Review: degradation-induced embrittlement in semi-crystalline polymers having their amorphous phase in rubbery state

Abstract

The literature dealing with degradation-induced embrittlement mechanisms in semi-crystalline polymers having their amorphous phase in rubbery state is reviewed. It is first demonstrated that the decrease of molar mass resulting from a quasi-homogeneous chain scission process is responsible for embrittlement. The main specificity of the polymer family under study is that embrittlement occurs at a very low conversion of the degradation process, while the entanglement network in the amorphous phase is slightly damaged. In these polymers, chain scission induces chemicrystallization. The analyses of available data on this process show that it is characterized by a relatively high yield: about one half entanglement strands integrate the crystalline phase after one chain scission. A simple relationship expressing the chemicrystallization yield for a given polymer structure is proposed. Chain scission and chemicrystallization can lead to embrittlement through two possible causal chains: (1) chain scission → molar mass decrease → chemicrystallization → decrease of the interlamellar spacing → embrittlement. (2) Chain scission → molar mass decrease → chemicrystallization → decrease of the tie-macromolecule concentration → embrittlement. At this state of our knowledge, both causal chains are almost undistinguishable.

Introduction

In a wide variety of chemical aging causes, for instance photo-oxidation, thermo-oxidation, and hydrolysis, linear polymers undergo random chain scission which induces embrittlement. The latter can be described as a transition from a ductile to a brittle behavior, which displays the following characteristics:

• 1. It occurs suddenly and abruptly. This abrupt character is especially marked when the mechanical behavior is characterized by a tensile test [1], but no doubt, embrittlement corresponds to a critical structural state.

• 2. It occurs at low conversion of the degradation process, generally when less than 1% of the monomer units have undergone a chain scission. These conversions are generally out of reach of common spectrochemical methods (IR, NMR, etc.). This fact explains, at least partially, why embrittlement mechanisms remained an opened question for a long time and why interest of research workers had been mostly focused on chemical aspects of aging at high conversions. It is noteworthy that a ductile–brittle transition can also occur without chemical degradation in polymers submitted to mechanical loads during aging, for instance in the case of polyethylene (PE) pipes under pressure [2]. This important case will not be considered here, our investigation being focused only on chemical degradation effect.

As far as embrittlement mechanisms are concerned, semi-crystalline polymers having their amorphous phase in rubbery state such as PE, polypropylene (PP), polyoxymethylene (POM), etc., constitute a specific family because their embrittlement occurs at especially low conversions of the chain scission process. One can suspect that the chain scission process induces morphological changes and that these latter play an important role as it will be seen next.

It is not uncommon to see articles reporting mechanical properties of aged polymers in the scientific and technical literature, but articles giving at the same time mechanical properties and pertinent analytical data needed for the understanding of embrittlement processes are very scarce. On the other hand, relatively little was known, until the end of the last century, on structure–fracture behavior relationships applied to aging cases. These factors can explain the relative under-development of the knowledge of embrittlement processes. The situation is now changing because there were decisive advances in the domain of polymer fracture properties in the last 15 years [3, 4]. It seemed to us interesting to use these advances to investigate on the links between the concerned domains, i.e. degradation-induced molar mass changes, morphological changes and consequences of these structural changes on the fracture behavior. Here, “degradation” is taken as synonym of “random chain scission”.

This article has been written having in mind a more or less remote objective: the elaboration of a non-empirical kinetic model leading to lifetime prediction, using the ductile–brittle transition as an endlife criterion. The point of departure of this approach would be a model based on chemical kinetics principles and able to predict the number s of chain scissions. Examples of such models with relatively strong predictive power have been recently published in the case of unstabilized polymers [5, 6]. Their extension to the case of stabilized polymers is now under study in our laboratory [7]. It will be supposed here that the step of kinetic modeling of chemical degradation is solved and that the remaining issue can be summed up in the following question: Is it possible to imagine a causal chain starting from chain scission and leading to embrittlement, in which the elementary links would have a mathematical expression?

It seemed to us interesting to formulate the main problem relative to each link as a question and to dedicate one section to each question, in the order corresponding to the presumed causal chain: chain scission → decrease of molar mass → chemicrystallization → decrease of the interlamellar spacing → change of key morphology parameters → embrittlement.

Is molar mass a pertinent variable?

According to Griffith [8], the critical conditions for crack propagation in a solid would be given by:

$$ \frac{{G_{\text{IC}} }}{{a_{\text{C}} }} = \frac{{ 2 {{\pi}}\nu {{\sigma}}_{\text{Y}}^{2} }}{E} = \varphi $$

(1)

where G _IC is the critical ratio of energy release in crack propagation, a _C is the half defect size, σ_Y is the applied stress equal to yield stress, ν and E are respectively the Poisson’s ratio and the Young’s modulus, so that φ is a material’s property. The crack propagates if G _IC/a _C < φ and does not propagate if G _IC/a _C > φ. This relationship is only valid for brittle and elastic materials, but it can be generalized as follows.

According to (1), there are only two ways leading to fracture for a given applied stress: toughness decrease (G _IC) or defect size increase (a _C). Both ways can be envisaged in the case of aging: the first one would correspond to homogeneous degradation, G _IC being an increasing function of molar mass, i.e. a decreasing function of the number of chain scissions. The second one would correspond to heterogeneous degradation, as evidenced, for instance, from chemiluminescence experiments by Celina and George [9]. In the first case (toughness decrease), use of molar mass–fracture property relationships would have a physical meaning. In the second case, average molar mass or any species concentration values would not be the relevant parameters. Although the coexistence of homogeneous and heterogeneous processes is not excluded, it will be tried here to demonstrate that, for the polymers under consideration, aging-induced embrittlement results from the first way, i.e. the toughness decrease associated to a homogeneous degradation. Two types of arguments will be used: (1) about defect size at the embrittlement point and (2) about the fact that degraded polymers behave in the same way as virgin ones showing a same molar mass value.

