Magneto-diffusion-viscohyperelasticity for magneto-active hydrogels: Rate dependences across time scales

Abstract

Soft materials have experienced an increasing interest from both scientific and industrial communities during the last years. This interest has become even stronger with the potential of such materials to mechanically respond to external stimuli. Among these materials, magneto-active hydrogels (MAHs) offer great opportunities for novel applications, especially within the biomedical field. However, the design of these stimuli-responsive materials is rather complex as they combine nonlinear mechanical behaviour, rate dependences, magneto-active responses and solvent diffusion processes. To help the understanding of the magneto-mechanical behaviour of these materials, their design and optimization, this work proposes a general continuum framework to couple magnetics, solvent diffusion and nonlinear mechanics. This framework is specialised to and implemented within the finite element method for 3D problems. Different rate-dependent responses of a free-standing MAH are analysed: (i) the strain rate dependency of the instantaneous response to magnetic fields applied at different rates; (ii) the viscous relaxation mechanisms occurring after the complete application of the magnetic field; and (iii) the long-time responses due to solvent diffusion. In addition, an evaluation of the interplay between magnetic fringing effects and solvent diffusion processes is performed. This work provides a flexible computational framework to model the coupled effects of different physical processes occurring within MAHs and highlights the importance of considering rate dependences. The computational framework has the potential to guide the future design of bioactive scaffolds and drug delivery systems based on MAHs.

Introduction

During the last decades, the scientific community has observed the emergence of smart materials that respond to external stimuli. These responsive materials have revolutionised the design and conceptualisation of the structures that, now, not only provide mechanical support but also specific functionalities. Some examples of these materials are thermally activated shape memory polymers (Zare et al., 2019), photo-activation polymers (Yu et al., 2019), electro-active polymers (Liao et al., 2020), and magneto-active polymers (MAPs) (Mehnert, Hossain, Steinmann, 2017, Zhao, Kim, Chester, Sharma, Zhao, 2019). Currently, responsive polymers can be found in soft robotics (Hu, Lum, Mastrangeli, Sitti, 2018, Liu, Gillen, Mishra, Evans, Tracy, 2019a), vibration controllers or dampers (Nadzharyan et al., 2018) and, especially, in bioengineering applications (Kim, Parada, Liu, Zhao, 2019, Zhao, Kim, Cezar, Huebsch, Lee, Bouhadir, Mooney, 2011). Within the bioengineering field, MAPs are of interest due to their fast activation responses and the stimulation possibilities offered in a remote and non-invasive manner. In this regard, MAPs allow for mechanical stimulation of biological tissues that overcome the limitations associated with the need for direct contact, strong alteration of biological functions (as in thermal or electrical activation systems) or low penetration depth of visible light in biological tissue (as in photo-activation systems). The concept of MAPs was motivated by the development of magnetorheological fluids, which consisted of magnetic particles dispersed within a fluid allowing for modifiable mechanical behaviour from a fluid like material to a viscoelastic solid by the application of an external magnetic field (Carlson et al., 1994). However, these materials presented important limitations due to particle sedimentation and low stiffness. To address these issues, MAPs were developed to provide an efficient solution by embedding magnetic particles within polymeric matrices (Zrnyi et al., 1996). In this regard, MAPs represent magneto-sensitive polymers that can adapt their shape, elasticity and motion by external magnetic stimuli (Odenbach, 2016). The magneto-mechanical behaviour of MAPs is determined by the interaction between magnetic forces and the internal stresses within the polymeric matrix.

Among the different MAPs, magneto-active hydrogels (MAHs) are proposed as novel smart materials that fill the gap, in terms of stiffness, between magnetorheological fluids and magnetorheological elastomers (Tang et al., 2018). MAHs consist of a soft polymeric matrix filled at the micro and nano levels with liquid solvent and magnetic particles (Li et al., 2013). The control of the polymeric network density and the magnetic particles’ volume fraction allows for mimicking the mechanical properties and deformation mechanisms of biological tissues (Li et al., 2013). In addition, they permit deformation and/or stiffness control by application of external magnetic fields. Several MAHs also possess excellent biocompatibility (i.e., alginate-based, polyacrylamide-based MAHs) which makes them ideal candidates for applications as microfluidic valves (Hwang et al., 2008), chemical adsorbent microspheres (Meng et al., 2019), and drug delivery (Hu et al., 2018b) or bioactive scaffolds (Santos et al., 2015). As a ground-breaking application, small MAH nanocapsules have been proposed to locally deliver drugs in the targeted region reducing the associated toxicity (Hu, Wang, Zhang, Xu, Zhang, Dong, 2018, Zhao, Kim, Cezar, Huebsch, Lee, Bouhadir, Mooney, 2011). In this regard, external magnetic fields can be applied to manipulate the MAHs towards the specific region to be treated and, then, such external magnetic fields can be used to induce compression states leading to the diffusion of the solvent within the MAHs, along with the drug, to the surrounding environment.

