2.1. Heat conduction model

As shown in Figure 2, when the laser beam irradiates the target (e.g., aluminum), the temperature in the target rises, and melting occurs when the surface temperature reaches the melting temperature Tm. Then a part of molten materials begins to vaporize when the surface temperature reaches the boiling temperature Tb. With sufficient laser fluence, the target temperature may approach the critical temperature Tc, where dielectric transition occurs, and the dielectric layer is formed near the surface. Furthermore, the ablation plasma expands in the opposite direction of the incident laser beam, and absorbs part of laser energy before the incident laser beam reaches the target surface. The absorption of the laser energy in the plasma accelerates the plasma expansion, at the same time, the shielding of the laser energy by the plasma significantly affect the heat conduction of the target.

Figure 2. Schematic of the physical model and coordinates.

The initial length of the aluminum target is labeled as δ, as shown in Figure 2. The locations of the melting phase interface, the interface between dielectric layer and liquid phase, as well as the exposed ablation surface are labeled as sm, d, and s, respectively. The length and the inner diameter of the ceramic tube are labeled as Lc and Dc.

The heat conduction equation in terms of the volumetric enthalpy H can be written as:

$$\begin{aligned} \stackrel{\circ}{O} \stackrel{\circ}{M} - \stackrel{\circ}{\partial} \stackrel{\circ}{\pi} \left( \mathcal{K}(\mathcal{T}) \stackrel{\circ}{O} \stackrel{\circ}{\pi} \right) + \stackrel{\circ}{\nu}\_{\text{a}\nu}(t) \stackrel{\circ}{O} \stackrel{\circ}{M} + \mathcal{S}(t) \end{aligned} \tag{1}$$

ð4Þ

193

ð5Þ

ð6Þ

ð7Þ

As shown in Figure 2, ξ = 0 (i.e. z = s(t)) indicates the ablation surface, and ξ = 1 (i.e. z = δ) indicates the rear surface of the target. Moreover, the chain rule can be obtained as

Plasma Generation and Application in a Laser Ablation Pulsed Plasma Thruster

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Combining with Eq. (5), the heat conduction Eq. (1) can be transformed into following

The initial temperature of the target is considered to be equal to the ambient temperature. The boundary conditions on the rear and side surfaces are considered to be thermal insulation, and the boundary condition on the ablation surface is obtained by energy conservation. Therefore,

where T<sup>0</sup> is the initial temperature, Lv is the latent heat of vaporization, and r represents the

The ablation plasma generates with the target ablation, then expands in the opposite direction of the incident laser beam, as shown in Figure 2. Part of laser energy deposits in the plasma through inverse bremsstrahlung (IB) absorption, which causes the decrease of the laser intensity reaching on the target surface [31]. In other words, the calculated result of the plasma expansion has an significant effect on the calculation of the target ablation. Moreover, the ionization in the plasma also affects the properties of the plasma. Therefore, the plasma expansion should be calculated in detail, considering the ionization and plasma

the initial and boundary conditions are written as:

2.2. Plasma expansion and ionization model

follows:

form:

density of the target.

absorption.

where K(T) represents the temperature-dependent thermal conductivity. vsurð Þt denotes the surface recession velocity, which can be calculated by Hertz-Knudsen [33, 34] and Clausius-Clapeyron equation [35]. The heat source term S is given by:

$$\begin{cases} S(t) - (1 - R\_{\omega, \nu}) I\_{\omega \nu}(t) \alpha(t) \exp \left( - \int\_{t(t)}^z \alpha(t) d\overline{z} \right), & s(t) \le z \le d(t) \\\\ S(t) - (1 - R\_{\omega}) I\_{\omega}(t) \alpha(t) \exp \left( - \int\_{t(t)}^z \alpha(t) d\overline{z} \right), & z > d(t) \end{cases} \tag{2}$$

where α is the absorption coefficient of the target, Rsur and Isur are the reflectivity and laser intensity on the target surface, respectively. d, Rd and Id are the location, reflectivity and laser intensity on the interface between dielectric layer and liquid, respectively. Herein, Isur and Id can be given by:

$$\begin{cases} I\_{\text{asr}}(\ell) - I\_0(\ell) \exp\left(-\int\_{\frac{\ell}{\delta\_\sharp}}^{\ell(\ell)} \beta(\ell) d\overline{z}\right) \\\\ I\_{\text{s}}(\ell) - (\mathbb{I} - R\_{\text{sur}}) I\_{\text{asr}}(\ell) \exp\left(-\int\_{\frac{\ell}{\delta\_\sharp}(\ell)}^{\ell(\ell)} d\ell\right) \end{cases} \tag{3}$$

where I0(t) is the initial laser intensity, and β is the absorption coefficient of the plasma.

