Pulsed laser surface melting (pLSM) is a tec hnique that offers efficient and effective way to modify the geometry surfaces without any addition or removal of material. The resultant surface geometry plays a critical role in several applications. This paper presents a steady state thin film approximation of the meltpool created by pLSM and the resulting semi -analytical solution for the evolved surface geometry. This predictions of the semi -analytical solution are then compare with a validated numerical solution. The c omparison demonstrate a good match with errors ranging from ~4% to ~25% across several pulse duration . Larger errors are observed at comparatively lower and higher pulse duration , and smaller errors are observed for intermediate pulse duration values. Over all, the thin film solution is a reasonable and useful approximation of the of the evolved surface geometry through pLSM process, thus saving significant computational costs. Keywords: Laser surface melting, surface tension, thin film approximation, surface geometry NOTATIONS AND NOMENCLATURE [∗]∗ Dimensional quantity 𝑃 Laser peak power, 𝑊 [∗]𝑇 Transpose operator 𝑅𝑏 Laser beam radius, 𝜇𝑚 [∗](0) Order 1 parameter 𝜏𝑡∗, 𝜏𝑛∗ Tangential and Normal stress tensor at free surface ∇∗ Gradient operator 𝜏∗ Viscous stress tensor 𝜌 Density, 𝑘𝑔/𝑚3 (𝜕𝛾∗/𝜕𝑇∗) Surface tension coefficient, 𝑁/𝑚−𝐾 𝑐𝑝′ Equivalent specific heat, 𝐽/𝑘𝑔−𝐾 𝐿 Latent heat of fusion, 𝐽/𝐾 𝑇∗ Temperature, 𝐾 𝑇𝑠 Solidus temperature, 𝐾 𝑡∗ Time, 𝑠 𝑇𝑚 Melting temperature, 𝐾 𝑉∗ Velocity vector, 𝑚/𝑠 𝑇𝑙 Liquidus temperature, 𝐾 𝑘 Thermal conductivity, 𝑊/𝑚−𝐾 𝑇𝑣 Evaporation temperature, 𝐾 𝑝∗,𝑝𝑐 Pressure and its characteristic value, 𝑃𝑎 𝑇𝑖 Initial temperature, 𝐾 𝜇 Dynamic viscosity, 𝑘𝑔/𝑚−𝑠 Preprint submitted to SME Journal of Manufacturing P rocesses 2 𝜈 Kinematic viscosity, 𝑚2/𝑠 Non-dimensional quantities 𝑡𝑝 Pulse duration, 𝜇𝑠 𝑟∗ Radial position, 𝜇𝑚 𝑟 Radial coordinate 𝑅 Computational domain radius, 𝜇𝑚 𝑧 Vertical coordinate 𝑧∗ Vertical position, 𝜇𝑚 𝑢𝑟 Velocity component in 𝑟- direction 𝑍 Computational domain height, 𝜇𝑚 𝑢𝑧 Velocity component in 𝑧- direction ℎ∗ Free surface height function, 𝜇𝑚 ℎ Free surface height function 𝑢𝑟∗,𝑈𝑅 Velocity and its characteristic value in 𝑟 direction, 𝑚/𝑠 𝑝 Pressure 𝑢𝑧∗,𝑈𝑧 Velocity and its characteristic value in 𝑧 direction, 𝑚/𝑠 𝑡 Time 𝑙𝑐 Characteristic value in 𝑟-direction, 𝜇𝑚 𝜃 Temperature 𝑧𝑐 Characteristic value in 𝑧-direction, 𝜇𝑚 𝑞 Heat flux 𝑡̂,𝑛̂ Unit tangent and normal vector at free surface 𝑅𝑒 Reynolds number 𝛼 Absorptivity 𝑃𝑒 Peclet number 𝑞∗ Heat flux, 𝑊/𝜇𝑚2 𝐶𝑎 Capillary number 𝛾∗,𝛾0 Surface tension and its characteristic value, 𝑁/𝑚 1 Introduction Laser surface melting is a non -traditional, effective and efficient manufacturing process that has been increasingly used for generating micro -scale surface features on metallic components without addition or removal of material [1]. Creating a near flat feature to reduce surface roughness , also known as polishing, is an application of laser surface melting that has been widely studied recently . Several experimental stud ies have successfully demonstrated that polishing via laser surface melting is an effective alternative to other polishing techniques such as coatings, chemical polishing, abrasive polishing, etc. [1, 2] . In laser surface melting (LSM), a continuous wave or pulsed laser source irradiates a surface. This heats the material and eventually melts a thin layer on the surface, creating a melt pool. Complex flows within the melt pool, primarily under viscous and surface tension forces , result in the dynamic evolution of the surface and upon resolidification results in new surface feature s. While conductive heat transfer is dominant in the initial stages of surface melting, convection heat transfer dominates the later stage s in molten state . Among continuous wave and pulsed sources , pulsed laser sources have the advantage of smalle r melt pool sizes while producing the desired final surface . Thus , there is minimal modification of material morphology and the bulk properties of the workpiece material are conserved. In pulsed laser surface melting (pLSM), the evolution of the surface a nd the geometry of the surface feature depends on process properties such as incident laser power, pulse duration and incident beam diameter. T here are few studies that try to capture the effects of various process parameters in pulsed laser surface melting in the context of determining final surface evolution. While most of the works are experimental [3–6], some have developed complex Preprint submitted to SME Journal of Manufacturing P rocesses 3 multi -physics numerical models to investigate the evolution of the final surface after melting [7– 9]. Numerical studies done in [10] demonstrate that buoyancy forces in the melt pool during laser surface melting can be neglected. They have considered non -deformable free flat surface to find the velocity and temperature distribution in melt pool of rectangular cavity shape. In this context, studies done at the University of Wisconsin Madison [3,11] are of particular interest as they have systematically demonstrated the effec ts of temperature gradients of surface tension on surface evolut ion due to laser micro melting. Ma el al. [11] have developed a numerical model and showed that the resultant surface geometry matches well with experimental values. Their work will be used to validate the work done in this paper. For axisymmetric flow under gaussian beam distribution, analytical solution s of the heat conduction equation is obtained by Woodard et al. [12] and provide initial estimates of temperature distribution and hardened spot size. Tokarev et al. [13] presented an analytic al estimation for melt depth using a one dimensional transient heat conduction in pulsed laser melting. They have given overestimate and underestimate values for melt depth which depends on various property values of material and laser pulse duration (𝑡𝑝). They have shown a good match with experiments. Their results will be used in the current current work. A limitation among a ll these studies is that the analysis was done under non -deformable free surface of the material and haven’t considered stre ss balance conditions at free surface. Subsequently, t wo dimensional numeric al simulations were performed [14] to showcase the deformation of free surface under steady state and transient condition. Analytic al similarity solution for velocity profile and temperature distribution was given by [15] for marangoni convected flow over flat surface . They have assumed the formation of boundary layer on flat free surface under imposed temperature gradient. While these a re useful in deriving insights into the process, no studies exist that can provide for a analytical or semi -analytical solution for the evolution of the surface geometry, which ulmitately play a key role in several applications ranging from bio -implants to micro -fluiding channels. The present study, therefore focusses to develop a semin -analytical solution for prediction of surface geometry. In this work, a two dimensional transient axisymmetric numerical model with a deformable free surface will be develo ped for pulsed laser surface melting. This model will include temperature dependent material properties and will act as a baseline model the semi -analytical solution. For rigor, it will be shown that this model compares well with the results presented in Ma et al. [11]. Next, a steady state semi -analytical experession will be derived for the surface geometry using a thin film approximation. It will be shown that the predictions of the this semi - analytical solution match closely with the numerical results with certain cavi ats. Finally, a detailed discussion on the results will be provided highlight the advantages and more importantly, the limitations of the semi -analytical solution . 2 Physics of Pulsed Laser Surface Melting 2.1 Mechanism of pulsed laser surface melting pLSM is a multi -physics process consisting heating of a micro scaled metal surface asperities beyond the melting and below the ablation temperature, using laser pulses. This complex process involves heat transfer primarily through conduction and convection , phase change of material Preprint submitted to SME Journal of Manufacturing P rocesses 4 and resulting flow of the melt. Once material is in molten state, it starts to redistribute under influence of surface tension, pressure and viscous forces until material is resolidifie d. The mechanism of pLSM can be easily understood w ith the following four stages: Stage 1: Heating In th is stage, the surface of the material is heated by the incident laser beam and the heat is conducted into the metal. As shown in Figure 1(a), conduction continues and the temperature at center point (shown in blue) of metal reaches the melting temperature. At this temperature, the material begins to melt by absorbing energy equi valent to its latent heat of fusion before changing phase. Note that in case of alloys, there is no single temperature at which the material melts, but the melting happens from solidus temperature, the temperature in metal below which the entire metal is i n solid phase, to the liquidus temperature, above which the entire metal is in liquid phase. Stage 2: Melting This stage begins when the surface temperature is beyond the liquidus temperature (melting temperature