Session S34: Mechanics and Self-Assembly of Knots and Tangles: From Knitted Fabrics to Polymer Melts

Mechanics of Head-Carrying Textile Rings

Throughout human history, the practice of using textile rings to facilitate the transport of goods (e.g. pots of water; harvested produce) atop the head has surfaced in cultures spanning vast time scales and geographic length scales. This technique is still used today predominantly by women in rural, agricultural communities. Textile rings are often fabricated by bending, twisting, and wrapping readily available materials (e.g. natural fibers in a plain weave), to form structures that distribute complex loads and conform to arbitrarily shaped objects. In this work, we study the mechanics of textile rings constructed from different fabrics and different wrapping strategies. We examine the effects of material (i.e. weaving pattern, fabric composition) and design (i.e. twists, folds) on the stiffness, conformability, and behavior of textile rings under quasistatic and dynamic loading conditions that mimic uneven terrain. We find that ring stiffness increases with the number of twists resulting in more rigid designs with low conformability which may be favorable for transporting solid objects with a curved base (e.g. baskets, containers). Softer ring designs exhibit increased conformability, which may be preferable for carrying and stabilizing non-unform loads (e.g. sugar cane, wood). These findings provide a mechanistic understanding of head-carrying textile rings, and illustrate the mechanical intuition and human ingenuity that enable women to carry up to 60% of their body weight atop their head.   

Self-Entagled States of Linear Catenanes in Poor Solvent: Knots and Threadings

We used molecular dynamics simulations to explore self-entanglements in collapsed linear catenanes, unveiling two types of topologically complex states. Firstly, we observed numerous and long-lasting knots in the catenane backbone. However, when compared to traditional polymers, these knots were notably less probable in catenanes. Secondly, we identified complex topological states that lack counterparts in polymers. In these states, concatenated rings were threaded by other nearby or distant rings, sliding through them. Unlike knots, these threaded states could disentangle by becoming fully tightened. We employed a thermodynamic and microscopic analysis of these entagled states, rationalizing their persistence and disentanglement modes.   

Tying Together Yarn Compression and Knit Fabric Jamming

The mechanical response of knitted fabrics is typically characterized by a soft, bending-dominated linear elastic response at low strain, followed by strain-stiffening due to yarn compression and contacts. However, when a fabric is very tightly knit, we see an additional response in the very low strain region where the modulus starts out stiff and then softens. This low-strain high-modulus behavior in knitted fabrics is known as jamming. Using computational simulations of stockinette fabric, we present a new energy marker for jammed knit fabrics that shows that the rest state of the fabric is not a minimum contact energy state. Instead, the stitches reach a contact energy minima when the fabric is stretched and leaves the jammed regime. This contact energy marker shows how the jammed regime changes as a function of yarn compressibility; yarns close to the incompressible limit show a sharp jamming transition while yarns further from the in-compressible limit have a soft transition. *This work is funded by NSF 1847172   

Topological entanglement in polymers

Predicting polymer material properties based on chemical composition, density, crosslinking, or architecture requires bridging the gap between the properties of a single chain and those of a collection of chains. At the core of models of polymer dynamics is entanglement, understanding of which remains elusive. In this talk, we will see how new advances in mathematical topology lead to new tools that can be used to measure topological entanglement in polymers of varying architectures (e.g., ring or linear). We will use Molecular Dynamics simulations of polymer melts to examine how the mathematical topology of polymers varies with molecular weight and stiffness of the chains, and how these can capture refined information about the conformation of the chains. Next we will show that the topological entanglement captured by mathematical methods indeed captures polymer entanglement effects in polymer melts and solutions. We will demonstrate this by using topology to predict a critical lengthscale in entangled polymers, the entanglement length, which is in agreement with experimental estimates. All of these results point to the advantages of topological parameters in analyzing polymers and the emergent topological framework of modeling and prediction of material properties that arises.   

Twisted Structure of Multifilament Bundles and Sheets

We will report on an experimental investigation of the shapes and strength of elastic multifilament bundles and sheets as a function of applied end-to-end twists in the limit of low tension. Two complementary geometries will be discussed. One consists of a set of filaments that are mounted parallel on linear clamps and then twisted, and compared with a twisted sheet with the same aspect ratio. In the other, the filaments are mounted corresponding to a cylindrical surface in circular end-clamps and then twisted axially to compare with a sheet which is rolled into a cylinder and then twisted. While the filaments deform corresponding to a hyperbolic hyperboloid before contact, deviations are observed from these shapes in the case of sheets even at vanishing twist. In both cases, a bundle forms at the center as the filaments come in contact with a cross section which evolves with an applied twist. We will discuss the analysis of the shapes and torques required in terms of the Geometric yarn model [1], and by generalizing the energetics derived for twisted two filament bundles [2] noting the important contributions of stretching, bending, and contact, besides twist.

[1]: Tensional twist-folding of sheets into multilayered scrolled yarns, J Chopin, A Kudrolli - Science Advances, 8, eabi8818 (2022).

[2]: Energetics of twisted elastic filament pairs, J Chopin, A Biswas, and A Kudrolli, arXiv preprint arXiv:2309.11344 (2023).   

