Multi-Cycle Process Signature of Laser-Induced Thermochemical Polishing

Abstract

Laser-induced thermochemical polishing (LCP) is a non-conventional processing technique that uses laser radiation to smooth the surface of self-passivated metallic parts by initiating a localized anodic material dissolution. This technology can be used to selectively micro-polish without the need for masking or thermal and mechanical stress. However, there is still a lack in understanding the surface quality depending on the applied laser machining parameters. This paper takes up the concept of Process Signatures and interprets the surface smoothing as result of multiple, recurring internal material loads of a constant energy amount. The laser-induced thermal impact is identified as the relevant internal material load and is correlated with the surface roughness. This derives an empirical-based functional relation as multi-cycle Process Signature. The experiment results show an exponential decay in surface roughness with increasing cycle loads for titanium, Ti6Al4V, Nitinol, Stellite 21, and metallic glass. The Process Signature of LCP is a solution to a differential equation with respect to the cycle loads. The paper demonstrates how the multi-cycle Process Signature helps determine suitable machining parameters to predict the surface roughness, as well as to scale the polishing rate.

1. Introduction

Surface finishing processes of metallic parts like grinding, manual polishing, and electropolishing are used to control the components’ surface integrity to improve their functional performance [1]. Among these are corrosion resistance, fatigue strength, and biocompatibility, as well as optical, tribological, and haptic properties [2]. Machining parameters for a given surface finish are often experience-based. Frequently, extensive iteration steps are necessary in order to generate a desired surface finish. This prompts development of correlations that describe material modifications by means of internal material loads [3] generalized of the explicit machining parameters and manufacturing processes [4]. Such a knowledge-based description of a manufacturing process is called a Process Signature (PS) [5].

Conventional polishing processes face a major challenge for filigree micro- and mesoscale metallic parts or for complex geometries [6] and multi-material composition [7] of additive manufactured parts [8]. Polishing such parts requires a minimum thermal and mechanical energy impact and a high grade of automatization [9]. This drives the need for new methods, e.g., laser macro [10,11] and micro [12,13] melting polishing, plasma electrolytic polishing [14], and electron beam irradiation [15], by which the surface can be polished gently and selectively.

Laser-induced thermochemical polishing (LCP) is a direct and selective material removal process of self-passivating metals (e.g., titanium and cobalt-based alloys) in an acidic electrolyte environment. It can effectively reduce the surface roughness by over 95%. Therefore, the surface is locally heated by focused laser radiation which shifts the corrosion potential [16]. According to the thermo-battery model [17,18], a local electric cell is generated, resulting in active anodic material dissolution of the irradiated surface. During material dissolution, the surface is smoothed over time [19]. Elevated surface parts like roughness peaks are dissolved faster compared to valleys. So far, the explicit relation between machining parameters and surface quality, as well as smoothing kinetics, are unknown. Understanding and controlling the surface finish is a major challenge in further development for potential industrial applications.

A novel multi-cycle PS approach for LCP is presented in this paper by first identifying the relevant internal material loads, and second, describing the surface roughness by an exponential decay. An analysis of the multi-cycle PS is conducted, focusing on the scaling and adjustment of the polishing rate, as well as the transfer of this method to other self-passivated materials.

2. Objective and Hypothesis

The objective of the present work is to derive an empirical-based functional relation between surface roughness and internal material loads during LCP. This correlation is interpreted as a multi-cycle Process Signature. For this purpose, the following research hypotheses are tested:

• The influence of various machining parameters on the material removal can be generalized by an internal thermal load.
• The surface roughness can be described by a functional correlation of the thermal load (multi-cycle Process Signature).
• The multi-cycle Process Signature is applicable to other self-passivating materials.

