Effects of sinusoidal temperature and pressure modulation on the structural relaxation of amorphous solids

Journal of Non-Crystalline Solids 281 (2001) 91±107

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E€ects of sinusoidal temperature and pressure modulation on the structural relaxation of amorphous solids J. Wang, G.P. Johari * Department of Materials Science and Engineering, McMaster University, 1280 Main St. West, Hamilton, Ont., Canada L8S 4L7 Received 23 March 2000; received in revised form 30 October 2000

Abstract Because molecular di€usivity in liquids and glasses varies as the inverse exponential of the reciprocal of the temperature, T , a sinusoidal change in T would change di€usivity unequally at the extremes of a T -cycle, and this would have a net e€ect on its di€usion-controlled structural relaxation towards their equilibrium states. A sinusoidal change in pressure P would have a similar e€ect. To elaborate on the expected e€ects of the T - and P -modulation on the spontaneous approach of an amorphous solid towards its equilibrium state, formalisms have been developed for two cases: (i) when an amorphous solid's T , or P , is sinusoidally modulated about its ®xed values; (ii) when the solid is heated at a constant rate and its T modulated about the increasing value of T . Similar calculations for the two cases have been done for conditions without T -modulation. The structural relaxation features computed for the T -modulated conditions show that the time- and T -dependent relaxation function, ®ctive temperature and normalized speci®c heat in the above-given two cases are quite di€erent from those for the unmodulated conditions, and that additional features in the speci®c heat appear on sinusoidal modulation for di€erent phase conditions. This also a€ects the features of glasssoftening endotherm on modulation. Sinusoidal T - or P -modulation leads to the determination of the second harmonic terms, which in turn may be used for determining the T -dependence of the di€usion process. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction One of the characteristic feature of an amorphous solid is that its physical properties change with time, a phenomenon known as isothermal structural relaxation or physical aging [1±11]. At a molecular level, structural relaxation is a consequence of self-di€usion, which brings spontaneously the solid's atomic structure to a lowerenergy, higher-density state. Heating increases the

* Corresponding author. Tel.: +1-905 525 9140; fax: +1-905 528 9295. E-mail address: [email protected] (G.P. Johari).

molecular di€usion coecient, D, and consequently the rate of structural relaxation. Cooling decreases D and thus the rate of structural relaxation. The magnitude of D decreases non-linearly with decrease in the temperature, and in its simplest form, D  exp… E=RT †, where E is the activation energy, R the gas constant and T the temperature. Therefore, if T is varied sinusoidally with time, D will increase to a maximum value when T is at the maximum and decrease to a minimum value when T is at the minimum of the cycle. The rate of structural relaxation would increase and decrease in a corresponding manner. Because of the non-linear dependence of D on T , the maximum increase in D at the maximum

0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 4 3 2 - 4

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J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

temperature of the sinusoidal cycle will be more than the maximum decrease in D at the minimum temperature of the sinusoidal cycle. Consequently, the value of D will become time-dependent through temperature, and therefore the e€ective D (and the rate of structural relaxation) will be more than D at the mean temperature Tmean ; D…Tmean † of the sinusoidal oscillation, and its magnitude will correspond to an average temperature higher than the mean temperature. For a material whose D and structural relaxation rate do not spontaneously decrease with time, there will be a net change in its measured physical property after one cycle of sinusoidal T modulation, which will result from the e€ective D being greater than D…Tmean †. This change will be di€erent from the net change which will result from the change in D during the sinusoidal cycle when the same material is kept isothermally at Tmean of the sinusoidal oscillation. The di€erence between the net change in the physical property resulting from di€usion would accumulate with both time and increase in amplitude of the T modulation. Qualitatively similar additional change will occur for a material whose D also changes with time. Here we provide a mathematical formalism for this e€ect, and calculate the change in a given physical property as a function of time for two cases: (i) time-invariant and (ii) time-variant. We also provide an analysis for the second-order harmonic terms in the non-linear e€ects of sinusoidal variations in temperature. Finally, we develop a formalism for the condition when the temperature is changed at a prescribed rate and describe how the T -modulations will affect the magnitude of properties such as enthalpy, volume and speci®c heat. Thus we simulate the usual di€erential scanning calorimetry scan, i.e., a plot of the speci®c heat against T , as a glass is heated through its glass-softening range until it has reached its equilibrium liquid state [1±11]. The e€ect is fundamental to all processes in which temperature, pressure or concentration may be sinusoidally modulated. Therefore, it has signi®cance for our current e€orts in incorporating temperature modulation in calorimetry [12±15], and studying the frequency-dependent speci®c heat [16±19] and thermal conductivity [19]. As

propagation of sound waves occurs by sinusoidal adiabatic compression and decompression, and is seen as equivalent to a sinusoidal temperature oscillation, we also provide the corresponding e€ects of pressure modulation. In addition to their use in monitoring irreversible processes by using ultrasonic waves and in studying spinodal decomposition in a liquid [19], temperature- and pressure-modulation may provide signi®cance information on the second-order phase transformation, particularly at the critical point of a liquid [20], for which formalisms are available (see citations in [20]) in terms of dynamic scaling [21,22]. In this paper a detailed discussion is provided only for the study of the dynamics of supercooled liquid and glasses in which there is currently great interest [1±3,6]. Subsequent papers will provide a discussion of the other phenomenon. An application of this e€ect to the chemical process of polymerization kinetics has been described before [23]. 2. Sinusoidal modulation and physical properties First, we consider in general terms how sinusoidal modulation of a variable, y, e€ects an intrinsic property, q, of a material. When the change of y with macroscopic time, t, is much slower than the response of the material's property, the measured value of q is the equilibrium value. For a sinusoidal variation in y for that condition, the magnitude of y at time t is given by y…t† ˆ y0 ‡ Dy sin…x0 t†;

