A Simple Model to Predict Machined Depth and Surface Profile for Picosecond Laser Surface Texturing

A simple mathematical model was developed to predict the machined depth and surface profile in laser surface texturing of micro-channels using a picosecond laser. Fabrication of micro-craters with pulse trains of different numbers was initially performed. Two baseline values from the created micro-craters were used to calculate the estimated simulation parameters. Thereafter, the depths and profiles with various scanning speeds or adjacent intervals were simulated using the developed model and calculated parameters. Corresponding experiments were conducted to validate the developed mathematical model. An excellent agreement was obtained for the predicted and experimental depths and surface profiles. The machined depth decreased with the increase of scanning speed or adjacent interval.


Introduction
In recent years, surface texturing has been one of the major research topics of interesting due to potential applications in improving surface tribological property [1][2][3], wetting behavior [4][5][6], adhesion performance [7][8][9], reflectivity [10][11][12], and others.For these applications, the dimensions of the created micro-features play a vital role in tailoring of final surface functional performances.Over the last decade or so, laser surface texturing (LST) is a commonly utilized technology to fabricate designed micro-features on surfaces owing to the tool-less nature, high precision, high flexibility and ease of use [13].During LST, the final geometry of the created micro-features is determined by plenty of process parameters, like laser power, frequency, wavelength, pulse diameter and overlap, etc.Hence, the laser process parameters should be rationally chosen to create designed micro-features.Developing predictive models for the geometry is an efficient means to choose optimal laser process parameters.
A considerable amount of literature has been published on constructing the predictive models for the machined depths or geometries during laser machining process.For example, Kim [14,15] employed boundary element method to develop two numerical models which can be used to predict groove shape during laser cutting process.Wang et al. [16] built a numerical model that considers the instant material removal to predict the surface recession and to obtain the ablation depth.Vora et al. [17,18] developed multiphysics-multistep computational models combining the heat transfer, hydrodynamic boundary conditions and thermomechanical properties to simulate the temperature and surface profile.For ultrafast laser micromachining, the twotemperature model has been successfully employed to simulate the evolution of the temperatures and to predict the final geometry [19][20][21][22][23].However, numerical modeling is computationally expensive, and has some difficulty when materials used do not have extensive property data readily available.Based on abundant experimental data, an artificial neural networks (ANN) model was developed by Ciurana et al. [24] to predict the T-shaped deep features in laser machining and to identify optimum process settings.Yousel et al. [25] built a multi-layered neural network to predict the machined depth and diameter of the groove during laser cutting process.Xing et al. [26] used response surface method (RSM) to develop a mathematical model to calculate the width, depth and surface roughness during the fabrication of micro-channels on polycrystalline diamond by nanosecond laser.Wang et al. [27] developed a second-order regression and a least squares support vector machines (LS-SVM) model to predict the machined depth and profile of micro-channels on medical needles using a picosecond laser.A lot of experiments need to be conducted to improve the prediction accuracy of these experiment-based models, however, so this process is time consuming.An analytical model combined with several experiments is also a promising method to obtain the proper results.For example, a similar model was built by Liang et al. [28] to predict the width of created micro-channels and to calculate the ablation threshold.Gilbert et al. [29] constructed a mathematical model to obtain the ablated depth and the texture of targeted surfaces by considering the surface gradient and focal length as well as dynamic and energetic parameters.
In this model, baseline experiments were initially conducted to calibrate the material response to laser energetic parameters.Micro-channels are the fundamental building elements in LST and can be arrayed to create patterns with different geometries.In contrast with the previous models characterised by complex multi-physics and/or abundant experimental data, a simple mathematical model is developed in this paper to predict the machined depth and profile of micro-channels by a picosecond laser.The only necessary step for rapid application of the model is to carry out two baseline experiments to calculate the estimated simulation parameters.Hence, fabrication of micro-craters with pulse trains of two different numbers is initially performed to provide baseline values.The estimated simulation parameters are calculated using the obtained baseline values.Subsequently, the depths and profiles of created micro-channels with various scanning speeds or adjacent intervals are simulated using the developed model and calculated parameters.Corresponding experiments have been conducted to validate the developed mathematical model.

