**2. The applicability of the Fourier heat equation for study of laser-nano particles clusters interaction**

Light has always played a central role in the study of physics, chemistry and biology. In the past century, a new form of light, laser light, has provided important contributions to medicine, industrial material processing, data storage, printing and defense [8] applications. In all these areas of applications, the laser-solid interaction played a crucial role. The theory of heat conduction was naturally applied to explain this interaction since it was well studied for a long time [9]. For describing this interaction, the classical heat equation was used in a lot of applications. Apart of some criticism [10], the heat equation still remains one of the most powerful tools in describing most thermal effects in laser-solid interactions [11]. In particular, the heat equation can be used for describing both of light interaction with homogeneous and inhomogeneous solids. In the literature, thus a special attention was given to cases of light interaction with multi-layered samples and thin films.

It is undertaken in the following treatment that it has a solid consisting of a layer of a metal such as Au, Ag, Al or Cu, respectively. Assuming that only a photo-thermal interaction takes place, and that all the absorbed energy is transformed into heat, the linear heat flow in the solid is fully described by the heat partial differential equation, Eq. (1):

$$\frac{\frac{\partial^2 T}{\partial \chi^2} + \frac{\partial^2 T}{\partial \chi^2} + \frac{\partial^2 T}{\partial z^2} - \frac{1}{\chi} \frac{\partial T}{\partial \chi} = -\frac{A(x, y, z, t)}{k} \tag{1}$$

where: T(*x, y, z ,t*) is the spatial-temporal temperature function, *γ* is the thermal diffusivity, *k* is the thermal conductivity and *A* is the volume heat source (per unit time). In general, one can consider the linear heat transfer approximation and using the integral transform method assume the following form for the solution of the above heat equation, Eq. (2):

$$\begin{aligned} T(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) &= \sum\_{i=1}^{n} \sum\_{j=1}^{n} \sum\_{k=1}^{n} f(\mu\_i, \nu\_j, \lambda\_k) \cdot \mathbf{g}(\mu\_i, \nu\_j, \lambda\_k, t) \\ &\times K\_{\times}(\mu\_i, \mathbf{x}) \cdot K\_{\times}(\nu\_j, \mathbf{y}) \cdot K\_{\times}(\lambda\_k, \mathbf{z}) \end{aligned} \tag{2}$$

where: *f* (*μi* , *ν<sup>j</sup>* , *<sup>λ</sup><sup>k</sup>* ) <sup>=</sup> <sup>1</sup> *<sup>k</sup>* <sup>⋅</sup> *Ci* <sup>⋅</sup> *Cj* <sup>⋅</sup> *Ck ∫* 0 *a ∫* −*b b ∫* −*c c A*(*x*, *y*, *z*, *t*)*Kx*(*μi* , *x*) ⋅ *Ky*(*ν<sup>j</sup>* , *y*) ⋅ *Kz*(*λ<sup>k</sup>* , *z*)*dxdydz* and

$$\begin{aligned} \mathbf{g}\left(\mu\_i, \nu\_j, \lambda\_k, t\right) &= \mathbf{l} \left(\mu\_i^2 + \nu\_j^2 + \lambda\_k^2\right) [\mathbf{l} - e^{-\beta\_{\mu}^2 t} - \\ \mathbf{g}\left(\mathbf{l} - e^{-\beta\_{\mu}^2 \left(t - t\_0\right)}\right) \cdot h(t - t\_0)] \end{aligned} \tag{3}$$

with *βijk* <sup>2</sup> <sup>=</sup>*γ*(*μi* <sup>2</sup> <sup>+</sup> *<sup>ν</sup><sup>j</sup>* <sup>2</sup> <sup>+</sup> *<sup>λ</sup><sup>k</sup>* 2 ).

that the heat equation has the same form in the case of irradiation with a laser beam or an electron beam, at sufficiently large beam intensities [6, 7]. There is, however, a disadvantage in this model as it cannot take into account simultaneously the variation with temperature of several thermal parameters involved in the interaction like, for example, the thermal conduc‐ tivity or thermal diffusivity. In consequence, the model should be regarded as a first approx‐ imation of the thermal field. The main advantage is that the solution is a series which converges rapidly. It is important to note that the integral transform technique, as it will be shown in the next sections, belongs to the "family" of Eigen functions and Eigen values-based methods.

