Complexity

Volume 2018, Article ID 1814082, 8 pages

https://doi.org/10.1155/2018/1814082

## Fractal Method for Modeling the Peculiar Dynamics of Transient Carbon Plasma Generated by Excimer Laser Ablation in Vacuum

^{1}Polymer Materials Physics Laboratory, “Petru Poni” Institute of Macromolecular Chemistry, 41 A Gr. Ghica Voda Alley, 700487 Iasi, Romania^{2}Department of Physics, Gheorghe Asachi Technical University, 700050 Iasi, Romania^{3}Univ. Lille, CNRS, UMR 8523, Physique des Lasers, Atomes et Molécules (PhLAM), Centre d’Etudes et de Recherches Lasers et Applications (CERLA), 59000 Lille, France

Correspondence should be addressed to P. Nica; or.isaiut@acinp

Received 1 March 2018; Accepted 8 July 2018; Published 12 August 2018

Academic Editor: Thach Ngoc Dinh

Copyright © 2018 C. Ursu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Carbon plasmas generated by excimer laser ablation are often applied for deposition (in vacuum or under controlled atmosphere) of high-technological interest nanostructures and thin films. For specific excimer irradiation conditions, these transient plasmas can exhibit peculiar behaviors when probed by fast time- and space-resolved optical and electrical methods. We propose here a fractal approach to simulate this peculiar dynamics. In our model, the complexity of the interactions between the transient plasma particles (in the Euclidean space) is substituted by the nondifferentiability (fractality) of the motion curves of the same particles, but in a fractal space. For plane symmetry and particular boundary conditions, stationary geodesic equations at a fractal scale resolution give a fractal velocity field with components expressed by means of nonlinear solutions (soliton type, kink type, etc.). The theoretical model successfully reproduces the (surprising) formation of V-like radiating plasma structures (consisting of two lateral arms of high optical emissivity and a fast-expanding central part of low emissivity) experimentally observed.

#### 1. Introduction

Plasma structures are often assimilated into complex systems, when considering their functionality and structures [1, 2]. The models commonly used for studying their dynamics are assuming the differential hypothesis of the physical quantities describing space and time evolution, such as energy, momentum, density, and so on, although such models are limited to large-enough plasma volumes where differentiability and integrability can be applied [3–5]. The differential methods become unsuccessful when considering the physical reality having nonintegral and nondifferentiable dynamics, for example, instabilities of complex systems that are generating chaos or patterns. Therefore, it is required to define explicitly a scale resolution in the expression of variables and thus implicitly in the equations governing the systems’ evolution. Consequently, the dynamic variables remain space and time dependent as in the classical description, but they become also dependent on the scale resolution. Otherwise, instead of using nondifferentiable functions, one can work with various approximations of them, obtained by averaging at various scale resolutions. Therefore, dynamic variables behave as the limit of a family of functions, which at zero scale resolution are nondifferentiable, while at nonzero scale resolution are differentiable [6–8].

The previous approximation is well applied for complex system dynamics, because the real measurements are done for a finite scale resolution. This implies building a physical theory for a new geometric structure, by introducing the scale laws (and thus the scale transformation invariance) into the motion laws (which already are invariant to transformations of space and time coordinates). These requirements are satisfied by the scale relativity theory (SRT) [6] and more recently by the SRT in an arbitrary constant fractal dimension [7], where the interaction complexity is replaced by nondifferentiability, and motions take place without constraints on continuous but nondifferentiable curves in a fractal space time. Otherwise, when the time scale resolution is large enough when compared with the inverse value of the highest Lyapunov exponent [9, 10], potential routes are replacing deterministic trajectories and positions characterized by probability densities are substituting the definite ones. Therefore, the motion curves are both geodesics (in a fractal space) and fractal fluid stream lines, so that the geodesics are selected through the external constraints.

When the scale covariance principle is functional, the transition from differentiable to fractal physics is achieved by replacing the usual time derivative with the fractal operator [7]
with
where is the fractal space coordinate, *t* is the nonfractal time coordinate having the role of an affine parameter, and is the complex velocity field, of real part , which is scale resolution () independent, and imaginary part , which depends on the scale resolution. The diffusion coefficient, , describes the fractal-nonfractal transition. For the fractal dimension of the motion curves, one can choose various definitions (Kolmogorov, Hausdorff-Besikovitch, etc. [9, 10]), but its value has to be kept constant during the analysis of the complex system dynamics.

The fractal operator (1) behaves as a scale covariant derivative, and it allows us to obtain the dynamics fundamental equations, similarly as in the differentiable case. Thus, applying the fractal operator (1) to the complex velocity field (4), it gives while for an external scalar potential , it results with

In the present paper, by means of (4), new solutions of fractal velocity fields are obtained in the hypothesis of constant density, null fractal force, and constant flux momentum per length unit. Then, the theoretical results are applied in the field of laser-produced plasma, to explain various expansion features of carbon plasma obtained by excimer laser ablation in various focusing conditions. Such transient plasmas play a significant role in the production of various nanostructured materials (carbon nanotubes, nanowires, graphene, etc.) [11–13], and controlling their properties (expansion velocities, density, and temperature) allows a significant enhancement of the deposition process [14–16].

#### 2. Theoretical Aspects

Separating the dynamics of the complex system on the resolution scale (on the differential one through the real part, , and on the fractal one through the imaginary part, ) leads to

At differential resolution scale, this separation implies the fractal force,

Nullifying its value, in the condition, induces a particular velocity field of explicit form given in the following.

Finding the solutions for these equations can be relatively difficult, due to the fact that this equation system is nonlinear [4, 5]. However, for the particular case of a stationary flow in an symmetry plane, an analytical solution of this system exists. With the previous assumption, (8) and (9) take the form where and are the velocities along the and axes, respectively.

Let us impose the boundary conditions of the flow and that the flux momentum per length unit is constant,

Such conditions imply that on the limit , the velocity is directed on the *x*-axis and that it is on the *y*-axis at large distances, while (12) is the conservation law of a specific kinetic energy density along the *x*-axis.

Using the method from [4, 5], the following solutions result with

For , we obtain in (13) the critical flow velocity (maximum value along the *x*-axis) in the form
while (12), taking into account (15), becomes
so that the critical cross section of the stream line tube (diameter of the flowing tube corresponding to the critical flow velocity) of the complex system flow is given by

Equations (13), (14), and (15) can be significantly simplified by introducing the normalized quantities, where are the specific lengths, is the specific velocity, and is the fractalization degree (similar to the one defined in [17]). It results that

Relation (20) suggests that, independent of the fractalization degree, the fractal velocity field is highly nonlinear: along the () flow axis, it is of soliton type (), while along the () flow axis, it is a “mixture” between a soliton type, a kink type (tanh), and so on [4, 18]. The 3D dependences of the normalized components of the fractal velocity field on the non-dimensional spatial coordinates are plotted in Figures 1(a) and 1(b), along with its norm (Figure 1(c)), for the fractalization degree . It results with a central symmetry axis of the flow and a decrease of the velocity norm from the left boundary. Meanwhile, the local central maximum is adjacent with two symmetrical regions of the minimum.