On the motive of O'Grady's ten-dimensional hyper-K\"ahler varieties

We investigate how the motive of hyper-K\"ahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of B\"ulles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-K\"ahler varieties of O'Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the Andr\'{e} motive of projective hyper-K\"ahler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford--Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-K\"ahler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-K\"ahler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the Andr\'e motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.Comment: Published version. Communications in Contemporary Mathematics (open access

A cooperative domain model for multiple phase transitions and complex conformational relaxations in polymers with shape memory effect

Shape memory polymers (SMPs) are thermo-rheologically complex materials showing significant temperature and time dependences. Their segments often undergo cooperative phase transitions and conformational relaxations simultaneously along with shape memory effect (SME). In this study, a cooperative domain model is proposed to describe the composition dependence, multiple phase transitions and conformational relaxations of SMPs within their glass transition zones. Variations in local-area compositions and cooperative domains of the amorphous SMPs cause significant differences in their segmental relaxation. At a fixed domain size, both intermolecular activation energy and relaxation time significantly influence the SME and thermomechanical properties of the SMPs. Finally, the model is successfully applied to predict the shape memory behavior of SMPs with one stage SME and triple-SME, and the theoretical results have been validated by the experimental ones. This model could be a powerful tool to understand the working mechanisms and provide a theoretical guidance for the designs of multi-SME in SMPs

Modeling strategy for dynamic-modal mechanophore in double-network hydrogel composites with self-growing and tailorable mechanical strength

Smart materials with self-growing and tailorable mechanical strength have wide-range potential applications in self-healing, self-repairing, self-assembly, artificial muscle, soft robots and intelligent devices. However, their working mechanisms and principles are not fully understood yet and mathematically and physical modeling is a huge challenge, as traditionally synthesized materials cannot self-grow and reconstruct themselves once formed or deformed. In this study, a phenomenological constitutive model was developed to investigate the working mechanisms of self-growing and tailorable mechanical strength in double-network (DN) hydrogel composites, induced by mechanochemical transduction of dynamic-modal mechanophore. An extended Maxwell model was firstly employed to characterize the mechanical unzipping of hydrogel composites, and then mechanochemically induced destruction and reconstruction processes of brittle network in the hydrogel composite were formulated. The enhanced mechanical strength of brittle network has been identified as the key driving force to generate self-growing and tailorable mechanical strength in the hydrogel composite. Finally, a stress-strain constitutive relationship was developed for the dynamic-modal mechanophorein the hydrogel composite. Simulation results obtained from the proposed model were compared with the experimental data, and a good agreement has been achieved. This study provides an effective strategy for modelling and exploring the working mechanism in the mechanoresponsive DN hydrogel composites with self-growing and tailorable mechanical strength

Anchoring-mediated topology signature of self-assembled elastomers undergoing mechanochromic coupling/decoupling

Soft elastomers with their ability to integrate strain-adaptive stiffening and coloration have recently received significant research attention for application in artificial muscle and active camouflage. However, there is a lack of theoretical understanding of their complex molecular dynamics and mechanochromic coupling/decoupling. In this study, a topological dynamics model is proposed to understand the anchoring-mediated topology signature of self-assembled elastomers. Based on the constrained molecular junction model, a free-energy function is firstly formulated to describe the working principles of strain-adaptive stiffening and coloration in the self-assembled elastomer. A coupled ternary “rock-paper-scissors” model is proposed to describe the topological dynamics of self-assembly, mechanochromic coupling and mechanoresponsive stiffening of the self-assembled elastomers, in which there are three fractal geometry components in the topology network. Finally, the proposed models are verified using the experimental results reported in the literature. This study provides a fundamental approach to understand the working mechanism and topological dynamics in the self-assembled elastomers, with molecularly encoded stiffening and coloration

A dynamic model of complexly mechanoresponsive chain-poly[n]-catenations in double-network polyampholyte hydrogels

Polyampholytes have been widely used to improve mechanical performance of double-network (DN) hydrogels, however, the complex mechanisms of electric charge reactions and chain catenations have not been well understood. In this study, a collective and cooperative model is developed to describe the dynamics and constitutive relationships of complexly mechanoresponsive chain-poly[n]-catenations in polyampholyte DN hydrogels. The freely jointed chain (FJC) model and Flory-Huggins theory are firstly employed to formulate mechanochemical behaviors of the DN hydrogels, in which the stretchable network undergoes a folding-to-unfolding transition and the brittle one undergoes a reversibly mechanochemical transition. The worm like chain (WLC) model is then introduced to describe the chain-poly[n]-catenations, of which the strong and weak ionic bonds have been modeled based on the entanglement and dangling effects, respectively. Finally, a free-energy equation is developed to describe their collective and cooperative dynamics. Effectiveness of the newly proposed model is verified by applying it to predict the experimental results of the polyampholyte DN hydrogels reported in literature