…1†

where y0 is the mean value of y, and Dy is its modulation amplitude. As y varies sinusoidally about y0 , the material's properties, such as volume, enthalpy, entropy, viscosity, relaxation time, diffusion coecient, surface tension, permittivity, mechanical modulus, refractive index, phonon-related properties, etc., would also change cyclically with t. Because both q and dq=dy are non-linear in y, the net change in q at any instant during the upper part of the cycle will not be equal and opposite to the net change in q at the corresponding instant during the lower part of the cycle. The oscillation in q would become asymmetric about y0 and consequently, the average value, hqi will not

J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

be equal to the value of q…y ˆ y0 †. As the oscillation of q will be asymmetric and only appear to be apparently sinusoidal, the magnitude of hqi will correspond to the new value of y, or hyi, which would be shifted from y0 by an amount dy. According to Eq. (1), the magnitude of dy would vary with the type of variable y (i.e., T or pressure P ), the magnitude of Dy and the manner in which q changes with y. When y is chosen to be T and the property q is such that both …dq=dy† and …d2 q=dy 2 † are positive, as for volume, enthalpy and entropy, hqi would be greater than q0 at y0 . This means that hyi would also exceed y0 Similarly, when q is chosen such that …dq=dy† is negative and …d2 q=dy 2 † positive, as for the variation of relaxation time, viscosity, permittivity, surface tension, and elastic modulus with temperature, hqi would still be greater than q0 at y0 but hyi will be less than y0 . If, on the other hand, y is chosen to be P , whose e€ect is opposite to that of T , hqi would always exceed q0 and hyi would be less than y0 for volume, enthalpy and entropy and more for relaxation time, viscosity, permittivity, surface tension and elastic modulus. The reason is that the ®rst set of properties decreases with increase in P , and the second set of properties increases with increase in P .

3. Harmonic terms in sinusoidal modulation Sinusoidal modulation is expected to produce one more e€ect, which may be expressed in terms of the second and higher harmonic terms, when the change in q is non-linear with change in y. Consequently, the observed magnitude of q itself may become a re¯ection of the magnitude of nonlinearity. We consider how this may occur, in general terms. As a non-linear function may be expressed as a Taylor expression near the zero point of the variable, the magnitude of q near q0 may be given by   dq …y y0 † q…y† ˆ q0 ‡ dy yˆy0  2  dq …y y0 †2 ‡


…2† ‡ dy 2 yˆy0 2!

On substituting for y from Eq. (1),   dq Dy sin x0 t q…y† ˆ q0 ‡ dy yˆy0  2  2 dq …Dy† sin2 x0 t ‡


‡ dy 2 yˆy0 2

93

…3†

The sin2 x0 t term in Eq. (3) may be replaced by …1 cos 2x0 t†=2 and, on rearranging,    2  2 dq …Dy† dq ‡ Dy q…y† ˆ q0 ‡ 2 dy yˆy0 4 dy yˆy0  2  2 dq …Dy† cos 2x0 t  sin x0 t dy 2 yˆy0 4 ‡




…4†

Hence, the average q over the modulation period of one cycle is given by Z 1 t‡…t0 =2† 0 0 hqi…t† ˆ q…t † dt ; …5† t0 t …t0 =2† where t0 …ˆ 2p=x0 † refers to the modulation period. The third- and further odd-order terms on the right-hand side of Eq. (4) do not contribute to hqi because their magnitude oscillates symmetrically near their zero value. Thus, after combining Eqs. (4) and (5) and integrating,  2   2 dq Dy hqi ˆ q0 ‡ ‡


…6† dy 2 yˆy0 4 The quantity hqi, as observed on modulation, di€ers from q0 observed without modulation. This di€erence is given by  2   2 dq Dy hqi q0 ˆ ‡


…7† dy 2 yˆy0 4 (Note that Eq. (7) is more generally obtained by the integration of sinm x0 t (where m is an integer), which gives zero for odd values of m and is equal to 1  3  5 


 …m 1†=…2  4  6 


 m† for even values of m.) According to Eq. (7), the di€erence will be positive or negative depending on the sign of the term …d2 q=dy 2 † at y ˆ y0 . As mentioned earlier here, when sinusoidal modulation is in T , d2 q=dT 2 is positive for enthalpy, entropy,

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J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

volume, viscosity, and relaxation time, irrespective of …dq=dT † being negative or positive. The reason is that the slope of the plot of these quantities is known to increase with increase in T . When sinusoidal modulation is performed in P , d2 q=dP 2 is also positive for properties such as enthalpy, entropy, viscosity, and relaxation time for which dq=dP > 0, and for the volume for which dq=dP < 0.

4. Temperature and pressure modulation without irreversible changes The above-given general description may now be made speci®c for the sinusoidal modulation of the variable T , or P , by using the structural relaxation time, s, as a di€usion-related, intrinsic property of a material. We may express the variation of s with T in its simplest form, by the Arrhenius equation   EA s ˆ s1 exp ; …8† RT where s1 is the magnitude of s at a formally in®nite temperature, EA is the Arrhenius energy and R the gas constant. A sinusoidal variation of T …ˆ T0 ‡ DT sin x0 t† will produce a variation of s, according to   EA s…t† ˆ s1 exp …9† R…T0 ‡ DT sin x0 t† and s…t† will reach a maximum value, when sin x0 t ˆ 1, at Tmin of the cycle,   EA smax …t† ˆ s1 exp : …10† R…T0 DT † It will reach a minimum value, when sin x0 t ˆ 1, at Tmax of the cycle,   EA smin …t† ˆ s1 exp : …11† R…T0 ‡ DT † During each sinusoidal cycle, hsi, the average value of s, will not correspond to T0 . Instead it will correspond to a temperature …T0 ‡ dT †, and its average magnitude will be given by

1 hsi…t† ˆ t0

Z

t‡…t0 =2†

t …t0 =2†

s…t0 † dt0 ;

