Reprinted from Macromolecules, 1983. 16, 898 Copyright g 1983 by the American Chemical Society and reprinted by permission of the copyright owner.

Effects of Annealing and Prior History on Enthalpy Relaxation in Glassy Polymers. 4. Comparison of Five Polymers

Ian M. Hodge

BFGoodrich Research and Development Center, Brecksville, Ohio 44141. Received October 27, 1982

ABSTRACT: Parameters for enthalpy relaxation in and below the glass transition temperature range were determined for poly(vinyl acetate) (PVAc), poly(methyl methacrylate) (PMMA), and bisphenol A polycarbonate (PCarb) and compared with those previously obtained for another PVAc, two polystyrenes (PS), poly(vinyl chloride) (PVC), the inorganic glasses As 2Se3 and B203 , and the simple organic glass 5-phenyl 2-ether. The four-parameter treatment of Moynihan and co-workers was found to give an adequate description of the relaxation component of the heat capacity, as a function of cooling rate, annealing temperature, and annealing time, for PVAc and PCarb. Significant deviations were found for PMMA. Strong correlations were observed between all four parameters for the five polymers, which were interpreted in terms of a cooperative relaxation mechanism.

Introduction Annealing of amorphous polymers below the glass

transition temperature range produces changes in many physical properties such as density, complex permittivity, enthalpy, complex mechanical modulus, and creep com-pliance. A particularly convenient property for the study of annealing is enthalpy, because of the availability of accurate and sensitive DSC instruments. The decrease in enthalpy during annealing is recovered during reheating to above the glass transition, and this recovery is usually manifested as a maximum in the heat capacity at tem-peratures ranging from well below to near the upper edge of the glass transition range. Recovery of the enthalpy lost during annealing has been observed in many polymeric materials, 1 '8 and a sufficiently large amount of experi-mental data has now been collected to permit several generalizations to be made:

(1) The temperatures, Tmax , at which the heat capacity maxima, Cp.., occur increase approximately linearly with annealing temperature, Te , and the log of the annealing time, t e, provided the aged glass is not too close to equi-librium. Also, T approximately linearly with the log of the heating rate, QH. It also appears that T is insensitive to the history before aging, such as cooling rate," vapor-induced swelling,' hydrostatic pressure ap-plied during cooling,5,9,11,15 or mechanical straining. 6,16,18

(2) The magnitude of Cp... also increases approximately linearly with Te and log te, again providing that the aged glass is not too close to equilibrium. In contrast to T , however, Cp.. is a strong function of prior history. Gen-erally speaking, those histories that elevate the enthalpy of the glass before annealing serve to increase C Similar Aug' behavior is observed for the enthalpy lost during aging, AH, of which Cp is a crude measure.

(3) As the agedrglass approaches equilibrium deviations from these linear relations are observed, until at equilib-rium no changes occur with annealing. At fixed t e, AH and Cp.. pass through maxima as a function of Te , often when Te is about 20 K below the center of the glass transition range, Tg, and decreases to zero when Te is well above Tg. At fixed Te , AH becomes constant at long t e as the aged glass approaches equilibrium.

The first reported observations of heat capacity maxima well below Tg in aged glasses appear to be those of niers.' The first theoretical account of this phenomenon was that of Kovacs et al.," given in terms of a multiparameter model for the glass transition which included the well-established nonlinearity and nonexponentiality (relaxation time dis-tribution) of the glass transition kinetics. Recently Hodge and Berens2° gave a physically similar, but mathematically

different, account in terms of the four-parameter treatment of the glass transition kinetics by Moynihan and co-workers" and demonstrated explicitly that all of the general features listed above could be reproduced by this approach. Good fits to data for poly(vinyl chloride , (PVC) were also obtained for several combinations of Te and t e. Hodge and Huvard22 demonstrated that the four param-eters of the Moynihan formulation could be optim ized by using a standard computer search routine, and applied this optimization procedure to data for two polystyrenes (PS). For the present work essentially the same technique was applied to data for poly(vinyl acetate) (PVAc), poly (methyl methacrylate) (PMMA), and bisphenol A polycarbonate (PCarb). The enthalpy relaxation parameters for these materials were compared with those already found for PVC,2° PS,22 another PVAc:," B203," As2Se3 ,24 and the lubricating oil C 6H5(0C6H4)30C6H5 ("5-phenyl 4-ether" or "5P4E") 24 by using the same formalism.

