Structural relaxation in thermorheologically complex materials

IOURNA L OF

YgClIRd E ELSEVIER

Journal of Non-Crystalline Solids 172 174 (1994) 541--553

Section 3. Non-equilibrium dynamics near Tg

Structural relaxation in thermorheologically complex materials J.-P. D u c r o u x a' ~, S.M. R e k h s o n b'*, F.L. M e r a t a a Case Western Reserve University, Cleveland, O H 44106, USA b General Electric Co., Nela Park. Bldg 336, Cleveland, O H 44112, USA

Abstract

Further development and testing of the thermorheologically complex (TC) model is reported. This five-parameter model deals with structural relaxation as a multi-relaxation process. A set of relaxation mechanisms is associated with the distributions of energy barriers and Kauzmann temperatures. The energy barriers decrease and the Kauzmann temperatures increase from slow to fast mechanisms. The TC model can be fully defined using equilibrium data in which case it has only three adjustable and two dependent parameters. The TC model successfully accounts for structural relaxation of glass plates quenched into molten salt and structural relaxation of rapidly cooled fibers the experimental data that defined the Tool-Narayanaswamy model.

1. Introduction When a glass-forming liquid undergoes a downward or upward jump of temperature, one observes an instantaneous change of its properties (density, enthalpy, index of refraction, etc.) due to the vibrational response, followed by a slow approach to equilibrium due to structural relaxation. The modeling of structural relaxation has been the subject of numerous studies. Scherer [1] classifies different approaches in four groups; rheological, kinetic, relaxation and phenomenological. Rheological theories deal with the temperature dependence of the viscosity and/or relaxation time. Kinetic theories predict the form of the relaxation function.

1Present address: Computer Simulators International, Inc., AMC, 1751 East 23rd Street, Cleveland, OH 44114, USA. * Corresponding author. Tel.: + 1-216 266 2784. Telefax.: + 1216 266 2936.

Relaxation theories explain both the form of the relaxation function and the temperature dependence of the relaxation time. The present work focuses on the phenomenology of the glass transition. A phenomenological model seeks to predict the behavior of glass in an arbitrary thermal history using a small set of parameters found through limited experimentation. The Tool-Narayanaswamy (TN) theory [2,3] effectively performs this function by enabling successful predictions of glass behavior [4,5], and thermal stresses in glass [6] and glassto-metal seals [7,8]. The T N model is a good example of the expediency of an insightful approximation. It treats glassy materials as thermorheologically simple (TS) although they are not. The TS behavior implies that the shape of the relaxation function is invariant with respect to temperature. This is only true for a narrow temperature region. However, then the glass transition phenomenon is observed in a

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542

J.-P. Ducroux et al. / Journal o f Non-Crystalline Solids 17~174 (1994) 541-553

narrow temperature range, so the approximation works. In a wider range, the shapes of the relaxation function and of the spectrum of relaxation times are known to broaden with decreasing temperature. This was demonstrated in various types of experiments with various materials, e.g., ultrasonic measurements for B203 [9] and specific heat spectroscopy for o-terphenyl [10]. Therefore, it was inevitable that the TN model would fail when a broader temperature interval was probed. This takes place during stabilization at very low temperatures [11,12], or when a rapidly cooled glass is slowly reheated [ 13,14]. Mazurin and Startsev [ 15] and Scherer [ 12] explored the potential of the thermorheologically complex (TC) approach, and declared it a limited success. Rekhson and Ducroux first introduced their TC model in 1992 [16]. The model is designed to account for both equilibrium and non-equilibrium behavior in a broad temperature region. In Ref. [16], the authors performed a cursory calibration of the model by using equilibrium viscosity and non-equilibrium structural relaxation data. This showed that the model is structurally sound. Detailed testing of the model revealed that more development work was needed. We now report the improved version of the model, and results of its testing.

function, Mp(t) = Mp(t, AT)ar_~o. Structural relaxation is linearized for an arbitrary thermal history by replacing the time, t, in Mp(t) by the reduced time [3], (" dt' = 3p, Jo %( T(t),---Tfp(t)) ' with the relaxation time, 3p, given by [3,19] 3p(V(t), rfp(t)) =

[xAH ( 1 - - x ) A H ] 3°exP|RT( t ) L - + RTfp(t) J '

The most recent (and excellent) reviews of the phenomenology of structural relaxation and solutions adopted by the TN model are given by Scherer [1,18]. The response of property, p, to a temperature step greater than a few degrees is defined by the structural relaxation function [3] 7"2) - p( 00, T2) Mp(t, AT = T1 - T2) - p(0, r2) - p( o% T2)" p(t,

(1)

The function, Mp, is non-linear in that it depends both on the direction and amplitude of the temperature jump. The TN model assumes the nonlinearity to stem from the change of the relaxation time due to the change of structure. As the temperature jump tends to zero, the relaxation function approaches the equilibrium structural relaxation

(3)

where 3o is a constant, AH is the activation energy, R is the ideal gas constant, x is a constant between 0 and 1 and Tfp is the fictive temperature for the property p. %r is the equilibrium value of % at an arbitrary reference temperature, T , With Mp(~) linearized, Boltzmann's superposition principle is used to calculate the fictive temperature, Trp, for any thermal history [3]: Tfp= T +

~ (To)

, dTd, M p ( ~ - ~ ) f i ~ ; ~.

