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The laser chemical machining is a non-conventional substractive processing method. It is based on the laser-activation of a material dissolution of metals in electrolyte ambient via local-induced temperature gradients and allows a gentle and smooth processing of especially temperature-sensitive metals. However, the material removal is characterized by a narrow process window and is restricted by occurring disturbances, which are supposed to be related to the localized electrolyte boiling. In order to control the removal quality and avoid disturbances, the correlation between the laser-induced temperatures and the resulting removal geometry has to be better understood. In this work an analytical modeling of the laser-induced temperatures at the surface of titanium based on a Green-function approach is presented. The main influencing factors (laser, electrolyte, material) as well as possible heat transfer into the electrolyte are included and discussed. To verify the calculated temperatures, single spot experiments are performed and characterized for titanium in phosphoric acid solution within laser irradiation of 1 s. The correlation between the temperature distribution and the resulting removal geometry is investigated based on a spatial superposition. Thereby, the bottom limit temperature is found to range between 63 °C and 70 °C whereas the upper limit is related to the nucleate boiling regime. Based on the performed correlation an indicator is identified to predict the ruling removal regime and thereby to reduce the experimental expenditure.

Within the ever-increasing trend of miniaturization, traditional mechanical machining reaches more and more its performance limits due to the enhanced complexity and amount of micro-components as well as occurring size effects [

Among others, these include the laser chemical machining (LCM) that unifies the advantages of laser machining with its precise and localized energy deposition and the electrochemical machining with its smooth processing without significant thermal impacts [

However, the dynamics of the laser light absorption, heat, chemical reactions, hydrodynamics and transport phenomena causes within a certain range of parameters a disturbance of material removal [

Within this work, an analytical modeling of the laser-induced surface temperatures is presented based on a Green-function approach. It takes into account the main influencing factors (laser, electrolyte and material) as well as possible heat transfer into the electrolyte and is used to identify the influence of single process parameters on the temperature distribution. In addition, the removal properties are experimentally characterized in order to validate the modeling results. Therefore, single spot experiments are performed for titanium in phosphoric acid solution within laser irradiation of 1 s. The correlation between the temperature distribution and the resulting removal geometry is investigated based on a spatial superposition. In dependence of laser power and beam diameter the starting temperature for a laser chemical removal as well as its upper limit temperature are determined. Furthermore, it is shown that once validated the presented model can be used to predict the process window for a disturbance-free LCM-process and there by helps reducing the experimental expenditure.

The laser irradiation is used in laser chemical machining as a localized and selective heat source that can induce suitable thermal impact for the activation of heterogeneous chemical reaction between a liquid ambient and a metallic surface and results in a temperature-induced electrochemical etching [

Me + 2 H + → Me 2 + + H 2 ↑ (1)

In laser chemical machining, the workpiece surface is in a direct contact to the electrolytic solution. In consequence of its heat impact the laser beam can induce or enhance reactions at the metal-liquid interface via changes in the electrochemical Nernst potential. The locally induced temperature gradients result in the generation of a thermobattery allowing a current flow within the metal between the center of the incident laser light and its periphery. Despite the low generated electromotive forces (some 0.1 V for a temperature rise of 100 K) the electric field strengths are very high due to the small battery dimensions [

Within the LCM-process the laser induced-temperatures define both, the proton activity within the redox-reaction and the electrochemical potential at the workpiece surface [^{−9} m/s). Thereby, the laser-induced temperatures depend on different factors that can be divided in:

1) Laser characteristics: laser power, irradiation duration, intensity distribution, spot size and feed velocity.

2) Material properties: wavelength-dependent absorption coefficient, chemical composition, thermal diffusivity, specific heat, density…

3) Electrolyte properties: acidic/basic solution, concentration, layer thickness, wavelength-dependent transmission coefficient.

Beside the temperature, the electrochemical potential is also depending on the chemical activity of the dissolved metal ions and on the mass transport limitation [

As already mentioned, the laser irradiation has the function of thermally activate the chemical etching reaction. Because of the high number of influence factors as well as the complex interaction nature (chemically, physically and flow dynamics) it is still a lack of knowledge about the mechanisms occurring during LCM- process.

The laser radiation propagates throughout the etchant solution, is absorbed at the workpiece surface, induces a temperature distribution that define the area and properties of the chemical dissolution reaction and thereby the quality of the material removal. Therefore, a determination of the temperature distribution over the workpiece surface would help to understand the influences of the main process parameters. To build up a theoretical description of the surface temperature following assumptions have been made:

1) The laser beam―as the heat source―is a TEM_{00}-mode with a Gaussian intensity distribution

I ( x , y ) = I 0 ⋅ e [ ( x r x ) 2 + ( y r y ) 2 ] (2)

where I 0 = P a b s / A with P_{abs} is the absorbed laser power and A is the laser beam area at the workpiece surface, A = π r x r y and r x = r y .