…12†

where t refers to the time at which the observation is made and t0 …ˆ 2p=x0 † is the T -modulation period. The corresponding equation for sinusoidal pressure variation at a constant temperature is given by   DV  P …t† ; …13† s…t† ˆ s…P ˆ 0† exp RT where s…P ˆ 0† is the magnitude of s at zero pressure, DV  the volume of activation, and R the gas constant. For a sinusoidal variation of P , P …t† ˆ P0 ‡ DP sin…x0 t†: On combining Eqs. (13) and (13a),   DV  …P0 ‡ DP sin x0 t† s…t† ˆ s…P ˆ 0† exp RT

…13a†

…14†

and s…t† will reach a maximum value, when sin x0 t ˆ 1, at Pmax of the cycle,   DV  …P0 ‡ DP0 † ; …15† smax …t† ˆ s…P ˆ 0† exp RT and a minimum value, when sin x0 t ˆ 1, at Pmin of the cycle,   DV  …P0 DP0 † : …16† smax …t† ˆ s…P ˆ 0† exp RT Therefore, during a sinusoidal P -cycle, hsi will not correspond to s at P0 . Instead, it will correspond to s at …P0 ‡ dP †, and its average magnitude at time t will be given by Z 1 t‡…t0 =2† 0 0 hsi…t† ˆ s…t † dt ; …17† t0 t …t0 =2† where t0 …ˆ 2p=x0 † is the P -modulation period. For simulating these e€ects, we use s1 ˆ 10 14 s, EA ˆ 500 kJ=mol and T0 ˆ 200 K and calculate the variation of s with t for both the unmodulated and T -modulated conditions, the latter with a DT of 1.5 K and an x0 of 2p  10 2 rad s 1 (or f ˆ 10 mHz). The calculated value of s is plotted against time t on the ordinate

J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

scale on the right-hand side in Fig. 1(a), for both the modulated (dashed line) and unmodulated (solid horizontal line) conditions, and the corresponding sinusoidal variation of temperature shown (solid line). The calculated hsi is plotted against t on the ordinate scale in Fig. 1(a) for the modulated condition (dotted line). The change on sinusoidal variation of pressure was calculated by using P0 ˆ 500 bar, s…P ˆ 0† ˆ 1 s, DV ˆ 100 ml mol 1 , x0 ˆ 2p  10 2 rad s 1 (f ˆ 10 mHz) and DP ˆ 100 bar, and the calculated value of s is plotted against t on the right-hand side in Fig. 1(b) for the modulated (dashed line) and the unmodulated (full horizontal line) condition. Here the

95

corresponding sinusoidal variation in P and modulated hsi are shown by the full line and the dotted line. At the outset of this paper, we considered that modulation would increase the rate of the property's spontaneous change with time if the property itself changed non-linearly with T and P . It is instructive to show how this increase may occur by using a formalism for the non-linear variation of the property with T or P . We illustrate it for the simplest case for a dynamics, say di€usivity, whose rate, k …ˆ 1=s†, varies with T in an Arrhenius manner k ˆ k0 exp… EA =RT †, where EA is the activation energy. The di€erence between the average value of hki and k at T0 may be obtained by combining Eqs. (7) and (8),  hki

k0 ˆ

d2 k dT 2



…DT † 4 T ˆT0

2

…18†

or 2

hki

…DT † k0 EA k0 ˆ 4RT03





EA RT0

2 exp



 EA ; RT0 …19†

where all notations are as described before. For the usual condition, EA =RT0 > 2, in Eq. (19), the term …hki k0 † is positive. Accordingly, …hki k0 † would increase linearly with (DT 2 ) for a given EA and T0 . It would of course be much more sensitive to the term exp… EA =RT0 † than to the cubic term in T0 in Eq. (19). Similarly, for pressure modulation, k ˆ k0 exp… DV  P =RT †;

…20†

or 

Fig. 1. (a) The variation of temperature against real time for sinusoidal modulation (full line) and the consequent change in the relaxation time; (b) the variation of pressure against real time for sinusoidal modulation (full line) and the consequent change in the relaxation time (dashed line). The full horizontal lines represent the mean temperature or pressure, T0 or P0 . The same lines in (a) and (b) indicate the relaxation time, s0 , at T0 or P0 on the right-hand side scale. The dotted line represents the average relaxation time, hsi as labeled. The frequency of oscillation is 1 rad s 1 . Parameters used for the calculation are given in the text.

hki

d2 k dP 2



2

…DP † 4 P ˆP0    2  2 …DP † k0 DV ˆ exp 4 RT

k0 ˆ

DV  P RT

 ; …21†

or the average rate of the process on P -modulation will be greater than its rate in the unmodulated conditions, and this di€erence may be calculated from the knowledge of DV  and k0 .

96

J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

To summarize, when the second derivative of k with respect to y is positive, i.e., …d2 k=dy 2 † > 0, sinusoidal modulation of y will increase the e€ective rate, and this increase may be calculated from the activation parameters for the variable y. 5. Simulation of a structural relaxation process It is recognized that the normalized relaxation function, /…t†, for molecular dynamics of a linear stretched exponential model in macroscopic time t is given by "  b # t /…t†jt!t0 ˆ exp ; …22† s…t ! t0 † where s is the characteristic structural relaxation time and b is an empirical parameter with a value between 0 and 1. More generally, by using the Hopkins±Narayanaswamy reduced time variable, Z t dt0 n…t† ˆ ; …22a† 0 0 s…t † the non-linear form of the stretched exponential relaxation function can be written as [1±4,24±26] ( Z b ) t dt0 b /…t† ˆ exp‰ n…t† Š ˆ exp : 0 0 s…t † …23† The magnitude of b determines both the rate of change in a physical property, e.g., enthalpy H , volume V , the index of refraction at a given instant, and the shape of their plots against t. The physical properties of an amorphous solid are described also by its ®ctive temperature, Tf which is de®ned as the temperature at which the material would correspond to the internal equilibrium state [1±4,24±26]. Incorporating the superposition principle, Tf can be calculated from ( " Z b #) Z T T dT 00 0 ; Tf ˆ T1 ‡ dT 1 exp 00 00 T1 T 0 q…T †s…T † …24†