For convenience we outline the treatment of the glass transition kinetics of Moynihan arid co-workers, 21 how annealing was introduced by Hodge and Berens, 2° and the optimization procedure used by Hodge and Huvarci. 22 The glass transition kinetics are characterized by two essential features—nonexponentiality (relaxation time distribution) and nonlinearity. The Moynihan approach linearizes the kinetics by using the method of Gardon and Narayana-swamy,25,26 considers cooling and heating as a series of temperature steps, and applies Boltzmann superposition of nonexponential responses to the temperature steps. As a matter of convenience, and good accuracy, the response function is chosen to be of the Williams-Watts 2' ,29 form

Oft) = exP[-- (t/Tor] (1)

where 1 > 0 and To is a relaxation time which depends on temperature, T, and fictive temperature, 29 Tf, as

thh* (1 _ x)Ah* To = A exp —

RT— RTf (2)

where A, x (1 x > 0), and Ah* are parameters and R is the ideal gas constant. The four parameters 0, A, x, and Ah* are assumed to be independent of T and T. Hodge and Berens 2° introduced annealing into the cooling cycle, with the annealing time divided into ten logarithmically even spaced subintervals to allow for nonlinearity during annealing. Hodge and Huvard 22 used an optimization technique based on the Marquardt search routine- 30 The thermal history (cooling rate, annealing temperaimre, an-nealing time, and reheating rate) and Ah* were input, and the best fit values of A, x, and 0 were output. The best-fit

Macromolecules, Vol. 16, No. 6, 1983

Table I Polymer Molecular Weights

material Mw Mr, MW /ME ref

PVAc -1.95 x 10 4.77 X 104 4.1 this work PVAc 2.0 x 106 3.4 x 10' 5.9 23 PVC 2.05 x 10 5 6.5 x 104 3.2 18 PS 3.21 x 10' 8.46 x 10 4 3.8 22 PS 2 x 10' 2 x 10' <1.06 17 PMMA 6.06 x 10 4 3.32 x 10 4 1.8 this work PCarb 3.38 x 10 4 1.34 x 10 4 2.5 this work

value of Ail* was obtained from the minimum in residual sum of squares deviation as a function of WO'. For the present work, Ah* was determined from the cooling rate dependence of the frozen in fictive temperature TI: 21

d in Qc

= -Ah* /R (3) da/M

Experimental Section All three materials were Aldrich Secondary Standards.

Weight-average and number-average molecular weights are given in Table I, together with those of other materials for which the enthalpy relaxation parameters are known. The materials studied here were dried for 2 h or more at about 30 °C above T. before being sealed into the DSC pans. Sample weights were about 10 mg. Each sample was left untouched in the DSC instrument between runs to reduce experimental scatter due to changes in thermal transfer between the instrument cell and sample pan. Three repeat scans were performed during the course of the annealing experiments to test for possible thermal decomposition and estimate experimental uncertainty. For these repeat scans cooling at 40 K min-1 from well above T. to well below T. was followed by immediate reheating at 10 K min -1 , with no inter-vening annealing. For convenience, we shall refer to this thermal history as -40/+10, with appropriate changes for other cooling rates. The upper and lower temperature limits for all thermal histories were 220 and 350 K for PVAc, 300 and 450 K for PMMA, and 350 and 510 K for PCarb. The heating rate was fixed at 10 K All measurements were made with a Perkin-Elmer DSC-2.

Experimental data C(T) were normalized with respect to the difference between liquid and glassy heat capacities, C ,, and Cpe respectively:

C(T) - Cp,(T) CpN = (4)

Cp,(T) - Cpg(T)

These normalized data are directly comparable with the calculated values. The temperature dependences of Cm and Cj,, were obtained by linear extrapolation of the liquid- and glassy-state data. The temperature dependence of C p1 was always linear, and extrapo-lation into the transition range was not subject to significant uncertainty. When C,,. was extrapolated, care was taken to ensure that only the lowest temperature range data were used (more than -40 K below T.), since relaxation effects associated with aging and the glass transition can make a significant contribution to Cp well below the transition range, according to calculations.

Three sets of experiments were performed. In the first, the cooling rate was varied (40, 20, 10, and 5 K min -1) with no an-nealing. In the second, the annealing time was fixed at 1 h at

Enthalpy Relaxation in Glassy Polymers 899

PVAc

1.0

C Np

T/K

290 300 310 320 330 290 300 310 320 550

1/K 1/K

Figure 1. Experimental points and fits for C i,N as a function of cooling rate for PVAc. Best fit parameters given in Table II.

annealing temperatures which were ca. 30, 20, and 10 K below T. (for present purposes T. is defined as the temperature at which CpN = 0.5, measured at 10 K min -1 after cooling at 40 K In the third set of experiments, the annealing temperature was fixed at ca. 30 K below T. for annealing times of 1 and -16 h. The cooling rate was fixed at 40 K min -1 for all annealing ex-periments, and annealing was performed in the cooling cycle.