Considering the regression of the ablation surface, it is more convenient to transform the coordinate (z, t) into the fixed coordinate system (ξ, τ) by the following expressions:

$$\begin{cases} \tau - \iota \\ \varsigma = \frac{z - \kappa(t)}{\delta - \kappa(t)} \end{cases} \tag{4}$$

As shown in Figure 2, ξ = 0 (i.e. z = s(t)) indicates the ablation surface, and ξ = 1 (i.e. z = δ) indicates the rear surface of the target. Moreover, the chain rule can be obtained as follows:

$$\begin{cases} \frac{\partial}{\partial \mathbf{r}} = \frac{\mathbf{l}}{\delta - s(t)} \frac{\partial}{\partial \boldsymbol{\zeta}}\\ \frac{\partial}{\partial t} = \frac{\partial}{\partial \mathbf{r}} - \frac{\frac{\partial s(t)}{\partial t} (\mathbf{l} - \boldsymbol{\xi})}{\delta - s(t)} \frac{\partial}{\partial \boldsymbol{\zeta}} \end{cases} \tag{5}$$

Combining with Eq. (5), the heat conduction Eq. (1) can be transformed into following form:

$$\frac{\partial \mathcal{H}}{\partial \boldsymbol{\sigma}} - \frac{1}{(\boldsymbol{\delta} - \boldsymbol{s})^2} \frac{\partial}{\partial \boldsymbol{\xi}} \left( \boldsymbol{\kappa} \frac{\partial \mathcal{T}}{\partial \boldsymbol{\xi}} \right) + \frac{\nu\_{\text{av}} (\boldsymbol{\mathcal{Z}} - \boldsymbol{\zeta})}{\boldsymbol{\delta} - \boldsymbol{s}} \frac{\partial \mathcal{H}}{\partial \boldsymbol{\xi}} + \mathcal{S} \tag{6}$$

The initial temperature of the target is considered to be equal to the ambient temperature. The boundary conditions on the rear and side surfaces are considered to be thermal insulation, and the boundary condition on the ablation surface is obtained by energy conservation. Therefore, the initial and boundary conditions are written as:

$$\begin{cases} \left. \mathcal{U} \left( \nu, z, t \right) \right|\_{\ast=0} - I\_{\diamond} \\ \left. \frac{\partial \mathcal{U}}{\partial z} \right|\_{z=\ast} - 0, \ K \frac{\partial \mathcal{T}}{\partial z} \bigg|\_{z=0} - I\_{\diamond} \mathcal{W}\_{\ast \ast} \end{cases} \tag{7}$$

where T<sup>0</sup> is the initial temperature, Lv is the latent heat of vaporization, and r represents the density of the target.

#### 2.2. Plasma expansion and ionization model

The initial length of the aluminum target is labeled as δ, as shown in Figure 2. The locations of the melting phase interface, the interface between dielectric layer and liquid phase, as well as the exposed ablation surface are labeled as sm, d, and s, respectively. The length and the inner

where K(T) represents the temperature-dependent thermal conductivity. vsurð Þt denotes the surface recession velocity, which can be calculated by Hertz-Knudsen [33, 34] and Clausius-

where α is the absorption coefficient of the target, Rsur and Isur are the reflectivity and laser intensity on the target surface, respectively. d, Rd and Id are the location, reflectivity and laser intensity on the interface between dielectric layer and liquid, respectively. Herein, Isur and Id

where I0(t) is the initial laser intensity, and β is the absorption coefficient of the plasma.

coordinate (z, t) into the fixed coordinate system (ξ, τ) by the following expressions:

Considering the regression of the ablation surface, it is more convenient to transform the

ð1Þ

ð2Þ

ð3Þ

The heat conduction equation in terms of the volumetric enthalpy H can be written as:

diameter of the ceramic tube are labeled as Lc and Dc.

192 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Figure 2. Schematic of the physical model and coordinates.

Clapeyron equation [35]. The heat source term S is given by:

can be given by:

The ablation plasma generates with the target ablation, then expands in the opposite direction of the incident laser beam, as shown in Figure 2. Part of laser energy deposits in the plasma through inverse bremsstrahlung (IB) absorption, which causes the decrease of the laser intensity reaching on the target surface [31]. In other words, the calculated result of the plasma expansion has an significant effect on the calculation of the target ablation. Moreover, the ionization in the plasma also affects the properties of the plasma. Therefore, the plasma expansion should be calculated in detail, considering the ionization and plasma absorption.

