A one-fluid approach is applied and the model is made of five partial differential equations governing mass, momentum, (thermal) energy, volume fraction of alloy, and electric potential. Presented in the same order as implemented, these equations are written as follows (Luo et al., 2019; Noori Rahim Abadi et al., 2021; Noori Rahim Abadi et al., 2022): ∂ α ∂ t + ∇ ⋅ α U ⃗ + C α ∇ ⋅ α 1 − α U ⃗ r = − m ̇ vap ρ 1 | ∇ α | , (1) ∂ ρ ∂ t + ∇ ⋅ ρ U ⃗ = 0 , (2) ∂ ρ U ⃗ ∂ t + ∇ ⋅ ρ U ⃗ U ⃗ − ∇ ⋅ μ ∇ U ⃗ + ∇ U ⃗ T − 2 3 μ ∇ ⋅ U ⃗ II + C D 1 − f l 2 f l 3 + ϵ D U ⃗ = − ∇ p + ρ m 1 − β T − T m g ⃗ + σ κ + d σ d T ∇ T − n ⃗ n ⃗ ⋅ ∇ T − P r n ⃗ | ∇ α | 2 ρ ρ 1 + ρ 2 , (3) ∇ ⋅ σ ϵ ∇ V = 0 , (4) ∂ ρ c p T ∂ t + ∇ ⋅ ρ c p T U ⃗ − ∇ ⋅ k ∇ T + ρ 1 h sf ∂ α f l ∂ t + ∇ α f l U ⃗ = σ ϵ | ∇ V | 2 + − m ̇ vap h f g − ϵ rad σ B T 4 − T amb 4 + A q ̇ laser | ∇ α | 2 ρ ρ 1 + ρ 2 , (5) where t is the time. The primary variables are the metal volume fraction, α; the one-fluid velocity vector, U ⃗ ; the pressure, p; the electric potential, V; and the one-fluid temperature T. The one-fluid density, ρ = ρ(α); dynamic viscosity, μ = μ(α); specific heat capacity, c _p = c _p (α, T); and thermal conductivity, k = k(α, T), are defined by the mixture model given in Section 2.2.

The remaining terms are now successively defined. Concerning Eq. 1, the third term at the left hand side was introduced by Berberovic et al. (2009) to sharpen the gas–metal free surface. It is active at the free surface alone and depends on the compression velocity, U ⃗ r = U ⃗ r ( U ⃗ ) , and the compression factor that is here set to C _α = 1 to satisfy conservation. The term at the right hand side holds for the net rate of loss of liquid alloy volume fraction due to vaporization at the free surface. The rate of mass vaporization, m ̇ vap = m ̇ vap ( T ) , and the vapor (or recoil) pressure, P _r = P _r (T), are given by (see e.g., Panwisawas et al. (2018) and Luo et al. (2019)): m ̇ vap = 2 1 − β r 1 + β r M 2 π R T P r , (6) P r = 1 + β r 2 P amb ⁡ exp M h f g R 1 T v − 1 T , (7) where M is the molar mass of the alloy, R (= 8.314 J/(mol.K)) is the universal gas constant, and h _fg and T _v = T _v (P _amb) are the vaporization enthalpy and temperature, respectively. β _r = β _r (P _amb) accounts for the re-condensation of the vaporized metal as a function of the ambient pressures, P _amb. This coefficient varies from 1 at a low vaporization rate (i.e., at high ambient pressure or low laser power density) to 0.18 at a high vaporization rate (corresponding to low ambient pressure or high laser power density) (Li et al., 2019; Luo et al., 2019). The values used in this study are given in Table 1. At the left hand side of Eq. 1, the third term is the viscous friction and II denotes the identity matrix. The fourth term is the Darcy-type source term for damping the fluid alloy velocity in the mushy zone where it re-solidifies. It depends on the fraction of liquid alloy, f _l = f _l (T), that ranges from zero in the solid region to one in the liquid region. This fraction is defined by a continuous function of temperature with continuous derivative (Rösler and Brüggemann, 2011), according to f l = 1 2 e r f 4 T − T m T l − T s + 1 , (8) where T _s is the solidus temperature, T _l is the liquidus temperature, and T _m = 0.5(T _s + T _l ) is the arithmetic averaged melting temperature. The constant C _D (which is inversely proportional to the permeability of the porous media) and ϵ _D (that is here to avoid division by zero in the solid phase) are set to 10⁹ and 0.001, respectively. At the right hand side, the second term is the buoyancy force, in which β is the volume expansion coefficient of the alloy in liquid state, ρ _m = ρ _m (T _m ) is the working material density at the melting temperature, and g ⃗ is the gravitational acceleration. The last momentum source term includes surface tension forces that are the capillary and thermocapillary forces, as well as the vaporization (or recoil) pressure force applied on the free surface (Panwisawas et al., 2018; Noori Rahim Abadi et al., 2022). These forces depend on the surface tension coefficient, σ = σ(T) (see Table 1), the local unit vector normal to the free surface, n ⃗ = ∇ α | ∇ α | , and the surface curvature, κ = − ∇ ⋅ n ⃗ . The last coefficient of Eq. 3 is a multiplier used to maintain the surface forces independent of the fluid density while liquid alloy and atmosphere have very disparate densities (Noori Rahim Abadi et al., 2021). This multiplier is also present in Eq. 5.