3. Materials and Methods

3.1. Experimental Setup

For the experimental investigation, the laser setup shown in Figure 1a was used. It consists of a beam source, optical components, a two-dimensional (2D) laser scanner, and a closed wet etching chamber where the specimens are mounted. The single mode fiber laser (JK400FL) emits continuous wave (cw) laser radiation of λ = 1080 nm wavelength in TEM₀₀ mode with a beam quality factor of M² = 1.3 and laser power between P_L = 400 to 15 W. The emitted intensity distribution I(r) is almost Gaussian-shaped, and the focus diameter d_f is defined by I(d_f/2) = I₀/e² and laser power by P_L = 8πd_f²I₀. The laser power is reduced between 0.1 and 15 W by tilting the angle of a transmission variable edge filter. To adjust the laser power, the laser radiation is partly reflected to a powermeter (Coherent PM150-50C, Santa Clara, CA, USA). The laser radiation can be reduced to a third in diameter by an inverse mounted telescope (Sill Optics S6EXZ5310, Wendelstein, Germany). This allows expansion of the beam diameter incrementally from d_f = 31 μm to 156 μm. The laser radiation is focused by a telecentric f-theta optic (Sill Optics S4LFT3162, Wendelstein, Germany) with a focus length of f = 163.5 mm and guided along the specimen surface using the 2D high-speed scanner system (Raylase Superscan III-15, Wessling, Germany) with a velocity between v = 2 mm/s to 10 m/s.

The specimens were machined in a closed wet etching chamber, which is flown by a 5 molar (28.7 vol. %) phosphoric acid (H₃PO₄). The electrolyte was pumped with 1 L/min through an interim tank to the etching chamber to minimize pulsation from the peristaltic pump. The electrolyte flowed with v_f = 2 m/s as a cross-jet over the specimen surface in a 25 mm × 2 mm flat channel. This ensured faster evacuation of emerging process gases from the focus area. This experimental setup enabled investigation of the influence of a wide range of laser powers P_L, focus diameters d_f, line distances b, scan velocities v, and multiple passes N on the surface quality, i.e., surface roughness and isotropy as illustrated in Figure 1b.

3.2 Materials and Surface Preparation

In this study, titanium, titanium alloys, cobalt–chrome-based alloys, and metallic glass were investigated. An overview of materials, their element composition, material properties, and labeling is shown in

Table 1. Overview of the conducted experiments and their machining settings.

Label	Grade 1 3.7024	Grade 2 3.7035	Ti6Al4V 3.7165	Nitinol	Metallic Glass	Stellite 21
Elements (%)	Ti: 99.5	Ti: 99.3	Ti: 89.4 Al: 6.1 V: 4.0	Ni: 55.8 Ti: 44.2	Zr: 62.5 Cu: 31.0 Al: 3.3 Ni: 3.2	Co: 62.3 Cr: 27.0 Mo: 5.5 Ni:2.8
Density ρ (kg/m3)	4510	4510	4430	6500	6900	8360
Therm. cond. K (W/mK)	25	25	7	10	13	15
Heat capacity cp (J/K)	523	523	560	320	419	404
Absorp. α (%)	39	39	37	30	26	27
Absorp. αA.b. (%)	68	68	72	61	51	46

and taken from [20,21,22,23]. Prior to machining, the materials were cut in 20 × 20 mm specimens and hot mounted in an epoxy thermosetting polymer. Then, they were cleaned with ethanol. The titanium (Grade 1) specimens were pre-machined with abrasive blasting (BHS, SKK 4i) and turning to investigate the influence of the initial topography on the polishing. The as-delivered rolled surface was abrasive blasted with a garnet mineral (grain size 70 μm, 5 bar) for 1 s, 3 s, and 6 s and turned with two machining settings to produce distinct turning grooves. The pre-machining of the specimens by 6 s abrasive blasting induced initial roughness between 2.5 μm and 3.4 μm, depending on the hardness of the material. After the surface treatment, the absorption coefficient α of the materials was measured with an integrating sphere at a wavelength of 808 nm as performed by Kügler [24]. The absorption coefficient of polished surfaces can differ up to 30% from that of abrasive blasted surfaces.

3.3. Methodology and Characterization

Two types of experiments were carried out, structuring individual line cavities and area polishing. Since typical laser-induced thermochemical material removal rates are about some micrometers per second, after one pass with velocities of millimeters per second, material removal takes place at the nanometer scale. For this reason, the LCP was performed as a multi-cycle process with up to 100 s of passes.