Gas-dynamical equation is used to simulate the plasma expansion, and it can be written as follows:

$$
\begin{bmatrix}
\mathcal{P} \\
\mathcal{E}t \\
\mathcal{E}t \\
E \end{bmatrix} \mid \begin{bmatrix}
\mathcal{P}u \\
\mathcal{E}u \\
\mathcal{E}t \\
(E+p)u
\end{bmatrix} \begin{bmatrix}
0 \\
p u^2 + p \\
S\_{p\text{meas}} \\
\end{bmatrix} \tag{8}
$$

ð13Þ

195

ð14Þ

) [22].

Where , and are the first, second and third ionization energy of aluminum,

Plasma Generation and Application in a Laser Ablation Pulsed Plasma Thruster

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IB process is considered to be the most important mechanism of the plasma absorption, for nanosecond laser ablation of aluminum [22, 34, 36]. The IB absorption coefficient is given by the sum of the processes between electron and neutral atomic species, and between electron

where vl and λ<sup>l</sup> are the laser frequency and wave length. Where denotes the averaged

Obviously, the absorption coefficient of the plasma β and the length of the plasma plume δ<sup>p</sup> are changing, during the laser ablation process. Therefore, it is necessary to calculate the target

In the numerical calculation, the length of target and plasma plume are assumed to be 5 and 200 μm. Meanwhile, 400 and 1000 uniform meshes are utilized, respectively, in the target and plasma region. The time-step is taken to be <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>10</sup>�<sup>12</sup> s, respecting the CFL condition in the

The heat conduction Eq. (6) is solved employing an explicit finite difference technique. And the gas-dynamical Eq. (8) is diverged utilizing AUSM+-up method [38, 39]. In the numerical calculation, the target ablation and plasma expansion are calculated simultaneously, and coupled in each time step. Since the plasma occurs due to the ablation of target, the ablation surface of target is the inlet of the plasma. Meanwhile, the instantaneous laser intensity

cross section for a collision between electron and neutral atom (10�<sup>46</sup> m<sup>5</sup>

ablation coupled with the plasma expansion in each time step.

reaching on the target surface is decided by plasma absorption.

respectively.

and ions, as follows [22, 37]:

2.3. Numerical method

entire computational domain.

where the total energy per unit volume E is the sum of internal energy and kinetic energy.

$$\mathcal{H} - \frac{3}{2}\rho\overline{K}\mathcal{V} - \sum\_{s\sim\sigma} \frac{\rho\_{\prec} R\_{\prec} \underline{\mu}\_{1}^{\iota\vee} \Theta\_{s\downarrow}^{\langle\alpha\rangle} \exp(-\Theta\_{s\downarrow}^{\langle\alpha\rangle} \wedge T\_{\prec})}{\sum\_{i=0}^{j} \underline{\nu}\_{i}^{\{\epsilon\}} \exp(-\Theta\_{s\downarrow}^{\{\iota\}} \wedge T\_{\prec})} - \frac{1}{2}\rho m^{\natural} \tag{9}$$

where r denotes density, , , cs and Ms are the mass fraction and molar mass of species s. and are the characteristic temperature and degeneracy for species s at electronic energy level i.

The pressure p can be obtained from the state equation:

$$p = \sum\_{s=1}^{4} \rho\_s R\_s T + \rho\_s R\_s T\_s \tag{10}$$

where Rs is molar gas constant of species s, Te is the temperature of electron.

The initial and boundary conditions of Eq. (7) are determined by the results of target ablation. The instantaneous laser energy deposition in the plasma Splasma can be calculated as:

$$\left(S\_{\mathfrak{p}\_{\text{primary}}}(x,t) - I\_0(t)\mathcal{J}(t)\exp\left(\int\_{\mathcal{S}\_{\mathfrak{p}}}^{x} \mathcal{J}(t)dt\right)\right) \tag{11}$$

where δ<sup>p</sup> is the length of the ablation plasma.