Defect size at the embrittlement point

Applications of Eq. 1 to PP or PE, for which G _IC typically ranges between 6 and 8 kJ m⁻² [3], would lead to a _C values of the order of 1 cm, that is, no doubt, excessive. Furthermore, it is well known that visible scratches do not affect significantly the strength of PP injection molded bumpers or PE rotomoulded tanks. In these ductile polymers, it is clear that defects must be relatively large to induce brittleness. However, this conclusion is not consistent with the fact that embrittlement occurs at an extremely low (average) aging conversion, for instance, in PP, before the appearance of a carbonyl peak in IR spectra [1].

In some cases, brown spots could appear during oxidation after an induction period [10]. This indicates the build-up of quasi-isotropic heterogeneities in PP thermal oxidation after the end of induction period. However, the extrapolation of their size to short times seems to indicate that they nucleate after embrittlement time [11]. When available, SAXS measurements do not indicate the existence of density fluctuations linked to an eventual defect build-up [12]. In the same way, positron annihilation measurements show no change in the size of free volume holes of PP after photo-oxidation [13]. Last but not least, steric exclusion chromatograms of oxidized samples do not reveal the polydispersity increase or, even, the build-up of a low molar mass component attributable to localized degradation. In contrast, progressive shift of molar mass distribution toward low values with a decrease of polydispersity index (when this latter is initially higher than 2), both characteristic of homogeneous random chain scission, is observed in PP photo-oxidation [14, 15], thermal oxidation in solid state [16–19], and probably also in radiochemical oxidation, although in this latter case the polydispersity index values were not reported [20]. Some observations were made in the case of PE thermal oxidation in solid state [21–23], although in some cases, anomalous stabilizer depletion could explain some anomalous behavior in stabilized samples [21]. Viebke et al. [21] mentioned that polybutene-1 behaves in the same way as PE. Photo-oxidation [24] and radio-oxidation [25] of PE led to similar observations. However, crosslinking can compete with chain scissions in PE oxidation, which can lead to a polydispersity increase, as it has been found by Fayolle et al. [26] by studying molar mass distribution of thermally oxidized metallocene-PE samples. But in all cases, M _W decreases as soon as the exposure begins, which indicates clearly the existence of a homogeneous degradation process. Molar mass data are also available for POM thermal oxidation at 130 °C [27]: they indicate clearly the existence of homogeneous degradation.

These observations can appear surprising since it is well known that oxidation is heterogeneous at the morphological scale: it occurs only in the amorphous phase because molecular reagents (oxygen, water…) are insoluble in the crystalline phase. This latter is organized in more or less regularly spaced lamellae with long periods typically few dozens of nanometers. There is a wide consensus (see below) on the fact that ductility is linked to the presence of macromolecules of which the gyration radius is higher than the long period, which allows the existence of tie-chains interconnecting lamellae. Two possible distributions of chain scission events can be distinguished. They are schematized in Fig. 1.

Fig. 1

Schematization of two possible distributions of scission events at the lamellae scale (a, b), and highly degraded state (c)

In the case (a) where scissions are homogeneously distributed in the amorphous phase, degradation will display the characteristics of a random chain scission at low conversions. At relatively high conversions, however, the molar mass distribution will display regularly spaced peaks corresponding to a small integer number of crystalline segments (Fig. 1c) [11].

“Critical” molar mass for ductile–brittle transition in virgin and degraded polymers

The most common parameter used to characterize the ductile–brittle transition in the polymer under study is the ultimate strain ε_R or the ultimate draw ratio λ_R = 1 + ε_R. Its change with molar mass M _W displays generally the shape of Fig. 2.

Fig. 2

Ultimate draw ratio determined from tensile testing (λ_R) versus weight average molar mass (M _W) from literature data [26]

Brittle and ductile regimes of rupture can be well distinguished by the absence in the former and the presence in the latter of a plastic deformation. In the ductile regime, λ_R is a slowly decreasing function of M _W, attributed to the increase of entanglement density. The ductile–brittle transition occurs in a more or less sharp molar mass interval. In a first approach, it will be considered that it corresponds to a critical molar mass $M_{{\text{C}}}^{\prime}$ (M _C being the critical rheological mass, $M_{{\text{C}}}^{\prime}$ ≥ M _C in all cases). In the case of PE, the graph λ_R = f(M _W) has been built from compiled literature data irrespective of the polymer microstructure, processing conditions, strain rate, aging history [26]. This graph displays the shape of Fig. 2, the ductile–brittle transition being spread over the 40–100 kg mol⁻¹ molar mass interval. One can conclude for the time being that $M_{{\text{C}}}^{\prime}$ = 70 ± 30 kg mol⁻¹. In this set of data, virgin samples cannot be distinguished from degraded ones.

The same analysis can be made for PP. For virgin samples, the ductile–brittle transition is always located between 170 and 250 kg mol⁻¹ [19, 28–30]. Concerning degraded samples, it has been found that $M_{{\text{C}}}^{\prime}$ = 200 kg mol⁻¹ in the case of photo-oxidation [31] and thermal oxidation [1], and 180 kg mol⁻¹ in the case of radiochemical oxidation [20].

From this short review, it is clear that if systematic differences exist between degraded and virgin samples of same average molar mass, they are too subtle to be observed in such compilations, owing to the multiple sources of scatter, especially those relative to molar mass measurements.

It can be concluded that, in the polymers under investigation, molecular weight data indicate that there is always a homogeneous random chain scission process, which does not exclude the coexistence of a heterogeneous one, and that embrittlement is due to the decrease of molar mass resulting from the homogeneous process. Embrittlement is linked consequently to a decrease of polymer toughness rather than from defects build-up. As a result, molar mass can be considered as the key parameter to characterize aging effects in the cases under study.

Why embrittlement occurs at so high molar mass?

