements at temperatures as low as 270 ◦ C, which may be thelowest temperature ever reported for silicate glasses [72]. They employed the fiber-bendingmethod, which was developed by Koide [77]. They performed measurements for a simulatedhigh-level radioactive waste (HLW) glass. The main difference between HLW and SLS glassis inclusion of B O in HLW. Hence, it is reasonable to observe that the viscosity of HLWglass was lower than that of SLS glass. If, for Kawamura’s data, log η is calculated to make η ( T ) constant below T g , the pre-exponential factor is obtained as log η = 10 .
4, as indicatedby the green line in Fig. 4.All the presented data were measured by independent authors and by different methods.Therefore, the agreement between the values of E b below as well as above T g validates thereliability of the obtained T dependence. The energy barrier E b becomes saturated below T g . By taking all the above results together, it is likely that E b of SLS glass is no morethan 1 eV. Furthermore, the present analysis shows that no divergence occurs at T . Thisconclusion agrees with the conclusions of recent studies on the glass transition [18, 78, 79]. B. Organic glass
The next class of glasses to be examined comprises organic glasses. Generally, the glassesbelonging to this class have low-melting temperatures that are near or lower than roomtemperature. Most of them are fragile glasses, and hence the apparent activation energy Q ∗ a strongly depends on T . When T approaches to T g from high temperatures, Q ∗ a increasesquickly, as shown in the Angell plot. Large values of Q ∗ a are reported at T g : e.g., Q ∗ a = 5.5 eVfor glucose and 1.1 eV for glycerol [21]. More examples are provided in the review by Hodge,who has presented many values of Q ∗ a in different ways depending on the definition of theactivation energy: apparent activation energies exceeding 10 eV are listed [23]. In this work, o -terphenyl is examined as an example of organic glasses. This is because the viscosity ofthis material was measured in a wide range from log η = − . T g = 239K with the width ∆ T g = 5K.For model calculations of the T dependence of o -terphenyl, see Ref. [24]. The inset of Fig. 515hows Q ∗ a of o -terphenyl as obtained by the conventional method using Eq. (4). The data E ( e V ) b T (K) g T (K)
250 350300 Q ( e V ) * a FIG. 5: Energy barrier E b of o -terphenyl glass. T g = 239K. The inset shows the apparent activationenergy Q ∗ a obtained using the traditional Arrhenius plot. The data are taken from Ref. [24]. η ( T ) were retrieved from Nemilov’s textbook (Fig. 44 of Ref. [24]): the plot data actuallyconsist of three sets of data provided by Laughlin [25], Greet [80], and Cukierman [81]. Theapparent activation energy Q ∗ a increases up to 4 eV, which is close to that of silicate glass.In the literature, this large Q ∗ a for organic glasses is at times explained by the collectivemodes of a large moving units. This explanation is not convincing. A collective mode canalter λ in the pre-exponential factor in Eq. (3). However, each atom constituting the movingunit receives equally thermal energy k B T . Hence, it is unlikely that the activation energyincreases with the increase in the size of moving units. This unreasonably large value can,however, be corrected by the present theory. By applying Eq. (8), a significant reductionfrom Q ∗ a is obtained. This is shown in Fig. 5. In this case, log η = − E b is 0.8 eV, which is within a reasonable range of the energy barrier. If the multiplicationfactor, in this case k = 50, is adopted, the value E b becomes as small as 0.1 eV, when E b,l is ignored. Now, it is clear that the large values of the apparent activation energy Q ∗ a thatare common for organic glasses can be ascribed to the narrow width ∆ T g .16 . Metallic glass The third example comprises metallic glasses. Metallic glasses have fragility intermedi-ate between that of strong covalent glasses and fragile organic glasses [82]. The apparentactivation energy Q ∗ a of the glass transition of metallic glasses ranges from 2 to 8 eV near T g [83]. The plastic deformation of crystalline materials is mediated by dislocations, whereasfor metallic glasses it is madiated by the so-called shear transformation zone (STZ) [56].The approach for describing the T dependence of the viscosity is opposite to that employedin Sec. II B, where the starting state is the liquid state. For the area of metallic glasses,the study starts from the solid state [84]. The formula for viscosity is formally the same asthat given in Eq. (2). The activation volume V a is considered to be that of an STZ. ForZrTiCuNi glass, a typical STZ includes 20–30 atoms [84].The temperature dependence of the viscosity has been gleaned from recent experimentson metallic glasses. A theoretical model was proposed by Johnson et al. , who considered aformula for the energy barrier based on the elastic moduli of the solid state and extendedit to the liquid state [83, 85]. They adapted the exponential decay for the T dependence ofthe energy barrier, obtaining a formula ηη = exp (cid:26) W g k B T exp (cid:20) n (cid:18) − TT g (cid:19)(cid:21)(cid:27) , (12)for T > T g . Here, W g is the energy barrier at T = T g , and n is a fitting parameter whosevalue is the order of unity [83]. When T approaches T g , the term exp [2 n (1 − T /Tg )] inEq. (12) can be expanded with respect to ∆ T = T − T g , resulting in ηη = exp (cid:26) W g k B T (cid:18) n − n TT g (cid:19)(cid:27) . (13)This has the same form as the linear approximation of Eq. (5) with a magnification factor k = 1 + 2 n . Numeric examples are given by them: for Pd . Cu Si . , T g = 634K, log η = − .
