Discharging dynamics of topological batteries

Topological constraints have long been known to provide efficient mechanisms for localizing and storing energy across a range of length scales, from knots in DNA to turbulent plasmas. Despite recent theoretical and experimental progress on the preparation of topological states, the role of topology in the discharging dynamics is not well understood. Here, we investigate robust topological energy release protocols in two archetypal soft systems through simulations of 238 knotted elastic fibers and 3D liquid crystals across a range of different topologies. By breaking the elastic fiber or switching the liquid crystal surface anchoring, such topological batteries can perform mechanical work or drive fluid flows. Our study reveals topologically resonant states for which energy release becomes superslow or superfast. Owing to their intrinsic stability we expect such tunable topological batteries to have broad applications to storage and directed release of energy in soft matter.

Topological constraints have long been known to provide efficient mechanisms for localizing and storing energy across a range of length scales, from knots in DNA to turbulent plasmas. Despite recent theoretical and experimental progress on the preparation of topological states, the role of topology in the discharging dynamics is not well understood. Here, we investigate robust topological energy release protocols in two archetypal soft systems through simulations of 238 knotted elastic fibers and 3D liquid crystals across a range of different topologies. By breaking the elastic fiber or switching the liquid crystal surface anchoring, such topological batteries can perform mechanical work or drive fluid flows. Our study reveals topologically resonant states for which energy release becomes superslow or superfast. Owing to their intrinsic stability we expect such tunable topological batteries to have broad applications to storage and directed release of energy in soft matter.
Topological protection provides a robust means for storing and controlling energy, an effect widely used in a variety of biological and physical systems [1][2][3][4][5][6]. On small scales, knotted topologies play important functional roles [7] in the behavior of tangled DNA, proteins and polymers [8][9][10][11][12]. In continuum systems, foundational work on topology has revealed the origin of helicity conservation in classical [2,13] and complex fluids [14], and the dynamo effect in turbulent plasmas [15,16]. Structured continua such as liquid crystal fluids are a rich source of emergent topological phenomena, from interacting defects [17,18] to knotted field configurations [3,19,20]. Recent experimental advances in mechanical lattices [21,22] and soft robotics [23], bring the question of topologically tunable designer materials into the experimentally accessible realm. Although the study of topological modes [24] has dramatically improved our understanding of soft matter, harnessing topology to perform useful work such as driving flows [17,25,26] continues to present fundamental challenges.
Here we study how topology affects energy release dynamics in knotted elastic fibers [27,28] and nematic liquid crystals [19], demonstrating two distinct realizations of a topological battery. In both cases, topology mediated buckling and instability phenomena underlie the discharging rates and functional capabilities of the batteries. Knotted filaments present an intuitive mechanical realization of a topological battery: By initializing closed elastic loops in tight knotted states [27,29,30] with varying twist, energy may be stored robustly (Fig 1). Cutting the knot at the point of maximum stress results in a controlled topology dependent energy discharge. Transferring this idea to liquid crystals, nematic batteries can be realized by imprinting topologically non-trivial field configurations, which are energetically stabilized through the anchoring of the nematic orientational director field to a colloidal surface [3,19,20]. Energy is then released by optically changing the anchoring profile. Using 3D numerical simulations, we explore the energy discharge dynamics in elastic fibers and liquid crystals across a range of different knot types and topological charges. For both systems, we find special topologically resonant states that are characterized by superslow or superfast energy release, exemplifying control of discharging dynamics through topology.
