Applied Physics A (2020) 126:82 https://doi.org/10.1007/s00339-019-3202-8

S.I. : CURRENT STATE-OF-THE-ART IN

Simulations of deep drilling of metals by using combined smoothed particle hydrodynamics and ray‑tracing methods

Deepak Shah1 · Alexey N. Volkov1

Received: 16 October 2019 / Accepted: 2 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract A robust numerical method based on smoothed particle hydrodynamics (SPH) is used to investigate the melt pool expul- sion mechanism during high-aspect ratio laser drilling of two diferent materials: aluminum and 316L stainless steel. The computational model is validated with previously published results which show extremely close matches. The ray-tracing method is combined with SPH method which makes the model capable of taking into account the multiple refections of laser radiations from the keyhole wall. The drilling velocity is found to be a non-linear function of time, which is dependent on the shape of the heated surface. For a fat surface, the drilling velocity is small, but after the cavity is formed and multiple refections of laser radiations become signifcant, the drilling velocity strongly increases. The main driving force of melt expulsion is the repulsive force produced by vapor pressure. The Marangoni stresses are found to provide marginal efect on melt expulsion during deep drilling.

1 Introduction (LPBF) additive manufacturing based on selective laser (SLM) and selective laser sintering (SLS) of metal Continuous wave (CW) lasers are broadly used in various powders. Most of the commercially available SLM and SLS industrial applications for welding, drilling, and cutting of systems use CW lasers [7]. Attempts have been made to metals and alloys [1]. Although CW laser drilling is more use pulsed lasers in LPBF; however, it was found that the than six-decades-old process, it is still a feld of active application of CW lasers provides much larger throughput research targeted at understanding of major physical mecha- and results in more stable melt pool, higher material deposi- nisms responsible for material removal from the laser key- tion rates, smaller porosity, and better mechanical properties hole and at optimization of the process parameters for new of fabricated parts compared to parts produced with pulsed technologies, applications, and materials. For instance, just lasers [7]. recently, CW lasers were used for highly precise drilling of The understanding of the keyhole formation process micro-holes in nickel-based superalloys, including Nimonic during laser melting of materials is crucial to improve the 263 [2], GH4049 [3], and Inconel 718 [4]. Another direction efciency and quality of outputs from both laser drilling of research is targeted at combining CW and pulsed lasers to and LPBF. The computational studies can provide valu- enhance the overall throughput of laser drilling. It has been able insights into the mechanisms responsible for transient also shown, in particular, that the efciency of CW laser processes of laser melting, melt fow, solidifcation, and drilling of aluminum, 316L stainless steel [5], and Q235B formation of various defects [8, 9]. Multiple experimental steel [6] can be substantially increased by assisting it with and theoretical studies, e.g., [10, 11], including high-speed picosecond and nanosecond laser pulses. X-ray imaging that was recently applied for in situ char- Recently, the growing interest towards understanding the acterization and time-resolved visualization of the keyhole CW laser–metal interaction has been raised by introduction formation process in LPBF [12, 13], showed that the process of 3D-printing technologies such as laser powder bed fusion of keyhole formation during SLM of metal powders shares multiple similarities with CW laser drilling of bulk targets. * Alexey N. Volkov Thus, further progress in understanding of CW laser drilling [email protected] may also contribute to the progress in LPBF technologies. The primary mechanism of CW laser drilling is the melt 1 Department of Mechanical Engineering, University expulsion from the center of the laser spot [14–16]. The of Alabama, Tuscaloosa, AL 35487, USA

