# The three successive stages of creep of PMMA between 55-degrees celsius and 90-degrees celsius. (poly(methyl methacrylate)).

INTRODUCTION

Viscoelasticity is the main behavior of polymers, when low stresses are applied on amorphous polymers especially near [T.sub.g]. In order to be able to predict long term behavior of creeping polymers, it is necessary to establish accurate mechanical models taking into account not only creep but also recovery. Usually the available data concern only creep. For example, the extensive review published by Struik (1) about the influence of physical aging on creep of amorphous polymers, gives a great deal of very precise data about creep but ignores the evolution of recovery during aging.

This study deals with a systematic comparison between creep and recovery of PMMA during physical aging at 55, 70, 84 and 90 [degrees] C. To be sure that the repeated creep tests made on the specimen during aging do not disturb the structure of this specimen, the applied creep stress is very low and the total creep strain in each test is less than [10.sup.-4]. It is more appropriate to use the word microcreep for this study.

The comparison between creep and recovery reveals that creep is not due to a single process but must be analyzed in three successive stages. Immediately after the application of the stress, and for a brief duration, creep follows a logarithmic law, then the well-known Kohlrausch-Williams-Watts creep in [t.sup.0.35] starts and at T [greater than] [T.sub.g] - 25 [degrees] C, a third nonrecoverable creep has been identified, following a power law in [t.sup.0.78]. This creep is in fact the beginning of the rubbery creep described by Plazek et al. (2).

A physical interpretation of these stages is given in terms of creation and extension of microshear bands by means of molecular motions.

EXPERIMENTAL

PMMA ([M.sub.n] = 49,200, [M.sub.w]/[M.sub.n] = 2.12) has been kindly supplied by CdF Chimie. The glass transition temperature determined by DSC is 111 [degrees] C. To eliminate water, each specimen is maintained 12 h at 80 [degrees] C under vacuum. Then it is fixed on the inverted torsion pendulum described previously (3) and annealed at 135 [degrees] C for 30 min. During this heat treatment no stress, neither longitudinal nor torsional, is applied.

The specimens are cooled from 135 [degrees] C to the test temperatures 55, 70, 84 or 90 [degrees] C in two different ways - either in oven at a cooling rate of -1 [degrees] C/min, the origin of the aging time is the passing of [T.sub.g], or by quenching in air to room temperature and then heating again to the test temperatures (heating rate of 6 [degrees] C/min), the origin of the aging time ta is taken when the temperature is 5 [degrees] C below the test temperature.

After cooling from 135 [degrees] C the specimen is kept for several days at test temperature. Following the experimental procedure proposed by Struik (1) a torsional microtorque is applied from time to time to the specimen inducing a maximal shear stress [less than]20 KPa. The duration of each microcreep test is the fifth of the aging time [t.sub.e] [ILLUSTRATION FOR FIGURE 1 OMITTED].

ANALYSIS OF CREEP DATA: EXISTENCE OF A LOGARITHMIC CREEP

For each test both the creep strain and the recovery are recorded [ILLUSTRATION FOR FIGURE 2 OMITTED].

Whereas at 55 and 70 [degrees] C the creep strain is entirely recovered, at 84 and 90 [degrees] C, beyond three times the creep duration the recovery rate becomes negligible even if the recovery is only partial. To account for this recovery, it has been attempted to fit the creep strain by the stretched exponential or KWW law:

J(t) = [J.sub.u] + [Delta]J(1 - exp - [(t/[Tau]).sup.[Beta]]) with 0 [less than] [Beta] [less than] 1 (1)

Assuming that the creep strain starts only after the end of the application of the torque that takes 0.2 s, [J.sub.u] is the unrelaxed compliance. As shown further, this assumption is only partly verified. [Delta]J is the intensity of the creep defined by the relaxation time [Tau] and the exponent [Beta]. Since t [much less than] [Tau] (as shown below, this is verified here always), this relation can be reduced to

J(t) = [J.sub.u] + B [t.sup.[Beta]] where B = [Delta]J/[[Tau].sup.[Beta]] (2)

Two different methods have been used to determine directly the three parameters [J.sub.u], B, and [Beta]; obviously [J.sub.u] is not measured directly on the recorded creep curve.

Method 1: 3 Points Approximation Scheme

Three experimental values of J(t) are taken at three values of t chosen in geometrical progression, namely J(t), J(10t) and J(100t).

Each of the three unknown parameters is given by a relation between these three values of J, for instance:

[Beta] = log J(100t) - J(10t)/J(10t) - J(t) (3)

Obviously, if the entire creep curve is fitted by a single power law, the value of the exponent [Beta] does not depend on the value of the variable t.

In Fig. 3 the calculated values of [Beta] are reported versus the variable t for two aging times, 1035 and 4225 min. The value of [Beta] is constant only beyond 30 and 110 s, respectively. Before these critical times [t.sub.c], [Beta] increases from lower values.

This method gives the mean value of [Beta] between t and 100t, so that it determines the exact value of [Beta] when this value does not depend on the time t, namely for t [greater than] [t.sub.c]. When 0 [less than] t [less than] [t.sub.c], the calculated [Beta] overestimates the actual value of [Beta] at time t. In fact it is possible to say that the exponent [Beta] is close to 0 at the beginning of the creep.

Method 2: Least Squares Method

This method is a classical linear regression, which assumes the knowledge of [Beta]. For each given value of [Beta], the values of [J.sub.u] and B as well as the error with the data are determined. The accepted value of [Beta] is that corresponding to the minimal error.

Figure 4 concerns a test made after an aging of 16,025 min at 84 [degrees] C.

In this Figure are reported the successive errors obtained for [Beta] varying from 0.3 to 0.7. The accepted value of [Beta] is 0.47 by taking into account all the recorded data.

Four analyses have been performed. The first one by considering all the recorded points; the following ones by considering only the data beyond 50, 200 or 384 s. The minimal error is obtained in the last case and for [Beta] = 0.46; this means that the KWW law is fitted by the creep curve only beyond 384 s.

Figure 5, where compliance is plotted versus [t.sup.0.46], illustrates the fact that the strain recorded before 667 s increases more rapidly than beyond.

Transient Creep Is Logarithmic

Both methods show clearly that at 84 [degrees] C, the KWW creep is preceded by a transient creep occurring from the application of the microtorque till the time [t.sub.c] corresponding to the compliance J([t.sub.c]). Thus the KWW creep, which exists beyond [t.sub.c], can be written

J(t) = J([t.sub.c]) + B.[t.sup.[Beta]] for t [greater than] [t.sub.c] (4)

In fact, we will rather use:

J(t) = J([t.sub.c]) + B.[(t - [t.sub.c]).sup.[Beta]]

The transient creep is more rapid than any power law, which suggests a logarithmic form:

0 [less than] t [less than] [t.sub.c] J(t) = [J.sub.o] + b log (t/[Tau] + 1) (5)

In this relation, the initial parameter [J.sub.o] is slightly different from the initial compliance [J.sub.u] used in the KWW law. Because of oscillations following the loading stage, the exact value of the "instantaneous" compliance can be extrapolated from the creep curve only to the short times.