Tying a knot in an elastic fiber and fusing together the free ends produces our first example of a topological battery. By twisting the free ends before fusing, knotted batteries can be studied along the two axes of knot type and twist density (Fig 1a, 5). We describe this system using the Kirchhoff model for elastic rods together with a contact condition forbidding self intersections [28,31]. The fibers have natural length L, radius h, and circular cross section with moment of inertia I = πh 4 /4 and cross sectional area A = πh 2 . The state of the fiber is defined by its centerline curve x(s), and an orthonormal material frame, {d 1 (s), d 2 (s), d 3 (s)} constrained by x × d 3 = 0, where s is the arc length parameter of the unstretched fiber. The elastic energy depends on the geometric curvature κ, twist density θ = d 1 · d 2 , and stretch |x | where E b is the bending modulus, E = γE b is the Young's modulus and ν = α −1 − 1 is the Poission's ratio. To describe the dynamics, we define Ω = ḋ 1 · d 2 d 3 . The governing equations follow from extremizing (1) and assuming viscous damping forces dominate inertial terms Charging knotted filaments with twist results in a series of topology mediated buckling instabilities (Fig 1a). The twist in the buckled state is typically lower than the initial twist supplied to the pre-buckled state [32]. The evolution of the total twist in the fiber, T w, follows from relating spatial and time derivatives of the frame When the fiber is closed, the first term on the right hand side vanishes. Although initializing a fiber with a specific twist requires a time dependent torque application protocol, the amount of buckling provides an indicator of the twist charge within the battery, for a given knot type. For example, twist produces initial battery states with varying post-buckled geometries (Fig 1a) and energy densities (Fig 1b). Additionally, twist is sensitive to certain underlying topological properties. In our simulations, chiral knots generally exhibit handedness-dependent buckling (Fig 1a, top), whereas achiral knots tend to buckle independently of twist handedness (Fig 1a, bottom). As knotted mechanical batteries unravel, the competition between bending and twisting energies reveals a landscape of topological resonances, where certain initial states lead to super slow energy discharge. After being broken at the point of maximum stress, different transient, metastable states occur (Fig 2a). The discharge dynamics separate into a bending dominated regime and a twist dominated regime, as evident from the initial state of the battery (Fig 2b). The crossover between the two regimes corresponds to the scaling of strains associated with twisting and bending. From (1), the characteristic strains for tightly knotted configurations are ε b ∼ hκ ∼ 1 for bending and ε tw ∼ hθ ∼ hT w/L for twisting. As a result, twist dominates the discharge dynamics at high twist densities with θ h > 1 (Fig 2c-e). Since the effects of varying knot type are more naturally related to the bending energy, twist can be thought of as washing out topology. Indeed, at high twist, θ h > 1, the batteries discharge quickly, independent of knot type (Fig 2d,e). By contrast, at low twist, knot topology essentially determines the discharging dynamics (Fig 2e). In particular, select knots exhibit extremely long discharge times (Fig 2d,e). We can explain these slow topological resonances by considering the mechanisms by which knots release twist and bending energy. Bending forces point in the d 3 direction, which lies in the fiber's local osculating plane, spanned by d 3 and d 3 . From (3), twist changes whenḋ 3 has a component in the d 3 × d 3 direction; twist relaxation therefore pushes the fiber out of plane. The topologically resonant slow knots can thus be thought of as maximally non-planar and therefore self-confining.
Nematic fluids containing spherical colloidal particles [19] enable another construction of a topological battery. The effective topological charge of the particle is determined by its prescribed liquid crystal anchoring profile, which may be optically changed [33,34] to release energy. We model the nematic fluid with a tensor order parameter Q. The top eigenvalue and eigenvector define the degree of order S and the director n. The free energy density describes the effective elastic behavior of Q-field with one elastic constant L, and A, B, and C are parameters that tune the nematic phase behavior. The model has intrinsic length scale ξ N = L/(A + BS eq + 9 2 CS 2 eq ) and time scale τ N = ξ 2 N /ΓL, where Γ is the rotational viscosity parameter. Our 3D simulations (SI) use the Beris-Edwards formulation of the Q-tensor dynamics [35,36].
Nematic topological batteries are charged by imposing a topologically non-trivial director field anchoring profile on the surface of a spherical particle given by n q (θ, φ) = (sin θ cos qφ, sin θ sin qφ, cos θ). Here, θ and φ are the polar and azimuthal angles, and the integer q sets the effective topological charge of the particle [37]. The system is placed in a cell with strong perpendicular anchoring of the nematic director on the top and bottom boundary (Fig 3a). The bulk nematic structure counteracts the topological charge of the particle by forming a defect ring [38], known as a Saturn ring for q = 1. The director field in the cross-sections of the defect ring varies continuously between +1/2, −1/2, and twisted profiles (Fig 3a). For small q, the defect ring lies on the equato-rial plane of the particle (Fig 3a,b). As q increases, the gradients of the director field increase, eventually causing the defect loop to buckle at a critical q c depending on colloid size. The resulting oscillatory defect profiles have wavelength matching that of the anchoring profile, and amplitude increasing with q (Fig 3b). The buckling transition is characterized by the average horizontal director field, n , which is zero for q < q c and positive for q ≥ q c (Fig 3c,d). The total free energy, F , however, strictly increases with q (Fig 3e). Topology thus determines the distribution of energy throughout different modes, along with the total energy.