Local conservation law of rubber elasticity in hydrogel networks undergoing microphase separation and toughening

Thermoresponsive polymer segments have been reported to induce lateral microphase separations due to their switching transitions from a hydrophilic state to a hydrophobic one in hydrogels, which result in shrinkage and collapse of the polymer networks and significantly improved mechanical strength. However, the route from which the hydrophobic segments are assembled into micelles during microphase separations, and their thermoresponsive toughening mechanisms are not fully understood. In this study, a local conservation law of rubber elasticity is firstly formulated to describe the micellization and collapse of polymer networks in hydrogels, during which the thermoresponsive segments undergo a microphase separation. Flory-Huggins theory, interfacial free-energy equation and the extended Maxwell model are then employed to model the thermodynamics of micellization and microphase separations in the hydrogel, in which the polymer networks are composed of both hydrophilic and thermoresponsive segments. The toughening mechanism is further explored and discussed based on the proposed models. Finally, the proposed models have been verified using the experimental results reported in the literature. This study provides a new mechanism of local conservation law for rubber elasticity in hydrogels and also critical insights into the physical principles which govern the molecular self-assembly

Cooperative dynamics of heuristic swelling and inhibitive micellization in double-network hydrogels by ionic dissociation of polyelectrolyte

In this study, a cooperative model has been proposed for the double network (DN) hydrogel, which synchronously undergoes heuristic swelling and inhibitive micellization by the ionic dissociation of polyelectrolyte. Flory-Huggins solution theory is initially employed to identify the working mechanism of dielectric constants on swelling behavior of the DN hydrogel. Then a free-energy function is introduced to formulate the constitutive relationship of the DN hydrogels, in which the first hydrotropic network undergoes a heuristic swelling and the second hydrophobic network undergoes an inhibitive micellization. Finally, the proposed model has been verified using the experimental results reported in the literature. A good agreement between the theoretical results and experimental ones has been achieved. This study provides a fundamental approach to formulate the constitutive relationship and to understand the cooperative dynamics of two types of networks in DN hydrogels induced by the polyelectrolyte

Understanding complex dynamics of interfacial reconstruction in polyampholyte hydrogels undergoing mechano-chemo-electrotaxis coupling

Polyampholyte (PA) hydrogels have attracted significant attention for their superior mechanical strength and toughness compared with other conventional hydrogels. In this study, we present a novel thermodynamic approach to understanding the mechano-chemo-electrotaxis coupling and interfacial dynamics in PA hydrogels. Flory–Huggins theory, carried out through an interfacial free-energy model, is the foundation for the quantitative study of the mechanically constitutive relationship of the PA gels. The proposed free-energy model is further extended to describe the mechano-chemo-electrotaxis switching and interfacial dynamics by co-relating the Williams–Landel–Ferry equation and scaling laws. It was concluded that the interfacial bonding strength is the key factor influencing the mechanical strength and reconstruction reversibility of the PA macromolecular gel system. The resulting analytical outcomes showed good agreement with the reported experimental data. We opine that the proposed model will guide the future application of PA hydrogels

Interfacial Confinement in Semi-Crystalline Shape Memory Polymer Towards Sequentially Dynamic Relaxations.

Sequential glass and melting transitions in semi-crystalline shape memory polymers (SMPs) provide great opportunities to design and generate multiple shape-memory effects (SMEs) for practical applications. However, the complexly dynamic confinements of coexisting amorphous and crystalline phases within the semi-crystalline SMPs are yet fully understood. In this study, an interfacial confinement model is formulated to describe dynamic relaxation and shape memory behavior in the semi-crystalline SMPs undergoing sequential phase/state transitions. A confinement entropy model is first established to describe the glass transition behavior of amorphous phase within the SMPs based on the free volume theory, where the free volume is critically confined by the crystalline phase. An extended Avrami model is then formulated using the frozen volume theory to characterize the melting and crystallization transitions of the crystalline phase in the SMPs, whose interfacial confinement with the amorphous phase has been identified as the driving force for the supercooled regime. Furthermore, an extended Maxwell model is formulated to describe the effect of dynamic confinement of two phases on the multiple SMEs and shape recovery behaviors in the semi-crystalline SMPs. Finally, the effectiveness of the newly proposed model is verified using the experimental data reported in the literature. This study aims to provide a new methodology for the dynamic confinements and cooperative principles in the semi-crystalline SMP towards multiple SMEs