The optimization procedure for obtaining best-fit parameters is outlined above and described in detail elsewhere. 22 Best-fit parameters were obtained for each thermal history and averaged. The averages were heavily weighted by the thermal histories that were most difficult to fit. These were for PVAc t e = 1 h, T6 = 300 K; for PMMA te = 17 h, Te = 350 K; for PCarb t e = 16 h, Te = 390 K.

Results The random experimental scatter in Cp...N (the maxi-

mum in CpN just above Tg) for the repeated -40/+10 scans was 15%. No systematic changes of more than ca. 5% were observed, indicating no significant decomposition. For PVAc and PCarb the estimated uncertainties in the best fit parameters were 10.05 in x and 0, 110% in Ah*, and 11.0 for ln A. For PMMA the best fits were poorer and the uncertainties were twice as large (see Discussion). The fits to experimental data for PVAc are shown in Figures 1 and 2, for PMMA in Figures 3 and 4, and for PCarb in Figures 5 and 6. Only two 1-h annealing ex-periments were performed for PVAc. Sets of best-fit pa-rameters for these three materials, together with those obtained previously for the other polymers listed in Table I, two inorganic glasses and a lubricating oil, are given in

1.01

-201+201 ci)

e° ri _ _ct

Table II Enthalpy Relaxation Parameters

material 1:1 A (s) ± 1 A h*110 3 /i ± 10%, K x ± 0.05 /3 ± 0.05 ref

P-VAc -275.4 88 0.28 0.53 this work PVAc -223.6 71.3 0.41 0.51 23 PVC -619.0 225 0.11 0.25 18 PS -216.4 80 0.43 0.68 22 PS -457.0 175 0.12 0.39 17 PMMA -355.7 138 (±20%) 0.22 (±0.1) 0.37 (±0.1) this work PCarb -353.6 150 0.22 0.54 this work As,Se, -85.5 40.9 0.49 0.67 24 B i O, -75.6 45 0.40 0.65 21 5134E -153.1 38.5 0.40 0.70 24

PVAc

2.2- te 1 Hour •- Te = 300K

2.0— 290

- 40 /+ 10

1.8 —

1.6 —

1.4 —

900 Hodge

cp

01 1 o ° i o e

1 1 300 310 320 330

1/K Figure 2. Experimental points and fits for CpN for the indicated combinations of T0 and te for PVAc. Parameters same as in Figure I.

PMMA

1 fo‘V=9--0 _

_ —51+10 f — —101+10 p 1 f _ I

- I I- oi - ....eled - c3-0

/

I 1 i I I I I I I 1 360 370 380 390 400 360 370 380 390 400

T/K T/K

- _

— —201+10

— —401+10 p

1.0 —

T/K T/ K

Figure 3. Experimental points and fits for C pN as a function of cooling rate for PMMA. Best fit parameters given in Table II.

Table II. These parameters were all obtained by using the Moynihan formulation, and are thus directly compa-rable.

Discussion

The fits for PVAc (Figures 1 and 2) and PCarb (Figures 5 and 6) deviate from the data by less than the experi-mental uncertainty, indicating that the model provides an adequate account of enthalpy relaxation in these materials. The fits to PMMA deviate from the experimental data by

Macromolecules, Vol. 16, No. 6, 1983

PMMA

70 = 350 1.0 — te I Hour

CpN

—

390 T/K

_ T. = 360

te = I Hour

N -

,o/.7cP _ 0

360 370 380 390 400

T/K T/K

Figure 4. Experimental points and fits for CpN for the indicated combinations of T. and t. for PMMA. Parameters same as in Figure 3.

PCarb

- -

- -

- - A Cp

N — 5 / + 10 L. —10/4-10 Ti _ _

° 8

/ . _ .6....1) _

0 _ . _ ...I

T/K T/K _

P ) -

1.0 — —201+10 —401+10 flLo—o p ea0 ..0.■.0.. —

N - 6 - 1 Cp _

/ - i

- o -

.1

- ..0.....61 - 0 - - 400 410 420 430 440 400 410 420 430 440

T / K T/K

Figure 5. Experimental points and fits for C pN as a function of cooling rate for PCarb. Best fit parameters given in Table II.

up to 17%, well outside experimental uncertainty, although the occurrence of sub- Tg shoulders and maxima in CpN in annealed glasses is qualitatively reproduced. Evidently there is at least one feature of the enthalpy relaxation kinetics for PMMA that is not being properly accounted for in the model. Accordingly, the uncertainties in the enthalpy relaxation parameters for PMMA are larger than those for the other materials (see Table II).