Let us compare, in Table 1, three amorphous glassy polymers and three semi-crystalline polymers having their amorphous phase in rubbery state [1, 3, 26, 27, 32–35].

Table 1 Critical molar mass for ductile–brittle transition (weight average) ($M_{{\text{C}}}^{\prime}$), entanglement molar mass (M _E), and ratio $M_{{\text{C}}}^{\prime}$/M _E for three glassy polymers and semi-crystalline polymers

For a polymer of number average molar mass M _N0, the concentration ν ₀ of entanglement strands is given by:

$$ \nu_{ 0} = \frac{ 1}{{M_{\text{E}} }}\left( { 1- \frac{{ 2M_{\text{E}} }}{{M_{\text{N0}} }}} \right) $$

(2)

with M _E being the molar mass between entanglements for the given polymer.

At the embrittlement point, the number of broken entanglement strands is given by:

$$ s_{\text{C}} = 2\left( {\frac{1}{{M^\prime_{\text{C}} }} - \frac{1}{{M_{\text{W0}} }}} \right) = \frac{2}{{M^\prime_{\text{C}} }}\left( {1 - \frac{{M^\prime_{\text{C}} }}{{M_{{{\text{W}}0}} }}} \right) $$

(3)

The fraction of broken entanglement strands at the ductile–brittle transition is therefore:

$$ \frac{{s_{\text{C}} }}{{{{\nu}}_{0} }} = \frac{{2M_{\text{E}} }}{{M^\prime_{\text{C}} }} \left[ {\frac{{1 - \frac{{M^\prime_{\text{C}} }}{{M_{{{\text{W}}0}} }}}}{{1 - \frac{{2{\text{IP}}_{0} \times M_{\text{E}} }}{{M_{{{\text{W}}0}} }}}}} \right] $$

(4)

where IP₀ is the initial polydispersity index. For high initial molar masses:

$$ \frac{{s_{\text{C}} }}{{{{\nu}}_{0} }} = \frac{{2M_{\text{E}} }}{{M^\prime_{\text{C}} }} $$

(5)

One sees that, in amorphous polymers of Table 1:

$$ \frac{{s_{\text{C}} }}{{{{\nu}}_{0} }} > 12\% $$

whereas for semi-crystalline polymers having their amorphous phase in rubbery state:

$$ \frac{{s_{\text{C}} }}{{{{\nu}}_{0} }}\,<\,7\% $$

In other words, in this latter polymer family, ductile–brittle transition occurs before the entanglement network has been significantly damaged, whereas in amorphous polymers, the destruction of the entanglement network can be reasonably considered as the direct cause of embrittlement.

There are two possible ways to explain the fact that $M_{{\text{C}}}^{\prime}$ ≫ M _E. The first one is that the lamellar morphology determines a critical chain dimension not related to the entanglement molar mass, but longer than this latter. Two important quantities have to be considered in this case: the lamella thickness l _c and the long period l _p. Some theories trying to establish a link between the lamellar structure and the critical molar mass have been recently reviewed by Plummer [36]. For instance, for Tervoort et al. [37] one could have:

$$ M_{\text{C}}^{{\prime}} \,\sim \,3M_{\text{L}} $$

where M _L is the molar mass of the chain segment corresponding to lamella thickness. For PE, M _L ~ 10–14 kg mol⁻¹, which leads to a $M_{{\text{C}}}^{\prime}$ value significantly lower than the experimental value: 3M _L ~ 30–40 kg mol⁻¹ against 40–100 kg mol⁻¹ experimental values. For Benkonski et al. [38], the onset of ductility would be reached for chains having their end-to-end distance of the order of the long period l _p, which would lead to:

$$ M^\prime_{\text{C}} = \frac{{l_{\text{p}}^{2} M_{\text{m}} }}{{C_{\infty } l^{2} }} $$

(6)

where M _m is the molar mass of an elementary chain unit of which the length is l and C _∞ is the chain characteristic ratio. In PE, for example, C _∞ = 6.8 and l = 0.154 nm. The long period is an increasing function of the molar mass which can vary between 15 and 40 nm in PE. The lowest value of $M_{{\text{C}}}^{\prime}$ would be thus close to 20 kg mol⁻¹, a value significantly lower than the experimental one.

The l _p value corresponding to $M_{{\text{C}}}^{\prime}$ = 70 kg mol⁻¹ is 28 nm. In reality, such long periods are found in PE samples of M _W higher than 200 kg mol⁻¹.

Many authors tried to interpret aging-induced embrittlement in terms of tie-molecules interconnecting lamellae. Embrittlement would correspond to a critical tie-molecules concentration. The problem here is that embrittlement occurs when a relatively small fraction of chain has been broken.

Some authors, for example Oswald and Turi [39] in the case of PP thermal oxidation, suggested that oxidation attacks selectively tie-chains. On the other hand, we know that tie-chains constitute only a relatively small fraction of the whole number of chains [40]. How to explain the selectivity of oxidation? The second and promising way of explanation of the fact that embrittlement occurs at very high molar mass in the polymer family under consideration is that chain scission induces morphological changes. The true critical quantity would be therefore a morphological parameter rather than a molar mass. The following sections are aimed to explore this way.

Does random chain scission modify crystallinity?

Molar mass changes

Let us recall that molar mass distribution can be modified by random chain scission or crosslinking. The average molar masses are linked to the number s of chain scission and x of crosslinking events (per mass unit) by Saito’s equations [41]:

$$ \frac{1}{{M_{\text{N}} }} - \frac{1}{{M_{\text{N0}} }} = s - x $$

(7)

$$ \frac{1}{{M_{\text{W}} }} - \frac{1}{{M_{{{\text{W}}0}} }} = \frac{s}{2} - 2x $$

(8)

In the following, only “pure” random chain scission will be considered (x = 0). The above equations lead then to the following expression for M _W and for the polydispersity index I:

$$ M_{\text{W}} = \frac{{M_{{{\text{W}}0}} }}{{1 + s \times \frac{{M_{\text{W0}} }}{2}}} = \frac{{M_{{{\text{W}}0}} }}{{1 + \frac{C}{2}}} $$

(9)

$$ I = I_{0} \times \frac{{1 + \frac{C}{{I_{0} }}}}{{1 + \frac{C}{2}}} $$

(10)

where C = s × M _W0 is the number of chain scission events per initial weight average chain. Equation 10 shows that I increases when I ₀ < 2 and decreases when I ₀ > 2 but tends toward 2 in both cases.