for pure metals). The heat is transferred into the material through conduction and convection modes. The metal continues to melt and melt pool size incre ases in both radial and axial directions. Figure 1(b) shows the ascending melt -front, as the liquidus temperature isothermal contour, as it moves into th e bulk of the material. This melt pool size continues to increase until the laser pulse is ON. The direction of the molten metal flow depends on the value of surface tension coefficient ( 𝜕𝛾∗/𝜕𝑇∗ ) which will be discussed in Section 2.2. Stage 3: Solidification Solidification begins once the incident heat source is switch ed off. In th is stage the conduction and convection of he at in the melt pool continue to slightly increase the melt pool size before solidification begins . At this point, the melt pool continues to lose energy as conduction into the bulk and a small amount as radiation to the environment. Figure 1(c) shows the solidification -front, as the liquidus temperature isotherm al contour, receding from the bulk of the material toward the surface. Stage 4: Cooling In this stage , the entire molten material has resolidified. The solid material continues to cool down by conducting heat into the bulk and losing little heat as radiati on. The stage is complete once the material returns back to its original temperature, usually at room temperature. Preprint submitted to SME Journal of Manufacturing P rocesses 5 2.2 Effect of surface tension gradient In Stage s 2 & 3 of pLSM, the metal is in a molten state of a metal. A large gradient of temperature exists across the melt pool surface. This creates a gradient in the surface tension and thus causing an force imbalance across the surface. This imbalance in s urface tension coupled with viscous forces causes the molten metal to flow. This phenomenon is called the Marangoni effect and the resulting flows are called thermocapillary flows . The nature of the flows and the behavipor of the molten material are dictat ed by the surface tension coefficient, 𝜕𝜎∗/𝜕𝑇∗, of the material . Typical power distribution of a laser beam is Gaussian , i.e., the intensity at the center of the beam is maximum and it exponential decays to zero towards the edges . Therefore, the temperature at center of the melt pool is much higher than that at the edge. Materials with a negative surface tension coefficient will have higher surface tension at the edge s than the center of the meltpool . This creates gradients of surface tension and flow of molten metal takes place from low surface tension to higher surface tension. This process develops the upwelling of material from center point towards the edge of molten pool as shown in Figure 2. Conversely, a positive surface tension coefficient results in upwelling of the material towards the center of the pool. (a) (b) (c) (d) Figure 1: Four stages of pLSM process Preprint submitted to SME Journal of Manufacturing P rocesses 6 Figure 2: Schematic of solidified surface geometry for materials with negative 𝝏𝝈∗/𝝏𝑻∗ 3 Numerical model of pLSM To capture the physics of the process, a numerical model is developed and is simulated using COMSOL commercial software for simulation purpose . Axisymmetric model is the natural choice of for numerical analysis , as w ithin each cycle of melting and solidification, the process is inherently symmetric about the central axis of the incident laser beam. Further, sufficient time is provided between two laser pulses to ensure the material returns to its room temperature. Therefore, studying the process under a single pulse is sufficient. is sufficient to study the process. This nume rical model will also act as a reference to validate the semi -analytical results to be discussed later in the paper. In this section, this model is detailed with the validation results. 3.1 Governing equations and a ssumptions For modelling simplicity and ease, the following assumptions are made: • The material is isotropic and homogeneous. • Chemical changes are ignored as the process is typically carried out in an inert environment. • Radiation and convection heat losses to the surroundings are neglected. • Newtonian flow is assumed in melt pool, i.e., constant viscosity. • The surface tension of the molten material is varying linearly with temperature. • The gravitational forces [3] and buoyancy forces [16] are neglected as they are considerably small as compared to the surface tension effects. • Ablation is not modeled as maximum temperature is ensued to be not beyond vaporization temperature of material. • There is no additional source irradiating on su rface. Under the above assumptions, the coupled temperature and velocity fields are given by