The order parameter model for structural relaxation in glass

Journal of Non-Crystalline Solids 29 (1978) 143-158 © North-Holland Pubfishing Company

THE ORDER PARAMETER MODEL FOR STRUCTURAL RELAXATION IN GLASS Cornelius T. M O Y N I H A N and Prabhat K. G U P T A * Vitreous State Laboratory, Chemical Engineering and Materials Science Department, Catholic University of America, Washington, DC 20064, USA Received 22 August 1977 Revised manuscript received 30 January 1978

The order parameter model assumes that the state of a glass or liquid depends on T, P and a number of order parameters Z i. Structural relaxation is due to the kinetically impeded evolution of the order parameters following a rapid change in T or P. The linear relaxation function for the evolution of property Q (V or/4) in response to a change in X (T orP) is of the form ¢~QX = Ei g~QX exp(-t/ri)" Expressions are derwed for the weighting coefficients gQX m terms of the dep.endences .of V and H on the various order parameters• It is shown that g~VT = gtttP and that g~vTgtHp/gtvpg~HT = rt, where II is the Prigogine-Defay ratio. The corresponding relations among the relaxation functions are cPVT = (PHP and qSVT~HP/f)VPfgHT <~l-I. The predictions of the order parameter model for structural relaxation are compared with and found generally to agree with existing literature data. A number of suggestions for future investigations to test this model are made.

1. Introduction C o n c e p t u a l l y the most satisfying a c c o u n t o f structural relaxation and the glass transition in glass-forming liquids is in terms o f the order parameter model. It is assumed that the state o f the system depends on t e m p e r a t u r e T, pressure P and a n u m b e r o f order or internal parameters, Z1, Z2 .... Z/v, so that the Gibbs free energy is o f the f o r m G = G(T, P, Z l , Z 2.... Z N ) = G(T, P,

Z),

(1)

where Z is a c o l u m n m a t r i x representing the entire set o f Z i. Physically the order parameters might represent such things as the n u m b e r o f holes or vacancies in the liquid, the n u m b e r o f b r o k e n or flexed bonds o f various types, the n u m b e r o f sites with a particular c o o r d i n a t i o n g e o m e t r y , etc. It is p r e s u m e d that e v o l u t i o n o f the order parameters toward equilibrium is kineticalty i m p e d e d . Hence, if the system at equilibrium (the liquid) is subjected to * Present address: Technical Center, Owens-Corning Fiberglas, Granville, Ohio 43023, USA. 143

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sufficiently rapid changes in T and/or P, a non-equilibrium state (the glass) is produced in which the order parameters have been "frozen-in" on the experimental time scale at values characteristic of the initial state. This leads in turn to measured differences between the liquid and glass in the second derivatives of the Gibbs free energy, namely, the specific heat Cp, the thermal expansion coefficient a, and the isothermal compressibility K. With time the order parameters will evolve toward their equilibrium values at the new T and/or P, accompanied by a corresponding time evolution of macroscopic observables such as volume V and enthalpy H. This last phenomenon is known as structural relaxation. Davies and Jones [ 1] presented the first general treatment of the thermodynamics of the order parameter model of glasses and liquids. Although DiMarzio [2] recently challenged their conclusions, subsequent papers [3-5] have shown that the original Davies-Jones treatment was correct. The kinetics of the structural relaxation are of somewhat more importance from a practical standpoint, since they govern such things as the annealing and thermal tempering of glass, stress and strain buildup and decay and aging of high polymers, the behavior of lubricants in high-pressure contacts, etc. A direct connection between the order parameter model thermodynamics and the kinetics of the structural relaxation was pointed out in a series of papers [4,6-9] from this laboratory and resulted in the experimentally verified prediction that different kinetics should be observed for the relaxation of different properties of the same glass in response to temperature changes. Roe [5] has extended this treatment to include predictions about the relative rates of response of the same property to temperature and pressure changes. It is the purpose of the present paper further to develop the order parameter treatment of structural relaxation to give a set of concrete predictions capable of furnishing a critical test of the correctness of this model. Some of these predictions have also been derived in a recent paper by Berg and Cooper [10].

