**3.3 Physical values and conditions**

In this calculation, Si was selected as the target for laser ablation. Physical properties of Si used in the calculations are shown in Table 1 (Weast, R. C., 1965; Touloukian, Y. S., 1967; AIST Home Page, 2006).

As parameters in the simulation, the atmospheric gas pressure, *P*atm, and target-wall distance, *L*TS, may be varied, but conditions of *P*atm = 100 Pa and *L*TS = 20 mm were most commonly used in the present study. To examine the confinement effect on the nanoparticle formation, however, parametric numerical experiments for *L*TS = 20, 40, 60, 80, and 200 mm were also conducted.


Table 1. Physical values of Si

The parameters for laser irradiation of the target, the surface, and the vapor conditions are shown in Table 2. Here, the Laser energy is the energy per single laser pulse, the Laser fluence is the energy density of laser beam having a diameter of 1 mm, the Surface temperature is the temperature of the target surface resulting from the thermal analysis, and the Vapor temperature and Vapor density at the Knudsen layer are the conditions resulting from the Knudsen layer analysis.

Thermodynamics of Nanoparticle Formation in Laser Ablation 131

Fig. 2. Typical flow field calculated using the methods and conditions presented in Section

Using the same conditions as discussed in the previous section, more detail on the time

Figure 3 (a) shows the time variation of the total mass of nanoparticles in the confined space. The horizontal axis is the elapsed time from laser irradiation. This axis is logarithmic to facilitate simultaneous description of the multiple phenomena occurring over several different time scales. The mass of nanoparticles increases between 0.001 μs and 0.1 μs (Figure 3(a)). After 0.1 μs, the mass becomes constant and begins to rise again at 10 μs. The time of the second mass increase is consistent with the moment at which the reflected shock wave collides with the plume. The time variation of the spatially averaged nucleation rate in the confined space is shown in Figure 3(b). The nucleation rate reaches a maximum value at 0.01 μs. The integrated value of nucleation also increases rapidly in the early stages and then becomes constant (Figure 3(c)), which means that the nucleation phenomenon is completed

The variation of nanoparticle size, which corresponds to the spatially averaged number of atoms constructing the nanoparticle, is shown in Figure 3(d). Since the nanoparticle size starts to increase at 10 μs, when the reflected shock wave arrives at the plume front, it substantially determines the final nanoparticle size, which indicates that the growth of the nanoparticles is facilitated by the effect of the reflected shock wave. Because the nucleation is completed at a very early stage, as already shown, the calculated results also show that

nanoparticle growth can be clearly separated from the nucleation process.

3.3.

very early on.

**3.5 Nucleation and growth** 

variation of the state variables is presented in this section.


Table 2. Parameters for laser irradiation

### **3.4 Typical results for 1D flow**

To substantially separate the nucleation and the growth of nanoparticles and facilitate the formation of uniform-sized nanoparticles, the behavior of the shock wave incidentally generated by laser ablation was investigated.

Nanoparticle evaporation is generally thought to be due to an increase in temperature during the passage of shock waves. Therefore, comparatively weak shock waves, which occur in soft laser ablation, were used to promote nanoparticle growth without the evaporation. When soft laser ablation in the confined space was studied, the shock wave and plume were generated, followed by the collision of the reflected shock wave into the plume front. For verification of these processes, a simulation was also carried out with the one-dimensional compressible fluid equations.

A typical flow profile in the calculation showing the change in densities of the Si vapor, helium gas, and nanoparticles between the target surface and the solid wall are shown in Figure 2. Figure 2(a) indicates these densities in the early stages following laser ablation. In general, the silicon vapor atoms in the plume generated by laser ablation are in the electronically excited state by the high energy of the laser. In the plume front, an emission has been observed with de-excitation based on collisions between the vapor atoms and helium gas. Pushing away helium gas by expansion, the plume gradually increases the density in the front region by reaction. Because the ablation laser pulse is limited to a very short time duration, the plume cannot continue to push away helium gas. The clustering of atomic vapors can thus be promoted in the compressed region of plume due to an increase in supersaturation. In front of the plume, it is clearly shown that a shock wave is formed and propagated in helium gas. A transition is observed wherein the plume propagation speed is greater than the speed of the shock wave (Figures 2(b) to 2(d)). On the other hand, while the peak height of plume density progressively decreases, the spatial density of the nanoparticles continues to increase. The shock wave crashes into the right side wall and reflects to the left (Figures 2(e) and 2(f)). In addition, the peak position of nanoparticle density is slightly shifted from the peak position of vapor density. The shock wave is strengthened by reflection to the right side wall, followed by collision with the plume (Figure 2(g)). Figure 2(h) shows the state just after the collision between the reflected shock wave and the plume. The shock wave penetrates into the plume, enhanced the plume density, and thus slightly pushes it back to the left (Figure 2(i)). When the shock wave has completely passed through the plume, the spatial density of nanoparticles effectively increases(Figure 2(j)).

