# Thermal Modelling and Experimental Validation in the Perspective of Tool Steel Laser Polishing
## 1 Introduction  
The laser polishing (LP) process of metals consists of re-melting a shallow layer of material at the surface of a rough workpiece. In the liquid state, the material will smoothen due to the surface tension and increase mobility in the liquid [1]. Then, the material will re-solidify with a decreased roughness. This present work concentrates on process in the conduction regime without vaporization. Still, it is important to mention that another LP process exists, which includes ablation of the material at much higher laser fluence [2]. LP is an interesting method to polish hard materials such as diamond [3] or tool steel [4] that would require a long time to polish mechanically or which contain difficult to access areas. Another major advantage is that LP is a non-contact process so is subjected to no or very little wear as compared to mechanical polishing using abrasive materials. In recent years, LP has received new interest as a way to improve the surface of workpieces produced by additive manufacturing of Co-Cr [5], Co-Cr-Mo Alloy [6], or 316 L stainless steel [7]. LP in metals is employed in the conduction regime, where a small volume of the material is melted by a laser beam. [PERSON] et al. [8] showed experimentally, and which was confirmed via modelling by [PERSON] et al. [9], that during the melting, the asperities are smoothed by the action of surface tension aided by an increased mobility of the liquid state. Two types of LP process are distinguished for metals depending on the initial roughness of the workpiece. For large sample roughness, continuous wave (CW) laser is employed and produced a larger, deeper continuous melt pool (10-80 \(\upmu\)m deep) and so it isreferred to as macro polishing as defined by [PERSON] et al. [1] and [PERSON] et al. [10]. In contrast, [PERSON] et al. [10] and [PERSON] et al. [11] referred to micro polishing when pulsed lasers are used for samples with small roughness and create for each pulse a much smaller, shallower melt pool (<5 um deep). Under such circumstances, [PERSON] et al. [10] showed that the duration of the liquid state is usually shorter than the repetition rate meaning that no continuous melt pool is present during this process. Despite the relatively long development, LP is still not extensively employed in industry. One of the reasons is that it is very difficult to produce a perfectly smooth surface with only one LP processing step. Indeed, [PERSON] et al. [10] reported that the polishing process itself creates some surface structures. The reason is that, while the surface tension tends to smoothen the liquid, convection inside the liquid or Marangoni flow can lead to the creation of ripples inside the liquid which can result in defects in the re-solidified layer. For this reason, the melted layer is always kept as low as possible. On the other hand, [PERSON] et al. [4] found that the depth of the melt pool must be greater than the peak-to-valley height of the surface. Consequently, in order to polish a very rough surface, several LP steps are needed to progressively decrease the roughness and so the depth of the melted layer [1]. From the beginning of LP, [PERSON] et al. [12] showed that the need for a model was apparent. Indeed, a good numerical model can help with choosing the right process parameters to keep the melted layer thickness as close as possible to the peak-to-valley height of the surface roughness. Actually, the same approach has been successfully applied for ablation of various materials by [PERSON] et al. for stainless steel [13] and silicon [14]. Recent developments in the modelling of LP can be found in the review by [PERSON] et al. [15]. For the present study, the model assumed a weak coupling between the thermal flow and convective flow. As a consequence, the thermal model is solved first and the solution of the thermal model can be employed to solve the flow equations inside the melt pool. The solution of the flow will be presented in Section 2.