Self-sustained rolling of a thermally responsive rod on a hot surface
Introduction
Self-sustained motions of different structures powered by steady stimuli have recently attracted much attention [1], [2], [3], [4], [5], [6], [7], [8], [9]. The main motivations of studying self-sustainable motion at least include: 1. most dynamic processes in biological systems are self-sustained and autonomous without needing external control; 2. a device with self-sustained motion may accomplish various tasks such as harvesting energy [10], [11], [12], locomotion [13], [14], [15] and transportation [16], [17], [18] without complex control, which, therefore, enables compact design and possible reduction of energy consumption.
Representative examples of self-sustained motion reported in the literature are self-oscillating gel driven by BZ reaction [19], continuous vibration [20], [21], torsion [21], rolling [22], [23] and buckling [24], [25] of various structures resulting from self-shading effect, periodic swelling–shrinking of gels caused by the coupling between large deformation, chemical reaction and solvent diffusion [26], [27], self-excited motion of a volatile drop on a swellable sheet [28]. One key feature of those structures is their capability of continuously harvesting energy from steady environment. Therefore, those structures are all made up of responsive materials which can convert energy from one form to another such as heat/optical energy to elastic energy or chemical energy to kinetic energy.
Compared to the intensive experimental work of the structures with self-sustainable motion, theoretical modeling has been very limited [2], [27], [29], [30], [31], [32]. The main challenges for the modeling include the interplay between different fields and the intrinsic nonlinearity of the system.
In this paper, we try to construct a theoretical model for autonomous rolling of a responsive rod on a flat surface, recently reported by us and other researchers [16], [17], [18], [29]. It was observed in the experiment that a polymer rod of cylindrical shape can automatically roll on a flat and hot surface at a homogeneous temperature. During the steady rolling, the shape of the cylinder remains almost unchanged. Though it has been clear that the driving force for such steady rolling mainly originates from the inhomogeneous thermal expansion of the rod and certain semi-quantitative analyses including simple scaling analysis have been conducted, a comprehensive quantitative modeling of such autonomous rolling phenomenon has not yet been available. In this article, we first compute the deformation of the rod during its steady rolling, and then calculate the moment driving the rod to roll. The rolling velocity of the rod is then obtained by the equilibrium condition. We find that for a certain range of the system parameters, the rod can be in a bistable state. The effects of the viscoelasticity of the rod on its rolling are also discussed in the end.
Section snippets
Self-sustained rolling of a heat-responsive rod on a hot surface
We first study heat-driven self-sustained rolling of an elastic cylindrical rod on a hot surface. When an initially straight rod rolls continuously on the hot surface, the rod spontaneously bends in the plane, and the distribution of the supporting force provided by the hot surface to the rod becomes inhomogeneous, which results in a net driving moment applied on the rod as shown in Fig. 1(a) and (b). In the following, we first provide the temperature field in the rod during its steady rolling,
Autonomous rolling of a viscoelastic rod on a hot surface
Viscoelasticity of polymer can be significant for certain ranges of frequency. It has been discovered that the viscoelasticity of the material can be a main mechanical dissipation mechanism (as compared to friction) during the self-sustained rotation of a Nylon torus [16], [17]. In this section, we further consider the viscoelasticity of the rod as sketched in Fig. 6 in the formulation. The temperature distribution in Eq. (1), the thermal strain in Eq. (3), the strain distribution in a cross
Conclusions
In this paper, we have established a model for a thermally responsive rod rolling on a hot surface. Although the system is extremely simple, we found that the phenomenon associated with it can be very rich. Due to the inhomogeneous thermal expansion in the rod and the coupling between the heat transfer and the rolling/deformation of the rod, self-sustained rolling of the rod can be achieved. In the model, we have computed the deformation and stress distribution in the rod undergoing steady
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
KL acknowledges support from the 2019 Outstanding Talents Cultivation Project of Universities in Anhui under Grant No. gxyqZD2019056, and the Chinese Natural Science Foundation under Grant No. 11402001. SC acknowledges the support from the National Science Foundation, USA through Grant No. CMMI-1554212.
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