Finite Extensibility Of Polymer Chains Research Articles

Dynamic modeling of hard-magnetic soft actuators: Unraveling the role of polymer chain entanglements, crosslinks, and finite extensibility

In recent years, there has been increasing interest in hard-magnetic soft materials (HMSMs) due to their ability to retain high residual magnetization and undergo large deformations under external magnetic loading. The performance of these materials in the dynamic mode of actuation is significantly influenced by internal properties, such as entanglements, crosslinks, and the finite extensibility of polymer chains. This article presents a theoretical framework for modeling the dynamic behavior of a hard-magnetic soft material-based planar actuator. A physics-based nonaffine material model is utilized to consider the inherent properties of polymer chain networks. The governing equation for dynamic motion is derived using Euler–Lagrange’s equation of motion for conservative systems. The devised dynamic model is utilized to examine the dynamic response, stability, periodicity, and resonance properties of a planar hard-magnetic soft actuator for different values of polymer chain entanglements, crosslinks, and finite extensibility parameters. The Poincaré maps and phase-plane plots are presented to analyze the stability and periodicity of the nonlinear vibrations of the actuator. The results reveal that transitions between aperiodic and quasi-periodic oscillations occur when the density of polymer chain entanglements and cross-linking changes. The findings from the present investigation can serve as an initial step towards the design and manufacturing of remotely controlled actuators for various futuristic applications.

Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators

Soft materials exhibiting large deformation under external stimuli have gained an increasing attention in the recent past because of their potential applications in soft transducers aimed at achieving biomimetic actuation. This paper theoretically analyzes the effect of internal properties, including entanglements, crosslinks and finite extensibility of polymer chains along with inherent viscoelastic properties of polymers on the performance of Dielectric Elastomer actuator (DEA) in dynamic modes of actuation. A physics based nonaffine material model proposed by Davidson and Goulbourne is used to model the polymer chains entanglements, crosslinks and finite extensibility. To incorprate the viscoelastic properties, a rheological material model based on the additive decomposition of the isotropic strain energy density into equilibrium and viscous parts is implemented. A computationally efficient method, which relies on the principle of least action is used for extracting the governing equation representing the dynamic motion of the DE actuator. The results demonstrate that DEAs with strong entanglements and crosslinks along with small finite extensibility of polymer chains exhibits lower deformation level in DC dynamic modes of actuation. It is inferred that the strong entanglements and crosslinks in polymer chains enhances the resonant frequency, but debilitate the intensity of vibration of viscoelastic DEAs. Further, the periodicity and stability of the nonlinear oscillations exhibited by the viscoelastic DEAs are assessed by employing the Poincare maps and phase portraits. The results of the present investigation can be applied effectively in bridging the mechanism between the microcosmic polymer chains and macroscopic dynamic behavior of viscoelastic DE actuators.

Electroelasticity of dielectric elastomers based on molecular chain statistics

The mechanical response of dielectric elastomers can be influenced or even controlled by an imposed electric field. It can, for example, cause mechanical stress or strain without any applied load; this phenomenon is referred to as electrostriction. There are many purely phenomenological hyperelastic models describing this electroactive response of dielectric elastomers. In this contribution, we propose an electromechanical constitutive model based on molecular chain statistics. The model considers polarization of single polymer chain segments and takes into account their directional distribution. The latter results from non-Gaussian chain statistics, taking finite extensibility of polymer chains into account. The resulting (one-dimensional) electric potential of a single polymer chain is further generalized to the (three-dimensional) network potential. To this end, we apply directional averaging on the basis of numerical integration over a unit sphere. In a special case of the eight-direction (Arruda–Boyce) model, directional averaging is obtained analytically. This results in an invariant-based electroelastic constitutive model of dielectric elastomers. The model includes a small number of physically interpretable material constants and demonstrates good agreement with experimental data, with respect to the electroactive response and electrostriction of dielectric elastomers.