2. Thermodynamics of structural relaxation The equilibrium liquid state is defined by the condition (OG/OZi) e ~ - A i e

=

0

for i = 1,2 .... N ,

(2)

where Ai is the affinity for order parameter Zi and subscript "e" indicates that a property or derivative of interest is evaluated for the equilibrium state. At any temperature and pressure the Gibbs free energy of the non-equilibrium glass must be greater than or equal to that of the equilibrium liquid. This leads to a thermodynamic stability requirement known as the Prigogine-Defay condition [1,4] II - A C p A K / T V ( A a ) z >~ 1 ,

(3)

where ACp is the difference between the liquid and glass specific heats, and simi-

C. T. Moynihan, P.K. Gupta I Order parameter model

145

lady for AK and As. The equality in eq. (3) applies when only one order parameter is needed to fix the state of the system (N = 1) or when two or more parameters are needed (N > 1) which fulfill the condition

OH/aZi) _ (0 V/aZi)

(4)

(aHlaZj) (avlazj) for all i and ]. The inequality applies in the case of two or more order parameters which do not satisfy eq. (4). We may obtain expressions for the temperature and pressure dependence of the order parameters if we consider the dependence of the affinities on T, P, and the order parameters [1,5]

Ai - - - a a / a z i = Ai(T, P, Z)

(5)

so that

= (aS)~ d T -

\-~i](aVldp+ ? \ ~(j a] dAZi lj .

(6)

If we change the temperature, keeping pressure constant (dP= O) and apply the constraint that the system remains in equilibrium (Ai = Aie = O, dAi = O) we obtain

from eq. (6)

( aAi~(aZj ] = _(aH/aZi)/T

(7)

j \aZ i]\aT/e where, because of eq. (2), we have been able to make the substitution (as/azi) = (aH/aZi)/T. There are N simultaneous equations of this form, which may be represented in matrix notation: a(aZ/aT)° = (1/T)(aH/aZ)

(8)

where a is a square matrix with elements

au

= -(aAdaZp = &G/aZ~aZp.

(9)

Solving eq. (8) for the temperature dependence of the order parameters at equilibrium gives (azlaT)e

=

a-l(aHlaZ)lT,

(10)

where a - I is the inverse of matrix a. Matrices a and a - I are both positive definite symmetric. In similar fashion the pressure dependence of Z at equilibrium is given by

(azlaJ%= - a - l D v / a z ) .

(I I)

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3. Kinetics of structural relaxation We shall confine ourselves here to a consideration of the rates of relaxation of those extensive thermodynamic observables that occur in the above expressions, namely, V and H.* The liquid at equilibrium is subjected to an instantaneous small change in temperature or pressure, AT or zSa°; this is followed by the kinetically impeded adjustment o f the order parameters to their equilibrium values in the new state. According to irreversible thermodynamics [11] the rate of change with respect to time t o f each order parameter is given by d ~ Z i / d t = ~ LijA] = ]

ELi/~

aikAZx,

(12)

i

where AZi(= Zi - Zie) is the deviation of an order parameter from its equilibrium value. The Lii are the phenomenological transport coefficients and satisfy the Onsager reciprocal relations (L 6 = Lfi). In eq. (12) we have assumed that AT and AP are sufficiently small that the relaxation processes are linear in two senses: (1) the A/ may be approximated by the linear terms in a Taylor series expansion about the equilibrium position, and (2) at a given T and P the L 6 are constants, i.e. they do not depend on the Z i. Writing eqs. (12) in matrix notation gives (daZ/dt) = - t a X Z ,

(13)

which may be subjected to the identity operation M - l(dAZ/dt) = - M - ILaMM- 1AZ.

(14)

We choose the matrix M and its inverse M-1 so that

M=ILaM = I / t ,

(15)

where 1/t is a diagonal matrix with elements 6ij/r i and 6 6 is the Kronecker delta. This defines a new set of order parameters and their time derivatives which are linear combinations of the old values: (d AZ*/dt) = M - l(d a Z / d t )

(16)

AZ* = M-1AZ

(17)

.

