# A Viscosity-Based Model for Bubble-Propelled Catalytic Micromotors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{2}) [5,13,14,15,16] or a Zn-based microtube driven in acidic water [17,18]. Bubble-propelled micromotors driven by chemical reactions are normally analyzed in two shapes: Janus microsphere motor and tubular micromotor. Other geometries like the maple tree fruit or samara are not the commonly used ones [19]. In the experiment, the frequently used tubular micromotors are conical rather than cylindrical. The inner wall of the tube is coated with catalytic metals such as Pt or Zn. The bubbles generated from the inner wall eject one way in the tube, driving the micromotor to move in the opposite direction, as shown in Figure 1. By depositing other functional metals, such as Fe and Ti, onto the outside surface of the tube, such a simple metal layer could be magnetized by an external magnetic field. The magnetic field orients the moving direction without altering the speed. Moreover, the motion can be stopped and initiated by modulating the magnetic field intensity. There are also other ways to control the motion of micromotor, like thermally-driven acceleration and chemical stimuli [20]. This kind of micromotor can be applied in biological medicine, the chemical industry, and environment engineering [20,21,22,23,24,25,26].

_{2}O

_{2}solution.

**.**The model takes into account of the effects of fluid viscosity, H

_{2}O

_{2}concentration, and shape parameters of micromotors. After verification of the published test results, the model is used to analyze the effects of the semi-cone angle, length radius ratio, and viscosity.

## 2. Materials and Methods

_{max}and a semi-cone angle δ.

_{2}O

_{2}solution, the bubble nucleates from the decomposition of H

_{2}O

_{2}into O

_{2}due to the Pt layer. The net surface tension force drives the bubble to move to the larger opening of the tube. When the bubble reaches the larger end, it ejects or bursts, which generates a force to push the micromotor forward. This driving force is shown in Figure 2a.

_{2}O

_{2}solution; ${v}_{b}$ and ${R}_{b}$ are the velocity and radius of the bubble. The velocity of bubble is not easy to measure, hence, the oxygen productivity q is here:

_{2}O

_{2}concentration CH

_{2}O

_{2}, so q can be expressed as [27]:

_{d}is the drag force caused by fluid environment. For a conical micromotor, F

_{d}can approximately be expressed as [39]:

_{1}is the shape coefficient:

_{j}is calculated based on the geometry:

_{b}in this equation is a time-dependent parameter and can be expressed in terms of the volume of the bubble, as follows:

_{0}(infinitely tending to zero). The instantaneous velocity of micromotor is shown below:

_{2}O

_{2}concentration CH

_{2}O

_{2}, geometric parameters of the micromotor, like L, R

_{max}, and δ, the mass of the micromotor m, and the fluid viscosity μ. The influence of these factors will be discussed in detail.

## 3. Results and Discussion

#### 3.1. Concentration of H_{2}O_{2}

_{2}O

_{2}concentration is listed below. Micromotors with two different shapes are immersed in solutions with two different concentrations. One with length L = 9.1 μm, ${R}_{max}$ = 1.16 μm, δ = 2.3° immersed in H

_{2}O

_{2}concentrations ranging from 1% to 5% and the other with length L = 100 μm, ${R}_{max}$ = 10 μm, δ = 0° immersed in H

_{2}O

_{2}concentrations ranging from 5% to 15%. The results obtained based on Equation (15) are shown in Figure 3. The speed increases rapidly with increasing solution concentration.

_{2}O

_{2}concentrations, as shown in Figure 3b. Results from our study are happened to be in good agreement with Li et al. [34]. A well-defined propulsion is observed over a higher peroxide concentration, with speed ranging from 62.7 μm/s to 1480 μm/s, when the H

_{2}O

_{2}concentrations are 1% (conical tube, inset a) and 15% (cylindrical tube, inset b), respectively. Raising the H

_{2}O

_{2}concentration increases the catalytic reactivity toward the decomposition of hydrogen peroxide. The driving force in Equation (4) strongly depends on oxygen gas productivity q. The propel efficiency is improved by the increase of the H

_{2}O

_{2}concentration. Thus, Equation (15) can provide a good prediction to the motion of the micromotor.

#### 3.2. Semi-Cone Angle

_{max}and L.

_{max}and length L. In order to find out the hydrodynamic behavior of two different shapes, a numerical simulation using the commercial software package Fluent 18.0 (ANSYS) has been carried out.

^{3}and viscosity μ = 1.003 mPa∙s. Since the fluid area, the micromotor, and the boundary conditions are all symmetric, a two-dimensional model was set up by using Fluent in Figure 5. A model of the domain fluid area 100 μm × 100 μm was built. On the left side, there is a velocity inlet boundary condition (100 μm/s). The other three sides are pressure outlet boundary conditions (Figure 5), and the static pressure at the outlet boundary is 0. The two different micromotors were immersed in the center of the fluid: the cylindrical one with length L = 50 μm, radius R = 5 μm, and wall thickness η = 1 μm; the conical one with length L = 50 μm, R

_{min}= 5 μm, R

_{max}= 10 μm, and wall thickness η = 1 μm (Figure 5). The no-slip boundary condition is enforced at the walls around the micromotor.