Vol.:(0123456789)1 3 82 Page 2 of 12 D. Shah, A. N. Volkov drilling velocity, shape of the molten pool, and geometry of steel targets. The combined SPH-RT method accounts for the free surface in the keyhole are determined by a trade-of all major interfacial efects in CW laser processing of met- between various interfacial efects, including surface ten- als. This method enables simulations of laser-induced fows sion, Marangoni stresses, recoil efect of and in the molten pool without any restrictions on the topologi- evaporative cooling, as well as volumetric processes dom- cal changes of the free surface in the course of a simulation inated by and thermal difusivity of the molten at relatively low computational costs. The major goals of material. In particular, the theoretical estimations [17] and our computational study are to reveal the efects of multiple direct numerical simulations [18] predict strong efect of refections of laser radiations inside the keyhole, to compare the recoil force due to evaporation of the target material contributions of Marangoni stresses and recoil pressure to at a surface subjected to laser heating. The recoil vapor the melt expulsion process, and to observe the diferences in pressure was identifed as the primary factor afecting the drilling of stainless steel and aluminum targets. The results melt expulsion process based on the analysis of experimen- of our simulations confrm the dominant role of the recoil tal results on drilling of a nickel-based superalloy [2]. The vapor pressure in the keyhole formation, as well as very hydrodynamic simulations of CW laser melting of aluminum strong efect of multiple refections, especially in the case flms also revealed the dominant role of the recoil efect of a highly refective aluminum target. compared to the Marangoni stresses [14]. At the same time, negligible efect of the recoil vapor pressure on the melt fow in laser melting of thin flms is reported in Refs. [19, 2 Combined SPH‑RT method for simulations 20]. The Marangoni stresses are found to dominate the fow of laser drilling of metals of molten material in Ref. [21], where the recoil pressure provides non-negligible, but relatively small contribution to In simulations, we consider a three-dimensional (3D) metal the interfacial stresses. In LPBF, the recoil efect is identifed workpiece of size L × W × H along x−, y−, and z-directions to be the major reason for keyhole formation and spattering in Cartesian coordinates (Fig. 1). The CW laser beam is of powders, while the Marangoni stresses are found to be assumed to propagate along the z-direction and to be normal responsible for pore migration in the melt pool [13]. On the to the top sample surface. The distribution of the laser inten- contrary, in Ref. [22], it was concluded that both the recoil sity I(r) in the laser beam is assumed to be Gaussian [26]: pressure and Marangoni stresses equally contribute to the 2 2r melt pool shape evolution in LPBF. I(r) = I exp −(ln 2) , L D (1) Mass and heat transfer in the molten pool also strongly  L   depends on the distribution of heat sources, associated with the deposited laser energy. The prediction of the heat source where IL is the laser intensity at the spot center, DL is the distribution in deep keyholes or cavities is a non-trivial prob- full-width at half-maximum (FWHM) spot diameter, and r lem, since the incident laser radiation is absorbed by the tar- is a radial coordinate counted along the irradiated surface get material during multiple refections from the keyhole. In from the spot center placed at x = y = 0. The lateral surfaces addition, the majority of metals and alloys are highly refec- parallel to the xz and yz planes and the bottom surface are tive, so that the intensity of radiation decreases relatively assumed to be thermally insulated. The spot diameter at 1/e2 slowly with increasing number of refections. Simulations level, D1∕e2 , is another way to characterize the laser spot size. D 2 D 2 = (ln 2)∕2D . of CW heating with shallow keyholes are often performed 1∕e is related to DL as 1∕e L without consideration of multiple refections [14, 18, 20, The mathematical model of √laser heating of the target 21]. For deep laser drilling, the multiple refections strongly material, its melting, and melt fow includes two-phase increase the fraction of laser energy absorbed in the cavity hydrodynamic equations and the model of geometrical and overall drilling velocity when the cavity aspect ratio optics describing propagation of laser radiation and its becomes greater than 0.42 [23, 24]. The classical method absorption on the target surface. To describe the target melt- of ray tracing (RT) [25] is one of the efective tools, which ing and solidifcation, an additional variable ϕ(T) equal to allows one to describe the propagations of laser beams in the volume fraction of liquid material is defned in every high-aspect ratio keyholes in the approximation of geometri- point occupied by the target material. In the initial state, cal optics. before onset of irradiation of the target with a CW laser, the In the present paper, a computational approach that con- whole material sample is solid and ϕ = 0 everywhere. Melt- sists of the smoothed particle hydrodynamics (SPH) method ing and solidifcation are described based on the enthalpy for predicting the laser melting of metal target, molten mate- formulation approach [27] as variation of the volume frac- rial fow and formation of keyhole, and the RT method for tion of the molten material. In the present work, we use a propagation and absorption of laser radiation is applied to simplifed approach, where the formation of the mushy zone study deep laser drilling of bulk aluminum and stainless is neglected, and it is assumed that the material at ϕ ≥ 0.5