the energy conservation equation and the Navier -Stokes equation and are given by the following, Preprint submitted to SME Journal of Manufacturing P rocesses 7 𝜌𝑐𝑝′(𝜕𝑇∗ 𝜕𝑡∗+𝑉∗⋅𝛻∗ 𝑇∗)−𝛻∗⋅(𝑘𝛻∗ 𝑇∗)=0 (1) 𝜌[ 𝜕𝑉∗ 𝜕𝑡∗+(𝑉∗⋅𝛻∗ )𝑉∗ ]= −𝛻∗ 𝑝∗+𝛻∗⋅{ 𝜇[(𝛻∗𝑉∗)+(𝛻∗𝑉∗)𝑇]} (2) where, 𝑐𝑝′ is equivalent specific heat which accounts latent heat of fusion, 𝑘 is thermal conductivity, 𝜇 is dynamic viscosity of a material. Further, under the assumption of steady flow of the molten material, the continuity equation is written as, 𝛻∗⋅𝑉∗=0 (3) 3.2 Interfacial and boundary conditions Normal and tangential stress is developed on the interface between affected surface area and surrounding environment. Stress balance in normal and tangential direction on interface is as follows, Normal Stress Balance −𝑝∗+(𝜏∗⋅𝑛̂)⋅𝑛̂=𝛾∗ (𝛻∗⋅ 𝑛̂) (4) Tangential Stress Balance (𝜏∗⋅𝑛̂)⋅𝑡̂=(𝜕𝛾∗/𝜕𝑇∗)(𝛻∗ 𝑇∗⋅ 𝑡̂) (5) where 𝛾∗ is the surface tension and ∇∗⋅𝑛̂ is the curvature of the surface. The laser beam source incident on the surface is modelled as a heat flux boundary condition on ini tially flat surface is given by, 𝑛̂⋅(𝑘𝛻∗𝑇∗)=𝛼𝑞∗ (6) where 𝑞∗ is heat flux and 𝛼 is the absorptivity . Gaussian distribution of heat flux within pulse duration is assumed and is given by, 𝑞∗=𝑃 𝜋𝑅𝑏2𝑒(−2𝑟∗2 𝑅𝑏2) (7) where 𝑃 is the peak power of gaussian distribution and 𝑅𝑏 is the laser beam radius. Boundaries 𝑟∗=𝑅 and 𝑧∗=0 in Figure 2 are considered stationary as material has no velocity at these boundaries. Therefore, no slip conditions are app ropriate to imposed at these boundaries. Further, adiabatic boundary conditions are assumed at 𝑟∗=𝑅 and 𝑧∗=0 as these are sufficiently far from the melt pool to ensure no heat or mass transfer with the surroundings. These are mathematically described a s, 𝑉∗(0,𝑟∗)=0 & 𝑉∗(𝑧∗,𝑅)=0 (8) 𝑛̂∙(𝑘𝛻∗𝑇∗)|(0,𝑟∗)=0 & 𝑛̂∙(𝑘𝛻∗𝑇∗)|(𝑧∗,𝑅)=0 (9) 3.3 Material p roperties Without loss of generality, Ti6Al4V alloy is the material of choice for all the work presented in this paper. Temperature dependent properties of Ti6Al4V , viz., density, thermal Preprint submitted to SME Journal of Manufacturing P rocesses 8 conductivity, spec ific heat and dynamic viscosity, are taken for the numerical simulations [17,18] . Effective viscosity method is used to enable the solid material to stationary position and molten metal to flow [16]. For convergence, the solid state viscosity is set to be approximately 30 million times of liquid state viscosity. The value of surface tension coefficient is used for pure titanium in air ( 𝜕𝜎∗/𝜕𝑇∗ = -0.26 𝑚𝑁/𝑚−𝐾 [19]), as it was not found in literature for Ti6Al4V alloy. As 𝜕𝜎∗/𝜕𝑇∗ plays an important role, this value would build some error in simulating final surface geometry . Absorptivity value is taken as 0.5 [12]. Table 1 shows some properties used in simulation. Table 1: Parameters and constants used in simulations 3.4 Meshing and solver Computational domain was meshed with free triangular and finer mesh element. As material will flow once the metal is in molten state, moving mesh was applied to meshed geometry and mesh element size was calibrated for fluid dynamics. Direct solver PARDISO used in simulation requires less RAM then other solvers and can store the solution out -of-core [21]. Lap lace mesh smoothing method ensures smooth deformations of mesh, given the constraints on boundaries of domain. 3.5 Model validation The model was validated by comparing the melt pool geometry and normalized peak to valley height (PVH) to those discussed in Ma et al. [11]. Although the application in [11] laser micro polishing of metallic surfaces, as discussed ear lier, it is a specific application of pLSM . Note that the model by Ma et al . was validated against experimental laser polishing results on Ti6Al4V and therefore, this model -to-model validation is acceptable . The results of both the models are within +7% of error. These are summarized in following Table 2. Table 2: Validation of numerical simulation model 𝑡𝑝 (𝜇𝑠) Normalized PVH (𝜇𝑚) %Error Ma et al [11] Current work 1 0.01 0.0105 -5 3 0.06 0.056 6.66 5 0.12 0.113 5.83 Computational dimensions ( 𝑅 x Z) (𝜇𝑚) 120 x 30 Solidus temperature, 𝑇𝑠 (K) [20] 1877 Melting temperature, 𝑇𝑚 (K) [20] 1903 Liquidus temperature, 𝑇𝑙 (K) [17] 1923 Evaporation temperature, 𝑇𝑣 (K) [18] 3500 Ambient temperature, 𝑇𝑎𝑚𝑏 (K) 298 Initial temperature, 𝑇𝑖 (K) 298 Latent heat of fusion, L (J/K) [17] 3.24 × 105 Preprint submitted to SME Journal of Manufacturing P rocesses 9 4 Non-dimensionalization of equations The mathematical model of the proce ss described in Sections 3.1 and 3.2 are non - dimesionalized for ease of analysis. Further, Non-dimensionalization results in dimens ionless numbers through which material independent insights into