Flow patterns induced by thermocapillary effect and resultant

We focus on the flow patterns and resultant structures of suspended solid particles in a hanging droplet caused by thermocapillary effect. A droplet i...
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Article Cite This: Langmuir 2017, 33, 13197-13206

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Flow Patterns Induced by the Thermocapillary Effect and Resultant Structures of Suspended Particles in a Hanging Droplet Tomohiko Takakusagi and Ichiro Ueno* Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan

ABSTRACT: We focus on the flow patterns and resultant structures of suspended solid particles in a hanging droplet caused by the thermocapillary effect. A droplet is hung on a heated cylindrical rod facing downward, and another cooled rod is placed just beneath the droplet to create the temperature difference between both ends of the droplet. As the temperature difference increases, the induced flow exhibits transitions from an axisymmetric time-independent steady state to three-dimensional timedependent oscillatory states. These flow states are judged through detecting spatiotemporal correlations between the behaviors of the particles and the variation of the temperature over the droplet surface. We find that the particle accumulation structures are realized in this geometry and that their structures vary as a function of the intensity of the thermocapillary effect.

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Ueno et al.5 were interested in the flow structure inside the noncoalescence droplets and indicated that the suspended particles in the heated droplet accumulate on a single line like a whip. That is, the particles flow down along the free surface of the droplet and rise up in the central region toward the heated rod. Each particle radially spreads with a constant phase difference of particle motion toward the free surface in each azimuthal plane. Then the structure by the particles does not change its global shape and seems to rotate like a rigid structure under a constant azimuthal velocity (Figure 1(I)). This unique behavior of the suspended particles is similar to a phenomenon known as the particle accumulation structure (PAS) in an HZ liquid bridge with a high Prandtl number (7 ) fluid after Schwabe et al.6 It has been known that the PAS emerges in a traveling-wave-type time-dependent oscillatory flow7,8 induced by the hydrothermal wave instability in a high-7 liquid bridge.9−12 Such unique particle behaviors and the induced flow patterns due to the thermocapillary effect, however, have not been reported in the geometry of the hanging droplet to the best of our knowledge. Elucidating particle behavior inside a droplet brings about important knowledge not only on controlling the dispersion of the particles but also on processes of crystal growth. Such geometry has been employed in the

INTRODUCTION A number of experiments on Marangoni convection, or thermocapillary-driven convection, have been conducted under microgravity as well as normal gravity conditions in order to understand its influences on crystal growth in the floating-zone (FZ) method.1 Half-zone (HZ) geometry has been employed for this fundamental research; coaxial cylindrical rods with a designated temperature difference are prepared, and a liquid is bridged between the rods by the surface tension and the wettability. A half portion of the fullzone liquid bridge is mimicked by this geometry. Napolitano et al.2 unexpectedly found a unique phenomenon during their onorbit experiment on thermocapillary convection in the HZ liquid bridge in the space shuttle. After an accidental breakage of the liquid bridge to form semispherical droplets sitting on the rods with a certain temperature difference, they tried to rebridge the liquid by moving one rod to make a contact between the droplets, but in vain. Some experimental and numerical research was carried out to indicate the mechanism of such a noncoalescing phenomenon between the face-to-face droplets3−5 following that report. This unique phenomenon is caused by the thermocapillary convection on the surface of the droplets; the thermocapillary effect on the heated droplet drags ambient gas into the region between the droplets’ heads, and that effect on the cooled droplet drags out the gas from that region. As a result, the entrained gas forms a lubricating gas film between the heads of the droplets to prevent their coalescing. © 2017 American Chemical Society

Received: August 11, 2017 Revised: October 2, 2017 Published: October 30, 2017 13197

DOI: 10.1021/acs.langmuir.7b02789 Langmuir 2017, 33, 13197−13206

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bottom rod, cooled to impose a designated temperature difference between the rods, is placed just beneath the droplet tip (Figure 1(II)). The flow fields in the hanging droplet and the behaviors of suspended particles are discussed as functions of the volume and shape of the droplet and of the temperature difference between both ends of the droplet. To detect the spatiotemporal correlation between the unique particle behavior and the induced flow fields, we conduct threedimensional particle tracking to reconstruct the time series of particle positions inside the droplet as well as the surface temperature observation with an infrared (IR) camera.