(4)

ug

J¢(T)

Eq. (3) gives a satisfactory description of zp(T, Tfp) only in a narrow temperature range. A more useful basis is provided by the Adam-Gibbs formula [20-22] 3p(T, Tfp) = 3o exp

2. The TS approach to structural relaxation

(2)

TASdTk, Tfp)

'

where 3o, A are constants and the configurational entropy ASc(Tk, Tfp) is given by T! J Tk

J Tk

where Cp~,Cp~rand C,g are the liquid, crystal and glassy heat capacities and Tk is the Kauzmann temperature. Curiously, Eqs. (5) and (6) do not provide for a better fit of experimental data than Eq. (3), but do lead to more meaningful physical parameters of the model [12,23,24]. In addition, Eq. (5) satisfies the requirement [21] that with, parameters (Zo, A and Tk) found from equilibrium (Tfp = T) measurements, it fully determines the non-equilibrium

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 17~174 (1994) 541-553

behavior. Unfortunately the measurements by Mazurin and co-workers [25,26] suggest that there may be a problem with the actual performance of Eqs. (5) and (6) in this respect. These equations require that Zp extrapolate to Zo as T ~ ~ both for Tfp = constant and for Tfp = T. However, this was not the case in direct measurements [25,26] of the isostructural and equilibrium viscosities of borate and germanate glasses. In order to account for this behavior a different equation for %, one with four adjustable parameters was proposed in Ref. [27]:

lOglo[Zp(T, Tfp)]=

A ) Tfp B+ Tfp~k k T -(~-~-

1)log,o Zo.

(7)

According to Eq. (7), % extrapolates to Zo as T ~ ~ and Tfp = constant, but it extrapolates to 10n as T-~ oo and Tfp = T. Structural relaxation is a non-exponential process. The equilibrium relaxation function of property p is commonly approximated either by a Prony series [3,49,17]

543

[33] had serious difficulties in achieving a similar goal. In practice, Eq. (9) is preferred to Eq. (8) since it has fewer parameters. Depending on the form of the relaxation time used along with this function, the TN model has four or five adjustable parameters (Zo, AH, x, fl with Eq. (3), Zo, A, [irk, fl with Eq. (5) and B, A, Tk, to, fl with Eq. (7)). The Kohlrausch function (right side of Eq. (9)) is often approximated by a Prony series (right side of Eq. (8)) to take advantage of the efficient numerical algorithms available [12,34]. Note that, since this approximation is internal to the model, the number of adjustable parameters remains unchanged: the 2N-dimensional space of the parameters of the Prony series is mapped onto the two-dimensional space associated with the parameters of the Kohlrausch function. Such a transformation is effective in permitting the introduction of thermorheological complexity. A thermorheologically complex model with five adjustable parameters is presented in the next section.

3. The TC approach to structural relaxation

N

Mp(t) = ~ wlexp[ - t/Zpir] ,

(8)

i=1

or by the Kohlrausch function [28,4] Mp(t) = exp[

-

(t/Tpr)#]

.

(9)

The equilibrium relaxation function is obtained by averaging the non-equilibrium relaxation functions measured in small upward and downward quenches [28]. In general, it is not necessary to use temperature jumps of opposite sign. Any two or more approach curves can be used and the equilibrium function built by a simple and ingenious method pioneered by Le Bris [29] (for details of its use, see Ref. [16]). DeBolt et al. [30] used a numerical iteration method to determine the linear relaxation function from rate heating and cooling enthalpy measurements. Moynihan et al. [23], using this technique, successfully recovered the zp, but not the fl of the equilibrium relaxation function derived by Birge and Nagel [31,32] from their AC calorimetry measurements. Priven and Startsev

The shapes of the equilibrium relaxation function (i.e., fl in the Kohlrausch formalism) and of the distribution of relaxation times of a thermorheologically complex material are temperature-dependent. Dixon and Nagel [10] fitted a Kohlrausch law to the equilibrium relaxation functions obtained from specific heat spectroscopy experiments on o-terphenyl. They established that the temperature dependence of fl is equally well approximated with a linear form in T and 1IT. In a later publication [35], they showed that both linear forms, used in combination with

zp(T) = to[Lo(T)/to] 1/~,

(lO)

and a proper choice for Lo(T) led to the experimentally well-supported Vogel-Fulcher-Tamman (VFT) form: rp(T) = roexp(Q/(T - To)).

