**3. Theoretical modeling**

imaging). By representing the intensity maximum for individual emission line as a function of time [20], we determined the expansion velocities for the individual elements, with 31 km/s for the first peak of Fe I and 10 km/s for the second one, while for the Mn I a velocity of 18 km/s. The expansion velocity estimated for the first peak of Fe I and the Mn atoms are similar with the values of the second plasma structure, while the velocities of the second group of Fe atoms are in line with the value determined for the first plasma structure. This concludes the fact that the two-plasma structure have uniformly distributed atoms and ions amongst them

We can take a broader view of the discussions made in the previous paragaphs for both investigated plasmas as the laser fluence and background pressure and are (expansion conditions) identical. The results are seen in **Figure 3**-right-hand side, where we can observe for a time-delay of 150 ns the spatial distribution of Fe and Mn in the Fe-Mn-Si plasma and Cu an Al in the Cu-Mn-Al plasma, respectively. We notice that for lighter elements we obtain a narrow spatial distribution, while the *heavier* ones (Cu and Fe) have a wider distribution. These differences can be seen as a separation of the composing elements based on their physical properties. The separation was previously discussed by our group in [7, 13] where the fractality of the components played a significant role, based on that the spatial distributions of different elements are reflecting the elevated degree of fractality. Lighter elements will have a higher collision rate and thus a higher fractality degree, whereas the

*Axial distribution of the main elements in the alloys as seen through OES measurements at a time-delay of 150 ns (left) and a schematic representation of the particle distribution with the plasma volume (right).*

with the fast structure having a slight depletion of Mn.

*Axial distribution of Fe (a) and Mn (b) atomic emission at various time-delays.*

**Figure 2.**

*Practical Applications of Laser Ablation*

**Figure 3.**

**124**

The fractal analysis approach for understanding the dynamics of complex physical systems was shown over the years to provide with some of the most promising results towards understanding multiparticle flow in fluids [21, 22] or plasmas [7, 13–14, 17].

For a laser ablation plasma, the nonlinearity and the chaoticity have a dual applicability being both structural and functional, with the interactions between the so-called plasma entities (structural components like electrons, ions, atoms, photons) determine reciprocal conditioning micro–macro, local–global, individualgroup, etc. In such a case, the universality of the laws describing the laser ablation plasma dynamics becomes obvious and it must be reflected by the mathematical procedures which are utilized. Basically, it makes use more and more often of the "holographic implementation" in the description of plasma dynamics. Usually, the theoretical models used to describe the ablation plasma dynamics are based on a differentiable variable assumption. Most of the notable results of the differentiable models must be understood sequentially, where the integrability and differentiability still apply. The differentiable mathematical procedures are limiting our understanding of more complex physical phenomena, such as the expansion of a laser produced plasma which implies various nonlinear behaviors, chaotic movement and self-structuring. In order to accurately describe the LPP dynamics and still remain tributary to differentiable and integral mathematics we must explicitly introduce the scale resolution. The scale resolution will be integrated in the expression of the physical variable, which describe the LPP, and implicitly in the fundamental equations, which govern these dynamics. This means that any physical variable becomes dependent on both spatial and temporal coordinates and the scale resolution. In other words, instead of using physical variables described by a nondifferentiable mathematical function, we will use different approximations of this mathematical function obtained through its averaging at various scale resolutions. As a consequence, the physical variables used to describe the LLP dynamics will act as a limit of functions family, which are non-differentiable for a null scale resolution and differentiable for non-null scale resolution.

This approach for describing LPP dynamics infers the building of novel geometric structures [23, 24] and probably new physical theories, in which the movement

laws invariant to spatio-temporal transformation, can be considered integrated on scale laws, invariant to scale resolution conversions. These geometric structures can be generated by the multifractal theory of movement in the form of Scale Relativity Theory (SRT) with a fractal dimension *DF = 2* [25] or in the form of SRT in an arbitrary fractal dimension [21, 22]. In both cases the "*holographic implementation*" of specific dynamics of LPP suggests a substitution of dynamics with limitations in an Euclidian space with dynamics without any restriction in a free-multifractal space. Thus, we will use only of the expansion of the plasma particles on continuous and non-differentiable curves in a multifractal space [25].