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EXPERIMENTS

The experimental apparatus is illustrated in Figure 2. Two sets of optical systems are employed in the present study: (1) to observe a flow pattern in the droplet with a large number of particles and (2) to reconstruct the spatiotemporal behavior of particles by threedimensional particle tracking velocimetry (3-D PTV) with a rather small number of particles. We employ the triple pattern matching algorithm developed by Nishino et al.18 in the tracking process of the particles. A droplet of a designated volume is formed on the surface of the top rod facing downward. The end-surface radius of the top rod, R, is 1.5 mm. The top rod is heated by an electrical heater to maintain its temperature at Th. The coaxial bottom rod is placed just beneath the droplet and is cooled by cold water through a cooling channel to keep the temperature at Tc. The temperatures of both rods Th and Tc are measured by thermocouples. Through the series of experiments, Tc is kept constant at 20 ± 0.5 °C. The temperature difference to which the droplet is exposed is defined as ΔT = Th − Tc. We notice that the temperature at the tip of the droplet must be different from the cooled rod temperature. The tip temperature, however, is hardly detected in this system. Thus, we define the temperature difference as this formula as a matter of practical convenience. The distance between the droplet tip and the end surface of the bottom rod is able to be varied by traversing the bottom rod vertically. The distance is kept as small as possible by checking from the side for any droplets of different volumes. The height of the droplet H is also evaluated from the side view. The volume of the droplet V is varied from 5.0 to 15.0 μL. Figure 3 indicates the correlation between the droplet height and droplet volume. The droplet height increases almost linearly as a function of the droplet volume. This correlation is used to evaluate V throughout the experiments by the side view. The shape of the droplet is described by aspect ratio Γ defined as H/R. The preparation process of the droplet on the heated disk is as follows. The droplet without any particles is formed by using a

Figure 1. (I) Time series of snapshots of PAS in face-to-face droplets (first four frames from the left) and its schematic view (right) corresponding to the second frame of the time series: (a) top view and (b) side view. (II) Target geometry of a hanging droplet shown as (1) snapshots and (2) their schematic views with the r−θ−z coordinate system: (a) top view and (b) side view. The PAS propagates azimuthally counterclockwise for both cases.

crystal growth processes of proteins and gels known as the hanging drop method.13−17 The spontaneous formation of unique structures by particle accumulation has potential applications not only in high-quality crystal growth but also in techniques of mixing and stirring particles without using external forces in a closed system of a hanging droplet. We conduct a series of experiments with a geometry of a single hanging droplet with a high-7 fluid in order to investigate the flow patterns induced by the thermocapillary effect and resultant particle behaviors. In this article, a hanging droplet formed on a heated rod facing downward is employed as the target geometry instead of face-to-face geometry; a

Figure 2. Schematics of the experimental apparatuses used (1) to observe flow patterns and (2) to realize 3-D PTV. 13198

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report indicated that the 10 cSt silicone oil is almost transparent to 9, 12, 13, and 14 μm wavelength light and is opaque to 8, 10, and 11 μm wavelength light. Thus, the information monitored by the IR camera does not indicate the actual value of the temperature but rather the qualitative variation of the temperature. In this article, we call this qualitative variation the temperature for the sake of brevity. The typical values of the absorption coefficient α [m−1] for light of these wavelengths are of 5.4 × 104, 1.8 × 105, and 3.3 × 104, respectively. That is, the intensity of the light becomes 1/e of the original value in about 19, 5.6, and 30 μm, respectively. Therefore, the temperature detected by the IR camera almost corresponds to the surface temperature in the present system except in the region near the hot disk with a sharp thermal boundary layer. The intensity of thermocapillary flow described as the Marangoni number is defined by the following equation

