Perversions with a twist

Perversions connecting two helices with symmetric handedness are a common occurrence in nature, for example in tendrils. These defects can be found in our day life decorating ribbon gifts or when plants use tendrils to attach to a support. Perversions arise when clamped elastic filaments coil into a helical shape but have to conserve zero overall twist. We investigate whether other types of perversions exist and if they display different properties. Here we show mathematically and experimentally that a continuous range of different perversions can exist and present different geometries. Experimentally, different perversions were generated using micro electrospun fibres. Our experimental results also confirm that these perversions behave differently upon release and adopt different final configurations. These results also demonstrate that it is possible to control on demand the formation and shape of microfilaments, in particular, of electrospun fibres by using ultraviolet light.


SM1: Videos Overview and Legends
For a clearer understanding of the phenomena reported in this work we display a set of gures from accompanying videos and respective video legends. SFig. 9: Release of a polymeric bre without UV irradiation. The bre bends due to gravity but shows no sign of intrinsic curvature.

SM2: Experimental procedure
In SFig. 10 the several steps involved in sample preparation are depicted.
SFig. 10: Schematic representation of the several experimental steps to produce polymeric bres with symmetric and antisymmetric perversions. 1 A solution of PU/PBDO dissolved in toluene is accelerated towards a suspended collector target consisting of two parallel metallic bars by action of an electric eld applied between the syringe tip and the rotating target. The gel point is achieved after the deposition of the bres. 2 To produce symmetric perversions, bres are irradiated during 24h on one side with UV light. To produce antisymmetric perversions, two cycles of UV irradiation are applied sequentially on each side; masks protect complementary regions of the bre from the irradiation. By the action of UV light in the presence of the air oxygen the double bonds of the PBDO were allowed to open and form an additional network on the top of the bre. 3 UVirradiated bres are swollen in toluene for 24h (Soxhlet extractor) to remove the sol fraction. 4 After being dried, bres are released at a controlled rate, with both ends constrained from rotating.

SM3: Simulations of perversions and relation to experiments
The model described in this paper provides a simple description of the dierent types of perversions. In order to provide additional evidence for the mechanism proposed to observe the dierent types of perversions we used LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) to model the dynamics of elastic laments.
Microlaments were formed using structures with cross-sections made of N w = 3 by N h = 3 beads along the width and height, respectively (see SFig. 11a). Beads were connected by two harmonic potentials, denoted a and b, with equilibrium bond distances, l a and l b , and illustrated in dierent colours in SFig. 11. For suciently unmatched bond distances (in our case, with l b < l a ), one side of the lament becomes stretched relatively to the other, and upon release the lament gains a helical shape with an intrinsic curvature K, as well discussed in Liu et al. To create perversions, laments were released starting with the initial length L = L a = (N − 1)l a . This means that in the initial conguration only b bonds were pre-stretched, which agrees with the experimental set-up since only one side of the lament is irradiated with UV light. During release, ends are approached at constant rate along the same direction and without allowing rotations. For suciently large mismatches between l a and l b , one or more symmetric perversions appear as discussed previously [?]. To obtain antisymmetric perversions, a and b bonds alternate positions as shown in SFig. 11c. The same release procedure is used. Simulations were performed with time steps of dt = 1 × 10 −3 in lj units.
In order to match the results obtained in simulations with the release experiments in Fig. 3, it was necessary to select three parameters as shown in Liu et al. [?]: the appropriate mismatch between l a and l b , the lament length L * and the cross-section width, w = √ 3/2(N w − 1)l a . According to elastic beam theory, the longitudinal strain at an arbitrary point on the cross section of two strips is given by: where L * = θ/K and θ are the length and angle of the curve, respectively. Mechanical equilibrium of forces and momenta requires: In a linear elastic material the axial stress is given by σ = E . In computational simulations we used h a = 2h and h b = h, thus equations 2 and 3 become: Neglecting the eect of the pre-strain on the width of bres, then w a = w b = w, and hence we can write: Solving for L * and θ, K = θ/L * can be obtained giving: Intrinsic curvature is controlled by the pre-strain χ = L a /L b −1 applied to one side of the simulated bre. SFig. 12 shows the variation of the curvature K with the pre-strain χ. Hence, for a given bre of length L * the number of loops can be selected by increasing or decreasing the curvature value matching experimental observations. SFig. 12: Curvature K as a function of pre-strain χ. Elastic laments were simulated by using pre-strained layers with χ = L a /L b − 1. Black markers represent the curvature obtained in numerical simulations. The red line corresponds to the prediction obtained from the linear model.