(11)

The connection between the temperature dependences of rp and fl was further investigated by

J.-P. Ducroux et al. / Journal o f Non-Crystalline Solids 172-174 (1994) 541 553

544

Rekhson and Ducroux [16]. They used the relaxation time given by the coupling model [36] with an Arrhenius behavior for zo(T):

tion defined by zp(T0 and fl(Tr) from Eqs. (12) and (13) is approximated by a Prony series: Mp (t) = exp [ - (tlzp (7",))p(r') ]

zp(T) = (fl~o~-p To(T)) 1/~

N

~, wi exp[ - (t/%i(Td)] . = (flco~ -p Zo exp(H/T)) 1/~ ,

(12)

where o~ is chosen to be equal to 101° s- 1 [37], and z0 is fixed at 1.4 × 10-14s, which approximately corresponds to the quasilattice vibration time [38]. H is an adjustable parameter. The authors [16] fitted the equilibrium relaxation time (estimated from the viscosity data of NBS 710 glass) to Eq. (12) with 13 given by

fl(T) =

1 b(1 - To~T)

T> T t T* >~ T >~ To,

(13)

The TC model assumes a temperature-independent distribution of the weighting coefficients wis. In solving for partial relaxation times, the following constraint must be satisfied: the relaxation function built from the w~s and Zpi(T)s m u s t be in agreement with the Kohlrausch relaxation function obtained from fl and Zp given by Eqs. (12) and (13) at any temperature below T+. In the present version of the model, this is partially achieved, between T~ and T t, with

Zpi(T) where b and To are adjustable parameters and T t is the temperature at which

(15)

i=1

=

"~piO

(16)

exp[Qi/TAS¢(Tki, T)],

where the configurational entropy follows Tt

fl-b(1-

To/T +)= 1.

(14)

Eq. (13)defines two temperature regions: above T+ where relaxation is single (fl = 1) and the temperature dependence of Zp (Eq. (12)) is Arrhenius, and below T t where relaxation is distributed and zp (Eq. (12)) follows an equation of the VFT form. This picture is consistent with experimental observations [9,39,40]. Note that, as fl changes from 1 at T t to ~ 0.5 in the glass transition region, the term a~ 1-p)/o changes from 1 to 101°! Thus the relaxation time, zp, is very sensitive to changes of ft. This dependence of zp on fl suggests that strong liquids [50], whose viscosity is Arrhenian, should demonstrate a TS behavior with fl close to 1. The fragile liquids are expected to be the TC materials which is manifest by a strong departure of their viscosity from the Arrhenius law. Through Eqs. (12) and (13), the TC model seeks to account for both the temperature dependence of Zp and that of ft. For T ~< T t, structural relaxation is treated as a multi-exponential process, which must be compatible with Eqs. (12) and (13). At T = T~, an arbitrary reference temperature in the interval [To,T*], the Kohlrausch relaxation func-

ASdTki, T ) = f T (Cp,(T') T,_Cpcr( ))dT, d Tki

f r (C

pl

(T') T C pg (T'))dT

p

(17)

J Tki

Note that, in general, Eq. (17) does not reduce to Eq. (11). The single Kauzmann temperature of the previous version of the model [16] is now replaced by a distribution of Kauzmann temperatures Tkis to satisfy the constraint on the relaxation times. The Kauzmann temperatures TkiS are given by

Tki = Ak exp

[

-- \ % ~ ) j

] + Bk.

(18)

A k and B k a r e two extra parameters of the model. The Q~s and "~plOs are computed by solving the system obtained by combining, for each rank i, Eq. (16) computed at T = Tr with Zpi(Tr) of Eq. (15), and Eq. (16) computed at T = T+ with zp(T+) of Eqs. (12) and (14) (see Appendix, step 5). Eq. (18) is empirical: it was devised by numerically studying of the internal consistency of the model [51]. The relaxation times derived at equilibrium are extended to non-equilibrium conditions by

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 172-174 (1994) 541-553

incorporating the time-dependent fictive temperature, Tfp: %i(T, Tfp) = %io exp[Qi/TASc(Tki, Tfp)],

(19)

where the configurational entropy is given by

aSc(Tk,,

rTt~

Tfp)

t

545

4. Tests of the TC model

In this section, the TC model is tested using the data of Hara and Suetoshi [11] studied in great detail by Scherer [12]. The TC model is then further tested using the data of Gupta and Huang [13,14] for a rapidly cooled fiber.

~ __Jr((C'(T ) - Cpg(T'))/T')dT'. 4.1. Volume relaxation

(20) The fictive temperature, Tfp, is computed from N Tfp = 2 w i T f p i, i-1

(21)

and dTfpi/dt = ( T -

(22)

Tfpi)/'[pi(r , rfp).

The property p is calculated from the fictive temperature by

p(T, rf.) = p(To, oo) +

dT' JT o

+ IT (Op'] dT' JTf.k~TA '

1

(23)

where subscripts 1 and g refer to the liquid and glassy states. For a derivative of the property p,

cnp /~P\ T .7dTfp

(24)

Hara and Suetoshi [11] measured the variation of the density of soda-lime-silica glass plates during isothermal holds after upward and downward quenches, and continuous cooling. Scherer [12] reported that the TN model could not fit all of their data with a single set of model parameters. The structural relaxation function given by Eq. (9) with fl = 0.64, commonly used to describe relaxation near the glass transition temperature, failed to fit the low temperature data [11]. By using the more flexible formula (8) and by adjusting the coefficients to decrease/3 down to 0.4 at shorter times, Scherer was able to fit the low temperature data, but only for temperature step quenches, not for continuous cooling. He also devised and tested a model with a thermorheological complexity attribute (see Appendix C in Ref. [12]) which similarly failed to fit all of the experimental data. The most difficult experiment to fit was the anneal of sample I, the sample quenched from 700°C into a bath of molten salt at 320°C. The thermal history of cooling for sample I is not precisely known, and this additionally complicates the analysis. Scherer considered case (A) of Newtonian cooling from 700°C to 320°C according to dT/dt = - h ( T -

is used. The TC model has five adjustable parameters b, To, H, Ak and Bk. The procedure of computing these parameters from a structural relaxation experiment is outlined in the Appendix. The model can also be completely defined using (good quality) equilibrium experimental data for zp(T) and fl(T). In this case, only three parameters (b, To and H) are the adjustable parameters. Once they are optimized using Eqs. 02)-(14), the dependent parameters A k and B k a r e calculated from Eqs. (16)-(18).