Indeed, both average molar mass values, M _N or M _W, can be, in principle, indifferently chosen to express degradation effects since they are interrelated. Here, however, M _W will be preferred, first because it is easier to determine, for instance from rheometric data, second because fracture properties seem to be governed mainly by M _W as shown by the example of tensile ultimate elongation for HDPE [42–46]. In a detailed study of crystalline morphology–molar mass relationships, Robelin-Souffaché and Rault [47] find that the key parameter is the weight average gyration radius that leads to consider a new average molar mass, M _R, defined by:

$$ M_{\text{R}}^{1/2} = \sum\limits_{i} {\left( {\frac{{N_{i} M_{i} }}{{\sum_{i} {N_{i} M_{i} } }}} \right)} M_{i}^{1/2} $$

(11)

where M _R is intermediary between M _N and M _W but closer to M _W so that it will be considered that, at least in a coarse grain approach, M _W is a pertinent quantity.

Effect on chain scission on crystallinity ratio

In quenched semi-crystalline polymers, crystallization is limited by the presence of chain entanglements in the melt [47]. Chain scissions occur in the amorphous phase and liberate chain segments, which will integrate the crystalline phase if they have sufficient mobility, which is the case in rubbery state. This process, called chemicrystallization, was observed one half century ago in PET hydrolysis [48] and in polyolefins oxidation [49–51]. In the following decades, results showing the existence of chemicrystallization were accumulated as shown by the non-exhaustive compilation summarized by Table 2 [52–70].

Table 2 Literature references about chemicrystallization

Crosslinking tends to inhibit crystallization. This is clearly put in evidence in case of gamma-irradiated PE where oxidative chain scission predominates in a superficial layer (because oxidation is diffusion controlled) whereas anaerobic crosslinking predominates in the core [70].

Quantitative studies of chemicrystallization are often complicated by the following processes:

• 1. The loss of small molecules by evaporation (in the case of oxidation) or extraction by water (in the case of hydrolysis). Since the loss exclusively comes from amorphous phase degradation, it contributes to an increase of crystallinity ratio.

• 2. In the case of hydrocarbon polymer oxidation, there is a density increase due to the incorporation of “heavy” oxygen atoms to the chains. The density can largely overtake the value ρ_c corresponding to 100% of crystallinity, as shown for instance for crosslinked PE [71].

• 3. In the case of irradiation by ionizing radiations, radiochemical events occurring into the crystalline phase induce a decrease of crystallinity ratio which can compensate, at least partially, the effect of chemicrystallization. In well oxygenated regions, however, the radiochemical yield of crystal destruction is low compared to the crystallization yield, so that crystallization is easily observable.

• 4. In the case of thermal aging, annealing effects can coexist with chemicrystallization ones when the samples are initially undercrystallized. The amplitude of annealing effects can be easily appreciated by comparing degraded samples with samples having undergone the same thermal treatment but with the absence of the degrading agent (oxygen in the case of thermal aging, water in the case of hydrolysis [54]).

Fortunately, crystallinity changes become often observable at low conversion of the degradation process, when all the above-mentioned problems, except indeed (4), are negligible. This means that chemicrystallization is characterized by a relatively high yield. The latter can be defined as the number y of monomer (or elementary chain segment) units entering in the crystalline phase per scission event.

The instantaneous value y(t) of this yield is given by:

$$ y(t) = \frac{1}{{M_{\text{m}} }} \times {{\frac{{{\text{d}}x_{\text{c}} }}{{{\text{d}}t}}} \mathord{\left/ {\vphantom {{\frac{{{\text{d}}x_{\text{c}} }}{{{\text{d}}t}}} {\frac{{{\text{d}}s}}{{{\text{d}}t}}}}} \right. \kern-\nulldelimiterspace} {\frac{{{\text{d}}s}}{{{\text{d}}t}}}} $$

(12)

where M _m is the molar mass of the monomer unit, x _c is the crystallinity ratio, and s is the number of chain scission event per mass unit.

Quantitative data on chemicrystallization are relatively scarce in the literature: In the case of PET hydrolysis at 100 °C, Ballara and Verdu [54] reported a value of 5–6 monomer units per scission. In the case of thermal PE thermal oxidation in the 70–105 °C temperature range, Viebke et al. [21] found a value of 45 methylenes per scission. About the same value was found by Fayolle et al. [27] in the POM thermal oxidation at 130 °C. These values can be compared with the length of entanglement strands in the melt, in Table 3.

Table 3 Entanglement molar mass (M _E) [34, 35], number of monomer units per entanglement strands (N _E), chemicrystallization yield y(t), and ratio y(t)/N _E

It can be seen that, in all cases, the length of the chain segments entering the crystalline phase after one chain scission is an important fraction of the entanglement strand, often ~50% or more. Is it possible to derive this result from structure–property relationships established on virgin polymer samples? From a theoretical analysis, Robelin-Souffaché and Rault [47] proposed:

$$ x_{\text{c}} = 1 - \psi \left( {\frac{1}{{r_{\text{WE}} }} - \frac{1}{{r_{\text{W}} }}} \right) $$

(13)

where r _W and r _WE are the weight average gyration radii for respectively the polymer under consideration and a polymer of which the molar mass is close to the entanglement threshold. ψ is a parameter depending on chemical structure and crystallization conditions.

