1 Introduction

Suspension plasma spray (SPS) is a novel spray technology (Fauchais, 2004). In the SPS process, the nano-sized particles in the feedstock usually have diameters between 100 nm and 1 µm. These particles are dispersed in suspension and then injected into a plasma flow to form droplets with diameters smaller than 100 µm by atomization devices (Fauchais et al., 2008a). The SPS process involves a series of complex phenomena, such as feedstock injection, suspension breakup (Meillot et al., 2013a; 2013b) and evaporation, and the discharge of particles or their agglomerates into the plasma jet. The solid particles will experience further heating, melting, and evaporation before impacting the substrate. Since the behaviors and states of the particles will affect their coating qualities, it is necessary to understand their dynamic behaviors.

Due to the complex behaviors of the micro- and nano-sized particles, the modeling of SPS becomes difficult. The dynamic behaviors of the particles are dominated by the forces exerted on them. These forces have been studied using various models (Faeth, 1987; Pfender, 1989; Huang et al., 1995; Jen et al., 2005; Pateyron et al., 2013), mainly using the drag force. Among these models, Jen et al. (2005) considered the Brownian force and the Saffman lift force but only in a special flow domain, which is not universal and which introduces errors when used for the spray distribution with sub-micro-sized or nano-sized particles. Thus, it is necessary to use a comprehensive mathematical model to describe the forces affecting the nano-sized particles as well as the micro-sized particles in the plasma field. In this study, we establish a mathematical model to describe the movement of the particles, considering simultaneously the drag force, the Brownian force, and the Saffman lift force for micro- and nano-sized particles during their whole flight in the plasma jet. With the Brownian force, the size effects on the dispersion of the particles are considered, and with the Saffman lift force, the effect of flow shear stress is considered.

In a plasma spray, particle size is a key factor. It will affect the trajectory and thermal state of the particles, especially when considering their interaction with the substrate wall (Jen et al., 2005; Xiong et al., 2005). In presence of the substrate, the flow field and stream traces turn and run parallel to the substrate surface, and thus sufficiently small particles will move with the gas flow and not reach the substrate, while large particles having sufficient inertia will penetrate the boundary layer and reach the substrate surface. The Stokes number is often used to identify the trajectory of particles of different sizes. Crowe et al. (1985) firstly studied particle dispersion in free shear flows and concluded that particle dispersion might be strongly dependent on the Stokes number, which is defined as the ratio of particle aerodynamic response time to the time of large scale vortex structures. Since then, the Stokes number has often been used by subsequent researchers to analyze the size effect on the particle movement in the plasma spray process (Crowe et al., 1988; Ye et al., 2000; Jordan et al., 2004; Delbos et al., 2006; Fauchais et al., 2008a; 2008b; Bacciochini et al., 2010). However, most of these researchers estimated the Stokes number according to the average velocity, and the effects of local velocity were not well considered.

In this study, we use the Stokes number formula proposed by Crowe et al. (1985), based on the particle size, the local gas mean velocity, and boundary layer thickness with consideration of the effects of variable plasma properties and non-continuum (Delbos et al., 2006). Our numerical simulation captures the vortex structure caused by the flow around the substrate, and provides for the local boundary layer thickness that the particles pass through before impinging on the substrate. By calculating the particle Saffman lift force, each individual particle can be tracked precisely in its local shear environment. Thus, our simulation results can quantitatively analyze the movement, dispersion, and deposition behaviors of individual particles. Finally, the critical size of particles that follow the plasma jet is deduced, which is consistent with our numerical results under the specific operating conditions.

The spray distance affects the particle parameters significantly in the SPS process. Researchers have studied the effects of spray distances on particle parameters and coating quality (Berghaus et al., 2006; Fazilleau et al., 2006; Fauchais et al., 2008b; Guittienne et al., 2012). But most of these results were chiefly based on experiments, and their conclusions are not mutually consistent. The main reason is that the mechanisms by which the spray distance affects the local flow field and particle movement are not well understood. The effects of spray distance on the flow structures, as well as the particles’ heating process, are studied in this paper. The mean velocity and temperature of the particles with their statistical distributions were given for different spray distances, and the proper spray distances are obtained for different micro- and nano-sized particles.