Figure 3. Correlation between droplet volume V and its height H. The solid line corresponds to a linear approximation. microsyringe. Then, we put a certain number of particles into the droplet under the designated temperature difference of ≥5 K. Under such conditions, one can realize that the particles travel in a dispersed manner inside the droplet as a result of the thermocapillary-driven flow. In case one, the test liquid is prepared with particles suspended before forming on the hot disk, and in case two, all of the particles descend to the bottom of the droplet before the droplet is set on the hot disk, which results in a nonuniform distribution of the particles inside the hanging droplet. That is why we carry out the series of experiments with a temperature difference larger than 5 K. The top rod is made of transparent sapphire in order to observe the inside of the droplet through the rod. The successive behaviors of particles in the droplet are captured by a high-speed CMOS camera of 512 pixels × 512 pixels at frame rates of up to 250 frame per second (fps) (FASTCAM- 512PCI, Photron, Inc., Japan). In system 1, two synchronized high-speed cameras are settled above the top rod and beside the droplet. An infrared (IR) camera of 320 pixels × 240 pixels (Thermography R300, NEC Avio Infrared Technologies Co., Ltd., Japan) with a closeup lens (TVC-2100UB, NEC Avio Infrared Technologies Co., Ltd., Japan) is also installed beside the droplet to detect the surface temperature of the droplet at a fixed frame rate of 60 fps with a temperature resolution of 0.05 K at 30 °C. The sensor in this IR camera detects infrared light in the range of 8 to 14 μm in wavelength. The side-view camera and the IR camera are located π/2 apart in the azimuthal direction (Figure 2). In the case of travelingwave-type flow with a fundamental frequency under the designated thermocapillary effect, the detected signals by both devices constantly involve a phase shift of π/2 or −π/2 depending on the traveling direction of the hydrothermal wave. We thus check the traveling direction and the fundamental frequency f 0 of the hydrothermal wave by the IR camera as well as the camera for the top view through the hot disk in the fully developed traveling flow regime. Then we make comparisons to both signals after this matching process of the time difference of 1/(4f 0) or −1/(4f 0). In system 2, a cubic beam splitter is placed above the top rod, which makes it possible to observe the particles inside the droplet with two synchronized high-speed cameras.19 5-cSt silicone oil (KF96L-5cs, Shin-Etsu Chemical Co., Ltd., Japan) whose Prandtl number 7 is 68.4 at 25 °C is used as the test fluid. Its properties at 25 °C are listed in Table 1. Noted that the value of ∂σ/ ∂T = σT is measured by Ueno et al.,20 and the other properties are the data provided by Shin-Etsu Chemical Co., Ltd.21 Unfortunately, we do not have any information on the absorption coefficient as the optical property of our own silicone oil, so a precise evaluation of the temperature measured by the IR camera cannot be made. It is noted, however, that we have a report on the optical properties of another silicone oil of 10 cSt = 10 × 10−6 m2/s in kinematic viscosity.22 This

4=

|σT|ΔT

V R2

ρνκ

(1)

where ν and κ are the kinematic viscosity and thermal diffusivity of the test fluid, respectively. Note that we introduce a characteristic length L defined as L = V/R2. The temperature dependence of the fluid viscosity is considered by using an empirical correlation20,21

⎛ ν(T ) 25 − T ⎞⎟ = exp⎜5.892 ⎝ ν0 273.15 + T ⎠

(2)

where ν0 is the kinematic viscosity of the test fluid at 25 °C and T is the temperature in Celsius. The kinematic viscosity of the droplet used to evaluate the Marangoni number is estimated as the following: νexp(T ) =

ν(Th) + ν(Tc) 2

(3)

We run the experiments with ΔT ≤ 60 K in order to prevent significant evaporation of the test fluid in the present system. It is noted that there exist both thermocapillary and buoyancy forces in this configuration in the terrestrial experiments. One can evaluate the relative importance of two different forces by considering the Richardson number ζ as follows:

. 92

ζ=

(4)

In the definition, . and 9 are the Grashof number and the thermocapillary Reynolds number, respectively, defined as

.= 9=

gβ ΔTL3 ν2

(5)

|σT|ΔTL ρν 2

(6)

where g is the gravitational acceleration and β is the thermal expansion coefficient. In the typical case of D = 3 mm, Γ = 1.5, and ΔT = 40 K, the system has a value of ζ ≈ 10−5. In the present configuration, the buoyancy-driven flow can be almost negligible. Gold-coated cross-linking acrylic particles of 30 μm in diameter dp and 1470 kg/m3 in density ρp are used as tracer particles (Soken Chemical and Engineering Co. Ltd., Japan). The Stokes number : in this system is defined as the following23 :=

ϱd p2 18H2

(7)

Table 1. Physical Properties of 5 cSt Silicone Oil at 25 °C ρ

σ

κ

(kg/m )

2

(m /s)

2

(m /s)

(N/m)

(N/(m K))

(−)

