# Quantitative Analysis of the Complex Time Evolution of a Camphor Boat
## 1 Introduction  
Self-motion of a camphor piece on the water surface is a spectacular example of a system in which the chemical energy of surface active molecules is transformed into kinetic energy of a floating object [1; 2; 3; 4]. The phenomenon of camphor \"self-motion\" has been attracting scientific attention for almost 200 years [3; 4; 5; 6]. In normal conditions, camphor evaporation is fast, and a piece of camphor located at a water surface can be regarded as an open system with a continuous flux of molecules from the source to the air through a surface layer. Camphor is amphiphilic, and the water surface tension is a decreasing function of the camphor surface concentration that generates Marangoni flows around the piece. An asymmetric profile of surface tension around a camphor piece generated, for example, by an inhomogeneous camphor concentration profile can produce a force on the order of 10 uN [7], which is sufficient to drive the piece. A camphor boat is made of a boat-shaped float with a piece of camphor located in the stern area. As a result of asymmetric camphor location, the directed motion appears [3; 8; 9; 10; 11; 12]. It is hard to tell where and when the first camphor boats were made, but they were sold as toys in Edo-period Japan. Typical speeds of toy camphor boats are in the order of a few cm/s. For a long time, researchers who investigated camphor boats were just satisfied with the observation of their motion. In the paper [13], the authors proposed a very simplemathematical model of the boat motion. A pioneering study on the complexity of the boat's motion was published in 2010 by [PERSON] and collaborators [14], who recorded the boat speed as a function of time for different distances \(d\) between the location of a camphor piece driving the boat and the stern (cf. Figure 1). Their results can be summarized as follows: when the camphor piece was located close to the stern, a continuous motion with a constant speed was observed. If the piece was distant from the stern, then the boat motion was pulsating, and it was characterized by narrow intervals of high-speed motion separated by long time intervals when the boat did not move. The experiments reported in [14] were performed with boats freely moving inside a Petri dish (155 mm), which means that the boat collided with dish walls a few seconds after it was placed on the water surface. In our recent experiments [15], we considered a boat cut as a piece of U-shaped angle aluminum profile. The profile width was 4 mm, and the height of the profile was 2 mm. We used such a profile to localize camphor surface dissipation to the boat ends only. In experiments, we used boats of lengths 10 and 15 mm. The boats were propelled by a camphor cylinder or cuboid, which were made in a pill maker. The camphor source was located at the distance \(d\) from the stern, where \(d\) was the experiment's control parameter. An example of 10 mm long and 4 mm wide boat with a 4 mm \(\times\) 4 mm square pill below is illustrated in Figure 1b. We used the boats that could rotate along a circle with the radius of 70 mm, whose center was set so that it met the center of the Petri dish (cf. Figure 1). The boat trajectory was one-dimensional, and we were able to evaluate the motion at stationary conditions for more than 1 h without collisions on the wall. Our experiments confirmed both types of previously reported motion: continuous when the camphor source was located at the boat edge and pulsating if the pill was close to the boat center. Moreover, at the intermediate distances between the pill and the boat edge, we observed a new type of motion characterized by an oscillating speed always larger than zero [15]. Videos illustrating continuous, oscillating, and pulsating motion have been included as supplementary material to this paper. We used a one-dimensional model of boat motion introduced in [14] and adjusted its dimensionless parameters such that all three types of motion mentioned above were reproduced in numerical simulations.