The parameters obtained for PVAc agree within un-certainty limits with those obtained by Sasabe and Moy-nihan,23 except for x, where the difference is slightly greater than the sum of uncertainties. The similarity of the en-thalpy relaxation parameters for these two PVAc materials

Macromolecules, Vol. 16, No. 6, 1983

Enthalpy Relaxation in Glassy Polymers 901

PCarb

T8 = 390 I Wells,

= 390

te = 16 Hours

• C p

• 5P4E o PVAc (Ref. 23) 0 PVC (Ref. 18)

- 4 A PVAc (This Work) • PMMA (This Work) • non optimum fits • PCorb (This Work) x PS (Ref. 221 • As2Se3 (Ref. 24)

3 • PS (Ref. 17) 7 8203 (Ref. 21)

- ,s te

—.Up El

- -g 300 s.......\\Clio — 300 - - 2 :

a: — r = -0.91 s= r = - 0.81

0 . i 200 200—

6-0-- 1 .., ,,,`" • 1 • Io

0 o • _ _ - I00 saia-: ... WO it..

- --..

T/K

2.0 - ft,

- T8 = 400 K I\

= 1 Hour I° ° N _

cp

0 a I.0_

0

00_ 0.0.

400 410 420 430 440 400 410 420 430 440 T/K T/K

Figure 6. Experimental points and fits for CpN for the indicated combinations of Te and te for PCarb. Same parameters as in Figure 5.

is in sharp contrast with the large differences in parameters for two PS materials studied earlier 22 (see Table II). It should be noted, however, that the parameters for one of these polystyrenes (obtained from data published by Chen and Wang') are somewhat uncertain because they were obtained for only two thermal histories, and no directly determined experimental value of Ah* was available. It is therefore possible that the parameters for this PS are not optimum for all thermal histories and that the dif-ferences between the two polystyrenes are not as large as the parameters in Table II indicate.

An inspection of the data shown in Table II reveals a strong correlation between the parameters 0, x, Ah*, and ln A. The correlations between Ah* and x, Alt* and 13, x and 0, and In A and Ah* are displayed in Figure 7. The corresponding linear correlation coefficients are —0.93, —0.91, +0.85, and —0.95, respectively. The correlations are robust with respect to uncertainties in the parameters: varying Ah* in the optimization program produces tetrads of Ah*, x, 0, and In A that move along all four correlation lines. To illustrate this pairs of parameters for the aver-aged —40/+10 scans for PVAc, in which Alt* /R was varied by ca. ±15% from its experimental value, are shown in each of the correlation plots. The observed correlations do not appear to be artifacts of the uncertainties, however, since the uncertainties are small compared with the total variation (with the possible exception of the PS material discussed above). We shall therefore accept the correla-tions as established for the materials considered here and proceed to discuss their implications.

The correlation between Ah* and A results from the relatively small range in 7', for the materials considered here. The value of To (eq 2) is approximately 100 s at T g , and since 7'f T at Tg eq 2 yields

ln A = ln (100) — OA* /RTd (5)

Thus —ln A is directly proportional to Ah* for similar

0.6 13

0.6 X InA (SeC)

Figure 7. Correlations between pairs of enthalpy relaxation parameters for the materials listed in Table II. The axes are (A) Ah*,(1? and x, (B) Ah*/R and 13, (C) 13 and x, and (D) Ah*/R and ln A. Lines are least-squares best fits. Correlation coefficients are denoted by r. The sets of three points for PVAc (present work) illustrate correlations of parameter uncertainties (see text).

values of Tg. Correspondingly, when Ah* is fixed (e.g., by eq 3) in A essentially determines T, (x and # have a rel-atively minor effect on Tg).

In discussing the other correlations we first observe that the enthalpy relaxation model used here is phenomeno-logical and that any molecular interpretation of its pa-rameters is made at some risk. With this in mind, we now suggest tentative molecular interpretations of 0, Ah*, and x whose chief merit is to make the correlations between these parameters self-consistent.