11, and n = 1 .
67. This leads to the magnification factor k of 4 .
3. Hence, the correctionfor E b is not too large compared with those of other classes of glasses.An interesting point regarding metallic glasses is that viscosity below T g is sometimesmeasured because of the practical interest in the creep phenomenon. Taub and Spaepenfound that the activation energy Q ∗ a of viscosity of Pd Si is almost constant at about 2.0eV in a temperature range from 420 to 540 K [86]; Pd Si is a metallic glass with T g = 63417 [87]. Interestingly, the value Q ∗ a does not change by annealing at various temperatures.This is consistent with the present assumption, Eq. (5). There is no sign to exhibit thedivergent behavior of the VFT law in this case too. V. DISCUSSION
Although phase transition was not the original focus of this study, the findings of thisstudy have further implication to this issue. The analyses in Sec. IV established that noneof the experimental data on viscosity below T g showed any sign of exponential divergenceat a finite temperature. The T dependence of η ( T ) seems to obey the Arrhenius law ratherwell. Indirect evidence for this non divergence was recently reported [18, 78, 79].Both T and T K are obtained by extrapolating viscosity and entropy, respectively, mea-sured at T > T g . This extrapolation loses its validity when one accepts the experimentaltransition occurring at T g as a genuine phase transition. Nobody considers that an extrap-olation of the T dependence of entropy ( S ∼ ln T ) of an ideal gas to T = 0 is meaningful:such extrapolation diverges negatively. The repeated assertion is that the observed glasstransition occurring at T g is not an intrinsic property of a glass because T g is affected bythe preparation conditions. This subject was already discussed in Ref. [33]. The essen-tial points are repeated herein. First, the kinetic nature of the transition is common toall liquid/solid phase transitions. Crystallization is determined by the competition of thekinetic effect of entropy and the potential effect on constraining atomic motions. Second,preparation dependence is common in every transition, though the effect is outstanding forglasses. Freezing of water to ice is a well-known example. The graphite-diamond transitiontemperature is largely affected by its kinetics. Any crystallization accompanies a kineticprocess of nucleation that can be altered by external conditions. The reference temperature T in the VFT formula is only a fitting parameter to reproduce the temperature dependenceof viscosity above T g . VI. CONCLUSION
In usual phase transitions, the energy barrier for atomic movement changes abruptly.In fact, the transition occurs at a fixed temperature, such as T m , which hides this change18rom observation. In contrast, for glass transition, the continuous structural change occursin a certain range of temperature, i.e., ∆ T g . This structural change causes a change in theenergy barrier E b in a continuous manner, and this causes η ( T ) to deviate from the Arrhe-nius law. The present theory predicts that the observed value of the activation energy Q ∗ a largely overestimates the energy barrier E b . This prediction has been validated by examiningavailable experimental data. The degree of deviation depends mainly on the magnificationfactor k = T g / ∆ T g . Generally, fragile glasses exhibit sharp transitions with narrow widths∆ T g . This explains the extraordinarily large values for the observed Q ∗ a of fragile glasses:e.g. the magnification factor is on the order of magnitude 50. This magnification also occursin strong glasses, whereas the magnification factor is not very large. In either case, thegenuine energy barrier E b is estimated to be less than 1 eV, which lies in a reasonable rangeof the chemical energy of liquids and solids. The analysis has provided another interpreta-tion for fragility of glass: fragility indicates how fast the energy barrier varies with varyingtemperature. Thus, the present result enables us to interpret the behavior of the viscosity ofglass on the same ground of energetic approach that is commonly used in solid-state physicsand chemistry. The results also provide a starting point of future study by first-principlescalculations.A shortcoming of the current status of this theory is lack of the concrete formula forthe pre-exponential factor η . Since the present assumption of Eq. (5)—the energy barrierchanges only within T g —is not very good, the lack of the concrete formula for η can causea serious problem. 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