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Fig. 1 Computational domain used in 3D simulations of CW laser are assumed to be thermally insulated, while the top surface (CD) is drilling of metal targets: a sketch with the side view of the com- irradiated by a laser. The blue regions represent the solid material putational domain, b 3D view of a real computational sample used and red regions represent the molten pool. The laser beam propagates in simulations. The initial computational sample has size L in the along the z-axis. The laser radiation propagation, absorption, and x-direction, W in the y-direction, and H in the z-direction. The bottom scattering at the top material surface are described by the RT method (AB) and side (BC and AD) boundaries of the computational sample with discrete rays shown schematically by green lines in (a) represents molten material, while the remaining part of the a number of SPH particles. All SPH particles are initially sample is solid. For solid material, only the heat conduction placed in nodes of a Cartesian mesh with cubic cells of size equation is solved. The method of calculation of the fuid Δx. The hydrodynamic equations reduce to the Lagrangian volume fraction as a function of the material equations of motion, mass, and energy transfer for individual T adopted in the present work is identical to the method SPH particles. The boundary conditions for hydrodynamics used in Ref. [14]. The equation accounts for the equations are included into the equations for SPH particles gravity force that is collinear with the laser beam, i.e., acts based on the continuous surface force approach [29]. The in the direction opposite to the direction of the z-axis, as heat conduction term in the energy equation is approxi- shown in Fig. 1. The boundary conditions on the top surface mated as suggested in [30]. Molten material is considered of the sample include kinematic conditions that account for as a weakly compressible fuid with the equation of the state 2 γ the variation of the interface shape due to melt fow and p(ρ) = ρ0c /γ[(ρ/ρ0) − 1], where p and ρ are the pressure and surface recession caused by evaporation, momentum condi- density, γ = 7, c is the speed of sound, and ρ0 is the refer- tions that account for normal stresses due to the recoil efect ence material density. Value of c is chosen to be sufciently of vapor pressure and , as well as the tan- large to keep the density variation below 1%. We performed gential Marangoni stresses due to the dependence of the preliminary simulations of CW laser drilling with variable coefcient of surface tension on temperature, and energy parameter c and found that the variation of the molten mate- conditions that account for the balance of energy fuxes rial density is below 1% if c = 1000 m s­ −1 in agreement caused by absorption of laser radiation, heat conduction, and with the compressibility analysis presented, e.g., in Ref. cooling efect of evaporation [14]. The dependence of the [31]. This value of c is used in all simulations described saturated vapor pressure pe on temperature T is described by below. We also found that in simulations with randomly the Clapeyron–Clausius equation, pe(T) = 0.54p0e exp [LbM/ distributed SPH particles representing molten material, the (kBNa)(1/T0e − 1/T)], where kB, Na, and M are the Boltzmann approach suggested in Ref. [30] underestimates heat fuxes constant, Avogadro constant, and molar mass of the target in about 30%. To resolve this issue, all simulations described material, respectively, Lb is the latent heat of boiling, and p0e in Sects. 3 and 4 are performed with thermal conductiv- is the saturated vapor pressure at a reference temperature T0e ity of liquid material increased in 30% compared to values [8]. The net mass fux density of evaporated material Fe is indicated in Table 1. The further details of the SPH method calculated based on the Hertz–Knudsen model of evapora- used in this work are given in Ref. [32]. tion as Fe(T) = 0.82pe(T)∕ 2 kBNaT∕M [8, 14]. The propagation of laser radiation is simulated based on This hydrodynamic model√ is implemented for numerical the RT method, where the laser beam is represented by a num- simulations in the form of the SPH method [28]. Follow- ber of discrete rays or energy packets that travel in space and ing the SPH approach, the material sample is divided into scatter at SPH particles that compose the target surface, as it

