Moving force of metal particle migration induced by laser irradiation in borosilicate glass

Hirofumi Hidai; Makoto Matsushita; Souta Matsusaka; Akira Chiba; Noboru Morita

doi:10.1364/OE.21.018955

1. Introduction

Surface tension generally decreases with increasing temperature [1]. The variation of surface tension at a droplet surface under a temperature gradient induces a driving force [2]. Using the driving force, the manipulation of a droplet has been demonstrated by local heating with a laser [1]. Light can move matter by other types of driving force, for example, optical tweezers [3], a photophoretic force [4], fluid flow generated by local melting, and a volume change due to a phase change [5].

We optically manipulated a metal particle in borosilicate glass [6]. The glass in the neighborhood of the laser-heated metal particle softened; hence, the metal particle was able to migrate in the softened glass. The metal particle was obtained by melting a metal foil deposited on the back of the glass by illuminating the foil with a laser through the glass. However, the driving force of particle migration has not been identified.

In this report, we reveal the driving force of metal particle migration toward the light source. The variation in the surface tension of the glass at the interface between the glass and the metal particle induced by the temperature gradient was calculated via a numerical temperature calculation and compared with the moving speed obtained experimentally.

2. Numerical calculation

First, the temperature distribution of the melted metal particle and the surrounding glass was calculated numerically. Second, the surface tension of the metal particle was estimated from the temperature distribution. Third, the total pressure applied by the surface tension was clarified to be the moving force.

Figure 1 shows a schematic image of the simulation model. Two-dimensional axisymmetric spherical coordinates were used because both the laser beam and the metal particle were symmetrical about the optical axis (z-axis in Fig. 1). The flow around and within a spherical fluid have been derived by Hadamard and Rybczynski under the assumption that the inertia terms are set identically equal to zero everywhere in the flow field [7].

v_{r} = - v_{0} {\frac{1}{2} - 2 (\frac{3 μ' + 2 μ}{4 μ' + 4 μ}) \frac{r_{0}}{r} + 2 (\frac{μ'}{4 μ' + 4 μ}) {(\frac{r_{0}}{r})}^{3}} \cos θ

v_{θ} = v_{0} {1 - (\frac{3 μ' + 2 μ}{4 μ' + 4 μ}) \frac{r_{0}}{r} - (\frac{μ'}{4 μ' + 4 μ}) {(\frac{r_{0}}{r})}^{3}} \sin θ

when

r \geq r_{0}

v_{r}' = 2 v_{0} {{(\frac{r_{0}}{r})}^{2} - 1} (\frac{μ}{4 μ' + 4 μ}) \cos θ

v_{θ}' = - 2 v_{0} {2 {(\frac{r_{0}}{r})}^{2} - 1} (\frac{μ}{4 μ' + 4 μ}) \sin θ

when

r < r_{0}

Fig. 1 Schematic drawing of the simulation model.

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Here, $v_{r}$ and $v_{θ}$ are the flow velocities in the radial and circumferential directions, respectively, $v_{0}$ is the velocity of the metal particle, $r_{0}$ is the radius of the metal particle, and $μ$ and $μ'$ are the viscosities of the surrounding glass and metal, respectively.

Thermal radiation from the surface of the metal particle and heat conduction were considered. The temperature $T (t, r, θ)$ at time t at location $(r, θ)$ was calculated utilizing the following two-dimensional heat conduction equation in the spherical coordinate system:

\begin{array}{l} c ρ \frac{\partial}{\partial t} T (t, r, θ) = \frac{k}{r^{2}} {\frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r} T (t, r, θ)) + \frac{1}{\sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ} T (t, r, θ))} \\ + Q_{l a s} (t, r, θ) - Q_{r a d} (t, r, θ), \end{array}

where c is the specific heat,

ρ

is the density,

k

is the thermal conductivity,

Q_{l a s} (t, r, θ)

is the heat supplied by the laser, and

Q_{r a d} (t, r, θ)

is the heat loss by radiation.