the process can be easily derived without the need to run complex numerical simulations. This section details the procedure of non-dimensionalization under the following assumptions: • Only steady state equations are considered i.e., the temporal variations of temperature, pressure, veloc ity are neglected. Although this is only true when the meltpool is in molten state for sufficiently long durations, it will be shown later in the the paper that this analysis will result in useful estimates of the surface geometry. • Temperature dependence of material properties is neglected and constant properties at liquidus temperature are used . These are tabulated in Table 3. Under the above assumptions, Equations (1) - (6) are non -dimensionalised and are summarized in Table 4. A detailed derivation of the th ese equations is given in Appendix A . Table 3: Constant properties at liquidus temperature Thermal conductivity, k ( 𝑊/𝑚−𝐾) 30 Surface tension coefficient, (𝜕𝛾∗/𝜕𝑇∗) (m𝑁/𝑚−𝐾) -0.26 Surface tension, 𝛾0 (𝑁/𝑚) 1.65 Density, 𝜌 (𝑘𝑔/𝑚3) 3886 Specific heat, 𝑐𝑝 (𝐽/𝑘𝑔−𝐾) 800 Dynamic viscosity, 𝜇 (𝑘𝑔/𝑚−𝑠) 0.005 Table 4: Steady State Dimensionless Equations for pLSM Navier -Stokes Equations R-momentum 𝜖2𝑅𝑒[(𝑢𝑟𝜕𝑢𝑟 𝜕𝑟)+(𝑢𝑧𝜕𝑢𝑟 𝜕𝑧)]=−𝜕𝑝 𝜕𝑟+𝜖2𝜕 𝜕𝑟(1 𝑟𝜕 𝜕𝑟(𝑟𝑢𝑟))+𝜕2𝑢𝑟 𝜕𝑧2 (10) Z-momentum 𝜖4𝑅𝑒[(𝑢𝑟𝜕𝑢𝑧 𝜕𝑟)+(𝑢𝑧𝜕𝑢𝑧 𝜕𝑧)]=−𝜕𝑝 𝜕𝑧+𝜖2[𝜖2(1 𝑟𝜕 𝜕𝑟(𝑟𝜕𝑢𝑧 𝜕𝑟))+(𝜕2𝑢𝑧 𝜕𝑧2)] (11) Energy Balance Equation 𝜖2𝑃𝑒[(𝑢𝑟𝜕𝜃 𝜕𝑟)+(𝑢𝑧𝜕𝜃 𝜕𝑧)]=[𝜖2(1 𝑟𝜕 𝜕𝑟(𝑟𝜕𝜃 𝜕𝑟))+(𝜕2𝜃 𝜕𝑧2)] (12) Continuity Equation 1 𝑟𝜕 𝜕𝑟(𝑟𝑢𝑟)+𝜕𝑢𝑧 𝜕𝑧=0, 𝑈𝑧=𝜖𝑈𝑅 (13) Preprint submitted to SME Journal of Manufacturing P rocesses 10 Interfacial and Bourndary Conditions Normal Stress Balance 𝑝=−1 𝛿(𝜕2ℎ 𝜕𝑟2) (14) Tangential Stress Balance 𝜕𝑢𝑟 𝜕𝑧+𝑂(𝜖2)=ϵ(𝜕𝛾∗/𝜕𝑇∗)Δ𝑇 𝜇𝑈𝑅[𝜕𝜃 𝜕𝑟+𝜕𝜃 𝜕𝑧𝜕ℎ 𝜕𝑟] (15) Heat Flux (−𝜖2𝜕ℎ 𝜕𝑟)𝜕𝜃 𝜕𝑟+𝜕𝜃 𝜕𝑧=𝑞 (16) In Table 4, ℎ(𝑟) is the non -dimensional geometry of the surface , 𝜖=𝑧𝑐 𝑙𝑐, is the ratio of characteristic length scales in the 𝑧 and 𝑟 directions, 𝑅𝑒=𝑈𝑅𝑙𝑐/𝜈 is the Reynolds number , 𝑃𝑒=𝑈𝑅𝑙𝑐/𝛼 is the Peclet number, 𝛿=𝐶𝑎/𝜖3 is a non -dimensional parameter resulting from Capillary number 𝐶𝑎=𝜇𝑈𝑅/𝛾0. 5 Thin film approximation A thin film is defined as the region in which the characteristic depth scale is comparatively very smalle r than the characteristic length scale. In case of pLSM, this translates to 𝜖=𝑧𝑐 𝑙𝑐≪1. An illustrative schematic of metpool in pLSM is shown in Figure 3. The resultant melt pool diameters are of the order of 30 -50 m and the resultant maximum melt pool are of the order of 3-5 m [3,11] , meaning that 𝜖~𝑂(−1). Therefore, approximation of the meltpool obtained in pLSM as a thin film is reasonable. The solution obtained thereof gives the general idea about velocity, and temperature distribution. A outcome of particular interest in this paper is the semi - analytical solution for evolved interfacial height function. Figure 3: Thin Film domain of dimensio n 𝒛𝒄 x 𝒍𝒄 Preprint submitted to SME Journal of Manufacturing P rocesses 11 In additional to small 𝜖, the following are assumptions are made for analysis. It is to be noted that some of these assumptions are invalid for all process conditions and the effects of the same will be discussed in later sections. • Rescaled Reynolds number , defined as 𝑅𝑒̂=𝜖𝑅𝑒 and rescaled Peclet number , 𝑃𝑒̂=𝜖𝑃𝑒 are assumed to be 𝑂(1). • All dependent properties such as 𝑝, 𝜃, 𝑢𝑟, 𝑢𝑧, and ℎ can be expressed in asymptotic expansion in the thin film limit of 𝜖≪1. This can be mathematically expressed as: 𝑝=𝑝(0)+𝜖𝑝(1)+𝜖2𝑝(2)+․․․․․ (17) 𝜃=𝜃(0)+𝜖𝜃(1)+𝜖2𝜃(2)+․․․․․ 𝑢=𝑢(0)+𝜖𝑢(1)+𝜖2𝑢(2)+․․․․․ ℎ=ℎ(0)+𝜖ℎ(1)+𝜖2ℎ(2)+․․․․․ Under these assumptions and neglecting all terms of the order 𝜖2 or higher in Equations (10) - (17), the simplified process equations can be rewritten as follows. More details are available in Appendix B . 1 𝑟𝜕 𝜕𝑟(𝑟𝑢𝑟(0))+𝜕𝑢𝑧(0) 𝜕𝑧=0, (18) 𝜕2𝜃(0) 𝜕𝑧2=0 (19) 𝜕2𝑢𝑟(0) 𝜕𝑧2−𝜕𝑝(0) 𝜕𝑟=0 (20) 𝜕𝑝(0) 𝜕𝑧=0 (21) 𝑝(0)=−1 𝛿(𝜕2ℎ 𝜕𝑟2) (22) 𝜕𝑢𝑟(0) 𝜕𝑧=𝜕𝜃(0) 𝜕𝑟+𝜕𝜃(0) 𝜕𝑧𝜕ℎ 𝜕𝑟 (23) 𝜕𝜃(0) 𝜕𝑧=𝑞 (24) 5.1 Lower boundary of thin film Usually, thin film analysis assumes a rectangular shape of the film, where both the top and the bottom boundaries are flat. In case of pLSM, the meltfront, which is boundary of the meltpool is rarely flat. This is especially true when the melt pool sizes a re large. Therefore, in this paper, both flat as well as curved meltpool boundairs are considered. Further, based on numerical and experimental observations [11,22] , the curved meltpool boundary is approximated as a par abolic curve. A typical melt front created observed in simulations is shown in Figure 4, along with a parabolic curve with suitable coefficients. The dimensional and no n-dimensional Preprint submitted to SME Journal of Manufacturing P rocesses 12 parabolic boundary , along with the corresponding boundary conditions are mathematically given by, 𝑇∗=𝑇𝑙 & 𝑉∗=0,@ℎ∗=𝑓∗(𝑟∗)={0, 𝑓𝑙𝑎𝑡 𝑚𝑒𝑙𝑡𝑓𝑟𝑜𝑛𝑡 𝐾𝑟∗2+𝛽, 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑚𝑒𝑙𝑡𝑓𝑟𝑜𝑛𝑡 (25) 𝜃=0 & 𝑢𝑟,𝑧=0,@ℎ=𝑓(𝑟)={0, 𝑓𝑙𝑎𝑡 𝑚𝑒𝑙𝑡𝑓𝑟𝑜𝑛𝑡 𝐾̂𝑟2+𝛽̂, 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑚𝑒𝑙𝑡𝑓𝑟𝑜𝑛𝑡 (26) where 𝐾 and 𝛽 are constant s. 