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Table 1 Material properties of Parameter Value aluminum [14, 21, 26, 37, 38] and 316L stainless steel [8, 32, Aluminum 316L stainless steel 39] used in simulations of CW −3 −3 laser drilling Density 2700 kg m 7500 kg m Viscosity 0.00135 kg m−1 ­s−1 0.00642 kg m−1 s−1 Specifc heat of solid material 900 J kg−1 K−1 462.656 + 0.1338 T J kg−1 K−1 Specifc heat of liquid material 1000 J kg−1 K−1 776 J kg−1 K−1 Conductivity of solid material 238 W m−1 K−1 9.248 + 1.571 × 10−2 T W m−1 K−1 Conductivity of liquid material 107 W m−1 K−1 12.41 + 3.279 × 10−3 T W m−1 K−1 Melting temperature 933 K 1700 K Latent heat of fusion 4.1 × 105 J kg−1 2.7033 × 105 J kg−1 Reference boiling temperature, 2730 K 3086 K T0e 5 5 Reference boiling pressure, p0e 10 Pa 10 Pa 6 −1 6 −1 Latent heat of boiling, Lb 10.75 × 10 J kg 7.46 × 10 J kg −4 Coefcient of surface tension, 1.05 − 2.74 × 10 × (T − 300) N m −1 3.282 − 8.9 × 10−4 T N m−1 σ Molar mass, M 0.027 kg ­mole−1 0.056 kg ­mole−1 Refectivity 0.97 0.66 Refractive index – 3.27 + 4.48i

Optical properties are chosen for a laser wavelength of 10.6 μm. T is temperature in K is schematically shown in Fig. 1. The attenuation and scatter- In the frst case, we consider laser drilling of a 0.2 mm- ing of individual rays from the material surface are described thick aluminum flm with a at DL = 0.2 mm, 7 −2 by the Fresnel equations [33] or calculated assuming constant and IL = 3 × 10 W cm . Since in Ref. [14], the coefcients material refectivity. Scattering of discrete rays on SPH parti- 0.54 in the equation for pe and 0.82 in the equation for Fe cles is described based on the approach suggested in Ref. [32]. were not used, these coefcients were also omitted in our In the course of a simulation, each ray is terminated when its simulations. In addition, a constant refectivity 0.97 of alu- intensity becomes smaller than 1% of its initial intensity. minum was adopted following to Ref. [14]. The cavity shapes At every time step of the computational algorithm, the and temperature felds obtained in the present work and in combined SPH-RT numerical method is used as a two-step Ref. [14] are compared in Fig. 2. One can conclude that the approach. First, the RT method is applied to calculate the results obtained based on SPH-RT numerical model are in energy absorbed by the target material based on the distribu- good quantitative agreement with results obtained in Ref. tion of SPH particles obtained at the end of the previous time [14] based on the two-dimensional axisymmetric numerical step. The RT calculations are followed by the SPH calcula- solution of two-phase hydrodynamic equations with a high- tions which result in updating the position, density, velocity, order upwind fnite-diference method. temperature, and liquid fraction of every SPH particle. In the second case, we consider laser melting and keyhole The simulations are performed for 316L stainless steel formation in a 316L stainless steel target heated by a moving 7 −2 and aluminum targets and a CO­ 2 laser of 10.6 µm wave- CW laser beam at DL = 31.8 μm, IL = 1.75 × 10 W cm , and length. The material properties adopted in simulations are a scan speed of 1.5 m s−1. In Fig. 3, the keyhole shape and tem- listed in Table 1. perature feld obtained in our simulations are compared with the computational results obtained in Ref. [11] based on the arbi- trary Lagrangian–Eulerian (ALE) computational framework. 3 Model validation As one can see, our simulations closely reproduce the results reported in Ref. [11] both in terms of the cavity shape and tem- To validate the combined SPH-RT computational model, perature distribution. In particular, in both cases, the predicted we apply it for simulations of laser melting with stationary depth of the keyhole at a time of 33 µs is equal to 40 μm. and moving CW lasers under conditions considered in Refs. [11, 14]. In both cases, our simulations are performed with material properties and simulation conditions that were used in the original simulations reported in Refs. [11, 14].

