Recoil from elongation using general network models
Authors
- First Online:
- Received:
- Accepted:
DOI: 10.1007/s00397-005-0433-8
- Cite this article as:
- Tanner, R.I., Zdilar, A.M. & Nasseri, S. Rheol Acta (2005) 44: 513. doi:10.1007/s00397-005-0433-8
Abstract
In this paper, we use two new models and the irreversible KBKZ model of Wagner (Rheol Acta 18:681–692, 1979) to describe the famous experiments of Meissner (Rheol Acta 10:230–242, 1971) on the recoil of polyethylene. The new models are based both on network and reptation-type ideas. One of the new models (PTT-X) is a member of the PTT family and shows good agreement with polyethylene data in shearing, elongation, and recoil from elongation.
Keywords
PolyethyleneRecoilPTT modelElongation rheologyIntroduction
- 1.
Starting from a rest state at some initial time, t_{o}, a purely elongational flow history, not necessarily at a constant elongation rate, is imposed up to time zero (t=0).
- 2.
At t=0 the stress state is made zero, and an instantaneous recoil occurs.
- 3.
For t>0, further recoil may occur slowly; the stresses remain at zero.
The reason for exploring this flow is that it was shown by Wagner (1979) that some plausible constitutive models of the KBKZ type failed to describe such recoil (they were too ‘elastic‘), although otherwise they were useful. Wagner (1979) then used his “irreversible” idea in a KBKZ framework to describe recoil quite successfully. Langouche and Debbaut (1999) investigated the recoil of high-density polyethylene (HDPE) experimentally and used a multi-mode exponential Phan-Thien-Tanner (expPTT) model (Tanner 2000) to fit the elongational recoil data. The maximum Hencky strains used were three or four, and strain rates of 0.01, 0.1 and 1 s^{−1} were used. Some overestimation of the recovery data at large initial strains was noted—again, the model showed excessive elasticity.
Joshi and Denn (2004) have surveyed failure and recovery in elongational flow. They concluded that the results for recovery were model-dependent, and cited the work of Wagner (1979). While the Wagner irreversible model appears to be successful, we believe the irreversibility device is somewhat mysterious, and we prefer to explore other models.
Many new models have been proposed, and often they describe steady and some transient shear and transient elongational flows quite well. However, there is little information about their elongational recoil behaviour. In double-step shear, Chodankar et al. (2003) have done some careful work, but the differences in behaviour between models are often not as marked in shear as in elongational flow, and in many ways elongation is simpler. Therefore, we shall concentrate on it here.
The constitutive relations to be explored are all of the general network theory type ; for discussions of the assumptions involved see the books of Larson (1988), Tanner (2000) and Huilgol and Phan-Thien (1997). Special cases of this model include the Lodge rubber-like model (1964) and various PTT models (Tanner and Nasseri 2003). We also consider the XPP model of Verbeeten et al. (2002). In this model, which is said to be derived using reptation theory ideas and is a development of the pom–pom model (Blackwell et al. 2000), we note that if the Giesekus-type term (1982) in it is set to zero, the model falls into the general network class (Tanner and Nasseri 2003). (Note that Lodge (1989) has insisted that the Doi-Edwards tube model gives zero recoil; this problem does not arise with the models discussed here). In the present paper, we concentrate on models that are expressible in differential forms (these are easier to incorporate in large-scale computer programs) and so the more recent models of Wagner,for example,are not included.
Hence, the present paper will address the behaviour of general network theory models as its principal concern. Inertia and compressibility will be ignored—it is believed they are of minor importance in the cases described here.
The general network model
Hence a dependence on the average square of h can be replaced by a dependence on tr τ.
For the XPP model, G_{1}=η_{o}/λ. In general, a single relaxation time λ is not sufficient to describe realistically the response of a typical polymer, and usually several λ-values will be needed. This will be considered next.
Model responses in simple flows
Before looking at recoil, we mention the capabilities of various forms of the general network model in steady and transient shear and elongational flows. Tanner and Nasseri (2003) have discussed four-mode models of the so-called XPP (Verbeeten et al. 2002) and PTT-X models, as well as the usual exponential PTT model (Tanner 2000). (In Tanner and Nasseri (2003), what we refer to here as the PTT-X model is called the PTT-XPP model, which is too long a name for comfort).
