*3.1.1. Absorption of laser radiation*

When laser beam strikes the surface of the substrate material, a portion of laser energy will be reflected from the interface due to the discontinuity in the real index of refraction, and the rest will be transmitted into the material. The reflectivity of a given material depends on the frequency of the light source through the dispersion relation of its index of refraction. **Figure 1** [9] presents the variation of reflectivity with the wavelength of some common metallic materials. As indicated in **Figure 1**, the reflectivity of the material generally increases with increasing wavelength. Thus, the laser energy is strongly absorbed by materials at shorter wavelengths.

**Figure 1.** Variation of reflectivity with wavelength for some metallic materials.

In addition to the reflected fraction, the laser irradiation on the surface of substrate material leads to the excitation of free electrons (in metals), vibrations (in insulators), or both (in semiconductors) [8]. The excitation results in an energy increase in the system, and this excitation energy is rapidly converted into heat for a very short time period. That is the fraction of laser irradiation absorbed by the substrate materials.

Once inside the materials, absorption causes the laser intensity to decay with depth at a rate determined by absorption coefficient of the substrate materials *α*. In general, *α* is a function of laser's wavelength and temperature. However, for constant *α*, the laser intensity, *I*, decays exponentially with depth *z* according to the Beer-Lambert law [10]:

$$I\left(z\right) = I\_0 e^{-\alpha z} \tag{1}$$

where *I*0 is the intensity just inside the surface after considering reflection loss. Considering the depth at which the intensity of the transmitted light drops to 1/*e* of its initial value at the interface, the optical penetration or absorption depth can be conveniently defined as *δ* = 1/*α*. **Figure 2** [11] shows the optical absorption depth as a function of wavelength for some common materials, such as metals and semiconductors. The important thing to note from **Figure 2** is that the absorption depths are quite small relative to bulk material dimensions. Thus, absorption of the laser energy only happens at the surface of materials.

**Figure 2.** Optical absorption depths of some materials for different wavelengths.

### *3.1.2. Heating*

the laser-material interactions are very complex, and only in some simple cases, the laser can be merely seen as a heat source. The laser-material interactions should be comprehensively

378 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

When laser beam strikes the surface of the substrate material, a portion of laser energy will be reflected from the interface due to the discontinuity in the real index of refraction, and the rest will be transmitted into the material. The reflectivity of a given material depends on the frequency of the light source through the dispersion relation of its index of refraction. **Figure 1** [9] presents the variation of reflectivity with the wavelength of some common metallic materials. As indicated in **Figure 1**, the reflectivity of the material generally increases with increasing wavelength. Thus, the laser energy is strongly absorbed by materials at shorter

considered from the physical aspects and chemical aspects.

**Figure 1.** Variation of reflectivity with wavelength for some metallic materials.

of laser irradiation absorbed by the substrate materials.

In addition to the reflected fraction, the laser irradiation on the surface of substrate material leads to the excitation of free electrons (in metals), vibrations (in insulators), or both (in semiconductors) [8]. The excitation results in an energy increase in the system, and this excitation energy is rapidly converted into heat for a very short time period. That is the fraction

**3.1. Physical aspects**

wavelengths.

*3.1.1. Absorption of laser radiation*

Once the laser energy is absorbed by the substrate materials, it is followed by various heat transfer processes such as conduction into the materials, convection, and radiation from the surface. The most significant heat transfer process is the heat conduction into the material. The heat conduction from the surface to the inner of materials establishes a temperature distribution in the material depending on the thermophysical properties of the material and laser parameters. Ignoring the convective and radiative energy transport, the temperature distribution *T* in a material can be given by the heat conduction equation, which, in 2D, can be written as [12],

$$\rho \left( T \right) C\_{\rho} \left( T \right) \frac{\partial T \left( \mathbf{x}, \mathbf{y}, t \right)}{\partial t} = \nabla \left[ k \left( T \right) \nabla T \left( \mathbf{x}, \mathbf{y}, t \right) \right] + Q \left( \mathbf{y}, t \right) \tag{2}$$

where *x* and *y* are the space coordinates and *k*, *Cp* and *ρ* are the thermal conductivity, specific heat at constant pressure, and mass density of the target material, respectively. Here, the source term *Q*(*y*,*t*) is the laser radiation absorbed by the substrate material and is expressed as [13],