The time evolution of each new order parameter is governed by a characteristic relaxation time Yi: dAZ~. /dt = - AZT/7 i .

(18)

* We might have chosen S instead of H as an observable. However, as shown in a subsequent section experiments in the literature in which evolution of heat has been monitored during relaxation have been reported in terms of H, not S.

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In changing to the representation corresponding to the new set of order parameters the matrix a is transformed into a new matrix, a*. We may find the relations between a and a* if we note that a scalar such as AG, the deviation of the Gibbs free energy from its equilibrium value, must remain invariant under any transformation of the order parameters. If we carry out a Taylor series expansion of G about its equilibrium value at constant T and P and retain terms only up to the second order in Z, we get, using eq. (2):

AG = ½ ~ ~. (OZG/SZiOZi)IZiZ~Z i = ½AZ'aAZ, i

(19)

/

where a primed matrix (e.g., AZ') is a transpose. To convert to the new order parameter representation we subject the third expression in eq. (19) to the identity operation AG = I(M-1AZ)' M'aMM-IAZ = ½AZ*'a*AZ*.

(20)

a* = M'aM .

(21)

Hence

Solving eq. (21) for a and substituting this into eq. (15) we get 1/r = M - 1 L M ' - l a * - L ' a * , where L* is the matrix L transformed into the new representation. Now, although matrices L and a are symmetric, their product La need not be. Hence the transformation matrix M in eq. (15) is, in general, not orthogonal. In Appendix A it is shown that a matrix M must exist which diagonalizes the La product and, moreover, that this matrix is such that eq. (21) yields matrix a* in diagonal form, i.e.

(23)

ai/= ai ~ij •

If this is so, then a *-1, the inverse o f a*, will also be diagonal with elements 6q/a ~. In the remainder of this paper we shall presume that the order parameters have been chosen to satisfy eqs. (18) and (23) and drop the superscript * on Z~, a*, and a *-1" Let Q stand for V or H and X for T or P. In relaxation experiments one imposes a step change ~ at initial time t = 0 then monitors the time variation of A Q ( ~ ) , the deviation of the property from equilibrium. The experimental results are expressed in terms o f a relaxation function AQ(ZLY') _

CQX - AQo(ZkX)

i

(24)

~ (OQ/3Zi) zSZio i

where subscript " o " refers to initial time t = 0. The relaxation function is presumed

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148

to be of the form

(kQX = .Z-~g'Qx exp(-t/zi) ,

(25)

l

where giQx is the weighting coefficient for the contribution of order parameter Z i to the relaxation of property Q in response to a change in X. That is, giQx specifies the apparent distribution or spectrum of relaxation times for a particular experiment. Note that since (gQX 1 at t = O, the g ~ x are normalized: =

~. giQx= 1.

(26)

l

From eq. (18) we see that

AZi = AZio exp(-t/ri) .

(27)

Comparing this with eqs. (24) and (25) we obtain:

giQx =

(OQ/OZi) AZio ( o a / o z i ) AZio "

(28)

i

We may make the expressions for the g ~ x more explicit if we note from eq. (10) that when AX is AT

ATaF 1 AZio =-AT(OZi/OT)

e-

- -

T

(3H/OZi) .

(29)

Note that in the third expression we have utilized eq. (23). Similarly, from eq. (11) when zLY is Aft' we obtain:

AZio

=

-AP(OZi/OP)e = APaF 1(0 V/OZi).