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Fig. 2 Cavity shapes and temperature felds in an aluminum target ric simulations in Ref. [14] (b). The black contour is the solid–liquid of thickness 0.2 mm heated by a stationary Gaussian laser beam at interface. b is reproduced from Ref. [14] with permission from Else- 7 −2 DL = 0.2 mm and IL = 3 × 10 W cm obtained at t = 222 μs in vier the 3D SPH-RT simulations in this work (a) and in 2D axisymmet-

Fig. 3 Cavity shapes and temperature felds in a 316L stainless steel t = 33 μs in the 3D SPH-RT simulations result in this work (a) and in bulk target heated by a Gaussian laser beam at DL = 31.8 μm and Ref. [11] (b) 7 −2 −1 IL = 1.75 × 10 W cm moving at a speed of 1.5 m s obtained at

4 Computational results and discussion to the saturated vapor pressure of ~ 15 bar, which provides strong repulsive efect on the melt pool in the center of 4.1 316L stainless steel target the laser spot. The large pressure diference between the center and periphery of the molten pool enhances tan- The simulations of deep laser drilling of a 316L stainless gential velocity of the molten material close to its free steel target are performed for a material sample of size surface. The temperature gradient across the free surface 0.2 × 0.2 × 0.4 mm heated by a CW laser with the beam induces the Marangoni stresses that also drive the melt D 2 = 54 diameter DL = 31.8 μm ( 1∕e μm) and intensities from the lower surface tension region in the spot center varying from 6.56 × 106 to 1.75 × 107 W cm−2. The target to the higher surface tension region at the spot periphery. is divided into 2 × 106 SPH particles which are initially With increasing intensity of laser radiation, the placed in nodes of a Cartesian mesh with cubic cells of temperature in the spot center also raises, e.g., up 7 −2 size Δx = 2 μm. The time step is equal to Δt = 10 ns, to ~ 3800 K at IL = 1.31 × 10 W cm and to ~ 3950 K at 7 −2 which is about four times smaller than the maximum time IL = 1.75 × 10 W cm . It increases the vapor pressure in step satisfying the stability condition of our SPH method the spot center, enhances the melt expulsion, and increases [34]. The laser beam is divided into 250,000 rays which the overall drilling velocity. Here, we defne the drilling are distributed with equal spacing in the cylinder of radius velocity as a rate of change of the keyhole depth at the spot t DL. center. At = 250 μs, the cavity depth is equal to 114 μm, 6 −2 The cavity shapes and temperature felds in the tar- 185 μm, and 249 μm for intensities 8.75 × 10 W cm , 7 −2 7 −2 get obtained at t = 250 μs in simulations with inten- 1.31 × 10 W cm , and 1.75 × 10 W cm . With increasing sities of 8.75 × 106 W cm−2, 1.31 × 107 W cm−2, intensity and rate of melt expulsion, the melt pool thickness and 1.75 × 107 W cm−2 are shown in Fig. 4a–c. At at the spot center decreases. 6 −2 IL = 8.75 × 10 W cm , the maximum temperature at the At the considered conditions, the direct removal of the center of the laser beam reaches ~ 3700 K. It corresponds target material remains relatively small even at the tip of the keyhole, where the temperature is maximum. The rate

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Fig. 4 Cavity shapes and temperature felds in a 316L stainless steel target heated by a CW laser of intensity 6 −2 IL = 8.75 × 10 W cm (a), 1.31 × 107 W cm−2 (b), and 1.75 × 107 W cm−2 (c) at a time of 250 μs

of evaporation on the major part of the keyhole surface is of evaporated material predicted by the Hertz–Knudsen negligible, since the surface temperature inside the cavity model. In this case, the average temperature along cavity drops quickly with approaching to the target surface. For surface is ~ 3200 K, which corresponds to an ablation rate 7 −2 −1 instance, at IL = 1.75 × 10 W cm , the surface temperature of 0.0061 μm μs or ablation depth of 1.525 μm at an expo- in the keyhole tip is ~ 3950 K which corresponds to an abla- sure time of 250 μs. Thus, the primary mechanism of the tion rate of 0.0914 μm μs−1 or ablation depth of 22.85 μm keyhole formation is the expulsion of the molten material. at an exposure time of 250 μs. Here, the ablation rate is The evaporation from the melt surface, however, remains calculated as Fe(T)/ρ, where Fe(T) is the mass fux density