Energy partitioning within the charged nematic batteries causes topological resonances in the discharge dynamics. The relaxation process is triggered by switching off the anchoring on the spherical particle, allowing the director field to take any orientation at its surface  ( Fig 4a). The initial reorientation of the director field is accompanied by the shrinking of the defect ring un-til it is annihilated on the particle surface. The continued reorientation of the director field towards the equi-librium homogeneous vertical structure drives a complex flow pattern (Fig 4a). During the typical director relaxation time, τ d , the energy decreases by several orders of magnitude (Fig 4b). While the initial energy release rate increases with initial battery energy and topological charge q (Fig 4c), the asymptotic release rates show characteristic resonance peaks (Fig 4d). In particular, we observe superfast discharge rates for q close to q c (Fig 4d). The return to slow relaxation for q q c is caused by the formation of the long-scale deformation mode with n > 0 (Fig 3c,d), which obstructs fast discharge.
In both studied systems, topology and mechanics interact to produce long-lived states that are central to the observed topological resonances. For elastic batteries, these states are the intermediate knots that can form as the fiber unties. At small or zero twist, these configurations trap bending energy, leading to long relaxation times. Similarly, at sufficiently high topological charge, nematic batteries can store energy in the slow n mode. Over long timescales, these modes make the pivotal contribution to the overall discharge rate. More generally, this principle of topologically activated slow modes can provide a conceptual framework for understanding topological resonances in other soft matter settings.
To conclude, knotted elastic fibers and topologically charged nematic fluids provide prime demonstrations of topological batteries. Both systems permit controlled triggering of energy release via fracture and photoalignment techniques. Our above analysis shows how elastic and nematic batteries can be topologically optimized to achieve slow or fast discharge. Owing to the inherent robustness of topological structures, the above ideas translate to a wide range of scales. The energy stored in topological batteries may be harnessed to drive flows or perform mechanical or electrical work, thus opening an avenue for topological control of soft systems.

Knotted elastic fibers
The full set of 238 different initial knotted states is given in Fig 5.

Nematodynamic model
We model the nematic system with a tensor order parameter Q. The top eigenvalue and eigenvector correspond to the degree of order S and the director n. The free energy, F = f dV , is given by Derivatives of Q ij in the free energy density describe the effective elastic behavior of the director field, where L is the elastic constant, and A, B, and C are parameters that tune the nematic phase behavior. Our 3D nematodynamic simulations [36] are based on the Beris-Edwards formulation of the Q-tensor dynamics [35] ( where u is fluid velocity, χ is the alignment parameter, and Γ is the rotational viscosity coefficient. The molecular field H ij drives the system towards equilibrium, H ij = −(δF /δQ ij ) tr , where (.) tr denotes the traceless part. D ij and Ω ij are the symmetric and antisymmetric parts of the velocity gradient tensor W ij = ∂ i u j . We model the nematic as an incompressible fluid with stress tensor where p is the fluid pressure and η is the isotropic viscosity. Defining the intrinsic length scale ξ N = L/(A + BS eq + 9 2 CS 2 eq ) and time scale τ N = ξ 2 N /ΓL, the particle radius is set to R = 52.5ξ N and the director field has relaxation timescale τ d = R 2 /ΓL = 2760τ N . The phase parameters are set to A = −0.19 L/ξ 2 N , B = −2.34 L/ξ 2 N and C = 1.91 L/ξ 2 N , The nematic is in the alignment regime χ = 1, with isotropic viscosity η = 1.38 ξ 2 N /Lτ N . Our 3D simulations were performed on a 200 × 200 × 150 mesh with grid resolution ∆x = 1.5 ξ N and time resolution ∆t = 0.057 τ N .