(30)

There are four possible relaxation experiments we can perform, namely, the relaxation of both V and H in response to changes in both T and P. The denominators AQo(~X) = 2 i (OQ/OZi)AZio in eqs. (24) and (28) for the four experiments are [1,5]: A Vo(AT) = VAaA T,

(31 a)

A V0(AP) = - VAtcAP,

(31 b)

2d/o(A7) = A C p A T ,

(31 c)

AHo(AP ) = - TVAe~AP,

(31 d)

The weighting coefficients for the four experiments, using eqs. (28)-(31), are thus:

• (O V/OZi)(OH/OZi) g~VT = -aiTVAc t '

(32a)

C. T. Moynihan, P.K. Gupta / Order parameter model giv P = _

v/ozi)(o v/ozi) ai VAK '

149

(3 2b)

g Hv = -- ( HI Z )OHI Z )

aiTACp

,

griP = -- (OH/OZi)(O V i l l i ) aiTVAo~

(32c) (32d)

The relative values of the weighting coefficients are determined by the value of the Prigogine-Defay ratio, H. If II = 1, either there is only a single order parameter, so that eqs. (25) and (32) contain only one term, or else eq. (4) is satisfied, so that (3H/OZi) = C(O V/8Zi) for all i, where C is a constant. In either event, in view of eq. (26), if II = 1 all four of the weighting coefficients g/Qx are identical, and the corresponding relaxation functions CQx of eq. (25) are the same for all four experiments. On the other hand, if II > 1 there is more than one order parameter and (OH/OZi) 4=C(O V/OZi) for all i. Hence when H > 1 one predicts that

(33a)

givr= but that the other five relations among the g/Qx are inequalities

g~T v~g~p , giv T 4: griT, etc.

(33b)

Similarly, when II > 1 C V T = ~)HP ,

(34a)

but for the other five relaxation function relations C V T 5t= dPVP , dPVT ::/= dPHr , etc.

(34b)

provided that the relaxation times r i are not all the same. (This last condition will be taken for granted, since experimental relaxation data for glasses in the glass transition region always exhibit a distribution of relaxation times.) For 1I > 1, attention has been called in previous papers [4,6-9] to inequalities of the form Cvx ~ ¢ H X and also [5] to inequalities of the form ~QT=fi ~)Qp.However, to the authors' knowledge the relations CVT = CHe and Cvp 4: CHT have not been predicted previously in the published literature, although Berg and Cooper [ 10] have recently derived them. For II > 1 one will also generally expect to see disagreements at a given T and P among the different relaxation experiments which have different CQX in the mean relaxation times: (35) i Equations (32) predict a quantitative relationship among the four weighting

C. T. Moynihan, P.K. Gupta / Order parameter model

150

coefficients for a glass:

givTg~lP ACpA. gvpgiH; - TV(Aa) 2 13.

(36)

This means that if any two of the non-equivalent relaxation functions are measured (i.e. not ~bvr and O/~p) the other two may be calculated. Note that in order to do this one does not need to know explicitly the order parameters and their corresponding relaxation times. Experimental relaxation functions can be described in terms of empirical probability density functions gQx(r) for the relaxation times:

OQX = ? gQx(7) exp(-t/r) d~-.

(37)

0

Provided that the gQx(r) are chosen so that the implied spectra of relaxation times overlap for all four relaxation functions, then eq. (36) may be written:

gvT(r) grip(7") g vP(r) griT(r)

13.

(38)

Combining eqs. (3), (25) and (32) we get a relation analogous to eq. (36) among the relaxation functions:

OVTOHP/OVpCkHT= 13(~) qiri)2/( ~ q ] ) ( ~ i

i

r]) ~< I I ,

(39)

i

where

qi = (~ V/OZi) aF 1/2 exp(-t/2ri) ,

(40a)

ri = (aH/aZi) aF 1/2 exp(_t/2ri) .

(40b).

The second relation in eq. (39) follows from the Schwarz inequality. The equal sign applies for II = I and the less than or equal sign for II > 1. At very short times and at very long times (PVT (~ne/Ckvp Onr appoaches definite limits. If we arrange the relaxation times in the increasing order rl < r2 ... < rN, then using eqs. (25) and (26) we have lim ((PVT(~Hp/C~VP(PHT) t<<~"

1

1.