9.12 × 102

5.0 × 10−6

7.31 × 10−8

19.7 × 10−3

−6.37 × 10−5

68.4

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σT

7

ν 3

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RESULTS AND DISCUSSION Overview of the Behavior of Particles in a Hanging Droplet. The flow regimes in the hanging droplet can be categorized into an axisymmetric steady flow state and threedimensional oscillatory flow states that can be separated further into a standing wave and a traveling wave, judging from the behavior of particles in the droplet as well as the temperature variation over the droplet surface. It is found that two kinds of particle accumulation structures are realized in the traveling wave; they are named PAS1 for the simpler structure and PAS2 for the more complicated structure. The PAS in the hanging droplet is always observed whenever the traveling hydrothermal wave occurs. This is the uniquely different point compared to the case of the half-zone liquid bridge. In the half-zone liquid bridge, the PAS emerges under certain limited conditions in the Marangoni number and in the aspect ratio even in the travelingwave states.8 Typical example of time series of snapshots of particle motions and surface temperature deviations from the averaged temperature in time during the fundamental period of the hydrothermal wave is shown in Figure 4(1−4). The top views and side views are shown in (a) and (b), respectively. The distributions of the surface temperature and its deviation from the averaged field are illustrated in (c) and (d), respectively. It is noted that the side views are detected at the azimuthal position θ = π/2 and the surface temperature is detected at θ = 0. When ΔT or V is relatively low, that is, 4 is lower than the threshold, an axisymmetric steady flow emerges. The threshold will be discussed later with Figure 5. It must be noted again that the particles settle to the bottom in the droplet under the condition ΔT = 0 because of normal gravity. Once ΔT is increased, the particles follow the thermocapillary-driven flow. The particles slide down near the free surface from the region around the edge of the top rod to the tip of the droplet. Then they rise up from the bottom toward the top rod through the central region of the droplet, and they spread radially from the central region of the droplet toward the free surface in a certain r − θ plane. This is the basic pattern of the particles due to the thermocapillary-driven basic flow inside the droplet. The streamlines near the free surface become denser all along the tip of the droplet, where the coldest part is over the free surface. In the steady flow state, some particles never exhibit azimuthal motion during one turnover motion in a certain r − z plane. The temperature deviation over the free surface exhibits quite tiny temporal variations; the maximum variation in |T̂ | of ≲0.03 K is smaller than the resolution of the IR camera of ±0.05 K. The temperature deviation shows negligible changes over time. As 4 is increased, a three-dimensional oscillatory standing wave emerges beyond a threshold ΔTc or 4 c . In the oscillatory flow regimes, there exist time-dependent thermal-flow components due to the hydrothermal instability in addition to the basic flow. In the standing wave state (row (2)), the particles start swinging alternatively from side to side. In this state, the surface temperature deviation exhibits alternate behavior: the colder spot and the hotter one appear alternately. There is a node−antinode pair in the hydrothermal wave realized in the droplet (that is, with an azimuthal mode number of m = 1). The dashed line in the schematic image of the top

Figure 4. Time series of snapshots of each flow regime in the droplet (whose volume and aspect ratio are 11 μL and of 1.5, respectively): (1) steady flow, (2) standing wave, (3) PAS1 and (4) PAS2 in traveling wave, (a) top view, (b) side view, (c, d) distributions of surface temperature and its deviation from an averaged field, respectively. Note that the camera for the side view is located at azimuthal position θ = π/2 and the IR camera is located at θ = 0. The experimental conditions of (ΔT [K], 4 [−]) in each flow regime are (1) (10.2, 0.99 × 104), (2) (20.3, 2.1 × 104), (3) (21.7, 2.3 × 104), and (4) (31.7, 3.6 × 104). Time intervals Δt [s] are (1) 0.80, (2) 0.178, (3) 0.172, and (4) 0.126, respectively. The solid line shown in the schematic of (3) is part of the PAS line inside the droplet, and the dashed line is that along the droplet surface. The solid lines shown in the schematic of (4) are a part of the PAS line located in the front half of the droplet, and the dashed lines are those in the rear half of the droplet. Combinations of (Tmin, T0, Tmax) in °C of the color bar for row (c) are (1) (24.0, 25.5, 27.0), (2) (29.5, 32.5, 35.5), (3) (30.2, 33.7, 37.2), and (4) (37.0, 42.0, 47.0), respectively.

view in Figure 4(2) indicates the plane of the node. As the particles travel in the droplet, they tend to gather in the middle region of the antinode over the free surface. This region 13200

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Figure 5. Flow-pattern maps as functions of temperature difference ΔT and droplet volume V (left) and the corresponding Marangoni number 4 and aspect ratio Γ (right). Marks indicate the flow patterns, and bars indicate the existing region of each pattern. Note that the maximum temperature difference available with the present apparatus is limited to 60 K to prevent the evaporation of the test fluid.