Regarding the laser parameters, a cylindrical Gaussian energy distribution was set. We assumed that the laser energy was completely absorbed on the surface because the absorption depths of metals are small. The heat supplied by the laser is expressed as

Q_{l a s} (t, r, θ) = {\begin{cases} \frac{\partial}{\partial r} (1 - R) I (r, θ) \cos θ (r = r_{0}, 0 \leq θ \leq π / 2) \\ 0 (o t h e r w i s e) \end{cases},

where R is the reflectivity of the metal particle. $I (r, θ)$ is the laser intensity, expressed as

I (r, θ) = \frac{2 P}{π w^{2}} \exp {\frac{- 2 {(r \sin θ)}^{2}}{w^{2}}},

where P is the laser power and w is the spot radius. The heat loss of radiative heat transfer is expressed as

Q_{r a d} (t, r, θ) = {\begin{cases} \frac{\partial}{\partial r} ε β (T^{4} (t, r, θ) - T_{0}^{4}) (r = r_{0}) \\ 0 (r \neq r_{0}) \end{cases},

where ε is the emissivity, β is the Stefan-Boltzmann constant, and

T_{0}

is the environmental temperature.

The initial and boundary conditions were set as

{T (t, r, θ) |}_{t = 0} = T_{0}

{T (t, r, θ) |}_{r = r_{i b}} = T_{0},

where

T_{0}

is the environmental temperature and

r_{i b}

is the value of

r

at the boundary of the calculated region 160 μm from the center of the metal particle, where constant-temperature conditions were applied. Heat Eqs. (5)-(8) were solved by an explicit finite difference method, and flow Eqs. (1)-(4) were included in the calculation using the following values:

v_{0} = 50

μm/s,

r_{0} = 40

μm,

μ = 8.6 \times 10^{- 3}

Pa∙s,

μ' = 6.0 \times 10

Pa∙s,

R = 0.6

,

w = 150

μm,

T_{0} = 293

K,

ε = 0.4

,

β = 5.67 \times 10^{- 8}

W/(m²∙K⁴),

Δ r = 2

μm, and

Δ θ = 7.5

°. The properties of the glass and melted metal are

c = 1.15 \times 10^{3}

J/(kg∙K) and

7.24 \times 10^{2}

J/(kg∙K);

ρ = 2.2 \times 10^{3}

kg/m³ and

7.93 \times 10^{3}

kg/m³; and

k = 2

W/(m∙K) and 49 W/(m∙K), respectively.

v_{0}

is set to 50 μm/s in this calculation under all conditions. The temperature increased upon laser illumination and then became constant after a certain time.

The steady temperature $T (t_{c}, r_{0}, θ)$ at the surface of the metal particle was obtained by the above calculation. The surface tension of the glass $σ (T)$ varies in accordance with the temperature. The pressure applied on the metal particle by the surface tension of the glass was calculated using the Young–Laplace equation

p (T) = 2 \frac{σ (T)}{r_{0}} .

Therefore, the total applied force

F_{s u r}

on the metal particle from the surrounding glass with a temperature distribution was expressed as

F_{s u r} = \int_{0}^{π} 2 π r_{0}^{2} \sin θ \cos θ p (T) d θ = 4 π r_{0} \int_{0}^{π} \sin θ \cos θ σ (T (t_{c}, r_{0}, θ)) d θ .

On the other hand, the viscous resistance force $F_{r e s}$ generated by the movement of the metal particle was expressed under the assumption that the viscosity of the glass was constant.

F_{r e s} = 6 π r_{0} μ V

Here, V is the velocity of particle migration. The temperature dependences of the surface tension

σ (T)

N/m and viscosity

μ (T)

Pa∙s were approximated as

σ (T) = - 5.38 \times 10^{- 5} T + 0.333

and

10 l o g μ (T) = 120740 {(T - 273)}^{- 1.441}

from the data for Glass No. 5 in Fig. 7 in [8], and Fig. 1 in [9], respectively.