Capturing corrugation, folding, and multi-stability in knit materials with an elastic continuum finite element model

Weft knitting is a method of fabric production in which a continuous horizontal yarn is interlooped with itself. Each loop is connected to its neighboring loops, forming a fabric. Notably, the individual stitches are asymmetric through the plane, which introduces internal stresses in the finished fabric and results in curling at the boundaries. Additionally, during manufacturing, stitches can either be added in the front-facing or back-facing directions, creating localized regions of opposite curvature. By designing where we use both types of stitches, we can realize a diverse array of resultant fabrics, some of which exhibit self-folding or multistability. While this complex behavior arises from interactions at the yarn level, continuum finite element simulations with well-known material models can accurately capture the experimentally observed behaviors, including out-of-plane curvature and a multi-welled energy landscape. We use this model to explore the vast geometric space, and expect it to allow us to design fabrics with particular target properties.   

Exploring the Design Space of Polycatenated Metamaterials

Polycatenation, a phenomenon originated from polymer chemistry, delineates the process of conjoining multiple enclosed molecular loops or knots to form a more expansive and complex ensemble. These polycatenated configurations, due to their unique topologies, inherently exhibit variant degrees of freedom among their constituent units, all the while preserving their overarching polyknotted structures intact. Translating this molecular paradigm to the realm of mechanical metamaterials presents a transformative opportunity for the metamaterial design. In this study, we report a rational design platform based on topologically interlocking wireframes, allowing the creation of polycatenated metamaterials with tailor-made global structural pliancy. Our comprehensive investigations show that these polycatenated metamaterials can be engineered to display highly customizable mechanical properties, determined by their topological and geometric designs, including fluid-like to solid-like phase transitions. The introduction of polycatenated structures adds a new dimension in the design of architected materials and thereby offers new functionalities for application in impact protection, wearable materials and soft robotics.   

Geometry and mechanics of densely-packed helical filaments

Filament assemblies are of profound importance in structural materials and are ubiquitous in both engineered (cables, ropes, textiles) and natural systems (biopolymers, plant tendrils, muscles and tendons), spanning a huge range of length scales. In densely-packed filaments, structure and mechanics derive from the interplay of both the inherent flexibility of backbones and the local and non-local constraints of contact. Here, we consider a elementary geometry of helical close-packing of a single filament, extending the pioneering study of Pryzbyl and Pieranski (EPJE 2001) on the helical close-packing of the ideal rope model. In particular, inspired by recent experiments on self-coiling, anisotropic microfilaments, we study the helical close-packing of anisotropic rods with eccentric (elliptical) cross-sections. We show that close-packed states of anisotropic rods exhibit a strong sensitivity to tilt of cross-section, leading to a rich spectrum of structural transitions with increasing eccentricity. We compare these in terms of the capillary packing fraction and show that optimally dense states become increasingly helical with rod anisotropy. Next we consider a minimal mechanical model for capillary confinement of elastic filaments, and determine the packing fraction vs. pressure equation of states, which are marked by mechanically driven transitions between distinct states of helical packing.   

Programming Knitted ExoSkins to Assist with Knee Joint Motion

Knitted fabrics, depending on how their constituent stitches are patterned, exhibit emergent elasticity with distinct anisotropies. By taking classic knitted patterns and manipulating their unique directed responses, one can construct specialized wearable fabrics, exoskins, that conform to the wearer and possess bespoke properties that assist or augment motion. In this study, we designed programmed knitted exoskins that can provide resistive torques to resist knee joint flexion (i.e., body weight support) and compared their capabilities with off the shelf knee braces currently in market. We investigated the mechanics of the knitted prototypes on a test rig that mimics knee joint motion and extracted the torsional stiffness experienced under motion. By studying how user-specific exoskins can be designed and programmed for a wearer's specific needs (for instance overcoming an injury or combating the effects of aging), we can steer away from bulky and cumbersome exoskeletons commonly made with heavy and uncomfortable components.   

Braiding, twisting, and weaving microscale fibers using capillary forces

Materials that consist of braided or woven fibers have properties that make them useful across a wide range of applications. These properties are influenced by the diameter of the fibers as well as the topology of the braids or weaves they form. In this talk, we will discuss how to braid microscale fibers into specific topologies using 3D-printed machines that harness capillary forces [1]. The machines can manipulate microscale fibers into many different topologies, including braids, twists, and weaves. We will show how the topology of the material is controlled by the pattern of vertical motion of the machine and why capillary forces are a useful means to manipulate micrometer-scale fibers.

Funding: NSF through the Harvard University Materials Research Science and Engineering Center, grant DMR-2011754

References:

 [1] Zeng, C., Faaborg, M.W., Sherif, A., Falk, M.J., Hajian, R., Xiao, M., Hartig, K., Bar-Sinai, Y., Brenner, M.P., Manoharan, V.N. 3D-printed machines that manipulate microscopic objects using capillary forces. Nature (2022)   

Knotting semi-flexible filaments with centrifugal forces

The sedimentation of elastic filaments is a classical problem in fluid mechanics. In this context, hydrodynamic phenomena play a key role in the filaments' deformation, rotation, and translation while competing with elastic forces and thermal motion. Consequently, the interplay between gravitational, elastic, and thermal effects determines the filament's settling dynamics, which may follow intriguing complex paths or sediment in stable configurations. From Brownian Dynamics numerical simulations, we found that at high centrifugal accelerations, the semi-flexible filaments may acquire long-stable configurations characterized by a knotted leading head and a trailing stretched tail, revealing that stable configurations can also be intriguing. The stability of this configuration is conferred by the tightness of the knotted structure due to a hydrodynamically induced tension in the filament. We conducted a systematic study to understand the role of centrifugal acceleration and filament stiffness on the topological structure of the knots. Although further investigation is required, such long-live metastable knotted structures might be strongly associated with the anomalous decrease of the sedimentation factor in long polymers at high rotational speeds commonly observed in ultracentrifuge systems.