Figure 5 indicates the flow pattern map in the range of the present experimental conditions as functions of (a) ΔT and V and corresponding (b) 4 and Γ. The bars in these graphs indicate the region appearing where the flow regime is concerned. Note that there exists a transition region between flow regimes in which a distinct flow regime is not definitely judged; therefore, the conditions shown in Figure 5 include a small vacant region between the bars. In a droplet of small V (5 and 6 μL) or corresponding Γ, the flow state is quite stable and never exhibits any oscillatory modes by changing ΔT up to 60 °C. In a droplet of moderate V (7 or 8 μL) or corresponding Γ, the flow exhibits transitions to oscillatory states. The threshold in terms of 4 , or 4 c , becomes smaller than that of small-V droplets. By increasing V (≥9 μL) or corresponding Γ, the thresholds of the onset of the hydrothermal wave and of the PAS in the traveling wave become very low and converge to a certain constant value in terms of ΔT and 4 . One can see almost constant thresholds in 4 to realize a standing wave and PAS1 in the traveling wave against Γ. The ranges of occurrence of the standing wave and PAS1 in the traveling wave become narrower, and that of PAS2 extends as Γ. Typical examples of the time series of the temperature deviation over the droplet surface and its Fourier spectrum in each flow regime are shown in Figure 6a,b, respectively. The temporal variation of the surface temperature is detected at the midheight of the droplet on the center line of the captured IR image. In (1), which is the steady flow regime, almost no oscillation of the surface temperature is detected, as shown in Figure 4(1) as well. Once the flow exhibits a transition to timedependent oscillatory states (2)−(4), the surface temperature indicates an oscillation with a fundamental frequency f 0. Upon increasing the ΔT or 4 , the amplitude of the surfacetemperature variation as well as f 0 increases. The correlation between the deviation of the surface temperature and the particle motion in the PAS1 state is discussed in the following text. Figure 7a,b indicates snapshots of the deviation of the surface temperature and the side view of the droplet, respectively. The phase difference between the images detected by the side-view CMOS camera and by the IR camera is adjusted as previously mentioned. We monitor the variation of the surface-temperature deviation and the variation of the brightness of the side view along a central longitudinal line of the droplet as a function of time, whose azimuthal position is denoted as θ = θ1 here. Frames (c) and (d) indicate typical examples of those for two cycles of the oscillation. Note that the hydrothermal wave (HTW in the figure) propagates

corresponds to the coldest spot over the free surface. Once the particles are gathered to flow in the coldest spot, they do not move azimuthally over the free surface in the fully developed state. The particles fall to the tip of the droplet and rise toward the hot disk almost along the center axis of the droplet following the flow of the relatively colder fluid. This returning flow causes the opposite side of the droplet to cool to the prior coldest spot and brings the particles to the opposite side of the droplet. Thus, the thermal wave behaves as the standing wave, and they do not move in the azimuthal direction. Through a series of experiments, the azimuthal position of the node plane appears randomly when ΔT exceeds the threshold, and it never changes its position while the standing wave emerges in the droplet. This is a reflection of the uniformity of the geometrical and thermal−fluid boundary conditions of the apparatus. As 4 is further increased, the flow field exhibits a transition to the traveling wave state (row 3). The rotating behavior of the particle accumulation structure (PAS) like a rigid structure without any dynamic deformation is quite similar to that in the liquid-bridge geometry.6−8 The surface temperature deviation also seems to rotate as a rotating wave with a propagation angle; that is, the azimuthal phase of the hydrothermal wave depends on the vertical coordinate and the dependence of the phase on z is approximately linear. In Figure 4(3), the curved particle line (PAS) and the thermal wave over the free surface travel in the counterclockwise direction when one observes from above the droplet. Such oscillating flow patterns beyond the threshold 4 c accompanying the thermal wave over the free surface correspond to the hydrothermal wave predicted by Smith and Davis24 in the liquid films and by others9−12 in liquid bridges as previously mentioned. It is noted that the azimuthal direction of the traveling behavior is not predictable through the conduction of the experiments. As 4 is increased more, the particle accumulation structure exhibits a transition from PAS1 to PAS2 in the regime of the traveling wave. In the PAS2 state, although the flow field remains periodic and the thermal wave over the free surface travels in the azimuthal direction without any significant difference compared to the PAS1 case, the particles disperse, and there seem to exist multiple pathlines inside the droplet. It is noted that, as discussed latter, this PAS2 has a single closed structure and never deforms its shape in rotation; that is, PAS2 seems to rotate as a rigid body, as does PAS1. If one imposes a much larger temperature difference, the flow field becomes a chaotic flow. This article does not aim to discuss this kind of flow state. 13201