Modeling of Laser Ablation in a Single Location
The energy distribution of a Gaussian beam can be approximately expressed as [30]: where I is the laser fluence at point (x, y), I max is the maximum fluence and σ is the beam spreading parameter (proportional to the beam diameter).The laser ablation occurs once the local fluence exceeds the threshold value of the material removal.Hence, the depth at any point of the laser irradiated zone for a single pulse can be estimated using the Gaussian function.
where D is the machined depth at any point, D max is the maximum depth of a single pulse, (x 0 , y 0 ) represents the laser pulse center, w 0 is the beam diameter, and k is a scaling factor for adjusting the practical profile of laser irradiated zone to the Gaussian shape.The total ablation depth in a single location after multi-pulses irradiation can be considered as the accumulation of multiple independent pulses.Equation ( 2) is rewritten as: where D max (N) is the accumulated depth for N pulses.For given laser process parameters, the D max (N) and k are both estimated simulation parameters, which are determined according to experimental data.

Modeling of LST Process
A rectangular raster trajectory was used to fabricate micro-channels, as shown in Figure 1.A single-channel is initially created with a certain scanning speed and laser frequency.Subsequently, single-channels are overlapped by shifting a small interval to form a multi-pass micro-channel.This pattern is overscanned layer by layer to create the final micro-channel with desired depth.
Appl.Sci.2018, 8, x FOR PEER REVIEW 3 of 10 where Dmax(N) is the accumulated depth for N pulses.For given laser process parameters, the Dmax(N) and k are both estimated simulation parameters, which are determined according to experimental data.

Modeling of LST Process
A rectangular raster trajectory was used to fabricate micro-channels, as shown in Figure 1.A single-channel is initially created with a certain scanning speed and laser frequency.Subsequently, single-channels are overlapped by shifting a small interval to form a multi-pass micro-channel.This pattern is overscanned layer by layer to create the final micro-channel with desired depth.During LST process, the scanning speed and frequency determine the laser pulse center along the longitudinal direction, while the adjacent interval determines the laser pulse center along the transverse direction.In order to predict the machined depth of LST, a fine mesh is developed with dimensions slightly larger than the length and width of the created micro-channel.The depth at each mesh point is calculated using a summation of all the depths created by each individual pulse at this location, expressed as: where m and n are the number of increments in the longitudinal and transverse directions at point (x, y) and (xi, yi) is the laser pulse center determined by the scanning speed, frequency and adjacent interval.

Experiments for Laser Modeling
AISI 304 stainless steel sample (Shengjili Shenzhen, China) with a size of 25 mm × 20 mm × 0.5 mm and with an as-received surface roughness of ~10 nm was used in this study.The sample was irradiated by a commercial Nd: YVO4 laser (PX50, Edgewave, Germany) with wavelength of 532 nm and pulse duration of 10 ps.A 50 mm focusing lens was used to focus the laser beam, and a diameter of ~16 μm with Gaussian profile was obtained at the focal plane, which was measured using Liu's method, as described in [31].An x-y-z motorized stage (PS-30, Borui, China) with a movement precision 500 nm was employed to move the sample in the x-y plane.During LST process, the scanning speed and frequency determine the laser pulse center along the longitudinal direction, while the adjacent interval determines the laser pulse center along the transverse direction.In order to predict the machined depth of LST, a fine mesh is developed with dimensions slightly larger than the length and width of the created micro-channel.The depth at each mesh point is calculated using a summation of all the depths created by each individual pulse at this location, expressed as: where m and n are the number of increments in the longitudinal and transverse directions at point (x, y) and (x i , y i ) is the laser pulse center determined by the scanning speed, frequency and adjacent interval.

