**1. Introduction**

122 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Tikhonova, L.; Goba, V.; Kovtun, M.; Tarasenko, Yu.; Khavryuchenko, V.; Lyubchik, S. &

Weber, *W.* & Morris, J*.* (1963*).* Kinetics of adsorption on carbon from solutions. *J. Sanit. Eng.* 

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*Div. Am. Soc. Civ. Eng.* Vol.89, pp. 31-60

Boiko, A. (2008). Sorption of Metal Ions from Multicomponent Aqueous Solutions by Activated Carbons Produced from Waste. *Russian Journal of Applied Chemistry,*

> Nanometer-sized particles, or nanoparticles, are smaller than conventional solid-state materials and possess great potential for new, useful properties due to peculiar quantum effects (Roco, M. C., 1998). Highly functional devices synthesized from nanoparticles have been studied for use in various fields, such as semiconductors (Liqiang, J., 2003; Lu, M., 2006), photocatalysis (Liqiang, J., 2004), secondary batteries (Ito, S., 2005; Kim, K., 2009, 2010), superconductors (Strickland, N. M., 2008), and bonding substances (Ide, E., 2005). In the present chapter, we discuss the thermodynamics related to nanoparticle formation.

> Cooling processes of expanding vapor evaporated from a solid surface, such as gas evaporation, arc discharge, sputtering, pulsed microplasma and pulsed laser ablation (PLA), have been applied as a method of nanoparticle formation in the gaseous phase (Wegner, K., 2006). The PLA method, under reduced atmospheric pressure, has been found to be especially promising since it provides the following capabilities (Chrisey, D. B., 1994): (i) ablation of target material regardless of melting point due to the high intensity and focused laser beam pulse, (ii) flexibility in choice of atmospheric gaseous species and pressure, (iii) ease of production of the non-equilibrium state of the highpressure field due to the formation of shock waves, (iv) ability to obtain many different structured materials, from thin films to micrometer-sized particles, by controlling vapor association and condensation, and (v) ease of synthesis of nano-compounds of nonstoichiometric composition by preparing target materials with desired compositional ratios. The PLA method has been widely used for nanoparticle formation because the formed nanoparticles have diameters smaller than 10 nm with low size dispersion and can be formed as basic materials for highly functional devices via effective utilization of these capabilities (Li, S., 1998; Li, Q., 1999; Patrone, L., 1999, 2000; Wu, H. P., 2000; Suzuki, N., 2001; Inada, M., 2003; Seto, T., 2006).

> To understand the process of nanoparticle formation by the PLA method, two perspectives are necessary: (i) the thermodynamics of the microscopic processes associated with the nucleation and growth of nanoparticles, and (ii) the thermodynamics of the macroscopic processes associated with the laser irradiated surface of the target supplying the raw

Thermodynamics of Nanoparticle Formation in Laser Ablation 125

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,

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

> 11 2 21 3

*AA A AA A*

 

*i i* 1 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*-

1 1

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*

1,

2

\*

3 exp <sup>4</sup> *n cvc W W <sup>I</sup> r kT kT* 

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

*i i*

*K*

1,

exp *<sup>i</sup> i i*

\* \*

(3)

*Q D*

*Q Q kT*

(1)

(2)

*AAA*

**2. Thermodynamics of nanoparticle formation** 

**2.1 Nucleation and growth of nanoparticles** 

and making the transition into large particles.

expressed as follows:

particle is

gaseous materials, combined with the surrounding atmosphere, to provide adequate conditions for nucleation and subsequent growth.

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.

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 example, continuous fluid dynamics with a classical nucleation model.

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 nanoparticle generator.

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, conclusions are given in Section 6.
