Bases of Mathematical Simulation of Modes Obtaining Ultrafine Powder by Laser Ablation of Metal Particles Vertically Falling under the Action of Gravity in an Inert Gas Medium

Abstract

The fundamentals of mathematical modeling of the modes of obtaining an ultrafine powder by metal particles laser ablation of vertically falling in an inert gas medium under the action of gravity, provided that these particles flow around the laminar flows of this gas and are outlined. Using the results of the research as an example, the practical possibility of using these foundations of mathematical modeling is shown when obtaining ultrafine powders with particles having a shape close to spherical by the proposed method of laser ablation from spherical titanium particles of a fraction of 45–63 μm falling vertically in an inert gas medium. It is recommended to use the developed foundations of mathematical modeling in the design and development of laser ablation-based technologies for obtaining ultrafine and nanosized powders, including those in demand in the additive manufacturing of powders.

1. Introduction

It is known [1] that due to the high chemical activity and lower sintering temperatures, ultradispersed (less than 1 μm) powders, which may include nanosized (less than 100 nm), as a feedstock are increasingly used in existing and newly developed emerging technologies for obtaining promising materials with unique and enhanced physical and mechanical properties, which are in demand not only in mechanical engineering and medicine, but also in the chemical industry and energy. One of the most frequently used and intensively developing methods for obtaining ultrafine powders in powder metallurgy is laser ablation of solids. It differs from other known methods in its relative ease of implementation and the absence of accompanying impurities, which inevitably appear when similar powders are obtained by chemical methods or high-energy grinding. Currently, laser ablation is understood not just as the process of a possible transition of a substance from a solid state to a gaseous state under the action of monochromatic coherent electromagnetic radiation in the wavelength range from near infrared to red and green. At the same time, the process of transition of a substance in a vapor–gas state to its solid-state state is also understood. In addition to the vacuum induction gas atomization method, the laser ablation method makes it possible to obtain ultrafine powders with particles that have a shape close to spherical. Ultrafine metal powders with such particles are a promising feedstock for developing additive manufacturing [1, 2, 3]. As a rule, in order to obtain ultrafine powders, one tends to use laser ablation on solid, immobile, evaporating targets. Another similar method [4] based on metal particles falling vertically in an inert gas medium under the action of its own gravity may also be promising and relatively easy to implement for obtaining such powders. The development of the theoretical foundations of mathematical necessary modes for obtaining ultrafine powders modeling under such conditions can be facilitated by the formulation of the corresponding problem with an indication of the way to solve it. The formulation of such a problem, indicating the way to solve it using an illustrative example, is the main goal of this work. At the same time, the purpose of the work is also laboratory research, on the example of which it is possible to show the possibility, on the one hand, of performing mathematical modeling of laser ablation modes and obtaining ultrafine powder from spherical particles of titanium with a fraction of 45–63 μm, and on the other hand, reflects the pattern the course of this process. It should be noted that the development of the proposed method of laser ablation with metal particles sheerly falling in the field of gravity can contribute to the improvement of technologies for obtaining, using laser ablation, not only ultrafine and nanosized powder particles that are promising for additive production, but also purification from surface impurities and spheroidization, as themselves, particles with a fraction of less than 100 μm, and their laser selective layer-by-layer surfacing used in 3D printers, which are structurally similar to DMT (Direct Metal Tooling) 3D printers of the South Korean international company InssTex MX.

1. Problem statement, determination of the movement nature, and modes of particles of metal powder supply to the laser ablation zone. Let metal powder particles have a monospherical shape and a smooth surface, and a metered feed device provides them with a consistent exit and a vertical fall without collisions in an inert gas environment. In this case, they fall inside a hollow optically transparent tube (Figure 1) with smooth inner surfaces. The inside tube diameter (D_T) is larger than the particle diameter (D₀). Then, the falling of particles under these conditions will be prevented by the emerging forces of buoyancy (F_A) and gas-dynamic resistance (F_r) [5, 6]:

Figure 1.

Forces acting on particles when falling in an inert gas.

$$m{}_{\mathit{pp}}a={\gamma }_{\mathit{pp}}\frac{\pi \cdot D_0^3}{6}\frac{\mathit{dv}}{\mathit{dt}}=m_{\mathit{pp}}g-F_A-F_r=\left({\gamma }_{\mathit{pp}}-{\gamma }_g\right)\cdot g\cdot \frac{\pi \cdot D_0^3}{6}-F_r,$$E1

where m_pp, γ_pp—mass and density of material (metal) particles; γ_g—gas density; v, t, а, g—speed, time, acceleration, and gravitational acceleration.