2) The workpiece/laser beam is moving with a constant speed v and in x-di- rection. In this work the feed velocity v = 0, as only the case of a static irradiation is studied.

3) The material is supposed to be isotropic with temperature-independent properties.

4) Phase changes do not occur. On the one hand laser melting cannot occur at the applied laser powers and on the other hand the formation of metallic salt layers is neglected.

5) The laser power is affected by the transmission coefficient τ E that describes the power loss during the propagation throughout the etchant solution and by the absorption coefficient α a b s of the metallic material. Both coefficients are assumed to be constant and temperature-independent. The absorbed laser power P_{abs} is then described as:

P a b s = P L ⋅ α a b s ⋅ τ E (3)

6) The thermal interaction takes place only between laser beam and workpiece. Additional contributions, such as the heat of reaction, are neglected.

7) Heat transfer into the etchant solution is considered. The heat equation includes heat loss rates dependent on constant heat transfer coefficients of the electrolyte.

8) For the description of the heat equation a rectangular coordinate system is used. This is fixed in the workpiece, where the origin is directly beneath the beam center, x-y-plane is the workpiece surface and the positive z-axis points into the workpiece. The laser beam interacts with the workpiece at z = 0. All relevant modeling parameters are listed in

The interaction between laser irradiation and metallic surface is assumed to be dominated by conduction during the electron-lattice heating. The conduction is related to the thermal diffusivity D, the volumetric heat capacity C_{p} and the material density ρ . The resulting temperature distribution T in the material is associated to the power density Q through the diffusion equation:

∂ T ∂ t = D ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + ∂ 2 T ∂ z 2 ) − Q ρ C p (4)

Here, Q represents the contribution of the heat source (laser beam) and can be described as:

Q = P a b s 2 π r ² ⋅ e − ( x − v t ) 2 + y 2 2 r 2 ⋅ e − α z (5)

The temperature rise T ( x , y , z , t ) is valid for x > − ∞ , y < ∞ , 0 ≤ z ≤ ∞ and t > 0 and follows the boundary conditions (see

T = 0 at t = 0 , T → 0 for x , y → ± ∞

T → 0 for z → ∞ , ∂ T ∂ t = H κ T at z = 0

For a simplified mathematical treatment a dimensionless formulation of Equation (4) has to be provided. Therefore, following dimensionless variables were defined:

X = x r , Y = y r , Z = z r , τ = 2 D t r ² , μ = v ⋅ r D , α = r l , Θ = H ⋅ r κ

and

Ψ = T Δ T ′ with Δ T = P a b s 4 π κ r

The dimensionless heat equation can be then formulated as [

∂ Ψ ∂ t = ∂ 2 Ψ ∂ X 2 + ∂ 2 Ψ ∂ Y 2 + ∂ 2 Ψ ∂ Z 2 − μ ∂ Ψ ∂ X − 2 α π e − X 2 − Y 2 − α Z (6)

where following boundary conditions are valid:

Ψ = 0 at τ = 0 , Ψ → 0 for X , Y → ± ∞

Ψ → 0 for Z → ∞ , ∂ Ψ ∂ τ = Θ Ψ at Z = 0

The solution of the dimensionless temperature distribution at the workpiece surface (Z = 0) is given by:

Ψ ( X , Y , 0 , τ ) = ∫ 0 τ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ 0 ∞ G ( X , Y , 0 , τ , X ′ , Y ′ , Z ′ , τ ′ ) × 2 α π e − X ′ 2 − Y ′ 2 − α Z ′ d Z ′ d Y ′ d X ′ d τ ′ (7)

Here, the general solution of the Green-function G ( X , Y , 0 , τ , X ′ , Y ′ , Z ′ , τ ′ ) is:

G = 1 4 [ π ( τ − τ ′ ) ] 3 / 2 e − ( X − X ′ ) 2 + ( Y − Y ′ ) 2 + ( Z − Z ′ ) 2 4 ( τ − τ ′ ) (8)

In accordance with Carslaw and Jaeger [

G = 1 [ 4 π ( τ − τ ′ ) ] 3 2 e − [ ( X − X ′ − μ ( τ − τ ′ ) ) 2 + ( Y − Y ′ ) 2 4 ( τ − τ ′ ) ] × { e − ( Z − Z ′ ) 2 4 ( τ − τ ′ ) + e − ( Z + Z ′ ) 2 4 ( τ − τ ′ ) − 4 π ( τ − τ ′ ) θ ⋅ e θ 2 ( τ − τ ′ ) + θ ( Z + Z ′ ) ⋅ erfc [ Z + Z ′ 2 τ − τ ′ + θ τ − τ ′ ] } (9)

Using the substitution ϕ = τ − τ ′ and taking into account the velocity related term µ; the dimensionless Equation (Equation (7)) at Z = 0 can be transformed into:

Ψ = 2 α π ∫ 0 τ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ 0 ∞ 1 [ 4 π ϕ ] 3 2 e − [ ( X − X ′ − μ ϕ ) 2 + ( Y − Y ′ ) 2 4 ϕ ] ⋅ e − ( Z − Z ′ ) 2 4 ϕ ⋅ e − X ′ 2 − Y ′ 2 − α Z ′ d Z ′ d Y ′ d X ′ d ϕ + 2 α π ∫ 0 τ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ 0 ∞ 1 [ 4 π ϕ ] 3 2 e − [ ( X − X ′ − μ ϕ ) 2 + ( Y − Y ′ ) 2 4 ϕ ] ⋅ e − ( Z + Z ′ ) 2 4 ϕ ⋅ e − X ′ 2 − Y ′ 2 − α Z ′ d Z ′ d Y ′ d X ′ d ϕ − 2 α π ∫ 0 τ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ 0 ∞ 1 [ 4 π ϕ ] 3 2 e − [ ( X − X ′ − μ ϕ ) 2 + ( Y − Y ′ ) 2 4 ϕ ] ⋅ { e − ( Z − Z ′ ) 2 4 ϕ + e − ( Z + Z ′ ) 2 4 ϕ − 4 π ϕ θ ⋅ e θ 2 ϕ + θ ( Z + Z ′ ) ⋅ erfc [ Z + Z ′ 2 ϕ + θ ϕ ] } ⋅ e − X ′ 2 − Y ′ 2 − α Z ′ d Z ′ d Y ′ d X ′ d ϕ (10)

In Equation (10) the first two integrals represent the lossless heat equation, whereas the third term includes the heat losses when considering heat transfer into the electrolyte (i.e. H > 0). This formulation is in good agreement with the work of Yung et al. [

With help of the closed-form approximation of the error function, described in [

e x ⋅ erfc ( x ) ≈ a ( a − 1 ) π x 2 + π x 2 + a (11)

where a = π / ( π − 2 ) , the surface temperature rise T(x, y, 0, t) can be finally simplified and described as:

T = P ⋅ α A b s ⋅ τ E 4 π ⋅ κ ⋅ l ⋅ ∫ 0 τ 1 1 + 4 ϕ ⋅ a ( a − 1 ) π α 2 ϕ + π α 2 ϕ + a 2 ⋅ e − [ ( X − μ ϕ ) 2 + Y 2 1 + 4 ϕ ] d ϕ − P ⋅ α A b s ⋅ τ E 4 π ⋅ κ ⋅ l ⋅ θ ∫ 0 τ 1 1 + 4 ϕ e − [ ( X − μ ϕ ) 2 + Y 2 1 + 4 ϕ ] ⋅ ∫ 0 ∞ a ( a − 1 ) π ( Z ′ 4 ϕ + Θ 2 ϕ + Θ Z ′ ) + π ( Z ′ 4 ϕ + Θ 2 ϕ + Θ Z ′ ) + a 2 ⋅ e − Z ′ 2 4 ϕ ⋅ e − α Z ′ d Z ′ d ϕ (12)

As laser source a cw-fiber laser JK400 (from JK Lasers) was used. Its TEM_{00} cw-laser radiation of 1080 nm is focused by a telecentric f-theta optic and guided using the galvanometer scanning system Raylase Superscan III-15 to the surface. Using an inverse telescope, the laser beam diameter and thus the focal spot size can be varied. Within the performed investigations the laser spot diameters 30.5 µm and 109 µm were applied. As sample material rolled titanium (Ti 3.7024) with a thickness of 0.8 mm was used. The Ti-workpiece was placed in a closed liquid-phase etching chamber, where a 5 molar (28.7% vol.) phosphoric acid (H_{3}PO_{4}) was pumped as a cross-jet through a 25 mm × 2 mm cross-section with velocity v_{f} of 2 m/s. The electrolyte layer between workpiece and protective glass has a thickness of 2 mm. A detailed description of the used setup can be found in [

Depending on laser power and spot diameter arrays of 4 × 4 single irradiation spots were structured, while the irradiation time was kept constant at 1 s. The main process parameters as well as the properties of the used material and electrolyte are listed in

To study the influence of laser parameters we consider the case of an absent heat transfer into the etchant solution. Here, Θ = 0 and ∂ ψ / ∂ z = 0 at Z = 0 . Thus, the temperature rise in Equation (12) can be reduced to its first term:

T ( X , Y , 0 , ϕ ) = P ⋅ α A b s ⋅ τ E 4 π κ l ∫ 0 τ 1 1 + 4 ϕ e − X 2 + Y 2 1 + 4 ϕ a ( a − 1 ) π α 2 ϕ + π α 2 ϕ + a 2 d ϕ (13)