where T1 is the temperature from which cooling of the equilibrium liquid begins, and q is the cooling

or the heating rate …q ˆ dT =dt†. The non-linearity of the change in s with t at a ®xed temperature is described by a commonly used equation [1± 8,14,15,26],  s ˆ A exp

 xDh …1 x†Dh ; ‡ RT RTf

…25†

where A is the pre-exponential term, with units of time, which is found to be much less than the characteristic time for the phonon modes [1±4], x is the non-linearity parameter, whose magnitude is between 0 and 1, Dh is the e€ective activation energy, R the gas constant and T is the ®xed temperature. (Note that non-linearity here refers to a departure from the Arrhenius equation, for which x ˆ 1.) According to Eq. (25), s of an amorphous solid increases on its structural relaxation as Tf decreases with time. Before using these equations for calculating the e€ects, it seems pertinent to provide a brief account of the vitri®cation and the glass-softening processes and their relevance to Tf , enthalpy or energy and volume. A glass is produced usually by cooling an equilibrium liquid at a certain rate to a temperature where it becomes rigid on the timescale of one's observations. During cooling, its Tf , di€usion coecient, energy and volume decrease continuously until a temperature is reached when the di€usion becomes too slow to contribute to the con®gurational energy and volume in the time period allowed by the cooling rate. At this temperature, the liquid is said to be vitri®ed and is seen as a glass. Further cooling of a glass does not decrease its Tf . This T -invariant value of Tf of the vitri®ed state is denoted by Tf …0†, which is the ®ctive temperature at the end of its cooling process (or the beginning of its isothermal relaxation). The energy, volume and Tf …0† of a glass are therefore high when the cooling rate is high, and low when the cooling rate is low. When kept at a T below the vitri®cation temperature, a glass undergoes structural relaxation which may be studied either at a ®xed T or with increasing T by heating at a ®xed rate. At a ®xed T , structural relaxation decreases its Tf monotonically with t at a rate that itself decreases with decrease in Tf as for a self-retarding

J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

process. The glassy state of a given material structurally relaxes faster when it has been produced by rapid cooling and its Tf …0† is high than when it has been produced by slow cooling and its Tf …0† is low. On heating at a certain rate, structural relaxation is a€ected by both t and T . As t increases, structural relaxation rate decreases at a ®xed T , and as T is increased at a (mathematically) ®xed t, the relaxation rate increases. Thus the rate of decrease of Tf is reduced as t increases and increased as T increases. The net change in Tf is determined by the partial cancellation of the two e€ects, and therefore by the heating rate. For computation of s from Eqs. (23)±(25), it is required that cooling and heating be done in consecutive, step-wise changes, DTj for T , after a certain time, Dtk for t. On including the DTj and Dtk steps for cooling and heating processes, we may write 2 /…t† ˆ exp 4  sn ˆ A exp

Tf;n ˆ T1 ‡

n X Dtk sk kˆ0

!b 3 5;

…26†

 xDh …1 x†Dh ‡ ; RTn RTf;n 1

n X jˆ1

8 <

DTj 1 :

2 exp 4

…27† n X DTk q k sk kˆj

!b 39 = 5 ; ; …28†

where n is the all-inclusive number of steps in cooling, isothermally keeping a glass and thereafter in the heating procedures. For procedures in which isothermal relaxation occurs, DTk =…qk sk † in Eq. (28) becomes Dtk =sk . The computational accuracy increases as the magnitude of DTj and Dtk in Eq. (28) is reduced but the total time for the computation increases. As a compromise between accuracy and the time required for computation, we used DTj  0:1±0:2 K for cooling, and 1/64 K (15.625 mK) for heating, with a Dtk of t0 =16±t0 =256 for T -modulation. The integration needed for these calculations and the related procedure have been generally accepted as valid and appropriate [1±4,27].

97

For the unmodulated condition, Eqs. (23) and (24) yield decrease in / and Tf with t. When T is modulated, / and Tf change with t also for the reason that T in the exponential term of Eq. (25) changes according to DT sin x0 t, producing a sinusoidal change in Tf , which is superimposed on the monotonic decrease of Tf with t. Thus / and Tf change with t at a ®xed T0 for the unmodulated condition, but change with t at an average hT i for the T -modulated condition. For computing the changes due to structural relaxation isothermally, a liquid may be cooled rapidly from its equilibrium state at a high temperature to a non-equilibrium metastable state of a glass. During the cooling, its molecular dynamics is much faster than the modulation period, and therefore it will remain at equilibrium at both extremes, Tmax and Tmin , of the modulation cycle, provided the cooling rate is itself much faster than x0 as long as T  Tg . Thus the e€ect of modulation during cooling is insigni®cant. When the temperature decreases to the vitri®cation range and transverse it, the e€ects become signi®cant and need to be included in the formalism. These e€ects may then be computed either without T -modulation, or with T -modulation using a selected value of DT . As the thermodynamic or optical properties are usually directly proportional to Tf , the calculations require only the determination of Tf , which may then be related to those properties. We may use several cooling rates and di€erent values of T0 to calculate the decrease in Tf with t. Since the ®ctive temperature and the relaxation function in a modulation condition vary with time during a cycle, only average values of Tf;mod and /mod could be determined. These average values of hTf;mod i and h/mod i were calculated from Eq. (5), where hqi…t† represents hTf;mod i…t† and h/mod i…t†. As an example, the material may be cooled from the equilibrium state at 470 K to the non-equilibrium state at 370 K at 720 K min 1 in DTj steps of 0.1 K, and decrease in Tf determined during cooling. The system may be kept at a T0 of 370 K and its / and Tf computed with increasing t using Eqs. (23)± (25), initially for the simplest condition, b ˆ x ˆ 1, with ln A ˆ 150 and Dh ˆ 590 kJ mol 1 . The calculated / and Tf for di€erent conditions are plotted against t in Figs. 2(a) and (b), respectively.

