Processing constraints resulting from heat accumulation during pulsed and repetitive laser materials processing

Rudolf Weber; Thomas Graf; Christian Freitag; Anne Feuer; Taras Kononenko; Vitaly I. Konov

doi:10.1364/OE.25.003966

1. Introduction

Short- and ultra-short pulsed lasers promise high-quality processing of virtually any material with superior quality presuming the correct processing parameters [1–5]. This is in particular also the case for materials which absorb in the surface layer and which have high thermal conductivity such as metals. However, in any laser material interaction at least a part of the absorbed laser energy is e.g. not removed together with ablated material, is thermalized and remains in the workpiece as residual heat. In this paper, the residual heat after the laser-material interaction and after all phase transitions back to solid state is of particular interest. The fraction η_Heat of this residual heat relative to the absorbed energy depends on the material properties, the process parameters (such as pulse energy pulse duration, fluence, intensity, laser frequency, intensity distribution), and the process itself (energy coupling, ablation, plasma formation), especially in the case of ultra-short pulsed laser processing with high intensities >10¹² W/cm² [6]. During repetitive processing this residual heat becomes significant if the material has not enough time to cool down sufficiently before the next heat input is applied. The importance of this so-called heat accumulation effect was reported in [7–11] where the consequence of heat accumulation between subsequent pulses on the same spot on the processed surface was identified as one of the major mechanisms that significantly decreased the resulting process quality. Figure 1 shows the typical evolution of the temperature at the origin of the beam on the surface of the processed material (CrNi-Steel, 300 kHz laser frequency, 3 mJ pulse energy, 37% absorptance, 34% of residual heat, and 1-dimensional heat flow) when heat accumulation occurs.

Fig. 1 A typical example for heat accumulation showing the evolution of the temperature as a function of time at the origin of the beam on the surface of the material which is processed with a pulsed laser. The second pulse arrives, before the surface has cooled down to the original ambient temperature (left arrow). In the case of metals a significant change of the process occurs, when the subsequent pulses are incident on the still molten surface (right arrow), which usually significantly decreases the resulting process quality.

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The time between the first and the second pulse is too short to allow the material to cool down to the initial temperature (left arrow). The resulting continuous increase of the remaining temperature when the following pulses arrive (marked with squares in Fig. 1) leads to a significant change of the process, and usually to a reduction in processing quality, when the local temperature exceeds a critical temperature T_crit, such as for example the melting temperature in metals (Fig. 1, right arrow), or the matrix evaporation temperature in carbon fiber reinforced plastics (CFRP) [7].

Heat accumulation scenarios

Two different heat accumulation scenarios have to be distinguished when considering process strategies.

(a) For short-pulse materials processing working with a static beam when percussion drilling or a quasi-static beam when trepanning very small holes, the heat accumulation effect is caused by the heat input from subsequent laser pulses on the same spot (HAP). In the case of percussion drilling, the total number of pulses required to complete the processing task is determined by the depth of the hole which has to be drilled, and the energy input required to remove the material.
(b) For processing with a moving beam, such as surface structuring or multi-scan cutting, the HAP effect is caused by the finite number N_Spot of subsequent pulses on the same spot of the moving beam. This number of pulses per spot is approximately given by N_Spot ≅□d_Spot · f_Laser / v_Feed, where d_Spot is the diameter of the laser beam on the surface, f_Laser is the pulse repetition rate, and v_Feed is the velocity of the moving beam on the surface. This means that HAP usually can be avoided by moving the laser beam fast enough across the surface, i.e. with a feed rate of v_Feed ≳ d_Spot · f_Laser. In addition to HAP, however, the same basic heat accumulation mechanism also occurs for the heat input resulting from subsequent scans over the same spot (HAS) during multi-scan processing [9]. Again, the total number of scans required to complete the process is given by the depth of the structure which has to be achieved or by the material thickness which has to be cut, which defines the total energy input required to remove the material. Hence, depending on the applicable pulse energy, the total number of scans is inherently given for any specific application and cannot be reduced. This might be up to 2100 scans as reported in [10] for cutting of 2 mm thick CFRP. In this case HAS becomes the dominant thermal effect reducing the process quality [9].