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direction so that almost all of the particles in the r − z planes at θ = θ1, on which the side view camera is located, are detected. One can distinguish the particles traveling near the free surface by judging the size of the black dot due to the distance from the focal point and the direction of the points flowing down toward the tip of the droplet. When a band of relatively high temperature (hotter region) emerges near the droplet tip, a group of particles rise toward the hot-end wall in the central region inside the droplet (e.g., 3π/2 ≲ θ ≲ 7π/3 in the figure). When a thermal band of relatively low temperature (colder region) appears near the hot-end wall on the free surface, the particles come closest to the free surface (π ≲ θ ≲ 2π in the figure) and then go down along the colder region toward the bottom end of the droplet (π/2 ≲ θ ≲ 3π/2). Such behavior is similar to that in the liquid bridge; it was reported that the particles on the PAS in the half-zone liquid bridge appear near the free surface with a band of relatively low temperature.7 It should be noted, however, that the spatial correlation between the particle behaviors and the surface temperature distribution due to the hydrothermal wave depends on 7 in the HZ liquid bridge.8,25 This article does not deal with the effect of 7 on the hydrothermal wave and PAS; further discussion would be needed to judge the 7 effect in this geometry. Spatiotemporal Behavior of Particles. In the following text, we focus on the motion of individual particles for different flow regimes. Figure 8 indicates typical examples of the reconstructed orbits of the particles in (1) steady flow, (2) PAS1, and (3) PAS2 in the traveling wave in (a) the bird-eye view and (b) the top view. The meshes show the droplet shape, and plots show reconstructed positions of different particles at a constant time interval in the figures. Column (c) indicates the temporal variations of the position of a particle from the reconstructed data shown in (a) and (b). In the case of steady flow, the behaviors of two different particles are illustrated (row (1)). It is clearly seen that both particles travel along almost the same trajectory in a certain r − z plane. The particles travel along the end surface of the rod (z ≈ 0) from the central region of the droplet toward the free surface. The particles slide down toward the droplet tip and change their direction to rise up along the center line (or z axis) toward the end surface as already mentioned. This is the basic turnover motion of the particles as described before. Note that there have existed no investigations giving a comprehensive explanation of why the particles gather in almost a certain r − z plane even under the steady flow state. In the PAS1 state, as described above, the particles disperse again in the droplet but form a whip-like structure that seems to rotate. The trajectory of an individual particle in the hydrothermal wave state accompanying PAS1 lies approximately in a plane of θ = const in a single turnover period of the particle motion. Such behavior is similar to that in the case of steady flow. However, upon rising from the cold bottom spot to the hot wall near the z axis, the particle can move to another plane at another azimuth. This behavior is most likely due to the symmetry breaking of the hydrothermal wave with azimuthal wavenumber m = 1. The essence of the rotating whip-like structure is that each particle steadily travels in each r − z plane but with a constant phase difference from neighboring particles as shown in Figure 9a. In this frame, the small black dots are particle positions as shown in Figure 8(2). The square marks indicate the positions of different particles at the same instance t [s] = t0, and the circles at t = t0 + Δt (Δt = 0.33 s). The solid red and blue lines indicate the

Figure 6. Typical examples of (a) the time series of the surface temperature deviation at z = H/2 for the droplet measured by IR camera and (b) their Fourier spectra in the flow regimes of (1) steady flow, (2) the standing wave, (3) PAS1 and (4) PAS2 in the traveling wave under the same conditions as shown in Figure 4.

Figure 7. Snapshot of the deviation of the surface temperature (a) and side view of the droplet (b). Frames (c) and (d) indicate typical examples of those for two cycles of the oscillation monitored along the central longitudinal line of the droplet (the dashed-dotted lines in (a) and (b)) after the phase-matching procedures. The color range in (c) is same as the color bar in (a). The black dots in (d) indicate the particles flowing in the hanging droplet.

from right to left in this case. In frame (d), a group of black dots indicate the particles inside the droplet. It is noted that the side view is obtained with a full illumination of the droplet, and large focal depth covers almost half of the droplet in the r 13202