In the energy equation, the fourth term is related to metal melting and re-solidification and h _sf is the latent heat of fusion. The first term at the right hand side is the resistive (or Joule) heating, whereas σ _ϵ = σ _ϵ (T) is the electrical conductivity. The following terms act at the free surface. In the radiative cooling term that follows the vaporization cooling term, ϵ _rad is the radiative emissivity, σ _B (= 5.67 × 10⁻⁸ W/(m⁻².K⁻⁴)) is the Stefan–Boltzmann constant, and T _amb (= 300 K) is the ambient temperature. The laser beam of total power P _laser moves at a travel speed U _laser along the positive direction x. It has a Gaussian power density distribution (Noori Rahim Abadi et al., 2021), q ̇ laser = 2 P laser π a b exp − 2 x − U laser t a 2 − 2 y b 2 , (9) which is circular here so that the laser spot radius in focus is the same along the x and y axis, thus a = b. The fraction of beam energy locally absorbed, A = A ( λ , θ i , T ) , is expressed as a function of the laser wavelength, λ, the angle between the incident beam wave vector and the local normal to the interface at the incidence point, θ _i , and the material surface temperature. It is calculated for an unpolarized laser beam wave made in equal proportions of s- and p-polarization waves, thus A = 1 2 ( A s + A p ) . Underlying assumptions and derivation steps are available in Noori Rahim Abadi et al. (2022) and therefore not repeated here. The fraction of beam power deposited by the s- and p-polarization waves into the metal surface is given as follows (Noori Rahim Abadi et al., 2022): A s = 1 − cos θ i − a 2 + b 2 cos θ i + a 2 + b 2 , (10) A p = 1 − a − sin θ i ⁡ tan θ i 2 + b 2 a + sin θ i ⁡ tan θ i 2 + b 2 1 − A s , (11) with a 2 = 1 2 n ϵ ′ 2 − n ϵ ′ ′ 2 − sin 2 θ i + n ϵ ′ 2 − n ϵ ′ ′ 2 − sin 2 θ i 2 + 4 n ϵ ′ 2 n ϵ ′ ′ 2 , (12) b 2 = 1 2 − n ϵ ′ 2 + n ϵ ′ ′ 2 + sin 2 θ i + n ϵ ′ 2 − n ϵ ′ ′ 2 − sin 2 θ i 2 + 4 n ϵ ′ 2 n ϵ ′ ′ 2 , (13)

The refractive index, n ϵ ′ , and the extinction coefficient, n ϵ ′ ′ , are defined as follows (Noori Rahim Abadi et al., 2022): n ϵ ′ = 1 2 1 − Q 2 + Q ω λ τ 1 / 2 − Q + 1 1 / 2 , (14) n ϵ ′ ′ = 1 2 1 − Q 2 + Q ω λ τ 1 / 2 + Q − 1 1 / 2 , (15) where Q = ω p 2 ω λ 2 + τ − 2 , ω p = N e 2 m ∗ ϵ 0 1 / 2 , a n d τ = m ∗ σ e T N e 2 . (16)

In these expressions, ω _p is the plasma frequency, τ is the electron relaxation time, N is the free electron density, e is the electron charge, m ^* is the optical effective mass of a conduction electron, and ϵ ₀ is the permittivity of vacuum. The beam wave frequency, ω _λ = 2πc/λ, depends on the beam wavelength, λ (= 1.06 μm), and the speed of light in vacuum, c. The material electrical conductivity, σ _ϵ = σ _ϵ (h(T)), is a function of the local temperature at the metal surface through the sensible enthalpy h, see Table 1.