**2. The applicability of the Fourier heat equation for study of laser-nano**

Light has always played a central role in the study of physics, chemistry and biology. In the past century, a new form of light, laser light, has provided important contributions to medicine, industrial material processing, data storage, printing and defense [8] applications. In all these areas of applications, the laser-solid interaction played a crucial role. The theory of heat conduction was naturally applied to explain this interaction since it was well studied for a long time [9]. For describing this interaction, the classical heat equation was used in a lot of applications. Apart of some criticism [10], the heat equation still remains one of the most powerful tools in describing most thermal effects in laser-solid interactions [11]. In particular, the heat equation can be used for describing both of light interaction with homogeneous and inhomogeneous solids. In the literature, thus a special attention was given to cases of light

It is undertaken in the following treatment that it has a solid consisting of a layer of a metal such as Au, Ag, Al or Cu, respectively. Assuming that only a photo-thermal interaction takes place, and that all the absorbed energy is transformed into heat, the linear heat flow in the solid

> *TTT T* 1 *Axyzt* ( , , ,) *t k xyz*

g ¶

(, , ,) ( , , ) ( , , ,)

mn l

*T xyzt f g t*

= ×

mn

´× ×

( ,) ( ,) ( ,)

*x i yj zk*

*K xK yK z*

*ijk ijk*

 mn l

 l

 ¶

where: T(*x, y, z ,t*) is the spatial-temporal temperature function, *γ* is the thermal diffusivity, *k* is the thermal conductivity and *A* is the volume heat source (per unit time). In general, one can consider the linear heat transfer approximation and using the integral transform method

+ + - =- (1)

ååå (2)

**particles clusters interaction**

414 Radiation Effects in Materials

interaction with multi-layered samples and thin films.

is fully described by the heat partial differential equation, Eq. (1):

222 222

assume the following form for the solution of the above heat equation, Eq. (2):

111

*i jk*

¥¥¥ = = =

¶¶¶

¶¶¶ Here, *t*0 is the light pulse length (assumed rectangular) and *h* is the step function. The functions *Kx*(*μi* , *x*), *Ky*(*ν<sup>j</sup>* , *y*) and *Kz*(*λk*, *z*) are the Eigen functions of the integral operators of the heat equation and *μi* , *ν<sup>j</sup>* , *λk* are the Eigen values corresponding to the same operators. Here, for example: *Kx*(*μi* , *x*) = cos(*μi* ⋅ *x*) + (*hlin*/*k* ⋅ *μi* ) ⋅ sin(*μi* ⋅ *x*), with *hlin*—the linear heat transfer coefficient of the solid sample along x direction.

The coefficients *Ci* , *Cj* and Ck are the normalizing coefficients where, for example: *Ci* = *∫* −*b b Kx* 2 (*μi* , *x*)*dx*. (*a*, *2b* and *2c* are the geometrical target dimensions, which are supposed to be a parallelepiped one).

It has used the thermal parameters of the Cu sample as given in Table 1.