To substantially separate the nucleation and the growth of nanoparticles and facilitate the formation of uniform-sized nanoparticles, the behavior of the shock wave incidentally

Nanoparticle evaporation is generally thought to be due to an increase in temperature during the passage of shock waves. Therefore, comparatively weak shock waves, which occur in soft laser ablation, were used to promote nanoparticle growth without the evaporation. When soft laser ablation in the confined space was studied, the shock wave and plume were generated, followed by the collision of the reflected shock wave into the plume front. For verification of these processes, a simulation was also carried out with the

A typical flow profile in the calculation showing the change in densities of the Si vapor, helium gas, and nanoparticles between the target surface and the solid wall are shown in Figure 2. Figure 2(a) indicates these densities in the early stages following laser ablation. In general, the silicon vapor atoms in the plume generated by laser ablation are in the electronically excited state by the high energy of the laser. In the plume front, an emission has been observed with de-excitation based on collisions between the vapor atoms and helium gas. Pushing away helium gas by expansion, the plume gradually increases the density in the front region by reaction. Because the ablation laser pulse is limited to a very short time duration, the plume cannot continue to push away helium gas. The clustering of atomic vapors can thus be promoted in the compressed region of plume due to an increase in supersaturation. In front of the plume, it is clearly shown that a shock wave is formed and propagated in helium gas. A transition is observed wherein the plume propagation speed is greater than the speed of the shock wave (Figures 2(b) to 2(d)). On the other hand, while the peak height of plume density progressively decreases, the spatial density of the nanoparticles continues to increase. The shock wave crashes into the right side wall and reflects to the left (Figures 2(e) and 2(f)). In addition, the peak position of nanoparticle density is slightly shifted from the peak position of vapor density. The shock wave is strengthened by reflection to the right side wall, followed by collision with the plume (Figure 2(g)). Figure 2(h) shows the state just after the collision between the reflected shock wave and the plume. The shock wave penetrates into the plume, enhanced the plume density, and thus slightly pushes it back to the left (Figure 2(i)). When the shock wave has completely passed through the plume, the spatial density of nanoparticles

Table 2. Parameters for laser irradiation

generated by laser ablation was investigated.

one-dimensional compressible fluid equations.

**3.4 Typical results for 1D flow** 

effectively increases(Figure 2(j)).

Fig. 2. Typical flow field calculated using the methods and conditions presented in Section 3.3.

### **3.5 Nucleation and growth**

Using the same conditions as discussed in the previous section, more detail on the time variation of the state variables is presented in this section.

Figure 3 (a) shows the time variation of the total mass of nanoparticles in the confined space. The horizontal axis is the elapsed time from laser irradiation. This axis is logarithmic to facilitate simultaneous description of the multiple phenomena occurring over several different time scales. The mass of nanoparticles increases between 0.001 μs and 0.1 μs (Figure 3(a)). After 0.1 μs, the mass becomes constant and begins to rise again at 10 μs. The time of the second mass increase is consistent with the moment at which the reflected shock wave collides with the plume. The time variation of the spatially averaged nucleation rate in the confined space is shown in Figure 3(b). The nucleation rate reaches a maximum value at 0.01 μs. The integrated value of nucleation also increases rapidly in the early stages and then becomes constant (Figure 3(c)), which means that the nucleation phenomenon is completed very early on.

The variation of nanoparticle size, which corresponds to the spatially averaged number of atoms constructing the nanoparticle, is shown in Figure 3(d). Since the nanoparticle size starts to increase at 10 μs, when the reflected shock wave arrives at the plume front, it substantially determines the final nanoparticle size, which indicates that the growth of the nanoparticles is facilitated by the effect of the reflected shock wave. Because the nucleation is completed at a very early stage, as already shown, the calculated results also show that nanoparticle growth can be clearly separated from the nucleation process.