Tb,,h),

(25)

where the heat transfer coefficient, h, is the adjustable parameter helping to secure the experimentally observed frozen fictive temperature of 608.3°C. For theoretical and illustration purposes, Scherer also considered thermal history (B) where the sample was cooled to equilibrium in air from 700°C to 608.3°C, and then instantaneously quenched. The actual thermal history must be between these two. Hara and Suetoshi kept their samples on a sliding rack in the furnace. This rack needed to be

546

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 172-174 (1994) 541-553

pulled out of the furnace, and samples dropped into the bath. It is reasonable to assume that the samples were exposed to the ambient for several seconds before they entered the bath. The analysis of cooling by convection, surface radiation and conduction (e.g., Ref. [46]) indicates that a 2 mm thick glass plate could cool down to 650°C in about 4 s. The quench in the bath would certainly have a different (faster) dynamics. In this work we probe both thermal histories looked at by Scherer (cases (A) and (B) above) and then another one, case (C). The latter seeks to narrow the fork containing the actual thermal history. In thermal history (B) above, we replace an instantaneous quench in the bath by a continuous conductive cooling at the fastest possible rate. Consider the bath as an infinitely large heat sink with a perfect surface contact with the glass sample. Let the glass surface instantly assume the bath temperature of 320°C. This heat transfer problem is now reduced to a one-dimensional conduction problem. Its solution for the average temperature, given in Ref. [47] is oo

1 .Wo (2n + 1)2

T(t) = Tbath + 8(Tair - - Tbath) ~'~

×exp[

K(2n + 1)2x2t 7 ~ _],

(26)

where t is the time and x = K / p C p , with K = 0 . 0 0 2 c a l c m - 1 K - i s -1, p = 2 . 5 g c m -3 and Cp = 0.271 cal g- 1 K - 1. /'bath = 320°C is the temperature of the bath and Tair is the average temperature of the sample at the instance of its entry into the bath. l = 0.1 cm is half the thickness of the plate. Similarly to Scherer, in thermal history (A) we use h as an extra adjustable parameter. In thermal history (C), the adjustable parameter is Tai r. Prior to the entry, the sample is assumed to have been cooled from 700°C to Tair in ambient. Further, it is assumed to be at equilibrium in this temperature range. There is one minor difference in our fitting procedures. Scherer treated the first measured point in annealing experiments of sample I as a fixed constraint. He thus required that, at t = 0, in annealing experiments Tfv(0) is precisely 608.3°C. We elected to treat this datum as the rest of them, assigning no special significance to it. The frozen fictive temperature that we obtained is equal to 605.5°C. The variance is small. Table 2 of Ref. [11] indicates p = 2.4865 g/cm 3 as the initial density of annealing of sample I. This corresponds to Tfv(0) = 608.3°C. On page 130 of Ref. [11], the same quantity is quoted as 2.4867g/cm 3 which corresponds to Tfp(0) = 606.9°C. The fictive temperature 605.5°C corresponds to density 2.4869 g/cm 3. This difference cannot be distinguished in the plots below and is probably within experimental error.

Table 1 Parameters of the TC model (7", = 500°C) b

To (K)

H (K)

Ak (K)

B, (K)

h (s-

1)

rair (of)

Fit to the data of Hara and Suetoshi [11]; thermal history (A)

1.9382

510.82

24 263.1

106.73

387.4

0.26

Fit to the data of Hara and Suetoshi [11] thermal history (B)

1.9628

517.60

24021.7

123.23

366.28

Fit to the data of Hara and Suetoshi [11] thermal history (C)

1.8533

519.19

23031.3

611.33

95.68

375.95

264.15

313.84

125.43

348.95

17A9

Fit to the fiber data of Huang and Gupta [13,14]

1.6332

465.23

25468.0

Fit to the bulk and fiber data of Huang and Gupta [13,14]

1.4115

349.86

28 584.7

257.1

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 172 174 (1994) 541-553

Since no information regarding the variation of the isostructural heat capacity is available, following Hodge [48] and Scherer [12], the dependence of ACp with temperature was assumed to be linear in 1/T:

ACp(T) = C/T,

(27)

where C is a constant. Since C is unknown, the equations (Eqs. (5), (6), (16), (17), (19) and (20)) involving the configurational heat capacity ACp are rewritten so that C does not appear directly. The parameters of the TC model are given in Table 1. The results of computations using the TC model are shown in Figs. 1 and 2. For each thermal history applied to sample I, a set of parameters is optimized using all data, i.e., including step cooling experiments. Therefore three curves are presented for each set of data in Figs. 1 and 2. The fit is quite good. The most satisfying result of the computation is a very accurate representation of the curvature of curve 350°C in Fig. I(A). This was unattainable for previous models. The relaxation of 450°C in Figs. I(B) and (C) seems to be somewhat faster in computations than in the experiment. The discrepancy is most likely to be within an experimental error since Hara and Suetoshi [11] indicated that the actual temperature in the furnace may have been different from nominal (e.g., 450°C) by several degrees. To return to the uncertainties of thermal history of sample I, we show Fig. 3. The dashed line shows thermal history (A) which is a limiting case of sample I spending zero time in ambient and entering the bath with its temperature equal to that of the holding furnace, namely, 700°C. The solid line shows cooling by conduction in the bath. In this case, the optimization established that the sample entered the bath with the temperature of 611.3°C. Sample I would need 7 s to cool in ambient to this temperature from 700°C; this is not unreasonable. However, it would require 1 min to attain equilibrium at all temperatures above and including 611.3°C. This seems to be unrealistically long. Recall, that the solid line in Fig. 3 is the fastest possible cooling in the bath, and the dashed line is likely to be the slowest (as it averages the bath cooling with the slower prior cooling in ambient).

2.502

~

2.500:

~

~ ' ~ " "500 °C

~

2.498 .! f

x" 2.496.": f

547

450 °C

-/

2.494 J ~ #

T~(0)=605,5°C or 608.3°C

"~ 2.492-:

350 °C

2.490.: 2.488 i-¢ 2.486 .i 0 2.502 2.501 2.500

(A) 200

400

600 800 Time (minute)

1000

:~-~-'-~

--

.... : ~ ' ~

~'~

'2.499 / 2.498 /- J . 2.497.

1200

500oC 450°C

T#(0)=563.1°C

4 2.496 2.495

350°C

2.494 ~ 2.493 0

(B) 200

400

600 800 Time (minute)

2.503

1000

1200

500°C

2.502 f

.25Ol

450°C

T~,(0)=539.7°C

•N 2.499 .:

2.498 (/"

350°C

2.496 -:

0

200

I 400

J P 6O0 800 Time (minute)

(c) I000

1200

Fig. 1. Evolution of density during isothermal holds at 350°C (x), 450°C (O) and 500°C (O) for a sample equilibrated at 700°C and then dropped into a sodium nitrate bath at 320°C (A) and samples with initial fictive temperature To(O) = 563.1°C (B) and To(0 ) = 539.7°C (C). Data from Hara and Suetoshi [11]. The TC model fit (Eqs. (12)-(23) and (25-(27)) for thermal history (A) (dotted line), (B) (solid line) and (C) (dashed line). Parameters are listed in Table 1.

Both thermal histories produced the same frozen fictive temperature 605.5°C for sample I, and a very similar fit in Figs. 1 and 2, but of course different entry temperatures, 611.3 vs. 700°C. It is thus most likely that an actual bath cooling, which is the one

548

J.-P. Ducroux et al. / Journal o f Non-Crystalline Solids 172 174 (1994) 541-553

2.503 ,~ rzdo)=500.6°c 2.502 2.501 r " " - - ' 1 1 ~ " 'o A"2.500 515.9°C 2.499 2.498. "~ 2.497. t~2.496. 2.495. T.0,(0)=559.8°C 2.494. (A) I I P 2.493. 0 100 200 300 400 500 600 700 800 900 Time (minute)

~

2.503 2.502 ~ , ~ (0)=~00'6°C 2.501

2.503..

. 2499

"~ 2.498 .: "~ 2.497.1:

O3)

200

550.

500.:

#

, '

45o

4003503000

', ""

~ 5

"'-I .... 10

15 20 Time (second)

25

30

4.2. Rapidly cooled fiber

250

s. ~'~,, 542.5°C

(c) 60 80 Time(minute)

600: '

O"

of the time necessary for cooling at equilibrium, i.e., 3-4 s.

z

~2"496"! i 2.495 -!T~(O): 2.494 ~ v =559. °C2.493 J I 20 40 0

/

I,

53~.2°C

2.5012'502.!~~..(I)=504.6°C ~" 2.500.!

650 .

Fig. 3. Cooling in the bath from 700°C to 320°C for Newtonian cooling (thermal history (A), Eq. (25) with h = 0.26 s- 1, dashed line) and cooling by conduction (thermal history (C), Eq. (26), solid line).

2.5oo

2.499 ~'~""""~ 2.498 n '~ 2.497 2.496 / 2.495 2.494 T~(0)=559.8°C 2.493 0 50 100 150 Time (minute)

70O

100

120

Fig. 2. Evolution of density during isothermal holds at 515.9°C (A), 531.2°C (B) and 542.5°C (C) for annealed (0) and quenched ((3) samples. Data from Hara and Suetoshi [11]. The TC model fit (Eqs. (12)-(23) and (25)-(27)) for thermal history (A) (dotted line), (B) (solid line) and (C) (dashed line). Parameters are listed in Table 1.

