A constitutive model for multi network elastomers pre-stretched by swelling

Abstract

Multi network elastomers (MNEs), composed of a single sacrificial network and other matrix networks, exhibit appealing mechanical properties. In this paper, we develop a constitutive model for MNEs prepared by multiple swellings. Firstly, the swelling process is analyzed. The free energy of a swollen elastomer consists of the strain energy of deformed polymer chains and the energy of mixing. We analyze the mechanical equilibrium and chemical potential equilibrium during the swelling of MNEs. The degree of swelling of MNEs is affected by the preparation conditions, like the component of the solution, as well as the microscopic physical quantities, such as the chain length and density of polymer chains of the networks. Secondly, the free energy of the completed MNE subjected to external loading is composed of the strain energy of each network. The chains of the sacrificial network (first network) can be destroyed gradually when the applied load increases. We adopt the network alteration theory to describe the above progressive damage of the sacrificial network. In addition, the matrix networks are fully elastic. We verify this model using the experimental data including stress–stretch curves of elastomers with double and triple networks, as well as the step cycle curves of triple network elastomers (TNEs). This model predicts well for the swelling-induced pre-stretches of each network under specified preparation conditions. It relates the stress to the stretch of MNEs under external loading, and indicates the stress contribution of each network, and the damage evolution of the sacrificial network. Our model is instructive for designing MNEs with desired mechanical properties.

Introduction

Due to the distinguished ability of large deformation and excellent ductility, elastomers play an irreplaceable role in traditional industries, like tires, dampers and adhesives. Their increasing uses are also found in emerging technologies, such as flexible electronic devices and soft robotics [1], [2], [3], [4]. Microscopically, monomers are connected through covalent bonds to form polymer chains, and the polymer chains are cross-linked into three dimensional elastomer networks. Combining elastomer network and water molecules results in a hydrogel, among which the water content can reach above 90% [5], [6]. The outstanding biocompatibility, flexibility, swelling behavior along with the ability to respond in specific environment, make hydrogel the ideal candidate for novel biomedical devices. Application examples include ionic skin [7], bioelectronics [8], soft adhesives [9], [10] and actuators [11].

However, the application of elastomer is often limited by their poor mechanical properties. For example, pure polydimethylsiloxane (PDMS) has a fracture toughness about 10–100 J/m² [12]; and the toughness of ethyl acrylate elastomer is $\sim$50 J/m² [13], which is much lower than that of natural cartilage ($\sim$1000 J/m²) [14]. In addition, an alginate hydrogel breaks when the stretch reaches about 1.2 [15]. A number of methods have been proposed to improve the mechanical properties of elastomers. Particles were added into elastomers to promote both the strength and toughness [16], [17], [18]. Significant energy is dissipated by the debonding between the particles and polymer network, enabling the high toughness of the elastomer. However, dispersing the particles uniformly is always a complicated process, and the intrinsic toxicity of nanoparticles presents security risks [19], [20]. In other studies [12], [21], fibers were added into elastomers to form stretchable composite materials, with high toughness and low hysteresis at the same time. Nevertheless, adding fibers would induce anisotropy to the material.

Designing elastomers with multi network structure is another effective way to enhance their mechanical properties. This design principle was firstly presented by Gong et al. in 2003 in designing tough hydrogel, by a combination of hydrophilic polymer networks and diffused water molecules inside [22]. In their design, the interpenetration structure of the first and second networks, with the first being highly cross-linked by covalent bonds and the second loosely cross-linked, amplified the mechanical strength tens to hundreds of times that of individual polymer network. The breakage of the first network (also called sacrificial network), which is pre-stretched close to its locking stretch, gives the high toughness, while the survival of the second network keeps the hydrogel intact. Later on, Sun et al. replaced the chemically cross-linked sacrificial network with a physically cross-linked one [15], endowing the double network tough hydrogel with recoverable toughness by heating or acoustic excitation [23]. Ducrot et al. [13] applied this idea to the invention of a tough elastomer with multi network structure. High toughness is achieved for the TNEs (5000 J/m² compared to the 50 J/m² of the corresponding single network). Furthermore, the breakage of the sacrificial network can be traced in situ by using chemoluminescent cross-linkers in this network. The schematic diagram of preparing MNEs is exhibited in Fig. 1.

In order to predict their mechanical properties, a number of mechanical models have been developed. Wang and Hong [24] proposed a phenomenological model for double network gels. By combining network alteration model and interpenetrating network model, Zhao [25] developed a physically based damage model for interpenetrating polymer networks. Later on, Mao et al. [26] presented a large deformation model for double network hydrogels which incorporating the rate-dependent response during loading. Recently, a continuum model for MNEs was developed by Lavoie et al. [27], considering chain length distribution and corresponding chain fracture. That model bears a clear physical meaning though it is somewhat complicated.

Previous models of MNEs were focused on the mechanical properties of the prepared MNEs, but neglected the quantitative analysis of the effect of preparation conditions on the mechanical properties. In this study, we discuss how the swelling-induced deformation is affected by the preparation conditions, like the composition of solution, as well as the physical quantities such as the chain length and chain density. Then based on the physical features of each network, we describe the stress–stretch behavior of prepared MNEs under external loading. The damage of highly pre-stretched first network is described by a damage function, which relates to the historical maximal chain stretch. The matrix networks are loosely cross-linked with long polymer chains, thus are assumed to be fully elastic. The current model shows a good agreement with the published experimental data.

Compared with the previous studies, the major features and contributions of this work includes:

• Construction of a new constitutive model for MNEs. Quantitative analysis of the swelling-induced pre-stretches of each network based on the mechanical equilibrium and chemical potential equilibrium is put forward for the first time.

• We decompose the elastomer into sacrificial network (Fig. 2(a)) and matrix networks (Fig. 2(b)). The mechanical response of matrix networks is reversible while the constitutive relationship of the sacrificial network is inelastic.

The paper is organized as follows. In Section 2, an energy-based method is adopted to derive the mechanical equilibrium and chemical potential equilibrium conditions for a swollen elastomer. Then the constitutive relationships between stress and stretch for both sacrificial network and matrix networks are constructed respectively. Comparisons with experimental data of double/triple network elastomers are shown in Section 3. In Section 4, we demonstrate how the preparation conditions and microscopic physical quantities affect the mechanical properties of MNEs, and we analyze the stress contribution of each network as well as the damage evolution of the sacrificial network. Section 5 provides the final remarks of the paper.