To maintain process quality it is crucial to know when the temperature increase reaches its critical value due to heat accumulation in both cases, (a) and (b). For a given average laser power (i.e. pulse energy times laser frequency) it is therefore of particular interest to know the maximum acceptable number of laser pulses or scans (in the following in a more general way called “maximum number of heat inputs”, N_HeatInp,maxt) after which the temperature increase in the interaction region has reached the specific, material-dependent temperature limit. If this limit is reached, a processing pause must be introduced in order to allow the material to cool down and to avoid detrimental effects resulting from heat accumulation.

Heat accumulation equation

Heat accumulation was modelled analytically in detail in [7]. These calculations serve as basis for the following considerations of the limitations imposed by heat accumulation. The model presented in [7] is based on the formalism of Rykalin, described in [12,13] which assumes, that the energy is applied in an infinitesimal short time and volume. This holds for pulsed laser processing of metals and CFRP when the considered time scale is much larger than the pulse duration. When considering heat accumulation the relevant time scales typically are in the range of Microseconds for lasers with MHz repetition rates up to Milliseconds for lasers with KHz repetition rates or multi-scan processing, which both are orders of magnitude larger than the pulse duration of fs-, ps-, or ns-lasers. Furthermore the model presumes processing of materials with a high absorptivity, where the optical penetration depth is much smaller than the thermal diffusion length between two subsequent heat inputs. With increasing time the heat conduction into the material leads to a temperature field which strongly depends on the geometry of the heat flow. Typical examples for heat flow in one dimension (1D) with a plane heat source at the surface, two dimensions (2D) with a line source inside the workpiece, and three dimensions (3D) with a point source on the surface are shown in Fig. 2. The 1D case is for example of interest for large laser beams and for long processing times in thin sheets as will be explained later. The 2D case might apply for drilling of deep holes and the 3D case for small laser beams and long processing times in bulk material. The arrows sketch the directions of the heat flow.

Fig. 2 Typical geometries for the heat flow in one dimension (1D) with an area on the surface as heat source, in two dimensions (2D) with a line inside the material as heat source and in three dimensions (3D) with a point on the surface as heat source.

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The resulting temperature fields of each pulse can be summed up in time and space when the material parameters are taken as constant. This summing up results in the mechanism of heat accumulation as shown as an example in Fig. 1 for 1D-heat flow. The squares mark the remaining heat-accumulation temperature in the origin just before the next heat input occurs, which is of particular interest in this paper. The evolution of this heat-accumulation temperature is described by the general, so-called “heat accumulation equation” [7]. For a dimensionality nD ∈ {1,2,3} it reads

Δ T_{H A, n D} (t = \frac{N_{H e a t I n p}}{f_{H e a t I n p}}) = \frac{Q_{n D}}{ρ \cdot c_{p} \cdot \sqrt{{(\frac{4 \cdot π \cdot κ}{f_{H e a t I n p}})}^{n D}}} \cdot \sum_{N = 1}^{N_{H e a t I n p}} \frac{1}{\sqrt{N^{n D}}} .

where ΔT_HA,nD is the temperature increase due to heat accumulation caused by a number of N_HeatInp heat inputs in a temporal δ-peak at the origin of the heat source Q_nD and with the repetition rate f_HeatInp. In contrast to the notation in [7], we modify the term “pulses” into the more general term “heat inputs” to account for the both heat accumulation mechanisms, HAP and HAS. ρ is the mass density of the solid or liquid material, c_p its specific heat capacity, κ = λ_th /(ρ c_p) the thermal diffusivity, λ_th the heat conductivity, and t is time. In this solution, the material properties ρ, c_p, and λ_th have to be taken as constant with respect to temperature. The heat sources Q₁_, Q₂_, and Q₃_, for 1-dimensional (1D), 2-dimensional (2D) and 3-dimensional (3D) heat flow are