(41)

(PQX=g~Qxexp(--t/rN),

(42)

=

,

Also

lira t>>r N

so that via eq. (36) we get lira (C~VTC~rlP/~VPCbHT)= 13.

t>>r N

(43)

The predictions of eqs. (33), (34), (36), (38) and (39) require experimental veri-

C. T. Moynihan, P.K. Gupta / Order parameter model

151

fication, since, of course, they are only as sound as the assumptions on which they are based. Of these assumptions the most questionable is that of eq. (12). It may well be that the rates of relaxation of the order parameters are not linearly related to their deviations from equilibrium, that is, the relaxation is inherently nonexponential, as was pointed out clearly by Goldstein [12] some time ago. Models which give rise to such non-exponential relaxation functions have been constructed, the best known of which is probably the defect-diffusion model of Glarum [13]. However, as explained in Appendix B, there is a class of such models, which includes the Glarum model, for which the predictions of this paper remain valid. In section 4 we discuss the literature data currently available to test our predictions. We conclude that this data, while extremely sparse, generally tend to support them. In section 5 we comment on the experiments which are needed for a critical test of our predictions.

4. Discussion of experimental data Experimental determinations of the Prigogine-Defay ratio II have been made for a number of glasses, the most reliable probably being the values reported recently for B203 [4,8], 0.4 Ca(NO3)2-0.6 KNO 3 [4,8], poly (vinyl acetate) [4,8] and selenium [14]. For these four glasses II > l well within any possible experimental uncertainty, and it is reasonable to conclude that this result will hold true for the most other glasses (although a possible exception is noted later). Consequently, the predictions of section 3 for II > 1 will usually apply. Three of the possible four relaxation e x p e r i m e n t s - the relaxation of AV(AT), AV(&P) and 2d-/(AT) - have been carried out for one glass, B203, for which it is known that 11 > 1. The AV(&P) experiments were made by Corsaro [15-17] using pressure jumps small enough to ensure linearity. The AV(AT) experiment was carried out by monitoring the response of the index of refraction, which is presumably proportional to the change in V, to large temperature jumps, while in the 2d/(AT) experiment [7] enthalpy was monitored while rate cooling and heating the sample over a large temperature range. These latter two experiments are thus nonlinear. However, they were analyzed by a method [7,8,18,19] that corrects for the non-linearity and allows extraction of the linear relaxation function from the nonlinear data. The method appears to be reliable in that, on analysis, data which are non-linear to varying degrees all yield the same linear relaxation function. This method has also been used in analyzing data for poly (vinyl acetate) and As2Se3 discussed later. The linear relaxation functions for all three B203 experiments were found to be of the form: (~Qx = exp [-(t/'t'Qx)~Q X] ,

(44)

where, as explained elsewhere [20], the J3Qx parameter defines a distribution of

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Table 1 Results of relaxation experiments on B203 glass at 270.2°C. AQ(AX)

#QX

r'QX(S)

AV(AT) AV(AP) AH(AT)

0.82 0.60 0.65

5.7 × 103 1.6 × 103 2.4 × 103

Reference [8118] [15] [7]

t

relaxation times centered on a reference relaxation time rQx. Hence, differences in g~x and (gO× for the three B20 3 relaxation experiments may show up as differt ences in either/3Qx or rQX or in both. The values of these parameters at 270.2°C P for the three relaxation experiments are listed in table 1. Unfortunately the 7"QX values may not be strictly comparable, since the three experiments were carried out on different samples and 7-~2x for B20 3 is highly sensitive to the presence o f small amounts of water [16]. However, ~QX is not sensitive to water content [16]. The fact that ~VT=/z~Vpand [3VT~[3rlT thus means that ~VT:/:(~V? and ~VT:/:~HT". Here, [3vp and [3HTare the same within experimental error [7,16], but r'v/, and r~/T are not. Hence, we may conclude that C~vpva ~HT only with the reservation that the two samples may have had different water contents. Thus, the relaxation functions of B203 definitely exhibit two and possibly three of the inequalities predicted in eq. (34b). In table 2 we have tabulated for several times for B203 at 270.2°C the quantity C~2VT/dPVP(gHTcalculated from the data of table 1 and eq. (44). According to eqs. (34a), (39), (41) and (43) this quantity should be ~ > r ~ x ) the ~bQx are small and relative errors in their values become large. At short times ( t < < r~2x) each of the ~bQx approaches unity and eq. (41) is automatically fulfilled. Sasabe and Moynihan [9] have compared relaxational data for ~ q ( A T ) and

