**2. Thermodynamics of nanoparticle formation**

### **2.1 Nucleation and growth of nanoparticles**

124 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

gaseous materials, combined with the surrounding atmosphere, to provide adequate

Due to its importance in both academia and industry, the chemical thermodynamics of nanoparticle formation in the gaseous phase have been studied extensively (Finney, E. E., 2008). Two processes are important in these studies: (i) homogeneous nucleation, whereby vapors generated in the PLA process reach super-saturation and undergo rapid phase change, and (ii) growth, during which the nanoparticles continue to grow by capturing surrounding atoms and nuclei in the vapor. The size and generation rate of critical nuclei are important factors for understanding the homogeneous nucleation process. To evaluate the generation rate of critical nuclei, we need to know the partition function of each size of nuclei. If an assembled mass of each size of nuclei can be regarded as a perfect gas, then the partition functions can be calculated using statistical thermodynamic methods. However, because it is generally difficult to directly calculate the nucleus partition function and incorporate the calculated results into continuous fluid dynamics equations, what has been used in practice is the so-called surface free energy model, in which the Gibbs free energy of the nanoparticles is represented by the chemical potential and surface free energy of the bulk materials. In contrast, a kinetic theory has been used for treating the mutual interference following nucleation, such as nanoparticle condensation, evaporation,

aggregation, coalescence, and collapse, in the nanoparticle growth process.

example, continuous fluid dynamics with a classical nucleation model.

nanoparticle generator.

conclusions are given in Section 6.

Since statistical thermodynamics is a valid approach for understanding the mechanisms of nanoparticle formation, microscopic studies have increased aggressively in recent years. In the case of using a deposition process of nanoparticles for thin-film fabrication for industrial use, however, it is necessary to optimize the process by regulating the whole flow field of nanoparticle formation. In cases in which several vapors (plumes) generated during laser ablation are identified as a continuous fluid, macroscopic studies are needed using, for

Some studies have evaluated the thermodynamics and fluid dynamics that are involved in nanoparticle formation by using tools such as numerical analysis with an evaporation model, a blast wave model, and a plasma model. However, the shock waves generated in the early stage of PLA result in extensive reflection and diffraction which increasingly complicate clarification of the nanoparticle formation process. Up to now, no attempt to introduce shock wave generation and reflection into the plume dynamics has been reported in relation to nanoparticle formation. We note in particular that thermodynamic confinement could occur at the points of interference between the shock wave and the plume, and that nanoparticles with uniform thermodynamic state variables subsequently could be formed in the confinement region, thus making such a system a new type of

In Section 2 of the present chapter, we review the thermodynamics and fluid dynamics of nanoparticle formation during PLA. After providing analytical methods and models of 1D flow calculation in Section 3, we present the calculation results for laser-irradiated material surfaces, sudden evaporation from the surfaces, Knudsen layer formation, plume progression, and shock wave generation, propagation, and reflection. Extensive 2D flow calculation results (without nanoparticle formation) are presented in Section 4 to explore the flow patterns inside the new type of nanoparticle generator. The experimental results for the various nanoparticles formed by the generator are presented in Section 5. Finally,

conditions for nucleation and subsequent growth.

In nanoparticle formation, the following stages must be considered: (i) homogeneous nucleation, where vapor atoms produced by laser ablation have been supersaturated, and (ii) particle growth, where the critical nuclei are growing, capturing atoms on their surfaces, and making the transition into large particles.

