Understanding the underline fundamental mechanism behind experimental and industrial technologies embodies one of the foundations of the advances and tailoring new materials. With the pulsed laser deposition being one of the key techniques for obtaining complex biocompatible materials with controllable stoichiometry, there is need for experimental and theoretical advancements towards understanding the dynamics of multi component plasmas. Here we investigate the laser ablation process on Cu-Mn-Al and Fe-Mn-Si by means of space-and time-resolved optical emission spectroscopy and fast camera imaging. In a fractal paradigm the space–time homographic transformations were correlated with the global dynamics of the ablation plasmas.
- shape memory alloy
- laser ablation
- transient plasma
- optical emission spectroscopy
- fractal model
The dynamics of the ejected particles as a results of high power laser and solid matter is not a trivial problem, as it was showcased in several papers [1, 2]. The problem of complex materials, as it is the case of metallic alloys, it consists in differences in the physical properties of the composing elements. Phenomena like heterogenous melting and vaporization  are commonly reported for ns laser ablation, with dire consequences for applications like pulse laser deposition. Target material heterogeneity should be reflected in the dynamics of the ejected particles, which is often difficult to observe in industrial applications like laser welding, cutting, surface cleaning, but is otherwise excellent showcased in applications like LIBS or plasma spectroscopy. The amalgam of plasma entities found in a transient plasma generated by laser ablation contains ions, atoms, molecules, electron and photons. The most often used technique extensively reported by the other groups  or even by our group are non-invasive ones that can differentiate between the contribution of each individual component of the plasma in particular conditions even reflect the complex local and global phenomena reported in recent years. These techniques are mainly concerning the optical emission spectroscopy. Understanding laser based technologies and the interaction between high energy laser beam and metallic alloys are now relevant for a wide range of applications with fast feedback and accurate predictions on the behavior of physical processes. The dual approach of experimental investigations and theoretical modeling has proven to be a successful method for understanding the dynamics of multi-element fluids [5, 6] or as it was showcased recently by our group for complex laser produced plasmas (LPP) . The study presented in this chapter expands our previous attempts for stoichiometric transfer and plasma chemistry in the case of laser ablation of complex alloys. We discuss here the ablation of metallic particles as a result of short laser ablation interaction with ternary alloys from both an experimental and theoretical point of view. To comprehend the ablated particle dynamics we implemented optical emission spectroscopy in conjecture with ICCD fast camera imaging to record global and local information about their spatial distribution within the ablated cloud and their individual kinetic and thermal energy. From a theoretical perspective we built on our model from  and focused on exploring under, a fractal paradigm of motion, the effect of the plasma thermal energy (temperature) and ion physical properties (mass) on the spatial distribution of complex alloy plasmas. Usual models used to simulate the dynamics of complex systems are based on an assumption of the physical variable differentiability (e.g. density, momentum, energy, etc. [8, 9, 10, 11, 12] and the processes which they define. The practicality of such methods can be accepted sequentially, on space–time domains for which the differentiability still respected. However, the differential approach often fails when confronted by the reality of complex physical system (i.e. plasma plume expansion in PLD). To better represent most of the interactions at both local and global scales, it is required to introduce explicitly the scale resolution dependence. This breathes a new physical system where the variable dynamic that previously were dependent only on space and time, will now contain explicitly the dependence on the scale resolution. This can be even more abstracted and instead of using non-differential function, admittedly rather difficult to implement, just utilize different approximations of these multifractal mathematical functions derived by means of averaging at various scale resolution. A paramount consequence of this approximation is that any dynamic variable will behave as a limit of specific function families, which are non-differentiable for a null scale resolution (multifractal functions).