Considering that, in a first approximation:

$$ r_{\text{W}} \propto M_{\text{W}}^{1/2} $$

(14)

Equation 13 becomes:

$$ x_{\text{c}} = a + b \cdot M_{\text{W}}^{{ -{1/2}}} $$

(15)

where a and b depend on the chemical structure and crystallization conditions.

It has been tried to check Eq. 15 in the case of HDPE with available data [42, 72–74] plotting x _c against $M_{\text{W}}^{{ -{1/2}}}$ (Fig. 3).

Fig. 3

Crystallinity ratio (x _C) versus $M_{\text{W}}^{{ -{1/2}}}$ for quenched (open symbols) and isothermally crystallized (closed symbols) HDPE samples. ■, □: from Kennedy et al. [42]; ◯: from Nitta and Tanaka [72]; ▲, △: from Voigt-Martin and Mandelkern [73]; ◆, ◊: from Jordens et al. [74]

Although the results are relatively scattered, a linear dependence can be assumed. The parameters a and b of the straightline and correlation coefficient R ² are given in Table 4. These data call for the following comments: These results are too scattered to be considered as good proofs of validity of the Robelin-Souffaché and Rault’s theory [47], but this is not very surprising since we know that crystallinity depends on precise cooling conditions and thermal history in molten state prior to crystallization [47]. Furthermore, due to the approximation made in Eq. 14 (M _R assimilated to M _W), the parameters must display some dependence with polydispersity. These scattering sources must be added to experimental ones, as well on x _c as on M _W measurements. According to the starting theory (Eq. 13), all the straightlines x _c = f($M_{\text{W}}^{{ -{1/2}}}$) must converge on a point of coordinates x _c = 1, M _W = M _WE. M _WE, the “entanglement threshold molar mass”, is given by:

$$ M_{\text{WE}} = \left( {\frac{b}{1 - a}} \right)^{2} $$

(16)

Table 4 Parameters of Eq. 15, coefficient of correlation for the linear regression, and entanglement threshold molar mass calculated from Eq. 15

The values of M _WE calculated from Eq. 16 are listed in Table 4. The series of isothermally crystallized samples from Kennedy et al. [42] has been put apart because no dependence of x _c with M _W was found. M _WE values range between 1100 and 6400 g mol⁻¹, which is the good order of magnitude, considering that the entanglement molar mass is close to 1400 g mol⁻¹ and that M _WE can be significantly higher owing to polydispersity effects. The results of Fig. 3 and Table 4 clearly show that the relationship between M _W and x _c cannot be described by a single curve, which is not very favorable to the project of including chemicrystallization into a kinetic model. A possibility would remain, however, if one assumes that the Robelin-Souffraché and Rault’s law [47] applies to a sample undergoing degradation. In this case, all the points ($M_{\text{W}}^{{ -{1/2}}}$, x _c) corresponding to distinct aging times would be located in a straightline of which two points are known: the departure point ($M_{\text{W0}}^{{ -{1/2}}}$, x _c0) and the point corresponding to entanglement threshold ($M_{\text{WE}}^{{ -{1/2}}}$, 1). The equation of the straightline can be then determined. By using Eqs. 15 and 16, the relationship between x _c and M _W can be expressed as:

$$ x_{\text{c}} = x_{{{\text{c}}0}} + \frac{{1 - x_{{{\text{c}}0}} }}{{\left[ {\left( {\frac{{M_{{{\text{W}}0}} }}{{M_{\text{WE}} }}} \right)^{1/2} - 1} \right]}} \left[ {\left( {\frac{{M_{\text{W0}} }}{{M_{\text{W}} }}} \right)^{1/2} - 1} \right] $$

(17)

where

$$ \frac{{M_{{{\text{W}}0}} }}{{M_{\text{W}} }} = \left( {1 + \frac{C}{2}} \right) $$

(18)

In the range of interest (0.5 ≤ C ≤ 10), calculation proves that the expression between brackets of Eq. 17 can be approximated by:

$$ \left( {1 + \frac{C}{2}} \right)^{1/2} - 1 \approx C/5 $$

(19)

So that:

$$ x_{\text{c}} = x_{\text{c0}} + BC $$

(20)

where

$$ B = \frac{1}{5} \frac{{1 - x_{{{\text{c}}0}} }}{{\left[ {\left( {\frac{{M_{{{\text{W}}0}} }}{{M_{\text{WE}} }}} \right)^{1/2} - 1} \right]}} $$

(21)

The crystallization yield is:

$$ y = \frac{1}{{M_{\text{m}} }}\frac{{{\text{d}}x_{\text{c}} }}{{{\text{d}}s}} = \frac{1}{{M_{\text{m}} }}\frac{{{\text{d}}x_{\text{c}} }}{{{\text{d}}C}}\frac{{{\text{d}}C}}{{{\text{d}}s}} $$

(22)

It comes:

$$ y = B\frac{{M_{{{\text{W}}0}} }}{{M_{\text{m}} }} $$

(23)

where M _m is the molar mass of the monomer unit or the elementary chain unit (methylene in the case of PE). A numerical application has been made in three cases: PE thermal oxidation with characteristics close to the ones studied by Viebke et al. [21], POM thermal oxidation with characteristics close to the ones studied by Fayolle et al. [27], and PET hydrolysis in conditions close to the ones studied by Ballara and Verdu [54]. The results are reported in Table 5. They call for the following comments:

• 1. Equation 23 predicts good order of magnitude for the chemicrystallization yield in POM and PET.

• 2. For PE, the calculated value is about three times higher than the experimental one. In fact, Viebke’s result was obtained for a copolymer grade for pipe extrusion in which branching limits crystallization. Our calculation was made for linear PE in which the crystallization yield must be, indeed, higher.

