**2. The formation process of nanoparticles by laser ablation**

#### **2.1 Laser irradiation analysis**

The process of laser ablation commences from the interaction of lights and solids. When a laser irradiates on the surface of a solid target, the electromagnetic energy of the laser beam firstly turns into the excitation energy of electron, although the process may vary depending on the thermodynamic state of the solid. The energy conversion in the solid from the electronic excitation to the lattice vibration completes within some picoseconds. If the laser has nanosecond pulse, therefore, it is not necessary to take into consideration the non-equilibrium of the laser irradiation. Then, when viewed as a one-dimensional problem in the depth direction, the solid temperature can be described as the usual one-dimensional unsteady-state heat conduction Equation [7].

$$\frac{\partial T}{\partial t} = \frac{\partial}{\partial \alpha} \left( \kappa \frac{\partial T}{\partial \alpha} \right) + \frac{\alpha}{c\_p \rho} I(\kappa, t) \tag{1}$$

**105**

the Knudsen layer.

*Nanoparticle Formation and Deposition by Pulsed Laser Ablation*

ably little [11] so as to take the following velocity distribution [12].

*<sup>f</sup> kT*

*s*

Maxwell equilibrium distribution as following [12]:

**2.3 Plume expansion and shock wave propagation**

*<sup>f</sup> kT*

*K*

±

+

If the vapor pressure on the surface of the solids during evaporation is equal to the ambient pressure, the Maxwell velocity distribution for both evaporating and recondensing gas takes half shape of the distribution *f* <sup>+</sup> or *f* <sup>−</sup> , being opposite in sign to each other, and the average value of velocity for these gases ought to be zero. In other words, if we consider the gases evaporating from the target surface and recondensing into it as a mixed gas, it is in an equilibrium state in terms of translational motion. However, when evaporation occurs drastically, as in the case of laser ablation, the balance of velocity distribution breaks down and a one-sided distribution appears for the evaporating gas, while the recondensing gas becomes remark-

*I xyz* ( )

*E mv v v*

<sup>222</sup> 2

+ ++ ∝ −

*<sup>x</sup> y z v vv* 0, , , −∞ < < ∞

where direction *x* is normal to the target surface, directions *y* and *z* are in parallel to the surface, as an index of kinetic velocity *v*. While *E*I is the accessible internal energy, *T*<sup>s</sup> is the surface temperature, *m* is the atomic mass and *k* is Boltzmann constant. The movement of the evaporating atoms in vacuum and atmosphere is shown in **Figure 1** [13], by way of comparison. In vacuum as **Figure 1(a)**, the flow velocity as a macroscopic fluid is zero, while the kinetic velocity of individual atoms which are without interferences from the other atoms to form a whole gas flow (free molecular flow) is very high. When the gas density becomes high, as shown in **Figure 1(b)**, the velocity vector of atoms which keeps positive direction near the surface, gradually changes to negative direction with increasing distance from the surface due to the presence of recondesing atoms.

Meanwhile, the atomic velocity transits to the fluid velocity and the overall gas flow velocity increases. Thus, the velocity distribution of the evaporating atoms is relaxed in a very thin layer near the solid surface and finally reaches a shifted

*I xK yz* {( ) }

*E mv u v v*

 + − ++ ∝ −

<sup>2</sup> 2 2 2

*xyz* −∞ < < ∞ *vvv* ,, ,

where *u* is flow velocity and index *K* represents the state value at the very thin layer, which has a thickness corresponding to several mean free path and is called

At the boundary of the Knudsen layer, the velocity of the gas *u*K is equal to the speed of sound *a*K, and the subsequent flow state will be determined by the temperature of the target surface, the type of ambient gas and its pressure [12, 14]. By solving the Boltzmann equation for the Knudsen layer, the velocity and angular directional distributions of evaporating gas near the solid surface have also been analyzed [15, 16].