At 55 [degrees] C, the [Alpha] molecular motions are extremely slow and the creep curves are perfectly fitted by a logarithmic law [ILLUSTRATION FOR FIGURE 6 OMITTED], at least in the restricted duration of the tests. This creep kinetics is similar to the logarithmic creep observed in metals at low temperatures when the vacancies diffusion does not exist (4). As shown below, the duration [t.sub.c] of this logarithmic creep is increased when the diffusional processes are slowered, namely, either at low temperature or at long aging times. This is the reason for the delay of the logarithmic creep onset at 84 [degrees] C. For short aging times this creep is too brief to be detected. Thus it can be claimed that logarithmic creep is the first stage in the creep behavior of the amorphous polymers between [T.sub.[Beta]] and [T.sub.g].

PROPERTIES OF LOGARITHMIC CREEP

The main characteristic of logarithmic creep is the similarity between the kinetics of creep and recovery. This similarity has been shown at 55 [degrees] C when creep is only logarithmic [ILLUSTRATION FOR FIGURE 6 OMITTED]. At 70 [degrees] C the two successive creep stages, logarithmic and KWW, are clearly present [ILLUSTRATION FOR FIGURE 12 OMITTED]. The logarithmic creep and its recovery are identical whereas the complete recovery of the KWW part is slower and needs about three times the duration of this creep.

This similarity shows that the internal stresses that induce the recovery are of the same nature as the external stresses at the origin of the logarithmic creep.

The time of the end of logarithmic creep, [t.sub.c], increases with aging time [t.sub.e]. From Fig. 7 it seems that at 70, 84 and 90 [degrees] C this increase is the same: [Mathematical Expression Omitted]. The exponent 0.7 explains the decreases of the relative duration of logarithmic creep compared with test duration during aging.

In contrast, the slope b of logarithmic creep is an increasing function of temperature showing the thermal activation of this process [ILLUSTRATION FOR FIGURE 8 OMITTED]. During aging this slope decreases. At 55 and 70 [degrees] C the relationship between b and [t.sub.e] is [Mathematical Expression Omitted], but at 84 and [Mathematical Expression Omitted]. This rapid decrease of b with aging at high temperature suggests that the limiting values of b reached after a complete aging could be close to 2 x [10.sup.-5] [MPa.sup.1] whatever the temperature.

As a consequence of the opposite variations of b and re, the compliance reached at the end of the logarithmic state J([t.sub.c]) is constant at 70 and 84 [degrees] C, respectively, 1.3 x [10.sup.-3] and 1.38 x [10.sup.-3] [MPa.sup.-1] [ILLUSTRATION FOR FIGURE 9 OMITTED].

The value of the compliance [J.sub.o] is determined by extrapolation at the time t = 0 taken at the maximum of the first oscillation [approximately]0.2 s after the application of the torque on the pendulum. While at 55 [degrees] C, [J.sub.o] remains constant at 1.03 x [10.sup.-3] [MPa.sup.-1] during aging, it decreases slightly with [t.sub.e] at 70, 84 and 90 [degrees] C where the tests have been performed for particularly long aging times: [J.sub.o] decreases from 1.32 x [10.sup.-3] [MPa.sup.-1] at [t.sub.e] = 3 x [10.sup.4] s down to 1.15 x [10.sup.-3] [MPa.sup.-1] at [t.sub.e] = 1.6 x [10.sup.6] s [ILLUSTRATION FOR FIGURE 10 OMITTED]. At this temperature the high slope value b, in other words the rapid creep rate of the beginning of the logarithmic creep, suggests that the creep strain occurring during the 0.2 s necessary for the stress application is a non-negligible contribution to the compliance [J.sub.o]. Therefore [J.sub.o] will slightly decrease during aging; it is not an intrinsic value of the compliance of the polymer.

To estimate the specific logarithmic compliance, the difference [Delta][J.sub.log] = J([t.sub.c]) - [J.sub.o] must be considered [ILLUSTRATION FOR FIGURE 11 OMITTED]. This compliance increases rapidly with temperature from 1.3 x [10.sup.-4] [MPa.sup.-1] at 70 [degrees] C to 3.3 x [10.sup.-4] [MPa.sup.-1] at 90 [degrees] C and seems not to be affected by physical aging.

STRETCHED EXPONENTIAL AND RUBBERY CREEP

As described above, the tests performed at 70 [degrees] C are totally recoverable and the recovery kinetics are slower than the creep kinetics [ILLUSTRATION FOR FIGURE 12 OMITTED]. Beyond the logarithmic state the creep is fitted by a power law with an exponent [Beta] = 0.36 [+ or -] 0.04 whatever the aging time [t.sub.e]. This value is in agreement with the value of 0.33 given by Struik (1).

At 84 [degrees] C the creep does not seem to be totally recoverable and the apparent value of the exponent [Beta] is 0.46. This value is too high, as shown below.

Analysis of Creep at 90 [degrees] C

At 90 [degrees] C, after aging during [t.sub.e] = 1.37 x [10.sup.6] s, the creep is first logarithmic, and then can be fitted by a power law with [Beta] = 0.6, but the recovery strain is lower than the total creep strain [ILLUSTRATION FOR FIGURE 13 OMITTED]. Thus a new mechanism is acting at this temperature, the strain of which is not recoverable. The analysis of the creep data recorded at 90 [degrees] C needs a new assumption. The first stage of this creep is the recoverable logarithmic creep, as at 55 and 70 [degrees] C. So it will be assumed that the second stage at 90 [degrees] C is, as at 70 [degrees] C, the recoverable KWW creep with an exponent of [approximately]0.36 and an intensity given by the recovery. This KWW mechanism is acting all along the creep straining, and beyond a time [t.sub.r] a third nonrecoverable mechanism starts [ILLUSTRATION FOR FIGURE 14 OMITTED], which is superimposed to the KWW stage.

To identify the three parameters of the reduced KWW law the experimental data available are

i) the origin (0 = [t.sub.c], J([t.sub.c])),

ii) the end ([t.sub.f] - [t.sub.c], [J.sub.Rmax]), where [t.sub.f] is the total duration of the creep and [J.sub.Rmax] the maximum of the experimental recovery, obtained after a recovery duration three times longer than the creep test,

iii) the first data obtained beyond [t.sub.c] [ILLUSTRATION FOR FIGURE 14 OMITTED].

In the considered test, [t.sub.c] = 500 s and the numerical law found by this way for the recoverable creep [J.sub.R](t - [t.sub.c]) is

[J.sub.R](t - 500) = 1.42 [multiplied by] [10.sup.-3] + 1.8 [multiplied by] [10.sup.-5] [(t - 500).sup.0.36] (6)

As shown in Fig. 14, this law fits the experimental curve only till the time [t.sub.r] = 4,920 s with J([t.sub.r]) = 1.8 x [10.sup.-3] [MPa.sup.-1]. Beyond [t.sub.r] the creep deformation is due to the addition of this recoverable creep [J.sub.R](t - [t.sub.c]) and a new nonrecoverable creep [J.sub.NR](t - [t.sub.r]).

t [greater than] [t.sub.r] J(t) = J([t.sub.c]) + [J.sub.R](t - [t.sub.c]) + [J.sub.NR] (t - [t.sub.r]) (7)

This creep [J.sub.NR] is determined by subtraction of the recoverable creep defined by Eq 6 from the total creep strain, and is fitted by a power law:

[J.sub.NR] (t - [t.sub.r]) = B[prime] [(t - [t.sub.r]).sup.[Gamma]] (8)

For the specimen it has been found:

[J.sub.NR](t - 4920) = 1.74 [multiplied by] [10.sup.-7][(t - 4920).sup.0.73] (9)

This method has been used to analyze all the creep tests performed at 90 [degrees] C.

