**4. A multifractal theoretical approach for understanding the separation of particle flow during pulsed laser deposition of multicomponent alloys**

The details of the model have been previously reported in [14]. Let us consider that the evolution of the plasma components (plasma entities) is defined by continuous but non-differential curves, in specific rage of values. This premits us to corelate the properties of plasma plume in a multifractal matrix and thus reducing the dynamics of the individual entities by integrating them with their respective multifractal trajectories (geodesics). Therefore, at extreme times scales with respect to the inverse of the maxim Lyapunov exponent [23], the classical trajectories (deterministic) are replaced by fractal geodesics (families of potential trajectories and the notion of defined spatial coordinates is replaced by that of probability densities.

In such a context, in agreement with the results from [21, 22] at a differentiable resolution scale the ablation plasma dynamics are driven by the specific fractal force:

$$\boldsymbol{F}\_{F}^{i} = \left[ \boldsymbol{u}\_{F}^{l} + \frac{\mathbf{1}}{\mathbf{4}} (d\boldsymbol{t})^{(2/D\_{F})-1} \boldsymbol{D}^{kl} \partial\_{\mathbf{k}} \right] \partial\_{\mathbf{l}} \boldsymbol{u}\_{F}^{i} \tag{8}$$

The introduction of this multifractal force in explicit manner is essential at and is responsible for the structuring of ablation plasma on each component, though a special velocity field. The functionality of our differential system of equations is given by:

$$F\_F^i = \left[ u\_F^l + \frac{1}{4} (dt)^{(2/D\_F)-1} D^{kl} \partial\_k \right] \partial\_l u\_F^i = 0 \tag{9}$$

$$
\partial\_l \boldsymbol{u}\_F^l = \mathbf{0} \tag{10}
$$

(9) represents that at a differential scale resolution the multifractal force becomes null, while (10) represents the state density conservation law at nondifferentiable scale resolution.

Generally speaking it is rather difficult to obtain an analytic solution for the system of equations considered here, taking into account its multifractal nature (through *u<sup>l</sup> F∂lui <sup>F</sup>* the multifractal convection and *<sup>D</sup>kl∂l∂ku<sup>i</sup> <sup>F</sup>* the multifractal type dissipation); also the fractalization type, introduced through multifractal type tensor *Dkl*, is left unknown purposefully in this particular representation of the model.

The continuous development of our multifractal model and its implementation for the simulation of *real* plasma like phenomena implies the definition of a threedimensional plasma like-fluid flow of a with a revolution symmetry around the *z* axis, and investigate its dynamics through a 2-dimensional projection of the plasma in the (x*,y*) plane.

Considering the symmetry plane (*x,y*), the (9) and (10) system becomes:

$$
\mu\_{F\_x} \frac{\partial \mathfrak{u}\_{F\_x}}{\partial \mathfrak{x}} + \mathfrak{u}\_{Fx} \frac{\partial \mathfrak{u}\_{F\_x}}{\partial \mathfrak{y}} = \frac{1}{4} (dt)^{(2/D\_F)-1} \, \mathcal{D}^{\mathcal{V}} \frac{\partial^2 \mathfrak{u}\_{F\_x}}{\partial \mathfrak{y}^2} \tag{11}
$$

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

$$\frac{\partial u\_{F\_x}}{\partial \mathbf{x}} + \frac{\partial u\_{F\_y}}{\partial \mathbf{y}} = \mathbf{0} \tag{12}$$

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

$$\lim\_{y \to 0} u\_{F\_y}(\mathbf{x}, y) = \mathbf{0}, \lim\_{y \to 0} \frac{\partial u\_{F\_x}}{\partial y} = \mathbf{0}, \lim\_{y \to \infty} u\_{F\_x}(\mathbf{x}, y) = \mathbf{0} \tag{13}$$

$$\Theta = \rho \int\_{-\infty}^{+\infty} u\_x \,^2 dy = \text{const.}$$

with:

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

**4. A multifractal theoretical approach for understanding the separation of particle flow during pulsed laser deposition of multicomponent**

The details of the model have been previously reported in [14]. Let us consider

continuous but non-differential curves, in specific rage of values. This premits us to corelate the properties of plasma plume in a multifractal matrix and thus reducing the

that the evolution of the plasma components (plasma entities) is defined by

dynamics of the individual entities by integrating them with their respective multifractal trajectories (geodesics). Therefore, at extreme times scales with respect to the inverse of the maxim Lyapunov exponent [23], the classical trajectories (deterministic) are replaced by fractal geodesics (families of potential trajectories and the notion of defined spatial coordinates is replaced by that of probability densities. In such a context, in agreement with the results from [21, 22] at a differentiable

resolution scale the ablation plasma dynamics are driven by the specific fractal

<sup>4</sup> ð Þ *dt* ð Þ� <sup>2</sup>*=DF* <sup>1</sup>

The introduction of this multifractal force in explicit manner is essential at and is responsible for the structuring of ablation plasma on each component, though a special velocity field. The functionality of our differential system of equations is