Fig. 5 Cavity depth (a) and drilling velocity (b) versus time W cm−2 (orange dash–double–dotted curve). In a, the magenta curve obtained in simulations of laser drilling of a 316L stainless steel is obtained assuming that every ray representing a part of the laser target with a CW laser at intensity 1.75 × 107 W cm−2 (red and beam undergoes only single refection from the targeted surface and magenta solid curves), 1.31 × 107 W cm−2 (greed dashed curves), then is excluded from further consideration, so that the efect of mul- 8.75 × 106 W cm−2 (blue dash–dotted curves), and 6.56 × 106 tiple refections of rays from the cavity wall is not accounted for

1 3 Simulations of deep drilling of metals by continuous wave lasers using combined smoothed particle… Page 7 of 12 82 important, since the recoil efect of evaporation is the major obtained in simulations with and without the Marangoni mechanism driving the material expulsion from the keyhole. stresses are small for all considered intensities, so that the The variation of the cavity depth and drilling veloc- melt expulsion is dominated by the recoil efect of evapo- ity with time for different laser intensities is shown in ration. At the same time, we found that the crown or rim Fig. 5. To investigate the effect of multiple reflections formed around the keyhole is much smoother when the efect inside the keyhole, we perform an additional simulation at of temperature on the coefcient of surface tension is not 7 −2 IL = 1.75 × 10 W cm , assuming that every discrete ray accounted for, since in this case the efect of surface tension representing the laser beam contributes to the laser heating at a relatively cold free surface of the rim at a periphery of only during a single interaction with the target surface and the laser spot is overestimated. then is excluded from further calculations (solid magenta The attenuation and scattering of individual rays from the curve in Fig. 5a). The simulations with and without multiple stainless steel surface are described by the Fresnel equations, refections result in the same cavity depth when the cav- when refectivity depends on the angle of incidence. The ity aspect ratio, i.e., the ratio of the cavity depth to DL, is Fresnel equations give a refectivity of about 0.66 for 316L smaller than 0.4. When the cavity aspect ratio becomes stainless steel in the range of angles of incidence from 0° larger, multiple refections of radiations increase the total to 85° [32]. We perform additional simulations with fxed absorbed laser energy and the drilling velocity becomes refectivity equal to 0.66 and fnd that the cavity depth and larger as well. the temperature fled in these simulations are only margin- The drilling velocity demonstrates fast increase during ally diferent from the case when refectivity is calculated initial ~ 50 µs of drilling independently of laser intensity, and based on the Fresnel equations. It indicates that simulations then it starts to decrease slowly (Fig. 5b). The simulations, of deep laser drilling can be performed with the material thus, predict the saturation of the cavity depth due to gradual refectivity independent of the angle of incidence. increase in the area of the keyhole surface with increasing depth and, correspondingly, reducing the fux density of the 4.2 Aluminum target absorbed laser energy. To reveal the effect of the Marangoni stresses, we Simulations of laser drilling of an aluminum target are per- perform additional simulations at laser intensities of formed for a sample of the same size and for the same spot 6.56 × 106 W cm−2 and 1.31 × 107 W cm−2 with a constant diameter that are used in simulations with the stainless steel value of the coefcient of surface tension equal to 1.769 N sample in Sect. 4.1 at the laser intensities 1.8 × 108 W cm−2 ­m−1 that corresponds to the melting temperature. The results and 3 × 108 W cm−2. In these simulations, the numeri- of these simulations are compared with the results obtained cal parameters are also identical to those used in simula- based on the temperature-dependent coefcient of surface tions with the stainless steel target. Based on the results of tension, as shown in Fig. 6. The diferences between the preliminary study of refectivity variation with the angle cavity depths and surface in the spot center

