Fracture of polymer networks with diverse topological defects

Shaoting Lin and Xuanhe Zhao

Phys. Rev. E 102, 052503 – Published 10 November 2020

Abstract

Polymer networks are pervasive in biological organisms and engineering materials. Topological defects such as cyclic loops and dangling chains are ubiquitous in polymer networks. While fracture is a dominant mechanism for mechanical failures of polymer networks, existing models for fracture of polymer networks neglect the presence of topological defects. Here, we report a defect-network fracture model that accounts for the impact of various types of topological defects on fracture of polymer networks. We show that the fracture energy of polymer networks should account for the energy from multiple layers of polymer chains adjacent to the crack. We further show that the presence of topological defects tends to toughen a polymer network by increasing the effective chain length, yet to weaken the polymer network by introducing inactive polymer chains. Such competing effects can either increase or decrease the overall intrinsic fracture energy of the polymer network, depending on the types and densities of topological defects. Our model provides theoretical explanations for the experimental data on the intrinsic fracture energy of polymer networks with various types and densities of topological defects.

Received 24 August 2020
Accepted 21 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052503

Physics Subject Headings (PhySH)

Biomimetic & bio-inspired materials Defects Fracture Functional materials Polymer behavior Structural properties

Condensed Matter, Materials & Applied PhysicsNetworksPhysics of Living SystemsPolymers & Soft Matter

Authors & Affiliations

Shaoting Lin¹ and Xuanhe Zhao^1,2,*

¹Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*zhaox@mit.edu

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Vol. 102, Iss. 5 — November 2020

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Images

Figure 1
(a) Schematic illustration of crack propagation in a polymer network with topological defects by fracturing active polymer chains. (b) Cyclic loops and dangling chains are the two most representative types of topological defects in polymer networks.
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Figure 2
Schematic illustration of the defect-network fracture model. (a) Crack propagation in a polymer network with topological defects. (b) The defect-network fracture model gives the same density of fractured polymer chains as that in (a) but effectively longer fractured chain than those in (a). Overall, the defect-network fracture model accounts for the energy in multiple layers of polymer chains adjacent to the fractured chains.
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Figure 3
(a) Experimentally measured force-extension curves of two single polymer chains [i.e., poly( $N, N^{'}$ -dimethylacrylamide) (PDMA) and poly( $N, N^{'}$ -diethylacrylamide) (PDEA)] by force spectroscopy [17]. (b) Schematic illustration of the force-extension curve of a single polymer chain, the enclosed area of which defines the fracture energy of the chain $U_{chain}$ . (c) Schematic illustration of a fractured polymer chain on the crack path (i.e., $i = 0$ ) and deformed neighboring chains (i.e., $| i | > 0$ ). (d) The energy relaxed by all $i th$ polymer chains $U_{i}$ normalized by the fracture energy of an ideal chain on the crack path $U_{0}$ in ideal polymer networks with functionality $f = 3$ , 4, 5, and 6. (e) The effective energy relaxed by fracturing a polymer chain in the ideal polymer network $U_{ideal}$ normalized by the fracture energy of the chain on the crack path $U_{0}$ as a function of the functionality $f$ .
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Figure 4
(a) Schematic illustration of a fractured second-order loop on the crack path and deformed neighboring chains. (b) The energy relaxed by all $i th$ polymer chains $U_{i}$ normalized by the fracture energy of a second-order loop on the crack path $U_{0}$ , compared with that of an ideal chain. (c) Schematic illustration of a fractured chain closest to a first-order dangling chain and deformed neighboring chains. (d) The energy relaxed by all $i th$ polymer chains $U_{i}$ normalized by the fracture energy of the most affected chain by a first-order dangling chain on the crack path $U_{0}$ , compared with that of an ideal chain. (e) Schematic illustration of a fractured chain closest to a second-order dangling chain and deformed neighboring chains. (f) The energy relaxed by all $i th$ polymer chains $U_{i}$ normalized by the fracture energy of the most affected chain by a second-order dangling chain on the crack path $U_{0}$ , compared with that of an ideal chain. (g) Schematic illustration of a fractured chain closest to a third-order dangling chain and deformed neighboring chains. (h) The energy relaxed by all $i th$ polymer chains $U_{i}$ normalized by the fracture energy of the most affected chain by a third-order dangling chain on the crack path $U_{0}$ , compared with that of an ideal chain.
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Figure 5
Normalized intrinsic fracture energy and normalized shear modulus of polymer networks containing only one type of topological defect. (a) Second-order loop with $γ_{2 l} = 2$ and $ɛ_{2 l} = \frac{1}{2}$ . (b) First-order dangling with $γ_{1 d} = \frac{9}{8}$ and $ɛ_{1 d} = \frac{8}{9}$ . (c) Second-order dangling with $γ_{2 d} = \frac{3}{2}$ and $ɛ_{2 d} = \frac{2}{3}$ . (d) Third-order dangling with $γ_{2 d} = \frac{9}{8}$ and $ɛ_{2 d} = \frac{8}{9}$ . (e) Fourth-order dangling with $γ_{2 d} = 0$ and $ɛ_{2 d} = 0$ . The intrinsic fracture energy and shear modulus are normalized by the corresponding values of defect-free polymer networks.
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Figure 6
(a) Ratios of the numbers of macromers with various orders of dangling chains over the total number of macromers that constitute the tetra-arm polymer network (i.e., $C_{1 d}$ , $C_{2 d}$ , $C_{3 d}$ , and $C_{4 d}$ ) as functions of the reaction efficiency $p$ . (b) Comparison of the normalized intrinsic fracture energy and the normalized shear modulus between the experiment results [9] and the defect-network models.
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