2 Mathematical model

The droplets or particles experienced accelerating and heating in the plasma jet, and a code named LAVA-P-3D was developed to simulate these processes. The plasma gas flow fields were obtained by solving the Navier-Stokes (N-S) equations by the Eulerian method, and tracking the particles as Lagrangian entities.

The N-S equations of the plasma jet were established by assuming that the plasma jets were continuum, multi-component, compressible, and chemical reacting, with temperature-dependent transport properties in local thermodynamic equilibrium. The turbulence was simulated using the k-ε model. Details of the plasma jet simulation may be found in reference (Xiong et al., 2004).

For particles injected into the plasma jet, their aerodynamic force, movement, trajectory, heating, melting, and evaporation were simulated during their flight. Different forces exerted on the particles were considered, including drag force, Saffman lift force, and Brownian force. The corrections for drag coefficient and Nusselt number Nu were also considered, taking into account Knudsen effects and the influence of mass evaporation. In a plasma flow field, the volume fraction of particles is less than 10−4, so the effect of collisions between particles was neglected.

2.1 Accelerating and tracking model of particles

For particles in plasma jets, the forces imparted on the particles are mainly the drag force, Saffman lift force, and Brownian force. For particles smaller than 100 μm, the drag force is prominent. For the particles near the jet edge and the substrate, where the flow shear stress is large, the Saffman lift force is significant. For the sub-micron or nano-particles, Brownian force is important. By accounting for these three forces, the acceleration rate of the particles can be expressed as

$$\begin{array}{*{20}c} {F = {F_{{\rm{drag}}}} + {F_{{\rm{Saffman}}}} + {F_{{\rm{Brownian}}}} = {m_{\rm{p}}}{{d{V_{\rm{p}}}} \over {{\rm{d}}t}},\quad }\\ {{F_{{\rm{drag}}}} = {m_{\rm{p}}}{3 \over 8}{{\bar \rho } \over {{\rho _{\rm{p}}}}}{{{C_{\rm{D}}}} \over {{r_{\rm{p}}}}}\left\vert {{V_{\rm{g}}} - {V_{\rm{p}}}} \right\vert ({V_{\rm{g}}} - {V_{\rm{p}}}),\quad \quad }\\ {{F_{{\rm{Saffman}}}} = {m_{\rm{p}}}{{2{K_{\rm{c}}}{{(\mu/{\rho _{\rm{g}}})}^{0{.}5}}{d_{ij}}{\rho _{\rm{g}}}} \over {{\rho _{\rm{p}}}{d_{\rm{p}}}{{({d_{lk}}{d_{kl}})}^{0{.}25}}}}({V_{\rm{g}}} - {V_{\rm{p}}}),}\\ {{F_{{\rm{Brownian}}}} = {m_{\rm{p}}}{G_0}\sqrt {{{\pi {S_0}} \over {\Delta t}}}, \quad \quad \quad \quad \quad \quad \quad \quad } \end{array}$$
(1)

where F is the total particle force that includes three parts, V is the velocity, m is the mass, d is the diameter, r is the radius, ρ is the density, \(\bar \rho\) is the turbulent mean density of gas, Vg is the gas velocity within the turbulent fluctuation calculated from the gas turbulence model, and μ is the viscosity. Subscript p means particles or agglomerates, and subscript g means plasma gas. Kc=2.594 is the constant in the Saffman lift force (Saffman, 1965; Aggarwal and Peng, 1995; Chung, 1998), d ij is the deformation tensor as well as d lk and d kl . G0 is a random 3D vector with each number between −1 and 1 and subjected to a Gaussian distribution. S0 is the spectral intensity, which can be expressed as \({S_0} = (216\mu {\sigma _{\rm{B}}}{T_{\rm{g}}})/(32{\pi ^2}r_{\rm{p}}^5\rho _{\rm{p}}^2{C_{\rm{c}}}),\;\;{\sigma _{\rm{B}}} = 1{.}38 \times {10^{ - 23}}\;{\rm{J}}/{\rm{K}}\) is the Boltzman constant, Tg is the gas temperature, and Cc is the Cunningham correction factor. During the particle tracking procedure, the turbulent dispersion of particles is calculated by integrating the trajectory equations for individual particles, using the instantaneous fluid velocity along the particle path. CD is the drag force coefficient expressed by (Chen and Pfender, 1983b)

