Sébastien Neukirch Institut Jean le Rond d'Alembert Centre National de la Recherche Scientifique Université Pierre et Marie Curie Paris, France tel: +33 1 44 27 72 61 fax : +33 1 44 27 52 59 e-mail: sebastien.neukirch (-atat-) upmc.fr

Drop-on-coilable-fibre systems exhibit negative stiffness events and transitions in coiling morphology

H. Elettro, F. Vollrath, A. Antkowiak, and S. Neukirch

Soft Matter, vol. 13 (2017) 5509-5517

Abstract : We investigate the mechanics of elastic fibres carrying liquid droplets. In such systems, buckling may localize inside the drop cavity if the fibre is thin enough. This so-called drop-on-coilable-fibre system exhibits a surprising liquid-like response under compression and a solid-like response under tension. Here we analyze this unconventional behavior in further detail and find theoretical, numerical and experimental evidence of negative stiffness events. We find that the first and main negative stiffness regime owes its existence to the transfer of capillary-stored energy into mechanical curvature energy. The following negative stiffness events are associated with changes in the coiling morphology of the fibre. Eventually coiling becomes tightly locked into an ordered phase where liquid and solid deformations coexist.

DOI: 10.1039/c7sm00368d

arXiv:1703.02614

Submitted: Feb. 21st, 2017

Referee: 1

The paper presents an original study of the mechanical properties of filaments spontaneously coiled around liquid droplets. This study follows previous works on “elasto-capillary” coiling recently conducted in different groups (eg the paper from Schulman et al. in Soft Matter).
In the first part of the paper, the authors propose to study in detail the buckling properties of such system. In particular they show that in contrast from standard super-critical Euler buckling of a rod, buckling may here be sub-critical. The force displacement curve indeed undergoes an undershoot as the fiber is progressively compressed. As a consequence, the system displays hysteresis in the vicinity of the buckling transition. The authors provide a theoretical description of this behavior in good agreement with experimental measurements.
In the second part, the authors focus on the structure. They nicely show how the initial non-organized coil undergoes a transition to an ordered state. I found this work original and interesting and I recommend its publication to Soft Matter. However I think that minor points should be clarified before publication.

In the introduction I found the paragraph related to Figure 2 a bit awkward. The authors indeed comment regimes displayed in a figure without presenting the figure before. It actually seems to me that this paragraph should come after the general background from the introduction (and not before a general introduction to coiling). I would instead base the introduction on Figure 1 and say that the aim of the present study is to understand the buckling properties of the structure. I would also present the outlines so that the reader know in advance that there will be a second part on the ordering of the coils.

In materials and methods, I would specify the role of possible twist, or at least say that this issue in negligible as will be justified later. It also seems that the effect of gravity is negligible but this could be clearly justified by giving the value of the “Bond” number $\rho g D^3/\gamma r$. In section 3.1, $D$ is defined at the same time as the diameter and as the length of the drop. Although both parameters are linked by the contact angle of the liquid on the fiber, the length of the drop seems to be the relevant definition. The caption of Figure 2 could also be improved with a description of the 3 cases a, b and c. One of the main points of the paper relies on equation 4. Although one can read in detail reference 31, it would be nice to have a qualitative explanation of this expression, which is far from obvious to me. From what I understand, it comes from the fact the drop tends to localize the buckling.

In section 4, the “rugby” (or American football) ball shape only works if the contact angle is high enough. In the case of a wetting liquid the longitudinal profile of the “onduloid” shape changes its curvature in the vicinity of the contact line and forms a cusp.

In Equation (8) I guess that V(n) is based on R(n)? I would also state before that the effect of twist is here negligible and will be described later. In Equation (11) the parameter $\eta$ is a bit obscure. Although the interested reader will get all the information in reference (23) a few words describing this quantity (and some order of magnitude) would be welcome. The notions of “writhe” and “link” are certainly standard in rod mechanics, but I think it would be worth describing these notions in a few words for the general reader of soft matter.

Referee: 2

The authors revisit the elegant drop-on-coilable-fiber system, introduced by them in a recent paper (Elettro et al. 2016), where a slender fiber can buckle/coil inside a liquid drop. In this current study, focus is given to the mechanical response (force versus end-shortening), with a special emphasis on the existence of regions of negative stiffness, the subcritical nature of the underlying transitions (with hysteresis), and the relation between coiling morphology (order versus disorder) and mechanical response.
The investigation combines technical experiments (with a remarkable level of precision/resolution, especially given the system’s small size), numerics and scaling arguments; all of which are in good agreement. A reduced model description rationalizes the undershoot of the tension in the fiber, the existence of both regions of positive/negative stiffness regions, and periodic oscillations in the force signal. It is noteworthy that the numerical simulations do *not* involve adjustable parameters, which make the agreement with experiments even the more impressive. A predictive quantitative description is also provided for the thresholds for activation and de-activation (with hysteresis) of coiling activity. In this hysteretic region, it is interesting that the fiber-drop system can be in a coiled state, or not, depending on the history of the preparation/loading.
This study and its results will surely be of great interest to the soft matter community, with an interest and potential applications that are broad. As such, I certainly recommend this paper for publication in Soft Matter. Still, prior to publication, I would encourage the authors to consider the following minor points:

* In the introduction, it would be helpful for the reader, to briefly review other instances of physical systems that exhibit negative stiffness. This will also help further highlight the originality of negative stiffness in the current fiber-drop system, with respect to other previous studies.

* Early in the paper (e.g. abstract, 2nd paragraph), the terms ‘liquid-like’ and ‘solid-like’ should be more thoroughly defined/explained, given that they are being used in a specific context that may confuse reader from other areas of soft matter research.

* In Section 2, descriptions such as “fairly reproducible”, “reasonably constant”, and “drifting slowly with time” should be made more quantitative.

* Page 3, column 2: “Consequently we globally adjust it so that the average value of the plateau tension corresponds to eq. (1).” To me, it is not totally clear how the experimental data is being adjusted/normalized to remove the effects of load drifting in the apparatus. Could the authors be more explicit in describing how the data was post-processed?

* In a few places throughout the manuscript, the authors make use of previously developed contributions. For example: in Page 3, column 1, reference is made to Ref. [31]; in Section 4, it is mentioned that the simulations were developed in Ref. [35]. As another example, I believe that the data in Fig. 3 of the current manuscript, was already presented in the PNAS (2016) paper (Fig. 4). There is plenty of novelty and new results/analyses in the current study but the manuscript would benefit in clarity by more detailed statements on what is new and what is being adapted or implemented from previous studies.

* Page 4, column 2. “Coiling activity” is classified as a binary quantity: either as being possible (1) or not possible (0). Was the experimental activity this clear cut, i.e. is the distinction sharp? Also, how is D being decreased in the data of Fig. 5?

* Page 6, column 1: How is the value of n^*=24 for the data in Fig. 10 being calculated? Still related to the data in Fig. 10, why does the theoretical prediction contain error bars? It would be helpful to specify what is the quantity that is providing the main source of uncertainty for the predictions. Finally, I appreciate that measuring the order-parameter S is challenging with the existing system, but I was not clear on how this quantity was actually being measured from the experiments. Can more details be provided?