Fig. 6 Cavity depth (a) and surface temperature in the spot center (b) (red curves) and 1.31 × 107 W cm−2 (blue curves) with (solid curves) versus time obtained in simulations of laser drilling of a 316L stain- and without (dashed curves) the Marangoni stresses 7 −2 less steel target with a CW laser of intensity IL = 6.56 × 10 W cm

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Fig. 7 Cavity shapes and temperature felds obtained in an aluminum target heated by a CW laser at laser intensities 8 −2 IL = 1.8 × 10 W cm (a–c) and 3 × 108 W cm−2 (d–f) for various times. g shows a shape of a high-aspect-ratio cavity obtained in experiments on CW laser drilling of aluminum. g is reproduced from Ref. [15] with permission from Elsevier

of incidence, a constant refectivity coefcient of 0.97 is The shape of the cavity in Fig. 7f is qualitatively simi- adopted for aluminum. lar to the cavity shape in the experimental picture of high- The cavity shapes and temperature felds obtained in sim- aspect ratio cavity in an aluminum target which is adopted ulations with the both considered laser intensities are shown from Ref. [15] and is shown in Fig. 7g. In both experimental 8 −2 in Fig. 7a–f. At IL = 1.8 × 10 W cm , the cavity depth is picture and simulation snapshots, the disturbances of the equal to 11 μm, 182 μm, and 342 μm at times 30 μs, 50 μs, cavity cross section induced by instability of the free-surface and 70 μs, correspondingly. At this laser intensity, the melt melt fow are clearly visible. In our simulations, the mag- temperature at the laser spot center reaches ~ 3200 K and the nitude of these disturbances increases with increasing laser 8 −2 vapor pressure raises up to 2 bar. At IL = 3 × 10 W cm , intensity. the cavity depth is equal to 12 μm, 193 μm, and 370 μm at The comparison of results shown in Figs. 4 and 7 reveals times 10 μs, 20 μs, and 30 μs. In this case, the melt tem- a diference between the cavity shapes for aluminum and perature in the spot center increases up to ~ 3500 K. The stainless steel targets. In the stainless steel target, the key- corresponding vapor pressure is equal to 7 bar. As a result, hole has a conical shape and the cavity diameter gradually the melt layer is very thin in the tip of the keyhole. The time decreases with depth. In the aluminum target, the keyhole required for the through hole formation in 0.4 mm-thick alu- is roughly cylindrical with wider opening close to the target 8 −2 minum flm is equal to 77.67 μs at IL = 1.8 × 10 W cm surface. Such cavity shapes qualitatively match the experi- 8 −2 and 32.75 μs at IL = 3 × 10 W cm . mentally observed cavity shapes in stainless steel [36] and

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Fig. 8 Cavity depth (a), absorbed energy fux (b), drilling veloc- assuming that every ray representing a part of the laser beam under- ity (c), and surface temperature in the spot center (d) versus time goes only one refection with the target surface and then excluded obtained in simulations of laser drilling of an aluminum tar- from further consideration, so that the efect of multiple refections 8 −2 get with a CW laser at intensity IL = 3 × 10 W cm (red solid of rays from the cavity wall is not accounted for. The red and blue curves and greed dash–dotted curves) and 1.8 × 108 W cm−2 (blue curves are terminated after the through hole is formed in the simula- dashed curves). In a, b, the green dash–dotted curves are obtained tions aluminum [15, 16] targets. The diference in the cavity in Fig. 8 reveals two distinct time intervals or regimes of shapes is attributed mainly to the diference in thermal dif- laser drilling. fusivities of these materials. With increasing thermal dif- In the frst regime, the cavity depth increases relatively fusivity, the temperature distribution along the cavity depth slowly. It happens, since during this time the target surface becomes more homogeneous and, consequently, the cavity remains roughly flat and multiple reflections are infre- diameter exhibits smaller variations. The thermal difusivity quent. Consequently, the absorbed laser power found in of aluminum is higher than the thermal difusivity of stain- calculations with and without multiple refections is nearly less steel, so that the cavity in the aluminum target attains the same and small, because only 3% of the incident laser more cylindrical shape. energy is absorbed. At both considered intensities, this The variation of the cavity depth, absorbed laser power, regime lasts until the cavity aspect ratio raises up to ~ 0.4. 8 −2 drilling velocity, and surface temperature in the spot It occurs at 9.9 μs at IL = 3 × 10 W cm and 28.8 μs at 8 −2 center with time is shown in Fig. 8. To make a conclusion IL = 1.8 × 10 W cm . During this initial time, the drill- about the efect of multiple refections, the simulation at ing velocity for the aluminum target is much smaller than 8 −2 IL = 3 × 10 W cm is repeated without multiple refections that for the stainless steel target at comparable irradiation (green dash–dotted curves in Fig. 8a, b). The results shown conditions, since the absorptivity of the aluminum target