In the first step, individual 0.8-mm-long lines were structured on rolled titanium (Grade 1) specimens using various laser powers, focus diameters, scan velocities, and passes. An overview of the parameters is given in

Table 2. Overview of the conducted experiments and their machining settings.

	First Step: Line Structuring	Second Step: Area Polishing	Third Step: Material Influence
Material	Ti-Grade 1	Ti-Grade 1	(Table 1)
Initial surface	Rolled	Rolled, Abrasive bla.: 1, 3, 6 s Turned: 4, 6 mm/min	Abrasive bla.: 6 s
PL (W)	0.5, …, 7.8	2, …, 3	1, …, 3
df (μm)	31, 68, 110, 156	110	110
N (1)	1, …, 600	1, …, 400	1, …, 400
v (mm/s)	2, 4, 8, 16	2	2
B (μm)	-	35	35

(first step). The cavities were topologically characterized with a confocal laser scanning microscope. 2D height profiles perpendicularly oriented to the laser scanning direction were randomly chosen. The depth of the cavity center was measured in a 2D height profile of the cavity and averaged.

In the second step, the influence of the machining parameters on the surface finish was investigated. On the pre-machined specimens, areas of 500 μm × 500 μm were polished for various laser powers and number of passes. A square zig-zag laser beam trajectory was used for area polishing with a scan velocity of 2 mm/s. The laser spot diameter was 110 μm with a line distance of 36 μm, resulting in a 66% line overlap. A list of the parameters is shown in Table 2 (second step). All matrix fields were topographically measured using the laser scanning microscope. An automated measurement was used to take one image in the center of each matrix field. Each surface topography was analyzed by the Sa value according to ISO 25178.

In the third step, areas of 500 μm × 500 μm were polished on the materials listed in Table 1 for varied laser powers and number of passes. All other machining parameters were kept constant, as listed in Table 2 (third step). The material specimens were pre-machined by 6 s abrasive blasting. Again, the surface finish was characterized by the roughness Sa.

After laser thermochemical machining, the specimens were cleaned for 5 min in an ultrasonic ethanol bath before analyzing the topography using the laser scanning microscope (LSM, Keyence VK-9710, Osaka, Japan). The topography was measured using a 50×-objective (1 px = 0.139 μm) covering a surface area of 285 μm × 215 μm. Measurement of the roughness and height was performed with the VK-Analyzer software (Keyence).

3.4. Definition of Loading

In order to describe roughness as a function of machining parameters, it is useful to define the “exposure time t_r” and the “processing time t_A”. The exposure time describes the average time each surface element is illuminated by the laser radiation after multiple passes, and is defined as:

$$t_r=t_N·N=\frac{\pi }{4}\frac{d_f^2}{b·v}·N$$

(1)

with the exposure time for one pass t_N (N = 1), the line distance b, scan velocity v, and passes N. Exposure time is related to processing time in the same ratio as the focus area to the polished area. The processing time is defined as:

$$t_A=\frac{A}{b·v}·N=\frac{A}{\frac{\pi }{4}{d}_f^2}·t_r$$

(2)

with the polished area A, the line distance b, scan velocity v, and passes N. As well as the geometrical machining parameters, the induced temperature distribution has a major impact on the modification. In the following, the “thermal load T_L” is defined as averaged temperature with respect to the focal diameter. Therefore, the temperature distribution T(x,y,z,t) induced by the Gaussian intensity distribution I(r) is calculated by solving the heat equation. Since the laser focus is small compared to the specimen, a semi-infinite substrate with constant material parameters and a constant surface heat source (defined by the absorbed intensity) is implied. A general solution of the boundary value problem is obtained analytically by the Green function technique [25]. This leads to a complex integral expression which is evaluated based on a Fast Fourier Transformation approach [26]. The temperature distribution is shown in Figure 2a. This is used to derive in first approximation an analytical value for the center temperature rise T_p (stationary). The thermal load T_L is defined as the temperature decrease to 1/e according to:

$$T_L=\frac{T_p}{e}=\frac{2·P_L·\alpha }{e\sqrt{2\pi }·K·d_f}$$

(3)

with the thermal conductivity Κ, heat capacity c_p, and the absorption coefficient α. The thermal load T_L for the materials is shown in Figure 2b and varies by 4-fold. In Ti6Al4V, the thermal load is significantly higher due to lower thermal conductivity K.