The exponents [Beta] and [Gamma] of the power laws do not depend on the aging time, [Gamma] = 0.78 [+ or -] 0.06, and moreover the value of [Beta] is the same at 70 and 90 [degrees] C, [Beta] = 0.36 [+ or -] 0.04. This result confirms the proposed analysis by suggesting that the KWW mechanism is the same at 70 and at 90 [degrees] C.

The value of the parameter B of the KWW creep (Eq. 2) is higher at 90 [degrees] C than at 70 [degrees] C, and at both temperatures it decreases with aging time following a power law [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 15 OMITTED]. The exponent -0.6 must be compared with the exponent 0.7 previously determined for the increase of the onset time [t.sub.c] of the KWW creep with aging. In both cases these exponents are directly related to the decrease of molecular mobility during aging.

During aging, the onset time [t.sub.r] of the nonrecoverable creep increases nearly linearly with aging time, [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 16 OMITTED].

The parameter B[prime] decreases following [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 17 OMITTED]. This exponent value, higher than the corresponding value of the KWW creep, shows that the reduction of molecular mobility is more efficient on the nonrecoverable creep.

Creep at 84 [degrees] C

The temperature 84 [degrees] C can be considered as a transition temperature, at which the nonrecoverable creep starts to be active.

Without the possibility to determine accurately the intensity of the recovered strain, a reliable creep data analysis as made at 90 [degrees] C cannot be done. Therefore, the analysis in terms of KWW can be used at 84 [degrees] C only as a first approximation.

The value 0.46 found for the exponent [Beta] can be easily explained by the superposition of the recoverable contribution ([Beta] = 0.36) found at 70 and 90 [degrees] C and the earliest steps of the nonrecoverable creep described by a time power law with the exponent 0.78 found at 90 [degrees] C.

DISCUSSION

A logarithmic creep was first observed by Wyatt (4) on copper, aluminum, and cadmium at -196 [degrees] C. At higher temperatures, the creep curves fit the Andrade law. In an intermediate temperature range, the curves fit:

[Epsilon] = [Alpha] log(t) + [[Beta]t.sup.1/3] + [Gamma] (10)

It seems that between 55 and 90 [degrees] C, PMMA exhibits the same behavior.

Logarithmic Creep

The theory of the logarithmic creep has been proposed for metals by Mott (5) and for amorphous polymers by Escaig (6) by using Eyring's model.

Indeed, by derivation of Eq. 5, the kinetics of the logarithmic creep can be written in terms of creep strain [Epsilon]:

[Epsilon][prime] = b.[Sigma]/[Tau] exp - [Epsilon]/[Sigma]b (11)

This macroscopic strain is due to molecular jumps governed by the Eyring relationship:

[Epsilon] = [[Epsilon][prime].sub.o] (exp - [Delta]G - ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT - exp - [Delta]G + ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT) (12)

in which [Delta]G is the activation energy of the barrier between the two sites, [Sigma] the applied stress, [[Sigma].sub.1] the local back stress, [V.sup.*] the activation volume, and

[[Epsilon][prime].sub.o] = N[[Epsilon].sub.o][Nu] (13)

with N the number of mobile elements, [[Epsilon].sub.o] the strain due to the jump of each of these elements and [Nu] the maximal frequency of the jump, close to the Debye frequency ([approximately equal to][10.sup.13] [s.sup.-1]).

It is possible to identify Eqs. 11 and 12 if:

i) the influence of the back jumps is negligible, namely ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT [greater than] 3; therefore the activation volume [V.sup.*] is large, [10.sup.5] [[Angstrom].sup.3].

ii) the internal back stress is proportional to the strain: [[Sigma].sub.i] = [Alpha]E[Epsilon], E being the Young's modulus and [Angstrom] a proportionality factor.

Under these conditions Eq. 12 becomes:

[Epsilon][prime] = [[Epsilon][prime].sub.o] exp [Delta]G - [Sigma][V.sup.*]/kT exp - [Alpha]E[V.sup.*][Epsilon]/kT (14)

Combining Eqs. 11 and 14, the slope of the logarithmic strain can then be written:

[Sigma].b = kT/[Alpha]E[V.sup.*] (15)

These assumptions imply that:

i) the maximal strain of the logarithmic creep is reached when [[Sigma].sub.i] = [Sigma]. Because [[Sigma].sub.i] = [Alpha]E[Epsilon], this means that [[Epsilon].sub.max] = [Sigma]/[Alpha]E, and thus the maximal strain is proportional to the applied stress. It must be noted that when [[Sigma].sub.i] is close to [Sigma], the back jumps can no longer be neglected and the creep kinetics become the usual exponential law with a single relaxation time as shown on aluminum (3, 7).

ii) after the stress removal, the recovery kinetics is:

[Epsilon][prime] = [[Epsilon][prime].sub.o]exp - [Delta]G - [Alpha]E[V.sup.*][Epsilon]/kT (16)

the value of the parameter b is identical for creep and recovery if the internal back stress [[Sigma].sub.i] has reached a value close to [Sigma].

iii) the high values of b when T approaches [T.sub.g] can be attributed to the increase of the free volume near [T.sub.g], which reduces the activation volume [V.sup.*]. The decrease of b with aging can be correlated with the decrease of free volume during aging.

From Eq. 14 it appears that the creep rate [Epsilon][prime] is not proportional to the applied stress [Sigma] if [V.sup.*] is constant. In fact it has been shown by Haussy (8) that [V.sup.*] is a rapid decreasing function of [Sigma] when [Sigma] [less than] [[Sigma].sub.y], [[Sigma].sub.y] being the yield stress of PMMA, so that no conclusion can be drawn from Eq. 14 concerning the influence of [Sigma] on the creep rate.

The molecular motion, which can be proposed to build up a back stress [[Sigma].sub.i] of the same nature as the applied stress [Sigma], is the formation of double kinks on the chains leading to the creation of micro shear bands. This mechanism of shear bands has already been proposed by many authors since Bowden and Raha (9). Escaig (6, 10) and subsequently Perez (11) have described these shear bands in detail.

KWW Creep

At 70, 84 and 90 [degrees] C, it has been shown that the KWW creep starts when the logarithmic strain reaches a value J([t.sub.c]) - [J.sub.o] specific of each temperature, this value corresponding to a well-defined level of internal stress. This accumulated stress will then be spread out in the volume surrounding the shear bands by means of molecular motions. In this temperature range, the contribution of the [Beta] relaxation is nearly instantaneous, while the [Alpha] molecular motions become preponderant with increasing temperature.