In addition, the ionization and IB absorption are considered in the model. In the plasma plume, a local thermodynamic equilibrium stage is assumed. Meanwhile, the plasma plume is considered nonviscous and electrically neutral, containing five species (Al, Al<sup>+</sup> , Al2+, Al3+, and e�). Otherwise, the electron temperature is considered to be equal to the ions and neutrals. Hence, the partition function of species Q<sup>s</sup> and Q<sup>e</sup> can be expressed as follows:

$$\begin{cases} \mathbf{Q}^s = V \left( 2 \pi m\_\epsilon k\_\mathcal{B} T / h^2 \right)^{3/2} \sum\_{i=0}^r \mathbf{g}\_i^{(s)} \exp\left( -\Theta\_{cl,i}^{(s)} / T \right) \\\\ \mathbf{Q}^c = V \left( 2 \pi m\_\epsilon k\_\mathcal{B} T\_\epsilon / h^2 \right)^{3/2} \end{cases} \tag{12}$$

where ms denotes the mass of species s, s = Al, Al<sup>+</sup> , Al2+, Al3+. V is the plasma plume volume, h is Planck constant.

The species number density ns can be solved in an equilibrium state [22]:

Plasma Generation and Application in a Laser Ablation Pulsed Plasma Thruster http://dx.doi.org/10.5772/intechopen.77511 195

$$\begin{aligned} \frac{n\_{\text{Al}^{\circ}}\cdot n\_{\text{s}}}{n\_{\text{Al}}} &= \frac{Q^{\text{Al}^{\circ}}Q^{\text{s}}}{V\underline{Q}^{\text{s}^{\text{Al}}}} \exp\left(-\frac{l P\_{\text{Al}}}{k\_{\text{g}}T}\right) \\ \frac{n\_{\text{Al}^{\circ}}\cdot n\_{\text{s}}}{n\_{\text{Al}^{\circ}}} &= \frac{Q^{\text{Al}^{\circ}}Q^{\text{s}}}{V\underline{Q}^{\text{s}^{\text{Al}^{\circ}}}} \exp\left(-\frac{l P\_{\text{Al}^{\circ}}}{k\_{\text{s}}T}\right) \\ \frac{n\_{\text{Al}^{\circ}\cdot\text{H}\_{\text{e}}}}{n\_{\text{Al}^{\circ}\cdot\text{s}}} &= \frac{Q^{\text{Al}^{\circ}}\underline{Q}^{\text{s}}}{V\underline{Q}^{\text{s}^{\text{Al}^{\circ}}}} \exp\left(-\frac{l P\_{\text{Al}^{\circ}\cdot\text{s}}}{k\_{\text{o}}T}\right) \\ n\_{\text{Al}^{\circ}} &+ n\_{\text{Al}^{\circ}} + n\_{\text{Al}^{\circ}} + n\_{\text{Al}^{\circ}} + n\_{\text{c}} = n\_{\text{r}} \\ n\_{\text{Al}^{\circ}} &+ 2n\_{\text{Al}^{\circ}} + 3n\_{\text{Al}^{\circ}} - n\_{\text{r}} \end{aligned} \tag{13}$$

Where , and are the first, second and third ionization energy of aluminum, respectively.

IB process is considered to be the most important mechanism of the plasma absorption, for nanosecond laser ablation of aluminum [22, 34, 36]. The IB absorption coefficient is given by the sum of the processes between electron and neutral atomic species, and between electron and ions, as follows [22, 37]:

$$\begin{split} \boldsymbol{\beta} &= \boldsymbol{\beta}^{\overline{\mathfrak{m}}} = \boldsymbol{\beta}\_{\boldsymbol{\varsigma}\boldsymbol{\varsigma}\boldsymbol{\lambda}}^{\overline{\mathfrak{m}}} + \boldsymbol{\beta}\_{\boldsymbol{\omega}\boldsymbol{\lambda}}^{\overline{\mathfrak{m}}} \\ \boldsymbol{\beta}\_{\boldsymbol{\varsigma}\boldsymbol{\omega}\boldsymbol{\lambda}}^{\overline{\mathfrak{m}}} &= \left[ 1 - \exp\left( -\frac{h\nu\_{\boldsymbol{\varsigma}}}{k\_{\boldsymbol{\varkappa}}T} \right) \right] \boldsymbol{n}\_{\boldsymbol{\epsilon}} \boldsymbol{n}\_{\boldsymbol{\omega}\boldsymbol{\lambda}} \boldsymbol{Q}\_{\boldsymbol{\epsilon}\boldsymbol{\omega}\boldsymbol{\lambda}} \\ \boldsymbol{\beta}\_{\boldsymbol{\varsigma}\boldsymbol{\lambda}}^{\overline{\mathfrak{m}}} &= \left[ 1 - \exp\left( -\frac{h\nu\_{\boldsymbol{\zeta}}}{k\_{\boldsymbol{\varkappa}}T\_{\boldsymbol{z}}} \right) \right] \frac{4\boldsymbol{\epsilon}^{\boldsymbol{\varsigma}}\boldsymbol{\lambda}\_{\boldsymbol{\imath}}^{\boldsymbol{\mathsf{z}}}}{3\boldsymbol{\mathsf{h}}\boldsymbol{\epsilon}^{\boldsymbol{\mathsf{L}}}\boldsymbol{m}\_{\boldsymbol{\varkappa}}} \sqrt{\frac{2\pi}{3\boldsymbol{m}\_{\boldsymbol{\varsigma}}\boldsymbol{k}\_{\boldsymbol{\mathsf{z}}}T\_{\boldsymbol{z}}}} \,\boldsymbol{n}\_{\boldsymbol{\epsilon}} \left( \boldsymbol{n}\_{\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{r}}} \,|\, \boldsymbol{\mathsf{A}}\boldsymbol{n}\_{\boldsymbol{\mathsf{z}}\boldsymbol{\mathsf{l}}} \,|\, \boldsymbol{\mathsf{N}}\_{\boldsymbol{\mathsf{z}}\boldsymbol{\mathsf{l}}} \,|\, \boldsymbol{\mathsf{N}}\_{\boldsymbol{\mathsf{z}}} \right) \end{split} (14)$$