2. Laser induced plasmas on memory shape alloys
When investigating the ejected cloud of particle, the
We observe that the plasma has a quasi-spherical shape and increases its volume as plasma evolves. The expansion velocity was estimated using the technique presented other previous papers , where it is discussed the effect of multi-element composition of the ablation process. When performing cross-section on the recorded images in axial and transversal directions, we notice different behaviors across the two directions. We also performed cross section across the main expansion axis (axial cross-section) reveals a splitting of the plasma cloud in multiple distinct structures (two or three). Some studies report on a specific terminology for these structures, the
We notice a significant difference in the overall emission and shape of the LPP generated on the two alloys. The global emission is noticeably larger for the Fe-Mn-Si plasma and with less inner structuring, while for the Cu-Mn-Al the global emission is reduced and presents more pronounced structuring. These differences are induced by the energetic distribution uniformity on the excitation process as opposed to other types of interactions (i.e. ionization). Fe-Mn-Si plasma has an uniform aspect which is attributed similarities in the melting points of the composing elements, which leads to a uniform and homogeneous ablation. For the Cu-Mn-Al plasma there are significant differences between the physical properties of Al and Mn or Cu, could lead to a more heterogenous ablation process. These statements will further be verified with the space and time resolved OES. We would like to also note that, the fractality of the laser produced plasmas will also be affected by the inner energy of the plasma and its distribution on the composing entities [7, 13]. We anticipate here another type of analysis (fractal analysis) which we will further use in this study, that could offer valuable information about the laser produced plasmas.
In Figure 2 we plotted the spatial distribution of atoms (Fe and Mn) from the Fe-Mn-Si plasma highlighting the discrepancies amongst the two elements. We would like to note that Si was not considered as the emission line intensity for its species insignificant (lower) than those of the other elements. The Fe atoms have a dual peak distribution, while the Mn one presents only a single peak distribution. This reads as Fe atoms can be excited throughout the whole plasma volume, especially at longer distance where the electron density is significantly lower. This assessment can also explain the elevated
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
However, given our set-up optical configuration, lighter elements strongly scattered during expansion will appear to have a narrower distribution at relative short distances, while heavier particles will have a broader distribution most likely covering the whole plasma plume. Translating these results into the expansion of a three-dimensional plasma, low-mass elements are scattered towards the edge of the plasma plume while the high-mass ones are the building blocks the plasma core. For industrial applications like PLD, the result is of paramount importance interest as the particular volumes of the plasma plumes lack stoichiometry or uniformity. These properties could induce a non-congruent transfer of multielement material and affect the physical properties of the subsequent thin film. Furthermore, the diagnostic system used here allowed to capture the complex nature of the plasma and present some meaning behind it. We will further attempt to unravel more information about the relation between the fractality of specific elements and their spatial distribution within the plasma volume in the following section.
3. Theoretical modeling
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, individual-group, 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
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:
In the aforementioned conditions, 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:
(1) takes the more simplified non-dimensional form:
In (5.3) and (5.4)
so that (5.1) becomes:
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
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
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 . 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 , 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.
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:
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 the multifractal convection and the multifractal type dissipation); also the fractalization type, introduced through multifractal type tensor , 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
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.
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
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
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 three-structure system. Another approach of understating this novel phenomenon is to assimilated then with
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 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
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, 27, 28, 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.
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 direction. This complex behavior affects the angular plasma expansion and subsequently affect the spatial distribution of the deposited film. The heterogeneity of the plasma plume velocity field is in good agreement with the theoretical assumption presented in the framework of the non-differential model.
ICCD imaging revealed the splitting of the laser produced plasmas into two different structures, expanding with different velocities. An angular distribution of the front velocity was reconstructed for each of the two plasmas. The specie velocities were correlated to the properties of the elements found in the target (mass and conductivity).
A novel theoretical approach based on multifractal physics was used to simulate the behavior of multi element plasmas. The model considers the relation between the scattering probability, collision frequency and the fractality degree of the plasmas. The angular distribution of the ejected particles was discussed with respect to the fractality of the system. The simulation results are in good agreement with the experimental data.
This work was supported by Romanian Ministry of Education and Research, Nucleu Program LAPLAS VI – contract n. 16 N/2019 and contract n. PD-145/2020.
Conflict of interest
The authors declare no conflict of interest.
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