Many models describe relaxation or creep due to diffusion, an extensive overview has been given by Blumen (12). In our case, the most appropriate models have been developed by Hellinckx (13) using a fracial rheological mechanism and Perez with an energetic approach (14). For these authors, creep of amorphous polymers follows a stretched exponential law, and their models take into account the recovery of the strain. The exponent [Beta] is attributed to the degree of hierarchy of correlated molecular motions, [Beta] decreases from 1 to 0 when hierarchy increases, [Beta] = 1 means individual motions and [Beta] [approximately equal to] 0 means that the motions involve a great deal of elementary segments.

When the local back stress is spread out by this diffusional process, the chains are slightly oriented around the shear bands. The expansion of this reorientation is now limited by the presence of entanglements in pure amorphous polymers like PMMA or by the entries of chains in crystallites in semicrystalline polymers.

Nonrecoverable Creep

The two stages of creep described above can be found either with PMMA or with semicrystalline polymers like PA11 (15) or PEEK (16) at temperatures between [T.sub.g] and [T.sub.[Beta]]. The third creep stage in [t.sup.0.78] can be observed only with PMMA. It is specific to amorphous polymers. The main difference between pure amorphous polymers like PMMA and semicrystalline polymers is the chain entries in crystallites, which restrict the displacements of chains. Those restricted motions close to crystallites can explain the nonexistence of the nonrecoverable third stage In semicrystalline polymers.

In contrast, in PMMA, the entanglements can move. leading to the rubbery behavior. This creep is not recoverable at 90 [degrees] C in the experimental time, or more exactly, its recovery is so slow that it seems to be not recoverable. This suggests that the acting back stresses are low and may be of entropic nature, resulting from displacements of entanglements.

It is now necessary to quote the study of Plazek (2) on the recoverable creep of PMMA between [T.sub.g] - 10 [degrees] C and [T.sub.g] + 50 [degrees] C. This creep is in fact the delayed rubbery strain of PMMA, which can be well fitted by a stretched exponential law with an exponent 0.8 [ILLUSTRATION FOR FIGURE 18 OMITTED]. At [T.sub.g] - 10 [degrees] C, creep starts during the first 30 s by a power law in [t.sup.0.33]. Beyond 30 s, the creep law is a power law in [t.sup.0.8], i.e., the beginning of the stretched exponential law described above. This creep is recoverable, but at temperatures close to [T.sub.g], Plazek must heat the specimen to achieve the recovery.

By analogy with the study of Plazek, the nonrecoverable microcreep detected at 90 [degrees] C is probably the beginning of the rubbery creep. It is nonrecoverable because 90 [degrees] C is a temperature too far below [T.sub.g]. Knowing that the rubbery strain in amorphous polymers is due to displacements of entanglements, it is now possible to attribute the "nonrecoverable" creep at 90 [degrees] C to such displacements. This is in accordance with the nonexistence of this creep in semicrystalline polymers. It is surprising that the influence of entanglements can be detected at strains as low as [10.sup.-4] [s.sup.-1]. Nevertheless, if this creep is actually the beginning of the delayed rubbery deformation of PMMA, it is difficult to imagine the existence of a limiting strain for this behavior.

Thus during the KWW stage, the back stress is progressively spread out of the shear band, and then it reaches the entanglements, which move during the last step of creep.

CONCLUSION

Microcreep of PMMA between 55 and 90 [degrees] C starts with logarithmic kinetics. During this stage, microshear bands are created and induce at their limit internal back stresses proportional to their dimensions. This logarithmic creep stops as soon as these internal back stresses are equal to the external stress. When the creep temperature is high enough to allow molecular diffusion, internal back stresses are progressively spread out around the shear bands by appropriate correlated molecular jumps. This is the recoverable KWW creep, which can be approximated by a power law in B.[t.sup.0.36] At 90 [degrees] C, a third stage corresponding to a nonrecoverable creep appears; it is described by the power law in B[prime].[t.sup.0.78]. This stage appears to be the beginning of the creep studied by Plazek above [T.sub.g]. It can be described by a stretched exponential law and corresponds to a delayed rubbery strain. Therefore, the nonrecoverable power law creep can be attributed to the beginning of displacements of entanglements.

NOMENCLATURE

J = Compliance = [Sigma]/S.

[J.sub.u] = Unrelaxed compliance extrapolated from the KWW law.

[J.sub.o] = Unrelaxed compliance extrapolated from the logarithmic law.

[J.sub.R] = Recoverable compliance.

[J.sub.NR] = Unrecoverable compliance.

B = Factor of the recoverable power law creep.

B[prime] = Factor of the unrecoverable power law creep.

[[Sigma].sub.i] = Internal back stress.

[Alpha] = Proportionality factor between the internal back stress and the microscopic strain.

[Beta] = Exponent of the recoverable KWW creep or power law approximation.

[Gamma] = Exponent of the unrecoverable KWW creep or power law approximation.

b = Slope of the logarithmic creep.

[t.sub.c] = Aging time.

[t.sub.e] = Beginning of the recoverable power law creep (end of the logarithmic creep).

[t.sub.r] = Beginning of the unrecoverable power law creep.

[t.sub.f] = Total duration of the creep.

REFERENCES

1. L. C. E. Struik, in Physical Aging in Amorphous Polymers and Other Materials, Elsevier Scientific Pub, Co. (1978).

2. D. J. Plazek, V. Tan, and V. M. O'Rourke, Rheol. Acta, 13, 367 (1974).

3. J. L. Gacougnolle, J. F. Pelletier, and J. de Fouquet, in Mechanical Testing for Deformation Model Development, Rohde-Swearengen, ed., ASTM STP 765, 67 (1980).

4. O. Wyatt, Proc. Phys. Soc., B66, 459 (1953).

5. N. F. Mort, Philos. Mag., 44, 742 (1953).

6. B. Escaig, in Plastic deformation of Amorphous and Semicrystalline Materials, pp. 187-225, B. Escaig and C. G'Sell, eds., Les Editions de Physique, Les Ulis (1982).

7. J. F. Pelletier, These Universite de Poitiers, pp. 23-24, France (1982).

8. J. Haussy, J. P. Cavrot, B. Escaig, and J. M. Lefebvre, J. Polym. Sci. Phys., 18, 311 (1980).

9. P. B. Bowden and S. Raha, Philos. Mag., 29, 149 (1974).

10. J. M. Lefebvre and B. Escaig, Polymer, 34, 518 (1993).

11. J. Perez, Physique et Mecanique des Polymeres Amorphes, p. 151, Techniques & Documentation, Ed. Lavoisier, Paris (1992).

12. A. Blumen, in Lecture Notes in Physics 277, Molecular Dynamics and Relaxation Phenomena in Glasses, 1, Springer Verlag (1985).

13. S. Hellinckx, N. Heymans, and J.-C. Bauwens, J. Non-Crystalline Solids, 172-174, 1058 (1994).

14. J. Perez, Physique et Mecanique des Polymeres Amorphes, p. 75, Techniques & Documentation, Ed. Lavoisier, Paris (1992).