𝐾̂=𝐾𝑙𝑐2/𝑧𝑐, 𝛽̂=𝛽/𝑧𝑐 are the coefficient that are functions of characteristic length scales . 5.2 Solution of surface height In this section, a steady state solution to the surface height, ℎ(𝑟), is derived under the thin film approximation of the melt pool created in pLSM. For conciseness, only the key details are provided in the section, while the detailed derivation is provid ed in Appendix B . To begin with the solution, Equation (19) for leading -order temperature, 𝜃(0) is solve d under heat flux boundary condition in Equation (24) and constant temperature boundary condition in Equation (26), 𝜃(0)=𝑞(𝑟)(𝑧−𝑓(𝑟)) (27) where 𝑓(𝑟) is the meltfront geometry as defined in Equation (26) and 𝑞(𝑟)=𝐶 𝑒−8𝑟2. The constant 𝐶 is defined in Appendix A . The solution for leading -order velocity , 𝑢𝑟(0) can be found from Equation s (20) and (23) under the boundary condition in Equation (26) and is given by, 𝑢𝑟(0)=𝜕𝑝(0) 𝜕𝑟𝑧2 2+(𝐴−𝜕𝑝(0) 𝜕𝑟ℎ)𝑧−𝜕𝑝(0) 𝜕𝑟𝑓(𝑟)2 2−(𝐴−𝜕𝑝(0) 𝜕𝑟ℎ)𝑓(𝑟) (28) Figure 4: Melt front shape with a parabolic fit Preprint submitted to SME Journal of Manufacturing P rocesses 13 where, 𝐴= 𝑞′(𝑟)(ℎ−𝑓(𝑟))+𝑞(𝑟)(−𝑓′(𝑟))+𝑑ℎ 𝑑𝑟 𝑞(𝑟) Given that there is no loss of material in the melt pool, a zero net volume flux condition can be imposed as follows, ∫𝑢𝑟(0)(𝑟,𝑧)𝑑𝑧=0ℎ(𝑟) 0 (29) Substituting from (28), Equation (29) gives the pressure gradient a s, 𝜕𝑝(0) 𝜕𝑟=𝐴(𝑓(𝑟)−ℎ 2) (−ℎ2 3−𝑓(𝑟)2 2+ℎ𝑓(𝑟)) (30) Finally, differentiating normal stress balance condition in Equation (22) with respect to 𝑟, 𝜕𝑝(0)/𝜕𝑟 from Equation (30) can be eliminated, resulting in , 𝑑3ℎ 𝑑𝑟3=𝛿[ 𝑞′(𝑟)(ℎ−𝑓(𝑟))+𝑞(𝑟)(−𝑓′(𝑟))+𝑑ℎ 𝑑𝑟 𝑞(𝑟)](𝑓(𝑟)−ℎ 2) (−ℎ2 3−𝑓(𝑟)2 2+ℎ𝑓(𝑟)) (31) A closed form solution to Equation (31) cannot be obtained. However, in the limit of 𝜖≪1, the deformation is small enough . Therefore, the 𝑂(𝜖2) and higher order terms can be neglected in the expression of ℎ in Equation (17) to obtain an approximate solution in the form of ℎ(𝑟)=ℎ(0)(𝑟)+𝜖ℎ(1)(𝑟) (32) Equations (31) and (32), yield two expression for the leading -order term, ℎ(0) and the first -order term, ℎ(1). The det ailed derivation of these expression s is in documented in Appendix B . At 𝑂(1), a trivial solution of ℎ(0) emerges as, ℎ0(𝑟)=1 (33) And t he first-order term, ℎ(1) becomes, 𝑑3ℎ(1) 𝑑𝑟3=6𝛿𝐶𝑒−8𝑟2[16𝑟(1−𝑓(𝑟))+𝑓′(𝑟)] (𝑓(𝑟)−0.5) (3𝑓(𝑟)2−6𝑓(𝑟)+2) (34) While there is no analytical solution for ℎ(1), Equation (34) can be numerically integrated for the two boundary functions 𝑓(𝑟). Substituting both in Equation (32), thus results in a solution to the surface height of the meltpool in pLSM. Solution for ℎ1(𝑟), combined with ℎ0(𝑟) gives a semi -analytical solution for interfacial height function ℎ(𝑟) as, ℎ(𝑟)=ℎ0(𝑟)+𝜖ℎ1(𝑟) (35) 5.3 Method of validation In order to validate the thin -film approximate solution for the steady state surface height of the meltpool, comparison are made with the detailed numerical model described earlier. Note that the numerical model itself was validated against experimental re sults from literature [11]. Preprint submitted to SME Journal of Manufacturing P rocesses 14 These are summarized in Table 2. Comparisons for validation will be made at nine different pulse duration on an initially flat surface . The pulse energy for each pulse dura tion is chosen such that maximum temperature in the numerical simulation is just below the ablation temperature. This is hypothesized to provide sufficient time for the meltpool flows to reach steady state. It will be shown in later discussion that this has limitations. To have a meaningful compar ision of the solution from thin film approximat ion with the numerical solution , non -dimensionaliz ing characteristic values, 𝑧𝑐 and 𝑙𝑐 are required. These characteristic values ha ve to be chosen diffrerently for the two solution methods to properly non - dimesionalize the results. For numerical results, 𝑧𝑐 is chosen as maximum melt depth defined as maximum penetration of the melt front from initial flat surface and 𝑙𝑐 is chosen the farthest radial reach of the meltfront. The values for each pulse duration are tabulated in Table 5. For thin film approximation , 𝑧𝑐 is chosen based on theoretical analysis done in [13], wherein the authors present given an analytic al solution of maximum underestimated value , ℎ11 𝑚𝑎𝑥 and overestimated value , ℎ12 𝑚𝑎𝑥 for pulsed laser induced melt depth, based on simple heat conduction model valid for any