The drag force F_r in expression (1) is influenced not only by the shape, size, and speed acquired by the particles, but also by the density and viscosity (η) of the inert gas, including the drag coefficient C_x = C_x(Re), which depends on the Reynolds number [5, 6]:

$$Re\le 0,1:F_r=3\cdot \pi \cdot \eta \cdot D_0\cdot v,$$E2

$$0,1<Re:F_r=\frac{{С}_x\cdot \pi \cdot D_0^2\cdot {\gamma }_g\cdot v^2}{8},$$E3

$$0,1<Re\le 2:{С}_x=\frac{24}{Re};2<Re\le 700:{С}_x=\frac{24}{Re}+\frac{4}{\sqrt[3]{Re}};700<Re:{С}_{х}=0,4–0,5.$$E4

and besides:

$$Re=\frac{v\cdot {\gamma }_g\cdot D_0}{\eta }.$$E5

Depending on the Reynolds number, most often [5, 6] the following types of motion of gas flows flowing around the surfaces of moving bodies are distinguished: laminar (Re < 0.1), close to it (0.1 < Re < 2), and transitional to turbulent (2 < Re < 700) and turbulent motion (700 < Re). Of these types, for free-falling particles of metal powders with fractions D₀ = 40–150 μm and D₀ = 10–40 μm, the most characteristic is laminar or close to it motion, in which Re < 2. Therefore, the expression of resistance forces (3) is reduced to its particular representation (2), and the limiting values for particle velocities, at which their motion can still be considered close to laminar, are determined by the expression:

$$v_{\max }=\frac{2\cdot \eta }{{\gamma }_g\cdot D_0}.$$E6

It is noted that in natural conditions inert gases behave as ideal gases. Therefore, their density and dynamic viscosity will depend both on the mass (m_0g), diameter (d_0g), effective cross section (σ), most probable velocity (v_0g) and mean free path (λ) of molecules between collisions, and their volume concentration (n) [7]:

$$\eta \sim \frac{{\lambda }_{0g}\cdot v_{0g}\cdot {\gamma }_g}{3};{\lambda }_{0g}=\frac{1}{n\cdot \sigma \cdot \sqrt{2}};\sigma =\pi \cdot d_{0g}^2\cdot \left(1+\frac{T_S}{T}\right);v_{0g}=\sqrt{\frac{8\cdot k_{\mathbf{B}}\cdot T}{\pi \cdot m_{0g}}};$$E7

where k_B—Boltzmann’s constant; T_S—Sutherland constant.

In contrast to the density (γ_g = n·m_0g) and pressure (p = n·k_B·T) of an inert gas, which directly depend on the volume concentration (n) of molecules, its dynamic viscosity, as followed from expressions (7), will depend only on temperature [7]:

$$\eta \left(T\right)={\eta }_0\left(T_0\right)\cdot \frac{T_0+T_S}{T+T_S}\cdot {\left(\frac{T}{T_0}\right)}^{\frac{3}{2}}.$$E8

If the free fall of particles in an inert gas is not limited by height, then their speed will increase along with the drag force. This, after a certain time (τ_рр), will lead to the appearance of a certain steady rate of fall (v_pp), after which they will acquire a suspended state. The transition to this state is determined by the values of τ_рр and v_pp, which are obtained based on the solution of expression (1):

$$v_{\mathit{pp}}=\frac{\left({\gamma }_{\mathit{pp}}-{\gamma }_g\right)\cdot g\cdot D_0^2}{18\cdot \eta \cdot {\chi }_D};{\tau }_{\mathit{pp}}=\frac{{\gamma }_{\mathit{pp}}\cdot D_0^2}{18\cdot \eta },$$E9

where ${\chi }_D=1+\frac{2,4\cdot D_0}{D_T}$.

Let us take the above as the condition of the problem, using the MathCAD-15 application package and known data [8, 9] for titanium, we will study the nature of the motion of vertically falling particles with a diameter D₀ = 50 μm of the powder of this metal in an argon medium with a constant pressure p = 100 KPa depending on from changes in its temperature within T = 0–500°C. Calculation studies have established that an increase in temperature leads to a nonlinear decrease in the Reynolds number, which can have values close to Re = 0.04 at T = 0°C (273 K) and Re = 0.006 at T = 500°C (773 K), and in the subrange Т = 300–500°С (Figure 2a)—0.04 and 0.006, respectively.

Figure 2.

The values of the Reynolds number (a) and the steady velocity of particles (b) depending on temperature.

The values of the steady velocity, as well as the Reynolds number, will decrease monotonically with increasing temperature. In particular, at T = 0°С, they can be close to v_pp = 0.2 m/s, and at T = 500°С–v_pp = 0.09 m/s, and in the subrange Т = 300–500°С (Figure 2b)–v_pp = 0.11 m/s and v_pp = 0.09 m/s, respectively.

Figure 3 shows the results of modeling the path traveled and the time it took particles to acquire a steady velocity for the case when, during precipitation, they experienced pressure from the overlying layers of other particles, the height (Н_р) of which was Н_р = D₀ and Н_р = 10⋅D₀. The changes in the consumption of powder (G_p, g/min) of titanium and the amount of ejected particles associated with it are shown in Figure 4.