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J. Wang, G.P. Johari / Journal of Non-Crystalline Solids 281 (2001) 91±107

Fig. 2. (a) The plots of the relaxation function, /, against real time for the T -modulated and unmodulated conditions; (b) the plots of the ®ctive temperature, Tf against real time for the T modulated and unmodulated conditions; (c) the plots of the di€erence d/ between the / for the two conditions; (d) the plots of the di€erence dTf between the Tf for the two conditions. The line passing through the oscillating curves for /mod ; d/, hTf;mod i and dTf is the average value in each case. Parameters used for the calculations are given in the text. The data refer to isothermal annealing.

For the T -modulated case, we use, DT ˆ 2:5 K and x0 ˆ p mrad s 1 for cooling the sample at 720 K min 1 , from 470 to 370 K in DTj steps of 0.1 K. The sample is then kept at a T0 of 370 K, with DT and x0 for T -modulation the same as during the cooling described above and its /mod …t† and Tf;mod …t† calculated by using the same values of b; x; ln A and Dh as for the unmodulated condition. These are also plotted in Figs. 2(a) and (b). The di€erence between the / values for the unmodulated and modulated conditions, d/ ˆ /unmod /mod , is plotted in Fig. 2(c) and that between the Tf values, dTf ˆ Tf;unmod hTf;mod i in Fig. 2(d). These show that /unmod and Tf;unmod are,

as expected, decreasing functions of t, but while /mod and Tf;mod remain overall decreasing functions, the decrease is superposed on another e€ect, which causes local oscillations in its slope in the plots against t. Values of h/mod i; hTf;mod i; hd/i and hdTf i are also determined and are plotted against t in Figs. 2(a)±(d), which show that h/mod i decreases faster than /, and hTf;mod i decreases faster than Tf;unmod initially and then slowly. This produces a crossover in the Tf and dTf curves at a certain time, tx . The oscillation envelope of h/mod i is initially large, and tends to vanish as t ! 1. Similarly, the oscillation envelope of Tf;mod is large initially, decreases up to a certain time tx and thereafter increases and becomes constant as t ! 1. The observation in Fig 2(b) that Tf;mod becomes higher than Tf;unmod as t ! 1 seems surprising at ®rst sight, and needs to be explained. The temperature increase to Tmax on modulation brings the sample to a high-Tf state where its structural relaxation is faster than at T0 , and Tf decreases more rapidly than at T0 . The temperature decrease to Tmin correspondingly causes Tf to decrease more slowly than at T0 . Thus, after one modulation, the sample is left with a Tf lower than the Tf without modulation at the same t. When t has increased such that the sample is close to the equilibrium state, an increase in T of the sinusoidal cycle to Tmax causes it to cross the equilibrium line. Here Tf now increases with t instead of decreasing with t. A decrease to Tmin of the cycle returns it to the condition where Tf still decreases. Since structural relaxation is faster at Tmax than at Tmin , the net e€ect is that hTf;mod i becomes equal to Tf;unmod . When the sample is closer still to the equilibrium state, hTf;mod i exceeds Tf;unmod and then remains so as t ! 1. The di€erence at t ! 1 is expected be less than DT =2, and would vary with the activation energy of the relaxation. The above calculations show the results for the conditions (i) / is a simple exponential, or b ˆ 1 in Eq. (22), and (ii) s varies in an Arrhenius manner, or x ˆ 1 in Eq. (24), and not additionally with Tf . We now perform calculations for the condition when / is a stretched-exponential, i.e., when 0 < b 6 1, and s varies additionally with Tf , i.e., 0 < x 6 1. For this purpose, we choose ln A ˆ 150 and Dh ˆ 590 kJ mol 1 , as in the preceding cal-

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culations, and three conditions, (i) b ˆ 1 and x ˆ 0:5, (ii) b ˆ 0:5 and x ˆ 0:5, and (iii) b ˆ 0:5 and x ˆ 1. The calculated values of hd/i for these conditions are plotted against t in Fig. 3(a) and of hdTf i in Fig. 3(b). The plots show that the peak in hd/i which appears at 7 ks shifts to a shorter time at 1 ks and its height decreases from 0.11 to 0.06 when b is decreased from 1 to 0.5. The e€ect of decreasing x or increasing non-linearity is qualitatively similar to a decrease in b, but is enhanced when b is decreased. The crossover time, Tx , of hdTf i increases when b is decreased at a constant x. It increases also when x is decreased at a constant b. Thus, in principle, a set of b and x pairs may be found for which Tx will remain constant. Next, we consider the e€ects of a change in the modulation frequency x0 on Tf . For that purpose we use x0 values as 2p=30, 10p=30 and 20p=30 rad s 1 , b ˆ 1, x ˆ 1, ln A ˆ 150 and

Dh ˆ 590 kJ mol 1 as before, and calculate hd/i and hdTf i. These are plotted against time in Figs. 4(a) and (b), respectively, which shows that the height of the peak in hd/i decreases on increasing the modulation frequency and the crossover time tx of hdTf i, which is shown by the crossing of the zero-line, shifts to longer time. The above-given calculations are based on the conditions that T ˆ T0 at t ˆ 0, or that T increases towards Tmax of the sinusoidal cycle as the structural relaxation begins from Tf …0†. Another condition of interest is when T decreases towards Tmin of the sinusoidal cycle as the structural relaxation begins from Tf …0†, i.e., the phase angle is shifted by p, or equivalently x0 becomes x0 . For this condition, we may write DT sin…x0 t ‡ p† in place of DT sin…x0 t† for the p-shifted phase starting at T ˆ T0 and calculate the values of h/i and hdTf i starting from Tf …0†. These values are also plotted

Fig. 3. (a) The plots of the average value for d/ against real time for four pairs of b and x, as noted; (b) plots of dTf against real time. Parameters used for the calculation are given in the text.