<sup>4</sup> ð Þ *dt* ð Þ� <sup>2</sup>*=DF* <sup>1</sup>

*∂lu<sup>l</sup>*

(9) represents that at a differential scale resolution the multifractal force becomes null, while (10) represents the state density conservation law at non-

Generally speaking it is rather difficult to obtain an analytic solution for the system of equations considered here, taking into account its multifractal nature

dissipation); also the fractalization type, introduced through multifractal type tensor *Dkl*, is left unknown purposefully in this particular representation of the model.

Considering the symmetry plane (*x,y*), the (9) and (10) system becomes:

*∂uFx <sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>1</sup>

The continuous development of our multifractal model and its implementation for the simulation of *real* plasma like phenomena implies the definition of a threedimensional plasma like-fluid flow of a with a revolution symmetry around the *z* axis, and investigate its dynamics through a 2-dimensional projection of the plasma

<sup>4</sup> ð Þ *dt* ð Þ� <sup>2</sup>*=DF* <sup>1</sup> <sup>D</sup>*yy <sup>∂</sup>*<sup>2</sup>

*<sup>F</sup>* the multifractal convection and *<sup>D</sup>kl∂l∂ku<sup>i</sup>*

*Dkl∂<sup>k</sup>*

*Dkl∂<sup>k</sup>*

*∂lui*

*<sup>F</sup>* ¼ 0 (10)

*uFx*

*<sup>∂</sup>y*<sup>2</sup> (11)

*∂lui*

*<sup>F</sup>* (8)

*<sup>F</sup>* ¼ 0 (9)

*<sup>F</sup>* the multifractal type

*<sup>F</sup>* þ 1

*<sup>F</sup>* þ 1

*Fi <sup>F</sup>* <sup>¼</sup> *ul*

*Fi <sup>F</sup>* <sup>¼</sup> *<sup>u</sup><sup>l</sup>*

*uFx ∂uFx ∂x*

þ *uFx*

differentiable scale resolution.

*F∂lui*

part of the core.

*Practical Applications of Laser Ablation*

**alloys**

force:

given by:

(through *u<sup>l</sup>*

in the (x*,y*) plane.

**128**

$$D^{\mathcal{Y}} = \mathfrak{a} \exp\left(i\theta\right) \tag{14}$$

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 be written as:

$$X = \frac{\boldsymbol{\varkappa}}{\boldsymbol{\varkappa}\_{0}}, \mathbf{Y} = \frac{\boldsymbol{y}}{\boldsymbol{y}\_{0}}, \mathbf{U} = \boldsymbol{u}\_{F\_{x}} \frac{4\boldsymbol{y}\_{0}^{2}}{\boldsymbol{\varkappa}\_{0}d}, \boldsymbol{V} = \boldsymbol{u}\_{F\_{y}} \frac{4\boldsymbol{y}\_{0}^{2}}{\boldsymbol{\varkappa}\_{0}d}, \frac{\left(\frac{\boldsymbol{\Phi}\_{0}}{\boldsymbol{\Phi}\boldsymbol{\rho}}\right)^{1/3}}{\left(\boldsymbol{\varkappa}\_{4}\right)^{2/3}} = \frac{\boldsymbol{\varkappa}\_{0}^{2/3}}{\boldsymbol{\varkappa}\_{0}}, \boldsymbol{\mu} = \left(\mathbf{d}\mathbf{t}\right)^{\left(D\_{\mathbb{E}\_{2}}\right)-1} \tag{15}$$

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

$$U(\mathbf{X}, Y) = \frac{3\slash\_2}{[\mu \mathbf{X}]^{\natural\_3} \exp\left(\frac{i\theta}{3}\right)} \cdot \text{sech}^2\left\{\frac{\omega \llcorner\_2 Y}{[\mu \mathbf{X}]^{\natural\_3} \exp\left(\frac{2i\theta}{3}\right)}\right\} \tag{16}$$

$$V(X,Y) = \frac{{\langle \gamma'\_{\gamma} \rangle}^{2\gamma\_{\beta}}}{[\mu X]^{\dot{\gamma}\_{\dot{\gamma}}} \exp\left(\frac{i\theta}{3}\right)} \left\{ \left[\frac{Y}{\mu X^{\dot{\gamma}\_{\dot{\gamma}}} \exp\left(\frac{2i\theta}{3}\right)}\right] \cdot \text{sech}\, h^2 \left[\frac{\nu\_{\dot{\gamma}} Y}{[\mu X]^{\dot{\gamma}\_{\dot{\gamma}}} \exp\left(\frac{2i\theta}{3}\right)}\right] - \tanh\left[\frac{\nu\_{\dot{\gamma}} Y}{[\mu X]^{\dot{\gamma}\_{\dot{\gamma}}} \exp\left(\frac{2i\theta}{3}\right)}\right] \right\} \tag{17}$$

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*).