The one-fluid properties are defined by the following mixture model: ϕ = α ϕ 1 + 1 − α ϕ 2 i f ϕ = ρ o r μ , (17) ϕ ̃ = α f l ϕ ̃ 1 , l + 1 − f l ϕ ̃ 1 , s + 1 − α ϕ ̃ 2 i f ϕ ̃ = k o r c p . (18)

The index s holds for the solid phase and l for the liquid one. The indices 1 and 2 stand for primary (Ti-6Al-4V) and secondary (argon gas) materials, respectively. To our knowledge, there is no database containing all the material properties needed here for the metal alloy. Most of its thermodynamic and transport properties are taken from Mills (2002). The thermal expansion coefficient (β) and the surface tension (σ) are taken from Rai et al. (2009). The properties extracted from other sources are the re-condensation coefficient (β _r ) (Li et al., 2019), electrical conductivity (σ _ϵ ) (Boivineau et al., 2006), radiation coefficient (ϵ _rad) (Fan, 2013), absorptivity-related parameters (Wieting and Schriempf, 1976), and latent heat of fusion (h _sf ) and vaporization (h _fg ) (Desai, 1987). They are given in Table 1 and Table 2 for the shielding gas.

The computational domain used for the simulations is shown in Figure 1. It spans 80 × 9 × 15 mm in the x, y, and z directions, respectively. A 3-mm-thick metal sheet made of alloy Ti-6Al-4V, which is used as a substrate, is located in the bottom of the domain to initialize the computations. The remaining upper part of the domain is filled with argon gas. A filler wire of same alloy and diameter D _w enters the domain from the top surface. It makes an angle, θ _w , with respect to the x direction and is fed at the velocity U _w . In addition, it is translated along the positive x-direction at the same speed U _laser as the laser beam. The laser beam of wavelength λ and total power P _laser is tilted by a small angle θ _laser with respect to the z-direction to avoid back reflection into the processing optics (an issue that typically has to be considered during LDEDw). At the workpiece surface, the center of the beam spot is at the (offset) distance of 2 mm from the tip of the filler wire. This offset value was selected based on previous experience since it led to a smooth metal transfer through the liquid metal bridge during the deposition at the on-Earth reference conditions of case C1.

At the domain boundaries, the alloy is in solid state, at rest, and cooled with natural convection (= 10 W/(m².K)) and radiation to the atmosphere at T _amb = 300 K. In the atmosphere sub-region, the domain boundaries let the argon gas enter (at T _amb) or exit freely based on the pressure field computed inside the domain. To predict the resistive heating of the filler wire, an electric potential difference is applied between the wire section at the top boundary and the workpiece lower surface. The boundary value of the electric potential is set to reproduce the potential difference (ΔV) experimentally measured in C1 between the wire upper section and its tip on the top surface of the workpiece. In addition, since the geometry has a symmetry about the xz-plane passing through y = 0, only half of the domain is simulated. The process parameters used identically in all the simulations are summarized in Table 3.

The computational domain is discretized into cuboid grid cells. The computational results presented in the next section were obtained with an adaptive mesh refinement technique to make the grid finer at the gas–metal interface. Based on the mesh study, the maximum grid size far from the free surface is 1 mm, while it is reduced to 0.25 mm close to the interface.

The solution method is as described in Noori Rahim Abadi et al. (2021) and Noori Rahim Abadi et al. (2022). To achieve a stable solution during solving the highly coupled set of Eqs 1–5, the CFL number is set to 0.04, which leads to a maximum time step size of 10⁻⁴ s. The iterations at each time step continue until the residual of each variable is less than 10⁻¹⁰. The simulation time is 4 s, at which the solution has reached quasi-stable condition. This can be confirmed through observing the evolution over time of the volume of liquid alloy (see Section 3) and the thermal flow fields.

