Power-efficient low-temperature woven coiled fibre actuator for wearable applications

A fibre actuator that generates a large strain with high specific power represents a promising strategy to develop novel wearable devices and robotics. We propose a new coiled-fibre actuator based on highly drawn, hard linear low-density polyethylene (LLDPE) fibres. Driven by resistance heating, the actuator can be operated at temperatures as low as 60 °C and uses only 20% of the power consumed by previously coiled fibre actuators when generating 20 MPa of stress at 10% strain. In this temperature range, 1600 W kg−1 of specific work (8 times that of a skeletal muscle) at 69 MPa of tensile stress (230 times that of a skeletal muscle) with a work efficiency of 2% is achieved. The actuator generates strain as high as 23% at 90 °C. Given the low driving temperature, the actuator can be combined with common fabrics or stretchable conductive elastomers without thermal degradation, allowing for easy use in wearable systems. Nanostructural analysis implies that the lamellar crystals in drawn LLDPE fibres are weakly bridged with each other, which allows for easy deformation into compact helical shapes via twisting and the generation of large strain with high work efficiency.

As preparation for the derivation of work efficiency η, we first derive a formula between the displacement of coil length ∆L and deformation of the fibre surface caused by thermal strain of the fibre. This derivation is the same as that discussed in ref. [5], except that we reserve Δ ⁄ because it will be used to arrange the formula for η. Although the derived formula is considered only for the deformation of the fibre surface, we can roughly estimate how the thermal strain affects ∆L and η.

Derivation of formula between ∆L and thermal expansion of fibre surface.
Fibre bias angle is defined by considering the twisted angle of the fibre surface with diameter d, as indicated in Fig. S4: where n, λ, and l are the number of twists around the fibre axis, the fibre length before twisting, and the fibre length after twisting, respectively. From the geometry shown in Fig. S3, λ is denoted as 2 = 2 + 2 2 2 (S2).
By ignoring the second orders of tiny variations and substituting equation (S1) and (S2), we obtain a formula for ∆n/n, which is also derived in ref. [5]: For comparison, equation (S4) can be converted to the formula denoted in ref.
[5] by replacing with 2 ⁄ − . When this twisted fibre is coiled, the total twist T applied to the fibre is denoted as When rotation at the end of the twisted fibres is held to keep them from untwisting, n and N are assumed to be constant and the change in total twist is expressed using equations (S5) and (S6): We then arrange this equation for ∆L by ignoring the second orders of tiny changes and substituting equation (S6): Next, when the change in the number of twists of the fibre is assumed to be coupled with the change in the number of the twists of the coiled fibre, ∆T/T is denoted as

From this equation, equation (S8) is derived as
where we assume that Δ ⁄ ≅ 0 and = 45°, as in [5]; ∆L is derived as This equation clarifies that negative thermal expansion of the fibre length Δ ⁄ and positive thermal expansion of the fibre diameter Δ ⁄ contribute to the large contraction of the coiled fibre actuator. The assumption of Δ ⁄ ≅ 0 implies that the deformation caused by thermal expansion is almost converted into the untwisting motion of the twisted fibre.
1.2 Work efficiency of coiled fibre actuator. Because equation (S10) is not explicitly affected by tensile stress, the work efficiency of the coiled fibre actuator increases with increasing tensile stress. However, the tensile stress P/πd 2 , where P is the load applied to the coiled fibre, is limited by the breakdown shear stress: where c is the spring index (c = D/d, where D is the mean diameter of the coil). The work of coiled fibre actuator A is defined as which reaches a maximum at a τ value close to the breakdown shear stress. Moreover, input energy W due to heating is defined as W = 2 4 Δ (S14).
The expected work efficiency η is calculated from η = A/W; that is, where we assume Δ ⁄ ≅ 0. Equation (S15) implies that the following factors contribute to high work efficiency: 1. Heat capacity, which reduces the heat capacity per volume of coiled fibre. 2. Thermal strain of fibre, which enlarges the anisotropic thermal expansion coefficient. 3. Shape of coiled fibre, which enlarges αc and reduces c. 4. Stiffness, which enlarges the breakdown shear stress. The coiled LLDPE fibre inherently has the features of a large thermal expansion coefficient, small c and large αc.
As discussed in the "Nanostructure analysis" section in the main text, these features originate from the nanostructure of the drawn LLDPE fibre, which contributes not only large thermal expansion but also affects the compact coil shape, thereby contributing to the high work efficiency. For comparison, a general stiff coil spring with a compact shape, e.g., c < 10, breaks at P values less than that described in equation (S12). The actual τ is known to be corrected to τ/κ, where κ is Wahl's empirical coefficient: In this case, because κ diverges when c is reduced to 1, the breakdown shear stress decreases with reducing c. For example, the breakdown shear stress decreases by approximately half at c = 2 (κ ≈ 2); the coils break at c = 1, even with no tensile stress applied. However, in the case of the coiled LLDPE fibre, whose diameter deforms into an ellipsoid because of applied stress, the shear stress applied to the inter-fibrils is relaxed; therefore, κ does not work well even for small c values. The apparent c of the LLDPE shown in Fig. 1b is 0.7. We confirmed that such extreme deformation does not occur for hard crystalline polymers except for LLDPE, as listed in [5]. HDPE and nylon6,6 broke down at c > 1 when large tensile stress was applied during twisting to yield a coiled fibre with small c. At this point, deformation of the diameter of the LLDPE fibre contributes to the high work efficiency of the coiled LLDPE fibre actuator. Figure S1. Thermal expansion coefficient of LLDPE fibres drawn in various drawing ratio λ/λ0 a) along the polar direction (α﬩) and b) along the fibre axis (α//), as measured by TMA. Figure S2. Drawn fibres fabricated from crystalline polymers. a) Thermal expansion of drawn fibres along the polar direction (α﬩) and b) along the fibre axis (α//), as measured by TMA.    Fig. 1d. The coil length is 10 mm. Table S2. Parameters used for estimation of work efficiency: conventional name, density of fibre , specific heat C, coil bias angle c, spring index c, operating temperature for actuation T, average linear thermal expansion coefficient // ave. in the operating temperature, average thermal expansion coefficient along fibre diameter  ave. in the operating temperature, tensile stress , estimated work efficiency cal., actual work efficiency exp. measured by experimental, and Figures referred to take exp.