1. Introduction
Microscopic periodic features provide surfaces with superior functionalities. In biological and medical applications, repetitive surface textures improve the biocompatibility of bone implants [
1], guide directional cell growth [
2] and inhibit bacterial adhesion and biofilm formation [
3]. Further advantages of periodic structured surfaces include enhanced light absorption [
4], reduced friction [
5,
6] and anisotropic wetting [
7]. These topographies are increasingly considered for the manufacture of functional surfaces, e.g., in biomedical and marine engineering, tribology, optics and aeronautics.
Direct laser interference patterning (DLIP) is a novel method that allows the production of periodic surface structures with feature sizes in the submicron to micron range in a single processing step. For this purpose, the primary beam of a pulsed laser with wavelength
is split into two or more partial beams. The periodic intensity distribution due to the interfering coherent beams is employed to treat a surface situated in the interference volume. Here, the interference of two beams is considered, resulting in a sinusoidal energy density distribution [
8]
where
is the fluence of each beam and
is the intersecting angle between the beams. Accordingly, two-beam interference patterning generates line-like surface structures with a spatial periodicity
the minimum achievable periodic distance being half of the used wavelength, i.e.,
for
. The fabrication of repetitive microstructures by means of DLIP, using nanosecond pulses of ultraviolet laser radiation, was demonstrated on ceramics [
1], polymers [
2,
3,
4,
7], non-metals [
5] and metals [
6].
A thorough understanding of the process is essential to the precise patterning of surfaces. However, an insight into the physical mechanisms effective during laser processing cannot be gained from experimental observation, especially owing to the short laser pulse duration and the microscopic scale of the surface modification. Nevertheless, numerical simulations enable the detailed investigation of the contemplated physical effects. This approach allows for parameter variations to identify suitable process conditions with regard to texturing a specific substrate, avoiding an excessive consumption of resources in experiments. Grid-based numerical techniques, such as the finite volume and the finite element method (FEM), are usually employed in the simulation of laser material processing [
9].
The mathematical model of the DLIP process originally consisted in the heat diffusion equation with the pulsed laser interference irradiation incorporated in the heat source term [
8] and sink terms accommodating the latent heat of involved phase changes [
10,
11]. Using this model, thermal simulations of DLIP were carried out by the FEM to predict the temporal evolution of the temperature distribution near the metal surface and to assess the effect of the laser fluence on the extent of molten and vaporised material regions [
10,
11]. For aluminium substrates, the significant increase of the absorptivity with temperature necessitates the consideration of a temperature-dependent reflectivity in the model [
12]. A detailed description of the FEM simulation of DLIP for metal substrates is presented in [
13].
In this work, the thermal modelling outlined so far is expanded for the first time, according to the best of the present authors’ knowledge, to comprise the molten bath convection during nanosecond pulsed DLIP of metal surfaces. The additional complexity of this modelling approach is accepted in the prospect of insight into the role of melt pool convection in surface patterning and a profound explanation of the structuring mechanism. Furthermore, the thermofluiddynamic simulations are performed using the mesh-free smoothed particle hydrodynamics (SPH) technique. The application of mesh-free methods, which permit a deformation or even disintegration of the computational domain, is comparatively novel in the simulation of laser processes.
Specifically, the use of SPH in the modelling of laser material interactions is still little explored. The SPH method was originally developed by Gingold, Lucy and Monaghan [
14,
15] to address astrophysical problems. Notwithstanding its continuing importance in theoretical astrophysics, SPH was subsequently applied to various problems, notably those involving fluid flow, as in turbomachinery [
16], coastal and hydraulic engineering [
17]. A detailed account of these advances can be found in systematic work [
18,
19,
20,
21]. Concerning the simulation of laser processing, Chen and Beraun first solved the coupled heat conduction equations governing ultrashort laser pulse action, i.e., of subpico- to picosecond duration, on metal films by the corrective smoothed particle method [
22].
The interaction of micro- to millisecond laser pulses or continuous laser irradiation with materials was modelled more frequently. Gross developed an SPH model for the laser cutting of metals [
23], which was extended by Muhammad et al. to simulate the micromachining of coronary stents [
24]. Considering microsecond pulses as well, Abidou et al. simulated the laser drilling of stainless steel [
25]. Earlier on, Tong and Browne used weakly compressible SPH (WCSPH) to model the melt pool flow and heat transfer during laser spot welding [
26], i.e., for millisecond pulses. Comparable laser spot welding simulations were performed for metallic workpieces in [
27,
28]. Regarding continuous irradiation, Yan et al. studied hydrodynamic interactions during laser underwater machining of alumina by SPH [
29]. Later on, Hu et al. simulated conduction mode [
27,
30] and deep penetration laser welding [
30] of aluminium. Russell et al. developed a comprehensive SPH methodology for laser based additive manufacturing and applied it to selective laser track melting [
28]. Tanaka et al. investigated also heat conduction due to a stationary or moving laser source using the moving particle semi-implicit method similar to SPH [
31]. In addition to [
28], selective laser melting was modelled by WCSPH in [
32,
33,
34].