Fig. 4. (a) The plots of the average value for d/ against real time for three cases with di€erent values of x0 as noted. The curve labeled x0 is for the condition when the phase angle was shifted by p. (b) The plots of dTf against real time. Parameters used for the calculation are given in the text.

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in Figs. 4(a) and (b). The plots show that up to a macroscopic time, t ˆ p=x0 ; hd/i remains negative and thereafter becomes positive. The position of the peak in hd/i is shifted to the right and its height decreases in comparison with the height for the unshifted phase conditions. This produces two crossover points in the corresponding plot of hdTf i in the case of the p-shifted phase angle. The second crossover point, tx;2 , which appears after the hdTf i peak, corresponds to the crossover point tx in the normal phase conditions, and its magnitude is less than that of tx . 6. Modulation during heating

375 K for 1 h with T -modulation using DT ˆ 0:637 K and x0 ˆ 2p=60 rad s 1 . It was ®nally heated from 375 to 400 K at 4 K min 1 with the same T -modulation at 375 K and using computational steps of 1/64 K. The calculations were performed for four pairs of b and x values, (i) b ˆ 1:0; x ˆ 1, (ii) b ˆ 0:5, x ˆ 1, (iii) b ˆ 1, x ˆ 0:5, and (iv) b ˆ 0:5, x ˆ 0:5. For each pair, this yields two curves against the temperature, one for Tf;unmod and the second for hTf;mod i (calculated by averaging Tf;mod using Eq. (5)), which are shown in Fig. 5(a). The di€erence, dTf …ˆ hTf;mod i Tf;unmod †, between each set of the two curves in Fig. 5(a) is plotted against T in Fig. 5(b).

When an amorphous solid is heated at a certain rate, its structural relaxation time changes in two ways: (i) it increases with t as the sample approaches its equilibrium state, and this increase occurs at a higher rate when T increases, and (ii) it decreases as T increases. A combination of these e€ects is expressed by Eq. (27), with 0 < x 6 1. For this purpose, the quantity determined experimentally is the speci®c heat, Cp . Since Tf is directly proportional to the enthalpy or energy, its di€erential with respect to T yields Cp . The calculations yield Cp values, already normalized by the di€erence between the Cp at the two extreme temperatures in the glass-softening temperature range. The normalized Cp is then compared against the normalized value of the measured Cp . We drop the notation for normalized, and write [1±4,27] Cp;n ˆ

dTf Tf;n ˆ dT Tn

Tf;n 1 : Tn 1

…29†

For simulation of the Cp values during the heating of a glass, we use ln A ˆ 355:7, Dh ˆ 1147:4 kJ mol 1 (the same values as for a typical polymer, polymethyl methacrylate [1±3]) and several values of b and x. The simulation was begun by cooling the equilibrium liquid from 400 K to T0 of 375 K at 20 K min 1 , using computational steps of 0.2 K. It was kept at 375 K for 1 h in the unmodulated condition and heated to 400 K at 4 K min 1 in the unmodulated condition using the computational steps of 1/64 K. The procedure was repeated for cooling to 375 K, and the sample was then kept at

Fig. 5. (a) The plots of Tf for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, dTf , between hTf;mod i and Tf;unmod against temperature. Values of b and x are as noted. The line labeled 1 is the equilibrium line when T ˆ Tf . Other parameters used for the calculation are given in the text. The horizontal lines are at zero values in each case, with the same vertical scale. The simulation was done for the sine wave function.

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Both hCp;mod i and Cp;unmod were calculated by di€erentiating the values of hTf;mod i and Tf;unmod in the curves of Fig. 5(a) with respect to T . These are plotted against T in Fig. 6(a) and the di€erence, dCp …ˆ Cp;unmod hCp;mod i† is plotted against T in Fig. 6(b). To examine the e€ect of the phase angle at the start of the modulation, hCp;mod i and dCp were calculated for the condition of modulation with a p-phase shift, or equivalently x0 ˆ 2p=60 rad s 1 . These values are plotted in thick full line in Figs. 6(a) and (b). To examine the e€ects of increase in annealing period, the calculations were repeated with the same parameters with the annealing period increased from 1 to 4 h. The plots of the calculated Cp;unmod and hCp;mod i and dCp are shown in Figs. 7(a) and (b).

Fig. 7. (a) The plots of Cp for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, dCp , between Cp;unmod and hCp;mod i. Values of b and x are as noted. Other parameters used are the same as for Fig. 5, except that the isothermal structural relaxation time is increased to 4 h. The horizontal lines are at zero values in each case, with the same vertical scale.

Fig. 6. (a) The plots of Cp for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, dCp , between Cp;unmod and hCp;mod i. Values of b and x are as noted. The thick full lines are the plots of hCp;mod i and dCp for T -modulation in which the phase had been p-shifted at the beginning. Other parameters used for the calculation are given in the text. The horizontal lines are at zero values in each case, with the same vertical scale.

Cp;unmod and hCp;mod i change also with change in the cooling rate, heating rate, and the magnitudes of ln A and Dh , and Cp;mod further varies with modulation frequency and DT . Therefore, there are virtually an in®nite number of shapes of the curves that may be generated to show these variations. However, ln A; Dh ; b and x are unique for a given liquid and glass, and when the cooling rate of the liquid and the amplitude of T -modulation are kept ®xed, e€ects of a variation of only two quantities need be shown here: (i) variation of the heating rate and (ii) variation of the modulation frequency. For that purpose, we recalculate Cp;unmod and hCp;mod i by using the same parameters as those used for the curves in Fig. 6(a), but decrease the heating rate from 4 to 1 K min 1 . The