4. Results

4.1. Line Structuring

Figure 3 shows the resulting cavity depth over the laser power after multiple passes for focus diameters of 31, 68, 110, and 156 μm. The material removal starts at a characteristic threshold power P_L,th and increases linearly in depth until disturbances in the cavity center occur at a laser power P_L,dist. In (1) and (2), exemplary microscope images of the undisturbed and disturbed line cavities in the top and cross-section view are shown. In the disturbed cavity, piled-up accumulations appear irregularly in the center. A white layer is visible in the cross-section images and consists of titanium phosphate [27]. The undisturbed cavity shows a flat smooth surface. For larger focus diameters, the same is observed in wider cavities.

Three categories of multi-pass line cavities are identified:

• No material modification (0 W to P_L,th)
• Undisturbed cavities (P_L,th to P_L,dist)
• Disturbed material removal (>P_L,dist)

The thresholds depend on the focus diameters and are determined by the cross-section of a linear fit with the x-axis. For P_L,th results: 0.66, 1.32, 2.22, and 3.21 W. Disturbances occur at laser powers P_L,dist: 1.25, 2.82, 4.46, and 6.23 W. The gradient of the removal depths and the cavity aspect ratio decreases steadily with larger focus diameters. Only the undisturbed cavities between P_L,th and P_L,dist, without any material deposition, are suitable for polishing.

Figure 4 shows removal depths depending on the number of passes for different scan velocities. Since the undisturbed and disturbed material removal is defined by exceeding a specific power, their occurrence is independent of the scan velocity and number of passes, as shown (1) and (2). The microscopy top images and cross-sections of the line cavity show disturbances. The removal depth increases in a linear way with the passes. Its gradient decreases as scan velocity increases. This means that for a higher scan velocity, the surface must be scanned with a significantly higher frequency to achieve the same removal depth. This shows that the removal depth of the multiple passes is the result N-fold superpositions of small nanometer-scaled material removal. Increasing the scanning velocity leads to a lower gradient, i.e., less material is removed with each pass.

4.2. Area Polishing

Figure 5 shows microscopic images of the pre-machined surfaces in both the initial and the final state (after 300 passes), as well as the surface roughness and absorptivity. The initial absorptivity α_ini on titanium varies between 0.42 and 0.68 depending on the initial surface topography. Independent of the pre-machined surface topography, the final roughness is about 0.2–0.3 μm, except for the 4 mm/s turned surface. There is tendency for higher absorptivity with higher roughness.

The systematic analysis of roughness for the pre-machined rolled and abrasive blasted specimens depends on the number of passes, as shown in Figure 6. The laser power is kept constant at 2.6 W during LCP. Depending on the initial roughness of 3.2 μm (dark red points), 2.2 μm (red points), 1.4 μm (pink points), and 0.6 μm (orange points), it takes 150, 100, 50, and 25 passes to polish to the minimum Sa_min ≈ 0.2 μm, respectively. No further significant surface improvement can be observed with processing up to 300 passes. The roughness follows an exponential fit (continuous lines) according to Equation (4):

$$Sa=a·e^{-c·N}+b$$

(4)

The obtained fit parameters a, b, and c are summarized in Figure 6. The result shows that higher initial roughness decreases more rapidly.

Figure 7 shows the influence of laser power on the decrease in roughness. The roughness decreases faster for higher laser powers of 2.59 W (red points) compared to 2.28 W (orange points) or 2.19 W (grey points) after 100, 200, and 300 passes to a minimum of 0.3 μm. Again, the experimental data follow an exponential fit (continuous lines) according to Equation (4). The exponent c of the fit parameter only changes depending on the laser power.

Furthermore, the roughness can have similar absolute value but of varying topography origin, as comparison of the 3 s abrasive blasted (red curve) surface to the 4 mm/min (dark blue curve) and 6 mm/min (light blue curve) turned surfaces shows. Figure 8 shows the roughness and the exponential fit (continuous line) of these surfaces with increasing number of passes. The 4 mm/min turned surface decreases two times slower compared to the 3 s abrasive blasted and 6 mm/min turned surfaces. While the latter two achieve the minimum roughness of 0.2 μm after 100 passes, the 4 mm/min turned surface achieves 0.5 μm after 300 passes. All fit parameters a, b, and c differ significantly between these two topographies. The fit coefficients of the 3 s abrasive blasted and 6 mm/min turned surfaces are similar.