On the contrary, SPH was rarely employed to address the effects of nanosecond laser pulses. The authors of this manuscript previously suggested a thermal model of DLIP for metallic substrates [
35]. Cao and Shin predicted the particle motion due to phase explosion during high fluence laser ablation of metals by SPH [
36]. Further, Alshaer et al. used SPH to simulate the thermal ablation of aluminium at elevated fluences, particularly the ejection of particles by the recoil pressure [
37]. However, the melt pool flow during nanosecond laser irradiation at moderate fluences was not studied to date using SPH. According to the best of the authors’ knowledge, an incompressible SPH (ISPH) model was not applied to the laser-induced molten bath flow before, in contrast to weld pool convection [
38].
6. Discussion
The results presented above suggest that stainless steel and aluminium represent two very dissimilar substrate materials with distinct behaviour when subject to laser interference irradiation. To begin with, the maximum surface temperature exhibits a more pronounced sensitivity to the laser fluence for the aluminium substrate, see
Figure 8, as compared to the results in
Figure 3 for stainless steel. This observation may be attributed to the high reflectivity of aluminium and the self-enhancing effect of its temperature-dependent absorptivity during laser heating. In addition, the thermal diffusivity of aluminium, being one order of magnitude higher than the one of stainless steel, facilitates conductive heat transfer, which leads to moderate maximum surface temperatures. On the other hand, the high thermal diffusivity of aluminium results in an enlarged melt pool, see
Figure 9, particularly in a larger melt pool depth, when compared with the melt pool dimensions predicted for stainless steel in
Figure 4.
Furthermore, the maximum velocity magnitude trends determined for the aluminium substrate, see
Figure 10, indicate a more distinct sensitivity to the laser fluence in comparison with the results for stainless steel presented in
Figure 5. The maximum horizontal velocity magnitudes presented in
Figure 10 for the aluminium melt, which notably exceed 10 m/s for elevated fluences, are at least twice as high as the ones in
Figure 5 predicted for stainless steel. Contemplating on the Marangoni boundary condition in Equation (
15), this difference is explained by the considerably smaller dynamic viscosity of liquid aluminium, which is only one fifth of the value for stainless steel according to
Table 3. The higher velocity values for the liquid aluminium are even bounded by the lower magnitudes of the temperature coefficient of surface tension, cf.
Table 3, and the horizontal temperature gradient, see
Figure 11c. In conjunction with the markedly deeper melt pool, the increased horizontal melt velocities suggest that aluminium is a delicate substrate material, which requires a careful selection of the fluence for DLIP. Actually, the high melt velocities predicted in this work provide an explanation for the irregular surface morphology observed for aluminium substrates, see
Figure 12c [
96], unlike the nearly perfect repetitive topography of stainless steel, see
Figure 12a [
35], after DLIP at moderate fluences.
The surface microstructure and profile reprinted in
Figure 12a,b were observed after a single pulse DLIP experiment with a periodicity of
µm and a fluence of 0.6 J/cm
2 on stainless steel. The width of the molten zone at the steel surface presented in
Figure 12a is approximately 3.2 µm [
35], which is in reasonable agreement with the maximum melt pool width of 2.9 µm, see
Figure 4a, in the simulation. The surface profile in
Figure 12b shows that melt is displaced from the interference maxima towards the interference minima [
35]. The height difference between the original surface and the resulting depression is approximately 200 nm [
35] and of the same order as the melt pool depth of 270 nm, see
Figure 4b, computed in this work.
Concerning the high-purity aluminium substrate, the surface morphology and profile shown in
Figure 12c,d, respectively, were obtained using a slightly smaller periodicity of
µm and a fluence of 1.015 J/cm
2. For this reason, only the height of the resulting surface structures is compared with the melt pool depth predicted for aluminium. A structure height of approximately 0.8 µm can be inferred from the cross section in
Figure 12d, which suggests that the volume of metal melt is considerably larger in the case of aluminium. The melt pool depth of 1.1 µm predicted in the present simulations for a fluence of 1.0 J/cm
2, see
Figure 9b, is in agreement with the measured depth of the surface microstructure on aluminium after DLIP with a periodicity of
µm, as reported in [
96].
The present results are in reasonable agreement with previous work on DLIP of metals [
12,
96,
97]. In detail, the maximum surface temperatures of aluminium during DLIP are in accordance with values computed using the FEM model, which were presented in [
97]. However, lower maximum surface temperatures were predicted in [
97] for stainless steel, probably owing to a considerably higher thermal diffusivity value employed. The magnitude of the horizontal temperature gradient at the surface of the stainless steel and aluminium substrates is in line with the thermal gradient determined in [
97]. Further, the melt pool depth during DLIP of stainless steel presented in
Figure 4b is of the same order of magnitude as the molten depth calculated in [
12], in spite of the fact that the latter results were obtained for a smaller periodicity.
