Entropic Elasticity Research Articles

Mark Warner. 26 January 1952 — 24 December 2021

Mark Warner was a prominent theoretical physicist working in the field of soft-matter and polymer science. He was a pioneer in discovering liquid crystalline elastomers as a concept and was the key figure in establishing this subject as a thriving area of research and development. The idea of a material that couples the entropic rubber elasticity with the orientational order of network elements was a complete novelty and a surprise to many. Largely driven by Mark's efforts, this concept, stemming from the coupled order parameters, has been now recognized as a universal phenomenon in several fields: from martensitic transition in shape-memory alloys to engineering and applied mathematics of quasi-convexification. ‘Liquid crystalline elastomers’ will firmly remain as his lasting legacy in science. Mark also made major contributions to physics education, his final legacy being to co-found the national physics education programme, Isaac Physics, in 2013. The ideology of the programme focuses uniquely on students ‘learning by doing’—reassuring students that making mistakes and trying again are fundamental and crucial for learning and developing confidence. Isaac Physics has demonstrated quantifiable impact in raising the aspirations and attainment of hundreds of thousands of students since its inception.

Understanding the Response of Poly(ethylene glycol) diacrylate (PEGDA) Hydrogel Networks: A Statistical Mechanics-Based Framework.

Thanks to many promising properties, including biocompatibility and the ability to experience large deformations, poly(ethylene glycol) diacrylate (PEGDA) hydrogels are excellent candidate materials for a wide range of applications. Interestingly, the polymerization of PEGDA leads to a network microstructure that is fundamentally different from that of the "classic" polymeric gels. Specifically, PEGDA hydrogels comprise PEG chains that are interconnected by multifunctional densely grafted rod-like polyacrylates (PAs), which serve as cross-linkers. In this work, we derive a microstructurally motivated model that captures the essential features which enable deformation in PEGDA hydrogels: (1) entropic elasticity of PEG chains, (2) deformation of PA rods, and (3) PA-PA interactions. Expressions for the energy-density functions and the stress associated with each of the three contributions are derived. The model demonstrates the microstructural evolution of the network during loading and reveals the role of key microscopic quantities. To validate the model, we fabricate and compress PEGDA hydrogel discs. The model is in excellent agreement with our experimental findings for a broad range of PEGDA compositions. Interestingly, we show that the response of PEGDA hydrogels with short PEG chains and long PA rods is governed by PA-PA interactions, whereas networks with longer PEG chains are dominated by entropy. To enable design, we employ the model to investigate the influence of key microstructural quantities, such as the length of the PEG and the PA chains, on the macroscopic properties and response. The findings from this work pave the way to the efficient design of PEGDA hydrogels with tunable properties and behaviors, which will enable the optimization of their performance in various applications.

Nonlinear Poisson effect in affine semiflexible polymer networks.

Stretching an elastic material along one axis typically induces contraction along the transverse axes, a phenomenon known as the Poisson effect. From these strains, one can compute the specific volume, which generally either increases or, in the incompressible limit, remains constant as the material is stretched. However, in networks of semiflexible or stiff polymers, which are typically highly compressible yet stiffen significantly when stretched, one instead sees a significant reduction in specific volume under finite strains. This volume reduction is accompanied by increasing alignment of filaments along the strain axis and a nonlinear elastic response, with stiffening of the apparent Young's modulus. For semiflexible networks, in which entropic bending elasticity governs the linear elastic regime, the nonlinear Poisson effect is caused by the nonlinear force-extension relationship of the constituent filaments, which produces a highly asymmetric response of the constituent polymers to stretching and compression. The details of this relationship depend on the geometric and elastic properties of the underlying filaments, which can vary greatly in experimental systems. Here, we provide a comprehensive characterization of the nonlinear Poisson effect in an affine network model and explore the influence of filament properties on essential features of both microscopic and macroscopic response, including strain-driven alignment and volume reduction.

