Volume fraction and width of ribbon-like crystallites control the rubbery modulus of segmented block copolymers

Matthias Nébouy; Ameur Louhichi; Guilhem P. Baeza

doi:10.1515/polyeng-2019-0222

Article

Volume fraction and width of ribbon-like crystallites control the rubbery modulus of segmented block copolymers

Matthias Nébouy , Ameur Louhichi and Guilhem P. Baeza

Published/Copyright: October 12, 2019

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From the journal Journal of Polymer Engineering Volume 40 Issue 9

Abstract

We discuss the origin of the plateau modulus enhancement (χ) in semi-crystalline segmented block copolymers by increasing the concentration in hard segments within the chains (X_HS). The message we deliver is that the plateau modulus of these thermoplastic elastomers is greatly dominated by the volume fraction (Φ) and the width (W) of crystallites according to χ–1 ~ ΦW in agreement with a recent topological model we have developed. We start by a quick review of literature with the aim to extract χ(Φ) for different chemical structures. As we suspected, we find that most of the data falls onto a mastercurve, in line with our predictions, confirming that the reinforcement in such materials is mainly dominated by the crystallite’s content. This important result is then supported by the investigation of copolymer mixtures in which Φ is fixed, providing a similar reinforcement, while the chains compositions is significantly different. Finally, we show that the reinforcement can be enhanced at constant Φ by increasing W for a given class of block copolymers. This can be done by changing the process route and is again in good agreement with our expectations.

Keywords: crystallites; reinforcement; rubbery modulus; segmented block copolymer; thermoplastic elastomer

Acknowledgments

All the authors thank Dimitris Vlassopoulos (FORTH) and Evelyne Van Ruymbeke (Univ. Louvain-la-Neuve) for enlightening discussions and technical support as well as Carel Fitié, Wilco Appel, Luna Imperiali, and Ashwinikumar Sharma (DSM Ahead) for providing the TPE. G.P.B. thanks Nino Grizzuti (Univ. Federico II, Naples) for his invitation to publish in the special issue of Journal of Polymer Engineering on industrial polymer rheology.

Funding: M.N. is thankful to the French Ministry of Research for funding his PhD. EU FP7 – ETN SUPOLEN, Grant Number: GA-607937.

Appendix 1: Details on the topological model – see Ref. [25]

The topological model to which we largely refer in this article considers the HS crystallites as local densifications of the amorphous network rather than independent objects. The reason for this resides in the fact that all the HS are directly connected to two SS through covalent bonds. Following this logic, we propose to count the extra-number of topological node induced by the crystallites and use the rubber elasticity theory to calculate the resulting reinforcement.

Because the rubbery modulus is driven by the density of entanglements in the amorphous melt, we assume that the extra reinforcement caused by the crystallites must be calculated from the same length scale. Our model lies in Appendix Figure 1 below in which:

R_e is the tube diameter (tube model).
H and W are, respectively the height (along the chain axis) and the width (perpendicular to the chain axis) of the crystallite.
R is the section of the TPE molecule (=cell parameter) that we assume equal along L→ and W→.
N_H, N_W, and N_Le are the number of HS along the three directions, with N_ce=N_WN_Le, (N_H=1 by construction).

$Appendix Figure 1: Schematic representation of the topological nodes present in a subspace of soft-phase crossed by a HS crystallite. By construction, the HS can stack along W→$\vec W$ and L→$\vec L$ only.$

Appendix Figure 1:

Schematic representation of the topological nodes present in a subspace of soft-phase crossed by a HS crystallite. By construction, the HS can stack along W→ and L→ only.

Considering then the volume fraction in “reinforced” rubber, i.e. Φ and the inter-crystallites distance (periodicity) d^*, our model leads to:

(I)χ=GNTPEGN0=1+Φ(Nce−1)

with:

(II)Nce=ReR2Φd*2μH=ReR2W

where μ=1 or 3 (see full article) and W=Φd^*2/μH. Then, by combining I and II and assuming N_ce≫1, we obtain the following expression (two possible forms displaying d^* or W):

(III-a)χ=1+Red*2R2μHΦ2

(III-b)χ=1+ReR2WΦ

Eq. (III) is the main output of our model describing the reinforcement as a function of the volume fraction in crystallites. It is also interesting to note that d^* is usually seen to decrease with Φ such as d^*~Φ⁻^λ (with14< λ <12 describing the crystallites arrangement) and therefore W~Φ¹⁻²^λ leading to:

(IV)χ−1 ∼ Φ2(1−λ)

We found Eq. (III) to be valid up to X_HS=15 wt%. For higher values another equation was developed by considering non-Gaussian conformations of the SS. However, both models could be unified through the use of an empirical approximation such as:

(V)χ=exp(AΦ)

with A an empirical value was found to be equal to 18.38 for T4T/PTHF samples. Note that keeping this value for the whole set of TPE in Figure 1 gives quite satisfactory results too.

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Received: 2019-08-09

Accepted: 2019-09-04

Published Online: 2019-10-12

Published in Print: 2020-10-25

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Articles in the same Issue

https://doi.org/10.1515/polyeng-2019-0222

Keywords for this article

crystallites; reinforcement; rubbery modulus; segmented block copolymer; thermoplastic elastomer