Figure 3.

The distance traveled (a) and the time of acquisition of a steady velocity (b) by particles.

Figure 4.

Consumption of precipitated particles depending on the height of the titanium powder layer.

The nature analysis of the change in the steady velocity of powder particles (titanium) falling vertically in an inert gas (argon) with increasing temperature (Figure 2) decreases and in the studied temperature range can differ by at least an order of magnitude (v_pp = 0.18 m/s at T = 0°C and v_pp = 0.09 m/s at T = 500°C). The time taken for the particle velocity to reach stable values depends on the pressure and temperature of the gas, but does not depend on the pressure of the powder layers pushing these particles out (Figure 3). It also follows from computational studies that in the temperature range Т = 0–500°С, the nature of motion corresponding to laminar or close to it, for all incident particles of 10 μm < D₀ < 100 μm of titanium powder in an inert gas, will be preserved. If we single out a subrange within T = 300–500°C from this temperature range, then for particles D₀ < 200 μm. In this temperature subrange, the nature of motion corresponding to close to laminar motion will be retained for particles with D₀ < 100 μm even if the pressure in argon increases to 300–500 kPa.

If (Figure 5) titanium particles falling in a hollow cylindrical tube 1 acquire a stable velocity into the zone of action of conical laser beams 2, which has a length of l_cb = 10 mm, then the number of particles (N_pp) squeezed out by a layer of powder with a height of Н_р = 10⋅D₀ and Нр = 10⋅D₀ in the indicated zone will correspond to the data given in Table 1 and in Figure 6.

Figure 5.

Scheme of laser ablation of powder particles. 1: Hollow optically transparent tube; 2: Laser radiation conical beam; 3: Laser ablation chamber; 4: Inert gas flow.

G, g/min	Npp, p.	vpp, m/s	t, s
0,5	2003	0,13	0,2
1,5	2146	0,38	0,07

Table 1.

The quantity, speed and time spent by particles in the laser action zone.

Figure 6.

The dependence of the particles number on the time of transition to a suspended state.

When the inert gas flow is set in the laser ablation chamber 3 (Figure 5), a pressure difference (Δp) will appear at the exit of the hollow optically transparent tube 2, which can create an additional force that contributes to the deceleration of the falling particles. Taking this into account and taking into account expression (1), it is possible to purposefully carry out additional adjustment of the steady velocity of these particles in the area of action of the conical laser beam:

$$v_{\mathit{pp}}=\frac{\left({\gamma }_{\mathit{pp}}-{\gamma }_g\right)\cdot g\cdot D_0^2}{18\cdot \eta \cdot {\chi }_D}-\frac{\mathrm{\Delta }p\cdot D{}_0}{12\cdot \eta }.$$E10

With a laminar flow of an inert gas through a laser ablation chamber 3 (Figure 5) with smooth walls, based on the laws of continuity and conservation of energy, such adjustment can be made using a pressure drop (Δр):

$$\mathrm{\Delta }p=p_2-p_1=\frac{p_4\cdot G_{g4}-p_3\cdot G{}_{g3}}{G_{g2}}-p_1,$$E11

where р₁–р₄—pressure at the inlet of a hollow optically transparent tube, in the area of action of the conical laser beam, at the outlet and inlet of the laser ablation chamber; G_g2–G_g4—volume flow rate of gas at the exit of the zone of action of the conical laser beam, at the exit and inlet of the laser ablation chamber.

On the example of studying the nature of the motion of titanium powder particles in an inert gas, limited by the inner diameter of a hollow optically transparent tube, and the results obtained, the following conclusions can be drawn.

1. In a free vertical fall under the action of gravity, the particles of the medium and fine fractions of metal powders located in an inert gas experience deceleration due to the emerging buoyancy forces and gas-dynamic resistance, which affects the nature of their movement. A feature of this movement is that the gas flows around the particles are essentially laminar or close to laminar. The speed of the particles, which increases with the fall, after a certain period of time, reaches the stable values.

2. The path traveled, the time of transition to a constant value of the falling velocity of particles, and the value of this velocity itself are influenced not only by the size of the particles, but also by the density, pressure, and temperature of the inert gas in which they move. If argon is an inert gas, then in the selected range of its temperatures T = 0–500°C and pressures p = 100–500 kPa, a change in the first parameter (temperature) will, with a high degree of certainty, lead to an inversely proportional change in the value of the steady velocity, and a change the second parameter (pressure), on the contrary, to a linear change in this value.

3. The steady velocity of the particles is also affected by the pressure of their ejection from the metered feeder. If the particles move in a hollow optically transparent tube into the zone of action of the conical laser beam, and from it into the laser ablation chamber, then it is possible to additionally adjust the time of their transition to the equilibrium state and the value of the steady velocity by varying the pressure, temperature, and flow rate of the supplied inert gas. These parameters, including the granulometric composition of the supplied powder and its flow rate, as well as the pressure that pushes out the particles, all these taken together determine the modes of powder supply to the laser ablation zone.