Investigating the influence of stress dependency on the evaluation of high-temperature deformation resistance in SBS-modified asphalt

The rheological responses of Styrene-Butadiene-Styrene (SBS) modified asphalt are stress-dependent, yet the underlying mechanisms and their relationship to the phase structures formed by SBS in the asphalt matrix remain unclear. To address this issue, this study utilized varying dosages of SBS to prepare binders (SBSMA), which were then characterized through fluorescence microscopy, FTIR, and rheological tests including temperature sweeps and Multi-Stress Creep Recovery (MSCR) at four stress levels. Additionally, rutting resistance of the mixtures was evaluated using wheel tracking tests, correlating the results with rheological data. The test results indicate that the formation of a three-dimensional SBS network structure within the asphalt significantly increases the Stress Sensitivity Index (SSI) by 69.5 %, thereby enhancing the binder's stress sensitivity. Based on the theory of entropic elasticity, the stress sensitivity of SBSMA is explained. The cross-linked network structure exhibits superelastic behavior under low stress but may be overstretched and reorganized under high stress, leading to weakened entropic elasticity. Furthermore, the Jnr values obtained at higher stress levels show stronger correlation (r > 0.9) with the Dynamic Stability (DS) from the wheel tracking test results. This finding suggests that entropic elasticity does not significantly improve the rutting resistance performance at the mixture level. Consequently, increasing the stress level in MSCR tests is recommended to more accurately predict the binder’s rutting performance by inhibiting the network's entropic elasticity.

Critical parameters governing elastocaloric effect in polyisoprene rubbers for solid-state cooling

The aim of this work is to study the critical parameters governing the eCe (elastocaloric effect) of poly-isoprene rubber (NR and IR) in use conditions of a cooling device, i.e. under partial cyclic loading. The effect of mechanical cyclic loading parameters (pre-extension ratio, waveform and frequency) on eCe was first studied. It shows that the eCe increases when the mimimum pre-extension ratio is increased and frequency is lowered from 1Hz to 0.001Hz, as it promotes strain-induced crystallization. However, it also leads to a decrease in potential cooling power, from 7 to 0.01 MW/m3 (i.e. 8 kW/kg to 0.01 kW/kg). At intermediate frequency (f ≈ 0.1 Hz), the comparison of square and triangular waveforms demonstrated that the former enables greater temperature variation. This is due to its holding step (at maximum extension ratio), which maximizes crystallization but also promotes stress relaxation, resulting in increased mechanical losses. Consequently, the square and triangular loadings have a COPmat of 12 and 25, with a temperature variation of 4.2K and 3.6K, respectively. Regarding the formulation, rubbers that do not have a great tendency to crystallize such as synthetic polyisoprene rubber seem to have the best COPmat, while to maximize ΔT, the best of our formulations are NR crosslinked with sulfur and having a crosslink density close to 1.5 × 10−4 mol/cm3, which combine large strain induced crystallization and entropic elasticity. This material has a COPmat of about 27 and allows a ΔT≈4K, i.e.performance comparable to that of the best Shape Memory Alloys (at equivalent COPmat).

Molecular Weight-Dependent Physiochemical Behaviors of Calcium Carbonate Chains.

The regulation of physiochemical behaviors by changing molecular weights is an important cornerstone of polymer physics. However, similar correlations between molecular weights and properties have not been discovered in inorganic ionic compounds. In this work, we prepared a calcium carbonate specimen with a semiflexible chain topology analogous to those of polymers. The molecular weights of the calcium carbonate chains, which ranged from 3400 to 54 100 Da, were directly correlated to their physiochemical behaviors, including gel point, zero shear viscosity, and plateau modulus. The calcium carbonate chains showed similar polymeric characteristics, including shear thinning, thixotropy, entropic elasticity, and viscoelasticity. These features agreed with recent theories and formulas in polymer physics textbooks. On the basis of this understanding, the mechanical properties of calcium carbonate-based gels could be altered by changing their molecular weights. This study could represent a fusion of inorganic chemistry and polymer physics with similar molecular weight-dependent behaviors and material properties, establishing an alternative pathway for designing future inorganic materials.

Structural and physical basis for the elasticity of elastin.

Elastin function is to endow vertebrate tissues with elasticity so that they can adapt to local mechanical constraints. The hydrophobicity and insolubility of the mature elastin polymer have hampered studies of its molecular organisation and structure-elasticity relationships. Nevertheless, a growing number of studies from a broad range of disciplines have provided invaluable insights, and several structural models of elastin have been proposed. However, many questions remain regarding how the primary sequence of elastin (and the soluble precursor tropoelastin) governs the molecular structure, its organisation into a polymeric network, and the mechanical properties of the resulting material. The elasticity of elastin is known to be largely entropic in origin, a property that is understood to arise from both its disordered molecular structure and its hydrophobic character. Despite a high degree of hydrophobicity, elastin does not form compact, water-excluding domains and remains highly disordered. However, elastin contains both stable and labile secondary structure elements. Current models of elastin structure and function are drawn from data collected on tropoelastin and on elastin-like peptides (ELPs) but at the tissue level, elasticity is only achieved after polymerisation of the mature elastin. In tissues, the reticulation of tropoelastin chains in water defines the polymer elastin that bears elasticity. Similarly, ELPs require polymerisation to become elastic. There is considerable interest in elastin especially in the biomaterials and cosmetic fields where ELPs are widely used. This review aims to provide an up-to-date survey of/perspective on current knowledge about the interplay between elastin structure, solvation, and entropic elasticity.