Table 5 Calculated (according to Eq. 22) and experimental chemicrystallization yields, M _WE being assimilated to the entanglement molar mass M _E

To conclude this section, Eq. 17 or its simplified version Eq. 23 valid at low conversions of the degradation process seems to be an interesting way to predict the chemicrystallization yield, because it is based on reasonable hypotheses, contains no adjustable parameters, and is very simple. However, its validity is limited to linear homopolymers. The chemicrystallization yield appears to be proportional to the initial amorphous content (1 – x _c0) and to the initial weight average degree of polymerization M _W/M _m. It is also a slowly increasing function of the entanglement molar mass.

The relatively good predictive value of Eq. 23 can appear surprising considering the multiple sources of scatter. It can be understood, at least in the case of thermal aging, because aging imposes a long duration thermal treatment during which the polymer can reach a quasi-equilibrium mainly determined by molar mass distribution and the initial morphology. From a practical point of view, however, consideration of x _c rather than M _W will not improve the accuracy of a kinetic model based on the above equations. Furthermore, as it will be seen next, the concept of critical crystallinity ratio corresponding to ductile–brittle transition would be questionable: for a given x _c value, it is possible to observe both mechanical behaviors.

Effect of chain scission on lamellar structure

There is now a wide consensus on the fact that fracture properties are mainly governed by the lamellar morphology [36, 75–77]. Although the lamella length, or better, its aspect ratio, plays a non-negligible role, at least in elastic properties [78], it is usual to focus on transverse dimensions of lamellae stacks, i.e. the long period l _p, the lamella thickness l _C, and the thickness of the amorphous layer separating two adjacent lamellae l _a. These quantities are interrelated as follows:

$$ l_{\text{p}} = l_{\text{c}} + l_{\text{a}} $$

(24)

$$ l_{\text{c}} = x_{\text{c}} \frac{{l_{\text{p}} {{\rho}}_{\text{a}} }}{{{{\rho}}_{\text{c}} - x_{\text{c}} \left( {{{\rho}}_{\text{c}} - {{\rho}}{}_{\text{a}}} \right)}} $$

(25)

where ρ_c and ρ_a are the densities of crystalline and amorphous phase respectively. l _p can be determined by SAXS [47], l _c can be determined by Raman spectroscopy [79] or from melting point T _M using Gibbs–Thomson equation:

$$ l_{\text{c}} = \frac{ 2\sigma }{{\rho_{\text{c}} \Updelta H_{\text{M}} }} \frac{{T_{\text{M0}} }}{{T_{\text{M0}} - T_{\text{M}} }} $$

(26)

where σ is the surface energy, T _M0 is the equilibrium melting point, and ΔH _M is the melting enthalpy of crystal.

As noticed by Gedde and Ifwarson [80], the parameters of this equation, especially σ and ΔH _M, can vary during oxidative aging so that this approach for estimating l _c can be used with caution. At least, l _p, l _c, and l _a can also be determined from micrographs obtained by electron microscopy when well resolved images are available [36, 73, 76].

Schematically, for virgin samples, l _c appears almost independent of molar mass whereas l _a is an increasing function of molar mass. l _c depends sharply on the crystallization temperature, it is almost proportional to the reciprocal of supercooling. In some polymers, however, l _c increases with M _W. For instance, for HDPE quenched samples. Nitta and Tanaka [72] find that l _c increases from about 10 nm at M _W ~ 10 kg mol⁻¹ to about 20 nm at M _W = 1000 kg mol⁻¹. The same trend was observed by Kennedy et al. [42]. All the authors find that l _a is an increasing function of M _W [42, 47, 72, 73, 77] as shown by Fig. 4 where l _a appears to be a linear function of the square root of M _W:

$$ l_{\text{a}} = 3.67 + 0.015 \times M_{\text{W}}^{1/2} $$

(27)

l _a being expressed in nanometers and M _W in g mol⁻¹.

Fig. 4

Interlamellar spacing l _a versus square root of the weight average molar mass. Compilation of literature data (see text). ◊: apparently aberrant points eliminated for the review

Robelin-Souffaché and Rault [47] found:

$$ l_{\text{a}} = 1.2 + 0.044 \times M_{\text{R}}^{1/2} $$

(28)

This equation cannot be valid in the whole full range of M _W values, otherwise it would disagree with Eq. 13 according to which x _c = 1, i.e. l _a = 0 at a molar mass close to the entanglement threshold. It will be supposed, here, that it remains valid in the molar mass domain of interest, relatively far above the entanglement threshold.

Since M _W/M _R > 1, the difference of slopes in Eqs. 26 and 27 is not very surprising. According to Nitta and Tanaka [70], l _a would be independent of the comonomer content in ethylene co hexene 1 copolymers, which could simplify the study of these materials, for instance in studies of PE for pipe extrusion [77].

Data relative to the aging effects on lamellar morphology are relatively scarce in the literature. In the case of crosslinked PE, Gedde and Ifwarson [80] studied the changes of melting characteristics during thermal aging and tried to apply the Gibbs–Thompson equation to determine l _c. l _c was found to increase significantly after aging but, as noticed by the authors, this method is complicated by changes of parameters such as the surface energy (which decreases). Zhang and Cameron [13] have performed SAXS and WAXS measurements on gamma-irradiated PP samples. The crystallinity ratio increases slightly: 2–3% for 50 kGy irradiation dose, independently of the initial morphology. The long period is almost constant, whereas l _a decreases slightly but significantly. At high doses, the phenomenon tends to be masked by the effect of crystal destruction due to the occurrence of radiolysis events within crystals, which is not the case in thermo or photo-oxidation. In the case of POM thermal oxidation at 130 °C, Fayolle et al. [60] found also a decrease of l _a after an initial period dominated by annealing effects. Here, the mass loss due to depolymerization can also contribute to the decrease of l _a.

Despite the scarceness of published data about aging effects on lamellar morphology, it seems reasonable to assume the mechanism illustrated by Fig. 5. Chemicrystallization would proceed by lamellae thickening at the expense of the amorphous phase, the long period remaining constant. In polymers such as PE, PP, or POM, chemicrystallization seems to occur exclusively through lamella thickening. In other polymers, especially those having a relatively low crystallinity ratio, chemicrystallization involving the nucleation of new crystals is not excluded. This mechanism is probably the cause of the decrease of long period in polyamides submitted to hydrolysis [81].