1 D Q_{1} = σ \cdot Q_{H e a t} / A (i n J / m^{2}),

2 D Q_{2} = σ \cdot Q_{H e a t} / ℓ (i n J / m), and

3 D Q_{3} = σ \cdot Q_{H e a t} (i n J),

respectively. A and ℓ are the area and the length where the energy Q_Heat is deposited in the case of 1D and 2D heat flow, respectively. The factor σ depends on the geometry under consideration. σ = 1 when the deposited heat can flow in the complete surrounding of the source, and σ = 2 when the hat can only flow in one half space. In the examples shown in Fig. 2, σ = 2 in the 1D, σ = 1 in the 2D, and again σ = 2 in the 3D case.

Equations (1) and (2) allow to conveniently calculate the temporal evolution of the heat-accumulation temperature resulting from a sequence of energy inputs. In order to be able to solve Eq. (1) for the number N_HeatInp of heat which yields a certain temperature increase ΔT_HA,nD, approximation formulas are derived in the following. This will then allow to formulate analytically expressed scaling laws for the maximum number of energy inputs or the maximum average laser power allowed to keep the temperature increase caused by heat accumulation below a given value.

2. Approximations for large numbers of heat inputs

Explicit analytical expressions for the sum term in (1) were found by approximating the sum with the corresponding integral, plus a constant C_nD, which depends on the heat flow geometry, i.e. by setting

\sum_{n = 1}^{N_{H e a t I n p}} \frac{1}{\sqrt{N^{n D}}} ≅ \int_{1}^{N_{H e a t I n p}} \frac{1}{\sqrt{N^{n D}}} d N + C_{n D} .

The three definite integrals, which result from (3) for the three dimensionalities, have the solutions

1 D \int_{1}^{N_{H e a t I n p}} \frac{1}{\sqrt{N}} d N + C_{1} ≅ 2 \sqrt{N_{H e a t I n p}} + C_{1},

2 D \int_{1}^{N_{H e a t I n p}} \frac{1}{N} d N + C_{2} ≅ \ln (N_{H e a t I n p}) + C_{2},

3 D \int_{1}^{N_{H e a t I n p}} \frac{1}{\sqrt{N^{3}}} d N + C_{3} ≅ \frac{- 2}{\sqrt{N_{H e a t I n p}}} + C_{3} .

The constants C₁ = −1.46, C₂ = 0.58, and C₃ = 2.61 were found numerically by setting N_HeatInp →□∞ and rounded to two digits for convenience. The relative accuracy of these approximations for the sum is shown in Fig. 3.

Fig. 3 Relative deviation of the integral approximation for the three dimensions (1D, 2D, and 3D) with respect to the original sum which is <10% for N_HeatInp >3 in any case.

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The relative deviation of the analytical approximation (right expressions in Eqs. (4)) from the exact values (left of Eq. (3)) is < 10% for all three dimensionalities, when N_HeatInp > 3. For N_HeatInp > 100 the error becomes negligibly small. Inserting the Eqs. (4) into (1) yields the following analytical approximations for the temperature increase due to heat accumulation for the three dimensionalities:

1 D Δ T_{H A, 1 D} ≅ \frac{Q_{1}}{ρ \cdot c_{p} \cdot \sqrt{\frac{4 \cdot π \cdot κ}{f_{H e a t I n p}}}} \cdot (2 \sqrt{N_{H e a t I n p}} + C_{1}),

2 D Δ T_{H A, 2 D} ≅ \frac{Q_{2}}{ρ \cdot c_{p} \cdot \frac{4 \cdot π \cdot κ}{f_{H e a t I n p}}} \cdot (\ln (N_{H e a t I n p}) + C_{2}),

3 D Δ T_{H A, 3 D} ≅ \frac{Q_{3}}{ρ \cdot c_{p} \cdot \sqrt{{(\frac{4 \cdot π \cdot κ}{f_{H e a t I n p}})}^{3}}} \cdot (\frac{- 2}{\sqrt{N_{H e a t I n p}}} + C_{3}),

Equations (5a-c) are used in the reminder of the paper to derive analytical constraints and scaling laws for the critical parameters of processes with a large number of repetitive heat inputs.