2. Conical laser beams effect on vertically falling particles of metal powders, and determination of modes for obtaining finer fraction powders from them. Let us assume (Figure 5) that conical laser beams intersect along the axis of a hollow cylindrical tube on a segment of a certain length (l_cb) and the total laser radiation flux formed on this segment is uniform. Then, the density (q_cb) of the flow when falling vertically and successively falling particles of metal powders will be determined by the expression:

$$q_{\mathit{cb}}=\frac{P}{S}=\frac{P}{\pi \cdot D_0\cdot l_{\mathit{cb}}}=\frac{1}{S}\cdot \frac{\mathit{dW}}{\mathit{dt}}=\frac{W_{\mathit{\text{puls}}}}{S\cdot {\tau }_{\mathit{\text{puls}}}};q_{Ф}=q_{\mathit{cb}}\cdot \frac{l_{\mathit{cb}}}{D_0},$$E12

where Р, W, W_puls, τ_puls—power, energy, pulsed energy, and pulse duration of laser radiation; S—the area of laser action on particles formed by conical beams; q_Ф—flux density of laser radiation projected onto the particle surface.

In this case, the density (q₀) of the laser radiation flux absorbed by the surface layer of particles and the light pressure exerted by it (p_red) will be determined based on the Bouguer (Beer–Lambert) law [4, 7]:

$$q_0\left(\xi \right)=q_{Ф}\cdot \left(1-R_{\mathit{ref}}\right)\cdot exp\left(-{\int }_0^{\xi }\chi \left(\xi \right)\mathit{d\xi }\right);p_{\mathit{red}}=\frac{2\cdot R_{\mathit{ref}}\cdot q_{Ф}}{c}\cdot exp\left(-{\int }_0^{\xi }\chi \left(\xi \right)\mathit{d\xi }\right)$$E13

where ξ—depth of penetration of laser radiation into the metal surface; R_ref—reflection coefficient; χ—linear absorption coefficient; с—light speed.

If the particles surfaces absorbed by the laser beam have a juvenile metal surface, then χ = Const, then expressions (13) will take a simpler form:

$$q_0\left(\xi \right)=q_{Ф}\cdot \left(1-R_{\mathit{ref}}\right)\cdot exp\left(-\chi \cdot \xi \right);p_{\mathit{red}}=\frac{2\cdot R_{\mathit{ref}}\cdot q_{Ф}}{c}\cdot exp\left(-{\int }_0^{\xi }\chi \cdot \mathit{d\xi }\right).$$E14

Let the laser radiation flux be uniform in the zone of action of the conical beam. Then, in accordance with expressions (13) and (14), in the surface layer of particles to a depth (ξ ∼ 10⁻⁸–10⁻⁷ m) of penetration of laser radiation into the metal,a heat source arises that generates a heat flux having a uniform density (q₀). This heat flux during the action of the laser radiation pulse can lead to the detachment (ablation) of some atoms from the particle surface by the evaporation mechanism and heat the material of the particle itself. The relationship between the heat flux density generated by a surface heat source and the heat flux densities causing ablation (q_abl) and heating (q_λ) is established by the heat balance equation: q₀ = q_abl + q_λ. Then, according to Refs. [4, 7], the thermal flows causing ablation (q_abl) and heating (q_λ) are determined by known transcendental equations:

$$q_{\mathit{abl}}=\frac{\left(H_{м}+2,5\cdot R_g\cdot T_{\chi }\right)\cdot \sqrt{{\gamma }_{\mathit{pp}}\cdot E_M}}{\mu \cdot exp\left(\frac{H_{м}}{R_g\cdot T_{\chi }}\right)},$$E15

$$q_{\lambda }=\pi \cdot \left(T_{\chi }-T_0\right)\cdot {\gamma }_{\mathit{pp}}\cdot c_{\mathit{pp}}\cdot D_0\cdot f_{Ф}\cdot \frac{B_1\left(\mathit{Pe}\right)}{B_0\left(\mathit{Pe}\right)},$$E16

where H_м—molar enthalpy of phase transition of a metal into a gaseous state; R_g—universal gas constant; Т₀, T_χ,—initial temperature and ablation temperature; E_M—elastic modulus; μ—molar mass; f_Ф—laser pulse frequency; В₀, В₁—zero and first-order Bessel functions; Ре—Peclet number.