15. L. Belec, These Docteur Science des Materiaux, pp. 97-121, Poitiers, France (1995).

16. X. Xinran, thesis, Free University of Brussels, pp. 89-92 (1987).

Viscoelasticity is the main behavior of polymers, when low stresses are applied on amorphous polymers especially near [T.sub.g]. In order to be able to predict long term behavior of creeping polymers, it is necessary to establish accurate mechanical models taking into account not only creep but also recovery. Usually the available data concern only creep. For example, the extensive review published by Struik (1) about the influence of physical aging on creep of amorphous polymers, gives a great deal of very precise data about creep but ignores the evolution of recovery during aging.

This study deals with a systematic comparison between creep and recovery of PMMA during physical aging at 55, 70, 84 and 90 [degrees] C. To be sure that the repeated creep tests made on the specimen during aging do not disturb the structure of this specimen, the applied creep stress is very low and the total creep strain in each test is less than [10.sup.-4]. It is more appropriate to use the word microcreep for this study.

The comparison between creep and recovery reveals that creep is not due to a single process but must be analyzed in three successive stages. Immediately after the application of the stress, and for a brief duration, creep follows a logarithmic law, then the well-known Kohlrausch-Williams-Watts creep in [t.sup.0.35] starts and at T [greater than] [T.sub.g] - 25 [degrees] C, a third nonrecoverable creep has been identified, following a power law in [t.sup.0.78]. This creep is in fact the beginning of the rubbery creep described by Plazek et al. (2).

A physical interpretation of these stages is given in terms of creation and extension of microshear bands by means of molecular motions.

EXPERIMENTAL

PMMA ([M.sub.n] = 49,200, [M.sub.w]/[M.sub.n] = 2.12) has been kindly supplied by CdF Chimie. The glass transition temperature determined by DSC is 111 [degrees] C. To eliminate water, each specimen is maintained 12 h at 80 [degrees] C under vacuum. Then it is fixed on the inverted torsion pendulum described previously (3) and annealed at 135 [degrees] C for 30 min. During this heat treatment no stress, neither longitudinal nor torsional, is applied.

The specimens are cooled from 135 [degrees] C to the test temperatures 55, 70, 84 or 90 [degrees] C in two different ways - either in oven at a cooling rate of -1 [degrees] C/min, the origin of the aging time is the passing of [T.sub.g], or by quenching in air to room temperature and then heating again to the test temperatures (heating rate of 6 [degrees] C/min), the origin of the aging time ta is taken when the temperature is 5 [degrees] C below the test temperature.

After cooling from 135 [degrees] C the specimen is kept for several days at test temperature. Following the experimental procedure proposed by Struik (1) a torsional microtorque is applied from time to time to the specimen inducing a maximal shear stress [less than]20 KPa. The duration of each microcreep test is the fifth of the aging time [t.sub.e] [ILLUSTRATION FOR FIGURE 1 OMITTED].

ANALYSIS OF CREEP DATA: EXISTENCE OF A LOGARITHMIC CREEP

For each test both the creep strain and the recovery are recorded [ILLUSTRATION FOR FIGURE 2 OMITTED].

Whereas at 55 and 70 [degrees] C the creep strain is entirely recovered, at 84 and 90 [degrees] C, beyond three times the creep duration the recovery rate becomes negligible even if the recovery is only partial. To account for this recovery, it has been attempted to fit the creep strain by the stretched exponential or KWW law:

J(t) = [J.sub.u] + [Delta]J(1 - exp - [(t/[Tau]).sup.[Beta]]) with 0 [less than] [Beta] [less than] 1 (1)

Assuming that the creep strain starts only after the end of the application of the torque that takes 0.2 s, [J.sub.u] is the unrelaxed compliance. As shown further, this assumption is only partly verified. [Delta]J is the intensity of the creep defined by the relaxation time [Tau] and the exponent [Beta]. Since t [much less than] [Tau] (as shown below, this is verified here always), this relation can be reduced to

J(t) = [J.sub.u] + B [t.sup.[Beta]] where B = [Delta]J/[[Tau].sup.[Beta]] (2)

Two different methods have been used to determine directly the three parameters [J.sub.u], B, and [Beta]; obviously [J.sub.u] is not measured directly on the recorded creep curve.

Method 1: 3 Points Approximation Scheme

Three experimental values of J(t) are taken at three values of t chosen in geometrical progression, namely J(t), J(10t) and J(100t).

Each of the three unknown parameters is given by a relation between these three values of J, for instance:

[Beta] = log J(100t) - J(10t)/J(10t) - J(t) (3)

Obviously, if the entire creep curve is fitted by a single power law, the value of the exponent [Beta] does not depend on the value of the variable t.

In Fig. 3 the calculated values of [Beta] are reported versus the variable t for two aging times, 1035 and 4225 min. The value of [Beta] is constant only beyond 30 and 110 s, respectively. Before these critical times [t.sub.c], [Beta] increases from lower values.

This method gives the mean value of [Beta] between t and 100t, so that it determines the exact value of [Beta] when this value does not depend on the time t, namely for t [greater than] [t.sub.c]. When 0 [less than] t [less than] [t.sub.c], the calculated [Beta] overestimates the actual value of [Beta] at time t. In fact it is possible to say that the exponent [Beta] is close to 0 at the beginning of the creep.

Method 2: Least Squares Method

This method is a classical linear regression, which assumes the knowledge of [Beta]. For each given value of [Beta], the values of [J.sub.u] and B as well as the error with the data are determined. The accepted value of [Beta] is that corresponding to the minimal error.

Figure 4 concerns a test made after an aging of 16,025 min at 84 [degrees] C.

In this Figure are reported the successive errors obtained for [Beta] varying from 0.3 to 0.7. The accepted value of [Beta] is 0.47 by taking into account all the recorded data.

Four analyses have been performed. The first one by considering all the recorded points; the following ones by considering only the data beyond 50, 200 or 384 s. The minimal error is obtained in the last case and for [Beta] = 0.46; this means that the KWW law is fitted by the creep curve only beyond 384 s.

Figure 5, where compliance is plotted versus [t.sup.0.46], illustrates the fact that the strain recorded before 667 s increases more rapidly than beyond.

Transient Creep Is Logarithmic

Both methods show clearly that at 84 [degrees] C, the KWW creep is preceded by a transient creep occurring from the application of the microtorque till the time [t.sub.c] corresponding to the compliance J([t.sub.c]). Thus the KWW creep, which exists beyond [t.sub.c], can be written

J(t) = J([t.sub.c]) + B.[t.sup.[Beta]] for t [greater than] [t.sub.c] (4)

In fact, we will rather use:

J(t) = J([t.sub.c]) + B.[(t - [t.sub.c]).sup.[Beta]]

The transient creep is more rapid than any power law, which suggests a logarithmic form:

0 [less than] t [less than] [t.sub.c] J(t) = [J.sub.o] + b log (t/[Tau] + 1) (5)

In this relation, the initial parameter [J.sub.o] is slightly different from the initial compliance [J.sub.u] used in the KWW law. Because of oscillations following the loading stage, the exact value of the "instantaneous" compliance can be extrapolated from the creep curve only to the short times.