Table 2 Test of eq. (39) for B203 glass at 270.2°C (I1 = 4.7) t(s)

c~2VT/dPVt:~HT

3 X 102 1.6 + 0.2

1 × 103 2.3 + 0.3

3× 103 4.2 +-0.9

1 × 104 10.6 +-9.0

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AV(AT) for another glass, poly (vinyl acetate), for which it is known that II > 1. These data are noteworthy in that the relaxation experiments and the measurement of II were all carried out on specimens from the same batch. The AV(AT) data were sparse, consisting of a single atmospheric pressure cooling curve, but were sufficient to establish that (THT) is about twice as long as (~'vr). This means in turn that (gt4T ~: (~VT, satisfying the prediction of eq. (34b). Relaxation measurements of AV(AT) and AH(AT) have been carried out and (~VT and (~HT found to be different for two glasses, As2Se 3 [8,21] and poly (vinyl chloride) [5], for which the value of II has not been determined. Assuming that II >1 for these two glasses, these results are in accord with our prediction. On the other hand Sasabe et al. [22] have recently determined ¢VT and ~HT for an alkalilime-silicate glass, the former quantity being evaluated by refractive index measurements. Here, ~ v r and tpnr were found to be the same and exhibited distributions of relaxation times. This suggests that II = 1 for this glass, a prediction that remains to be verified. Weitz and Wunderlich [23] have reported results of experiments which bear on the relative rates of relaxation of volume and enthalpy for five glasses: sucrose, phenolphthalein, poly (methyl methacrylate), polystyrene, and 0.38 Ca(NO3) 2 0.62 KNO3. For the latter two glasses it is known that II > i [1,4,8]. In their experiment the glass was slowly cooled through the transition region under pressure, depressurized at room temperature (well below the glass transition region), taken to a somewhat higher temperature still below the transition region, and the evolution of V and H monitored during annealing at that temperature. It was found in all cases that the rate of approach of H to equilibrium was markedly different from the rate of approach of V. There is some difficulty in interpreting these experiments, since they monitor relaxation in response to both T and P changes. (That part of the relaxation is due to an effective temperature change may be seen if one notes that, even if the pressure changes were omitted in this experiment, relaxation would still be observed at the final annealing temperature. This is because the ticrive temperature attained by the glass on rate cooling would be different from the final annealing temperature.) It is impossible to tell whether the observed relaxations are due primarily to the pressure change or to the temperature change, but this is not too important for the present purposes. The simple fact that H and V relax at different rates is in accord with our predictions.

5. Conclusions and suggestions for future work The results cited in section 4 support our predictions, but the results are few. Plainly, what is needed for a critical test of the order parameter model of structural relaxtion are measurements on a given glass of II and of all four relaxation functions, so that the predicted equalities eqs. (33a), (34a), (36) and (38) may be tested.