The configuration of plume (vapor atomic mass) formed by the vapor atoms emitted from the laser-irradiated area and shock wave formed by the piston effect

*K*

exp ; <sup>2</sup> (3)

*s*

exp ; <sup>2</sup> (2)

*DOI: http://dx.doi.org/10.5772/intechopen.95299*

**2.2 Knudsen layer**

in the above equation, *T* stands for temperature, while *κ* for thermal conductivity, *α* for optical absorption coefficient, *c*p for the specific heat, *ρ* for the density of the target material, and *I* for the value of laser energy, according to Lambert's law, along the depth direction after deduction of the loss caused by surface reflection.

On the other hand, a boundary condition is applied on the surface of target, where quantity of heat is removed from the surface corresponding to the heat of melting and evaporation. In addition, since the physical properties such as thermal conductivity are generally different between the solid and liquid phases, the equation should be applied separately in each phase [8].

As will be discussed in the next section, since the evaporating atoms during laser ablation are highly directional, the one-directional analysis is effective in most cases. However, in a certain case, one-dimensional analysis fails to take into consideration some factors, such as the thermal conductivity in radial direction which becomes dominant factor when the laser irradiation time becomes longer. Formerly, Houle and Hinsberg [9] applies a two-dimensional axisymmetric model with probabilistic algorithms to the analysis of laser ablation. This allows phenomena with different characteristic time such as absorption, melting, evaporation, and thermal conduction to be analyzed at the same time marching [10].

## **2.2 Knudsen layer**

*Practical Applications of Laser Ablation*

ductor, energy and so on.

self-ordering on the substrate.

**2.1 Laser irradiation analysis**

unsteady-state heat conduction Equation [7].

tion should be applied separately in each phase [8].

conduction to be analyzed at the same time marching [10].

nanoparticles like core-shell type have been conducted. The laser ablation in nanoparticle research is regarded as the promising method at present which is possible to fabricate a novel function device in the area of electricity, semicon-

**2. The formation process of nanoparticles by laser ablation**

The process of laser ablation commences from the interaction of lights and solids. When a laser irradiates on the surface of a solid target, the electromagnetic energy of the laser beam firstly turns into the excitation energy of electron, although the process may vary depending on the thermodynamic state of the solid. The energy conversion in the solid from the electronic excitation to the lattice vibration completes within some picoseconds. If the laser has nanosecond pulse, therefore, it is not necessary to take into consideration the non-equilibrium of the laser irradiation. Then, when viewed as a one-dimensional problem in the depth direction, the solid temperature can be described as the usual one-dimensional

( )

(1)

*p*

ρ

α

*T T I xt t x xc* ∂ ∂∂ = + ∂∂ ∂ ,

in the above equation, *T* stands for temperature, while *κ* for thermal conductivity, *α* for optical absorption coefficient, *c*p for the specific heat, *ρ* for the density of the target material, and *I* for the value of laser energy, according to Lambert's law, along the depth direction after deduction of the loss caused by surface reflection. On the other hand, a boundary condition is applied on the surface of target, where quantity of heat is removed from the surface corresponding to the heat of melting and evaporation. In addition, since the physical properties such as thermal conductivity are generally different between the solid and liquid phases, the equa-

As will be discussed in the next section, since the evaporating atoms during laser ablation are highly directional, the one-directional analysis is effective in most cases. However, in a certain case, one-dimensional analysis fails to take into consideration some factors, such as the thermal conductivity in radial direction which becomes dominant factor when the laser irradiation time becomes longer. Formerly, Houle and Hinsberg [9] applies a two-dimensional axisymmetric model with probabilistic algorithms to the analysis of laser ablation. This allows phenomena with different characteristic time such as absorption, melting, evaporation, and thermal

κ

We have known PLD as a process during which a solid target is vaporized with laser irradiation, and then the nanoparticle formation in the gas phase is occurred, followed by its soft-landing on a substrate. But each process remains unclear to us. In order to apply the laser ablation method for the fabrication of functional materials, it is important to understand each step of the process so that we can put them to practical use. In this chapter, we divide the formation process of nanoparticle films as following steps for further studies: (1) temperature increase of solid surface by laser irradiation, (2) evaporation of the surface and its conversion to kinetic energy, (3) plume expansion with shock wave propagation, (4) supercooling of evaporated gas, (5) uniform sized nanoparticle formation, and (6) nanoparticle deposition and