**Figure 5.**

*3D representation of the total fractal velocity field of a multi-fractal plasma flow for various complex phases (0.5 – (a) 1 – (b) 1.5 – (c)).*

different flow velocities as the distance between the maxima of the three structures does not remain constant during expansion. Supplementary investigations were performed by implementing similar data treatment for the Y direction. For the cross section on the Y axis (at X = 0) we can report a more fuzzy separation. This result can be explained as the structuring phenomena of the plasma is not limited to a unique flow axis, being observed in all directions. Moreover, the fractality of the our multifractal system, defined here through ξ and *μ*, is directly related to the trajectory of the plasma *particles*. As such, for a more complex plasma model (multielement, multi-structured) the intrinsic dynamic within the plasma structures will

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

This rather complex multifractal theoretical approach manages to simulate the structuring of a multielement (complex) plasma flow. Nevertheless, this is remains an abstracted view to a real dynamics in various technological application. For the validation of the conceptual and mathematical approach we chose to perfume comparisons with our experimental investigations of laser produced plasmas in quasi-identical conditions to those generally used for pulse laser deposition. Our model is suitable for the description of PLD physical phenomena as in past years various groups have shown [26–29] that in the case of multi-element plasmas there is axial and lateral segregation of the plasma particles during expansion based on their physical properties (mass, melting temperature), which damages the quality

The dynamics of a complex multi element plasma was investigated in the framework on a non-differential, multifractal theoretical model. Structuring into multiple plasma fragments were observed for the multi-fractal fluid like system containing structural units with various physical properties. The formation of complex plasma structures during expansion is corelated to the interaction between the transient plasma structural units and it is defined here by the complex phase of the velocity field and the fractalization of the particle geodesics. The multifractal system of equations was simplified by analyzing only two main directions. The plasma

The multifractal theoretical model was compared with empirical investigations of transient plasmas generated by laser ablation of a multielement metallic targets. The expansion of the plasma plume was monitored by means of ICCD fast camera photography and optical emission spectroscopy. The ICCD fast camera imaging showcased the formation of two or three main plasma structures in the main expansion direction, coupled with a similar phenomenon in the transversal

splits in multiple structures symmetrically to the main expansion axis.

induce to a separation on the main expansion axis (at X = 0).

*Transversal cross section for ξ = 0.3 (a), ξ = 1 (b) and ξ = 5 (c).*

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

and properties of the deposited film.

**5. Conclusion**

**131**

**Figure 7.**

**Figure 6.**

*Total fractal velocity field evolution on the two main directions (X,Y) for a multi fractal system with ξ values of 0.1 (a), 0.4 (b), 0.7 (c).*

For small values of the fractalization degree, which from here on further will be considered as *control* parameter, we define a plasma flow containing only one type of particles thus on plasma component. In **Figure 6(a)**, there can be seen only one plasma structure along the main expansion axis. The increase of this fractalization degree, control parameter, subsequently leads to changes in the homogeneity of the structural units of the plasma (i.e. our model of plasma becomes more heterogeneous in term of plasma particle mass and energy). It is also noticeable the formation of two symmetrically positioned secondary structures (lateral with respect to the main expansion axis). We conclude that those volume of plasma contain mainly components defined by a small physical volume and low kinetic energy. A subsequent increase in the fractalization degree and thus of the heterogeneity of the plasma leads to the formation of lateral-symmetrically situated plasma *structures* each defining different families of particles with physical properties.

An important conclusion extracted from our simulation is that the plasma structuring process is gradual one. For values of ξ = 0.3 1, we can obtain three main lateral structures which are also followed by a continuous internal structuring visible for fractalization degrees ξ > 1. Within the framework of our multifractal model this is reversible transition as the distribution often returns to the threestructure system. Another approach of understating this novel phenomenon is to assimilated then with *breathing modes of the fractal system* (oscillatory behavior). The evolution of our plasma model in a multifractal framework attempts a complete transition towards a completely separate flow but the interactions of the multifractal forces between the individual plasma structures are then in charge for the unification of the plasma structures.