The parameter 0 may be regarded as reflecting the breadth of distribution of relaxation times or as a direct measure of the departure from an exponential decay function (single relaxation time). Although these two in-terpretations are mathematically equivalent, they reflect different physical emphases. When a relaxation is de-scribed in terms of a relaxation time distribution it is not unusual to identify individual relaxation time components with physically distinct processes. For example, a change in shape of the distribution with temperature is sometimes regarded as reflecting different activation enthalpies for the processes corresponding to the different relaxation time components. When viewed as a measure of nonexponen-tiality, 0 can be interpreted as a measure of degree of cooperativity of the relaxation process. In this case, changes in 0 with temperature correspond to changes in the degree of cooperativity. Here we interpret as a measure of cooperativity, which can be usefully defined in terms of the number of chain segments involved in a particular relaxation event. Low values of 0 correspond to a high degree of cooperativity and a large number of chain segments. If it is assumed that the activation en-thalpy per segment is a weak function of polymer type, the involvement of a large number of chain segments would result in a large activation enthalpy for the relaxation event (Ah*). The observed inverse correlation between 0 and Ah* (Figure 7B) is consistent with this, suggesting that the

0 0.2 x 0.4 0.6

0 8

7/ ;2 re OA

0

,N 0.4 I ?"/ P

0.2 ° r = 0.85

902 Hodge Macromolecules, Vol. 16, No. 6, 1983

activation energy per segment is indeed a weak function of polymer type. Some support for this reasoning is found in the theory recently proposed by Bendier and Ngain for volume recovery in amorphous polymers. This theory provides theoretical support for the Williams—Watts re-sponse function, with 0 determined by the number of correlated low energy states in the heat bath around a reference molecule and by the strength of coupling between the heat bath and conformational states of the molecule. A result of this theory is that the apparent activation energy, corresponding to Ah*, is given by EA/#, where EA is the barrier height between conformations of adjacent segments. In terms of this theory, the observed inverse correlation between 0 and Aft* suggests that Ah* is de-termined more by the strength of coupling between con-formational states and the heat bath (i.e., degree of co-operativity) than by the intramolecular energy barrier between conformational states. We also observe that the clustering of parameters for B 203, As2Se3, and 5P4E around the maximum observed values of x and # and minimum value of Ait* (Table II) supports the idea that enthalpy relaxation in these materials is less cooperative than in polymers.

The parameter x is a measure of the relative importance of macroscopic structure (as defined by T f) and tempera-ture in determining the average relaxation time, and it is natural to extend this to the molecular level by interpreting "structure" in molecular terms such as chain conformations and their correlations. A more cooperative relaxation mechanism involving many chain segments would be ex-pected to be more strongly influenced by molecular structure (smaller x), in agreement with the observed correlations x, 13, and All*. Also, the inverse correlation between x and Ah* results in the product xAh* being a weak function of polymer type. (For the polymers con-sidered here the standard deviation for xAh* is 5% of the mean, compared with 40% for Ah*.) Thus the activation enthalpy for chain diffusion above 7', (Ah*) is a stronger function of polymer type than the activation enthalpy for molecular motion in the glassy state (xAh*). The glassy state behavior is consistent with relatively localized mo-tions dominating the temperature dependence of relaxation processes in the glassy state and with the suggestion of-fered above that the activation enthalpy per chain segment does not depend strongly on polymer type. On the other hand, the isothermal structural dependence of the average relaxation time in the glassy state, which determines the rate of physical aging and of which (1 — x),11-t* is a direct measure, is an even stronger function of polymer type than the temperature dependence above 7', (the standard de-viation for (1 — x).6J-t* is 72% of the mean). This is con-sistent with the expectation that highly cooperative mo-tions involving a large number of chain segments (low #, high AM) are strongly affected by polymer type and mo-lecular structure in the glassy state.

Summary and Conclusions The correlations shown in Figure 7 appear to be real and

not artifacts of correlated uncertainties in the parameters, although the latter possibility cannot be completely ruled out. The correlations are all consistent with the well-es-

tablished cooperativity of the glass transition kinetics. To the extent that the five polymers considered here are representative of polymers in general, it also appears that the phenomenological enthalpy relaxation parameters 0, Ah*, and x reflect cooperative molecular relaxation pro-cesses, although their relationship to specific molecular details of different polymers is obscure at this time. Characterizations of enthalpy relaxation in more polymers are needed to establish how general the correlations are and to what extent the enthalpy relaxation parameters can be identified with specific molecular features.

Acknowledgment. This work was supported, in part, by NSF Grant No. CPE-7920740 under the Industry/ University Cooperative Research Program. It is a pleasure to acknowledge the experimental assistance of D. Puraty, useful discussions with H. Hopfenberg, and comments and suggestions from A. R. Berens. We thank J. T. Bendler for providing a preprint of ref 31. We thank The BFGoodrich Co. for permission to publish.

Registry No. PVAc, 9003-20-7; PMMA, 9011-14-7; PCarb, 24936-68-3; carbonic acid—bisphenol A copolymer, 25037-45-0.

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