material/alloy . In the current work, 𝑧𝑐 is chosen as an average of ℎ11 𝑚𝑎𝑥 and ℎ12 𝑚𝑎𝑥. It can be seen from Table 5, that the analytical approximation of Tokarev et al. [13] and the numerical solution sufficiently close. There are no suitable analytical approximations for the c haracteristic length , 𝑙𝑐. Therefore, the full -width -half-maximum of the assumed gaussian beam is chosen as a constant 𝑙𝑐 for all pulse durations . In reality, this is a poor choice as the melt pool size inceases w ith increasing pulse duration . However, addressing this is beyond the scope of the current work. Table 5: Characteristic length values used for non -dimensionalize the results at different pulse durations 𝑡𝑝(𝜇𝑠) Pulse Energy (𝑚𝐽) Length scales used to nondimensionalize numerical solution Length scales used to nondimensionalize thin film approximation results 𝑧𝑐(𝜇𝑚) 𝑙𝑐(𝜇𝑚) 𝑧𝑐(𝜇𝑚) 𝑙𝑐(𝜇𝑚) 0.5 265 2.30 62.10 2.29 62.18 0.8 336 2.88 65.46 2.90 62.18 1.0 380 3.05 67.96 3.24 62.18 1.2 408 3.46 67.49 3.55 62.18 1.5 473 3.94 66.80 3.97 62.18 2.0 564 4.72 70.63 4.58 62.18 3.0 720 5.87 74.43 5.61 62.18 4.0 860 6.95 77.51 6.48 62.18 5.0 975 7.80 77.24 7.24 62.18 Preprint submitted to SME Journal of Manufacturing P rocesses 15 6 Results and discussion Figure 5 shows nine different cases with final interfacial surface profile of simulation height (black color) and thin film approximation height for flat melt fron (red color) and parabolic melt - front (blue color) under various pulse duration values. Figure 5 also shows the percentage error plots for thin film predictions with respect to the numerical results. The error is quantified using two metrics namel, Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE) , for the two melt -front geometries. In general, the solution with parabolic melt front matches c losely with the solution with a flat melt front. However, there are few caveats and are discussed in the following. Observation 1 Figure 5 (a), (b) and (c) compare the thin film surface geometry estimation to surface evolution from the numerical solution for smaller pulse durations ( 𝑡𝑝=0.5,0.8 and 1.0 𝜇𝑠). The RMSE errors are 11%, 7% and 5% respectively with a flat meltfront geometry and 15%, 10% and 8% for a parabolic meltfront. It can be seen that the solution obtained by thin film overestimates the surface height as compared to the numerical results. This is because thin film is a steady state solution while numerical model accounts for transient s of the flows. Further, the thin film does not account for phase change. It assumes that the melt pool is always in liquidus state. On the other hand, phase change is accounted for in the numerical solutions. At smaller pulse durations, it is possible that the melt pool resolidifies before rea ching a steady state. In other words, the flows do not have sufficient time to reach a steady state. Therefore, the steady state thin film solution over - estimates the surface geometry as compared to the transient numerical solution. To support the above di scussed argument, further analysis and investigation is required which is beyond the scope of the current work. Nonethless, the errors in these three cases range from 5% to 15% RMSE value across the two geometries and quite useful in predicting the overall surface features and roughness of the surface . Observation 2 For longer pulse durations, 𝑡𝑝=3,4 and 5 𝜇𝑠, shown in Figure 5 (g), (h) and (i), The RMSE errors are ~17%, ~24% and ~26% respectively with a flat meltfront geometry and ~15%, ~23% and ~25% for a parabolic meltfront. Here, t he thin film solution underesti mates the numerical solution. At these longer pulse durations, dominance of thermocapillary flows result in more upwelling of surface from center towards outwards. In thin film approximation study, only the 𝑂(1) solution was considered , neglecting higher order terms. Further, the rescaled Peclect number, 𝑃𝑒̂ and the rescaled Reynolds number, 𝑅𝑒̂ are assumed to be of 𝑂(1). The values of 𝑃𝑒̂ and 𝑅𝑒̂ are tabulated in Table 6. While the values of 𝑃𝑒̂ are of 𝑂(1), 𝑅𝑒̂ values are of 𝑂(2). At higher pulse durations, deeper melt pools coupled with dominating thermocapillary flows result in significantly higher velocities of flows. However, the velocity scale assumed for thin film does not capture this effect and therefore violates the assumption of 𝑅𝑒̂~𝑂(1). Further, Table 6 also shows the value of parameter 𝜖 increases with pulse duration. It is also assumed that 𝜖≪1 ensures the thin film approximation validity. Therefore, the high value of 𝜖 indicates that, at the high pulse duration Preprint submitted to SME Journal of Manufacturing P rocesses 16 values, thin film cannot approximate the surface height , and