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Of the four possible relaxation experiments only the measurement of (grip has not been reported previously. Probably the easiest way to do this is by modifying the procedure of Weitz and Wunderlich [23] to make it similar to the method used to measure the isothermal evolution of H following a rapid temperature change in the equilibrium liquid [21]. The glass would be allowed to equilibrate in the lower end of the annealing range at a T and P where the rate of relaxation is much slower than the rate at which T and P changes can be imposed. The glass would then be subjected to an isothermal pressure jump zkP, held for a time t at the new pressure, quenched to a temperature low enough to preclude any further relaxation and depressurized. A differential scanning calorimetry (DSC) scan through the transition region would then measure the relative enthalpy change in time t due to the isothermal pressure jump at time 0. In this regard it is of interest to investigate the magnitude of the isothermal pressure jump needed to give an accurately measurable relaxational enthalpy increment. For B203 glass, as an example, using eq. (31d) and the data of refs. [4] or [8], we find that a pressure increase z3d~ of 1 kbar in the glass transition region would give an initial enthalpy increment zX/-/o(2U°) of - 2 . 6 cal/g. This is equivalent via eq. (31c) to the relaxational enthalpy change ~t/o(AT) that would be produced by an isobaric decrease in temperature of 17 K, an enthalpy increment easily measured accurately by DSC [6,7]. For investigators ambitious enough to undertake the measurement of II and the four relaxation functions, we close with a few caveats. (1) Since small composition changes can often produce large changes in relaxation rates of a glass, all experiments should be carried out on glass samples from the same batch. For the same reason, the glass composition chosen should not be one which is readily altered by interactions with the surroundings, e.g., the glass should not be hygroscopic. (2) All relaxation experiments must start with the glass in an equilibrium state. The analysis of many past structural relaxation experiments, particularly in the polymer literature, has been made impossible because the sample was not allowed to stabilize prior to the start of the experiment. (3) A number of relaxation experiments in which the magnitude and/or direction of the initial excursion from equilibrium is varied should be carried out in order to detect and, if necessary, correct for non-linear effects [7,18,24]. It is worth noting here that experiments with relatively complicated T or P schedules (e.g. two or more T or P steps [8,17,22,25], rate cooling followed by rate heating [7-9,21,22]) give complicated relaxation curves for which it is much easier to separate the effects of non-linearity from the effects of a distribution of relaxation times than for experiments which monitor the response to only a single T or P jump [7,21,22]. (4) To avoid complications in extracting the different relaxation functions from the data, experiments in which the sample responds to both T and P changes should similarly be avoided.

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Acknowledgement This research was supported by Contract No. N00014-75-C0856 from the Office of Naval Research. The authors wish to thank Drs Berg and Cooper for communicating to them the results of ref. [10] in advance of publication, and for pointing out a mistake in an earlier version of this paper. Appendix A Let us assume that the non-orthogonal matrix M of eq. (15) is of the form: M = OiD102 ,

(A1)

where 01 and 02 are orthogonal matrices (O1-1 = O'1, O~-1 = O~) and D 1 is diagonal with elements Dli~i/. Matrix a, being symmetric, can be diagonalized via similarity transformation using an orthogonal matrix. Let us choose matrix Ol to do this: O1-1a01 = a ,

(A2)

where ~ is diagonal with elements oqSq. Solving eq. (A2) for a and substituting this together with eq. (A1)into eq. (15) gives 1/t =

0 2 lD(101-1LO I~D 102

,

(A3)

Here, L is symmetric, so that if 1/~ is diagonal, this must represent a similarity transformation of L with an orthogonal matrix. Hence, O21D1-1Oll = (Ol~tDiO2)' = O21DlotO1-1

(A4)

D1

(A5)

and = ~-

1/2 ,

where a -1/2 is a diagonal matrix with elements ai-1/28ii. Substituting eq. (A5) back into eq. (A3) we obtain: 1/r = O~- I al/2Oi- ILO xal/202 ,

(A6)

where a l / 2 O l l L O l e d / 2 is symmetric and may be diagonalized by proper choice of the orthogonal matrix O 2. Finally, substituting eqs. (A1), (A2) and (A5) into eq. (21) we obtain: a *= 1 ,

(A7)

where 1 is the unit matrix. Inserting this result in eq. (22) we also have L* = (1/'t).

(A8)

This completes the proof that there exists a matrix M which diagonalizes via eqs. (15) and (21) both La and a.

C T. Moynihan, P.K. Gupta / Order parameter model

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We note finally that a matrix M which leads to eq. (A7) is not the only matrix which will diagonalize both La and a. Rather there exist an infinity of such matrices of the form: M = O1DlO2D2,

(A9)

where O1, D1 and 02 are chosen to satisfy respectively eqs. (A2), (A5) and (A6), and D 2 is a diagonal matrix with a non-zero determinant. In these alternative representations one would obtain via eqs. (21) and (22):

a* = D~,

L* = D~-2(1/~).