At the first stage of homogeneous nucleation, the nucleation rate and the size of critical nuclei are important factors. The nucleation rate, *I*, is the number of nuclei that are created per unit volume per unit time. To evaluate the nucleation rate, the number density of nanoparticles at equilibrium is needed. In the present case, it is assumed that the nanoparticles are grown only in the capture of a single molecule without causing other nuclei to collapse. That is, when a nanoparticle consisting of *i* atoms is indicated by *A*<sup>i</sup> (hereinafter, *i*-particle), the reaction process related to the nanoparticle formation is expressed as follows:

$$\begin{aligned} A\_1 + A\_1 &\leftrightarrow A\_2\\ A\_2 + A\_1 &\leftrightarrow A\_3\\ \dots &, \dots &, \dots \dots \\ A\_{i-1} + A\_1 &\leftrightarrow A\_i \end{aligned} \tag{1}$$

If the molecular partition functions of the various sizes of nanoparticle are derived by statistical mechanical procedure, the equilibrium constants for each equation are known. As a result, the number density of the nanoparticles at equilibrium can be inferred assuming ideal gas behavior. Namely, the equilibrium constant *K*i-1,i between (*i*-1)-particle and *i*particle is

$$K\_{i-1,i} = \frac{Q\_i}{Q\_{i-1}Q\_1} \exp\left(\frac{D\_{i-1,i}}{kT}\right) \tag{2}$$

Here, *Q*i is the *i*-particle partition function, *D*i-1,i is the dissociation energy of one atom for the *i*-particle, *k* is the Boltzmann constant, and *T* is the temperature of the system. In general, to explicitly calculate the Gibbs' free energy change from the molecular partition function of nanoparticles and to incorporate these into a continuous fluid dynamics equation are extremely difficult. Therefore, the so-called surface free energy model, where Gibbs' free energy change is represented by the surface tension and chemical potential of bulk materials, can be adopted. Furthermore, when assuming a steady reaction process for nanoparticle formation, the critical nucleation rate, *I*, is represented as (Volmer, M., 1939)

$$I = \frac{n^2 c \upsilon\_c}{4 \pi \epsilon} \sqrt{\frac{3 \mathcal{W} \mathcal{N}\_\*}{\pi kT}} \exp\left(-\frac{\mathcal{W} \ast}{kT}\right) \tag{3}$$

where *n* is the number density of species in the vapor, *c* is the average relative speed between nanoparticles and atomic vapor, *v*c is the volume per atom in the vapor, *r*\* is the radius of the critical nuclei, *W*\* is the energy of formation for critical nuclei, *k* is Boltzman constant, and *T* is the temperature of the system. The exponential term appeared in the above formula seems to be an essential factor for thermodynamic considerations in

Thermodynamics of Nanoparticle Formation in Laser Ablation 127

In the view of gas dynamics, the PLA process can be classified into (i) evaporation of the target material and (ii) hydrodynamic expansion of the ablated plume into the ambient gas. We make the approximation herein of a pure thermal evaporation process and neglect the interaction between the evaporated plume and the incident laser beam. For the fairly short laser pulses (∼10 ns) that are typical for PLA experiments, it is reasonable to consider the above two processes as adjacent stages. The energy of the laser irradiation is spent heating, melting, and evaporating the target material. The surface temperature of the target can be computed using the heat flow equation (Houle, F. A., 1998). For very high laser fluences, the surface temperature approaches the maximum rapidly during the initial few nanoseconds of the pulse. The evaporation process becomes important when the surface temperature of target approaches the melting point. With the laser fluence and pulse duration we considered, thermally activated surface vaporization can reasonably be used to describe the evaporation due to pulsed laser irradiation of the target. The saturated vapor pressure, *p*v, in equilibrium at the target surface can be calculated using the Clausius–Clapeyron equation from the surface

temperature, *T*s. The flux of vapor atoms leaving the surface can be written as

<sup>0</sup> 1

*n T g T*

0 0

0

*u*

*T*

calculated by using the above equations (Han, M., 2002).