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curves obtained are shown in Figs. 8(a) and (b). In the second calculation, we increase x0 from 2p=60 to 8p=60 rad s 1 and obtain the curves shown in Figs. 9(a) and (b). Finally, we determine the e€ect of the change in the shape of the modulation curve on Tf ; dTf ; Cp;unmod and hCp;mod i. For that purpose the T modulation was changed from a sinusoidal wave to a square wave of the same frequency while keeping other conditions the same as in Figs. 5 and 6 and Tf ; dTf ; Cp;unmod and hCp;mod i and dCp were recalculated. These values are plotted against temperature in Figs. 10 and 11. (Note that Fig. 10 was obtained for the square wave modulation and Fig. 5 for the sine wave modulation. The plots in

Fig. 9. (a) The plots of Cp for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, dCp , between Cp;unmod and hCp;mod i. Values of b and x are as noted. Other parameters used are the same as for Fig. 5, except that the frequency of sinusoidal modulation has been increased to 8p=60 rad s 1 . The horizontal lines are at zero values in each case, with the same vertical scale.

these two ®gures di€er in detail, although they may look similar.)

7. Discussion

Fig. 8. (a) The plots of Cp for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 1 K min 1 ; (b) the plots of the di€erence, dCp , between Cp;unmod and hCp;mod i. Values of b and x are as noted. Other parameters used are the same as for Fig. 5, except that the heating rate is decreased to 1 K min 1 . The horizontal lines are at zero values in each case, with the same vertical scale.

First, it is necessary to determine whether the results of T -modulation described above are consistent thermodynamically. To do so, we determined the net heat evolved on isothermal structural relaxation and during the heating plus the heat absorbed during heating through the glass-softening range for both the unmodulated and modulated conditions using the data given here. It is required by the ®rst law of thermody-

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Fig. 10. (a) The plots of Tf for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 . The line labeled 1 is the equilibrium line for Tf ˆ T . (b) The plots of the di€erence, dTf , between hTf;mod i and Tf;unmod against temperature. Values of b and x are as noted. Other parameters used for the calculation are given as for Fig. 5, except that the modulation is according to a square wave. The horizontal lines are at zero values in each case, with the same vertical scale.

namics that the Cp dT integral between two temperatures, one for the glass and second for the equilibrium liquid, be identical, irrespective of the values of DT ; x0 ; b, x, ln A; Dh , cooling rate or heating rates, and the time for isothermal structural relaxation. This was tested by numerical integration of the Cp against T plots shown in Figs. 6±11, and several more not shown here. The difference between the Cp dT integrals for the various conditions was found to be less than 0.04%, which is due to the errors in the averaging procedures and the integration used here. Therefore, we conclude that the calculations given here are thermodynamically consistent internally. As most of the physical aspects have already been described here,

103

Fig. 11. (a) The plots of Cp for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, dCp , between Cp;unmod and hCp;mod i. Values of b and x are as noted. Other parameters used are the same as for Fig. 5, except that the T -modulation is in the form of a square wave and not a sine wave. The horizontal lines are at zero values in each case, with the same vertical scale.

we only discuss the modi®cations in the physical properties resulting from T modulation as follows. 7.1. Structural relaxation's modi®cation on net enthalpy and entropy The ultimate Tf for the unmodulated condition is the temperature at which the glass is isothermally kept. The ®rst e€ect of modulation during isothermal structural relaxation is that the e€ective Tf is raised above the isothermal or mean temperature, T0 after the crossover time, tx . Since Tf is directly proportional to enthalpy, entropy and volume, and inversely proportional to the elastic modulus, refractive index, and phonon frequencies,

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this means that the ®rst set of properties will be higher for the modulated condition and the second set of properties lower. This will appear as a lessthan-expected loss in the magnitude of the properties in a structural relaxation experiment. The discrepancy in the properties would correspond to an increase in the temperature by dTf . As dTf itself approaches the magnitude of DT as t ! 1, the discrepancy observed ultimately would correspond to the di€erence between the magnitudes of the property at T0 and at T0 ‡ Tmax . This appears to be a natural consequence of a non-linear change in the property with change in T .

As the endothermic features change on T -modulation, it is clear that Tg determined from the T modulated experiment would be di€erent from that determined from the unmodulated experiment. This is shown by the lines drawn on the curve for b ˆ 1; x ˆ 0:5, in Fig. 7(a), and which yields a lower Tg for the T -modulated heating than for unmodulated heating. More signi®cantly, the extra features observed on T -modulation may lead to a misinterpretation of the dynamics of the glass and supercooled liquids.

7.2. The time- and temperature-dependent heat capacity and Tg

When T -modulation is done by using a square wave instead of a sine wave, the e€ects observed become more prominent, because the glass stays at its Tmax and Tmin for a longer duration than it does in the sinusoidal modulation. Also, instead of the smooth curves which are observed for sinusoidal modulation for the same heating rate and x0 , abrupt changes in Cp appear at the time (and temperature during heating at a ®xed rate) when T in the square wave suddenly increases or suddenly decreases. So, though the features become less smooth, the increase in the magnitude of the e€ect clearly shows that the modulation e€ects are substantial.

Parts (a) of Figs. 6±10 clearly show that extra features in Cp appear when a glass is heated with T -modulation. When b ˆ x ˆ 1, the modulation produces a second sigmoid-like feature before the equilibrium state of the liquid is reached as shown in Fig. 6(a). This feature remains even when the glass is structurally relaxed isothermally for a longer period prior to heating (Fig. 7(a)). The decrease in heating rate from 4 to 1 K min 1 tends to reduce this feature, as shown in Fig. 8(a), as does an increase in x0 in Fig. 9(a), where the shape of the curve is not smooth but is de®ned by a wave whose further averaging would reveal the curve. A decrease in the magnitude of either b or x or both, from unity, changes the shape of the feature but more so for the modulated conditions. The maximum change is found when x is reduced and b is kept ®xed, as for the case b ˆ 1; x ˆ 0:5 in Figs. 6(a) and 7(a). Here T -modulation produces the appearance of a second peak at high temperatures, only partially merged with the initial sigmoidshape rise. Its magnitude decreases with decrease in heating rate and with increase in x0 as seen in Figs. 8(a) and 9(a), respectively, where the curves apparently require further averaging for smoothing. The glass-softening temperature, Tg , is determined from the intersection temperature of two straight lines: (i) the tangent to the point of in¯ection on the sigmoid-shaped endotherm, and (ii) the extrapolated line from Cp of the glass [1±11].