Effects of entanglements and finite extensibility of polymer chains on the mechanical behavior of hydrogels

Polymer networks usually have many entanglements as well as a finite extensibility due to the uncrossability and the full stretching of the network chains. In this paper, the nonaffine model proposed by Davidson and Goulbourne is adopted to characterize the influence of entanglements and finite extensibility of the polymer chains on the mechanical behavior of hydrogels. The Davidson–Goulbourne model only contains three material parameters and has a simpler mathematical form than the Edwards–Vilgis slip-link model, which is the only model that has been adopted to characterize the influence of entanglements on the swelling deformation of polymeric gels so far. Some benchmark problems such as free swelling, constrained swelling, uniaxial tension, equibiaxial tension in the immersed state and uniaxial compression in the isolated state are analyzed, respectively, based on the new constitutive equations. The results show that entanglements as well as chain extensibility have important influences on the mechanical response of hydrogels.

Molecular structure of self-healing polyampholyte hydrogels analyzed from tensile behaviors.

Recently, charge balanced polyampholytes (PA) have been found to form tough and self-healing hydrogels. This class of physical hydrogels have a very high equilibrated polymer concentration in water (ca. 40-50 wt%), and are strongly viscoelastic. They are synthesized by random copolymerization of equal amounts of oppositely charged monomers at a high concentration, followed by a dialysis process of the small counter-ions and co-ions in water. The randomly distributed, opposite charges of the polymer form multiple ionic bonds of intra- and inter-chains with strength distribution. The strong inter-chain bonds, stabilized by topological entanglement, serve as quasi-permanent crosslinks, imparting the elasticity, while the weak bonds, both inter- and intra-chains, reversibly break and re-form to dissipate energy to toughen the materials. In this work, we intend to clarify the structure of the physical PA hydrogels from the tensile behaviors of the PA hydrogels. To clarify the structure and its formation mechanism, we analysed the tensile behaviors of the samples before and after the dialysis. We separated the quasi-permanent crosslinking of strong inter-chain bonds and the dynamic crosslinking of weak inter-chain bonds by using a combined model that consists of the Upper Convected Maxwell model and the Gent strain hardening model. The model fitting of the tensile behaviors extracts quantitative structural parameters, including the densities of weak and strong inter-chain bonds and the theoretical finite extensibility of polymer chains. Based on the fitting results of the combined model, the structural parameters of partial chains at a fixed observation time, including the Kuhn number, Kuhn length, and chain conformation, are determined using the scaling theory. The effects of monomer concentration at preparation, the effect of dialysis and the initial strain rate on the dynamic structure of PA gels, are discussed based on these analyses.

Asymptotic mechanical fields at the tip of a mode I crack in rubber-like solids

Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth and failure. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. In the present work, we perform an asymptotic analysis of the mechanical fields in the vicinity of a propagating mode I crack in rubber. Plane deformation is assumed, and the material model is based on the Langevin function, which accounts for the finite extensibility of polymer chains. The Langevin function is approximated by a polynomial, and only the term of the highest order contributes to the asymptotic solution. The crack is predicted to adopt a wedge-like shape, i.e. the crack faces will be straight lines. The angle of the wedge and the order of the stress singularity depend on the hardening of the strain energy function. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. Hence, there is no upper theoretical limit to the crack speed.

Stress-Strain Relationship of Highly Stretchable Dual Cross-Link Gels: Separability of Strain and Time Effect.

We studied the stress-strain relation of model dual cross-link gels having permanent cross-links and transient cross-links over an unusually wide range of extension ratios λ and strain rates ϵ̇ (or time t = (λ - 1)/ϵ̇). We propose a new analysis method and separate the stress into strain- and time-dependent terms. The strain-dependent term is derived from rubber elasticity, while the time-dependent term is due to the failure of transient cross-links and can be represented as a time-dependent shear modulus which shows the same relaxation as in small strain. The separability is applicable except for the strain stiffening regimes resulting from the finite extensibility of polymer chains. This new analysis method should have a wide applicability not only for hydrogels but also for other highly viscoelastic soft solids such as soft adhesives or living tissues.