Thermodynamics of Nanoparticle Formation in Laser Ablation 133

The time variation of the nanoparticle size and the positions of the shock wave and the plume, which were defined above, are shown in Figure 4. The left vertical axis is the nanoparticle radius, the right vertical axis is the position in the calculation region, and the horizontal axis is the elapsed time from laser irradiation. The dashed line, thick solid line, and the shaded area represent the nanoparticle size, the position of the shock wave, and the plume front, respectively. The shock waves are propagated backward and forward in the space by reflecting on the target surface and the opposed wall. The width of the shaded area, which represents the plume front, gradually broadens. In addition, the first, Tc1, and second, Tc2, times when the shock wave interferes with the plume front are shown. This interference can be seen as opportunities to enhance the growth rate of nanoparticles. The slope of the dashed line in Figure 4 represents the nanoparticle growth rate, which changes from 17.5 to 52.0 μm/s at Tc1, and from 16.0 to 34.2 μm/s at Tc2. Referring back to Eq. (7), the growth rate of the nanoparticles was determined by a kinetic balance between the condensation rate of nanoparticles, which is based on a macroscopic collision cross-section of the ambient vapors, and the evaporation rate of nanoparticles corresponding to the nanoparticle temperature. Therefore, the fact that the nanoparticle growth rate increases when the shock wave and plume collide means that the shock wave effectively increases the

Fig. 4. Increase of nanoparticle radius from interference between shock wave and plume

before the condensation rate balances the evaporation rate.

To investigate the effect of the distance between the target surface and the solid wall on the rate of nanoparticle growth enhanced by the shock wave passage, the numerical simulation was performed under the following conditions: *L*TS = 20, 40, 60, 80, and 200 mm. The calculated results for the increase of nanoparticle radius are indicated in Figure 5 against the elapsed time from laser irradiation. Nanoparticle growth was promoted by the passage of the shock wave under all of these conditions. The nanoparticle radius, *r*, increased with time and eventually reaches a constant value. A balance between the evaporation rate and condensation rate is reached at the maximum radius, and the growth rate of nanoparticles asymptotically approaches zero. When the radius of the nanoparticle is compared among the various distances between the target surface and the solid wall, the shorter *L*TS resulted in a larger value of *r*. Therefore, larger nanoparticles can be obtained with smaller distances because there are more opportunities for the shock waves to pass through the plume front

macroscopic collision cross-section.

Fig. 3. Time variation of nucleation and growth of nanoparticles

### **3.6 Influence of confinement**

The change in nanoparticle size over time was also examined; nanoparticle size increased when the shock wave hit the plume front. Before examining this process further, however, the typical nanoparticle size, as well as the locations of the plume front and the shock wave, must be clearly defined.

There is a definite relationship between the size and spatial density of nanoparticles. The nanoparticle size generally has a distribution, which is especially large in the region of the plume front. The width of the nanoparticle density distribution is smaller than the spread in nanoparticle size and has a sharper distribution profile. The peak positions of the two distributions are almost identical. This means that the maximum nanoparticle size is placed at the location where the nanoparticle density is also at a maximum. Therefore, the typical nanoparticle size in the space can be regarded as the maximum nanoparticle size.

The location of the shock wave propagating through the ambient gas is defined as the maximum value of the derivative for the change in gas density. On the other hand, the plume front is defined as the compression region in the atomic vapor, which comes into contact with the atmospheric gas and high-density area.

Fig. 3. Time variation of nucleation and growth of nanoparticles

contact with the atmospheric gas and high-density area.

The change in nanoparticle size over time was also examined; nanoparticle size increased when the shock wave hit the plume front. Before examining this process further, however, the typical nanoparticle size, as well as the locations of the plume front and the shock wave,

There is a definite relationship between the size and spatial density of nanoparticles. The nanoparticle size generally has a distribution, which is especially large in the region of the plume front. The width of the nanoparticle density distribution is smaller than the spread in nanoparticle size and has a sharper distribution profile. The peak positions of the two distributions are almost identical. This means that the maximum nanoparticle size is placed at the location where the nanoparticle density is also at a maximum. Therefore, the typical

The location of the shock wave propagating through the ambient gas is defined as the maximum value of the derivative for the change in gas density. On the other hand, the plume front is defined as the compression region in the atomic vapor, which comes into

nanoparticle size in the space can be regarded as the maximum nanoparticle size.