In summary, roughness Sa can be described appropriately by an exponential function of number of passes. Thereby, the exponential decrease is influenced by the machining parameters. However, the minimum roughness is independent of the initial roughness.

4.3. Material Influence

Figure 9a–e shows the roughness over the number of passes for titanium (red points), Ti6Al4V (orange points), Nitinol (purple points), Stellite 21 (blue points), and metallic glass (green points), as well as the corresponding laser powers and exemplary microscope images. The surface roughness of these materials can be significantly reduced (>90%) for specific machining parameters. The laser power threshold P_L,th necessary for thermochemical material removal depends on the material. For example, Ti6Al4V can be effectively polished at 1.03 W (orange triangle symbol), while titanium requires 2.32 W (red triangle symbol) for a comparable roughness decrease. The general trend shows that initial roughness Sa_ini decreases exponentially with the number of passes to a minimum value between 0.1 μm and 0.3 μm. For higher laser powers, this decays faster. At least two negative outcomes from this result can be observed:

• An incomplete surface finish, shown by the red and purple unfilled circle symbols in Figure 9a,c.
• For higher laser powers, reinforcing corrosion and secondary chemical disturbances occur, as shown by the purple, blue, and green filled circle symbols in Figure 9c–e.

These effects stop the smoothing and increase the roughness. The entire surface can be covered by these structures, as shown in the microscope images. The optimal power range and process window for effective surface polishing is therefore restricted between the threshold power P_L,th and occurrence of disturbances at P_L,dist. Each of these values depend on the specific material–electrolyte combination.

5. Discussion

5.1. Notation for Process Signatures of Multi-Cycle Loads

As outlined by Brinksmeier et al. [3], the concept of Process Signatures seeks to describe material modifications M in terms of internal material loads L rather than machining parameters (Correlation I), as shown in Figure 10a. This correlation is defined as the Process Signature (PS), represented by functional performing Process Signature Components (PSC) according to M = f(L). The aim of the following discussion is to derive an empirical-based functional relation that describes roughness, in terms of recurring material loads (caused the multiple passes), as a multi-cycle PSC. To formulate these correlations, a physics-based knowledge of internal loading is necessary. It is expected that the anodic material dissolution [16] due to the laser-induced local “thermo-battery effect” [17] is at least partly proportional to the induced temperature distribution [18]. For area polishing (zig-zag trajectory with 66% line overlap), it is shown that despite the non-uniform temperature distribution, isotropic surface smoothing is achieved. Therefore, it is sufficient to describe the material load during one-cycle by the thermal load T_L (averaged temperature) and exposure time t_N. For a single-cycle, the roughness modification can be described according to:

$$M_{Sa}{=f}_{Sa}\left(T_L,t_N\right)$$

(5)

Since surface polishing is the result after multiple, recurring material loads of a constant energy amount, the overall material modification ${\tilde{M}}_{Sa}$ should be a potentiation of one-cycle loads, as shown in Figure 10b. For the functional correlation follows:

$$\begin{array}{c}{\tilde{M}}_{Sa}=\underset{Ntimes}{\underbrace{f_{Sa}\left(T_L,t_N\right)·f_{Sa}\left(T_L,t_N\right),\dots ,·f_{Sa}\left(T_L,t_N\right)}}=f_{Sa}{\left(T_L,t_N\right)}^N\\ \stackrel{\mathrm{def}}{=}{f}_{Sa}\left(T_L,N·t_N\right)=f_{Sa}\left(T_L,t_r\right)\end{array}$$

(6)