Role of Relaxation on Strain Crystallization of Uniaxially Stretched Poly(ethylene naphthalate) Films: A Mechanooptical Study

The large deformation mechanooptical behavior of amorphous films of PEN in the rubbery temperature exhibits three-regime behavior. Regime I corresponds to classical stress−optical behavior of amorphous materials where PEN remains completely amorphous during stretching. Oriented crystallization and long-range connected network formation accompany the positive slope change in stress−optical behavior into regime II. In the last regime, III, the formation of “muscle bound” structure begins to level off birefringence as it approaches the finite extensibility of polymer chains. Relaxation of uniaxially stretched PEN was found to depend exclusively on the regime from which the relaxation is initiated. If the relaxation is initiated from within regime I, the stress and birefringence decrease traces back the linear stress−optical behavior while the material remains amorphous. The relaxation of stress and birefringence from the early stages of regime II deviates from the linear stress−optical behavior. At long enough relaxation times, birefringence eventually begins to increase while stress continues to decline. This marks the formation of highly localized highly oriented ordered regions with low translational ordering along the chain axis in the material. This unexpected behavior during relaxation was attributed to the formation of a long-range physical network that helps arrest the relaxation of orientations gained during stretching. If the relaxation is initiated toward the end of regime II, a fast increase of birefringence is observed while the stress decreases. This behavior is found to be the result of a spontaneous deformation process as observed by the increase of local true strain in the absence of macroscopic deformation. However, this mechanism is not as a result of relaxation alone but a combination of relaxation coupled with continuation of the deformation process that started during stretching and did not stop when the relaxation process started. In regime III, stress relaxation accompanies little or no increase in birefringence. In regime III other than establishment of three-dimensional order in crystalline regions, no other significant change is observed.

Fast and accurate closure approximations for bead-spring models of dilute polymer solutions

Dilute polymer solutions, besides being widely used in the chemical industry, are used to understand the physical processes that govern the dynamics of isolated macromolecules in solution. Several recent studies used Brownian dynamics simulations of mesoscale bead-spring chain models to demonstrate that the finite extensibility of polymer chains and the existence of fluctuating hydrodynamic interactions between different parts of the chains play a key role in determining the behaviour of polymer solutions in strong flows. Three closure approximations which incorporate these phenomena are presented which offer considerable gains in computational efficiency. The predictions of these approximate models are in good agreement with the results of simulations in strong extensional flows.

Nonlinear dynamics of melted polymer layers

AbstractA theory for non‐linear rheology of molten polymer layers between solid surfaces in the Rouse regime is discussed. It is shown that the effect of finite extensibility of polymer chains leads to the characteristic 1/3 power law for the shear stress vs. shear velocity in the regime of high velocities. It is also shown that bridging polymer fragments connecting the two surfaces play an important role for the rheology if the effective monomer friction in the immediate vicinity of the surfaces is much higher than far from the surfaces. In particular we predict that shear stress is decreasing with shear velocity u in a limited range between u1 and u, min. This effect results in a possibility of stick‐slip periodical dynamics of the layer under a constant imposed velocity.

Effect of the finite extensibility of polymer chains and of the finite size of added electrolyte ions on polyelectrolyte brushes

Our earlier polyelectrolyte brush theory [Macromolecules 22, 4173 (1989)] was derived by equating the sum of the nonelectrostatic and electrostatic potentials of the polymer segment to an overall parabolic potential shown to exist in the brush by Milner, Witten, and Cates (MWC) [Macromolecules 21, 2610 (1988)]. Here we correct the theory to take into account the finite extensibility of the chains by adopting the non-Gaussian stretch (or entropy) term of Shim and Cates [J. Phys. France 50, 3535 (1989)]. This correction leads to brush heights which will never exceed the chain length; an aspect that was not accounted for in the earlier work of MWC. In addition, the ions of the added symmetric electrolyte are considered to be hard spheres, the size of the cation and anion assuming different values in general. The total electrochemical potential of the ions, therefore, includes a volume exclusion term as well. The ions are assumed to ‘‘view’’ the solvent as a continuum (characterized only by its dielectric constant) and the brush layer as an obstacle course of randomly distributed spherical hard segments. The treatment is made simple by assuming a regime where the volume fraction of ions is much lower than the segment or solvent volume fractions.