The book gives an overview of the self-propelled motion of chemical objects far from their thermodynamic equilibrium at various spatial scales and its applications. The book will discuss theoretical aspects, the characteristics of the motion, and design procedures of such systems from the viewpoint of nonlinear dynamics. The book is suitable for graduate students and researchers interested in physical and theoretical chemistry as well as soft matter.
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Theoretical and Experimental Design of Self-propelled Objects Based on Nonlinearity
S. NAKATA AND M. NAGAYAMA,
In this chapter, we explain the coupling of two phenomena, i.e. self-propelled motion and nonlinear phenomena. Several types of self-propelled objects have been observed to move a material or themselves in a small space, like bacteria, having the function of taxis. However, most self-propelled objects exhibit random or unidirectional motion without an external force, or the direction of the motion is controlled by an external force (e.g. electromagnetic force or light irradiation). This means that the autonomy of self- propelled objects is clearly lower than that of bacteria, because bacteria can characteristically change their behaviour according to internal and external conditions. However, oscillation, synchronization, pattern formation and bifurcation are typical nonlinear phenomena that are often observed in living organisms and nature. Previous experimental and numerical studies aiming to clarify the mechanisms of nonlinear phenomena suggest novel insights into their design and control. Thus, if one can introduce nonlinearity into self-propelled objects, their autonomy is enhanced, i.e. they can exhibit characteristic motions, such as oscillatory and synchronized motion, while sensing the environment.
In this chapter, we explain how novel self-propelled systems, like bacterial motion, can be designed using simple experimental systems considering nonlinear science and chemical information (e.g. chemical structure and reaction–diffusion dynamics), as shown in Figure 1.1.
1.2 Camphor Boat Driven by the Difference in Surface Tension
In this section, we introduce a camphor boat in which the driving force is the difference in surface tension as a simple self-propelled object, as illustrated in Figure 1.2. The camphor boat is prepared by gluing a camphor particle to a plastic plate. If only the plastic plate is floated on water, it doesn't move because the surface tension is balanced around the plate. If the camphor boat is floated on water, it can move in the direction of the boat edge further away from the camphor particle (see the movement of the camphor boat on the left side of Figure 1.2). As camphor molecules are released onto the water surface, the surface tension around the camphor-containing end of the plastic plate is reduced (~55 mN m-1). Thus, the camphor boat moves from the edge of the plastic plate with the lower surface tension towards that with a higher surface tension. The released camphor molecules are sublimated or dissolved in the water. Therefore, the difference in surface tension around the camphor boat is maintained. Hence, uniform motion of the camphor boat can continue for at least 30 minutes. If a soap particle is used instead of the camphor particle, a soap boat is created. The soap boat can only move for a reduced amount of time, because the released soap molecules remain on the water surface, lowering the surface tension around the whole boat. Therefore, the camphor boat is more convenient for experimentally studying the nature of self-propelled objects modelling inanimate systems. To induce a uniform motion of the soap particle, the soap molecules adsorbed at the interface should be dissolved in the bulk liquid phase. Uniform motion of soap particles is realized at an oil–water interface because soap molecules dissolve in the oil phase. Thus, purification of the interface by the desorption of the adsorbed soap molecules can maintain a uniform motion.
1.2.1 Literature on the Research of Camphor Motion
According to our personal search, the oldest study on the subject of camphor motion was reported by Charles Tomlinson in 1862. Lord Rayleigh also reported on camphor motion in 1890. Charles V. Boys published a textbook titled 'Soap Bubbles' in 1920, containing demonstrations for younger generations at the Royal Institution Christmas lectures, including an introduction to camphor motion. There is a book about a self-propelled object on water called a 'floating doll', used as a toy in the Edo Period (1603–1868) in Japan. The camphor boat was sold as a toy in Japan until the 1970s. Thus, camphor motion has been used as a basic tool to study interfacial science, and also as a toy. In this chapter, we mainly describe the combination of camphor motion and nonlinearity to produce characteristic features of motion similar to living organisms.