Recently Huang and Gupta [13,14] presented new experimental data on rapidly cooled soda lime silicate (NBS 710) glass fibers. Glass fibers of diameters ranging from 8 to 12 gm were drawn from the melt and cooled to room temperature. Bulk relaxation data were also generated by cooling glass samples, annealed for 2 h at 973 K, to room temperature at constant rates ranging from 2K min- 1 to 40 K min- 1. Fiber and bulk samples were then reheated in a Perkin-Elmer differential scanning calorimeter (DSC) through the glass transition region at 40 K min- 1 from 373 to 973 K. The uncertainty in the heat capacity was determined to be about + 1.1%. The liquid heat capacity, Cp~,is assumed to be temperature-independent at 1.457 J g- 1 K - 1. The temperature dependence of the glass heat capacity, Cpg, follows the quasi-harmonic model recommended by Huang and Gupta [41]:

[" T x~3 ~O(T)/T x4eX C , , ( T ) = a T + 9R~o-~) ) Jo ie;,~-l)2dx, (28) between the solid and dashed lines, would be optimized to the same frozen fictive temperature and the entry temperature somewhere in between 611 and 700°C, say 650°C. A temperature close to the latter would be quite realistic from the point of view

where 3 R = 1 . 1 5 6 J g - l K -1 and for T>Oo~/2n, O(T) is given by

O(T) = 0oo[1 -- IA(Ooo/T)2 + p'(Oo~/T)4 . . . . ]1/2. (29)

549

J.-P. Ducroux et al. / Journal o['Non-Crystalline Solids 172 174 (1994) 541-553

The quasi-harmonic parameters were determined by Huang and Gupta [13] by fitting glass heat capacity data of bulk samples (310K < T < 650 K). They found 0~ = 1340.7 K, p = 0.0483, /£=.0.00126 and a = 1 4 . 9 x 1 0 - S j g ~K -2. The configurational entropy follows from a series of approximations proposed by Huang [42] and revised by Ducroux and Rekhson [43]. Two sets of data are investigated: the enthalpy recovery curve for a 10 lain diameter fiber as measured by Gupta and Huang [14], and the bulk relaxation data for a sample cooled at 20 K min 1; the latter set of data was digitized from Fig. 4 of Ref. [14]. The actual thermal history of the fiber is unknown. Therefore the fiber is assumed to cool from 1200 K to room temperature according to a Newtonian law given by dT/dt

=

-

(30)

TRT),

h(T-

the fiber is set to 1200 K. This is justified by a preliminary study with the T N model that shows that the system stays at equilibrium between 1473.15 K and 1200 K. The fit of the fiber data alone obtained by us with the TN model (dashed line) and the TC model (solid line) are compared in Fig. 4. The result of the best simultaneous fit of the fiber and bulk data is shown in Fig. 5 with the same convention. The values of the parameters are reported in Table 1 for the TC model and Table 2 for the T N model.

1.6

1.5' ~ 1.4 .~

/ "

1.2-

where TRT = 295.15 K is the room temperature and h is an additional adjustable parameter. In the experiment of Gupta and Huang, the initial temperature of the fiber is 1473.15 K. To accommodate for T t of the TC model which appears to demand a value below 1473.15 K, the initial temperature of

~1.1:

0.8 350

I 450

550

""

650 750 Temperature (K)

ii 950

850

1050

1.7 ¸

1.61.5-

1.6.

1.5.:

=

1.1-

i

N 1.1-_

0.8 .... 350 0.9-::¢,-550

650 750 Temperature (K)

850

950

1050

Fig. 4. Heat capacity relaxation data of Gupta and Huang [14] for the rapidly cooled fiber of 10 p.m diameter during reheating at 40 K rain - ~ ( • ). The solid line is obtained with the TC model (Eqs. (12) (22), (24) and (28)-(30)). The dashed line is obtained from the TN model (Eqs. (1), (2), (5), (6) and (28)-(30)). Parameters are listed in Tables 1 and 2.

f

_ J

~F

0.9- ~

,'

450

f

]

iI

¢j

'~ 1.2-i

0.8-: 350

'

..~ 1.3

~ 1.4-: ~ ~,~ 1.3-:

! l

I I

450

550

650 750 Temperature(K)

850

950

1050

Fig. 5. (A) Heat capacity relaxation data of Gupta and Huang [14] for the rapidly cooled fiber of 10 lam diameter during reheating at 40 K m i n - 1 ( . ) . (B) Heat capacity relaxation data digitized from fig. 4 of Ref. [14] for a bulk sample cooled at 20 K min- 1 and reheated at 40 K min- ~ ( • ). Simultaneous fit of the bulk and fiber data by the TC model (Eqs. (12) (22), (24) and (28)-(30), solid line) and by the TN model fit (Eqs. (1), (2), (5), (6) and (28)-(30), dashed line). Parameters are listed in Tables 1 and 2.

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 172-174 (1994) 541-553

550

539.7°C

Table 2 Parameters of the T N model

3500C

5-! ATkC (K)

ln(zo)

A ( J g -1)

Tk (K)

h(s -~)

0-:

Fit to the data of Hara and Suetoshi [11]; thermal history (A) 0.56

- 32.96

13701.5

-

460.74

log(900,rain.).