1 3 82 Page 10 of 12 D. Shah, A. N. Volkov is in order of magnitude smaller than that for the stainless target, the simulation results do not demonstrate distinct steel target and the efect of multiple refections during the two-regime behavior with order-of-magnitude changes in considered initial time is marginal. the drilling velocity when the cavity aspect ratio exceeds In the second regime of drilling of the aluminum tar- 0.4. This diference between stainless steel and aluminum get, the cavity depth increases with an order-of-magnitude targets is explained by the diference in refectivity of these higher rate. It is caused by rapid increase in the amount of materials. multiple refections and, correspondingly, in the fraction The through hole is formed when the cavity depth in of laser energy absorbed in the cavity as the cavity aspect Fig. 8a increases up to the thickness of the considered mate- ratio increases (solid and dashed curves in Fig. 8b). It occurs rial sample. The formation of the through hole is illustrated 8 −2 when the cavity becomes deep enough to enable multiple in Fig. 9 for IL = 3 × 10 W cm . First, a thin layer of refections of laser radiation at least near the cavity tip. In melt with concave shape is formed. The recoil force due to simulations without multiple refections, the absorbed laser evaporation pushes this layer down and breaks it up between energy only marginally depends on the cavity aspect ratio 32 μs and 33 μs. Breaking of this layer results in the fast and remains constant through the simulation (dash–dotted drop of temperature in the bottom part of the through hole curves in Fig. 8b). The drilling velocity increases rapidly up wall. Tapered through hole is formed with larger entry hole −1 8 −2 −1 to 20.2 m s for IL = 3 × 10 W cm and 10.05 m s for of diameter 67.62 μm and smaller exit hole of diameter 8 −2 IL = 1.8 × 10 W cm when the cavity aspect ratio increases 34.86 μm. Formations of such tapered cavities and tapered from 0.4 to 3.1. As the cavity aspect ratio further increases, holes is observed experimentally [15, 35, 36]. As the process the laser energy is distributed over larger surface area of of laser heating continues, the upper diameter of the hole the keyhole. It induces a moderated decrease in the drilling continues to slowly increase. velocity (Fig. 8c). The surface temperature in spot center remains roughly constant in this regime (Fig. 8d), which continues until the though hole is formed. Our simulations show that the efect of multiple refec- 5 Conclusion tions becomes strong when the cavity aspect ratio equal to 0.4 for both stainless steel and aluminum targets. This value The computational methodology combining the SPH method is in agreement with the threshold cavity aspect ratio, cor- for molten material fow and RT method for the radiation responding to the onset of strong enhancement of absorption propagation and absorption is found to be a computation- of laser energy due to multiple refections of laser radiation ally efective and robust tool to study deep drilling of met- from the cavity wall, which is found to be equal to 0.42 in als with CW lasers. It is found that the melt expulsion from Ref. [21]. At the same time, in the case of the stainless steel high-aspect ratio cavities in aluminum and stainless steel

Fig. 9 Cavity shapes and tem- perature felds in an aluminum target heated by a CW laser with 8 −2 intensity IL = 3 × 10 W cm at t = 32 µs (a), 33 µs (b), and 34 µs (c). In this simulation, the through hole is formed between 32 µs and 33 µs

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