**104**

If the vapor pressure on the surface of the solids during evaporation is equal to the ambient pressure, the Maxwell velocity distribution for both evaporating and recondensing gas takes half shape of the distribution *f* <sup>+</sup> or *f* <sup>−</sup> , being opposite in sign to each other, and the average value of velocity for these gases ought to be zero. In other words, if we consider the gases evaporating from the target surface and recondensing into it as a mixed gas, it is in an equilibrium state in terms of translational motion. However, when evaporation occurs drastically, as in the case of laser ablation, the balance of velocity distribution breaks down and a one-sided distribution appears for the evaporating gas, while the recondensing gas becomes remarkably little [11] so as to take the following velocity distribution [12].

$$f\_s^\* \propto \exp\left[-\frac{2E\_l + m\left(\nu\_x^2 + \nu\_y^2 + \nu\_z^2\right)}{2kT\_s}\right];\tag{2}$$

$$
\upsilon\_x \gg 0, -\infty < \upsilon\_y, \upsilon\_x < \infty,
$$

where direction *x* is normal to the target surface, directions *y* and *z* are in parallel to the surface, as an index of kinetic velocity *v*. While *E*I is the accessible internal energy, *T*<sup>s</sup> is the surface temperature, *m* is the atomic mass and *k* is Boltzmann constant. The movement of the evaporating atoms in vacuum and atmosphere is shown in **Figure 1** [13], by way of comparison. In vacuum as **Figure 1(a)**, the flow velocity as a macroscopic fluid is zero, while the kinetic velocity of individual atoms which are without interferences from the other atoms to form a whole gas flow (free molecular flow) is very high. When the gas density becomes high, as shown in **Figure 1(b)**, the velocity vector of atoms which keeps positive direction near the surface, gradually changes to negative direction with increasing distance from the surface due to the presence of recondesing atoms.

Meanwhile, the atomic velocity transits to the fluid velocity and the overall gas flow velocity increases. Thus, the velocity distribution of the evaporating atoms is relaxed in a very thin layer near the solid surface and finally reaches a shifted Maxwell equilibrium distribution as following [12]:

$$f\_K^+ \propto \exp\left[-\frac{2E\_I + m\left\{\left(\boldsymbol{\upsilon}\_x - \boldsymbol{\mu}\_K\right)^2 + \boldsymbol{\upsilon}\_y^2 + \boldsymbol{\upsilon}\_z^2\right\}}{2kT\_K}\right];\tag{3}$$

$$-\infty < \boldsymbol{\upsilon}\_x, \boldsymbol{\upsilon}\_y, \boldsymbol{\upsilon}\_z < \infty,$$

where *u* is flow velocity and index *K* represents the state value at the very thin layer, which has a thickness corresponding to several mean free path and is called the Knudsen layer.

At the boundary of the Knudsen layer, the velocity of the gas *u*K is equal to the speed of sound *a*K, and the subsequent flow state will be determined by the temperature of the target surface, the type of ambient gas and its pressure [12, 14]. By solving the Boltzmann equation for the Knudsen layer, the velocity and angular directional distributions of evaporating gas near the solid surface have also been analyzed [15, 16].

#### **2.3 Plume expansion and shock wave propagation**

The configuration of plume (vapor atomic mass) formed by the vapor atoms emitted from the laser-irradiated area and shock wave formed by the piston effect of the plume is shown in **Figure 2** [17]. From the laser irradiation zone shown in the leftmost part of the figure, since the vapor atoms emerge from the Knudsen layer in a certain angular distribution of velocity, the plume spreads out to the lateral

#### **Figure 1.**

*(a) Schematic of representation of vapor atoms emitted from a target surface which enter immediately into free flight, in the case of evaporation in vacuum. (b) Schematic representation of Knudsen layer formation followed by an unsteady adiabatic expansion and free fight, in the case of evaporation in high density regime [13].*

**107**

region.

in this region [24].

**2.4 Formation of nanoparticles**

mechanics and is expressed as follows [26].

theory of nanoparticle formation.