$${C_{\rm{D}}} = \left({{{24} \over {R{e_{\rm{p}}}}} + {6 \over {1 + \sqrt {R{e_{\rm{p}}}} }} + 0{.}4} \right)f_{{\rm{prop}}}^{ - 0{.}45}f_{Kn}^{0{.}45},$$
(2)

where the Reynolds number of the particle is defined as

$$R{e_{\rm{p}}} = 2{\rho _{\rm{g}}}{r_{\rm{p}}}\left\vert {{V_{\rm{g}}} - {V_{\rm{p}}}} \right\vert\Bigg/\mu,$$
(3)

fprop represents the effects of variable plasma properties in the boundary layer surrounding the particle, and can be expressed as fprop=ρcμc/(ρwμw) (Lee et al., 1981), where subscript c means the cell where the particle is located, and subscript w means the vicinity of particle surface. f Kn is the factor representing Knudsen effect, which can be expressed by

$${f_{Kn}} = {\left[ {1 + \left({{{2 - a} \over a}} \right)\left({{{{\gamma _{\rm{w}}}} \over {1 + {\gamma _{\rm{w}}}}}} \right){4 \over {Pr}}Kn} \right]^{ - 1}},$$
(4)

where a is the thermal accommodation factor, usually with the value of 0.8 (Chen and Pfender, 1983a), γw is the specific heat ratio of gas, Pr is the Prandtl number, and Kn is the Knudsen number based on the effective mean free path λ, Kn=λ/dp, where λ=2μ/(ρgvw), vw is the mean molecular speed that is dependent on the gas temperature near the particle surface Tw as well as the average molecular weight W of the gas mixture, which can be given as vw =[8RTw/(πW)]1/2 where R is the gas constant. For nano-particles, f Kn ranges from 0.005 to 0.1. For the agglomerates and micro-sized particles, f Kn changes from 0.994 to 0.996 (Xiong and Lin, 2009).

Eq. (1) is used to depict the particle velocity and trajectory. The local gas conditions around the particle are employed to calculate its heating.

2.2 Solvent evaporation of droplets

As shown in Fig. 1, the micro- or nano-sized solid particles were assumed to be suspended in the micro-sized droplets randomly. Thermal conduction was neglected in the heating model of the suspension droplets and a lumped thermal capacity model was used, because the Biot number was less than 3×10−3 in the present calculation.

Fig. 1
figure 1

Schematic of nano-particle injection, discharge, and tracking (Delbos et al., 2006)

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The droplet temperature Td can be expressed by

$$\begin{array}{*{20}c} {{T_{\rm{d}}} = {T_{{\rm{d}},0}} + {{{Q_{\rm{d}}}} \over {{m_{\rm{d}}}{c_{{\rm{p}},{\rm {d}}}}}}t, \;\;{\rm{if}}\; {T_{\rm{d}}} < {T_{{\rm{m}},{\rm {d}}}},\quad \quad \quad \quad \quad }\\ {{T_{\rm{d}}} = {T_{{\rm{m}},{\rm {d}}}}{{,}}\;\; \;{\rm{if }}\;{m_{\rm{d}}}{C_{{\rm{p}},{\rm {d}}}}({T_{{\rm{m}},{\rm {d}}}} - {T_{{\rm{d}},0}}) \leq {Q_{\rm{d}}}\quad \quad \quad }\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leq {m_{\rm{d}}}{C_{{\rm{p}},{\rm {d}}}}({T_{{\rm{m}},{\rm {d}}}} - {T_{{\rm{d}},0}}) + {m_{\rm{d}}}{\alpha _{{\rm{sl}}}}{L_{{\rm{v}},{\rm {sl}}}},} \end{array}$$
(5)

where Td,0 is the initial droplet temperature, Q and Cp are the heat gain and specific heat, respectively, Tm is the melting point, α is the weight fraction, and Lv is the latent heat of vaporization. Subscript d means suspension droplet, and subscript sl means solvent. Qd can be calculated by Qd=Qconv–Qrad, where Qconv represents the convection heat, and Qrad is the radiation heat loss of the particles. Cp,d is calculated based on the average of the mass fraction of the solid particle and solvent as Cp,d=Cp,p(1-αsl)+Cp,slαsl.