At 55 [degrees] C, the [Alpha] molecular motions are extremely slow and the creep curves are perfectly fitted by a logarithmic law [ILLUSTRATION FOR FIGURE 6 OMITTED], at least in the restricted duration of the tests. This creep kinetics is similar to the logarithmic creep observed in metals at low temperatures when the vacancies diffusion does not exist (4). As shown below, the duration [t.sub.c] of this logarithmic creep is increased when the diffusional processes are slowered, namely, either at low temperature or at long aging times. This is the reason for the delay of the logarithmic creep onset at 84 [degrees] C. For short aging times this creep is too brief to be detected. Thus it can be claimed that logarithmic creep is the first stage in the creep behavior of the amorphous polymers between [T.sub.[Beta]] and [T.sub.g].

PROPERTIES OF LOGARITHMIC CREEP

The main characteristic of logarithmic creep is the similarity between the kinetics of creep and recovery. This similarity has been shown at 55 [degrees] C when creep is only logarithmic [ILLUSTRATION FOR FIGURE 6 OMITTED]. At 70 [degrees] C the two successive creep stages, logarithmic and KWW, are clearly present [ILLUSTRATION FOR FIGURE 12 OMITTED]. The logarithmic creep and its recovery are identical whereas the complete recovery of the KWW part is slower and needs about three times the duration of this creep.

This similarity shows that the internal stresses that induce the recovery are of the same nature as the external stresses at the origin of the logarithmic creep.

The time of the end of logarithmic creep, [t.sub.c], increases with aging time [t.sub.e]. From Fig. 7 it seems that at 70, 84 and 90 [degrees] C this increase is the same: [Mathematical Expression Omitted]. The exponent 0.7 explains the decreases of the relative duration of logarithmic creep compared with test duration during aging.

In contrast, the slope b of logarithmic creep is an increasing function of temperature showing the thermal activation of this process [ILLUSTRATION FOR FIGURE 8 OMITTED]. During aging this slope decreases. At 55 and 70 [degrees] C the relationship between b and [t.sub.e] is [Mathematical Expression Omitted], but at 84 and [Mathematical Expression Omitted]. This rapid decrease of b with aging at high temperature suggests that the limiting values of b reached after a complete aging could be close to 2 x [10.sup.-5] [MPa.sup.1] whatever the temperature.

As a consequence of the opposite variations of b and re, the compliance reached at the end of the logarithmic state J([t.sub.c]) is constant at 70 and 84 [degrees] C, respectively, 1.3 x [10.sup.-3] and 1.38 x [10.sup.-3] [MPa.sup.-1] [ILLUSTRATION FOR FIGURE 9 OMITTED].

The value of the compliance [J.sub.o] is determined by extrapolation at the time t = 0 taken at the maximum of the first oscillation [approximately]0.2 s after the application of the torque on the pendulum. While at 55 [degrees] C, [J.sub.o] remains constant at 1.03 x [10.sup.-3] [MPa.sup.-1] during aging, it decreases slightly with [t.sub.e] at 70, 84 and 90 [degrees] C where the tests have been performed for particularly long aging times: [J.sub.o] decreases from 1.32 x [10.sup.-3] [MPa.sup.-1] at [t.sub.e] = 3 x [10.sup.4] s down to 1.15 x [10.sup.-3] [MPa.sup.-1] at [t.sub.e] = 1.6 x [10.sup.6] s [ILLUSTRATION FOR FIGURE 10 OMITTED]. At this temperature the high slope value b, in other words the rapid creep rate of the beginning of the logarithmic creep, suggests that the creep strain occurring during the 0.2 s necessary for the stress application is a non-negligible contribution to the compliance [J.sub.o]. Therefore [J.sub.o] will slightly decrease during aging; it is not an intrinsic value of the compliance of the polymer.

To estimate the specific logarithmic compliance, the difference [Delta][J.sub.log] = J([t.sub.c]) - [J.sub.o] must be considered [ILLUSTRATION FOR FIGURE 11 OMITTED]. This compliance increases rapidly with temperature from 1.3 x [10.sup.-4] [MPa.sup.-1] at 70 [degrees] C to 3.3 x [10.sup.-4] [MPa.sup.-1] at 90 [degrees] C and seems not to be affected by physical aging.

STRETCHED EXPONENTIAL AND RUBBERY CREEP

As described above, the tests performed at 70 [degrees] C are totally recoverable and the recovery kinetics are slower than the creep kinetics [ILLUSTRATION FOR FIGURE 12 OMITTED]. Beyond the logarithmic state the creep is fitted by a power law with an exponent [Beta] = 0.36 [+ or -] 0.04 whatever the aging time [t.sub.e]. This value is in agreement with the value of 0.33 given by Struik (1).

At 84 [degrees] C the creep does not seem to be totally recoverable and the apparent value of the exponent [Beta] is 0.46. This value is too high, as shown below.

Analysis of Creep at 90 [degrees] C

At 90 [degrees] C, after aging during [t.sub.e] = 1.37 x [10.sup.6] s, the creep is first logarithmic, and then can be fitted by a power law with [Beta] = 0.6, but the recovery strain is lower than the total creep strain [ILLUSTRATION FOR FIGURE 13 OMITTED]. Thus a new mechanism is acting at this temperature, the strain of which is not recoverable. The analysis of the creep data recorded at 90 [degrees] C needs a new assumption. The first stage of this creep is the recoverable logarithmic creep, as at 55 and 70 [degrees] C. So it will be assumed that the second stage at 90 [degrees] C is, as at 70 [degrees] C, the recoverable KWW creep with an exponent of [approximately]0.36 and an intensity given by the recovery. This KWW mechanism is acting all along the creep straining, and beyond a time [t.sub.r] a third nonrecoverable mechanism starts [ILLUSTRATION FOR FIGURE 14 OMITTED], which is superimposed to the KWW stage.

To identify the three parameters of the reduced KWW law the experimental data available are

i) the origin (0 = [t.sub.c], J([t.sub.c])),

ii) the end ([t.sub.f] - [t.sub.c], [J.sub.Rmax]), where [t.sub.f] is the total duration of the creep and [J.sub.Rmax] the maximum of the experimental recovery, obtained after a recovery duration three times longer than the creep test,

iii) the first data obtained beyond [t.sub.c] [ILLUSTRATION FOR FIGURE 14 OMITTED].

In the considered test, [t.sub.c] = 500 s and the numerical law found by this way for the recoverable creep [J.sub.R](t - [t.sub.c]) is

[J.sub.R](t - 500) = 1.42 [multiplied by] [10.sup.-3] + 1.8 [multiplied by] [10.sup.-5] [(t - 500).sup.0.36] (6)

As shown in Fig. 14, this law fits the experimental curve only till the time [t.sub.r] = 4,920 s with J([t.sub.r]) = 1.8 x [10.sup.-3] [MPa.sup.-1]. Beyond [t.sub.r] the creep deformation is due to the addition of this recoverable creep [J.sub.R](t - [t.sub.c]) and a new nonrecoverable creep [J.sub.NR](t - [t.sub.r]).

t [greater than] [t.sub.r] J(t) = J([t.sub.c]) + [J.sub.R](t - [t.sub.c]) + [J.sub.NR] (t - [t.sub.r]) (7)

This creep [J.sub.NR] is determined by subtraction of the recoverable creep defined by Eq 6 from the total creep strain, and is fitted by a power law:

[J.sub.NR] (t - [t.sub.r]) = B[prime] [(t - [t.sub.r]).sup.[Gamma]] (8)

For the specimen it has been found:

[J.sub.NR](t - 4920) = 1.74 [multiplied by] [10.sup.-7][(t - 4920).sup.0.73] (9)

This method has been used to analyze all the creep tests performed at 90 [degrees] C.