-5-"

Mr

0,6

Fit to the fiber data of Huang and Gupta [13,14] 0.82

- 25.931

-

4812.70

538.21

-15-:

13.65

Fit to the bulk and fiber data of Huan# and Gupta [13,14] 0.62

- 36.54

-

6947.3

520.83

375.59

-25-: 0

2

4

8

6

12

10

14

16

18

I04/T(K) 5. Discussion To understand the reason for a different performance of the TC and T N models, compare Figs. 6 and 7, which show the equilibrium and isostructural relaxation times for the T N and TC models, respectively. For the T N model, the former follows from Eq. (5):

Fig. 6. Equilibrium (Eqs. (31) and (32) with Tfp = T) and isostructural (Eqs. (31) and (32) with Tfp = 539.7°C) relaxation times for the T N model. P a r a m e t e r s used are from the fit of H a r a and Suetoshi data [11] (see Table 2).

12-

Tt

539.7°C

350"C

i

10-

Ti(T) = Toiexp [ A / T A S e ( T k ,

Tfp)'],

(31)

8-

with Tfp = T and the ZoiS given by

64-

Mp(T) = exp [ - (t/ro) ~]

v

2"

I

•

N

,~ ~ w/exp[ -

t/~oi].

(32)

i=l

The configurational entropy ASc(Tk, Tfp) is given by Eq. (6). For the TC model, the equilibrium relaxation time is calculated from Eqs. (16) and (17). The variation of the isostructural relaxation times below T = 539.7°C is calculated using Eq. (31) for the T N model and Eqs. (19) and (20) for the TC model; in both cases with Tfp = 539.7°C. A simplified analysis will shed some light on different predictions of the T N and TC models for the isothermal hold of 900 min at T = 350°C when the initial fictive temperature is 539.7°C. Consider the structural relaxation as a binary process: only a mechanism whose relaxation time is smaller than the observation time of 900 rain contributes to the relaxation process. The mechanisms whose relaxation times are greater than the observation time are frozen-in. Fig. 6 shows that seven mechanisms contribute to the relaxation process for the T N

. 4 .~

¢~

-6.: 9

~

i 10

I

11

12

I

13

14

15

16

17

10art(K) Fig. 7. Equilibrium (Eqs. (16) and (17)) and isostructural (Eqs. (19) and (20) with Tfp = 539.7°C) relaxation times for the TC model. Parameters used are from the fit of Hara and Suetoshi data I11] with thermal history (A) (see Table 1). model. This seems to be only slightly less than the eight mechanisms for the TC model (Fig. 7). The main difference is in the sums of the weights, wis, of active mechanisms. For the T N model this sum is 8.24x10 -3, which is significantly less than 2.07 x 10 -2, the total contribution of active mechanisms of the TC model. Fig. 7 also illustrates some of the unique features of the TC model. From a single relaxation time at T = T t, the spectrum spreads out as the temperature decreases. The isostructural relaxation times

J.-P. Ducroux et al. / Journal of Non-Crystalline Solids 172-174 (1994) 541-553

show the required Arrhenius behavior with shallower slopes for faster mechanisms. The dimensionless quantities QdC, proportional to the energy barriers Qi, are plotted in Fig. 8. C is the unknown constant in Eq. (27). The energy barrier decreases from slow to fast mechanisms. Such a variation is quite reasonable as fast mechanisms associated with smaller rearranging groups require smaller energy barriers. More peculiar is a sharp drop in energy barriers of the fastest relaxation times, which control the relaxation far from equilibrium. Another original feature of the TC model is the distribution of Kauzmann temperatures that increases from slow to fast mechanisms. Referring to Eq. (19), it also means that slow mechanisms are more 'Arrhenius' than the fast mechanisms. The distribution of Kauzmann temperatures is not introduced on theoretical grounds but has been necessary to accurately represent the temperature dependence of fl given by Eq. (13). However, despite the distribution of Kauzmann temperature, there are still discrepancies between fl(T) given by Eq. (13) (solid line in Fig. 9) and that (dashed line in Fig. 9) obtained from Kohlrausch functions reconstructed from a weighted sum of exponential functions (wis from Eq. (15) and rpiS from Eq. (16)). While the multi-exponential representation is not perfect, both curves are very close to the value determined by Rekhson and Mazurin ~28] from temperature jump experiments for a similar glass (e in Fig. 9)! Similarly, discrepancies between the Kohlrausch relaxation time calculated with Eq. (12) and that obtained from the Kohlrausch relaxation function reconstructed from Prony series (w~s from Eq. (I5) and rpiS from Eq. (16)) are observed. Analogous conclusions can be drawn when the TN and TC models are applied to the rapidly cooled fibers. The TN model gives a very poor fit of the fiber relaxation data (Fig. 4). Sructural relaxation in a glass fiber starts at temperatures much lower than in the slowly cooled bulk sample of the same glass. At such low temperatures, each mechanism of the TN model is still frozen-in and the model yields the glassy heat capacity. The TC model allows relaxation to occur at lower temperature by assigning lower energy barriers to the fast mechanisms. The TC model gives a fit that is within an experimental error (Fig. 4). When both fiber and

551

50-_

,o-

i

35-:: 30 - - -

~.

~1 ~ 25-

20-

-,

"----_

!

~0~

I

5-.

0-!