LDEDw induces different heating zones in the workpiece, which result in specific microstructure, and therefore can be used to test the model. Figure 2 compares a macrograph image (left) and computed isotherms (right) obtained with the process conditions C1 defined in Section 1 and in Table 3. The results are presented in a cross section selected in the region where the process is fully-developed. Three different heating zones can be observed. Away from the heat source, the unaffected zone (UZ) has the same microstructure as the original substrate since its temperature did not exceed a value of the order of the α − β transition temperature of the alloy, T _α−β = 1268 K (see the red isoline in Figure 2). On the contrary, close to the heat source, the fusion zone (FZ), delimited in the right side of Figure 2 by the computed melting temperature isoline (in black), results after re-solidification in large grain size that can be seen in the macrograph image (left). In the intermediate region (T _α−β ≲ T ≤ T _m ) or heat-affected zone (HAZ), the microstructure is distinct from the UZ and the grain size is smaller compared to the FZ. It can be seen that the results show an excellent agreement between the predictions and experimental data. In addition, the reinforced bead profile after re-solidification is predicted satisfactorily (in blue) compared to the experiment, which confirms that the deposition process is properly simulated by the solver.

Changing the environment from on-Earth to space conditions could change the material ability to absorb beam energy as well as to loose thermal energy through vaporization when performing LDEDw. These two aspects are now analyzed. Figure 3 shows the evolution over time of the average fraction of beam energy absorbed by the alloy. It compares the results computed with each of the process conditions C1–C4. The plots show no significant difference. When t ≥ 1.5 s, the average absorptivity is stabilized and equal to about 0.44 in each of the cases. It implies that, for the reference case C1 studied, the lowering of the atmospheric pressure down to 11 Pa did not result in the transition from conduction to the keyhole mode. In other words, changing the process conditions from on Earth to in-space has an impact on the temperature field and on the deformation of the melt pool free surface (see next sections) that is sufficiently localized and limited in intensity to have little consequences on the average absorptivity.

The repartition between the wire and the workpiece of the absorbed laser beam power is now considered. Figure 4 visualizes the evolution over time of the rate of beam energy absorbed by the workpiece and the filler wire for the computed test cases C1–C4. It can be seen that this rate is almost three times larger in the workpiece than in the filler wire. A very similar result is obtained in each of the four cases.

Figure 5 shows the evolution over time of the computed rate of mass loss due to vaporization for the different operating conditions C1–C4. It can be seen that the gravitational acceleration does not have any noticeable effect on vaporisation. This result is expected since the primary effect of changing the gravitational acceleration is on the weight and on the buoyancy force, and not on the thermodynamic properties related to vaporisation. In addition, when present (as on-Earth) the forces related to the gravitational acceleration are known to be of small intensity compared to, for example, the thermocapillary force. Therefore, if they have any indirect effect on vaporisation, this one should be weak. In addition, Figure 5 shows that the ambient pressure has a clear effect on vaporization. At on-Earth pressure no vaporization takes place. Indeed, at this condition, the metal temperature does not reach to the vaporization temperature (= 3637 K), as can be seen in the next section. By reducing the ambient pressure, the vaporization temperature decreases while the latent heat of vaporization increases. Hence, the metal can start vaporizing sooner and result in a larger rate of heat removal per unit mass vaporized at the lower ambient pressure. When the ambient pressure is reduced to 11 Pa, the vaporization temperature is 2200 K, which is low enough for vaporization to occur in the simulations, as can be seen in Figure 5. It can also be observed that at process start, the rate of mass loss due to vaporization first increases sharply up to the time t ≈ 0.3 s, then it decreases suddenly before increasing again. This abrupt variation is related to the distance between the wire and the workpiece. Initially, the wire is not in contact with the substrate and during the simulations multiple beam reflection is observed in the space between them. When the wire and the substrate enter into contact, this space disappears as well as the secondary Fresnel reflections. It results in a sudden lowering of the energy absorbed by the wire that moreover starts to shadow the workpiece. Thus, the variations in power absorbed can be seen in Figure 4 at t ≈ 0.3 s, which in turn affect vaporization.