therefore significant underestimation is observed. Observation 3 For intermediate pulse durations, 𝑡𝑝=1.2,1.5 and 2 𝜇𝑠 values i.e., from Figure 5 (c), (d), (e), thin film approximation results closely estimat e of final evolved surface height as predicted by numerical results. The RMSE errors are ~5%, ~4% and ~8% respectively with a flat meltfront geometry and ~6%, ~5% and ~7% for a parabolic meltfront. This demonstrates the immediate applicability of thin film solution at these intermediate pulse durations. Figure 6 shows the trend of both errors MAPE and RMSE for flat and parabolic melt front with respect to 𝑡𝑝. Figure confirms that MAPE and RMSE in the range of pulse duration of 1.2 to 2 𝜇𝑠 are mini mum. The maximum RMSE of 26.43% is captured at 5 𝜇𝑠 pulse duration and minimum MAPE 3.53% is captured at 1.2 𝜇𝑠 pulse duration for flat shape thin film from Figure 5. It can be seen from the Figure 6 (a) and (b) that the solution with flat melt front approximation has a smaller error as compared to the parabolic melt front approximation. This is quite intuitive as at small pulse durations, the pools are very shallow, and the melt front is much closer to being flat than parabolic. Figure 6 (c) and (d) are also in agreement with lowest errors at intermediate pulse durations 1.2,1.5 and 2 𝜇𝑠. (a) 𝑡𝑝=0.5 𝜇𝑠 (b) 𝑡𝑝=0.8 𝜇𝑠 Preprint submitted to SME Journal of Manufacturing P rocesses 17 (c) 𝑡𝑝=1 𝜇𝑠 (d) 𝑡𝑝=1.2 𝜇𝑠 (e) 𝑡𝑝=1.5 𝜇𝑠 (f) 𝑡𝑝=2 𝜇𝑠 (g) 𝑡𝑝=3 𝜇𝑠 (h) 𝑡𝑝=4 𝜇𝑠 Preprint submitted to SME Journal of Manufacturing P rocesses 18 (i) 𝑡𝑝=5 𝜇𝑠 Figure 5: Comparisons for simulation and semi -analytical solution at different pulse durations Table 6: Parameter 𝝐, Peclet, Reynolds number from TFA analysis 𝒕𝒑(𝝁𝒔) 𝝐=𝒛𝒄/𝒍𝒄 𝑷𝒆̂ 𝑹𝒆̂ 0.5 0.037 0.716 5.37 0.8 0.046 1.150 8.62 1 0.052 1.434 10.76 1.2 0.057 1.722 12.92 1.5 0.064 2.153 16.15 2 0.074 2.866 21.50 3 0.090 4.301 32.26 4 0.104 5.738 43.04 5 0.116 7.163 53.72 (a) (b) Preprint submitted to SME Journal of Manufacturing P rocesses 19 (c) (d) Figure 6: MAPE and RMSE for parabolic melt -front profile with respect to pulse duration 7 Conclusion and future work This pa per presents a steady state semi -analytical solution to evaluate the evolved surface geometry under pulsed laser surface melting . The solution models the meltpool as a thin film motivated by the observation that typical ratio of meltpool diameters to melt depths , 𝜖 are very small. T he thin film solution also assumes temperature independent material properties. Further, the thin film solution accounts for two geometries of the melt pool shapes, namely a flat lower boundary and a parabolic lower boundary. The paper also discusses a tra nsient axisymmetric numerical solution for pulsed laser surface melting that accounts for temperature dependent properties to study the evolution of surface geometry. To validate the numerical model, the evolved surface heights were compared with results f rom literature and the results closely match each other. Finally, the thin film solution of evolved surface geometry was compared with that of the numerical model . The comparisons were carried for different pulse duration s and pulse energ ies on initially f lat surface surface under similar process conditions. The final evolved surface height from both the models demonstrate a reasonable match with the root mean square errors ranging from ~4% to ~25%. Larger errors are observed at comparatively lower and hig her pulse duration and smaller errors are observed for intermediate pulse duration values. It was observed that at lower pulse duration, the steady state assumption may be violated as the time required for the flows to reach a steady state may be larger th an the time required for resolidification. At larger pulse durations the melt depths become larger, thus resulting in higher values of 𝜖. Further, it was also observed that thin film solution with a flat boundary predicts better at lower pulse duration and that with a parabolic shape is more suitable for higher pulse durations. Overall, the thin film solution is a a reasonable and useful approximation of the of the evolved surface geometry through pLSM process, thus saving significant computational costs. This paper clearly articulates the limitations of thin film approximations for pLSM and thus motivates for further modelling improvement s and considerations of higher orders of 𝜖. Preprint submitted to SME Journal of Manufacturing P rocesses 20 Acknowledgment The authors thank Prof. Uddipta Ghosh, Assistant Professor, IIT Gandhinagar for his valuable inputs and IIT Gandhinagar for support ing this work. Funding This work was supported by the Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India .