The exponents [Beta] and [Gamma] of the power laws do not depend on the aging time, [Gamma] = 0.78 [+ or -] 0.06, and moreover the value of [Beta] is the same at 70 and 90 [degrees] C, [Beta] = 0.36 [+ or -] 0.04. This result confirms the proposed analysis by suggesting that the KWW mechanism is the same at 70 and at 90 [degrees] C.

The value of the parameter B of the KWW creep (Eq. 2) is higher at 90 [degrees] C than at 70 [degrees] C, and at both temperatures it decreases with aging time following a power law [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 15 OMITTED]. The exponent -0.6 must be compared with the exponent 0.7 previously determined for the increase of the onset time [t.sub.c] of the KWW creep with aging. In both cases these exponents are directly related to the decrease of molecular mobility during aging.

During aging, the onset time [t.sub.r] of the nonrecoverable creep increases nearly linearly with aging time, [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 16 OMITTED].

The parameter B[prime] decreases following [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE 17 OMITTED]. This exponent value, higher than the corresponding value of the KWW creep, shows that the reduction of molecular mobility is more efficient on the nonrecoverable creep.

Creep at 84 [degrees] C

The temperature 84 [degrees] C can be considered as a transition temperature, at which the nonrecoverable creep starts to be active.

Without the possibility to determine accurately the intensity of the recovered strain, a reliable creep data analysis as made at 90 [degrees] C cannot be done. Therefore, the analysis in terms of KWW can be used at 84 [degrees] C only as a first approximation.

The value 0.46 found for the exponent [Beta] can be easily explained by the superposition of the recoverable contribution ([Beta] = 0.36) found at 70 and 90 [degrees] C and the earliest steps of the nonrecoverable creep described by a time power law with the exponent 0.78 found at 90 [degrees] C.

DISCUSSION

A logarithmic creep was first observed by Wyatt (4) on copper, aluminum, and cadmium at -196 [degrees] C. At higher temperatures, the creep curves fit the Andrade law. In an intermediate temperature range, the curves fit:

[Epsilon] = [Alpha] log(t) + [[Beta]t.sup.1/3] + [Gamma] (10)

It seems that between 55 and 90 [degrees] C, PMMA exhibits the same behavior.

Logarithmic Creep

The theory of the logarithmic creep has been proposed for metals by Mott (5) and for amorphous polymers by Escaig (6) by using Eyring's model.

Indeed, by derivation of Eq. 5, the kinetics of the logarithmic creep can be written in terms of creep strain [Epsilon]:

[Epsilon][prime] = b.[Sigma]/[Tau] exp - [Epsilon]/[Sigma]b (11)

This macroscopic strain is due to molecular jumps governed by the Eyring relationship:

[Epsilon] = [[Epsilon][prime].sub.o] (exp - [Delta]G - ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT - exp - [Delta]G + ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT) (12)

in which [Delta]G is the activation energy of the barrier between the two sites, [Sigma] the applied stress, [[Sigma].sub.1] the local back stress, [V.sup.*] the activation volume, and

[[Epsilon][prime].sub.o] = N[[Epsilon].sub.o][Nu] (13)

with N the number of mobile elements, [[Epsilon].sub.o] the strain due to the jump of each of these elements and [Nu] the maximal frequency of the jump, close to the Debye frequency ([approximately equal to][10.sup.13] [s.sup.-1]).

It is possible to identify Eqs. 11 and 12 if:

i) the influence of the back jumps is negligible, namely ([Sigma] - [[Sigma].sub.i])[V.sup.*]/kT [greater than] 3; therefore the activation volume [V.sup.*] is large, [10.sup.5] [[Angstrom].sup.3].

ii) the internal back stress is proportional to the strain: [[Sigma].sub.i] = [Alpha]E[Epsilon], E being the Young's modulus and [Angstrom] a proportionality factor.

Under these conditions Eq. 12 becomes:

[Epsilon][prime] = [[Epsilon][prime].sub.o] exp [Delta]G - [Sigma][V.sup.*]/kT exp - [Alpha]E[V.sup.*][Epsilon]/kT (14)

Combining Eqs. 11 and 14, the slope of the logarithmic strain can then be written:

[Sigma].b = kT/[Alpha]E[V.sup.*] (15)

These assumptions imply that:

i) the maximal strain of the logarithmic creep is reached when [[Sigma].sub.i] = [Sigma]. Because [[Sigma].sub.i] = [Alpha]E[Epsilon], this means that [[Epsilon].sub.max] = [Sigma]/[Alpha]E, and thus the maximal strain is proportional to the applied stress. It must be noted that when [[Sigma].sub.i] is close to [Sigma], the back jumps can no longer be neglected and the creep kinetics become the usual exponential law with a single relaxation time as shown on aluminum (3, 7).

ii) after the stress removal, the recovery kinetics is:

[Epsilon][prime] = [[Epsilon][prime].sub.o]exp - [Delta]G - [Alpha]E[V.sup.*][Epsilon]/kT (16)

the value of the parameter b is identical for creep and recovery if the internal back stress [[Sigma].sub.i] has reached a value close to [Sigma].

iii) the high values of b when T approaches [T.sub.g] can be attributed to the increase of the free volume near [T.sub.g], which reduces the activation volume [V.sup.*]. The decrease of b with aging can be correlated with the decrease of free volume during aging.

From Eq. 14 it appears that the creep rate [Epsilon][prime] is not proportional to the applied stress [Sigma] if [V.sup.*] is constant. In fact it has been shown by Haussy (8) that [V.sup.*] is a rapid decreasing function of [Sigma] when [Sigma] [less than] [[Sigma].sub.y], [[Sigma].sub.y] being the yield stress of PMMA, so that no conclusion can be drawn from Eq. 14 concerning the influence of [Sigma] on the creep rate.

The molecular motion, which can be proposed to build up a back stress [[Sigma].sub.i] of the same nature as the applied stress [Sigma], is the formation of double kinks on the chains leading to the creation of micro shear bands. This mechanism of shear bands has already been proposed by many authors since Bowden and Raha (9). Escaig (6, 10) and subsequently Perez (11) have described these shear bands in detail.

KWW Creep

At 70, 84 and 90 [degrees] C, it has been shown that the KWW creep starts when the logarithmic strain reaches a value J([t.sub.c]) - [J.sub.o] specific of each temperature, this value corresponding to a well-defined level of internal stress. This accumulated stress will then be spread out in the volume surrounding the shear bands by means of molecular motions. In this temperature range, the contribution of the [Beta] relaxation is nearly instantaneous, while the [Alpha] molecular motions become preponderant with increasing temperature.

Many models describe relaxation or creep due to diffusion, an extensive overview has been given by Blumen (12). In our case, the most appropriate models have been developed by Hellinckx (13) using a fracial rheological mechanism and Perez with an energetic approach (14). For these authors, creep of amorphous polymers follows a stretched exponential law, and their models take into account the recovery of the strain. The exponent [Beta] is attributed to the degree of hierarchy of correlated molecular motions, [Beta] decreases from 1 to 0 when hierarchy increases, [Beta] = 1 means individual motions and [Beta] [approximately equal to] 0 means that the motions involve a great deal of elementary segments.

