Journal Club for March 2020: Molecular Simulation-Guided and Physics-Informed Multiscale Modeling of Polymer Viscoelasticity

Journal Club for March 2020: Molecular Simulation-Guided and Physics-Informed Multiscale Modeling of Polymer Viscoelasticity

Ying Li, Department of Mechanical Engineering, University of Connecticut

1.       Introduction

       Polymers are long chain organic molecules. Elastomers consist of long polymer chains joined together by chemical bonds through cross-linkers. They are usually capable of recovering their original shapes after finite deformation due to covalent cross-linkages. Under equilibrium conditions, long polymer chains making up an elastomer are irregularly coiled together. However, when the elastomer is under tension, polymer chains tend to stretch out and straighten along the pulling direction. Upon unloading, the chains return to their original compact and random arrangement. During such a loading-unloading process, energy dissipates due to friction between polymer chains and the elastomer exhibits viscoelastic behavior, as illustrated in Fig. 1. Examples of elastomers include polyisoprene (natural rubber), polybutadiene (butadiene rubber), polychloroprene (chloropene rubber) and the copolymer composed of butadiene and acrylonitrile (nitrile rubber). These materials have a broad range of applications in engineering and industry, such as tires, seals, adhesives, soles for shoes, and damping and insulating elements.

Fig. 1 (a) Deformation behavior of elastomers under tension; Experimental results on uniaxial stress–strain curves during loading and subsequent un-loading of (b) un-vulcanized and (c) vulcanized natural rubber (NR).

      Viscoelasticity characterizes the most important mechanical behaviors of polymers and elastomers, represented by the storage (elastic) modulus G’ and loss (viscous) modulus G”. Their ratio G”/G’, so-called tan(δ), denotes the energy dissipation during the deformation process. We have tan(δ)=0 for a metallic spring, indicating that it loses very little energy when it is stretched and let go. However, other materials, like carbon-black filled rubber, or silica filled rubber, will have a larger value of tan(δ) due to the molecular friction between polymer chains. In the tire manufacturing and related industry, tan(δ) is used as the gold standard in design and evaluation the performance of tire materials. In the temperature range of 40–80 °C, the rolling resistance of tire correlates directly with tan(δ): the smaller tan(δ), the lower rolling resistance. A higher tan(δ) at lower temperature will lead to good wet grip. It has been recognized that 10% of fuel used in the average car is taken for overcoming the rolling resistance of tires. By itself, the automobile rolling resistance is responsible for the astonishing 4% of worldwide carbon dioxide emissions from fossil fuels [1]. Thus, it is another way to improve the fuel economy by lowering the rolling resistance (or tan(δ)).

      However, in the tire world, designers are constrained tradeoffs in which an improvement to rolling resistance has to sacrifice the wet-road grip and/or tread-wear of tire, so-called the “magic triangle” [2,3]. For example, in the temperature range of 30-60 °C, rolling resistance correlates directly with tan(δ): The lower the tan(δ), the lower the rolling resistance. Reducing the tan(δ) in this temperature range, however, also reduces the tan(δ) at lower temperatures. Good wet grip corresponds to a higher tan(δ) at lower temperatures. This is why the wet-grip and rolling-resistance corners of the magic triangle are hard to pull apart. To overcome this challenge, we need to develop a predictive multiscale modeling framework to link the chemical constituents, microstructure and viscoelasticity of polymers and elastomers. Here we will discuss our recent progress in understanding and predicting the polymer viscoelasticity from their chemical structures, through a molecular simulation-guided and physics-informed multiscale modeling scheme [4-6].

2.       Mechanics of Polymer Viscoelasticity

       Viscoelasticity, by definition, consists of elastic and viscous parts. In the past, we know that elasticity is usually the result of bond stretching along crystallographic planes in an ordered solid. While, viscosity is the result of the diffusion of atoms or molecules inside an amorphous material. The macroscopic mechanical response of elastomers originates from their microscopic molecular structure: long randomly oriented polymer chains are joined together via cross-linkers, forming a cross-linked polymeric network; free chains are randomly distributed within the cross-linked network. Upon loading/unloading, the cross-linked polymer network is stretched and the free chains diffuse within the network. The cross-linked polymer network acts as the backbone and is usually capable of recovering its original shapes after unloading, which results in a nonlinear hyperelastic mechanical response. In contrast, reptation of free chains within the cross-linked network is irreversible and causes energy dissipation (i.e. the mechanical response exhibits hysteresis upon cyclic loading), leading to a viscous mechanical behavior. Therefore, to develop a constitutive law that accurately describes the mechanical response of elastomers, their stress response needs to be decomposed into two parts: a hyperelastic part and a viscous part, as shown in Fig. 2.

Fig. 2 Decomposition of an elastomer into a cross-linked polymer network with free chains.