0

I

I o.1

0.2

0.3

0.4 0.5 0.6 0.7 Degree of relaxation

0,8

0.9

1

Fig. 8. Dimensionless QI/C, proportional to the energy barriers Qi, as a function of the degree of relaxation (Eq. (15) evaluated at %dTO). Parameters used are from the fit of Hara and Suetoshi data [11] with thermal history (A) (see Table 1). C is the unknown constant in Eq. (27). The points are connected for clarity.

1

0.85-

~

0.8 ~" 0.75- - -

~ ~

0'7- - - - - - 0.65 0.6 0.55 . . 0.5 450

iIf

. I T

.

[ I ~-i

.

I J ]

. 550

600 650 Temperature (°C)

700

750

7

Fig. 9. Variation of the fl coefficient of the TC model, from Eq. 03) (solid line) and computed from Kohlrausch functions reconstructed from weighted sum of exponential functions (weights wis from Eq. 05) and relaxation times %is from Eq. (16), dashed line). Parameters used are from the fit to Hara and Suetoshi data [11] with thermal history (A) (see Table l). The Rekhson and Mazurin [28] fl value at T = 520°C from the temperature jump experiments for a similar glass is also shown

to).

bulk data are fitted together (Fig. 5), the TN model presents a good fit for the bulk data, but fails to account for experimental data for the fiber. The TC model on the other hand gives reasonably good fits of the fiber and bulk data. The variation of the energy barrier and the distribution of Kauzmann

552

J.-P. Ducroux et al. / Journal o f Non-Crystalline Solids 172-174 (1994) 541-553

temperatures are consistent with those obtained from the fit of Hara and Suetoshi's data. Deviations are again observed between the values of the shape factor, fl, and the equilibrium relaxation time, zp, computed with Eqs. (12) and (13) and the values deduced from Kohlrausch functions built from Prony series (wis from Eq. (15) and ZpiS from Eq. (16)). The cooling rates obtained by fitting the fiber data with the TC model and the fiber and bulk data with the TN model are larger than that computed by Stehle and Brfickner [44] and that suggested by Fotheringham [52]. However, a similar fit can be obtained by fixing the cooling rate to a value that is closer to the estimation of Stehle and Brfickner.

The financial support of General Electric Co. and Saint-Gobain is greatly appreciated. The authors are thankful to Dr George W. Scherer of E.I. Dupont de Nemours and Co. for helping them to understand his paper [12] and for his critical comments of the first draft of this work. The authors also thank Dr Ulrich Fotheringham of Schott Glasswerke for sending them his computation of the cooling rate of a fiber, and Dr Jianzhong Huang of Ohio State University for his comments on the manuscript.

6. Conclusions

This appendix lists the steps for optimization of the parameters of the TC model when the latter is used with structural relaxation data. (1) Select five parameters b, To, H, Ak and Bk. (2) Compute T* (Eq. (14)) and zp(T t) (Eq. (12)). (3) Compute the Kohlrausch relaxation time (Eq. (12)) and fl (Eq. (13)) at T = Tr, an arbitrary reference temperature (To < Tr < T*). (4) Fit the Kohlrausch relaxation function obtained with a weighted sum of exponential functions (Eq. (15)). This gives the wis and the %i(Tr)s. (5) Find ZpioS and Qis by rewriting Eq. (16):

An improved version of the TC model is presented. It is successfully tested using two sets of relaxation data that defied the TN and previous TC models. Structural relaxation is treated as a multirelaxation process. Each mechanism is assigned its own energy barrier and Kauzmann temperature. The distribution of energy barriers decreases from slow to fast mechanisms. The distribution of Kauzann temperatures increases from slow to fast mechanisms. As of today, there is no independent experimental or theoretical evidence of this variation of Kauzmann temperatures. Five adjustable parameters are used to define the model by fitting the structural relaxation data. The model can also be defined (and tested) by equilibrium data for temperature dependences of the Kohlrausch relaxation time, %, and of the shape constant, ft. In this case, only three adjustable parameters are used. The other two are dependent parameters and are found by calculation. The TN Adam-Gibbs model ties together both the equilibrium and non-equilibrium behavior of the relaxation time. The TC model features the same capability but also incorporates the shape factor, ft. The connection between Zp and fl is established by the coupling equation. Since the model in its present form is still not fully consistent, it is an advantage that it can be tested by both the equilibrium and structural relaxation experiments for further improvement of the model.

Appendix: the optimization procedure for the parameters of the TC model

In Zpi(T) = In Zpio + QiX,

(33)

and solving this equation of a straight line using its two points

(1/Tt AS¢(Tki, T*), "rp(T*)) and (1/Tr AS~(Tk,, Tr), Zp,(Tr)). The configurational entropy ASc(Tki, T) is given by Eq. (17) and the Kauzmann temperatures Tkls are given by Eq. (18). (6) Solve Eq. (22) with Eq. (19), e.g., by using the Scherer [12] algorithm. (7) Compute the fictive temperature (Eq. (21)). (8) Evaluate the quantity given by Eqs. (23) or (24). (9) Compute the root mean square (RMS) of the deviation between the calculated and experimental values.

J.-P. Ducroux et al. / Journal o( Non-Crystalline Solids 172-174 (1994) 541 553

(10) Minimize RMS by using direct-search algorithm [45].

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[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16]

[17] [18]

[19] [20] [21] [22] [23]

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