Linear and non-linear parameters for fitting of the DSM Stamylan LD 2008 XC43 LDPE melt
Mode | Maxwell parameters | XPP and PTT-X parameters | ||
---|---|---|---|---|
i | G_{i} (Pa) | λ_{i} (s) | q_{i}=2/ν_{i} | r_{i}=λ_{b,i}/λ_{s,i} |
1 | 7.2006×10^{4} | 3.8946×10^{−3} | 1 | 7.0 |
2 | 1.5770×10^{4} | 5.1390×10^{−2} | 1 | 5.0 |
3 | 3.3340×10^{3} | 5.0349×10^{−1} | 2 | 3.0 |
4 | 3.0080×10^{2} | 4.5911×10^{0} | 10 | 1.1 |
Recoil with a single relaxation time
Here ε^{−1} is the ratio of the length just before recoil to the length after recoil. Thus the prediction of the instantaneous recoil can be done using only the results from the continuous history.
In the case of the Wagner (1979) relation, where strain measures are included in the kernel, no simple resolution of the equations occurs, and the recoil must be found by a numerical iterative process.
Some examples of instantaneous recoil
We now assume a single relaxation time model of the type given in Eq. 1. We can make this equation dimensionless by letting t~t/λ, L~λL, G_{1}~G_{1}/G (where G is a modulus equal to η_{o}/λ), τ ~ τ/G; H is already dimensionless.
The asymptotic results for τ_{1} have been given for several specific H-forms in our previous paper (Tanner and Nasseri 2003).
ε_{r} as function of dimensionless \(\dot \varepsilon \) (λ \(\dot \varepsilon \))
| Exp-PTT | PTT-X | XPP |
---|---|---|---|
\({\dot \varepsilon}\) | ε^{*}=0.25 | ν=1, r=3 | ν=1, r=3 |
10 | 0.875 | 1.145 | 2.14 |
100 | 1.106 | 1.526 | 3.29 |
1,000 | 1.241 | 1.789 | 3.55 |
In Table 2, we have chosen the moderate values ν=1, r=3 for illustrative purposes (compare with Table 1). Surprisingly, there is a tremendous range of recoil shown by these very similar models. The XPP model recoils about two to three times as much as the other models. These results may be confirmed by calculation; and we will return to the presentation of Wagner (1979) in the following figures. The exp PTT model, if compared at the same stress to the PTT-X model, will give the same recoil (Eq. 26) for high strain rates. Thus one should not compare the two PTT models directly in Table 2.
For the Lodge model, τ_{1}/τ_{2}=(1+ \(\dot \varepsilon \))/(1−2 \(\dot \varepsilon \)) which rises indefinitely as \(\dot \varepsilon \to 0.5\); it can only be considered for transient (start-up from −t_{o}) flows. We now consider the effect of multiple relaxation times.
Multiple relaxation times
This ratio is clearly less than one, except for a single mode. Thus, generally the sudden jump decreases as a percentage of the total recoil when multiple modes are assumed.
We can compare the total recoil of the multi-mode result to an equivalent single-mode model; clearly the effective average relaxation time here is \(\left({\sum {\eta _i \lambda _i} /\sum {\eta _i}} \right),\) which is heavily biased toward the longer relaxation times.
The mode parameters used for the fitting of IUPAC A/LDPE
Mode no., i | Modulus, g_{i} (Pa) | λ_{i} (s) | q_{i} | r_{i} |
---|---|---|---|---|
1 | 1.520×10^{5} | 1.0×10^{−3} | 1.0 | 2.0 |
2 | 4.005×10^{4} | 5.0×10^{−3} | 1.0 | 2.0 |
3 | 3.326×10^{4} | 2.8×10^{−2} | 2.0 | 2.0 |
4 | 1.659×10^{4} | 1.4×10^{−1} | 2.0 | 2.5 |
5 | 8.690×10^{3} | 7.0×10^{−1} | 4.0 | 2.0 |
6 | 3.151×10^{3} | 3.8×10^{0} | 7.0 | 2.0 |
7 | 8.596×10^{2} | 2.0×10^{1} | 8.0 | 1.5 |
8 | 1.283×10^{2} | 1.0×10^{2} | 12.0 | 1.0 |
9 | 1.8495×10^{0} | 5.0×10^{2} | 30.0 | 1.0 |
The results of Wagner’s irreversible hypothesis (1979) are also plotted in Fig. 3 (dashed lines). These lie somewhat closer to the experimental data than the PTT-X model results, but the differences are small, especially for large total strain and large recoverable strain.
Conclusion
The above-mentioned results confirm the value of the PTT-X model, at least for low-density polyethylene modelling. While it has previously been shown to give good results in shearing, we show here that these extend to unsteady elongation and recoil flows.
Its obvious drawback from the user’s point of view is the larger number of parameters involved—as opposed to the exponential PTT model, say. However, since it is based on both a molecular model and network theory, it is probably suitable for both branched and unbranched polymers, but further work will be needed to confirm this.
Acknowledgements
We thank the Australian Research Council, the Polymer Cooperative Research Centre and the University of Sydney for supporting this investigation.