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Figure 8. Particle behaviors in (1) steady flow, (2) PAS1 and (3) PAS2 in the traveling wave reconstructed by 3D-PTV (colored dots) inside a droplet of V = 12 μL and Γ = 1.6: (a) bird’s-eye view, (b) top view, and (c) temporal variation of position components corresponding to the blue dots in (a) and (b). The mesh indicates the droplet shape. The experimental conditions in (ΔT [K], 4 [−]) for flow regimes are (1) (17.4, 1.7 × 104), (2) (23.2, 2.4 × 104), and (3) (28.8, 3.1 × 104). The time interval of each dot is 1/125 s. The star marks in (a) and (b) indicate the particle position at a certain instant, and the star mark in (c) corresponds to the blue star in (a) and (b).

apparent structures with the particles at t = t0 and t = t0 + Δt, respectively. At a certain instant, particles exist on a curved structure (also see the square marks and the solid line in the schematics of the top view illustrated in Figure 9b). After the elapsed time, the particles travel almost straight outward to retain the shape of the apparent structure (the circle marks and the solid line in the same frame). That is, a particle on the r − z plane at θ1 + Δθ in the azimuthal direction exhibits the same motion in the r − z plane as the other particle at θ1 with a time difference of Δθ/(2πf 0), where f 0 is the fundamental frequency of the traveling hydrothermal wave. Thus, the PAS seems to rotate even though each particle travels in each r − z plane. In the PAS2 state, the particle behaviors become much more complicated. As seen in the top view in Figure 8(3)b, the particles exhibit azimuthal motion in a region of r > 0 in addition to the basic circular movement in the r − z plane. We find that the apparent structure of PAS2 consists of a single closed line such as that in the PAS1. A single particle on PAS2 is extracted from Figure 8(3) and is shown in Figure 10(1).

This particle on PAS2 travels in the inner area of the droplet between the surface and the center line in addition to the basic trajectory of the PAS1. Let us track this particle shown in frame (1) in a rotating frame of reference with respect to the hydrothermal wave (Figure 10(2)). One can see a folded structure in which the particle comes close to the free surface several times at different θ values before rising up near the center line of the droplet. It must be noted that any particles shown in Figure 8(3) indicate the same structure as shown in this frame (2) with a phase difference of θ in the rotating frame. Figure 10(3) indicates the top view of the reconstructed structure from the motion of a single particle in the rotating frame of reference for 2.92 s as shown in Figure 10(2). And the top view of the snapshot of the PAS2 state, that is, the view detected at an instant in the absolute coordinate, is shown in Figure 10(4). The structure composed of multiple particles observed at an instant (frame (4)) seems to consist of several lines simultaneously, but this corresponds to a single closed line as 13203

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Figure 9. Essence of the rotating whip-like structure: (a) bird’s eye view of the particle path lines as shown in Figure 7(2) and corresponding PAS1 at different instants (squares at t = t0 and circles at t = t0 + Δt (Δt = 0.33 s)). Each particle steadily travels in each r − z plane but with a constant phase difference from neighboring particles. The small black dots are particle positions as shown by the colored dots in Figure 7(2). (b) Schematic top view of typical particles traveling near the end surface (at z ≈ 0) on the PAS1 at two different instants.

Figure 10. (1) Typical trajectory of a particle on the PAS2 in the absolute coordinate in the droplet (whose volume and aspect ratio are 11 μL and 1.5, respectively) conducted under the experimental condition of (ΔT = 10.2 K, 4 = 4.3 × 104 ), (2) trajectory of the same particle on the rotating frame of reference with respect to the traveling hydrothermal wave (1.3 Hz), and (3) top view of the particle trajectory in the rotating frame of reference (same as for frame (2)), and (4) snapshot of PAS2 observed from above. Time interval between particles: 1/125 s. Blue arrows in (1) indicate the direction of the particle motion, and the curved arrow shows the rotating direction of the structure of PAS. Points A and B in frames (1) and (3) at t = t0 and t = t0 + 2.92 are marked as guides to make a comparison with the reconstructed trajectory of a single particle in the rotating coordinate system in frame (4). The circle, triangle, and square points indicate the particle position on the trajectory at moments t = t0 + 0.584, t = t0 + 1.392, and t = t0 + 2.2, respectively.

shown in frame (3). That is, it is found that PAS2 has an elongated structure and is deformed from PAS1 under a higher thermocapillary effect. The particle movement in regime PAS2 is illustrated in more detail via Figure 10(1)−(3). Point A

indicates the position near the tip of the droplet, and the instance when the particle is located at position A is defined at t [s] = t0. In t < t0, the particle travels near the free surface toward the tip. Point B corresponds to the position at t = t0 + 2.92. 13204

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correlations between the behavior of the particles suspended in the droplet and the variation of the temperature over the droplet surface. The particle behavior is monitored through a transparent top rod via a high-speed camera, and the variation of the surface temperature is monitored via an infrared (IR) camera. We find that unique structures by suspended particle accumulation, named PAS1 and PAS2, are realized in travelingwave-type flow regimes in this geometry. We illustrate their three-dimensional structures by applying three-dimensional particle tracking velocimetry (3D-PTV) and indicate that those structures vary as a function of the intensity of the thermocapillary effect.