Experiments for Laser Modeling
AISI 304 stainless steel sample (Shengjili Shenzhen, China) with a size of 25 mm × 20 mm × 0.5 mm and with an as-received surface roughness of ~10 nm was used in this study.The sample was irradiated by a commercial Nd: YVO4 laser (PX50, Edgewave, Germany) with wavelength of 532 nm and pulse duration of 10 ps.A 50 mm focusing lens was used to focus the laser beam, and a diameter of ~16 µm with Gaussian profile was obtained at the focal plane, which was measured using Liu's method, as described in [31].An x-y-z motorized stage (PS-30, Borui, China) with a movement precision 500 nm was employed to move the sample in the x-y plane.
The LST experiments were performed in air at room temperature.The pulse frequency was fixed at 1 kHz to reduce the heat accumulation, while the pulse energy was fixed at 8 µJ.Pulse trains with numbers of 2, 5, 10, 15 and 20 were firstly fired to create micro-craters on the sample.Then, micro-channels with a width of 50 µm and a length of 300 µm were selected to cretate on the sample.The scanning speed ranging from 1 mm/s to 5 mm/s, and the adjacent interval ranging from 2 µm to 5 µm, were employed to create micro-channels.The raster trajectory was overscanned 8 times to achieve a larger depth.After LST, the sample was cleaned in an ultrasonic bath by alcohol and distilled water to remove the debris.A confocal laser scanning microscope (OLS-400, Olympus, Japan) was used to observe the profiles of the micro-craters and micro-channels.The averaged depth of the micro-craters and micro-channels were obtained after performing measurements at five different micro-craters or locations.

Laser Ablation in a Single Location
Figure 2 shows the relationship between the ablation depth of the micro-craters and the pulse number.As the figure indicates, the mean depth increases almost linearly from 0.13 µm to 1.24 µm with the pulse number increasing from 2 to 20.The deviation of the measured depth is in the range of 15%, representing the high stability during picosecond LST process.It is important to note that the total ablation depth after multi-pulses irradiation is almost the accumulation of multiple independent pulses.This is consistent with prior estimation done in modeling of LST.In this study, the depths for 2 and 20 pulses were used as baseline values to determine the value of D max (N) from Equations ( 3) and ( 4) by linear fitting, and a slope of 0.062 and an intercept of 0.007 were obtained.The LST experiments were performed in air at room temperature.The pulse frequency was fixed at 1 kHz to reduce the heat accumulation, while the pulse energy was fixed at 8 μJ.Pulse trains with numbers of 2, 5, 10, 15 and 20 were firstly fired to create micro-craters on the sample.Then, microchannels with a width of 50 μm and a length of 300 μm were selected to cretate on the sample.The scanning speed ranging from 1 mm/s to 5 mm/s, and the adjacent interval ranging from 2 μm to 5 μm, were employed to create micro-channels.The raster trajectory was overscanned 8 times to achieve a larger depth.After LST, the sample was cleaned in an ultrasonic bath by alcohol and distilled water to remove the debris.A confocal laser scanning microscope (OLS-400, Olympus, Japan) was used to observe the profiles of the micro-craters and micro-channels.The averaged depth of the micro-craters and micro-channels were obtained after performing measurements at five different micro-craters or locations.