Fig. 5

Hypothetical changes of lamellar dimensions after crystallization: a initial sample and b aged sample

To conclude this section, it is well established that chain scission induces an increase of the crystallinity ratio and a decrease of the interlamellar spacing l _a. These morphological changes become significant at low conversion of the degradation process. It remains to determine if these changes can be responsible for embrittlement and which is the most pertinent quantity to use in kinetic modeling.

How to choose an embrittlement criterion?

The embrittlement mechanism is not clearly elucidated so that the embrittlement criterion cannot theoretically be deduced from this mechanism. Schematically, two main ways of explanation seem possible:

• 1. A molecular interpretation in which, for instance, the entanglement density or the tie-macromolecule concentration would be the key quantities.

• 2. A micromechanical interpretation in which lamellar dimensions would play the key role through stress concentration effect, control of cavity radius in cavitation processes [75] or simply from the fact that small amorphous domains cannot sustain large deformations [42].

Concerning the nature of the embrittlement criterion, it could be “single”, e.g. a critical value of a given property: crystallinity ratio, molar mass, etc., or “composite”, e.g. a boundary in a multidimensional space, for instance a curve in the molar mass-crystallinity ratio map, separating “ductile” and “brittle” regions. Some “single” criteria can be set aside in a first approach: molar mass and entanglement density for already seen reasons, or crystallinity ratio despite the existence of favorable arguments [82]. As a matter of fact, for samples of close molar mass, M _W = 51 and 54 kg mol⁻¹, it is possible to obtain a ductile or a brittle behavior depending on thermal treatments, but the ductile–brittle transition was located at x _c = 0.62 by Voigt Martin et al. [73] and 0.70 ≤ x _c ≤ 0.78 by Jordens et al. [74]. Such differences, as well as the scatter observed in x _c–M _W relationships, indicate that if the overall crystallinity ratio (or amorphous phase content) is a pertinent quantity, it cannot be established from a compilation of literature data. Anyhow, there are many reasons to suppose that the fracture behavior depends rather on factors such as the lamellar dimensions or the tie-macromolecule concentration.

First, concerning the lamellar dimensions, one can expect the same data scatter for the long period l _p or the lamella thickness l _c as for the crystallinity ratio, because they sharply depend on thermal history. In contrast, the interlamellar spacing l _a appears to be less dependant on thermal history, as illustrated for instance, by the data of Table 6 from Voigt-Martin and Mandelkern [73].

Table 6 Lamellar dimensions for two HDPE samples isothermally crystallized (C) or quenched (Q) [73]

It is clear that l _a varies considerably less than l _p or l _c with the crystallization conditions but is sharply linked to molar mass (Fig. 4). In their extensive study of tensile properties of HDPE, Kennedy et al. [42] have compared samples differing by their molar mass distribution and their crystallization conditions. They put in evidence the fact that the ductile–brittle transition occurs always when the interlamellar spacing reaches a critical value: l _a = l _aC = 6–7 nm. A recent study on POM thermal oxidation led to a similar observation, with a close value for l _aC [60].

These results militate in favor of the choice of interlamellar spacing as “single” criterion for ductile–brittle transition. According to Eq. 27, l _a values of 6 and 7 nm would respectively correspond to 24 and 49 kg mol⁻¹. Is it possible to reach a better precision?

Tie-macromolecules

As soon as the concept of tie-macromolecule appeared [83], there were authors to attribute embrittlement to their scission [39]. This interpretation reappeared then regularly in the literature, for instance to explain embrittlement of PP submitted to radiochemical aging [63, 64]. The problem is that, especially in PP, embrittlement occurs when a very small fraction of chains has been broken [40]. Indeed, random chain scission occurs preferentially on the longest chains, i.e. on the fraction of chains able to establish interlamellar links. Tie chains have thus a higher probability to be broken than “average chains” but is this probability high enough to compensate the scarceness of chain scission events? The problem here is to translate this question into quantitative relationships.

In his extensive critical review, Seguela [84] has compared the experimental and theoretical approaches of the tie-macromolecule concentration. No one has applied these approaches to the study of aging-induced embrittlement to our knowledge. For instance, the experimental method of Kriegbaum et al. [85] based on analysis of strain hardening during plastic flow, using rubber elasticity concepts to determine the concentration of mechanically active chains, seems interesting because there is an impressive number of aging studies reporting tensile curves, e.g. allowing in principle to quantify strain hardening. Unfortunately, only engineering tensile curves are reported in these studies, making thus this analysis impossible.

Among theoretical analyses, the one proposed by Huang and Brown [86] with the improvements proposed by Seguela [84] appears especially interesting. According to the basic theory, the probability p of a given chain to become a tie-macromolecule would be given by:

$$ p = \frac{1}{3} \frac{{\int_{L}^{\infty } {r^{2} \exp \left( { - \frac{3}{2}\frac{{r^{2} }}{{\left\langle {r_{0}^{2} } \right\rangle }}} \right){\text{d}}r} }}{{\int_{0}^{\infty } {r^{2} \exp \left( { - \frac{3}{2}\frac{{r^{2} }}{{\left\langle {r_{0}^{2} } \right\rangle }}} \right){\text{d}}r} }} $$

(29)

where r ₀ is the end-to-end distance and L is given by:

$$ L = 2l_{\text{c}} + l_{\text{a}} = 2l_{\text{p}} - l_{\text{a}} $$

(30)

In principle, this approach must be applied to every macromolecule of the distribution. Here, it will be assumed, in a first approach, that Eq. 29 applied to the weight average chain gives a good order of magnitude for p. In this case, one can use:

$$ \left\langle {r_{0}^{2} } \right\rangle = \alpha M_{\text{W}} $$

(31)

where α is characteristic of the chemical structure, for instance: α = 1.42 Å mol g⁻¹ for PE and α = 0.694 Å mol g⁻¹ for PP.