Although these materials are promising candidates as relevant solutions for current biomedical challenges, their use is still limited. One of the main bottlenecks that prevents their exploitation in these applications relates to their extremely complex multi-physics behaviour. In this regard, MAHs combine numerous nonlinearities and coupled responses at different time scales. At short-time scales, MAHs present nonlinear large deformations with rate dependences that determine their instantaneous mechanical response (Nam, Stowers, Lou, Xia, Chaudhuri, 2019, Yang, Shao, Meng, 2019). Then, at intermediate-time scale, MAHs show relaxation processes associated with viscoelastic mechanisms leading to a continuous decrease in stiffness until complete relaxation (Hu, Zhou, Daniel, Vatankhah-Varnoosfaderani, Dobrynin, Sheiko, 2017, Nam, Stowers, Lou, Xia, Chaudhuri, 2019, Wang, Wiener, Fukuto, Li, Yager, Weiss, Vogt, 2019). Finally, at long-time scales, mechanical deformations can lead to heterogeneous distributions of solvent concentration that evolve via long-term solvent diffusion process with subsequent alterations in the MAH’s deformation and stress states (Delavoipire, Tran, Verneuil, Chateauminois, 2016, Esteki, Alemrajabi, Hall, Sheridan, Azadi, Moeendarbary, 2019). Additionally, the presence of the magnetic particles introduces a coupled magneto-mechanical response that, in turn, interacts with the deformation mechanisms at the short-, intermediate- and long-time scales (Safronov et al., 2012). To overcome these difficulties, theoretical and computational models can help to elucidate the deformation mechanisms underlying the mechanical alterations within the MAH during magnetic stimulation and in the later deformation stages.

To model MAHs, different physics need to be accounted for in a coupled manner: magnetics, solvent diffusion, and mechanics. Regarding the diffusion-mechanical coupling in hydrogels, relevant works can be found in the literature. Hong et al. (2008) developed a nonequilibrium thermodynamics theory for coupled diffusion and large deformation in polymeric gels. This work has served as basis for further developments in this field (Bouklas, Landis, Huang, 2015, Lucantonio, Nardinocchi, Teresi, 2013, Zhang, Zhao, Suo, Jiang, 2009). More recently, these theories have been extended to incorporate viscoelasticity (Bacca, McMeeking, 2017, Wang, Hong, 2012), crack growth (Bouklas, Landis, Huang, 2015, Yu, Landis, Huang, 2018) and the analysis of instabilities in hydrogels (Dortdivanlioglu and Linder, 2019). Moreover, the magneto-mechanics of MAPs has been modelled using hyperelastic theories based on the fundamental theory published by Pao (1978) along with further developments by Dorfmann, Ogden, 2003, Dorfmann, Ogden, 2004. More recent works incorporate additional nonlinearities (Danas, 2017), viscous and thermal effects (Mehnert, Hossain, Steinmann, 2017, Saxena, Hossain, Steinmann, 2013), and the modelling of hard-magnetic MAPs (Garcia-Gonzalez, 2019, Zhao, Kim, Chester, Sharma, Zhao, 2019). In the last three years, models of MAHs where both diffusion-mechanics and magneto-mechanics formulations are coupled have been proposed. In this regard, Liu et al. (2017) formulated a multiphysics model considering magneto-chemo-hydro-mechanical coupled fields. These authors extended this modelling framework to study hydrogel-based drug targeting systems (Liu et al., 2018) and to include pH sensitivity (Liu, Li, Lam, 2019b, Liu, Liu, Li, Lam, 2019c). Another relevant work to model the response of MAHs was published by Gebhart and Wallmersperger (2019), where a thermodynamically consistent model based on Biot’s consolidation theory is proposed. However, these frameworks are based on equilibrium theories regarding the diffusion effects and do not incorporate inelastic deformation mechanisms associated to the polymeric network. The incorporation of these effects is essential to understand the short-time response of MAHs to mechanical and/or magnetic loading. In this regard, several studies have found a strong impact of hydrogel stress relaxation or creep on cell behaviours such as cell spreading, proliferation, and differentiation of mesenchymal stem cells (Chaudhuri, 2017, Chaudhuri, Gu, Klumpers, Darnell, Bencherif, Weaver, Huebsch, Lee, Lippens, Duda, Mooney, 2016, Nam, Stowers, Lou, Xia, Chaudhuri, 2019). These viscous rate-dependences remain absent in the modelling of MAHs and are thought to be highly relevant for the aforementioned biomedical applications.

This work aims to provide a modelling framework to account for the most relevant physics of MAHs across different time scales. To this end, a general continuum model is presented accounting for magnetics, solvent diffusion, and nonlinear visco-mechanics. To the best of the authors’ knowledge, this is the first work to couple all of these deformation mechanisms together to model the mechanical behaviour of MAHs. This model is framed within an implicit finite element formulation and, then, implemented for 3D problems. We use this computational framework to analyse different rate-dependent responses and remark on their influence at three time scales: (i) short-time scale: strain rate dependency on the instantaneous response to magnetic fields applied at different rates (stiffening due to rate dependency); (ii) intermediate-time scale: relaxation mechanisms associated to the viscoelastic response of the polymeric network occurring after the complete application of the magnetic field; (iii) long-time scale: deformation mechanisms arising from solvent diffusion processes. Finally, we provide a discussion on potential applications of the proposed modelling framework within the bioengineering field. We therefore provide a flexible computational framework that accounts for the main coupled physics governing the behaviour of MAHs, highlighting the importance of rate dependences at different time scales. This computational model establishes the bases for further developments to aid in the design and optimisation of MAH-based applications such as bioactive scaffolds, drug delivery systems, and microfluidic valves.