Laser Ablation in a Single Location
Figure 2 shows the relationship between the ablation depth of the micro-craters and the pulse number.As the figure indicates, the mean depth increases almost linearly from 0.13 μm to 1.24 μm with the pulse number increasing from 2 to 20.The deviation of the measured depth is in the range of 15%, representing the high stability during picosecond LST process.It is important to note that the total ablation depth after multi-pulses irradiation is almost the accumulation of multiple independent pulses.This is consistent with prior estimation done in modeling of LST.In this study, the depths for 2 and 20 pulses were used as baseline values to determine the value of Dmax(N) from Equations ( 3) and ( 4) by linear fitting, and a slope of 0.062 and an intercept of 0.007 were obtained.Using the experimental profile from 20 pulses, the scale factor k was estimated via Gaussian fitting, and a value of 1.5 was obtained.Based on the calculated Dmax(N) and k, the predicted results using Equation (3) compared with the experimental data, as illustrated in Figure 3.The length and width of the computational domain were both chosen as 20 μm, and the mesh sizes in both directions were set as 50 nm.The simulation was performed via Matlab.The experimentally measured surface topology and the predicted 3D micro-crater are shown on the left and right, respectively, while the comparison between the predicted and experimental cross-sectional profiles are plotted on the center.As shown, the predicted profile agrees very well with the experimental data.Using the experimental profile from 20 pulses, the scale factor k was estimated via Gaussian fitting, and a value of 1.5 was obtained.Based on the calculated D max (N) and k, the predicted results using Equation (3) compared with the experimental data, as illustrated in Figure 3.The length and width of the computational domain were both chosen as 20 µm, and the mesh sizes in both directions were set as 50 nm.The simulation was performed via Matlab.The experimentally measured surface topology and the predicted 3D micro-crater are shown on the left and right, respectively, while the comparison between the predicted and experimental cross-sectional profiles are plotted on the center.As shown, the predicted profile agrees very well with the experimental data.Furthermore, the depths of 2, 5, 10 and 15 pulses were validated using the developed model.The calculated k of 1.5 was used, while the Dmax(N) was calculated according to the above linear fitting and the number of pulses, N. The comparison between the predicted and the experimental results is demonstrated in Figure 4.As it can be seen, the model provides a very good prediction of the ablation depth for a varying number of pulses in a single location.The ablation depth increases quickly with Furthermore, the depths of 2, 5, 10 and 15 pulses were validated using the developed model.The calculated k of 1.5 was used, while the D max (N) was calculated according to the above linear fitting and the number of pulses, N. The comparison between the predicted and the experimental results is demonstrated in Figure 4.As it can be seen, the model provides a very good prediction of the ablation depth for a varying number of pulses in a single location.The ablation depth increases quickly with the increasing pulses.Furthermore, the depths of 2, 5, 10 and 15 pulses were validated using the developed model.The calculated k of 1.5 was used, while the Dmax(N) was calculated according to the above linear fitting and the number of pulses, N. The comparison between the predicted and the experimental results is demonstrated in Figure 4.As it can be seen, the model provides a very good prediction of the ablation depth for a varying number of pulses in a single location.The ablation depth increases quickly with the increasing pulses.