2.1   Hyperelasticity for cross-linked polymer network.

      The nonlinear elasticity of a cross-linked polymer network is typically characterized by a hyperelastic constitutive model, which is governed by a strain energy density function W. Depending on the approach followed by the authors to develop the strain energy function W, the hyperelastic models can be classified into phenomenological and physics-based models. The phenomenological models are mainly obtained from invariant-based continuum mechanics treatment of W and can be used to fit experimentally observed mechanical behaviors. The strain energy density W must depend on stretch via one or more of the three invariants of the Cauchy-Green deformation tensor. Model parameters in phenomenological models usually carry no explicit physical meaning and can only be calibrated by fitting experimental data. On the other hand, physics-based models are derived based on polymer physics and statistical mechanics which begins by assuming a structure of randomly-oriented long molecular chains. The strain energy density function W for physics-based models depends on the microscopic structure of the polymer network, in which model parameters are related to the underlying polymer physics and can be calibrated by using information of the microscopic structure of elastomers. Along this line, the most widely used phenomenological models include but are not limited to the Mooney Model [7], the Mooney-Rivlin Model [8], the Ogden Model [9], the Yeoh Model [10] and the Gent Model [11]. An excellent review of the detailed mathematical formulation of aforementioned phenomenological hyperelastic models is given by Marckmann and Verron [12]. For the physics-based models, there are the Neo-Hookean Model [13], the 3-Chain Model [14], the Arruda-Boyce (8-Chain) model [15], the slip-link model [16], the extended tube model [17], the non-affine micro-sphere model [18], and the non-affine network model [19]. Details about these physics-based models have been summarized and compared by Davidson and Goulbourne [19] recently. The Neo-Hookean Model, the 3-Chain Model and the Arruda-Boyce Model are widely deployed to characterize the hyperelastic response of elastomers because of the compact mathematical form. However, these models cannot capture the stress softening behavior observed in elastomers with low cross-linking density since they ignore the effect of polymer entanglements [19]. The extended tube model and the non-affine microsphere model can capture the stress softening but contains some parameters without clear physical connections or hard to be calculated [19]. Slip-link model and the non-affine network model can capture the stress softening and all their material parameters are connected to molecular quantities. It is worth noting that the non-affine network model is mathematically compact compared to the slip-link model, which facilitates its application in capturing stress-stretch response of elastomers [5, 19].

2.2   Viscosity for free polymer chains.

       The viscosity of polymers originates from their chain dynamics. If the chains are very short, i.e., they are oligomers; their dynamics are dominated by the friction between monomers. According to the Rouse theory [20], the viscosity, η, of these oligomers has a simple scaling relationship with chain length N, as η∼N, and the self-diffusion coefficient scales as D∼N-1. These phenomena have also been observed both in molecular dynamics (MD) simulations [4,21,22] and experiments [23]. However, when the chain length is larger than the entanglement length (N>Ne), due to chain connectivity and uncrossability, the dynamics of these long chains will be greatly hindered by topological constraints, referred to as entanglements. These entanglements are commonly assumed to effectively restrict the lateral motion of individual polymer chains into a tube-like region with diameter app. Thus, a chain will slither back and forth, or reptate, along the tube, instead of moving randomly through three dimensional space.

       This picture for entangled polymer chain dynamics constitutes the so-called tube model, the most successful theory from the field of polymer physics in the past forty years. The central axis of the tube-like region defines the primitive path (PP). The PP can be considered as the shortest path remaining, if one holds the two ends of the chain in space and continuously shrinks its contour without violation of the chain’s uncrossability with its neighboring chains. De Gennes [24] and Doi and Edwards [25] performed the pioneering works on the theoretical study of rheological properties of entangled polymer melts following from the tube concept. The dynamics of entangled polymer chains was considered in terms of the one-dimensional diffusion of a tracer chain along its PP in a mean-field approach (i.e., the constraints formed by neighboring chains are considered static). The PP was treated as a random walk in space with a constant step length, app. Thus, the degree of the topological interactions between different chains is also defined through the effective tube diameter, app. From the tube theory [25], D∼N-2 and η∼N3, which agree reasonably well with the experimental observations.

      To investigate the finite-strain viscoelastic properties of unfilled or filled elastomers, phenomenological viscoelastic constitutive models have been developed by Simo [26], Govindjee and Simo [27], Lion [28], Lubliner [29], Keck and Miehe [30] based on stress/strain variables. These phenomenological constitutive models are mainly used to fit existing experiment data since microscopic details about the underlying physics are not considered in the formulation of these constitutive models. Moreover, most of these reported models only capture a subset of the experimentally observed phenomena. In an attempt to overcome the disadvantage of the phenomenological models, Le Tallec et al. [31], Govindjee and Reese [32], Bergström and Boyce [33], Miehe and Göktepe [34] have developed their constitutive models for finite-strain viscoelasticity based on the tube model developed by Doi and Edwards [25]. However, these models still have several internal variables or model parameters without physical significance, and thus have to be classified as physics-inspired phenomenological models.