$$\mathcal{Q}(\mathbf{y}, t) = I\_s(1 - R)a \exp(-a\mathbf{y}) \tag{3}$$

where *α* and *R* are the absorption and reflectivity coefficient of the substrate material, respectively, *y* is the spatial coordinate in the direction normal to the sample surface, and *Is* is the laser irradiance at the sample surface. According to Bulgakova and Bulgakov [14], *Is* is expressed as,

$$I\_s(t) = I(t) \exp\left[-\alpha(t)\right] \tag{4}$$

Based on the above equations and thermophysical properties of the materials, the temperature distribution caused by the laser energy absorption at the material surface and subsequent conduction into the body material can be calculated. Finite element analysis and some numerical calculation methods can be used to solve these equations. For composite materials, the temperature distribution plays a more important role to analyze the ablation behavior and mechanism. The phases in the composites have different thermophysical properties, which cause a quite different temperature distribution in different phases, and consequently, the phases may present different ablation behavior and mechanism.

### *3.1.3. Melting*

Irradiated by laser beams with relatively large laser power densities, surface temperatures of the materials may reach their melting temperatures (without surface evaporation). The melting of materials significantly depends on their melting points. **Figure 3** shows the melting points of some high-temperature materials [15]. The surface of pure metals melts once the surface temperature reaches their melting temperatures. Nevertheless, the surface melting of composite materials is greatly influenced by the phase composition of the materials. The phases in the composites have different melting points, which can get melting in turn with increasing laser power densities or impulse time. The former melted phases may react with some other phases or the atmosphere. New substances may be produced and affect the surface melting and phase composition of the materials.

**Figure 3.** Melting points of some high-temperature materials.

### *3.1.4. Vaporization*

( ) ( ) ( ) ( ) ( ) ( ) , , ,, , *<sup>p</sup> T xyt TC T k T T xyt Q yt*

> *Q yt I R y* ( , 1 ) =- - *<sup>s</sup>* ( )a

> > *I t It t <sup>s</sup>* ( ) = é- ù ( )exp

phases may present different ablation behavior and mechanism.

and phase composition of the materials.

w

Based on the above equations and thermophysical properties of the materials, the temperature distribution caused by the laser energy absorption at the material surface and subsequent conduction into the body material can be calculated. Finite element analysis and some numerical calculation methods can be used to solve these equations. For composite materials, the temperature distribution plays a more important role to analyze the ablation behavior and mechanism. The phases in the composites have different thermophysical properties, which cause a quite different temperature distribution in different phases, and consequently, the

Irradiated by laser beams with relatively large laser power densities, surface temperatures of the materials may reach their melting temperatures (without surface evaporation). The melting of materials significantly depends on their melting points. **Figure 3** shows the melting points of some high-temperature materials [15]. The surface of pure metals melts once the surface temperature reaches their melting temperatures. Nevertheless, the surface melting of composite materials is greatly influenced by the phase composition of the materials. The phases in the composites have different melting points, which can get melting in turn with increasing laser power densities or impulse time. The former melted phases may react with some other phases or the atmosphere. New substances may be produced and affect the surface melting

where *x* and *y* are the space coordinates and *k*, *Cp* and *ρ* are the thermal conductivity, specific heat at constant pressure, and mass density of the target material, respectively. Here, the source term *Q*(*y*,*t*) is the laser radiation absorbed by the substrate material and is expressed as

where *α* and *R* are the absorption and reflectivity coefficient of the substrate material, respectively, *y* is the spatial coordinate in the direction normal to the sample surface, and *Is* is the laser irradiance at the sample surface. According to Bulgakova and Bulgakov [14], *Is* is

 a

¶ =Ñé Ñ ù+ ë û ¶ (2)

exp( ) (3)

( ) ë û (4)

*t*

380 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

r

[13],

expressed as,

*3.1.3. Melting*

If the incident laser intensity is sufficiently high, the surface of materials can be heated to its boiling temperature, and material removal by evaporation is caused by the laser ablation. The melting depth reaches the maximum value once the surface temperature reaches the boiling point. Further increase of laser power density or pulse time causes the more severe evaporative material removal from the surface without any further increase in the melting depth. The maximum melting depth (*z*max) at which the surface reaches the boiling point can be calculated as follows [16]:

$$\text{Verfc}\left(\frac{H\varpi\_{\text{max}}}{kT\_b\sqrt{\pi\pi}}\right) = \frac{T\_m}{T\_b\sqrt{\pi}}\tag{5}$$

where *Hz*max is the maximum melting depth, *Tb* and *Tm* are the boiling and melting temperatures of the substrate materials, and *k* is the Boltzmann constant.