The results show that at P _amb = 11 Pa the rate of mass loss reaches to a quasi-stable condition when t > 3.5 s and oscillates about 10⁻⁶ kg/s. Considering now quasi-stable conditions and the simulated half part of the workpiece, the computed rate of heat loss by vaporization is almost 10 W, which is negligible compared to the total absorbed energy (≈900 W) by the alloy (wire and workpiece) from the laser beam. Therefore, on average, the studied non-Earth working conditions do not significantly change either the trends or the values of rate of net absorbed heat during the LDEDw process compared to on-Earth operation. It is however anticipated that a different thermal environments, such as the interstellar space or lunar ground rather than the interior of a tempered space vehicle, would change this result.

The LDEDw process conditions C1–C4 thus lead to very similar mean laser beam heat input into the metal alloy. The near-vacuum conditions C2 and C4 also experience some heat loss by vaporisation, contrary to the Earth’s standard pressure condition.

The local effect of changing the ambient pressure and/or the gravitational acceleration on the temperature and velocity fields is now examined at the free surface. Figure 6 shows the melt pool top view at time t = 2 s (i.e., during the transient development) and at t = 4 s (fully developed melt pool). It compares isotherms computed at the two extreme conditions C1 (on-Earth P _amb and g) and C4 (near-vacuum P _amb and 0 g). Looking at the evolution from t = 2–4 s, it is observed that the extent of the melting temperature isotherm T _m (therefore the FZ) shrinks along the travel direction while the isotherms at lower T (thus the HAZ) expand. The former evolution is due to the establishment of re-solidification, and the latter to the development of the diffusion of thermal energy from the liquid to the solid alloy. Comparing at given time the plotted isotherms between the cases C1 and C4, it can be seen that they differ. Differences are also observed with the cases C2 and C3 (as seen in Figure 7 as the melt pool contour is related to the T _m -isoline). They show that a reduction in the ambient pressure (at constant g), results in an elongation of the isotherms toward the negative x-direction. The maximum temperature computed in the melt pool at quasi-steady condition is given in Table 4 for each of the process conditions C1–C4. These data show that when the ambient pressure is lowered while the gravitational acceleration is constant (i.e. C1 → C2 and C3 → C4), the maximum temperature is lowered by ≈ 60 K. This weak cooling is consistent with a weak vaporization. However, it might seem to be in contradiction with the observed melt pool elongation. A second effect of vaporisation is to apply a recoil pressure force on the free surface. The computations show that this force is maximum around the front side of the wire tip, where the temperature is highest (see Figure 6). Its influence on the flow of liquid alloy is discussed with the next figure. In addition, the results show that when the gravitational acceleration is reduced while the ambient pressure is constant (i.e. C1 → C3 and C2 → C4), the isotherms are also elongated in the negative x-direction. Furthermore, the maximum temperature is increased by ≈ 30 K (see Table 4). Possible reasons are highlighted below when examining the results of Section 3.4.

Figure 7 presents the top view of the melt pool with the velocity field and vectors at the quasi-stable stage (time t = 4 s) for the four different operating conditions. The results show that the flow pattern is very similar when lowering the gravitational acceleration at constant pressure. If instead the pressure is reduced at constant gravitation, some differences are observed in the flow direction, especially in the regions that are marked in the figure. It can be seen that at the atmospheric pressure the flow is stronger toward the lateral edge of the melt pool, while at near-vacuum condition the direction is tilted toward the melt pool rear. Furthermore, the maximum velocity in the liquid alloy is slightly increased when the ambient pressure is decreased at constant g (i.e., C1 → C2 and C3 → C4), as can be seen in Table 4. These differences are explained by the recoil pressure that results from the metal alloy vaporization occurring when the processes are operated at reduced ambient pressure. The recoil pressure force is an import momentum source term at the right hand side of Eq. 3 governing the u ⃗ field. It can explain the melt pool elongation obtained at near-vacuum condition compared to ambient Earth pressure.

When the gravitational acceleration is reduced while the ambient pressure is constant (i.e., C1 → C3 and C2 → C4), the maximum fluid velocity behaves differently depending on the ambient pressure: it is unchanged at Earth-pressure, while it varies with g at near-vacuum condition. A first effect of gravity on convection in the melt pool is through the momentum transferred with metal transfer. The related potential energy is here believed to be negligible as the wire tip is in contact with the pool. A second possible effect could be through the buoyancy force. At standard Earth-pressure, this upward force is known to be negligible compared to the other forces present (Cho et al., 2009). Thus, the unchanged maximum velocity in C1 and C3 is observed. In addition, at near-vacuum condition, the computational results C2 show that the buoyancy force is much lower than the recoil pressure force. It is thus believed that the buoyancy force has no significant contribution to accelerating the liquid alloy.