*where T* and *T*<sup>0</sup> being the specific temperatures and *M* and *M*<sup>0</sup> the specific mass,

, *<sup>θ</sup>* <sup>¼</sup> *<sup>M</sup> M*<sup>0</sup>

*:* (6)

*:* (7)

*τ* � �<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>θ</sup> τ* � �<sup>2</sup>

" #

*<sup>τ</sup>* <sup>¼</sup> *<sup>T</sup> T*<sup>0</sup>

*Dynamics of Transient Plasmas Generated by ns Laser Ablation of Memory Shape Alloys*

*τ*

� �<sup>2</sup> � �3*=*<sup>2</sup> exp � *<sup>ξ</sup>* � *<sup>θ</sup>*

The fundamental transient plasmas dynamics induced by laser ablation can be corelated with a multifractal medium for which its fractality degree is echoed by the elementary processes (collision, excitations, ionization or recombination, etc. -for other details see [7, 17]). In such a context (1) defines both the normalized state intensity and it is also measure of the optical emission of each plasma structure, case for which its spatial distribution of mass type is quantified through our

The results of our simulations are presented in **Figure 4(a,b)**. One can see that plasma entities with a fractality degree *μ* < 1 are defined by a narrow distribution centered around small values of ξ, while for a fractality degree *μ* > 1 the distribution is wider and is centered around values of one order of magnitude higher that the low fractality ones. Therefore, we can formulate an particular image of the plasma plume dynamics in a multifractal mathematical formalism as follows: a core of entities with low fractality and a relative low plasma temperature as well a *shell* of

In order to perform some comparison between our results and find if they can be correlated with the classical view of the LPP we have effectuated supplementary simulations on the plasma emission distribution over the particles mass for a plasma with an overall *μ* factor of 5 at an arbitrary distance (ξ = 5.5). The plasma entities with a lower mass are described by a higher relative emission for a particular temperature, and with the increase of the plasma temperature the emission of highmass elements increases as well. The obtained data is in accordance with some of our previous results from [7, 13, 17], where we correlated the plasma temperature with the plasma fractal energy. These results can have real implications for some technological applications: for low plasma excitation temperature the distribution is

*Spatial dependence of the simulated optical emission of plasma entities with various fractal degrees (a) and*

*mass distribution of the optical emission for various plasma temperature (b).*

*<sup>I</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*2*<sup>ξ</sup> <sup>θ</sup>*

mathematical model and corelated with our data.

<sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>θ</sup> τ*

high energetic particles described by a higher fractality degree.

we can further note:

**Figure 4.**

**127**

so that (5.1) becomes:

*DOI: http://dx.doi.org/10.5772/intechopen.94748*

In the following we will analyze some specific dynamics of a transient plasma generated by laser ablation, therefore postulating that the plasma particles are moving on multi-fractal curves. The mathematic procedure implies the usage of the following set of multifractal hydrodynamics equations. In such a context let us consider the density current:

$$\mathfrak{S}(\mathbf{x},t,dt) = \rho(\mathbf{x},t,dt)V(\mathbf{x},t,dt) \sum = \frac{\sum}{\pi^{1/2}} \frac{V\_0 a^2 + \frac{4t^2 (\operatorname{d}t)^{\frac{4}{\rho(\mathbf{r})} - 1}}{a^2} \mathbf{x} t}{\left[a^2 + \frac{4t^2 (\operatorname{d}t)^{\frac{4}{\rho(\mathbf{r})} - 1}}{a^2} t^2\right]^{3/2}} \exp\left[-\frac{\left(\mathbf{x} - V\_0 t\right)^2}{a^2 + \frac{4t^2 (\operatorname{d}t)^{\frac{4}{\rho(\mathbf{r})} - 2}}{a^2} t^2}\right],\tag{1}$$

where *Σ* is a surface which ℑ crosses, the other parameters have the meaning given in [21, 22].

In the aforementioned conditions, ℑ is invariant with respect to the coordinates transformation group and to the scale resolutions transformation group. Since these two groups are isomorphs, between them we can unravel various isometries like: compactizations of the spatial and temporal coordinates, compactization of the scale resolutions, compactizations of the spatio-temporal coordinates and scale resolutions, etc. Following this we can perform a compactization between the temporal coordinate and the scale resolution, which is given by the relation:

$$
\varepsilon = \frac{E}{m\_0} = 2\lambda (dt)^{\frac{2}{\Gamma(\sigma)} - 1} \nu, \quad \nu = \frac{1}{t}, \tag{2}
$$

where *ε* corresponds to the specific energy of the ablation plasma entities. Once admitted such an isometry by means of substitutions:

$$I = \frac{\mathfrak{T}\pi^{1/2}a}{V\_0\sum}, \quad \xi = \frac{x}{a}, \ u = \frac{e}{\varepsilon\_0}, \ \varepsilon\_0 = \frac{2\lambda V\_0 (dt)^{\frac{2}{\Gamma(\sigma)}-1}}{a}, \ \mu = \frac{2\lambda (dt)^{\frac{2}{\Gamma(\sigma)}-1}}{aV\_0},\tag{3}$$