LST of Micro-Channels via Changing Scanning Speed
In order to reduce the calculation time, the computational length of the micro-channel was chosen as 110 µm, while the width was set as 60 µm.The mesh sizes in both directions were set as 50 nm.The effect of scanning speed on the fabrication of the micro-channel was investigated using Equation (4) by changing scanning speed from 1 mm/s to 5 mm/s with a fixed adjacent interval of 3 µm.For a fixed pulse energy of 8 µJ and an overscan number of 8, D max (N) utilized was 0.50 µm which was determined by linear fitting based on the ablation depths from 2 and 20 pulses.The scale factor k was equal to 1.5, which was obtained on the above simulation of micro-craters.The predicted depths were compared to the actual experimental results, as shown in Figure 5.As can be observed, excellent agreement is obtained for the predicted depths from 1 mm/s to 5 mm/s, and the error is negligible.A low scanning speed of 1 mm/s leads to higher overlapping of consecutive craters in longitudinal direction, resulting in a greater mean depth up to 11.1 µm.Then, an increase in scanning speed from 1mm/s to 5 mm/s significantly decreases the longitudinal overlap, leading to the decrease of the machined depth of the micro-channel.
were compared to the actual experimental results, as shown in Figure 5.As can be observed, excellent agreement is obtained for the predicted depths from 1 mm/s to 5 mm/s, and the error is negligible.A low scanning speed of 1 mm/s leads to higher overlapping of consecutive craters in longitudinal direction, resulting in a greater mean depth up to 11.1 μm.Then, an increase in scanning speed from 1mm/s to 5 mm/s significantly decreases the longitudinal overlap, leading to the decrease of the machined depth of the micro-channel.To show the advantage of the developed model, not only the machined depths of the created micro-channels but also the profiles were compared.The computationally predicted and experimental results of micro-channels with three different scanning speeds are presented in Figure 6.The experimentally measured surface topologies are shown in Figure 6a, while the predicted 3D channels are presented in Figure 6c.The comparison between the experimental and simulated crosssectional profiles are plotted in Figure 6b.As the figure indicates, the machined depth of the microchannel decreases with the increasing scanning speed, while the width keeps almost unchanged at around 50 μm.The profile made with lower scanning speed has a shaper transition on the edges and has a higher aspect ratio.Moreover, the experimental and simulated profiles are in very close agreement for the scanning speed of 3 mm/s and 5 mm/s.However, for scanning speed of 1 mm/s there is a little discrepancy, which is likely due to the material redeposition resulting from heat accumulation at a high overlap.The random position of resolidification leads to the asymmetric profile at the bottom of the created micro-channel, as shown in the upper panel of Figure 6a,b.To show the advantage of the developed model, not only the machined depths of the created micro-channels but also the profiles were compared.The computationally predicted and experimental results of micro-channels with three different scanning speeds are presented in Figure 6.The experimentally measured surface topologies are shown in Figure 6a, while the predicted 3D channels are presented in Figure 6c.The comparison between the experimental and simulated cross-sectional profiles are plotted in Figure 6b.As the figure indicates, the machined depth of the micro-channel decreases with the increasing scanning speed, while the width keeps almost unchanged at around 50 µm.The profile made with lower scanning speed has a shaper transition on the edges and has a higher aspect ratio.Moreover, the experimental and simulated profiles are in very close agreement for the scanning speed of 3 mm/s and 5 mm/s.However, for scanning speed of 1 mm/s there is a little discrepancy, which is likely due to the material redeposition resulting from heat accumulation at a high overlap.The random position of resolidification leads to the asymmetric profile at the bottom of the created micro-channel, as shown in the upper panel of Figure 6a

LST of Micro-Channels via Changing Adjacent Interval
The influence of adjacent interval on the fabrication of the micro-channel was also investigated using Equation ( 4) by changing the interval from 2 μm to 5 μm with a constant scanning speed of 3 mm/s.The other parameters used in simulation were the same as those used in the previous section.Figure 7 shows the predicted and experimental depths of the micro-channels with various adjacent

LST of Micro-Channels via Changing Adjacent Interval
The influence of adjacent interval on the fabrication of the micro-channel was also investigated using Equation ( 4) by changing the interval from 2 µm to 5 µm with a constant scanning speed of 3 mm/s.The other parameters used in simulation were the same as those used in the previous section.Figure 7 shows the predicted and experimental depths of the micro-channels with various adjacent intervals.As can be seen, the machined depth decreases with an increase of the interval from 2 µm to 5 µm.An increase in the adjacent interval leads to a decrease of transverse overlapping, resulting in a decrease of the machined depth, which is consistent with previous research [32].Similar trends are observed for experimental and simulated results, and the differences between them are in the range of 7%.