The temperature distribution inside the particle material when heated by a heat flux formed by laser action is determined by a well-known equation, which in differential form has the following form:

$$\frac{\partial T}{\partial t}-a_{\mathit{pp}}\cdot {\mathrm{\Delta }}_L\left(T\right)=\frac{q_{\mathit{\lambda V}}}{{\gamma }_{\mathit{pp}}\cdot c_{\mathit{pp}}}\approx \frac{q_{\lambda }}{{\gamma }_{\mathit{pp}}\cdot c_{\mathit{pp}}\cdot \mathrm{\Delta }h},$$E17

where а_pp—coefficient of thermal diffusivity of the particle material; q_λV—specific thermal power; Δ_L—Laplace operator; с_рp—heat capacity of the particle material; Δh—heating thickness of the surface layer during the time (τ_puls) of the action of the laser pulse, and:

$$a_{\mathit{pp}}=\frac{{\lambda }_{\mathit{pp}}}{c_{\mathit{pp}}\cdot \gamma {}_{\mathit{pp}}},\mathrm{\Delta }h=\sqrt{a_{\mathit{pp}}\cdot {\tau }_{\mathit{\text{puls}}}},$$

λ_pp—thermal conductivity coefficient of the particle material.

If we assume that laser ablation is associated with the instantaneous evaporation of material from the particle surface, then the ablation pressure (p_abl) will counteract the surface tension pressure (p_L). These pressures are at least an order of magnitude higher than the static and dynamic pressures of the surrounding gaseous medium [7]. Therefore, we will assume that the resulting pressure drop Δp = p_L–p_abl and the volume of material removed by ablation (ΔV_pp) of particles are interconnected according to the Claperon-Clausius law [7], which in differential form has the form:

$$\frac{\partial p}{\partial T}=\frac{H_{м}}{T\cdot \mathrm{\Delta }{V}_{\mathit{pp}}}\approx \frac{\mathrm{\Delta }p}{T_{\chi }-T_0}=\frac{H_{м}}{T_{\chi }\cdot \mathrm{\Delta }{V}_{\mathit{pp}}}.$$E18

If we assume that Δp = р₀, and the molar enthalpy of the phase transition of the metal to the gaseous state is a constant value, then expressions (19) are transformed into the known transcendental equation:

$$ln\left(\frac{p_0}{p}\right)=\frac{H_{м}}{R{}_g}\cdot \left(\frac{1}{T_{\chi }}-\frac{1}{T_0}\right),\text{or}p\left(T\right)=p_0\cdot exp\left(-\frac{H_{м}}{R{}_g}\cdot \left(\frac{1}{T_{\chi }}-\frac{1}{T_0}\right)\right)$$E19

Eq. (19) reflects the change in the vapor pressure formed by the evaporated metal, and the velocity (v_mg) of movement of metal atoms forming vapors is the Hertz-Knudsen equation:

$$v_{\mathit{mg}}\left(T\right)=\frac{0,82\cdot p_0}{{\gamma }_{\mathit{pp}}}\cdot {\left(\frac{{м}_{\mathit{pp}}}{R_g\cdot T_{\chi }}\right)}^{\frac{1}{2}}\cdot exp\left(-\frac{H_{м}}{R{}_g}\cdot \left(\frac{1}{T_{\chi }}-\frac{1}{T_0}\right)\right)$$E20

Visually, laser ablation is accompanied by a rapidly expanding torch filled with vapor-forming metal atoms. If its expansion is taken as free self-similar, characteristic of a polytropic ideal gas [7] with a known polytropic index (ζ_p), then, as follows from [4, 7], the maximum velocity (v_F) of the expanding flame front will be determined by the expression:

$$v_F\left(max\right)=\frac{2}{{\zeta }_p-1}\cdot \sqrt{\frac{{\zeta }_p\cdot R_g\cdot T_{\chi }}{\pi \cdot \gamma {}_{\mathit{pp}}\cdot D_0^2\cdot \mathrm{\Delta }h}}.$$E21

The flame removed from the surface of the particle, due to the high speed of the front and rapid expansion, will not fully carry out heat exchange with the environment, which is unlimited in relation to it, practically immobile with pressure and temperature that are not significantly different from normal ones. This allows laser ablation and the rapidly expanding torch accompanying it to be considered as interconnected isolated thermodynamic systems with the polytropic index ζ_p = 5/3, to which the superposition principle is applicable. Then, during laser ablation of particles and flame expansion, evaporation and condensation of metal droplets occur, which is determined by expressions (18) and (19). In this case, droplet condensation occurs due to the removal of a heat flux of uniform density (q_v-l) at the “saturated vapor-droplet surface” interface in a known way:

$$q_v={\alpha }_v\cdot \left(T_v-T_s\right),{\alpha }_v=\frac{2\cdot B_l\cdot c_d}{D_d}\cdot {\left(\frac{{\gamma }_d^4}{\mu }\right)}^{\frac{1}{3}}$$E22

where α_v—heat transfer coefficient; T_v, T_s—vapor and drop surface temperatures; B_l—proportionality coefficient to the velocity of propagation of elastic waves in liquid metal; D_d—drop diameter; γ_d—density of liquid metal in a drop.