Fabrication and Photothermal Actuation Performances of Electrospun Carbon Nanotube/Liquid Crystal Elastomer Blend Yarn Actuators.

Liquid crystal elastomers (LCEs) are a kind of polymer network that combines the entropic elasticity of polymer networks and the mesogenic unit by means of mild cross-linking. LCEs have been extensively investigated in various fields, including artificial muscles, actuators, and microrobots. However, LCEs are characterized by the poor mechanical properties of the light polymers themselves. In this study, we propose to prepare a carbon nanotube/liquid crystal elastomer (CNT/LCE) composite yarn by electrospinning technology and a two-step cross-linking strategy. The CNT/LCE composite yarn exhibits a reversible shrinkage ratio of nearly 70%, a tensile strength of 16.45 MPa, and a relatively sensitive response speed of ∼3 s, enabling a fast response by photothermal actuation. The research disclosed in this article may provide new insights for the development of artificial muscles and next-generation smart robots.

HIV-1 capsid shape, orientation, and entropic elasticity regulate translocation into the nuclear pore complex

Nuclear import and uncoating of the viral capsid are critical steps in the HIV-1 life cycle that serve to transport and release genomic material into the nucleus. Viral core import involves translocating the HIV-1 capsid at the nuclear pore complex (NPC). Notably, the central channel of the NPC appears to often accommodate and allow passage of intact HIV-1 capsid, though mechanistic details of the process remain to be fully understood. Here, we investigate the molecular interactions that operate in concert between the HIV-1 capsid and the NPC that regulate capsid translocation through the central channel. To this end, we develop a "bottom-up" coarse-grained (CG) model of the human NPC from recently released cryo-electron tomography structure and then construct composite membrane-embedded CG NPC models. We find that successful translocation from the cytoplasmic side to the NPC central channel is contingent on the compatibility of the capsid morphology and channel dimension and the proper orientation of the capsid approach to the channel from the cytoplasmic side. The translocation dynamics is driven by maximizing the contacts between phenylalanine-glycine nucleoporins at the central channel and the capsid. For the docked intact capsids, structural analysis reveals correlated striated patterns of lattice disorder likely related to the intrinsic capsid elasticity. Uncondensed genomic material inside the docked capsid augments the overall lattice disorder of the capsid. Our results suggest that the intrinsic "elasticity" can also aid the capsid to adapt to the stress and remain structurally intact during translocation.

Magnetic tweezers characterization of the entropic elasticity of intrinsically disordered proteins and peptoids.

Understanding the conformational behavior of biopolymers is essential to unlocking knowledge of their biophysical mechanisms and functional roles. Single-molecule force spectroscopy can provide a unique perspective on this by exploiting entropic elasticity to uncover key biopolymer structural parameters. A particularly powerful approach involves the use of magnetic tweezers, which can easily generate lower stretching forces (0.1-20pN). For forces at the low end of this range, the elastic response of biopolymers is sensitive to excluded volume effects, and they can be described by Pincus blob elasticity model that allow robust extraction of the Flory polymer scaling exponent. Here, we detail protocols for the use of magnetic tweezers for force-extension measurements of intrinsically disordered proteins and peptoids. We also discuss procedures for fitting low-force elastic curves to the predictions of polymer physics models to extract key conformational parameters.

Liquid crystal polymer actuators with complex and multiple actuations.

Deformable liquid crystal polymers (LCPs), which exhibit both entropic elasticity of polymer networks and anisotropic properties originating from ordered mesogens, have gained more and more interest for use as biomedical soft actuators. Especially, LCP actuators with controllable mesogen alignment, sophisticated geometry and reprogrammability are a rising star on the horizon of soft actuators, since they enable complex and multiple actuations. This review focuses on two topics: (1) the regulation of mesogen alignment and geometry of LCP actuators for complex actuations; (2) newly designed reprogrammable LCP materials for multiple actuations. First, basic actuation mechanisms are briefly introduced. Then, LCP actuators with complex actuations are demonstrated. Special attention is devoted to the improvement of fabrication methods, which profoundly influence the available complexity of the mesogen alignment and geometry. Subsequently, reprogrammable LCP actuators featuring dynamic networks or shape memory effects are discussed, with an emphasis on their multiple actuations. Finally, perspectives on the current challenges and potential development trends toward more intelligent LCP actuators are discussed, which may shed light on future investigations in this field.