3. Heat accumulation limits

When the heat inputs are incident at a repetition rate f_HeatInp, and with an absorbed part η_abs of the average incident power f_HeatInp · E_Inc, = P_Inc,av, each single residual heat input remaining in the workpiece is given by

Q_{H e a t} = η_{H e a t} \cdot η_{a b s} \cdot E_{i n c} = \frac{η_{H e a t} \cdot η_{a b s} \cdot P_{I n c, a v}}{f_{H e a t I n p}} .

As η_Heat is a function of both, the laser parameters (such as pulse duration, fluence above threshold, intensity distribution, and wavelength) and the processing parameters (such as material, structure size, and structure depth) it has usually to be determined experimentally. It is noted that the assumption of a constant η_Heat might restrict the parameter range where the model is valid, especially in the case of ultrashort-pulse laser processing. Furthermore it is noted that P_Inc,av equals the incident average laser power only in the case of HAP. For HAS P_Inc,av is the time average of the laser power that is incident on the processed contour and is smaller than the average laser power in the case of open contours due to the positioning time.

The thermo-mechanical properties of the processed material as well as η_Heat, and η_abs, can be combined in the expression

C_{M a t, n D} = \frac{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ)}^{n D}}}{η_{H e a t} \cdot η_{a b s}} = \sqrt{{(ρ \cdot c_{p})}^{2 - n D}} \cdot \frac{\sqrt{{(4 \cdot π \cdot λ_{t h})}^{n D}}}{η_{H e a t} \cdot η_{a b s}} .

It can be seen that C_Mat,nD decreases with increasing η_Heat and η_abs and increases with increasing thermal conductivity λ_th. It is further seen that

C_{M a t, 1} \propto \sqrt{c_{p} ρ}

, C_Mat,2 is independent of ρ and c_p, and

C_{M a t, 3} \propto 1 / \sqrt{c_{p} ρ}

.

The limiting average incident power P_Inc,Limit,nD at which a given maximum acceptable temperature increase ∆T_Limit is reached for a given total number N_tot of heat inputs, is found by inserting Eqs. (2a-c), (6), and (7) into Eqs. (5a-c), setting N_HeatInp = N_tot, and solving for the average incident power, yielding

1 D P_{I n c, L i m i t, 1} ≅ \frac{C_{M a t, 1} \cdot Δ T_{L i m i t}}{σ} \cdot \frac{A \cdot \sqrt{f_{H e a t I n p}}}{2 \sqrt{N_{t o t}} + C_{1}},

2 D P_{I n c, L i m i t, 2} ≅ \frac{C_{M a t, 2} \cdot Δ T_{L i m i t}}{σ} \cdot \frac{ℓ}{\ln (N_{t o t}) + C_{2}}, and

3 D P_{I n c, L i m i t, 3} ≅ \frac{C_{M a t, 3} \cdot Δ T_{L i m i t}}{σ} \cdot \frac{\sqrt{N_{t o t}}}{\sqrt{f_{H e a t I n p}} \cdot (\sqrt{N_{t o t}} C_{3} - 2)} .

It is seen, that in the 1D-case the limiting average power is increasing with the root of the pulse frequency, while it is independent from the frequency in the 2D-case and decreasing with the root of the pulse frequency in the 3D case, if the pulse energy is kept constant.

If, however, the applied average incident power P_Inc,av > P_Inc,Limit,nD, pauses have to be introduced after a certain number N_Limit of heat inputs to keep the temperature increase below the acceptable limit. Hence, for given processing parameters N_Limit is the number of heat inputs, at which the temperature increase accumulated until just before the next heat input reaches ∆T_Limit. N_Limit is found from Eqs. (8a-c) by replacing P_Inc,Limit,nD with P_Inc,av and N_tot with N_Limit, and solving for N_Limit, yielding