Taking into account the results of metal sputtering presented in [1, 10, 11], as well as the above assumptions and dependences (19) and (20), the dimensions (D_p) of new powder particles obtained from condensed metal drops will largely correspond to the normal logarithmic distribution F(D_р):

$$F\left(D_p\right)=\frac{1}{D_p\cdot {\sigma }_p\cdot \sqrt{2\pi }}\cdot exp\left(-\frac{{\left[ln\left(D_p\right)-ln\left(D_{p0}\right)\right]}^2}{2\cdot {{\sigma }_p}^2}\right),$$E23

$${\sigma }_p=\sqrt{\frac{1}{n-1}\cdot \sum _{i=1}^n{\left[ln\left(D_{\mathit{pi}}\right)-ln\left(D_{\mathit{pp}}\right)\right]}^2},D_{p0}=\frac{1}{n-1}\cdot \sum _{i=1}^{\infty }\left(D_{\mathit{pi}}\right),$$E24

where σ_p—dispersion; D_p0—average or expected particle diameter.

If necessary, D_p0 can also be determined taking into account expression (22):

$$D_{p0}=\frac{2\cdot B_l\cdot c_d}{r_v}\cdot {\left(\frac{{\gamma }_d}{\mu }\right)}^{\frac{1}{3}}\cdot \left(T{}_{\chi }-T_l\right)\cdot \sqrt{\frac{\pi \cdot {\mu }_{\mathit{pp}}}{8\cdot R_g\cdot T_l}},$$E25

where r_v—vaporization specific heat.

Taking the above as the next condition of the problem and using the application package MathCAD-15, as well as the known data [8, 9] for titanium, we study the course of laser ablation under the action of conical beams (Figure 5) on powder particles, taking into account the data of Tables 1 and 2. If, in this case, we use not only dependences (14)–(17), but also take into account the method of heat sources, as well as the boundary conditions of the first and third kind [7, 12], then the temperature distribution inside the particle at the initial moment of exposure to the laser beam radiation will have the form shown in Figure 7.

Pulse duration, ms	Pulse repetition rate, Hz	Pulse energy, J	Radiation wavelength, nm
3	50	up to 3	1060

Table 2.

Characteristics of the pulsed laser radiation source.

Figure 7.

The temperature inside the particle at the initial moment of exposure to the laser radiation flux.

When using the temperature distribution inside the particle obtained by the method of heat sources as a calibration function, then the temperature spread obtained taking into account the boundary conditions of the first and second kind will look as shown in Figure 8.

Figure 8.

Possible spread of temperature inside the particle at the initial moment of exposure to the laser radiation flux.

When conducting virtual studies, it was taken into account that ablation from the surface of particles occurs at temperatures significantly higher than the critical Debye temperature; therefore, the electronic and atomic thermal conductivities (Figure 9a) were reduced to an average value that determines the thermal diffusivity as an inseparable concept (Figure 9b).

Figure 9.

Temperature effect on the thermal conductivity (a) and thermal diffusivity (b) of titanium.

At the same time, the effect of the thickness of powder pressure layer on the surface temperature of vertical falling particles and its distribution in the surface layer at the depth of r = 5 μm after the effect of a single pulse of laser radiation was taken into account (Figure 10a). The character of variation of particle surface heating temperature from this pulse is shown in Figure 10b.

Figure 10.

Temperature change inside (a) and at the boundary of the near-surface layer (b) of the particle under the influence of a single pulse of laser radiation.

Based on Figures 6–10 and Table 1, it follows that the value of the relative ablation of the material from the particle surface (Figure 11) will depend on the amount of powder supplied to the laser impact zone (on its consumption). If the energy of the laser radiation pulse is W = 3 J, then at a flow rate G = 0.5 g/min, it will be 45–47% of the entire mass of the particle, and at a flow rate of the studied powder G = 2 g/min it will be only 16–18%.

Figure 11.

Influence of the energy of a single pulse of laser radiation on the relative ablation.

Based on those shown in Figures 7–11 results, it is assumed that at normal (р = 100 kPa) pressure of an inert gas (argon), a complete transition to the vapor–gas phase of titanium particles with a diameter of D_х = 50 μm that fell into the laser ablation zone (Figure 5) can occur at a speed powder supply of G = 0.2 g/min., 50%—at G = 0.5 g/min., and 20%—at G = 1.8 g/min. At this flow rate, the granulometric distribution F(D_x) of the condensed particles of the new phase will be as shown in Figure 12. If at G = 0.2 g/min. Stepwise increases the pressure of the inert gas, then the granulometric distribution will have the form shown in Figure 13, and the relative mass distribution (M) will be as shown in Figure 14.

Figure 12.

Granulometric distribution of condensed particles depending on the consumption of the initial powder at p = 100 KPa.

Figure 13.

Granulometric distribution of condensed particles depending on the pressure of the inert gas at G = 0.2 g/min.

Figure 14.

Relative mass distribution of condensed particles depending on the pressure of the inert gas at G = 0.2 g/min.