2.3   Physics-informed constitutive model for polymer viscoelasticity

Recently, we have developed a physics-informed constitutive model by combining the non-affine network model  for hyperelasticity and a modified tube model for viscosity with following key assumptions [5]: a) Change of primitive chain length versus deformation gradient; b)   Change of tube diameter versus deformation gradient; c)  Change of chain orientation during deformation; d) Fractional order viscoelasticity for polymer chain dynamics, consider the polydispersity of molecular weight and cross-linking effect.

All the parameters in this constitutive model carry physical significance and the model is able to capture the mechanical response of a wide range of elastomers reasonably well, as summarized in Fig. 3. In contrast to most existing finite-strain viscoelastic model for elastomers, all of the 9 material parameters in this constitutive model carry physical significance and thus each parameter can be directly obtained through molecular dynamics simulations or measured by experiments [4,35,36]. This enables a bottom-up approach to predict and design the macroscopic mechanical behavior of an elastomer from its molecular structure, enabling a multiscale simulation methodology to investigate the viscoelastic material behavior of elastomers [4].

Fig. 3 Summary of the molecular simulation-guided and physics-informed constitutive model for viscoelasticity of elastomers.

3.       Molecular Simulations of Polymer Viscoelasticity

       Despite that we can develop a physics-informed constitutive models for the polymer viscoelasticity, based on the modified tube theory, there are many assumptions in this theory. It is not clear how reliable and accurate of these assumptions are. Due to the current limitations of experimental techniques, it is not feasible to directly observe and track the entanglement of polymers, and further understand its contribution to the mechanical properties (e.g. viscoelasticity) of polymers. Instead, with the advancements in computer simulations, in particular, molecular dynamics (MD) simulations, we can use MD simulations as “computational microscopy” to further understand and quantify the molecular mechanisms underpinning these important mechanical behaviors. Here we will demonstrate three cases that MD simulations can provide direct evidence on entanglement effects on elasticity, dynamics and viscosity of polymers.

3.1   How entanglements between polymer chains contribute to the elasticity of polymer network?

       In the classical mechanics field, the cross-linking between different polymer chains is considered to play the most important role in the hyperelasticity response of a cross-linked polymer network. However, according to the polymer physics theory, the entanglements (topological constraints between different chains) should also play another important role. To further understand this effect, we adopted a highly coarse-grained model, so-called FENE (finite extensible nonlinear elastic) model [36], to study the hyperelastic response of a cross-linked polymer without and with entanglements between different polymers. The stress–strain curves for samples under uniaxial tension, simple shear and equibiaxial tension with polymerization degree N=20 and 500 are presented in Fig. 4. The non-affine network model, developed by Davidson and Goulbourne [19], has been used to extract the cross-link modulus Gc, entanglement modulus Ge, and chain extensibility. Very interestingly, when the cross-linked chain length N is as small as N=20, there are basically no entanglements, and accordingly, the corresponding contribution Ge tends to vanish, as shown in Fig. 4. With the polymerization degree N increasing, Ge is rapidly increasing and approaches to the plateau value as predicted by the tube theory from polymer physics. These MD simulations indeed confirm that the entanglement between different polymer chains contribute to the hyperelasticity of a cross-linked polymer network, following the classical tube theory.

Fig. 4 Molecular dynamics simulation results on cross-linked network with (a) short (unentangled) chain N=20 and (b) long (entangled) chain N=500 under uniaxial tension, simple shear and equibiaxial tension. Calibrated material parameters (c) cross-link modulus Gc and (d) entanglement modulus Ge.

3.2   How can we quantify the entanglement and its influence on polymer chain dynamics?

       The topological confinement/constraint effect induced by neighboring polymer chains can be studied through the primitive path (PP) analysis (Z1 code), pioneered by my longtime collaborator, Dr. Martin Kroger from ETH Zurich. The original Z1 code [37-40] constructs the PP network of a polymeric system by fixing the ends of all polymer chains. Hereafter, each polymer chain is replaced by a series of infinitesimally thin, impenetrable and tensionless straight lines. The length of these multiple disconnected paths is monotonically reduced, subject to chain-uncrossability. Upon iterating the geometrical procedure, each multiple disconnected path converges to a final state, the shortest disconnected path, for example, an individual primitive path for each chain, as shown in Fig. 5. The convergence of the Z1 code is achieved once the difference between two successive iterations is smaller than a preset small numerical tolerance. In short, the primitive path can be considered as the shortest path remaining when one holds chain ends fixed, while continuously reducing (shrinking) the chain's contours without violating topological constraints (polymer chain uncrossability). A single primitive path is often characterized by its conformational properties such as primitive path length Lpp, number of interior kinks (or entanglements) per chain Z, and the end-to-end distance R2ee. Thus, the tube diameter can be obtained by app= R2ee /Lpp  [25]. More details on the implementation and application of Z1 code in primitive path study of polymers can be found in Refs. [37,38,41]. In particular, the PP network analysis can be further combined with the long-time MD simulation results to directly probe the dynamics (or diffusion) of polymer chains. The segment/tube survival probability function, which is central in the Doi-Edwards tube theory as it is behind all viscoelastic properties of the polymer, can be directly quantified [42,44]. This survival probability function can be further used in our physics-informed constitutive model to predict the viscoelasticity of polymers and elastomers.