The multi-structuring of the laser produced plasma was highlighted by executing cross sections in X direction (see **Figure 7**). In X direction the separation is more obvious first moments of expansion. Each of the new plasma structure is defined by *Dynamics of Transient Plasmas Generated by ns Laser Ablation of Memory Shape Alloys DOI: http://dx.doi.org/10.5772/intechopen.94748*

**Figure 7.** *Transversal cross section for ξ = 0.3 (a), ξ = 1 (b) and ξ = 5 (c).*

different flow velocities as the distance between the maxima of the three structures does not remain constant during expansion. Supplementary investigations were performed by implementing similar data treatment for the Y direction. For the cross section on the Y axis (at X = 0) we can report a more fuzzy separation. This result can be explained as the structuring phenomena of the plasma is not limited to a unique flow axis, being observed in all directions. Moreover, the fractality of the our multifractal system, defined here through ξ and *μ*, is directly related to the trajectory of the plasma *particles*. As such, for a more complex plasma model (multielement, multi-structured) the intrinsic dynamic within the plasma structures will induce to a separation on the main expansion axis (at X = 0).

This rather complex multifractal theoretical approach manages to simulate the structuring of a multielement (complex) plasma flow. Nevertheless, this is remains an abstracted view to a real dynamics in various technological application. For the validation of the conceptual and mathematical approach we chose to perfume comparisons with our experimental investigations of laser produced plasmas in quasi-identical conditions to those generally used for pulse laser deposition. Our model is suitable for the description of PLD physical phenomena as in past years various groups have shown [26–29] that in the case of multi-element plasmas there is axial and lateral segregation of the plasma particles during expansion based on their physical properties (mass, melting temperature), which damages the quality and properties of the deposited film.

## **5. Conclusion**

For small values of the fractalization degree, which from here on further will be considered as *control* parameter, we define a plasma flow containing only one type of particles thus on plasma component. In **Figure 6(a)**, there can be seen only one plasma structure along the main expansion axis. The increase of this fractalization degree, control parameter, subsequently leads to changes in the homogeneity of the structural units of the plasma (i.e. our model of plasma becomes more heterogeneous in term of plasma particle mass and energy). It is also noticeable the formation of two symmetrically positioned secondary structures (lateral with respect to the main expansion axis). We conclude that those volume of plasma contain mainly components defined by a small physical volume and low kinetic energy. A subsequent increase in the fractalization degree and thus of the heterogeneity of the plasma leads to the formation of lateral-symmetrically situated plasma *structures*

*Total fractal velocity field evolution on the two main directions (X,Y) for a multi fractal system with ξ values of*

*3D representation of the total fractal velocity field of a multi-fractal plasma flow for various complex phases*

each defining different families of particles with physical properties.

transition towards a completely separate flow but the interactions of the

the unification of the plasma structures.

**Figure 5.**

**Figure 6.**

**130**

*0.1 (a), 0.4 (b), 0.7 (c).*

*(0.5 – (a) 1 – (b) 1.5 – (c)).*

*Practical Applications of Laser Ablation*

multifractal forces between the individual plasma structures are then in charge for

The multi-structuring of the laser produced plasma was highlighted by executing cross sections in X direction (see **Figure 7**). In X direction the separation is more obvious first moments of expansion. Each of the new plasma structure is defined by

An important conclusion extracted from our simulation is that the plasma structuring process is gradual one. For values of ξ = 0.3 1, we can obtain three main lateral structures which are also followed by a continuous internal structuring visible for fractalization degrees ξ > 1. Within the framework of our multifractal model this is reversible transition as the distribution often returns to the threestructure system. Another approach of understating this novel phenomenon is to assimilated then with *breathing modes of the fractal system* (oscillatory behavior). The evolution of our plasma model in a multifractal framework attempts a complete

The dynamics of a complex multi element plasma was investigated in the framework on a non-differential, multifractal theoretical model. Structuring into multiple plasma fragments were observed for the multi-fractal fluid like system containing structural units with various physical properties. The formation of complex plasma structures during expansion is corelated to the interaction between the transient plasma structural units and it is defined here by the complex phase of the velocity field and the fractalization of the particle geodesics. The multifractal system of equations was simplified by analyzing only two main directions. The plasma splits in multiple structures symmetrically to the main expansion axis.

The multifractal theoretical model was compared with empirical investigations of transient plasmas generated by laser ablation of a multielement metallic targets. The expansion of the plasma plume was monitored by means of ICCD fast camera photography and optical emission spectroscopy. The ICCD fast camera imaging showcased the formation of two or three main plasma structures in the main expansion direction, coupled with a similar phenomenon in the transversal