When the droplet solvent is totally vaporized, the solid particles contained in the droplet will be discharged into the plasma jet. They might be released as micro-sized agglomerates containing many nano-sized particles or individual nano-sized particles depending upon the evaporation of the solvent and the aerodynamic forces of the plasma gas (Fauchais and Montavon, 2010). These individual micro- or nano-sized particles are treated as new Lagrangian entities with the current parameters of their mother particle, including position, droplet velocity, and temperature.

2.3 Heating and melting of particles

A 1D model was adopted for the heating and melting of the particle, in which the spherical shape of the particle was assumed as shown in Fig. 2.

Fig. 2
figure 2

Schematic of particle heat transfer

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Internal convection within the molten part of the particle was not considered. The temperature (Tp) distribution inside the particle was described as

$${\rho _{\rm{p}}}{C_{\rm{p}}}{{\partial {T_{\rm{p}}}} \over {\partial r}} = {1 \over {{r^2}}}{\partial \over {\partial r}}\left({{k_{\rm{p}}}{r^2}{{\partial {T_{\rm{p}}}} \over {\partial r}}} \right),$$
(6)

where k is the thermal conductivity.

Zero temperature gradient was assumed in the particle center. The particle surface was subjected to the energy conservation law, expressed as (Wan et al., 1999):

$$\begin{array}{*{20}c} {{{\left. {{{\partial {T_{\rm{p}}}} \over {\partial r}}} \right\vert }_{r = 0}} = 0,\;\;\;{\rm{and}}\quad \quad \quad \quad \quad \quad \quad \quad }\\ {{{\left. {4{{\pi }}r_{\rm{p}}^2\left({{k_{\rm{p}}}{{\partial {T_{\rm{p}}}} \over {\partial r}}} \right)} \right\vert }_{r = {r_{\rm{p}}}}} = {{\dot Q}_{{\rm{conv}}}} - {{\dot Q}_{{\rm{vap}}}} - {{\dot Q}_{{\rm{rad}}}},} \end{array}$$
(7)

where the convection, evaporation latent, and radiation heat rates (\({\dot Q_{{\rm{conv}}}}\), \({\dot Q_{{\rm{vap}}}}\), and \({\dot Q_{{\rm{rad}}}}\)) are expressed as \(4\pi r_{\rm{p}}^2{h_{\rm{f}}}({T_{\rm{f}}} - {T_{\rm{s}}}),\;\;{\dot m_{\rm{v}}}{L_{\rm{v}}}\) and \(4\pi r_{\rm{p}}^2{\varepsilon _{\rm{p}}}{\sigma _{\rm{s}}}(T_{\rm{s}}^4 - T_\infty ^4)\) respectively. Ts is the particle surface temperature, \({\dot m_{\rm{v}}}\) is the vaporization rate, εp is the particle surface emissivity coefficient, σs is the Stefan-Boltzmann constant, and T∞ is the temperature outside boundary layer. The film temperature Tf was defined as (Ts+Tg)/2, as shown in Fig. 2, which is introduced to deal with the steep temperature gradient in the boundary layer around the particle. Only the radiation between the particle surface and the environment was considered in the case of optically thin plasma gas. The heat transfer coefficient, hf, can be calculated from (Chen and Pfender, 1982)

$$Nu = {{2{h_{\rm{f}}}{r_{\rm{p}}}} \over {{k_{\rm{f}}}}} = \left({2{.}0 + 0{.}6Re_{\rm{p}}^{1/2}P{r^{1/3}}} \right){f_{{\rm{prop}}}}{f_{Kn}}{f_{\rm{v}}},$$
(8)

where kf is thermal conductivity of the film around the particle, fv accounts for the effect of mass transfer due to evaporation, which can be found in reference (Xiong and Lin, 2009). Additional constraints of energy balance between the heat conduction and latent heat at the melting interface rm were also considered:

$${\left. {{{\left. {\left({{k_{\rm{p}}}{{\partial {T_{\rm{p}}}} \over {\partial r}}} \right)} \right\vert }_{r = r_{\rm{m}}^ - }} - \left({{k_{\rm{p}}}{{\partial {T_{\rm{p}}}} \over {\partial r}}} \right)} \right\vert _{r = r_{\rm{m}}^ + }} = {L_{\rm{m}}}{\rho _{\rm{p}}}{{{\rm{d}}{r_{\rm{m}}}} \over {{\rm{d}}t }}{.}$$
(9)