It has been seen that the ambient pressure and the gravitational acceleration have some effect on the thermal flow field inside the melt pool during the studied LDEDw process. Their impact on the thermal zones and the free surface profile are now considered. The latter is an important aspect for process control as it might affect the stability of the process. Figure 8 presents a top view of the thermal zones during the quasi-stable stage at t = 4 s for the different operating conditions. The plotted outer edge of the HAZ corresponds to the α − β transition isotherm. The results confirm that the length and shape of FZ and the extent of the HAZ within the workpiece increase when the ambient pressure or the gravitational acceleration decrease. These variations in HAZ, which reveal variations in thermal history, might induce small variations in material microstructure.

In Figure 9, the evolution over time of the total volume of liquid alloy at the different operating conditions is shown. This volume becomes almost unchanged after t > 3.5 s, confirming that at this time the process reaches a quasi-stable state. Then its value differs by at most 3% between the different cases simulated. This is consistent with the results of Section 3.2, as the laser beam power entering the alloy shows no significant variation with the process conditions and is it much larger than the vaporization cooling (if any).

In Figure 10, the computed bead profile is presented after re-solidification for different operating conditions. The results show that the bead width is seemingly unchanged, while the bead surface curvature is slightly more downward at the lateral sides. As expected, the bead height is increased by lowering the ambient pressure or the gravitational acceleration. The highest elevation is reached at near-vacuum and no-gravity condition.

Figure 11 compares side view plots showing the gas–metal interface at the quasi-steady state for the different operating conditions. At the near-vacuum ambient pressure, vaporization occurs causing the appearance of the recoil pressure effect by which the liquid metal surface is pushed downward, as can be seen in the zoomed image of Figure 11. The displaced liquid metal then moves to the rear region of the melt pool (see Section 3.3) causing the longer melt pool and the relatively higher bead profile. The role of gravity and the resulting buoyancy force is not remarkable in a melt pool, as known from former studies (Cho et al., 2009). However, in the presence of metal transfer from a filler wire, the weight might explain the small increase in curvature observed at the front side of the wire tip (at x ≈ 46.5 mm) at near-vacuum pressure compared to standard on-Earth atmospheric pressure. This small change in curvature results in a small increase of the absorptivity that explains the higher temperature seen at the wire tip when comparing the isotherms between C1 and C4 in Figure 6. Thus, it could explain the small increase in maximum temperature noticed in Table 4 when changing the gravitation from on-Earth to non-terrestrial condition.

Another important effect observed at the reduced ambient pressure and in the absence of gravity is the thinner width of the liquid metal bridge in the region where the wire tip joins the melt pool free surface. It is most probably a consequence of the lowering of the free surface by the recoil pressure that was previously observed in front of the wire tip. As a result, the distribution of the current density flowing through the contact section for resistive heating of the wire is more constricted, which can lead to a larger Joule heating. The thinner liquid metal bridge implies that the LDEDw process is moving toward more unstable operating conditions. This is supported by the slightly larger irregularities of the volume of liquid alloy observed in Figure 9 at P _amb = 11 Pa and | g ⃗ | = 0 m / s 2 . There might then be a risk of breaking the liquid bridge and changing the metal transfer mode to drop-wise mode which is not desirable (Heralic, 2012).

As seen in Figure 11, this CFD-based feasibility study shows that the metal bridge between the wire and the workpiece is maintained in all the cases. This is the most important finding from a process standpoint since the continuous metal transfer is a prerequisite for high material quality and the ability to build multilayer parts successfully (Heralic, 2012). The variations in metal bridge width, build height, and thermal profile are all similar to what an LDEDw control system is made to handle for conventional on-Earth applications (Hagqvist, 2015). This indicates that, by using an appropriate process control to maintain the liquid metal bridge (Kisielewicz et al., 2021), LDEDw can be used for manufacturing metal parts in-space at tempered atmosphere.