2 *v s <sup>p</sup> <sup>J</sup> kT m* 

where η(≈1) denotes the sticking coefficient of surface atoms and *m* is the atomic mass of the vapor atom. The total number of ablated atoms is an integration of *J* over time and

To obtain the initial condition for vapor expansion problem, we can perform a Knudsen layer analysis to get the idealized states of the gas just leaving the Knudsen layer (Knight, C. J., 1979). The local density, *n*0, mean velocity, *u*0, and temperature, *T*0, of the vapor just outside the Knudsen layer can be calculated from the jump conditions and may be deduced

2

 

8 8 *<sup>s</sup> T g g*

2 2 <sup>0</sup> <sup>2</sup>

0

*kT*

*<sup>m</sup>* 

where *n*s is the saturated vapor density at the target surface g is a function of Mach number and *κ* is the adiabatic index. The idealized states just beyond the Knudsen layer are

Since the processes described above for nanoparticle formation arise in the high temperature plume generated by laser ablation, it is important to know the thermodynamic state of the

1 1 <sup>1</sup> 2 2 *s s g g*

*<sup>g</sup> e erfc g ge erfc <sup>g</sup> n T <sup>T</sup>*

(8)

(9)

(10)

(11)

2

**2.2 Thermal analysis and Knudsen layer analysis** 

surface area.

very simply using

*s*

**3. One dimensional flow problems 3.1 Fluid dynamics of laser ablated plume** 

nucleation process. The supersaturation, *S*, which is implicitly included in the variable *W*\*, is a dominant factor which significantly affects nucleation rate, *I*.

Once the Gibbs' free energy change, *G*, is known, the critical nucleus radius, *r*\*, can be easily obtained. For this, an assumption of capillary phenomena (capillarity assumption) is used as a condition for mechanical equilibrium of the particles and the extreme value at *dG*=0 may be considered. When the surface tension of the nanoparticle is depicted by σ, the radius of critical nucleus is

$$n\_\* = \frac{2\sigma v\_c}{kT \ln S} \tag{4}$$

Here, as in the case of nucleation rate, the degree of supersaturation, *S*, is what determines the size of the critical nucleus.

Next, it was assumed for convenience that the nanoparticle growth first occurred after its nucleus reached the critical nucleus size. In other words, the Gibbs' free energy of nanoparticle formation begins to decrease after it reaches maximum value at the critical nucleus size. At this time, the number of atomic vapor species condensing per unit area of particle surface per unit time, *β*, can be determined using the number density, *Nr*, of the species in the atomic vapor near the surface of a nanoparticle possessing radius, *r*, and assuming the equilibrium Maxwell-Boltzman distribution,

$$
\beta = \xi \mathbf{N}\_r \sqrt{\frac{kT}{2\pi m}} \tag{5}
$$

Here, *ξ* is the condensation coefficient, which represents the ratio of the number of condensing atoms to colliding atoms, and *m* is the mass of the vapor species. When the vapor species are in equilibrium with the nanoparticles, the number density is represented by *N*r,eq and the number of atoms evaporating, *α*, from the nanoparticle surface per unit time and area is given by

$$a = \xi \mathbf{N}\_{r, \alpha \eta} \sqrt{\frac{kT}{2\pi m}} \tag{6}$$

Therefore, the growth rate of the nanoparticle radius is

$$\frac{dr}{dt} = (\beta - a)v\_c \tag{7}$$

In this equation, the variable *α* is the equilibrium value corresponding to the temperature of the nanoparticle, while the kinetic parameters of the surrounding vapors, which affect significantly the variable *β*, are dominant.

As mentioned above, when the two processes of nanoparticle nucleation and growth are considered, each parameter governing the processes is different. That is, the degree of supersaturation dominates as a non-equilibrium thermodynamic parameter for nucleation, while the state variables related to the surrounding vapors are important as molecular kinetic parameters for particle growth. Thus, separating the nucleation and growth processes in time by using the difference, could hypothetically lead to the formation of nanoparticles of uniform size.