On the other hand, the molten depths determined from the thermal simulations for an aluminium substrate in [
12] are notably lower than the present results in
Figure 9b, although a comparable periodicity was considered. This deviation may be attributed to a higher thermal diffusivity of aluminium employed in [
12], which entails a more homogeneous temperature field with less pronounced maxima and, therefore, a reduced absorption of the incident radiation. Nevertheless, the present depths of the aluminium melt pool are in the same size range as the structure depths measured after DLIP experiments on aluminium substrates using a similar periodicity, which were reported in [
96].
Furthermore, the validity of the predicted velocity magnitudes is confirmed in the following. Considering the molten material as a shear layer, a characteristic horizontal velocity of the thermocapillary motion is obtained from the Marangoni boundary condition in Equation (
15) as [
98,
99]
where
denotes a height related to the melt pool. If the height
is specified as half of the molten film thickness,
in Equation (
79) represents an average horizontal velocity of melt displacement [
12,
98]. For the present simulations, excessive horizontal velocity values would result if the melt pool depth was employed in Equation (
79). Instead, it is suggested to substitute the thickness of the velocity boundary layer near the surface for
. In addition, a time scale for thermocapillary convection is given by the ratio of half the interference pattern periodicity and the characteristic velocity magnitude [
12,
100]
where the horizontal temperature gradient is often assessed by relating a temperature difference to a width, e.g.,
[
12], the resulting expression being proportional to
[
100].
Considering the simulations presented in detail in
Section 5, the horizontal velocity magnitudes at the time
ns are estimated by Equation (
79) using the material properties from
Table 3. For stainless steel subject to the laser fluence
J/cm
2, the average temperature gradient
K/µm and
nm result in
m/s. In case of the high-purity aluminium substrate and the energy density
J/cm
2, the average temperature gradient
K/µm and
nm yield
m/s. The maximum surface velocity magnitudes computed in the ISPH simulations, as may be identified from
Figure 6c and
Figure 11c, are 32% and 15%, respectively, lower than these values. These deviations may be attributed to the transient character of the surface temperature distribution and the short duration of melt pool convection. Employing
µm and the aforementioned velocities in Equation (
80), the time scale for thermocapillary flow equals
ns for stainless steel and
ns for aluminium.
7. Conclusions
The present work demonstrates the use of SPH in the simulation of heat transfer and fluid flow during nanosecond pulsed DLIP of metallic substrates on the basis of a dimensionless formulation of the governing equations. For single-pulse treatment, the simulation results reveal a distinct behaviour of the dissimilar substrates stainless steel and high-purity aluminium. Particularly in the case of processing aluminium, the predictions confirm that thermocapillary convection is characterised by substantial velocity magnitudes beyond 10 m/s even at moderate laser fluences. Consequently, this outward flow from the centre of the melt pool surface, at the interference maximum, towards its edges is a conceivable structuring mechanism effective during DLIP of aluminium using a nanosecond pulse at low and moderate energy densities.
On the contrary, the higher absorptivity and lower thermal conductivity of stainless steel lead to high surface temperatures up to the vapourisation point near the interference maximum, even at moderate laser fluences. In addition, thermocapillary convection is less pronounced in liquid steel owing to its high dynamic viscosity. Therefore, it is more difficult to distinguish between the effects of the thermocapillary flow and the recoil pressure induced by vapourisation on the melt displacement during DLIP of stainless steel at moderate energy densities. The further investigation of the roles of thermocapillary convection and vapourisation-induced recoil pressure in the structuring of metal surfaces by DLIP ultimately necessitates the modelling of the melt pool surface deformation.
The numerical results presented in this work are compared with surface microstructures obtained by material characterisation after DLIP experiments on stainless steel and high-purity aluminium. In agreement with the observed surface morphologies, the simulations indicate a significantly deeper molten bath and a more effective melt displacement mechanism for the aluminium substrate. The predicted velocity magnitudes, which suggest a notable outward flow due to thermocapillary forces at the surface of the aluminium melt pool, are confirmed by a theoretical estimation.
In this work, an ISPH approach is deliberately employed to simulate the melt flow during DLIP of metals. This choice is motivated by virtue of the meaningful, physical pressure field computed using the ISPH technique. It is expected that an accurate pressure field is essential for the further study of melt displacement during DLIP, particularly for modelling the effect of the recoil pressure. The present model could serve as a basis for the prospective investigation. Although the SPH method is inherently capable of describing the evolution of a free surface in fluid flow, the application of a projection-based ISPH approach involves fundamental difficulties in this context. Provided that these issues can be eliminated, the method presented herein can be appropriately extended to simulate the melt displacement during DLIP of metals as well.