#### 3.3. Length-Radius Aspect Ratio

_{2}O

_{2}= 5%. Figure 7a shows that the velocity decreases when ξ increased from 5 to 19 (keeping L = 9.1 μm unchanged, the radius ${R}_{max}$ varies from 1.82 μm to 0.479 μm), the velocity is especially high when ξ is around 5. There are two major influencing factors: the drag force and inner surface area.

^{2}to 2 μm

^{2}. The larger surface brings higher oxygen productivity, which produces a stronger driving force in return. Both the drag force and inner surface area decrease with the increase of ξ (Figure 8a). Synthesizing both factors, it shows that surface area displays greater influence than the drag force. This is the reason why velocity decreases when the length-radius aspect ratio ξ increases.

#### 3.4. Fluid Viscosity

_{2}O

_{2}, but also for self-reacted or other micromotors. We extend the discussion of the foregoing results to other factors like fluid viscosity. In this part, 24 different viscosities have been calculated based on Equation (15). According to our results, the velocity decreases dramatically along with the increasing of viscosity from 0.1 mPa·s to 4 mPa·s. The results also show that there is a non-linear relationship below 1.5 mPa·s and nearly linear above 1.5 mPa·s. The results show good agreement with the experimental results from Li et al. [42]. This indicates that the velocity is rather sensitive to fluid viscosity.

_{2}O

_{2}is 0.9 mPa·s, but the velocity is only 5 μm/s when the viscosity is 4 mPa·s irrespective of the chemical reaction that may occur. A slight change in viscosity brings a significant decrease/increase in velocity. Gao et al. [29,31] reported the observation regarding motion of catalytic micro/nano motors in biological environments, such as cell culture media and human serum. The speed exhibits a reduction by 50% and 68%, respectively, than the speed in water under similar conditions. The decrease is attributed to high viscosity, as well as passivation of the catalytic Pt surface. As far as our knowledge is concerned, there are very few investigations referring to the effect of viscosity, especially experimental results.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the geometry size and force acting on micromotor: (

**a**) F

_{jet}and F

_{d}are the driving and drag force caused by bubbles and fluid separately; and (

**b**) geometric size of the micromotor model.

**Figure 3.**The velocity of micromotor versus H

_{2}O

_{2}concentration: (

**a**) at low concentration (<5%) with length L = 9.1 μm, ${R}_{max}$ = 1.16 μm, δ = 2.3°, CH

_{2}O

_{2}= 1–5%; and (

**b**) at high concentration (>5%) with length L = 100 μm, ${R}_{max}$ = 10 μm, δ = 0°, CH

_{2}O

_{2}= 5–15%.

**Figure 4.**The velocity versus the semi-cone angle of the micromotor, the model used here is length L = 100 μm, ${R}_{max}$ = 10 μm, CH

_{2}O

_{2}= 5%, δ = 0°–5°.

**Figure 6.**The pressure contours of two differently shaped micromotors with all of the same boundary conditions: (

**a**) cylindrical micromotor; and (

**b**) conical micromotor.

**Figure 7.**The velocity versus length-radius aspect ratio: (

**a**) change the radius ${R}_{max}$, keeping $L=9.1$ μm constant; and (

**b**) change the length L, keeping ${R}_{max}=1.16$ μm constant.

**Figure 8.**The inner surface area and drag force versus length-radius aspect ratio $\mathsf{\xi}$: (

**a**) change the radius ${R}_{max}$, keeping L = 9.1 μm constant; and (

**b**) change the length L, keeping ${R}_{max}$ = 1.16 μm constant.

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**MDPI and ACS Style**

Wang, Z.; Chi, Q.; Liu, L.; Liu, Q.; Bai, T.; Wang, Q.
A Viscosity-Based Model for Bubble-Propelled Catalytic Micromotors. *Micromachines* **2017**, *8*, 198.
https://doi.org/10.3390/mi8070198

**AMA Style**

Wang Z, Chi Q, Liu L, Liu Q, Bai T, Wang Q.
A Viscosity-Based Model for Bubble-Propelled Catalytic Micromotors. *Micromachines*. 2017; 8(7):198.
https://doi.org/10.3390/mi8070198

**Chicago/Turabian Style**

Wang, Zhen, Qingjia Chi, Lisheng Liu, Qiwen Liu, Tao Bai, and Qiang Wang.
2017. "A Viscosity-Based Model for Bubble-Propelled Catalytic Micromotors" *Micromachines* 8, no. 7: 198.
https://doi.org/10.3390/mi8070198