The applied definition prerequisite is that f_Sa has exponential properties. On the bottom line, the N-fold modification ${\tilde{M}}_{Sa}$ is described by the same functional correlation as the one-cycle, but only with an N-fold longer exposure time t_r = t_N·N. This implies that the individual passes are stationary. The overall roughness modification ${\tilde{M}}_{Sa}$ is then proportional to the number of passes N. By differentiation of Equation (6) after N and inserting Equation (5), one receives the following ordinary differential equation for a multi-cycle PSC according to:

$${\tilde{M}}_{Sa}=c_1\frac{{\tilde{M}}_{Sa}}{dN}+c_2$$

(7)

5.2. Internal Material Load

The duration for each surface part illuminated after multiple passes is described by the exposure time t_r. In the first experiment, the cavities are separated and have no lateral overlap. Equation (1) can be simplified to t_r = (π/4)·(d_f/v)·N. This describes the duration required (for the laser spot) to pass the distance d_f. Figure 11 shows the removal depth over the exposure time for the results of Figure 4. Cavities structured with increasing scan velocity achieve the same depth after a proportionally increased number of passes, as shown in example in (1)–(4). The linear deepening after 100s of passes shows that, first, the material dissolution is little influenced by changes of the cavity geometry. Second, the material removal rate does not decrease with increasing depth due to mass transport limitation such as saturation.

The influence of laser-induced temperature distribution on the material removal rate and quality can be described by the thermal load. Figure 12 shows the cavity depth depending on the thermal load for the results of Figure 3. Regardless of the focus diameter and laser power, the cavity depth follows a linear increase over the thermal load until it drops abruptly. At this characteristic thermal load T_dist ≈ 190 °C, disturbances occur in the cavity center caused by emerging gas [29] or material deposition [27]. The cross section of the linear regression (dotted line) with the x-axis determines the thermal threshold T_th = 113 °C. This thermal threshold defines the minimum activation temperature for laser thermochemical material removal. It is a characteristic value for one specific material–electrolyte combination. Between the upper and lower thermal limit, the surface finish is of high quality and low roughness. The findings confirm the previous hypothesis that thermochemical material removal can be generalized by means of the thermal load and exposure time.

5.3. Multi-Cycle Process Signature of LCP

The results show that surface roughness Sa follows an exponential decrease with increasing number of passes. The roughness is described by the exponential fit according to Equation (4) with fit parameter a, b, and c. By comparing the fit parameters with the initial roughness Sa_ini (μm), minimum roughness Sa_min (μm), and thermal load T_L (°C), the following correlations are identified:

$$\begin{array}{c}a=\Delta {Sa=Sa}_{ini}-{Sa}_{min}\\ b={Sa}_{min}\\ c=-\frac{3}{4}\sqrt{\frac{T_L-T_{th}}{T_L·{Sa}_{ini}}·\frac{\mathsf{\mu }\mathrm{m}}{1}}·t_N\frac{1}{\mathrm{s}}\end{array}$$

(8)

The correlations of a and b are evident as the comparison of Figure 6, Figure 7 and Figure 8 show. First, for N = 0, the e-function is 1 and the roughness must to be equal the initial state Sa_ini. Second, for N → ∞, the e-function approaches 0 so that the roughness must become Sa_min. The assumption of exponent c depending on T_L, T_th, and Sa_ini is motivated by roughness decrease for varying laser powers (and by this thermal loads), as shown in Figure 7. At lower thermal loads, the exponential decrease is slower. Surface smoothing must stagnate if the thermal load verges on the thermal threshold. Figure 12 shows that no material removal occurs for thermal loads below T_th, and therefore, the function must be undefined. This is ensured by the square root of (T_L–T_th), thus negative values are not defined in $\mathbb{R}$. Further, it ensures that smoothing takes place faster for increasing thermal differences. The division by square root of Sa_ini considers that the smoothing rate increases with the initial roughness, as shown in Figure 6. This is because the absorption coefficient α (and thus thermal load) drops from 68% to 39% during polishing. Assuming a constant absorption coefficient in the calculation of the thermal load is simplified, since the absorption is a function of roughness. With Equation (8), the exponential fit (4) results in an empirical model to describe the surface roughness Sa (μm) by:

$${Sa(T}_L,t_N,N)=\mathsf{\Delta }Sa·e^{-\frac{3}{4}\sqrt{\frac{T_L-T_{th}}{T_{th}^{}·{Sa}_{ini}}·\frac{\mathsf{\mu }\mathrm{m}}{1}}·t_N·N\frac{1}{\mathrm{s}}}+{Sa}_{min}$$

