A Co-Condensation Model for In-Flight Synthesis of Metal-Carbide Nanoparticles in Thermal Plasma Jet

Abstract

We present a theoretical analysis of the formation, growth, and transport of two-component nanoparticles in thermal plasma jet. The approach of the aerosol science and the idea of multicomponent co-condensation are employed for the analysis. The processes of homogeneous nucleation, heterogeneous growth, and coagulations due to Brownian collisions are considered in combination with the convective and diffusive transport of particles and the reacting gases within an axisymmetric domain. As a particular example, the authors study multicomponent co-condensation of metal-carbide nanoparticles from various precursors in a DC plasma gun operating in an argon atmosphere.

Appendix

Physical Properties

The referenced points for Ta, C, and Si gases used for the simulations are summarized in Table 1. Between the data points, the vapor pressure was approximated by the Clausius-Clapeyron equation:

$$ p = p_{{{\text{ref}}}} \,\exp {\left( { - \frac{Q} {R}{\left( {\frac{1} {T} - \frac{1} {{T_{{{\text{ref}}}} }}} \right)}} \right)}. $$

Table 1 Vapor pressure reference points

Here p _ref and T _ref are the values given in Table 1, Q is the latent heat of vaporization per mole, and R is the universal gas constant. For the latent heat of vaporization the following values were used: for Ta, Q = 3.65 × 10⁵ J mol⁻¹ (Ref 24), for C, Q = 3.56 × 10⁵ J mol⁻¹ (Ref 25), and for Si, Q = 3.59 × 10⁵ J mol⁻¹ (Ref 24).

For the homogeneous nucleation of Ta gas, the value of the surface tension coefficient σ = 2.07 N m⁻¹, reported in Ref 26, was used. In the calculations in the current study, the temperature variations exceed the reported measurement interval. The authors chose to extrapolate the constant value for the surface tension coefficient on the entire temperature range where the particles exist. Following the common assumption of the classical nucleation theories the dependence of the surface tension on the particle size is not considered. To describe the process of nucleation of Si-C, the surface tension of C is needed. No measurements of the surface tension for carbon were found in the literature, which is likely because of the fact that carbon is not observed in a liquid form under atmospheric pressure (Ref 21). To estimate the value of the surface tension, the results of measurements of the free surface energy of solid carbon fibers (Ref 27), σ = 0.046 J m⁻², were used.

The density of TaC in solid form is 1.39 × 10⁴ kg m⁻³, and the density of SiC is 3.22 × 10³ kg m⁻³. To calculate the size of particles, the following diameters of Ta and C atoms were assumed, 2.7 × 10⁻¹⁰ m and 2.6 × 10⁻¹⁰ m; the diameter of Si atoms was taken as 2.2 × 10⁻¹⁰ m.