When the local back stress is spread out by this diffusional process, the chains are slightly oriented around the shear bands. The expansion of this reorientation is now limited by the presence of entanglements in pure amorphous polymers like PMMA or by the entries of chains in crystallites in semicrystalline polymers.

Nonrecoverable Creep

The two stages of creep described above can be found either with PMMA or with semicrystalline polymers like PA11 (15) or PEEK (16) at temperatures between [T.sub.g] and [T.sub.[Beta]]. The third creep stage in [t.sup.0.78] can be observed only with PMMA. It is specific to amorphous polymers. The main difference between pure amorphous polymers like PMMA and semicrystalline polymers is the chain entries in crystallites, which restrict the displacements of chains. Those restricted motions close to crystallites can explain the nonexistence of the nonrecoverable third stage In semicrystalline polymers.

In contrast, in PMMA, the entanglements can move. leading to the rubbery behavior. This creep is not recoverable at 90 [degrees] C in the experimental time, or more exactly, its recovery is so slow that it seems to be not recoverable. This suggests that the acting back stresses are low and may be of entropic nature, resulting from displacements of entanglements.

It is now necessary to quote the study of Plazek (2) on the recoverable creep of PMMA between [T.sub.g] - 10 [degrees] C and [T.sub.g] + 50 [degrees] C. This creep is in fact the delayed rubbery strain of PMMA, which can be well fitted by a stretched exponential law with an exponent 0.8 [ILLUSTRATION FOR FIGURE 18 OMITTED]. At [T.sub.g] - 10 [degrees] C, creep starts during the first 30 s by a power law in [t.sup.0.33]. Beyond 30 s, the creep law is a power law in [t.sup.0.8], i.e., the beginning of the stretched exponential law described above. This creep is recoverable, but at temperatures close to [T.sub.g], Plazek must heat the specimen to achieve the recovery.

By analogy with the study of Plazek, the nonrecoverable microcreep detected at 90 [degrees] C is probably the beginning of the rubbery creep. It is nonrecoverable because 90 [degrees] C is a temperature too far below [T.sub.g]. Knowing that the rubbery strain in amorphous polymers is due to displacements of entanglements, it is now possible to attribute the "nonrecoverable" creep at 90 [degrees] C to such displacements. This is in accordance with the nonexistence of this creep in semicrystalline polymers. It is surprising that the influence of entanglements can be detected at strains as low as [10.sup.-4] [s.sup.-1]. Nevertheless, if this creep is actually the beginning of the delayed rubbery deformation of PMMA, it is difficult to imagine the existence of a limiting strain for this behavior.

Thus during the KWW stage, the back stress is progressively spread out of the shear band, and then it reaches the entanglements, which move during the last step of creep.

CONCLUSION

Microcreep of PMMA between 55 and 90 [degrees] C starts with logarithmic kinetics. During this stage, microshear bands are created and induce at their limit internal back stresses proportional to their dimensions. This logarithmic creep stops as soon as these internal back stresses are equal to the external stress. When the creep temperature is high enough to allow molecular diffusion, internal back stresses are progressively spread out around the shear bands by appropriate correlated molecular jumps. This is the recoverable KWW creep, which can be approximated by a power law in B.[t.sup.0.36] At 90 [degrees] C, a third stage corresponding to a nonrecoverable creep appears; it is described by the power law in B[prime].[t.sup.0.78]. This stage appears to be the beginning of the creep studied by Plazek above [T.sub.g]. It can be described by a stretched exponential law and corresponds to a delayed rubbery strain. Therefore, the nonrecoverable power law creep can be attributed to the beginning of displacements of entanglements.

NOMENCLATURE

J = Compliance = [Sigma]/S.

[J.sub.u] = Unrelaxed compliance extrapolated from the KWW law.

[J.sub.o] = Unrelaxed compliance extrapolated from the logarithmic law.

[J.sub.R] = Recoverable compliance.

[J.sub.NR] = Unrecoverable compliance.

B = Factor of the recoverable power law creep.

B[prime] = Factor of the unrecoverable power law creep.

[[Sigma].sub.i] = Internal back stress.

[Alpha] = Proportionality factor between the internal back stress and the microscopic strain.

[Beta] = Exponent of the recoverable KWW creep or power law approximation.

[Gamma] = Exponent of the unrecoverable KWW creep or power law approximation.

b = Slope of the logarithmic creep.

[t.sub.c] = Aging time.

[t.sub.e] = Beginning of the recoverable power law creep (end of the logarithmic creep).

[t.sub.r] = Beginning of the unrecoverable power law creep.

[t.sub.f] = Total duration of the creep.

REFERENCES

1. L. C. E. Struik, in Physical Aging in Amorphous Polymers and Other Materials, Elsevier Scientific Pub, Co. (1978).

2. D. J. Plazek, V. Tan, and V. M. O'Rourke, Rheol. Acta, 13, 367 (1974).

3. J. L. Gacougnolle, J. F. Pelletier, and J. de Fouquet, in Mechanical Testing for Deformation Model Development, Rohde-Swearengen, ed., ASTM STP 765, 67 (1980).

4. O. Wyatt, Proc. Phys. Soc., B66, 459 (1953).

5. N. F. Mort, Philos. Mag., 44, 742 (1953).

6. B. Escaig, in Plastic deformation of Amorphous and Semicrystalline Materials, pp. 187-225, B. Escaig and C. G'Sell, eds., Les Editions de Physique, Les Ulis (1982).

7. J. F. Pelletier, These Universite de Poitiers, pp. 23-24, France (1982).

8. J. Haussy, J. P. Cavrot, B. Escaig, and J. M. Lefebvre, J. Polym. Sci. Phys., 18, 311 (1980).

9. P. B. Bowden and S. Raha, Philos. Mag., 29, 149 (1974).

10. J. M. Lefebvre and B. Escaig, Polymer, 34, 518 (1993).

11. J. Perez, Physique et Mecanique des Polymeres Amorphes, p. 151, Techniques & Documentation, Ed. Lavoisier, Paris (1992).

12. A. Blumen, in Lecture Notes in Physics 277, Molecular Dynamics and Relaxation Phenomena in Glasses, 1, Springer Verlag (1985).

13. S. Hellinckx, N. Heymans, and J.-C. Bauwens, J. Non-Crystalline Solids, 172-174, 1058 (1994).

14. J. Perez, Physique et Mecanique des Polymeres Amorphes, p. 75, Techniques & Documentation, Ed. Lavoisier, Paris (1992).

15. L. Belec, These Docteur Science des Materiaux, pp. 97-121, Poitiers, France (1995).

16. X. Xinran, thesis, Free University of Brussels, pp. 89-92 (1987).

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Title Annotation: | French Research on Structural Properties of Polymers |
---|---|

Author: | Cheriere, J.M.; Belec, L.; Gacougnolle, J.L. |

Publication: | Polymer Engineering and Science |

Date: | Oct 1, 1997 |

Words: | 5010 |

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