(9)

According to Equations (1) and (3), the thermal load T_L and exposure time t_r imply the relevant machining parameters, laser power P_L, focus diameter d_f, scan velocity v, line distance b, and number of passes N. This empirical correlation of surface roughness depending on thermal load, single-cycle exposure time, and number of passes is interpreted as multi-cycle Process Signature Component of LCP. It is confirmed within the shown experimental data of Figure 6, Figure 7, Figure 8 and Figure 9 and shown in Figure 13 by plotting the “relative roughness change (Sa − Sa_min)/∆Sa” over the e-functions exponent defined as “thermal exposure 3/4·((T_L − T_th)/(T_th·Sa_ini))^0.5·t_r”.

The surface smoothing follows an exponential decay over the thermal exposure. This shows that for all pre-machined surfaces, the same physical mechanism is responsible for smoothing and for preferred removal of surface peaks [19]. Only the 4 mm/min turned surface of Figure 8 (dark blue diamonds) could not be accurately described by Equation (9). The discrepancy from the exponential decay suggests a change of mechanism. The smoothing efficiency depends on the lateral size of the surface features d_λ. Turning ridges larger than the focus diameter d_f ≤ d_λ are smoothed significantly slower. That is because the smoothing mechanism is localized and confined to the laser-heated area. The insufficient coverage of the turning ridges by the laser spot results in the slower smoothing characteristic.

In order to apply Equation (9) to other material–electrolyte combinations, it is necessary to determine their specific thermal thresholds T_th. Figure 14a shows the results of Figure 9 by plotting the relative roughness over the thermal exposure using the thermal threshold of titanium (Grade 1) (T_th = 113 °C). The continuous blue line describes the ideal exponential decay. Only titanium (Grade 2) (red data points) is adequately described. While using an individual thermal threshold T_th for each material, the relative roughness is described independent of the machining parameters and material, as shown in Figure 14b. The thermal threshold can be interpreted as a property of the initial state that combines various physical and chemical interactions during LCP. It considers that electrochemical corrosion potential varies depending on material–electrolyte combination. However, the thermal threshold of a specific material–electrolyte combination cannot be derived from other material properties and must be determined experimentally. Discrepancies to the ideal exponential decay (blue line) are an indication of further unknown mechanisms.

This may include the following two mechanisms. First, an incomplete surface smoothing (unfilled circle symbols). During polishing, the absorption coefficient is reduced from 0.68 down to 0.39. The induced thermal load decrease below the thermal threshold. The smoothing process stops itself after several cycles. Second, process disturbances can be induced for higher thermal loads. Among these are discussed: Corrosion, secondary chemical effects like salt deposition [27], and a shielding by emerging gas [29].

As motivated above, the multi-cycle PSC must solve the ordinary differential Equation (7). This can be verified by derivate the PSC (Equation (9)) after the number of passes N by:

$$\frac{{dSa(T}_L,t_N,N)}{dN}=-\frac{3}{4}\sqrt{\frac{T_L-T_{th}}{T_{th}·{Sa}_{ini}}}·t_N·\underset{{=Sa(T}_L,t_N,{N)-Sa}_{min}}{\underbrace{\mathsf{\Delta }Sa·e^{-\frac{3}{4}\sqrt{\frac{T_L-T_{th}}{T_{th}·{Sa}_{ini}}}·t_N·N}}}$$

(10)

$$\underset{:=c_1}{\underbrace{-\frac{8\alpha 3}{3\pi 4}\sqrt{\frac{T_{th}·{Sa}_{ini}}{T_L{-T}_{th}}}·t_N}}·\frac{dSa\left(T_L{,t}_N,N\right)}{dN}=Sa\left(T_L,t_N,N\right)-\underset{:=c_2}{\underbrace{{Sa}_{min}}}$$

(11)

$$\frac{dSa\left(N,\dots \right)}{dN}=\frac{1}{c_1}Sa\left(T_L,\dots \right)-c_2$$

(12)

Apart from the integration constant c₂, this is the same ordinary linear differential equation as Equation (7). Thus, the same functional PSC is a solution for the single-cycle and multi-cycle problem, only with an exponent multiplied by the number of passes N. This means that roughness decreases stepwise with each cycle load at a rate proportional to its current value. In this context, roughness can be interpreted as a quantity of surface features. The exponential decay constant c₁ can be understood as a mean cycle number. This describes the average number of loads for which the surface features remain. It is the number of passes after which the roughness is reduced to 1/e.