*Nanoparticle Formation and Deposition by Pulsed Laser Ablation*

direction along the central axis. In general, under the conditions of laser ablation, the pressure of the plume developed from the target is extremely higher than the ambient pressure, which is the so-called under-expansion jet in term of high-speed fluid dynamics (region 1 in **Figure 2**). Thereafter, the plume continues to expand, then it overinflates and forms a mushroom-like vortex (region 2 in **Figure 2**). The pressure of the overinflating plume becomes negative in comparison to the surrounding gas, which causes the plume to contract, resulting in a minimum diameter of the plume (region 3 in **Figure 2**). After this, the plume expands again and gradually decays (region 4 in **Figure 2**). At this stage, the resistance of the ambient gas is strong for the progress of the plume. The ambient gas atoms diffuse into the plume's front edge (hatched area in **Figure 2**) and form a diffusion region there. Since radicals of vapor atoms are generated in this diffusion region and plunge in collisional relaxations with the ambient gas atoms, luminescence can be likely observed in the

Since the atoms fall on a substrate from the luminescence zone, photogenic property in this zone would have a powerful effect on the characteristics of the film produced by laser ablation. The phenomenon has been analyzed through direct imaging by Intensified Charge Coupled Device (ICCD) and Laser Induced Fluorescence (LIF) [18–23]. The formation of nanoparticles is also thought to occur

As described in the previous section, the formation process of the nanoparticle

is highly related to the thermodynamic state in the dilution zone of the vapor atoms and ambient gas atoms. Hagena and Obert [25] conducted experiments using a supersonic nozzle to summarize the relationship between the size of atomic nanoparticles and the thermodynamic state as similitude rules in which parameters of molecular movement such as an interatomic potential are primary variables. Thus, estimating the equilibrium concentration of a nanoparticle from the thermodynamic conditions of the surrounding environment is useful for practical purposes and has been discussed in many publications. The rate constant for the nanoparticle equilibrium concentration is essentially based on the assumptions of statistical

( )

Here, *K*g is the rate constant of the equilibrium concentration in a nanoparticle consisted of atoms of the number *g*, while *Q* is the partition function of the nanoparticle with the subscripts *t*, *r* and *v* representing the translation, rotation and oscillation respectively. The variable *q* stands for the partition function of a mono-atom, *E*g for the formation energy of the nanoparticle, *k* for the Boltzmann's constant, and *T* for the temperature of the system. The above equation describes the process in which the translational kinetic energy of a mono-atom transformed into the internal energy of the nanoparticle when it is captured by the nanoparticle. But there does not seem to be any detailed descriptions of this process in the classical

*g g t*

*K*

*g trv*

*q kT* <sup>=</sup> <sup>−</sup>

exp (4)

*QQQ E*

On the other hand, Eq. (4) refers to the nanoparticle concentration based on the steady-state theory. Generally speaking, the length of time it takes to form nanoparticle is not considered in this equation. However, for high-speed phenomena such as laser ablation, it is necessary to estimate the nanoparticle formation time to determine if the balance between condensation and evaporation on the surface of

*DOI: http://dx.doi.org/10.5772/intechopen.95299*

**Figure 2.** *Schematic drawing of laser plume expansion from the viewpoint of high-speed hydrodynamics [17].*

#### *Nanoparticle Formation and Deposition by Pulsed Laser Ablation DOI: http://dx.doi.org/10.5772/intechopen.95299*

*Practical Applications of Laser Ablation*

of the plume is shown in **Figure 2** [17]. From the laser irradiation zone shown in the leftmost part of the figure, since the vapor atoms emerge from the Knudsen layer in a certain angular distribution of velocity, the plume spreads out to the lateral

*Schematic drawing of laser plume expansion from the viewpoint of high-speed hydrodynamics [17].*

*(a) Schematic of representation of vapor atoms emitted from a target surface which enter immediately into free flight, in the case of evaporation in vacuum. (b) Schematic representation of Knudsen layer formation followed by an unsteady adiabatic expansion and free fight, in the case of evaporation in high density regime [13].*