Mechanical criticality of fiber networks at a finite temperature.

At zero temperature, spring networks with connectivity below Maxwell's isostatic threshold undergo a mechanical phase transition from a floppy state at small strains to a rigid state for applied shear strain above a critical strain threshold. Disordered networks in the floppy mechanical regime can be stabilized by entropic effects at finite temperature. We develop a scaling theory for this mechanical phase transition at finite temperature, yielding relationships between various scaling exponents. Using Monte Carlo simulations, we verify these scaling relations and identify anomalous entropic elasticity with sublinear T dependence in the linear elastic regime. While our results are consistent with prior studies of phase behavior near the isostatic point, the present work also makes predictions relevant to the broad class of disordered thermal semiflexible polymer networks for which the connectivity generally lies far below the isostatic threshold.

Deformation and failure mechanisms in spider silk fibers

Spider silk fibers are protein materials that exhibit high strength and toughness thanks to a unique microstructure. In this work, a microscopically motivated model that sheds light on the underlying mechanisms behind the mechanical response of silk fibers is developed. The governing deformation mechanisms are as follows: initial stretching is enabled by the distortion of intermolecular hydrogen bonds that restrict the mobility of polypeptide chains. Once a sufficient force is applied, these bonds dissociate and the external load is transferred to the chains. Next, intramolecular β-sheets in the chains and/or the crystalline domains dissociate to provide additional chain length, thereby resulting in a macroscopic softening. Further deformation is enabled by the entropic elasticity of the chains, which stiffen with stretch. Based on the model, an algorithm to determine the overall constitutive response of silk fibers as a function of the initial distribution of chains and their composition is introduced. Experiments have shown that these two quantities can be controlled by supercontracting and dehydrating fibers under load. The model is validated through a comparison to various experimental findings at different alignment parameters. The merit of the model is three-fold: (1) it captures the microstructural evolution of the network as the fiber experiences stretch and reveals the role of key microstructural quantities such as chain-density, chain alignment, and chain composition, (2) it enables to compare between the microstructures of silk fibers produced by different spider species, and (3) it provides a platform for the microstructural design of biomimetic synthetic fibers with tunable properties.

Pincus blob elasticity in an intrinsically disordered protein.

Understanding the dynamic structure of intrinsically disordered proteins (IDPs) is important to deciphering their biological functions. Here, we exploit precision entropic elasticity measurements to infer the conformational behavior of a model IDP construct formed from the disordered tail of the neurofilament low molecular weight protein. The IDP construct notably displays a low-force power-law elastic regime, consistent with the Pincus blob model, which allows direct extraction of the Flory exponent, [Formula: see text], from the force-extension relationship. We find [Formula: see text] increases with added denaturant, transitioning from a nearly ideal chain to a swollen chain in a manner quantitatively consistent with measurements of IDP dimensions from other experimental techniques. We suggest that measurements of entropic elasticity could be broadly useful in the study of IDP structure.

Particle simulations for inquiry-based teaching of polymer shape and entropic elasticity using computational thinking

An inquiry-based approach is proposed that allows students to develop and understand computer codes for simulations at the particle level. Computational thinking is employed as a low-threshold approach for students without programming experience. The resulting computer code cores are concise and traceable. Here, the topic of the simulations concerns polymer molecules. A special property of polymers, coiling, and the related entropic elasticity is suitable for the setup of a simulation in class. Students at the upper secondary level were introduced into the topic by an activity. This was a stochastic game serving as a basis to for the development of the corresponding algorithm. Once students are involved in the development of the code, they can meaningfully investigate the model by running the program, changing its parameters, or altering the code. With these simulations, students can prove wrong the early belief that polymer molecules exhibit a stretched chain structure. Furthermore, the confinement of the system leads to a decrease in entropy, which in turn results in a force acting on the walls that can be elaborated comprehensively by students through such simulations.

Solvent-Induced Negative Energetic Elasticity in a Lattice Polymer Chain.

The negative internal energetic contribution to the elastic modulus (negative energetic elasticity) has been recently observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the n-step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We theoretically demonstrate the emergence of negative energetic elasticity based on an exact enumeration up to n=20 and analytic expressions for arbitrary n in special cases. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer-solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.

Stretchable, Ultratough, and Intrinsically Self-Extinguishing Elastomers with Desirable Recyclability.