An alternative interpretation is to ignore the off-time between cycle loads. In other words, the exposure time t_N·N = t_r is considered to be continuous in time. The reciprocal exponential coefficient describes then the mean lifetime τ of surface features. Based on this, the “polishing exposure time τ_p” is defined as the time after which the initial roughness is reduced to 1/e³ ≈ 5% by:

$${\tau }_p=4\sqrt{\frac{T_{th}·{Sa}_{ini}}{T_L-T_{th}}·\frac{1}{\mathsf{\mu }\mathrm{m}}}·s$$

(13)

5.4. Polishing Time

The processing time defined by Equation (2) is proportional to area A, exposure time and anti-proportional to the square of the focus diameter. Considering the mean polishing exposure time τ_p, the “polishing rate r_A (s/mm²)” is calculated based on Equation (2) according to:

$$r_A{(d}_f,T_L)=\frac{16}{\pi }·\frac{1}{d_f^2}·\sqrt{\frac{T_{th}·{Sa}_{ini}}{T_L-T_{th}}·\frac{1}{\mathsf{\mu }\mathrm{m}}}·s$$

(14)

with the initial roughness Sa_ini (μm), focus diameter d_f (mm), and thermal load T_L (°C). This shows, that faster polishing can only be achieved by increasing the focus diameter or thermal load. The polishing time shortens with the square of focus diameter and with the square root of thermal load. Figure 15 illustrates the solution according to Equation (14) for a roughness reduction down to 5% of the initial roughness for a common set of machining parameters (I).

Polishing a surface area of 1 mm² takes 28 min. Doubling the thermal load up to 270 °C would reduce the polishing time by factor 2.7, while doubling the spot diameter to 220 μm would by factor 4. First, increasing the laser power, and thereby the thermal load, above 200 °C leads to undesired process disturbances. Second, an increase in the spot diameter coincides with a reduction of the thermal load. By increasing the spot diameter, the laser power must be increased to keep the thermal load constant. Figure 16a compares the common parameters (I) of 110 μm spot diameter with a defocused spot (II) of 1 mm. Both settings ensure a constant thermal load and an exposure time above the polishing exposure time of τ_p ≈ 10 s. The parameters for the defocused spot are inversely calculated with the multi-cycle PSC. According to Figure 15, the polishing time reduces along the isothermal line, as depicted in Figure 16b. The polishing time of 1 mm² reduces from 28 min down to 21 s by factor 81 using the defocused spot. This makes it possible to polish macroscopic-scaled surfaces in reasonable time. Figure 16c shows the result of a 1 cm² polished area with the defocused beam spot in 34 min. Both high precision and fast polishing can be achieved by scaling and shaping the laser intensity distributions.

6. Conclusions

This study provides a comprehensive understanding of the interaction of various machining parameters and surface finish during LCP. The empirically-derived multi-cycle PSC is an effective tool to calculate and scale suitable machining parameters for precise control of roughness and selective polishing of micro- and macroscopic-scaled surfaces. The following findings were outlined:

• The effect of machining parameters and initial roughness are described by the “thermal load T_L” and “exposure time t_r”. These quantities are identified as essential material loads that determine the removal depth, roughness changes, and surface finish during LCP.
• The smoothing of titanium during LCP is described as a multi-cycle PSC by an exponential decay over exposure time. The decay rate is proportional to the square root of the thermal load.
• Applying the multi-cycle PSC on titanium, Ti6Al4V, Nitinol, Stellite 21, and metallic glass determined their thermal thresholds to: T_th = 113, 170, 127, 115, and 76 °C
• The polishing rate r_A is reciprocally proportional to the square of focus diameter and square root of thermal load. The increase of the spot diameter results in an increase of the polishing rate by factor 80.