**3.6 Influence of confinement** 

must be clearly defined.

The time variation of the nanoparticle size and the positions of the shock wave and the plume, which were defined above, are shown in Figure 4. The left vertical axis is the nanoparticle radius, the right vertical axis is the position in the calculation region, and the horizontal axis is the elapsed time from laser irradiation. The dashed line, thick solid line, and the shaded area represent the nanoparticle size, the position of the shock wave, and the plume front, respectively. The shock waves are propagated backward and forward in the space by reflecting on the target surface and the opposed wall. The width of the shaded area, which represents the plume front, gradually broadens. In addition, the first, Tc1, and second, Tc2, times when the shock wave interferes with the plume front are shown. This interference can be seen as opportunities to enhance the growth rate of nanoparticles. The slope of the dashed line in Figure 4 represents the nanoparticle growth rate, which changes from 17.5 to 52.0 μm/s at Tc1, and from 16.0 to 34.2 μm/s at Tc2. Referring back to Eq. (7), the growth rate of the nanoparticles was determined by a kinetic balance between the condensation rate of nanoparticles, which is based on a macroscopic collision cross-section of the ambient vapors, and the evaporation rate of nanoparticles corresponding to the nanoparticle temperature. Therefore, the fact that the nanoparticle growth rate increases when the shock wave and plume collide means that the shock wave effectively increases the macroscopic collision cross-section.

Fig. 4. Increase of nanoparticle radius from interference between shock wave and plume

To investigate the effect of the distance between the target surface and the solid wall on the rate of nanoparticle growth enhanced by the shock wave passage, the numerical simulation was performed under the following conditions: *L*TS = 20, 40, 60, 80, and 200 mm. The calculated results for the increase of nanoparticle radius are indicated in Figure 5 against the elapsed time from laser irradiation. Nanoparticle growth was promoted by the passage of the shock wave under all of these conditions. The nanoparticle radius, *r*, increased with time and eventually reaches a constant value. A balance between the evaporation rate and condensation rate is reached at the maximum radius, and the growth rate of nanoparticles asymptotically approaches zero. When the radius of the nanoparticle is compared among the various distances between the target surface and the solid wall, the shorter *L*TS resulted in a larger value of *r*. Therefore, larger nanoparticles can be obtained with smaller distances because there are more opportunities for the shock waves to pass through the plume front before the condensation rate balances the evaporation rate.

Thermodynamics of Nanoparticle Formation in Laser Ablation 135

volume method using the MUSCL-type total variation diminishing (TVD) scheme with a

curvilinear generalized coordinate (Yaga, M., 2005, 2008; Fukuoka, F., 2008)

Fig. 6. Behavior of plume and shock wave in an ellipsoidal cell.

The contours of the wall are calculated by the following equation:

where, *a* and *b* are constants with a relation of *a b* 1 52 .

2 2 2 2 <sup>1</sup> *<sup>x</sup> <sup>y</sup> a b*

For the boundary conditions, non-slip conditions are applied to the cell wall, except for the position of the plume ejection. Outgoing flow conditions are applied to the boundaries outside the cell. The position of the plume ejection is set at one of the focal points of the ellipsoidal cell. The sudden ejection generates a traveling shock wave which is converges at the other focal point. The cell exit, through which the flow passes during the propagations of the shock and pressure waves, connects the inside and outside of the cell. During the focusing process of the propagating shock wave, the interaction between the converging shock wave and plume plays an important role in the growing nanoparticle size. An ejected jet of gas is shut off after a certain period so that the calculation can be used for a basic reference for PLA techniques. Then, the ejected gas is considered to be a plume traveling toward the exit of the cell on the right side wall. It is clear that many parameters are involved in this process. We have chosen the three main parameters to be the Mach number, jet duration, and diameter of the exit hole, because, in related experiments, the controllable

(16)

**4.2 Boundary and initial conditions** 

Fig. 5. Influence of distance between target and wall on nanoparticle growth. The circles indicate the arrival time of the shock wave.