### **2.2 Thermal analysis and Knudsen layer analysis**

126 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

nucleation process. The supersaturation, *S*, which is implicitly included in the variable *W*\*, is

Once the Gibbs' free energy change, *G*, is known, the critical nucleus radius, *r*\*, can be easily obtained. For this, an assumption of capillary phenomena (capillarity assumption) is used as a condition for mechanical equilibrium of the particles and the extreme value at *dG*=0 may be considered. When the surface tension of the nanoparticle is depicted by σ, the radius of

> 2 ln *c v*

Here, as in the case of nucleation rate, the degree of supersaturation, *S*, is what determines

Next, it was assumed for convenience that the nanoparticle growth first occurred after its nucleus reached the critical nucleus size. In other words, the Gibbs' free energy of nanoparticle formation begins to decrease after it reaches maximum value at the critical nucleus size. At this time, the number of atomic vapor species condensing per unit area of particle surface per unit time, *β*, can be determined using the number density, *Nr*, of the species in the atomic vapor near the surface of a nanoparticle possessing radius, *r*, and

> 2 *<sup>r</sup> kT <sup>N</sup>*

, 2 *r eq kT <sup>N</sup>*

*<sup>c</sup>*

In this equation, the variable *α* is the equilibrium value corresponding to the temperature of the nanoparticle, while the kinetic parameters of the surrounding vapors, which affect

As mentioned above, when the two processes of nanoparticle nucleation and growth are considered, each parameter governing the processes is different. That is, the degree of supersaturation dominates as a non-equilibrium thermodynamic parameter for nucleation, while the state variables related to the surrounding vapors are important as molecular kinetic parameters for particle growth. Thus, separating the nucleation and growth processes in time by using the difference, could hypothetically lead to the formation of

*m*

*v*

Here, *ξ* is the condensation coefficient, which represents the ratio of the number of condensing atoms to colliding atoms, and *m* is the mass of the vapor species. When the vapor species are in equilibrium with the nanoparticles, the number density is represented by *N*r,eq and the number of atoms evaporating, *α*, from the nanoparticle surface per unit time

 

 

*dr*

*dt*  *m*

*kT S* 

(4)

(5)

(6)

(7)

\*

*r*

a dominant factor which significantly affects nucleation rate, *I*.

assuming the equilibrium Maxwell-Boltzman distribution,

Therefore, the growth rate of the nanoparticle radius is

significantly the variable *β*, are dominant.

nanoparticles of uniform size.

critical nucleus is

and area is given by

the size of the critical nucleus.

In the view of gas dynamics, the PLA process can be classified into (i) evaporation of the target material and (ii) hydrodynamic expansion of the ablated plume into the ambient gas. We make the approximation herein of a pure thermal evaporation process and neglect the interaction between the evaporated plume and the incident laser beam. For the fairly short laser pulses (∼10 ns) that are typical for PLA experiments, it is reasonable to consider the above two processes as adjacent stages. The energy of the laser irradiation is spent heating, melting, and evaporating the target material. The surface temperature of the target can be computed using the heat flow equation (Houle, F. A., 1998). For very high laser fluences, the surface temperature approaches the maximum rapidly during the initial few nanoseconds of the pulse. The evaporation process becomes important when the surface temperature of target approaches the melting point. With the laser fluence and pulse duration we considered, thermally activated surface vaporization can reasonably be used to describe the evaporation due to pulsed laser irradiation of the target. The saturated vapor pressure, *p*v, in equilibrium at the target surface can be calculated using the Clausius–Clapeyron equation from the surface temperature, *T*s. The flux of vapor atoms leaving the surface can be written as

$$J = \frac{\eta p\_v}{\sqrt{2\pi kT\_s m}}\tag{8}$$

where η(≈1) denotes the sticking coefficient of surface atoms and *m* is the atomic mass of the vapor atom. The total number of ablated atoms is an integration of *J* over time and surface area.