Once the vaporization is initiated at the surface of the material, the continued laser irradiation causes the liquid-vapor interface to move inside the material with the evaporative removal of material from the surface above the liquid-vapor interface. Assuming the thermal ablation as vaporization, the flow of material vaporized from the surface of a body at temperature *T* can be calculated according to the Hertz-Knudsen equation [13], leading to an ablation rate *ϑ* as [17],

$$\mathcal{A}\mathcal{B}\left(T\right) = \left(1 - \beta\right) \sqrt{\frac{m}{2\pi mk\_B T}} \frac{p\_0}{\rho} \left[\frac{L\_r}{k\_B} \left(\frac{1}{T\_b} - \frac{1}{T}\right)\right] \tag{6}$$

where *Tb* is the boiling temperature at pressure *p*0, *kB* is the Boltzmann constant, *β* is the back flux coefficient, and *LV* is the latent heat of vaporization of the material.

### *3.1.5. Plasma formation*

Irradiated by the laser beams with sufficiently large intensity, a great amount of surface evaporation occurs as mentioned in the previous sections. Once the vaporization takes place, the interactions between the as-produced vapor and the incident laser beam become important in determining the overall effect of the laser irradiation on the substrate material. Interaction of the laser with the vapor can lead to the ionizing of the vapor. The highly ionized vapor is termed as plasma. In dynamic equilibrium, the degree of ionization *ε* in the vapor can be expressed by the Saha equation [18]:

$$\frac{\varepsilon^2}{1-\varepsilon} = \frac{2\,\mathrm{g}\_{\mathrm{i}}}{\mathrm{g}\_{\mathrm{a}}N\_{\mathrm{g}}} \left(\frac{2\pi mk\_{\mathrm{g}}T}{h^2}\right)^{\frac{3}{2}} \exp\left(-\frac{E\_{\mathrm{i}}}{k\_{\mathrm{g}}T}\right) \tag{7}$$

with *ε* = *Ne*/*Ng* and *Ng* = *Ne* + *Na*. Here, *Ne* and *Na* are the number densities of electrons and atoms/molecules, respectively; *gi* and *ga* are the degeneracy of states for ions and atoms/ molecules; and *Ei* is the ionization energy.

The vapor and the plasma can absorb and scatter the laser radiation, which changes the actual flux received by the substrate surface. Recoil from the vapor and plasma can also generate shock waves in the substrate material, which may cause plastic deformation and work hardening [19]. Expulsion of any remaining molten material as well as initiate shock waves can be further caused by the recoil as well. In this chapter, laser ablation method is used to characterize the ablation-resistance performance of materials. It is not reasonable to choose too large laser intensity. Therefore, we make very little consideration of plasma formation.

### *3.1.6. Recondensation and resolidification*

Irradiated by the laser beams with large power densities, the material surface is heated to a rather high temperature, and significant surface evaporation and sometimes plasma take place, which makes a positive pressure over the ablated surface. Surface temperature increment from 300 to 3500 K can lead to an enormous vapor pressure increase from 10 bar to almost 160 bar for a time interval comparable with the pulse duration (pulse D 55 ns). The vapor and plasma may eject from the ablated center under the driving of the positive pressure and subsequently recondensate on the surface of the material. Besides, laser ablation is usually carried out in the air atmosphere or some reactive atmospheres. Some new phases may be formed due to the chemical reaction between the vapor and the ambient gas. The new reaction-formed chemical compounds may also condensate on the surface of the materials.

As we mentioned above, recoil of the vapor and plasma can cause further expulsion of the remaining molten material. The molten materials expulsed from the ablated center and remaining in the ablated center resolidify and also form some new phases on the surface of materials.

Resolidification of the molten materials and condensation of the vapor and plasma are very important to analyze the morphologies of the ablated surface of the materials. They may form some thin films and nanoparticles and alter the topography at the rim and surrounding areas of the ablated region.