where vl and λ<sup>l</sup> are the laser frequency and wave length. Where denotes the averaged cross section for a collision between electron and neutral atom (10�<sup>46</sup> m<sup>5</sup> ) [22].

Obviously, the absorption coefficient of the plasma β and the length of the plasma plume δ<sup>p</sup> are changing, during the laser ablation process. Therefore, it is necessary to calculate the target ablation coupled with the plasma expansion in each time step.

#### 2.3. Numerical method

Gas-dynamical equation is used to simulate the plasma expansion, and it can be written as

where the total energy per unit volume E is the sum of internal energy and kinetic energy.

where r denotes density, , , cs and Ms are the mass fraction and molar mass of species s. and are the characteristic temperature and degeneracy for

The initial and boundary conditions of Eq. (7) are determined by the results of target ablation.

In addition, the ionization and IB absorption are considered in the model. In the plasma plume, a local thermodynamic equilibrium stage is assumed. Meanwhile, the plasma plume is considered nonviscous and electrically neutral, containing five species (Al, Al<sup>+</sup>

Al2+, Al3+, and e�). Otherwise, the electron temperature is considered to be equal to the ions and neutrals. Hence, the partition function of species Q<sup>s</sup> and Q<sup>e</sup> can be expressed as

> s <sup>i</sup>¼<sup>0</sup> <sup>g</sup> ð Þs

<sup>i</sup> exp �Θð Þ<sup>s</sup>

el,i =T � �

, Al2+, Al3+. V is the plasma plume volume, h

ð8Þ

ð9Þ

ð10Þ

ð11Þ

,

(12)

follows:

follows:

is Planck constant.

species s at electronic energy level i.

The pressure p can be obtained from the state equation:

194 Plasma Science and Technology - Basic Fundamentals and Modern Applications

where δ<sup>p</sup> is the length of the ablation plasma.

8 < :

where ms denotes the mass of species s, s = Al, Al<sup>+</sup>

where Rs is molar gas constant of species s, Te is the temperature of electron.

<sup>Q</sup><sup>s</sup> <sup>¼</sup> <sup>V</sup> <sup>2</sup>πmskBT=h<sup>2</sup> � �<sup>3</sup>=<sup>2</sup> <sup>P</sup><sup>j</sup>

<sup>Q</sup><sup>e</sup> <sup>¼</sup> <sup>V</sup> <sup>2</sup>πmekBTe=h<sup>2</sup> � �<sup>3</sup>=<sup>2</sup>

The species number density ns can be solved in an equilibrium state [22]:

The instantaneous laser energy deposition in the plasma Splasma can be calculated as:

In the numerical calculation, the length of target and plasma plume are assumed to be 5 and 200 μm. Meanwhile, 400 and 1000 uniform meshes are utilized, respectively, in the target and plasma region. The time-step is taken to be <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>10</sup>�<sup>12</sup> s, respecting the CFL condition in the entire computational domain.

The heat conduction Eq. (6) is solved employing an explicit finite difference technique. And the gas-dynamical Eq. (8) is diverged utilizing AUSM+-up method [38, 39]. In the numerical calculation, the target ablation and plasma expansion are calculated simultaneously, and coupled in each time step. Since the plasma occurs due to the ablation of target, the ablation surface of target is the inlet of the plasma. Meanwhile, the instantaneous laser intensity reaching on the target surface is decided by plasma absorption.