In case of aging, Eq. 31 can be transformed by substituting M _W by Eq. 9 into:

$$ \left\langle {r_{0}^{2} } \right\rangle = \frac{{\alpha M_{{{\text{W}}0}} }}{{1 + \frac{C}{2}}} $$

(32)

If, as suggested above, l _p remains constant during chemicrystallization, Eqs. 27 and 29 give:

$$ L = L_{0} - 0.015M_{{{\text{W}}0}}^{1/2} \frac{1}{{\left( {1 + \frac{C}{2}} \right)^{1/2} }} $$

(33)

where

$$ L_{0} = 2l_{\text{p}} - 3.67\quad({\text{in min}}) $$

(34)

It is then possible to express all the parameters of Eq. 29 in function of the number of chain scissions per initial weight average macromolecule. Indeed, the limit L of integration varies with the crystallization conditions through l _p. We have compared p values in three cases, using data obtained by Nitta and Tanaka [72] on quenched linear PE samples. The results are shown in Fig. 6. The sample characteristics interpolated for two p values, 0.05 and 0.03, have been reported in Table 7.

Fig. 6

Probability for a given macromolecule to be a tie-macromolecule (P) versus number of chain scissions per initial weight average chain (C) for three quenched linear PE samples (see text)

Table 7 Molar mass, interlamellar spacing and crystallinity ratio for p = 0.05 and p = 0.03, using Eqs. 30–36 for three quenched linear PE samples having the initial characteristics reported by Nitta and Tanaka [72]

The 100–715 kg mol⁻¹ molar mass range covers most of the usual PE grades displaying initially a ductile behavior except ultra high molar weight ones. p _F values have been chosen in order to have M _WF values coinciding with experimentally found ones, i.e. 40 kg mol⁻¹ ≤ M _WF ≤ 100 kg mol⁻¹. As hypothesized, the long period remains constant during degradation, whereas the interlamellar spacing decreases to reach a value l _aF at p = p _F. l _aF tends to increase slightly with the initial molar mass but this dependence could be drawn into the experimental scatter in most of the cases. The corresponding crystallinity ratio x _cF has been calculated from l _cF = l _p − l _aF using Eq. 25, which can be translated into:

$$ x_{\text{cF}} = \frac{\varphi }{ 1+ \varphi } $$

(35)

where

$$ \varphi = \frac{{l_{\text{cF}} {{\rho}}_{\text{c}} }}{{l_{\text{aF}} {{\rho}}_{\text{a}} }} $$

(36)

Here, also, a slight dependence of x _cF with M _W0 is expected but it would be observable only in very careful experiments.

Discussion

At this state of our knowledge, we can envisage the use of a “single” embrittlement criterion, e.g. a critical value of the interlamellar spacing, l _a = l _aF, or “composite” criterion, e.g. a critical value of the tie-macromolecule probability p = p _F which corresponds to a boundary in (l _a, M _W) space. At first sight, there is no way to decide between both criteria, owing to the scatter of available experimental data. The problem can be illustrated by the scheme of Fig. 7 representing aging trajectories in (l _a, M _W) map.

Fig. 7

a Schematic representation of ageing trajectories in the (l_a, M_W) map for four samples of which the initial characteristics are the coordinates of the points A, B, C, and D. Two embrittlement criteria are represented: (-·-·-·-) l_a = l_aF; (- - - -) p = p_F. b Zoom of the critical zone for HDPE with representation of both embrittlement criteria taking p_F = 0.03 for the critical tie-macromolecule probability

All the aging trajectories are expected to converge in a more or less sharp bottleneck, eventually a single curve if l _a depends only on M _W. The embrittlement criterion will be a horizontal straightline l _a = l _aF if the critical property is the interlamellar spacing and a curve corresponding to p = p _F if the critical property is a tie-macromolecule concentration. The example of HDPE (Fig. 7b) shows the difficulty to decide between both criteria.

Conclusion

It has been tried to demonstrate in the above review that when a linear polymer undergoes random chain scission aging, its embrittlement results from a decrease of its toughness rather than from defect build-up. In the case of semi-crystalline polymers having their amorphous phase in rubbery state, embrittlement occurs at an especially low conversion of the degradation process, at which the entanglement network in the amorphous phase has undergone only very little damage. Since these polymers undergo chemicrystallization, it has been hypothesized that this latter could occur by three ways: decrease of the whole amorphous content, decrease of interlamellar spacing or destruction of tie-macromolecules. Relatively abundant data are available on the eventual relationships between ductile–brittle transition and the crystallinity ratio, but their high scatter indicates the existence of important underlying parameters. Very scarce data are available on the role of interlamellar spacing l _a but they favor the hypothesis that embrittlement could correspond to a critical value of l _a, for instance l _a = 6–7 nm for PE and for POM. No data are available, to our knowledge, on the role of tie-macromolecule concentration, but applications of the theories of Huang and Brown [86], using a tie-macromolecule probability of 0.03–0.05 would lead to results quantitatively compatible with literature data available on HDPE.

At present time, it seems not possible to decide between the causal chains schematized in Fig. 8.

Fig. 8

Schematization of the possible causal chains for degradation-induced embrittlement

In the case where the interlamellar spacing would be the pertinent quantity, the embrittlement criterion would be a critical value l _aF of l _a. But since l _a is strongly linked to M _W, one can use the corresponding value M _WF as an embrittlement criterion that simplifies the approach of lifetime prediction.

In the case where the tie-macromolecule concentration would be the pertinent criterion, the theory of Huang and Brown [86], eventually improved by Seguela [84], could be a way to establish a bidimensional embrittlement criterion. However, the presently available experimental data are too scarce and fuzzy to benefit by such refinements.