3 Experimental and numerical setup

This study was conducted for a direct-current (DC) suspension plasma spray system with axial injection of feedstock as shown in Fig. 3. The injection point was at x=0, y=0.1, and z=0, and the injection direction was given by a random distribution from 0° to 20° with regard to y-axis. Gas mixtures of argon and hydrogen were ionized in a plasma gun to form a high temperature and high velocity plasma jet. The droplets and particles were accelerated and heated in the plasma field. Finally, the melting particles formed coatings on the substrate. Operating parameters of this gun are listed in Table 1.

Fig. 3
figure 3

Schematic of the DC plasma spray

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Table 1 Baseline operating conditions of single particle injection
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Different diameters of ZrO2 particles from 10 nm to 1 µm were suspended in the ethanol solvent as shown in Fig. 1. The particle properties are listed in Table 2 and the solvent properties in Table 3.

Table 2 Particle properties (ZrO 2 ) (Xiong and Lin, 2009)
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Table 3 Solvent properties (ethanol) (Xiong and Lin, 2009)
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At the nozzle exit, the velocity and temperature can be expressed by the empirical formulae (Ramshaw and Chang, 1992) v(r)=Vcl[1−(r/Ri)1.2] and T(r)=(Tcl−Tw)[1−(r/Ri)6]+Tw, where Vcl and Tcl are the velocity and temperature along the nozzle exit with the orifice radius Ri=0.4 cm, respectively, and are calculated from the total amount of momentum and thermal energy transferred to the plasma jet. Tw is the wall temperature with an initial value of 300 K, and the velocity at the wall boundary is 0. The downstream of the jet flow is open.

The plasma flow field was solved using a cylindrical coordinate system. The radial distance was 6 cm with 57 grid points, and the axial distance was 15 cm with 66 grid points. The droplets and particles were tracked in the plasma flow field. The particle temperature distribution and location of the melting interface were calculated using 50 grid points in spherical coordinates. The droplets had an initial temperature of 300 K and a velocity of 50 m/s. A typical mean droplet size of 50 µm was assigned based on the commonly used pneumatic atomizer (Xiong and Lin, 2009).

4 Validation of model predictions

To verify the numerical code LAVA-P-3D for the plasma spray process, the centerline gas temperature profiles were calculated and compared with published numerical results (Jabbari et al., 2014) and experimental data (Brossa and Pfender, 1988) in Fig. 4. A good agreement between the simulation results and experimental data was obtained. The deviation between experiment and our modeling result is within 10%. Both experiment and simulation used the Sulzer 3MB plasma gun with the same operating parameters as in reference (Jabbari et al., 2014).

Fig. 4
figure 4

Comparison of centerline gas temperature profiles with published numerical results (Jabbari et al., 2014) and experimental data (Brossa and Pfender, 1988)

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5 Discussion

5.1 Plasma jet

Fig. 5 shows the plasma field velocity and temperature contours as well as the streamlines. The hot core was downstream of the nozzle exit within an axial spray distance of about 1 cm, where the gas temperature was as high as 10 000 K and the velocity was up to 3000 m/s. Beyond this region, the gas velocity decelerated and the temperature cooled down continuously along the spray distance. The dark plate located at the distance of 5 cm in Fig. 5 represents the substrate wall. The gas flowed around the substrate and formed two vortexes behind the substrate and another two vortexes on the wall shoulders. The vortexes on the shoulders were formed because of a steep shear layer at the outer edge of the jet, and the flow separation induced those two vortexes behind the substrate. The vortexes behind the substrate had low gas velocity, where small particles might get entrained on the back of the substrate.

Fig. 5
figure 5

Gas velocity contours with streamlines (a) and temperature contours in the mid-plane of the plasma jet (b)

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5.2 Motion of the particles

Particle motion is complex in an SPS. Fig. 6 depicts the snapshots of particles in the mid-plane at different times from 0.01 ms to 0.06 ms. It shows how the particles move along the spray distance after injection. From 0.01 ms to 0.03 ms, the particles near the axis have higher axial velocity and smaller radial velocity than those in the periphery. Therefore, the particle traces form a cone shape that accords well with the streamlines of a jet flow field as shown in Fig. 5. At 0.04 ms, the spatial distribution of the particles becomes flatter. This is because in the downstream of the plasma field, the axial particle velocity becomes zero due to deceleration by the substrate. From 0.05 ms to 0.06 ms, the particles begin to move along the substrate, and the radial distribution becomes wider and wider, especially for the peripheral particles. This is because the radial particle velocity near the substrate becomes larger and larger as the gas flow separates at the substrate edge.