LST of Micro-Channels via Changing Adjacent Interval
The influence of adjacent interval on the fabrication of the micro-channel was also investigated using Equation ( 4) by changing the interval from 2 μm to 5 μm with a constant scanning speed of 3 mm/s.The other parameters used in simulation were the same as those used in the previous section.Figure 7 shows the predicted and experimental depths of the micro-channels with various adjacent intervals.As can be seen, the machined depth decreases with an increase of the interval from 2 μm to 5 μm.An increase in the adjacent interval leads to a decrease of transverse overlapping, resulting in a decrease of the machined depth, which is consistent with previous research [32].Similar trends are observed for experimental and simulated results, and the differences between them are in the range of 7%.Two simulated and experimental micro-channels with adjacent intervals of 2 µm and 4 µm are also presented in Figure 8.As can be observed, the machined depth of the micro-channel decreases with an increase of adjacent interval, while the width remains nearly constant at around 50 µm.The profile made with lower interval has a higher aspect ratio.Furthermore, the simulation provides a very good prediction of the profiles for the experimental results.Two simulated and experimental micro-channels with adjacent intervals of 2 μm and 4 μm are also presented in Figure 8.As can be observed, the machined depth of the micro-channel decreases with an increase of adjacent interval, while the width remains nearly constant at around 50 μm.The profile made with lower interval has a higher aspect ratio.Furthermore, the simulation provides a very good prediction of the profiles for the experimental results.

Conclusions
In this paper, a mathematical model was developed to predict the machined depth and surface profile for picosecond LST.Experiments of pulse trains with different pulse numbers were initially conducted to provide baseline values and to calculate the estimated simulation parameters.The

Figure 1 .
Figure 1.Schematics for the formation of micro-channel.

Figure 1 .
Figure 1.Schematics for the formation of micro-channel.

Figure 2 .
Figure 2. Effect of pulse number on machined depth of micro-craters.

Figure 2 .
Figure 2. Effect of pulse number on machined depth of micro-craters.

10 Figure 3 .
Figure 3.The comparison between experimental and predicted micro-crater for 20 pulses.

Figure 3 .
Figure 3.The comparison between experimental and predicted micro-crater for 20 pulses.

Figure 3 .
Figure 3.The comparison between experimental and predicted micro-crater for 20 pulses.

Figure 4 .
Figure 4. Comparison between experimental and predicted micro-craters for different pulses: (a) measured micro-craters, (b) comparison of cross-sectional profiles, and (c) predicted micro-craters.

Figure 4 .
Figure 4. Comparison between experimental and predicted micro-craters for different pulses: (a) measured micro-craters, (b) comparison of cross-sectional profiles, and (c) predicted micro-craters.

Figure 5 .
Figure 5.Effect of scanning speed on machined depth of micro-channels.

Figure 5 .
Figure 5.Effect of scanning speed on machined depth of micro-channels. ,b.

10 Figure 6 .
Figure 6.The comparison between experimental and predicted micro-channels for different scanning speeds: (a) measured micro-channels, (b) comparison of cross-sectional profiles, and (c) predicted micro-channels.

Figure 6 .
Figure 6.The comparison between experimental and predicted micro-channels for different scanning speeds: (a) measured micro-channels, (b) comparison of cross-sectional profiles, and (c) predicted micro-channels.

Figure 6 .
Figure 6.The comparison between experimental and predicted micro-channels for different scanning speeds: (a) measured micro-channels, (b) comparison of cross-sectional profiles, and (c) predicted micro-channels.

Figure 7 .
Figure 7. Effect of adjacent interval on machined depth of micro-channels.Figure 7. Effect of adjacent interval on machined depth of micro-channels.

Figure 7 .
Figure 7. Effect of adjacent interval on machined depth of micro-channels.Figure 7. Effect of adjacent interval on machined depth of micro-channels.

Figure 8 .
Figure 8.The comparison between experimental and predicted micro-channels for different adjacent intervals: (a) measured micro-channels, (b) comparison of cross-sectional profiles, and (c) predicted micro-channels.

Figure 8 .
Figure 8.The comparison between experimental and predicted micro-channels for different adjacent intervals: (a) measured micro-channels, (b) comparison of cross-sectional profiles, and (c) predicted micro-channels.