Fig. 5 Primitive path network of highly entangled polymer chains. The original chains (beads) and their corresponding primitive paths (thick lines) are given on the right. An individual polymer chain (beads) with its primitive path (thick line) has been enlarged on the left, without showing the neighborhood of chains. The different colors denote different polymer chains. The kinks of primitive paths represent the entanglements between different chains. For clarity, the configuration is shown in ‘unwrapped’ condition.

3.3   How can we understand the tube deformation and its contribution to finite viscoelasticity?

       In the above theoretical formulation (Section 2.3), we have made several important assumptions regarding the changes of the primitive chain length and tube diameter versus deformation gradient. To further verify these assumptions, we have performed large scale MD simulations on highly entangled polymers and have measured primitive chain length and tube diameter through Z1 code. These MD simulation results have been compared with our theoretical assumptions, as shown in Fig. 6. We find that the change of tube diameter is in good agreement with our theoretical formulation, regardless of the deformation state or temperature. However, for the primitive chain length, the theoretical prediction only agrees with MD simulations when the temperature is close to glass transition temperature. Note that since the number of entanglements per chain is still quite small (about 10 in this case), the polymer chain can easily slide during the deformation at the high temperature. Therefore, the affine deformation assumption is not applicable in this case. When the temperature of the system is reduced sufficiently close to the glass transition temperature, the polymer is more close to its solid state. The chain is constrained by the entanglements and cannot slide. Therefore, the change of primitive chain length is very well described by the affine deformation assumption. Overall, the theoretical assumptions made in Section 2.3 are physically relevant and reasonable, according to our MD simulations.

Fig. 6 Primitive path analysis results on changes of (a) contour length of the primitive chain Lpp and (b) tube diameter app during uniaxial tension.

4.       Summary and future opportunities

       Here we present a molecular simulation-guided and physics-informed constitutive model for polymer viscoelasticity. The hyperelastic and viscous behaviors of elastomers are attributed to the nonlinear deformation of a cross-linked network and diffusion of free chains, respectively. For the cross-linked network, its stress response is attributed to the cross-linking and entanglement. Therefore, the classical Arruda–Boyce model is not applicable as it does not include the effect of entanglements. The recently developed non-affine network model (developed by Davidson and Goulbourne in 2013), simultaneously considering the contributions of cross-linking and entanglement, is identified as the most proper continuum model to describe the hyperelastic behaviors of elastomers. For the free chains, the primitive chain length and tube diameter are found to be dependent on the applied deformations in the MD simulations, which can be captured by our theoretical formulations based on an affine deformation assumption. Based on these observations, the viscous contribution of free chains can be reformulated according to the classical tube model proposed by Doi and Edwards [25], which is named as the updated tube model. Combining the non-affine network model with the updated tube model, this new constitutive model has been proposed for studying the finite strain viscoelastic behavior of elastomers, for which all its parameters have direct physical meaning.

       Furthermore, we would like to emphasize the important role played by the MD simulations during the development of the proposed constitutive model. These molecular simulations are very useful for determining the material law parameters, such as relaxation time and tube diameter, as shown in our previous study [4]. Besides, these MD simulations have been used to verify the underlying assumptions and reveal new physical mechanisms of the continuum model, which are the most important elements during the development of the constitutive model. For example, the capability of the non-affine network model has been tested by our large scale molecular simulations. Through these simulations, the contribution of polymer chain entanglements in the non-affine network model has been confirmed, which has not been appreciated before. Note that it is difficult to confirm the entanglement contribution through experiments, as it requires precisely controlled microstructures of elastomers, such as cross-link density, polymerization degree between cross-linkages and entanglements per chain. In short, we believe that the current MD simulations play very important roles in understanding and developing the state-of-the-art continuum theories for hyperelasticity and viscosity.

       In addition to above aspects, the theoretical modeling and computer simulation of polymers also highlight a number of challenges and opportunities in the field of solid mechanics, such as hydrogels [45-47], dielectric elastomers [48-50], magnetorheological elastomers [51-53], liquid crystal elastomers [54-56], self-healing polymers [57-59], and many others. You are very welcome to share your thoughts and comments below. 

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