Of those presented in Figures 12 and 13 results of the normal-logarithmic granulometric distribution, it follows that in the case of laser ablation under consideration, the condensed particles of the new phase form a powder, the fractional composition of which will be fifty times less (∼ 1 μm) than the original one (∼D₀ = 50 μm). If an ideal gas filter from dust with particles smaller than 250 nm is placed at the exit of the laser ablation chamber (Figure 5), then after it the granulometric distribution of condensed particles of the new phase will be as shown in Figures 15 and 16.

Figure 15.

Granulometric distribution of condensed particles after an ideal filter at 250 nm, depending on the feed rate of the initial powder at p = 100 kPa.

Figure 16.

Granulometric distribution of condensed particles after an ideal filter at 250 nm, depending on the pressure of an inert gas at G = 0.2 g/min.

From the presented Figures 14–16 of the data it follows that within an hour at G = 1.8 g/min, nine times more particles of the initial titanium powder will pass through the laser ablation zone than at a flow rate of 0.2 g/min. In this case (Figure 17), the number of particles of a new phase formed after laser ablation with a size of less than 250 nm will also be nine times greater.

Figure 17.

Distribution of condensed particles by mass after an ideal filter at 250 nm.

Analysis of the model studies results (Figures 12–17) allows us to draw the following conclusions.

1. On the whole, the granulometric composition and the number of particles of the new phase obtained after ablation are generally affected not only by the energy, pulse duration, and frequency of laser radiation, but also by the pressure of the inert gas in the zone of action of this radiation, as well as the feed rate of the initial metal powder of the medium fraction (10–100 μm). In this case, an increase in the pressure of an inert gas in the ablation zone has a two to five times less effect on the relative amount of particles of the new phase obtained than the feed rate of the initial powder. If the feed rate of the initial powder is related to the height of the layer that puts pressure on the particles falling out of the dispenser, then the rate of their movement in the laser ablation zone will be the lower, the lower the height of this layer.

2. For spherical titanium particles of fraction D₀ = 45–63 μm, moving at a set speed in the laser ablation zone with a length lpr. = 10 mm, in which, according to the initial conditions, the pulse energy W_pulse = 3 J, its duration τ_pulse = 3 ms, and the frequency laser radiation following f = 50 Hz, at a feed rate of the initial powder at a level of G = 0.2 g/min after laser ablation, the highest relative (up to 50%) obtaining of particles of a new phase is possible, in which the arithmetic mean diameter has a value of D_x = 800 nm. If under these conditions, due to gas filtration, it is necessary to isolate the largest number of particles of a new phase with a diameter of D_х ≤ 250 nm, then the feed rate of the initial titanium powder into the laser ablation zone should be set at the level G = 1.8 g/min. Then, the number of particles of the new phase separated after filtration of the required fraction will be nine times greater than at a feed rate of G = 0.2 g/min.

3. Laboratory studies of the possibility of obtaining ultrafine powders of fraction ≤ 250 nm using filtration and laser ablation of vertically falling spherical particles of titanium with a fraction of 45–63 μm. When conducting laboratory studies, a powder laser stereo lithography unit (analogous to a 3D printer) was used as the main technological equipment [13]. The appearance of this analogue of this 3D printer is shown in Figure 18, and its characteristics are shown in Table 3. As the main equipment in the design of a 3D printer analog, the following were used: a pulsed solid-state laser 1, an optical-mechanical system 2 for focusing and positioning on the plane of the laser beam, a sealed technological chamber 3 with a vertically moving construction platform, a powder dispenser and a supply and exhaust system supplying working gas, as well as a control computer 4. To accommodate additional equipment, a modular compartment 5 is provided here.

Figure 18.

Appearance of metallurgical 3D printer analogue. 1: Solid-state laser; 2: Optical-mechanical system; 3: Technological chamber; 4: Control computer; 5: Modular compartment.

Characteristics	M. unit	Value
Laser wavelength	nm	1060
Duration of laser radiation	ms	1–3
Laser frequency adjustment range	Hz	1–50
Laser pulse energy adjustment range	J	0,3–3
Operating gas	—	Argon, Nitrogen

Table 3.

Technical characteristics of the metallurgical 3D printer.

Due to the block modular design, when conducting research in the technological chamber of a 3D printer analogue according to the scheme shown in Figure 19, opposite the lens of the opto-mechanical system focusing on the laser radiation stream 1, a conical laser beam shaper 2 was installed, as well as a powder feeder 3, a dispenser 4, an ultrafine powder sampling device 6, and an FT-3-1109 microfiber filter (Munktell) 7, a hopper for collecting unevaporated powder 8, and a bubbling filter 9.

Figure 19.

Scheme of laboratory research. 1: Laser ray; 2: Shaper of conical laser beams; 3: Powder feeding device; 4: Dispenser; 5: Laser ablation zone; 6: Ultrafine powder sampling device; 7: Microfiber filter; 8: Hopper for collecting unevaporated powder; 9: Sparge filter; 10: Argon; 11: To the outlet of the supply and exhaust system supplying working gas.