The velocity of the particle was calculated using

F_{s u r} - F_{r e s} = m \frac{d V}{d t},

where m is the mass of the particle.

3. Experimental procedure

The laser illumination setup and side-view microscope system used to observe particle migration were similar to those described in [6]. Briefly, borosilicate glass with a thickness of 10 mm (Pyrex®, Corning 7440, Corning Inc.) was used. SUS304 foil (753173 Nilaco Corp.) with a thickness of 10 μm was sandwiched between the glass sample and another glass plate. A Nd:YAG laser beam (Awave1064-50W50K, Advanced Optowave Corp.) with a wavelength of 1064 nm and continuous wave oscillation was focused on the foil through the sample glass by a convex lens with a focal length of 170 mm. The beam mode of the laser was TEM₀₀ (M² < 1.3). The theoretical beam radius was calculated to be ~150 μm from the specifications of the laser oscillator by assuming that M² was 1.3. The laser power was varied in the range of 17.3 – 26.7 W. The laser focus was fixed during laser irradiation. Side-view shadowgraph images under white-light illumination were obtained to monitor the process in situ. Scattered laser light and thermal emission were filtered by a band-pass filter (10BPF10-440, Newport) placed in front of the CCD camera used to obtain images of the particle.

4. Results and discussion

After the start of laser irradiation, a black particle moved backward (toward the light source) from the SUS304 foil. The particle migration stopped when laser illumination ceased. The particle started moving again when the laser illumination was restarted, as shown in Media 1. Figure 2 shows snapshots from Media 1 taken 8 s (a) and 13 s (b) after the start of laser illumination. The particle moved ~360 μm.

Fig. 2 Time-lapse photographs taken during particle migration (Media 1). The image in (b) was taken 5 s after the image in (a). The laser power was 21.3 W.

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The black particle was investigated with an energy-dispersive X-ray spectroscopy system (EDX, Link ISIS 300, Oxford Instruments Plc.) installed with a field-emission scanning electron microscopy system (FE-SEM, JSM-6340, JEOL Ltd.) after cutting to expose the particle. As a result, the elements comprising stainless steel, i.e., iron, nickel, and chrome, were detected from the particle. Therefore, the black particle was an implanted stainless-steel particle.

The trajectory of metal particle migration shown in Fig. 2 could be observed because heating and quenching by laser illumination caused a change in the refractive index [6]. It is noteworthy that the melting of the stainless-steel particle was confirmed, because two particles spontaneously fused upon contact with one another.

The temporal temperature change at the hottest point (r = 40 μm and θ = 0°) was calculated and is shown in Fig. 3. The temperature increases rapidly until ~0.02 s, then gradually increases, then becomes constant ~0.17 s after the start of laser illumination. Figure 4(a) shows the distribution of the temperature at a power of 21.3 W after the temperature became constant. Arrows show the flow within and around the particle. The color scale indicates the temperature. Figure 4(b) shows the temperature on the surface of the particle. The temperature at $θ = 0$ ° (with the optical axis corresponding to the normal line) is the highest (~2270 K) and the temperature on the other side ( $θ = 180$ °) is the lowest (~1950 K) on the surface of the particle. The temperature difference on the surface is ~320 K.

Fig. 3 Temporal behavior of temperature at r = 40 μm, θ = 0°. The laser power was 21.3 W.

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Fig. 4 Temperature distributions (a) in stainless-steel particle and glass and (b) on the surface of the particle. The laser power was 21.3 W. Arrows show the flow within and around the particle. The broken line shows the interface of the particle and glass.