To obtain the initial condition for vapor expansion problem, we can perform a Knudsen layer analysis to get the idealized states of the gas just leaving the Knudsen layer (Knight, C. J., 1979). The local density, *n*0, mean velocity, *u*0, and temperature, *T*0, of the vapor just outside the Knudsen layer can be calculated from the jump conditions and may be deduced very simply using

$$\frac{T\_0}{T\_s} = \left[\sqrt{1 + \pi \left(\frac{\mathcal{S}}{8}\right)^2} - \sqrt{\pi} \frac{\mathcal{S}}{8}\right]^2\tag{9}$$

$$\frac{m\_0}{m\_s} = \sqrt{\frac{T\_s}{T\_0}} \left[ \left( \text{g}^2 + \frac{1}{2} \right) e^{\text{g}^2} \text{erfc}\left( \text{g} \right) - \frac{\text{g}}{\sqrt{\pi}} \right] + \frac{1}{2} \frac{T\_s}{T\_0} \left[ 1 - \sqrt{\pi} \text{g} e^{\text{g}^2} \text{erfc}\left( \text{g} \right) \right] \tag{10}$$

$$
\mu\_0 = \sqrt{\kappa \frac{kT\_0}{m}}\tag{11}
$$

where *n*s is the saturated vapor density at the target surface g is a function of Mach number and *κ* is the adiabatic index. The idealized states just beyond the Knudsen layer are calculated by using the above equations (Han, M., 2002).

### **3. One dimensional flow problems**

#### **3.1 Fluid dynamics of laser ablated plume**

Since the processes described above for nanoparticle formation arise in the high temperature plume generated by laser ablation, it is important to know the thermodynamic state of the

Thermodynamics of Nanoparticle Formation in Laser Ablation 129

plume is formed forward. With the expansion of the plume, the ambient gas that originally

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;

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

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

filled the space is pushed away to the right and towards the solid wall.

Fig. 1. Calculation model for 1D flow

**3.3 Physical values and conditions** 

AIST Home Page, 2006).

were also conducted.

Table 1. Physical values of Si

from the Knudsen layer analysis.

species in the plume. The one-dimensional unsteady Euler compressible fluid equation can be obtained using the numerical scheme in order to solve the thermodynamic state of the plume species, as well as to understand the nanoparticle nucleation and growth. Discretization of the system equation was driven by a finite volume method in which a total variation diminishing (TVD) scheme for capturing the shock wave was adopted as a numerical viscosity term. In the present study, because the time evolution of the plume and shock wave interference need to be considered, a three-order precision Runge-Kutta scheme was used as the accurate time calculation.

The conservation equations of mass, momentum, and energy, which describe the behavior of the laser plume in an ambient gas, are as follows (Shapiro, A. H., 1953),

$$\frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial \mathbf{E}}{\partial \mathbf{x}} = \mathbf{W} \tag{12}$$

where

$$\mathbf{Q} = \begin{bmatrix} \rho\_v & \rho\_g & \rho\_m u & e & \mathbf{C}\_1 & \mathbf{C}\_2 & \mathbf{C}\_3 & \mathbf{C}\_4 \end{bmatrix}^T \tag{13}$$

$$\mathbf{E} = \begin{bmatrix} \rho\_v \mu & \rho\_g \mu & p + \rho\_m \mu^2 & (e+p)\mu & \mathbf{C}\_1 \mu & \mathbf{C}\_2 \mu & \mathbf{C}\_3 \mu & \mathbf{C}\_4 \mu \end{bmatrix}^\mathrm{T} \tag{14}$$

$$\mathbf{W} = \begin{bmatrix} -\dot{\rho}\_c & 0 & 0 & \lambda \dot{\rho}\_c & I & \dot{r} \mathbf{C}\_1 & 2\dot{r} \mathbf{C}\_3 & 4\pi \dot{r} \mathbf{C}\_3 \end{bmatrix}^\mathrm{T} \tag{15}$$

Here, *x* and *t* are distance and time, respectively, and the variables *ρ*, *u*, *p* and *e* are the density, velocity, pressure, and the total energy per unit volume, respectively. The subindices for the vapor, the ambient gas, and the gas mixture are expressed respectively as *v*, *g* and *m*. Moreover, *λ* is latent heat for the bulk material of the naonoparticle. In addition, the dotted variables and *r* represent the time derivative related to the density and the radius of nanoparticle, respectively. *C*1, *C*2, *C*3, and *C*4 are transient intermediate variables; among these, the last variable, *C*4, also represents the nanoparticle density, *ρ*c.

