Topology, Geometry, and Mechanics of Strongly Stretched and Twisted Filaments: Solenoids, Plectonemes, and Artificial Muscle Fibers

Soft elastic filaments that can be stretched, bent, and twisted exhibit a range of topologically and geometrically complex morphologies. Recently, a number of experiments have shown how to use these building blocks to create filament-based artificial muscles that use the conversion of writhe to extension or contraction, exposing the connection between topology, geometry, and mechanics. Here, we combine numerical simulations of soft elastic filaments that account for geometric nonlinearities and self-contact to map out the basic structures underlying artificial muscle fibers in a phase diagram that is a function of the extension and twist density. We then use ideas from computational topology to track the interconversion of link, twist, and writhe in these geometrically complex physical structures to explain the physical principles underlying artificial muscle fibers and provide guidelines for their design.

Figure 1

Geometry and topology of soft extensible filaments. (a) The filament centerline $\overline{\mathit{x}}(s,t)$ and local orthogonal frame $\{{\overline{\mathit{d}}}_1,{\overline{\mathit{d}}}_2,{\overline{\mathit{d}}}_3\}$. Shear and extension are defined by the vector $\mathit{\sigma }=\mathit{Q}{\overline{\mathit{x}}}_s-{\mathit{d}}_3$, while curvature and twist are defined by the vector $\mathbf{k}=\mathrm{vec}({\mathbf{Q}}^{\prime }{\mathbf{Q}}^{\mathrm{T}})$. (b) Writhe (Wr) equals the centerline’s average oriented self-crossing number computed in terms of the integral of the solid angle $d\mathrm{\Omega }$ determined by the infinitesimal centerline segments $\overline{\mathit{x}}(s_1)$ and $\overline{\mathit{x}}(s_2)$ (left-handed intersections are negative). (c) Twist (Tw) is the integral of the infinitesimal rotations $d\phi$ of the auxiliary curve $\overline{\mathit{a}}$ around ${\overline{\mathit{x}}}_s$. Here, the vector $\overline{\mathit{a}}$ traced out by ${\overline{\mathit{d}}}_1^{\perp }$ (i.e., the projection of ${\overline{\mathit{d}}}_1$ onto the normal-binormal plane) is shown in red while the curve associated with $-{\overline{\mathit{d}}}_1$ is shown in yellow (see Fig. 2). For a closed curve $\mathrm{Lk}=\mathrm{Tw}+\mathrm{Wr}$, where Lk (link) is the average oriented crossing number of $\overline{\mathit{x}}(s)$ with $\overline{\mathit{a}}(s)$.

Figure 2

Variation of the link, twist, and writhe as a function of the dimensionless twist density $\mathrm{\Phi }a$ ($a$ is the filament radius in the rest configuration). (a) To replicate the experimental observations in Ref. [13], we use a constant vertical load $F\approx 25F_C$ to produce a plectoneme ($F_C={\pi }^{2}EI/L_0^2$ is the buckling force for an inextensible rod; see Video S1 in the Supplemental Material [19] and Ref. [37]). (b) We repeat the simulation with $F\approx 90$$F_C$, stretching the filament to deformed length $L\approx 1.16L_0$. Increased stretching leads to an overall similar conversion of twist to writhe leading to tightly packed solenoidal structures (see Video S2 and the Supplemental Material [19] for plots of filament energy). Simulation settings (see Supplemental Material [19]): length $L_0=1\mathrm{m}$, $a=0.025L_0$, Young’s modulus $E=1\mathrm{MPa}$, shear modulus $G=2E/3$, $\mathit{S}=\mathrm{diag}(4GA/3,4GA/3,EA)\mathrm{N}$, $\mathit{B}=\mathrm{diag}(E{I}_1,E{I}_2,G{I}_3)\mathrm{N}{\mathrm{m}}^2$.