**106**

**Figure 2.**

**Figure 1.**

direction along the central axis. In general, under the conditions of laser ablation, the pressure of the plume developed from the target is extremely higher than the ambient pressure, which is the so-called under-expansion jet in term of high-speed fluid dynamics (region 1 in **Figure 2**). Thereafter, the plume continues to expand, then it overinflates and forms a mushroom-like vortex (region 2 in **Figure 2**). The pressure of the overinflating plume becomes negative in comparison to the surrounding gas, which causes the plume to contract, resulting in a minimum diameter of the plume (region 3 in **Figure 2**). After this, the plume expands again and gradually decays (region 4 in **Figure 2**). At this stage, the resistance of the ambient gas is strong for the progress of the plume. The ambient gas atoms diffuse into the plume's front edge (hatched area in **Figure 2**) and form a diffusion region there. Since radicals of vapor atoms are generated in this diffusion region and plunge in collisional relaxations with the ambient gas atoms, luminescence can be likely observed in the region.

Since the atoms fall on a substrate from the luminescence zone, photogenic property in this zone would have a powerful effect on the characteristics of the film produced by laser ablation. The phenomenon has been analyzed through direct imaging by Intensified Charge Coupled Device (ICCD) and Laser Induced Fluorescence (LIF) [18–23]. The formation of nanoparticles is also thought to occur in this region [24].

#### **2.4 Formation of nanoparticles**

As described in the previous section, the formation process of the nanoparticle is highly related to the thermodynamic state in the dilution zone of the vapor atoms and ambient gas atoms. Hagena and Obert [25] conducted experiments using a supersonic nozzle to summarize the relationship between the size of atomic nanoparticles and the thermodynamic state as similitude rules in which parameters of molecular movement such as an interatomic potential are primary variables. Thus, estimating the equilibrium concentration of a nanoparticle from the thermodynamic conditions of the surrounding environment is useful for practical purposes and has been discussed in many publications. The rate constant for the nanoparticle equilibrium concentration is essentially based on the assumptions of statistical mechanics and is expressed as follows [26].

$$K\_{\varrho} = \frac{Q\_{\iota} Q\_{\iota} Q\_{v}}{(q\_{\iota})^{\varepsilon}} \exp\left(-\frac{E\_{\varrho}}{kT}\right) \tag{4}$$

Here, *K*g is the rate constant of the equilibrium concentration in a nanoparticle consisted of atoms of the number *g*, while *Q* is the partition function of the nanoparticle with the subscripts *t*, *r* and *v* representing the translation, rotation and oscillation respectively. The variable *q* stands for the partition function of a mono-atom, *E*g for the formation energy of the nanoparticle, *k* for the Boltzmann's constant, and *T* for the temperature of the system. The above equation describes the process in which the translational kinetic energy of a mono-atom transformed into the internal energy of the nanoparticle when it is captured by the nanoparticle. But there does not seem to be any detailed descriptions of this process in the classical theory of nanoparticle formation.

On the other hand, Eq. (4) refers to the nanoparticle concentration based on the steady-state theory. Generally speaking, the length of time it takes to form nanoparticle is not considered in this equation. However, for high-speed phenomena such as laser ablation, it is necessary to estimate the nanoparticle formation time to determine if the balance between condensation and evaporation on the surface of

nanoparticle is kept or not. If we are assuming that the velocity of vapor atoms is represented in the equilibrium Maxwell distribution, we can estimate the approximate nanoparticle formation time *τ* with the following equation.

$$
\tau = \frac{\rho\_\circ r \sqrt{2\pi kT}}{p\sqrt{m}} \tag{5}
$$

In this equation, *ρ*c stands for the internal density of the nanoparticle, while *r* the radius of the nanoparticle, *p* the vapor pressure, and *m* the mass of the vapor atoms. In order to obtain an exact solution, Gillespie [27] analyzed the process of nanoparticle formation as a random walk problem. However, no matter how rigorous the probabilistic analysis is, it is still based on classical nucleation theory. To address the problem related to the formation of nanometer-sized particle in non-equilibrium, the following issues should be considered.


In recent years, with the development of computers, molecular dynamics or Direct Simulation Monte Carlo analyses which take into consideration the internal degrees of freedom of nanoparticles have been made [28], through which the Gibbs free energy and nanoparticle concentrations are elucidated under more realistic conditions.