1.2.2 Oscillatory Motion of a Camphor Boat Based on the Diffusion of Camphor Molecules
When a camphor disk is glued to the bottom of a plastic plate at its centre, oscillatory motion occurs. The mechanisms of this oscillatory motion are explained by the following three states (Figure 1.3(a)). State I (Rest): Camphor molecules dissolve from the disk into the bulk water, but the camphor boat doesn't move because the camphor molecules are distributed evenly under the plastic plate, resulting in a balanced surface tension around the boat. State II (Acceleration): The diffusion of camphor molecules progresses at the bottom of the plate with time and the camphor concentration at the edge of the plate reaches a threshold value to achieve acceleration. Then, the camphor molecules are developed from the edge of the plate to the water surface, and the camphor boat accelerates towards the opposite direction of the developed camphor molecules. State III (Return to State I): After the acceleration, the camphor boat decelerates and rests at a different position because the camphor concentration at the base of the plate decreased due to the movement of the camphor boat. Thus, State III ends by returning to State I. Hence, States I to III are repeated, i.e. oscillatory motion occurs. The period of oscillation increases with an increase of the camphor disk distance from the plastic plate edge. When the location of the camphor disk is changed, oscillatory motion is bifurcated to uniform motion as a function of the diffusion distance, d1 (Figure 1.3(b)). In particular, reciprocating oscillatory motion occurs when the camphor disk is located at the centre of the plate and the motion of the camphor boat is restricted to one dimension using a long and slender water chamber. Thus, oscillatory motion and mode bifurcation can be realized by changing the internal conditions of the camphor boat based on the diffusion of camphor molecules, which plays a role in creating the driving force.
1.2.3 Oscillatory Motion and Mode Bifurcation with the Addition of Surfactants
The surface tension of a saturated camphor aqueous solution (γc) is ~55 mN m-1. This suggests that a camphor disk cannot move when the surface tension of water is decreased below γc by the addition of surfactants. However, actual camphor motion surpasses expectations. For example, sodium dodecyl sulfate (SDS), which is widely used in everyday life and in laboratories, induces characteristic motion and bifurcation. Figure 1.4 shows a schematic illustration of (a-1) a phase diagram of the self-propelled motion of a camphor disk and (a-2) the surface tension depending on the concentration of SDS in the water (CSDS). The speed of the uniform motion decreases with increasing CSDS in a concentration range of CL, where γ>γc. In the concentration range of OL, where γ~γc, oscillatory motion with a higher amplitude of 15–20 mm s-1 occurs and the oscillation period decreases as CSDS increases. However, uniform motion is realized again in the CH range, where γm<γ<γc (γm: the surface tension of SDS at the critical micelle concentration (cmc)). At the OH range, where γ~γm, oscillatory motion with a smaller amplitude (~7 mm s-1) is realized, and no motion is observed in the N range.
We explain the mechanisms of the oscillatory motion in the OL range. In the resting state, there is no difference in the surface tension around the camphor disk because γ~γc (State I in Figure 1.4b-1). However, the camphor disk is accelerated as camphor molecules accumulated at the base of the disk are spilled out to the water surface and SDS molecules adsorbed on the water are dissolved into the bulk water phase together with camphor molecules (State II in Figure 1.4b-1). Thus, oscillatory motion occurs between States I and II.
Next, we explain why the uniform and oscillatory motions are regenerated in the CH and OH ranges, respectively. The mixture of the SDS micelle and camphor plays a key role in the process, because the concentration range of CH~cmc of SDS and that of OH>cmc. The Marangoni flow is completely inhibited around the cmc of SDS. However, it is regenerated above the cmc due to the increase in the dissolution rate of camphor into the SDS micelle aqueous solution. The existence of a camphor–SDS mixture was confirmed using C NMR and mass spectrometry. These results suggest that the uniform and oscillatory motion in the CH and OH ranges, respectively, are regenerated by the difference in the surface tension around the camphor disk resulting from the dissolution of camphor molecules into the SDS micelle in the aqueous phase (see Figure 1.4b-2). Such characteristic motion of the camphor disk was reported using other surfactants as well.
The camphor boat is also sensitive to chemicals. When a camphor boat that exhibits uniform motion is floated on an annular water channel, its uniform motion stops via oscillatory motion depending on the position of an ester droplet placed on the outside of the channel. This motion is controlled by the surface tension, i.e. the surface tension of water is decreased by the local adsorption of the ester vapour. Oscillatory motion occurs when the surface tension of the saturated camphor solution is equal to that of the ester solution. The features of the motion depending on the kind of ester used are discussed based on the physicochemical properties, e.g. enthalpy of vaporization.