**Table 1.** Thermal parameters of Cu.

It has used the heat equation for a configuration where the layers are assumed to have a thickness of 1 mm onto which are included clusters of nano-spheres. The heat term for such a system can be represented by the following equation:

$$A(\mathbf{x}, \mathbf{y}, z, t) = \sum\_{n, n, p} I(\mathbf{x}, \mathbf{y}, z) ( (a\_1 + r\_3 \delta(z) + a\_{mp} (\delta(\mathbf{x}\_n) \cdot \delta(\mathbf{y}\_n) \cdot \delta(z\_p))) \cdot (h(t) - h(t - t\_0)) \tag{4}$$

where, *m,n,p* denote the positions of the nano-particles-clusters, *α*1—the optical absorption coefficient, *I* the incident plane wave radiation intensity incoming from the top –z direction, *rS*—the surface absorption coefficient, *αmnp*—the nano-particles optical absorption coefficients, *x,y,t* represent the space and time coordinates on the layer surface and *h* is the step time function.

For the simulation, we have to consider:

$$a\_{mnp} >> a\_l + r\_3 d(z) \tag{5}$$

Inserting groups or clusters of nano-particles-clusters on top of a layer exposed to irradiation gives a detectable increase of temperature in comparison with the bulk material in pure form. This result can be seen in the following simulations.

For *m, n =* 1, 2 and *p =* 1, we have plotted in **Figures 1**–**3**, the thermal field of 1, 2 and 4 nanoparticles-clusters for the case of a Cu layer.

The present chapter continues the numerous ideas developed in the past few years with the integral transform technique applied to classical Fourier heat equation [1, 2].

From practical point of view, consider the formula (2), that*: i* varies from 1 to 100*; j* varies from 1 to 100, and *k* varies from 1 to 100. In consequence, the solutions will be like a sum of 1 million functions. In this way, these solutions become from semi-analytical into analytical one. The solutions are easy to compute in MATHEMATICA, or other package software.

In conclusion, it is considered that the method of integral transform technique is a serious candidate in competition with: Born approximation, Green function method or numerical methods. In **Figure 44** is represented the "geometrical" situation for **Figure 3**.

The nano-particles-clusters should be of the order of magnitude of 20 nm, which is the limit of availability of Fourier model [12].

**Figure 1.** The thermal field produced by one nano-particle-cluster on a Cu substrate. The nano-particle is situated at *x* = 0 and *y* = 0, and 200 nm depth inside Cu sample.

<sup>1</sup> ( ) *mnp <sup>S</sup>*

Inserting groups or clusters of nano-particles-clusters on top of a layer exposed to irradiation gives a detectable increase of temperature in comparison with the bulk material in pure form.

For *m, n =* 1, 2 and *p =* 1, we have plotted in **Figures 1**–**3**, the thermal field of 1, 2 and 4 nano-

The present chapter continues the numerous ideas developed in the past few years with the

From practical point of view, consider the formula (2), that*: i* varies from 1 to 100*; j* varies from 1 to 100, and *k* varies from 1 to 100. In consequence, the solutions will be like a sum of 1 million functions. In this way, these solutions become from semi-analytical into analytical one. The

In conclusion, it is considered that the method of integral transform technique is a serious candidate in competition with: Born approximation, Green function method or numerical

The nano-particles-clusters should be of the order of magnitude of 20 nm, which is the limit

**Figure 1.** The thermal field produced by one nano-particle-cluster on a Cu substrate. The nano-particle is situated at *x* =

integral transform technique applied to classical Fourier heat equation [1, 2].

solutions are easy to compute in MATHEMATICA, or other package software.

methods. In **Figure 44** is represented the "geometrical" situation for **Figure 3**.

>> + *rd z* (5)

 a

a

This result can be seen in the following simulations.

particles-clusters for the case of a Cu layer.

416 Radiation Effects in Materials

of availability of Fourier model [12].

0 and *y* = 0, and 200 nm depth inside Cu sample.

**Figure 2.** The thermal field produced by two nano-particles-clusters on a Cu substrate. The two nano-particles-clusters have the coordinates symmetric in rapport with the heat source. The two nano-particle-clusters are also 200 nm inside Cu sample.

**Figure 3.** The thermal field produced by four nano-particles-clusters on a Cu substrate. The depth is also 200 nm.

**Figure 4.** The "geometrical" situation for **Figure 3**.