Advanced elastomers are increasingly used in emerging areas, for example, flexible electronics and devices, and these real-world applications often require elastomers to be stretchable, tough and fire safe. However, to date there are few successes in achieving such a performance portfolio due to their different governing mechanisms. Herein, a stretchable, supertough, and self-extinguishing polyurethane elastomers by introducing dynamic π-π stacking motifs and phosphorus-containing moieties are reported. The resultant elastomer shows a large break strain of ≈2260% and a record-high toughness (ca. 460MJ m-3 ), which arises from its dynamic microphase-separated microstructure resulting in increased entropic elasticity, and strain-hardening at large strains. The elastomer also exhibits a self-extinguishing ability thanks to the presence of both phosphorus-containing units and π-π stacking interactions. Its promising applications as a reliable yet recyclable substrate for strain sensors are demonstrated. The work will help to expedite next-generation sustainable advanced elastomers for flexible electronics and devices applications.

Dilute polymer solutions under shear flow: Comprehensive qualitative analysis using a bead-spring chain model with a FENE-Fraenkel spring

Although the nonequilibrium behavior of polymer solutions is generally well understood, particularly in extensional flow, there remain several unanswered questions for dilute solutions in simple shear flow, and full quantitative agreement with experiments has not been achieved. For example, experimental viscosity data exhibit qualitative differences in shear-thinning exponents, the shear rate for the onset of shear-thinning, and high-shear Newtonian plateaus depending on polymer semiflexibility, contour length, and solvent quality. While polymer models are able to incorporate all of these effects through various spring force laws, bending potentials, excluded volume (EV) potentials, and hydrodynamic interaction (HI), the inclusion of each piece of physics has not been systematically matched to experimentally observed behavior. Furthermore, attempts to develop multiscale models (in the sense of representing an arbitrarily small or large polymer chain) which can make quantitative predictions are hindered by the lack of ability to fully match the results of bead-rod models, often used to represent a polymer chain at the Kuhn-step level, with bead-spring models, which take into account the entropic elasticity. In light of these difficulties, this work aims to develop a general model based on the so-called FENE-Fraenkel spring, originally formulated by Larson and co-workers [J. Chem. Phys. 124 (2006)], which can span the range from rigid rod to traditional entropic spring, as well as include a bending potential, EV, and HI. As we show, this model can reproduce, and smoothly move between, a wide range of previously observed polymer solution rheology in shear flow.

High-Entropy Microdomain Interlocking Polymer Electrolytes for Advanced All-Solid-State Battery Chemistries.

All-solid-state polymer electrolytes (ASPEs) with excellent processivity are considered one of the most forward-looking materials for large-scale industrialization. However, the contradiction between improving the mechanical strength and accelerating the ionic migration of ASPEs has always been difficult to reconcile. Herein, a rational concept is raised of high-entropy microdomain interlocking ASPEs (HEMI-ASPEs), inspired by entropic elasticity well-known in polymer and biochemical sciences, by introducing newly designed multifunctional ABC miktoarm star terpolymers into polyethylene oxide for the first time. The tailor-made HEMI-ASPEs possess multifunctional polymer chains, which induce themselves to assemble into micro- and nanoscale dynamic interlocking networks with high topological structure entropy. HEMI-ASPEs achieve excellent toughness, considerable ionic conductivity, an appreciable lithium transference number (0.63), and desirable thermal stability (Td > 400°C) for all-solid-state lithium metal batteries. The Li|HEMI-ASPE-Li|Li symmetrical cell shows a stable Li plating/stripping performance over 4000h, and a LiFePO4 |HEMI-ASPE-Li|Li full cell exhibits a high capacity retention (≈96%) after 300 cycles. This work contributes an innovative design concept introducing high-entropy supramolecular dynamic networks for ASPEs.

Mechanically robust, instant self-healing polymers towards elastic entropy driven artificial muscles

Despite the substantial developments, it remains a huge challenge for the fabrication of soft actuators to achieve the systematic integrations of good actuating performance, high mechanical strength and self-healing ability at ambient temperature. Herein, we report the preparation of robust self-healing polymeric materials by introducing strong coordination bonds to the polymers with multiple hydrogen bonds and anisotropic chain structures. Due to the synergistic effect between the weak hydrogen and strong coordination bonds, the resultant polymer exhibits excellent mechanical strength (12.68 MPa) and instant (30 s) self-healing ability with a high healing efficiency of ca. 95 % at ambient temperature. Thanks to the entropic elasticity trapped by the anisotropic polymer chains, the resultant anisotropic polymer can be used as thermal-triggered artificial muscles with a tensile stroke of ca. 12 %. Even the healed sample can still achieve a tensile stroke of ca. 11 % repeatedly.