1.2.4 Hysteresis and Memory of Camphor Motion
Hysteresis and memory can be created using camphor systems (Figure 1.5). To induce hysteresis, the order of placing the floating objects on the water surface is reversed. These objects are a camphor grain with an asymmetric shape and a circular plastic ring. If the camphor grain is introduced onto the water surface first, followed by placing the plastic ring around it (Condition I), unidirectional rotation of the grain is achieved. Meanwhile, the ring rotates in the opposite direction. However, by reversing the order in which the objects are introduced (Condition II), the grain exhibits oscillatory motion together with the ring. Thus, hysteresis is realized when the order of placing the objects on water is changed. The hysteresis mechanisms are described as follows. Under Condition I, the Marangoni flow around the camphor grain is helically generated on the water surface by the rotation of the camphor grain. As the plastic ring is driven by the helical flow, it rotates in the direction opposite to that of the grain. On the other hand, the surface tension around the camphor grain is decreased due to the presence of the ring under Condition II, so it does not move. However, the surface concentration of the camphor adsorbed on water increases with time and the camphor molecules are spilled out from the inside of the ring. The ring is accelerated together with the camphor grain, followed by a return to the resting state. Thus, oscillatory motion is maintained, similar to the one described by the system in Section 1.2.2.
When a camphor boat that exhibits oscillatory motion is floated on an annular water channel, motion with memory is generated, i.e. its resting positions are almost the same in the individual cycles. The memory effect is controlled by the relationship between the period of the oscillatory camphor boat motion and the resting time between each of its movements. The camphor concentration is locally increased around the position where the camphor boat rested in the previous cycle. Therefore, the camphor boat approaching this resting position is decelerated by the remaining camphor molecules due to the lowered surface tension. Thus, motion with memory effects does not occur if the resting time is not sufficiently long for the period of one cycle. Backward motion observed during the oscillatory motion in this experiment was reported separately.
1.2.5 Characteristic Motion of a Camphor Disk Depending on the External Boundary
Features of the camphor motion are characteristically changed depending on the shape and size of the water channel (Figure 1.6). This means that the spread of the camphor molecules influenced by the shape and size of the boundary can change the features of the motion. For example, reciprocating motion is generated when a camphor disk is placed on a linear water channel (Figure 1.6(a)). Reciprocating motion is caused by the reversal of the concentration gradient of camphor along the channel. That is, the surface concentration of camphor around the edge of the camphor disk closer to the channel wall is high when the disk approaches the wall. In contrast, the surface concentration of camphor is low around the other side of the disk, farther away from the wall. This is caused by the camphor molecules released onto the water sublimating and dissolving into the water. The camphor disk can turn in the direction of higher surface tension. Therefore, the periodic reversal of the concentration gradient induces a reciprocating motion.
The rotational behaviour of a camphor disk is changed depending on the symmetry breaking of the shape of a water channel. In this experiment, two half-disks were joined to prepare the channel, shifting one of them relative to the other (Figure 1.6(b)). Three types of camphor disk motions were observed, depending on the shift distance between the midpoints of the two half-disks, d2. When the channel has a circular shape (d2=0), i.e. a symmetric boundary, the camphor disk exhibits either a clockwise or counter-clockwise rotation, determined by its initial state. The symmetry of the chamber can be broken by increasing d2. A unidirectional orbital motion of the disk is observed for moderate values of d2. However, the direction of the rotation is no longer determined by the initial rotational direction, but the shape of the chamber. By further increasing d2, the closed trajectory of the rotation is broken, leading to an irregular motion. The selection of the rotation type can be explained in terms of the surface tension around the camphor disk and the channel boundary. This is due to the gradient of the surface concentration of camphor being dependent on the distance between the camphor disk and the boundary, which is changed by the disk trajectory. Mode change in the rotation is qualitatively reproduced by numerical calculation based on the dynamics of the developed camphor molecular layer around a self-propelled disk on water.(Continues…)
Excerpted from "Self-organized Motion"
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Table of Contents
Theoretical and Experimental Design of Self-propelled Objects Based on Nonlinearity;Mathematical Model and Analyses on Spontaneous Motion of Camphor Particle; Coupled Convective Instabilities: Autonomous Motion and Deformation of an Oil Drop on a Liquid Surface; Dynamical Deformation of Interfaces Induced by Aggregate Formation; Synthetic Approaches to Control Self-propelled Motion of Micrometre-sized Oil Droplets in Aqueous Solution; Physical Chemistry of Energy Conversion in Self-propelled Droplets Induced by Dewetting Effect; Tactic Droplets at the Liquid–Air Interface; Chemotactic Droplets Serving as ‘Chemo-Taxis’; Collective Behaviour of Self-propelled Objects on a Water Surface; Chemo-mechanical Effects for Information Processing with Camphor Particles Moving on a Water Surface; Collective Behaviour of Artificial Microswimmers in Response to Environmental Conditions; Nonlinear Dynamics of Active Deformable Particles; Active Particles Propelled by Chemical Reactions; Theory of Active Particles and Drops Driven by Chemical Reactions: The Role of Hydrodynamics on Selfpropulsion and Collective Behaviours