The studies were carried out using the modes recommended on the basis of the simulation results and contained in the conclusions of sections 1 and 2. According to this, the energy of laser radiation pulses in the ablation zone was W_puls = 3 J with their duration τ_puls = 3 ms and repetition rate f = 50 Hz. The temperature in the technological chamber filled with argon was Т = 50 ⁰С at a pressure close to normal (120 kPa). The mass flow rate of vertically falling spherical particles of titanium powder into the laser ablation zone was G = 1.8 g/min. The granulometric composition of spherical particles of titanium powder after sieve classification corresponded to a fraction of 45–63 μm. The result of the studies was the isolation (Figure 20) of particles of the required fraction of ultrafine titanium powder, the relative mass of which in relation to the initial one did not exceed 14–16%.

Figure 20.

Electronic diffraction pattern of titanium particles isolated after filtration.

The results of laboratory studies of the possibility of isolating by filtration of an ultrafine powder of a fraction ≤250 nm obtained from vertically falling spherical particles of titanium of 45–63 μm fraction using laser ablation in the modes recommended by modeling, concerning the relative amount (14–16%) of the separated fraction with respect to the initial one, are also consistent with the forecast indicators (Figure 16) obtained by modeling.

2. Conclusions

1. Using the above research results as an example, we show the fundamental possibility of using the developed foundations of mathematical modeling to determine the necessary modes for obtaining an ultrafine powder by laser ablation of metal particles vertically falling in an inert gas medium under the action of gravity, provided that these particles flow around laminar flows of this gas.

2. Along with this, the following is also shown:

   1. II-1. In a free vertical fall under the action of gravity, the particles of the medium and fine fractions of metal powders located in an inert gas experience deceleration due to the emerging buoyancy forces and gas-dynamic resistance, which affects the nature of their movement. A feature of this movement is that the gas flows around the particles are essentially laminar or close to laminar. The speed of the particles, which increases with falling, after a certain period of time reaches the established values.

   2. II-2. The path traveled, the time of transition to a constant value of the particle fall velocity, and the value of this velocity itself are influenced not only by the size of the particles, but also by the density, pressure, and temperature of the inert gas in which they move. If argon is an inert gas, then in the selected range of its temperatures T = 0–500°C and pressures p = 100–500 kPa, a change in the first parameter (temperature) will, with a high degree of certainty, lead to an inversely proportional change in the value of the steady velocity, and a change in the second parameter (pressure), on the contrary, to a linear change in this value.

   3. II-3. The steady velocity of the particles is also influenced by the pressure of their ejection from the metered feeder. If the particles move in a hollow optically transparent tube into the zone of action of the conical laser beam, and from it into the laser ablation chamber, then it is possible to additionally adjust the time of their transition to the equilibrium state and the value of the steady velocity by varying the pressure, temperature, and flow rate of the supplied inert gas. These parameters include the granulometric composition of the supplied powder and its flow rate, as well as the pressure that pushes out the particles; all these taken together determine the modes of powder supply to the laser ablation zone.

   4. II-4. On the whole, the particle size distribution and the amount of titanium particles of the new phase obtained after ablation are generally affected not only by the energy, pulse duration, and frequency of laser radiation, but also by the pressure of the inert gas in the zone of action of this radiation, as well as the feed rate of the initial powder into it. In this case, an increase in the pressure of an inert gas in the ablation zone has a two to five times less effect on the relative amount of particles of the new phase obtained than the feed rate of the initial powder. If the powder feed rate is related to the height of the layer that puts pressure on particles falling out of the dispenser, then the rate of their movement in the laser ablation zone will be the lower, the lower the height of this layer.

   5. II-5. For spherical titanium particles of medium fraction D₀ = 45–63 μm, moving at a set speed in the laser ablation zone with a length l_pr.=10 mm, in which, according to the initial conditions, the pulse energy W_imp.=3 J, its duration τ_pulse = 3 ms, and the repetition frequency of laser radiation is f = 50 Hz, at a feed rate of the initial powder at a level of G = 0.2 g/min after laser ablation, the highest relative (up to 50%) obtaining of particles of a new phase is possible, in which the arithmetic mean diameter has a value of D_x = 800 nm. If under these conditions, due to gas filtration, it is necessary to isolate the largest number of particles of a new phase with a diameter of D_х ≤ 250 nm, then the feed rate of the initial titanium powder into the laser ablation zone should be set at the level G = 1.8 g/min. Then, the number of particles of the new phase separated after filtration of the required fraction will be nine times greater than at a feed rate of G = 0.2 g/min. These simulation results are consistent with a high degree of confidence with the practical results of laboratory studies.

3. The developed foundations of mathematical modeling can also be used [13] in the design and development of advanced technological processes using laser ablation to obtain advanced powders of fine and ultrafine fractions for additive production, consisting of particles with a spheroidized surface, including the possibility of cleaning the surface of these particles from contaminants.