Fig. 6
figure 6

Snapshots of 100 nm particles in the mid-plane of the spray at different times

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Fig. 7 represents the radial velocity of single particles with different diameters of 10 nm, 100 nm, and 1 µm. From the picture, we can see the radial velocity of the 10 nm particle fluctuates sharply due to the effect of Brownian force, while that of the micro-sized particle shows less fluctuation.

Fig. 7
figure 7

Radial velocity of particles with different diameters of 10 nm, 100 nm, and 1 µm

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For nano-sized particles, the Brownian force affects their velocities and trajectories significantly at each instantaneous moment. In this study, we considered the effect of Brownian force on the motion history of an individual particle (Shao et al., 2015). The distributions of multiple particles on the substrate with and without Brownian force were also studied, and the results for the 10 nm particles are shown in Fig. 8.

Fig. 8
figure 8

Distribution of nano-sized particles in the vertical plane with (a) and without (b) Brownian force

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The spatial ranges were wider when considering the Brownian force compared with the case without Brownian force. Moreover, more particles were distributed in the periphery of the spray, and particles were scattered more evenly in the center of the spray. This is because the Brownian force leads the nano-particles to distribute more homogeneously.

5.3 Stokes number and critical particle diameter

Particles with different diameters move along different trajectories when impinging on the substrate. The Stokes number (St) is an important parameter in characterizing the trajectory of particle.

Using the formula by Crowe et al. (1985; 1988) as \(St = {\rho _{\rm{p}}}d_{\rm{p}}^2U/(18\mu L)\) with gas viscosity of about 5×10−5 kg/(m·s) for the mixture of argon and hydrogen near the substrate boundary layer, with gas velocity near the boundary layer of U=24 m/s, boundary layer thickness of L=0.1 mm, and consideration of the Reynolds number near the substrate, we calculated that the critical particle diameter was 780 nm (about 800 nm), corresponding to St=1. This means that particles smaller than 800 nm will not have enough inertia to penetrate the viscous boundary layer and will fly away with the flow. Particles larger than 800 nm are more likely to be captured by the substrate. The above Stokes number formula including the information on local gas velocity and local boundary layer can be used to estimate the critical diameter for a particle that follows the flow.

To demonstrate the critical particle diameter of 800 nm, we tracked numerically three particles around the substrate, with diameters of 10 nm, 800 nm, and 10 µm, as shown in Fig. 9. The 10 nm particle moved along the axis of the flow field at first. As it moved toward the substrate, it was entrained and kept spinning inside the vortexes behind the substrate. The scale of the spinning circle was comparable to the vortex size of the gas flow, and the trajectory reflected the flow structures near the substrate. The 800 nm particle followed the gas flow and moved over the substrate, but it would not enter the vortex as the 10 nm particle because of its larger inertia. The 10 µm particle could not flow over the substrate and, instead, moved near the axis and impacted directly on the substrate. Therefore, our numerical results of particle movement around the substrate agree well with the analysis using the Stokes number.

Fig. 9
figure 9

Trajectories of particles with different diameters of 10 nm, 800 nm, and 10 µm

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5.4 Effects of the substrate wall

The spray distance from the nozzle exit to the substrate is a key parameter that affects the flow structure, as well as the acceleration and heating of the particles, and the substrate also affects their distribution. Nano-particles are easily centrifuged away from the center region by the plasma flow near the substrate. Because of the existence of the substrate, there was a hollow area at the center. This phenomenon also indicated that the substrate affects the distribution of particles downstream.