### **3.2 Calculation model for 1D flow**

Figure 1 shows a numerical calculation model of nanoparticle formation during laser ablation. The one-dimensional computational domain, also called the confined space in the present study, is surrounded by a solid wall on the left and a laser target on the right (Takiya, T., 2007, 2010). The confined space is initially filled with ambient gas. The figure represents the initial state of the flow field immediately after laser irradiation. The target surface is melted by laser irradiation and then saturated vapor of high temperature and pressure is present near the surface. Outside it, the Knudsen layer, the non-equilibrium thermodynamic region where Maxwell-Boltzmann velocity distribution is not at equilibrium, appears. Following the Knudsen layer is the initial plume expansion, which is the equilibrium thermodynamic process. In this case, the high temperature and high pressure vapor, which is assumed to be in thermodynamic equilibrium, is on the outer side of the Knudsen layer and is given as the initial conditions for a shock tube problem. In the calculation, the high temperature and high pressure vapor is suddenly expanded, and a

species in the plume. The one-dimensional unsteady Euler compressible fluid equation can be obtained using the numerical scheme in order to solve the thermodynamic state of the plume species, as well as to understand the nanoparticle nucleation and growth. Discretization of the system equation was driven by a finite volume method in which a total variation diminishing (TVD) scheme for capturing the shock wave was adopted as a numerical viscosity term. In the present study, because the time evolution of the plume and shock wave interference need to be considered, a three-order precision Runge-Kutta scheme

The conservation equations of mass, momentum, and energy, which describe the behavior

**Q E <sup>W</sup>** (12)

T

(15)

*vgmueC C C C* <sup>1234</sup> **<sup>Q</sup>** (13)

<sup>T</sup> <sup>2</sup>

*vg m u u p u e pu Cu Cu Cu Cu* <sup>1234</sup> **<sup>E</sup>** (14)

and *r* represent the time derivative related to the density and the radius

 <sup>T</sup> 13 3 0 0 2 4 *c c* **W**

*I rC rC rC*

Here, *x* and *t* are distance and time, respectively, and the variables *ρ*, *u*, *p* and *e* are the density, velocity, pressure, and the total energy per unit volume, respectively. The subindices for the vapor, the ambient gas, and the gas mixture are expressed respectively as *v*, *g* and *m*. Moreover, *λ* is latent heat for the bulk material of the naonoparticle. In addition, the

of nanoparticle, respectively. *C*1, *C*2, *C*3, and *C*4 are transient intermediate variables; among

Figure 1 shows a numerical calculation model of nanoparticle formation during laser ablation. The one-dimensional computational domain, also called the confined space in the present study, is surrounded by a solid wall on the left and a laser target on the right (Takiya, T., 2007, 2010). The confined space is initially filled with ambient gas. The figure represents the initial state of the flow field immediately after laser irradiation. The target surface is melted by laser irradiation and then saturated vapor of high temperature and pressure is present near the surface. Outside it, the Knudsen layer, the non-equilibrium thermodynamic region where Maxwell-Boltzmann velocity distribution is not at equilibrium, appears. Following the Knudsen layer is the initial plume expansion, which is the equilibrium thermodynamic process. In this case, the high temperature and high pressure vapor, which is assumed to be in thermodynamic equilibrium, is on the outer side of the Knudsen layer and is given as the initial conditions for a shock tube problem. In the calculation, the high temperature and high pressure vapor is suddenly expanded, and a

*t x* 

of the laser plume in an ambient gas, are as follows (Shapiro, A. H., 1953),

 

these, the last variable, *C*4, also represents the nanoparticle density, *ρ*c.

 

was used as the accurate time calculation.

where

dotted variables

**3.2 Calculation model for 1D flow** 

plume is formed forward. With the expansion of the plume, the ambient gas that originally filled the space is pushed away to the right and towards the solid wall.

Fig. 1. Calculation model for 1D flow