(1) takes the more simplified non-dimensional form:

$$I = \frac{\mathbf{1} + \mu^2 \frac{\xi}{u}}{\left(\mathbf{1} + \mu^2 \frac{\xi}{u}\right)^{3/2}} \exp\left[-\frac{\left(\xi - \frac{1}{u}\right)^2}{\mathbf{1} + \left(\frac{\mu}{u}\right)^2}\right]. \tag{4}$$

In (5.3) and (5.4) *I* is assimilated to the normalized state intensity, *ξ* to the normalized spatial coordinate, *μ* to the normalized multifractalization degree, *u* to the normalized specific energy of the ablation plasma structures. The energy *ε* and the reference energy *ε*<sup>0</sup> can be written as:

$$
\varepsilon \approx \frac{T}{M}, \varepsilon\_0 \approx \frac{T\_0}{M\_0}, \tag{5}
$$

*Dynamics of Transient Plasmas Generated by ns Laser Ablation of Memory Shape Alloys DOI: http://dx.doi.org/10.5772/intechopen.94748*

*where T* and *T*<sup>0</sup> being the specific temperatures and *M* and *M*<sup>0</sup> the specific mass, we can further note:

$$
\pi = \frac{T}{T\_0}, \quad \theta = \frac{M}{M\_0}.\tag{6}
$$

so that (5.1) becomes:

laws invariant to spatio-temporal transformation, can be considered integrated on scale laws, invariant to scale resolution conversions. These geometric structures can be generated by the multifractal theory of movement in the form of Scale Relativity Theory (SRT) with a fractal dimension *DF = 2* [25] or in the form of SRT in an arbitrary fractal dimension [21, 22]. In both cases the "*holographic implementation*" of specific dynamics of LPP suggests a substitution of dynamics with limitations in an Euclidian space with dynamics without any restriction in a free-multifractal space. Thus, we will use only of the expansion of the plasma particles on continuous

In the following we will analyze some specific dynamics of a transient plasma generated by laser ablation, therefore postulating that the plasma particles are moving on multi-fractal curves. The mathematic procedure implies the usage of the following set of multifractal hydrodynamics equations. In such a context let us

*<sup>V</sup>*0*α*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*λ*2ð Þ *dt* <sup>4</sup>

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*λ*2ð Þ *dt* <sup>4</sup>

where *Σ* is a surface which ℑ crosses, the other parameters have the meaning

<sup>¼</sup> <sup>2</sup>*λ*ð Þ *dt* <sup>2</sup>

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>1</sup>

where *ε* corresponds to the specific energy of the ablation plasma entities. Once

, *<sup>ε</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*λV*0ð Þ *dt* <sup>2</sup>

� �<sup>3</sup>*=*<sup>2</sup> *exp* � *<sup>ξ</sup>* � <sup>1</sup>

In (5.3) and (5.4) *I* is assimilated to the normalized state intensity, *ξ* to the normalized spatial coordinate, *μ* to the normalized multifractalization degree, *u* to the normalized specific energy of the ablation plasma structures. The energy *ε* and

*<sup>M</sup>* , *<sup>ε</sup>*<sup>0</sup> <sup>≈</sup> *<sup>T</sup>*<sup>0</sup>

*M*<sup>0</sup>

*<sup>υ</sup>*, *<sup>υ</sup>* <sup>¼</sup> <sup>1</sup> *t*

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>1</sup>

*u* � �<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>μ</sup> u* � �<sup>2</sup>

" #

*<sup>α</sup>* , *<sup>μ</sup>* <sup>¼</sup> <sup>2</sup>*λ*ð Þ *dt* <sup>2</sup>

In the aforementioned conditions, ℑ is invariant with respect to the coordinates transformation group and to the scale resolutions transformation group. Since these two groups are isomorphs, between them we can unravel various isometries like: compactizations of the spatial and temporal coordinates, compactization of the scale resolutions, compactizations of the spatio-temporal coordinates and scale resolutions, etc. Following this we can perform a compactization between the temporal coordinate and the scale resolution, which is given by the relation:

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>2</sup> *<sup>α</sup>*<sup>2</sup> *xt*

� �<sup>3</sup>*=*<sup>2</sup> exp � ð Þ *<sup>x</sup>* � *<sup>V</sup>*0*<sup>t</sup>* <sup>2</sup>

2 6 4

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*λ*2ð Þ *dt* <sup>4</sup>

, (2)

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>1</sup>

*:* (4)

, (3)

*αV*<sup>0</sup>

, (5)

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>2</sup> *<sup>α</sup>*<sup>2</sup> *t*<sup>2</sup>

(1)

3 7 5,

*<sup>F</sup>*ð Þ *<sup>σ</sup>* �<sup>2</sup> *<sup>α</sup>*<sup>2</sup> *t*<sup>2</sup>

P *π*<sup>1</sup>*=*<sup>2</sup>

and non-differentiable curves in a multifractal space [25].

*<sup>ε</sup>* <sup>¼</sup> *<sup>E</sup> m*<sup>0</sup>

admitted such an isometry by means of substitutions:

*<sup>α</sup>* , *<sup>u</sup>* <sup>¼</sup> *<sup>ε</sup> ε*0

(1) takes the more simplified non-dimensional form:

*<sup>I</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>ξ</sup>*

<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>ξ</sup> u*

*<sup>ε</sup>*<sup>≈</sup> *<sup>T</sup>*

*u*

consider the density current:

given in [21, 22].

*<sup>I</sup>* <sup>¼</sup> <sup>ℑ</sup>*π*<sup>1</sup>*=*<sup>2</sup>*<sup>α</sup> V*<sup>0</sup>

**126**

<sup>P</sup> , *<sup>ξ</sup>* <sup>¼</sup> *<sup>x</sup>*

the reference energy *ε*<sup>0</sup> can be written as:

<sup>ℑ</sup>ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>*, *dt <sup>ρ</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>*, *dt V x*ð Þ , *<sup>t</sup>*, *dt* <sup>X</sup> <sup>¼</sup>

*Practical Applications of Laser Ablation*

$$I = \frac{1 + \mu^2 \xi \frac{\theta}{\tau}}{\left(1 + \left(\mu \frac{\theta}{\tau}\right)^2\right)^{3/2}} \exp\left[-\frac{\left(\xi - \frac{\theta}{\tau}\right)^2}{1 + \left(\mu \frac{\theta}{\tau}\right)^2}\right].\tag{7}$$

The fundamental transient plasmas dynamics induced by laser ablation can be corelated with a multifractal medium for which its fractality degree is echoed by the elementary processes (collision, excitations, ionization or recombination, etc. -for other details see [7, 17]). In such a context (1) defines both the normalized state intensity and it is also measure of the optical emission of each plasma structure, case for which its spatial distribution of mass type is quantified through our mathematical model and corelated with our data.

The results of our simulations are presented in **Figure 4(a,b)**. One can see that plasma entities with a fractality degree *μ* < 1 are defined by a narrow distribution centered around small values of ξ, while for a fractality degree *μ* > 1 the distribution is wider and is centered around values of one order of magnitude higher that the low fractality ones. Therefore, we can formulate an particular image of the plasma plume dynamics in a multifractal mathematical formalism as follows: a core of entities with low fractality and a relative low plasma temperature as well a *shell* of high energetic particles described by a higher fractality degree.

In order to perform some comparison between our results and find if they can be correlated with the classical view of the LPP we have effectuated supplementary simulations on the plasma emission distribution over the particles mass for a plasma with an overall *μ* factor of 5 at an arbitrary distance (ξ = 5.5). The plasma entities with a lower mass are described by a higher relative emission for a particular temperature, and with the increase of the plasma temperature the emission of highmass elements increases as well. The obtained data is in accordance with some of our previous results from [7, 13, 17], where we correlated the plasma temperature with the plasma fractal energy. These results can have real implications for some technological applications: for low plasma excitation temperature the distribution is

#### **Figure 4.**

*Spatial dependence of the simulated optical emission of plasma entities with various fractal degrees (a) and mass distribution of the optical emission for various plasma temperature (b).*

strongly heterogenous and it aides particles with an elevated fractalization degree leading to a non-congruent transfer in case of PLD and the lighter elements are predominantly in the outer regions of the plume, while the heavier ones are mainly part of the core.