Along the trajectory, we identify the circle, triangle, and square marks corresponding to the positions at t = t0 + 0.58, t0 + 1.39, and t0 + 2.20, respectively. The particle travels upward inside the droplet starting from point A. The path of the particle is offset from the z axis of the droplet. The particle changes its traveling direction to the free surface near the heated disk. When the particle comes close to the free surface (the circle mark), the path bends abruptly and the particle goes down along the free surface. In the case of PAS1, the particle keep going down along the free surface to the tip of the droplet and rises again inside the droplet. In the case of PAS2, on the other hand, the particle penetrates the interior of the droplet before reaching the droplet tip. The particle then changes its direction of motion to rise again between the free surface and the central axis and turns to the free surface again at about 0.2H (the triangle mark). The particle flows along the free surface and then travels inward and rises inside the droplet (the square mark) to form another loop. After reaching the free surface near the heated disk, the particle travels along the free surface all the way toward the tip. Finally, the particle rises toward the heated disk (point B). The set of these behaviors is a single cycle of the particle in the case of PAS2 so that the particle comes closer to the free surface three times in a single cycle but only once in the case of PAS1. The path of a single particle in the rotating frame of reference with respect to the hydrothermal wave (frames (2) and (3)) exhibits a structure similar to the PAS line composed of multiple particles observed in the snapshot (frame (4)). Such a feature is also seen in the cases of the PAS1 in the hanging droplet and the PAS in the half-zone liquid bridge.8,19 To illustrate the similarity between the structure in the rotating frame of reference and the one in absolute coordinates, we put in frame (4) A′ and B′ corresponding to the positions shown as A and B, respectively. It is emphasized that the behavior of the particle on PAS2 is similar in quality to that of the particle on SL-2 PAS8 as reconstructed by Niigaki and Ueno.19 Knowledge of the flow structure and accompanying particle behaviors driven by the higher thermocapillary effect, however, has not been intensively accumulated. In the case of SL-1 PAS in the half-zone liquid bridge, Muldoon and Kuhlmann26,27 illustrated the Kolmogorov−Arnold−Moser (KAM) tori in the flow field, and Kuhlmann et al.28 indicated a remarkable correlation between the SL-1 PAS and the KAM tori. We could reach a better understanding of the transition mechanism between PAS1 and PAS2 if further research accumulated knowledges on the transition mechanism of KAM tori by increasing the intensity of the thermocapillary effect in the hanging-droplet geometry.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 (0)4 71241501. Fax: +81 (0)4 71239814. ORCID

Ichiro Ueno: 0000-0003-1616-3683 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Prof. Koichi Nishino and Dr. Taishi Yano (Yokohama National University) and Prof. Masahiro Motosuke (Tokyo University of Science) for invaluable support in carrying out a series of experiments with three-dimensional particle tracking velocimetry. We also acknowledge Prof. Hendrik C. Kuhlmann (TU Wien) for a fruitful discussion. Mr. Takumi Watanabe at Tokyo University of Science is acknowledged for support in carrying out the experiments. This study was financially supported by a Grant-in-Aid for Scientific Research (B) (24360078) and by a Grant-in-Aid for Challenging Exploratory Research (16K14176) from the Japan Society for the Promotion of Science (JSPS). One of the authors, I.U., acknowledges support by Tokyo University of Science through the Fund for Strategic Research Areas.

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REFERENCES

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CONCLUDING REMARKS We conducted a series of terrestrial experiments with the geometry of a single hanging droplet of a high-Prandtl-number fluid in order to investigate flow patterns induced by the thermocapillary effect and resultant particle behavior. A droplet is hung on a heated cylindrical rod facing downward, and another rod that is cooled is placed just beneath the droplet before we expose the droplet to a designated temperature difference between the rods in order to induce thermocapillarydriven convection inside the droplet due to the nonuniform temperature distribution over the free surface. As the temperature difference increases, the induced flow exhibits transitions from an axisymmetric time-independent steady state to three-dimensional time-dependent oscillatory states. We judge those flow states by detecting spatiotemporal 13205

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