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The force applied to the metal particle from the surrounding glass $F_{s u r}$ was calculated using Eqs. (10) and (12) to be ~100 μN. The temperature and viscosity of the surrounding glass (at r = 54 μm and θ = 90°) were calculated to be 1440 K and 3.8 × 10³ Pa∙s, respectively, from Eq. (13). The migration speed of the particle was calculated to be 34 μm/s by assuming the viscous resistance force $F_{r e s}$ to be equal to the force applied from the surrounding glass $F_{s u r}$ . The experimental particle speed was measured to be ~72 μm/s and corresponded reasonably well with the calculated speed (34 μm/s).

Figure 5 shows the speed of the moving particle after the speed became constant, which was measured for various laser powers, and the calculated speed obtained by the method described above. The migration speed increased with the power, and the maximum speed was ~100 μm/s. When the power was higher than 26.7 W, the glass itself absorbed the light, a phenomenon that was also reported in [10, 11], and the particle did not migrate. The calculated particle speed also increases with increasing laser power, as shown in Fig. 5. However, the variation was larger than that in the experimental results, and the maximum speed was calculated to be 700 μm/s at a power of 26.7 W. The reason for this was considered to be as follows. The viscous resistance force was calculated under the assumption that the particle migrated in a medium with constant viscosity. However, the viscosity increased with increasing distance from the particle, hence only the glass around the particle was fluid. The viscous resistance force was calculated to be smaller than the actual force, particularly when particle speed was high. Therefore, the calculated particle speed was higher than the actual speed.

Fig. 5 Speeds of stainless-steel particle migration under laser illumination at various powers.

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Figure 6 shows the velocity of the particle, which was measured by plotting the position of the particle in Media 1 every second. The particle starts moving ~1 s after the start of illumination, the velocity increases gradually, and then the velocity becomes constant ~8 s after the start of illumination.

Fig. 6 Speed of stainless-steel particle migration under laser illumination. The laser power was 21.3 W.

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The speed of the particle was calculated using Eq. (14) by assuming that the stationary particle was accelerated immediately upon laser illumination with a constant force of 100 μN. The speed of the particle exceeded 90% of its terminal speed in less than 10 ns. Therefore, the speed of the particle became constant within ~0.17 s after the start of laser illumination, considering that the temperature distribution became constant after ~0.17 s as mentioned before.

The calculated duration of the laser illumination after which a particle reached a constant speed was shorter than the experimental result. This is considered to be because the calculated duration after which a constant temperature was reached was much shorter than the experimental duration, since the dependence of the material properties was not considered. In particular, the reflectivity of the solid metal is greater than that of the melted metal; therefore, the calculated temperature increased faster than the experimental temperature.

Finally, the migration force was considered. As discussed in [6], the illumination of light generates other types of force. The force generated by the surface tension gradient was calculated to be ~100 μN. In contrast, for example, the dipole force is on the order of nano-Newtons [3]. Therefore, the force generated by surface tension gradient was much larger than the dipole force; thus, the force generated by the surface tension is believed to be the dominant force causing the particle migration in glass.

5. Conclusion

The driving force of a metal particle toward a light source in glass upon laser illumination was investigated. The variation in the surface tension of the glass at the interface, induced by the temperature gradient, was calculated via a numerical temperature calculation. As a result, the temperature difference on the surface of a stainless-steel particle with a radius of 40 μm was calculated to be ~320 K. The force applied to the metal particle from the surrounding glass was calculated to be ~100 μN, which was approximately equal to the viscous resistance force. In addition, the experimental and numerically calculated speeds of the moving particle were discussed. Surface tension was the dominant force driving the particle toward the light source.

Acknowledgment

Support from Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research (24656096) is gratefully acknowledged.

References and links

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Moving force of metal particle migration induced by laser irradiation in borosilicate glass

Abstract

Corrections

1. Introduction

2. Numerical calculation

3. Experimental procedure

4. Results and discussion

5. Conclusion

Acknowledgment

References and links

Supplementary Material (1)

Cited By

Figures (6)

Equations (16)

Optics Express