*∂uFx ∂x* þ *∂uFy*

*Dynamics of Transient Plasmas Generated by ns Laser Ablation of Memory Shape Alloys*

*y*!0

þ ð∞

�∞

*uFy* ð Þ¼ *x*, *y* 0, lim

Θ ¼ *ρ*

tions

with:

be written as:

*<sup>X</sup>* <sup>¼</sup> *<sup>x</sup> x*0

*V X*ð Þ¼ , *<sup>Y</sup>* ð Þ <sup>9</sup>

**129**

, Y <sup>¼</sup> *<sup>y</sup> y*0

> *=*2 2*=*3

½ � *<sup>μ</sup><sup>X</sup>* <sup>1</sup>*=*<sup>3</sup> exp *<sup>i</sup><sup>θ</sup>* 3 � �

, U ¼ *uFx*

*U X*ð Þ¼ , *Y*

4*y*<sup>2</sup> 0 *xoa* ,*<sup>V</sup>* <sup>¼</sup> *uFy*

> 3*=*2

½ � *<sup>μ</sup><sup>X</sup>* <sup>1</sup>*=*<sup>3</sup> exp *<sup>i</sup><sup>θ</sup>* 3

*Y*

is given according to the method from [21, 22]:

*μX*<sup>2</sup>*=*<sup>3</sup> exp <sup>2</sup>*i<sup>θ</sup>* 3 � � " # � sec *<sup>h</sup>*<sup>2</sup> <sup>1</sup>

lim *y*!0

*DOI: http://dx.doi.org/10.5772/intechopen.94748*

Let us solve the equation system (11) and (12) by imposing the following condi-

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0, lim

*dy* ¼ const*:*

*∂uFx*

*ux* 2

Let us highlight that existence of a complex phase can be the pathway to a hidden temporal evolution of the system. The variation of a complex phase defines a time-dependence in an implicit manner. This means that for multifractal system can describe both spatial and temporal evolutions. Thus, the choice for Dyy gives the possibility of a both spatial and temporal investigations on the LPP plasma dynamics. The solution of (11) and (12), in their general form in normalized quantities can

> 4*y*<sup>2</sup> 0 *xoa* ,

To verify the validity of such unusual approach we obtained 3D (**Figure 5**) representations of the transient plasma flow developed based on the solution given by our multifractal system of equations. The transient plasma is *generated* in the framework of our multifractal model as a mixture of various particles with different physical properties (electron, ions, atoms, nanopatricles). This implies that the some multifractal parameters such as the complex phase, fractal dimension or specific length (*x0, y0*) will include within their values the properties of each individual component. In **Figure 5** we represented the angular separation of the plasma low for different values of the complex phase leading to the appearance of preferential *expansion directions for various elements of the plasma* for *Θ > 1.5*.

In **Figure 6** we have represented the 2D distribution portraying various plasma flow scenarios with respect to the structure of the laser ablation plasma, starting form a pure, single ionized plasma (only atoms, ions and electrons) towards a multi-component flow (including nanoparticles, molecules or clusters). There is a separation into multiple structures in the two expansion directions (across *X* and *Y*).

Φ<sup>0</sup> 6*ρ* � �1*=*3

*a=*<sup>4</sup> ð Þ<sup>2</sup>*=*<sup>3</sup> <sup>¼</sup> *<sup>x</sup>*

� � � sech<sup>2</sup> <sup>1</sup>

½ � *<sup>μ</sup><sup>X</sup>* <sup>2</sup>*=*<sup>3</sup> exp <sup>2</sup>*i<sup>θ</sup>* 3 � � " # � tanh <sup>1</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>0</sup> (12)

*<sup>D</sup>yy* <sup>¼</sup> *<sup>a</sup>* exp ð Þ *<sup>i</sup><sup>θ</sup>* (14)

2*=*3 0 *y*0

*=*<sup>2</sup>*Y*

½ � *<sup>μ</sup><sup>X</sup>* <sup>2</sup>*=*<sup>3</sup> exp <sup>2</sup>*i<sup>θ</sup>* 3 � � ( ) (16)

� � ( ) " #

*=*2*Y* , *<sup>μ</sup>* <sup>¼</sup> ð Þ dt *DF*

*=*ð Þ<sup>2</sup> �<sup>1</sup> (15)

> *=*2*Y*

> > (17)

½ � *<sup>μ</sup><sup>X</sup>* <sup>2</sup>*=*<sup>3</sup> exp <sup>2</sup>*i<sup>θ</sup>* 3

*<sup>y</sup>*!<sup>∞</sup>*uFx* ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* 0 (13)