(A10)

Appendix B The rate of relaxation at constant T and P of a non-exponentially relaxing order parameter can be written in the general form:

dAZi/dt =f(AZ 1 , AZ z ... AZ N, t) = f(AZ, t ) ,

(B1)

where the functional form on the right-hand side specifically excludes eq. (12). In this case the procedures [eqs. (15)-(17),(21)] which lead to a new set of independently relaxing order parameters cannot, in general, be implemented, and the predictions of the present paper, which follow from the use of these procedures, are not necessarily valid. There is, however, a special case of non-exponentially relaxing order parameters for which these predictions do remain valid. This is when

dAZi/dt = f(AZi, t)

(B2)

for all i, so that it is unnecessary to invoke a special procedure to obtain a set of independently relaxing order parameters. A common type of rate law that falls into the category of eq. (B2) is of the form:

dAZi/dt = f ' ( A Z i, t) f ' ( Z , t)

(B3)

Each order parameter may be written: z j --

+

(B4)

In the limit of linear relaxation in response to sufficiently small imposed T and P changes, AZi becomes negligible compared to Zje, and eq. (B3) becomes

dAZi/dt = f'(AZi, t) f"(Ze, t)

(BS)

This is of the form of eq. (B2), since at a given T and P f'(Ze, t) depends only on time. If eq. (B2) is valid, one may always describe the relaxation of Zi in terms of a

C. T. Moynihan, P.K. Gupta / Order parameter model

157

weighted sum of Ni exponentials: Ni

2xZi = AZ m ~ gi,i exp(--t/ri,/) , /=1

(B6)

Zi may then be divided into a subset of Ni order parameters:

Zi, i -=gi,/Zi

(B7)

each of which relaxes independently and exponentially according to

dAZi,//dt = - ~xZi, j/Ti, j .

(B8)

If each original non-exponentially relaxing order parameter is subdivided in this fashion, one obtains a collection of independently and exponentially relaxing order parameters whose number N' (= Ni Ni) is larger than the number N of original order parameters. This collection may then be renumbered 1 to N' and subjected to the procedures of eq. (24) and following to yield the predictions of eqs. (32)-(43). The previously mentioned defect diffusion model of Glarum [13] falls into the category of eq. (B2). This model was originally proposed to account for the orientation of dipoles following imposition of an electric field, but we can restate it here in a form appropriate for structural relaxation in response to T or P changes. There are two order parameters, Z1 and Z2. Here, Z1 is the number of local structural configurations of a certain form which can undergo rearrangement to a different form via processes such as bond breaking or flexing, change of local coordination number, etc. (In the original model Z1 was the number of dipoles oriented in the electric field direction.) The parameter Z 2 is the number of defects or holes in the glass. In the absence of defects Z1 relaxes exponentially with a relaxation time rl. If, however, a defect diffuses into the vicinity of one of the local configurations comprising Z 1, it can relax to the other form immediately. The time dependence of ALZ1 is given by the expression [13]

AZ1 = AZloexp(-t/rl)[1 -- p(Z2, t)]

(B9)

p ( Z 2 , t) is the probability that a defect will have diffused into the vicinity of one of

the Z 1 by time t and clearly depends both on time t and the number of defects Z2. Differentiating with respect to time, we obtain: d2xZ1/dt = _ 2 x z l l l + (dp(Z2, \-d~ t ) ) ( 1 - p(Z2, 1 t) )]

(BIO)

In the limit of linear relaxation p(Z2, t) becomes p(Zze, t) and eq. (BIO) is of the form of eq. (B5) and, hence, of eq. (B2). The number of defects Z 2 will also change in response to changes T and P, and d~Zz/dt will contribute in turn to the rates of relaxation of V and H. Glarum [ 13 ] was not concerned with a computation of dA~Zz/dt, since he presumed that the defects did not contribute to electrical polarization. To develop an explicit expres-

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158

sion for dAZ2/dt would thus require additions to Glarum's model, which we do not propose to do here. Suffice it to say that d A Z z / d t presumably depends on Z1 and in the linear limit would also end up in the form of eq. (B2).

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