7.3. The e€ect of step modulation

7.4. The e€ect of the beginning phase of modulation When the phase angle at the start of the sinusoidal modulation was shifted by p, the extra feature of hCp;mod i caused by modulation showed a shift for all the four cases, as seen in Fig. 6(a). It is evident that, in comparison with the results for the condition when the phase angle is zero at the start of the modulation, the p-shift in the phase has a lesser e€ect when b ˆ 0:5 and x is either 0.5 or 1. The e€ect is more when b ˆ 1 and x is either 0.5 or 1. The di€erence, dCp , also shifts and becomes opposite in sign to that of the zero phase at the start for the condition when b ˆ 0:5 and x is either 0.5 or 1. It shifts by a relatively small amount when b ˆ 1 and x is either 0.5 or 1. These conditions are more evident in the region where the effects of glass-softening overwhelm the modulation e€ects when b ˆ 1 and x is either 0.5 or 1. So the

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phase of the sinusoidal oscillation for the modulation also determines the shape of the Cp against T curve. When b ˆ 1 and x ˆ 1; hCp;mod i with zero phase at the beginning of modulation has qualitatively similar features to the unmodulated condition. This similarity is lost when the phase is p-shifted. This may be useful in determining the optimum modulation results when changes of DT and x0 are undesirable. 7.5. Relaxation time evolution with temperature We now consider how the t-, and T -dependent structural relaxation time, s, evolves when the glass is heated at a ®xed rate. To illustrate, we select the conditions used for the simulations described already in Figs. 5 and 6, and the same three pairs of b and x. The calculated s for the T -modulated and unmodulated conditions are plotted against T in Fig. 12(a), and the ratio, …ds=sunmod †, is plotted in Fig. 12(b). The values of s di€er most for the unmodulated and modulated pair, when b ˆ 1; x ˆ 0:5 and less when 0 < b 6 1; x ˆ 1 or b ˆ 0:5; x ˆ 0:5. The di€erence may reach values as high as a factor of 7.2 in the glass-softening region. 7.6. The linear response, and the physical aspects of the simulations

Fig. 12. (a) The plots of s for the T -modulated and unmodulated conditions against temperature during heating of the glass sample at 4 K min 1 ; (b) the plots of the di€erence, ds divided by sunmod , against temperature. Values of b and x are as noted. Curves 3 and 4 in (a) and curve 2 in (b) are also valid for b ˆ 1; x ˆ 1. The parameters used are the same as for Fig. 5, and are given in the text.

There is an ongoing discussion on the limitations of linear response theory [28] in its application for relaxation phenomenon generally [29±31], and particularly for those that lead to con®gurational freezing on vitri®cation of liquids discussed here. These limitations are of course as valid for dielectric and mechanical responses as they are for thermal responses. In such cases, the magnitude of the sinusoidal variation of the electrical ®eld, strain and temperature is kept small enough to minimize the non-linear e€ects, but they do not entirely vanish. In particular these e€ects in a calorimetric experiment increase (i) the amplitude of the heat capacity change at Tg , and (ii) the fraction of higher harmonics in the periodic heat ¯ow. The upper limit for the modulation amplitude of 1.5 K is seen to be within the linear response for

the dynamic heat capacity measurements of polystyrene [29], which is more than the amplitude of 0.637 K for simulation during heating and identical with that for isothermal annealing here. Therefore, e€ects from the non-linear response in DT have been ignored in our calculations, and this is consistent with the recommendation for modulated scanning calorimetry that DT be equal to the ratio of the heating rate to the modulation period, which is 0.637 K, as used here. Moreover, the experimental accuracy available with the commercial equipment is unable to detect non-linear response up to DT < 2 K, which is 1.4 K more than the DT used here. The advantages of pressure modulation are yet to be recognized, but it seems that the non-linear e€ects in sinusoidal pressure experiments would be considerably less, and more tractable than in the temperature modulation.

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Finally we discuss the physical implications of the study. It is evident from our discussion that (i) changes in the modulation period and modulation amplitude have a large e€ect on the apparent heat capacity and the calorimetric signal, and (ii) the magnitude of the molecular dynamics-controlling properties calculated from such experiments may be used to test the theories of the dynamics itself. Such experimental investigations have not been possible so far, although the e€ects of modulation on chemical reaction processes have been tested [23]. Also, since pressure increase has an opposite e€ect on the molecular dynamic properties of liquid and glass than temperature increase, the combined modulation of the two would provide us a way of maintaining conditions at which the molecular relaxation time would not change in a modulation cycle, thus revealing the change in the static properties corresponding to the equilibrium and the molecular relaxed states. It is hoped that this study will stimulate interest in new experiments in the currently widespread use of modulated calorimetry and ultrasonics. 8. Conclusions Mathematical simulation and numerical computation of the temperature-modulation e€ects on the structure relaxation process during structural relaxation and heating show that the modulation has substantial e€ects on the observed enthalpy decrease and heat capacity change with time and temperature. The e€ect of pressure modulation, as in an ultrasonic experiment, is similar. The shape of the heat capacity against temperature curve and the glass-softening endotherm are remarkably changed by the modulation, as is the shape of the enthalpy curve with time. This results from the non-linear response of a material's di€usion or structural relaxation rate on temperature, and not necessarily on its stretched-exponential and nonlinear relaxation characteristics changes used in describing the structural relaxation process of a glass. The latter two features add to the changes already caused by the sinusoidal modulation. These e€ects are important in the interpretation of the data obtained by the currently developing

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