Figure 3

Morphological phase space. We simulate a filament prestretched to $L/L_0$ by a constant axial load and twisted by an angle $\mathrm{\Phi }a$, as in Fig. 2. By computing centerline relative alignment in neighboring loops, we find four phases: straight, plectoneme, solenoid, and plectoneme-solenoid combinations. Plectoneme alignment $\approx -1$, solenoid alignment $\approx 1$, and transition configuration alignments approach 0 (dark green). For $L/L_0\gtrsim 1.1$, solenoids are preferred. We expect ${\mathrm{\Phi }}_{\text{critical}}$ to scale linearly with $L/L_0$ at high extension, in agreement with this plot. Our results agree qualitatively with experiments [13] (shown in black dots; see Supplemental Material [19] for details). Hollow symbols denote plectoneme transitions, while solid points denote solenoid transitions; different shapes correspond to different filament parameters (Supplemental Material [19]). Simulation settings (Supplemental Material [19]): $L_0=1\mathrm{m}$, $a=0.025L_0$, $E=1\mathrm{MPa}$, $G=2E/3$, $\mathit{S}=\mathrm{diag}(4GA/3,4GA/3,EA)\mathrm{N}$, $\mathit{B}=\mathrm{diag}(E{I}_1,E{I}_2,G{I}_3)\mathrm{N}{\mathrm{m}}^2$.

Figure 4

Actuation of fiber-based artificial muscles that use the straight-solenoid transition. (a) Passive extension via solenoid loss. We displace the unclamped end ${\overline{\mathit{x}}}_n$ of a solenoid formed as in Fig. 3 with a load $\approx 99F_C$ a distance $\mathrm{\Delta }U$ in the direction ${\overline{\mathit{x}}}_n-{\overline{\mathit{x}}}_0$ and plot force on ${\overline{\mathit{x}}}_n$ qualitatively reproducing experiments [13] (inset; see Video S13 in the Supplemental Material [19]). Simulation settings (Supplemental Material): $L_0=1\mathrm{m}$, $a=0.025L_0$, $E=1\mathrm{MPa}$, $G=2E/3$, $\mathit{S}=\mathrm{diag}(4GA/3,4GA/3,EA)\mathrm{N}$, $\mathit{B}=\mathrm{diag}(E{I}_1,E{I}_2,G{I}_3)\mathrm{N}{\mathrm{m}}^2$. (b) Active work done by changing the temperature, which effectively increases filament rigidity, here simply modeled by increasing the Young’s modulus $E$ of the material. This leads to the formation of a solenoidal loop in a stretched twisted filament as in Fig. 3 with a load $\approx 116$ $F_C$ as $E_0$ increases gradually from 1 MPa, showing displacement $\mathrm{\Delta }U$ of ${\overline{\mathit{x}}}_n$ and increase in writhe $\mathrm{\Delta }\mathrm{Wr}$ from initial coil writhe, reproducing experiments [14] (inset; see Video S3 in the Supplemental Material [19]). (c) Contraction of twisted and coiled nylon polymer muscle formed by inserting twist and annealing into a helix. Filament radius doubles from initial radius $a_0=0.01\mathrm{m}$, while twist decreases to keep $a{k}_3$ constant. Numerical slope and onset of self-contact (shown as a point) agree closely with experimental results [14] (see Supplemental Material [19] for details). Beyond self-contact, radial growth pushes adjacent loops farther apart leading to helix elongation. Note that $\mathrm{\Delta }\mathrm{Tw}+\mathrm{\Delta }\mathrm{Wr}<0$ in the inset. Indeed, link escapes from the free boundary due to revolution of the free filament end point around the helix axis, reducing the number of loops in the helix (see Supplemental Material [19] Fig. S7 and Videos S4 and S5). Simulation settings (Supplemental Material [19]): $L_0=1\mathrm{m}$, $a=0.025L_0$, $E=30\mathrm{GPa}$, $G=2E/3$, $\mathit{S}=\mathrm{diag}(4GA/3,4GA/3,EA)\mathrm{N}$, $\mathit{B}=\mathrm{diag}(E{I}_1,E{I}_2,G{I}_3)\mathrm{N}{\mathrm{m}}^2$. Note that pitch $P$, $\alpha =100$, number of loops, and helix radius determine $L_0$.