Particle distributions at different spray distances are shown in Fig. 10. These particles had the same diameter of 10 nm, and the locations of the substrate were 5 and 8 cm from the nozzle exit. More particles were collected at the center of the substrate located at 5 cm. There were 10% more particles at the center in the small distance case, and the spread domain was smaller than that with a substrate at 8 cm. The reason for this difference is that when the substrate is located near the nozzle, the gas velocity is larger, and more particles can penetrate the stagnation flow near the substrate and adhere to it. In addition, for the substrate located at 5 cm, the average temperature of the particles when they impacted on the substrate was 3215 K, which was higher than the melting temperature of ZrO2, whereas, when the spray distance was 8 cm, the highest particle temperature was 2465 K, and particles were thus not molten when they impacted on the substrate. So the deposition rate was higher when the substrate was located at a spray distance of 5 cm, than when it was located at 8 cm. If the spray distance is too short, particles may not be sufficiently heated, while if the distance is too long, the particles will lose some of their momentum and energy before impact, due to the cooling of the jet. Plasma spray processes under the above two conditions produce poor quality coatings.

Fig. 10
figure 10

Distribution of nano-sized particles in the vertical plane at spray distances of 5 cm (a) and 8 cm (b)

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5.5 Effects of particle size

The sizes of particles will affect their temperature and velocity. Fig. 11 shows the average temperature and velocity for four different particle diameters ranging from 100 nm to 100 µm. From Fig. 11, we can see that these four diameters of particles experienced different motion processes. The 100 nm particles got heated and accelerated more rapidly than the micron particles. However, in the downstream of flow field, the micron particles had greater velocity and higher temperature. This is because the micro-sized particles had larger inertia and heat capacity than the nano-particles. The 10 and 100 µm particles were always accelerating and getting heated. Besides, the larger the particle was, the longer the process lasted. Under the chosen conditions, the 10 µm particles could reach their melting temperature.

Fig. 11
figure 11

Effects of particle sizes on their average temperature (a) and velocity (b)

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The radial distributions of multiple nano-particles and micro-sized particles are shown in Fig. 12. Their diameter ranges were from 100 nm to 1 µm, and from 1 µm to 100 µm for nano-particles and micro-sized particles, respectively. The radial distribution range of micron particles was smaller than that of the nano-particles by 0.1 cm. This is because the Brownian force plays a more significant role for nano-particles than for micron particles. The percentage of micro-sized particles gathered near the center (radial distance smaller than 0.1 cm) was 87%, compared with 71% for nano-particles.

Fig. 12
figure 12

Radial distribution of nano-particles (a) and micro-sized particles (b)

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The temperatures and velocities of different sized particles are shown in Figs. 13 and 14, respectively. The average temperature and velocity of micro-sized particles are larger than those of nano-particles. The reason is that the micron particles have larger inertia and heat capacity than the nano-particles. The standard deviations (σ) of the radial distribution of temperature and velocity of the micron particles were larger, because the micron particles covered diameters from 1 µm to 100 µm, while the nano or sub-micrometer contained only two different size orders.

Fig. 13
figure 13

Distribution of temperatures for nano-particles (a) and micro-sized particles (b)

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Fig. 14
figure 14

Distribution of axial velocities for nano-particles (a) and micro-sized particles (b)

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6 Conclusions

In this study, a comprehensive model has been developed to simulate the trajectories and acceleration of different sized particles, and their heating, melting, and evaporation processes. This model is suitable for simulating the behaviors of both nano-particles and sub-micro-sized particles. Both Stokes number and the spray distance as key parameters affecting the particle motion were investigated. Besides, the dynamic processes of multiple particles were studied. The following conclusions can be drawn:

  1. 1.

    For the nano-sized particles, the Brownian force affects their distribution, and the present model can simulate the behaviors of those small particles.

  2. 2.

    There is a critical particle diameter corresponding to Stokes number equal to 1, to identify the trajectory of the particle around the substrate. In this study, the critical particle diameter that will follow the flow field is about 800 nm.

  3. 3.

    In the downstream of the plasma jet, the substrate affects the flow structures and spatial distribution of the particles. In this study, we identify the spray distance that achieves the ideal deposition quality, which is about 50 mm for nano-particles, and should be larger for micron particles.

  4. 4.

    The spatial distributions of nano-particles and micron particles are different. The nano-particles have wider distribution ranges than those of the micron particles. The Brownian motion is more significant for nano-particles than that for micron particles.

  5. 5.

    The acceleration and heating processes are different for nano-particles and micron particles. The nano-particles have smaller inertia and heat capacity than micron particles, so their velocity and temperature change more rapidly.