1 D N_{L i m i t, 1} ≅ {(\frac{C_{M a t, 1} \cdot Δ T_{L i m i t}}{σ} \frac{A \cdot \sqrt{f_{H e a t I n p}}}{2 \cdot P_{I n c, a v}} - \frac{C_{1}}{2})}^{2},

2 D N_{L i m i t, 2} ≅ e^{\frac{C_{M a t, 2} \cdot Δ T_{L i m i t}}{σ} \cdot \frac{ℓ}{P_{I n c, a v}} - C_{2}}, and

3 D N_{L i m i t, 3} ≅ {(\frac{2 \cdot P_{I n c, a v} \cdot \sqrt{f_{H e a t I n p}}}{P_{I n c, a v} \cdot \sqrt{f_{H e a t I n p}} C_{3} - \frac{C_{M a t, 3} \cdot Δ T_{L i m i t}}{σ}})}^{2} .

The required duration of the pause depends on the cooling rate. In a first rough approximation, the pauses have to reduce the overall average processing power to P_Inc,Limit,nD as given in Eqs. (8). With this assumption the duration of these pauses is

t_{P a u s e, n D} = \frac{N_{t o t}}{f_{H e a t I n p} \cdot n_{P a u s e s, n D}} (\frac{P_{I n c, a v}}{P_{I n c, L i m i t, n D}} - 1),

where the number of pauses is given by

n_{P a u s e s, n D} = ⌊ N_{t o t} / N_{L i m i t, n D} ⌋ .

It should be pointed out that in practice N_tot is the total number of energy inputs which is required for the desired application, e.g. the total number of pulses needed for drilling a hole, or the total number of scans needed to cut the material. It is given by

N_{t o t} = \frac{V_{P r o c} \cdot h_{P r o c}}{η_{P r o c} \cdot E_{I n c}} = \frac{V_{P r o c} \cdot h_{P r o c} \cdot f_{H e a t I n p}}{η_{P r o c} \cdot P_{I n c, a v}},

where V_Proc is the volume of the processed material, h_Proc is the volume specific energy for the desired process and η_Proc is the process efficiency (fraction of incident laser energy that contributes to the process) as defined in [14]. Replacing P_Inc,av with P_Inc,Limit,nD in (12), inserting this into (8), and solving for P_Inc,Limit,nD therefore allows to calculate the maximum average incident power that is applicable to prevent a temperature increase from exceeding ∆T_Limit for any given process. Hence, with materials properties, geometry, volume specific process energy, and process efficiency known and with a given frequency of the heat inputs this uniquely defines the maximum applicable average incident laser power allowed for the process to keep the temperature increase below ∆T_Limit.

4. Frequency of heat inputs and average incident power

In the case of HAP, the frequency of the heat inputs and the average incident laser power simply equal the laser pulse frequency and the average laser power, respectively, i.e.

f_{H e a t I n p} = f_{L a s e r} and P_{I n c, a v} = P_{L a s e r} = E_{P u l s e} \cdot f_{L a s e r} .

In the case of HAS, when processing contours, the frequency of the heat inputs is given by f_HeatInp = (t_Scan + t_Pos)⁻¹, where t_Pos is the positioning time between subsequent scans, if the contour is not closed, and t_Scan = ℓ_Contour / v_Feed is the time required to scan the contour given by the length of the contour ℓ_Contour, and the feed velocity v_Feed, i.e.

f_{H e a t I n p} = \frac{v_{F e e d}}{ℓ_{C o n t o u r} + v_{F e e d} \cdot t_{P o s}} .

The average incident power in the case of HAS is P_Inc,av = f_HeatInp · E_Contour, where E_Contour = E_Pulse ·f_Laser · ℓ_Contour / v_Feed is the total energy incident during one scan along the complete contour, yielding an average incident power

P_{I n c, a v} = P_{L a s e r} \cdot f_{H e a t I n p} \cdot \frac{ℓ_{C o n t o u r}}{v_{F e e d}} = P_{L a s e r} \cdot \frac{ℓ_{C o n t o u r}}{ℓ_{C o n t o u r} + v_{F e e d} \cdot t_{P o s}} .

P_Inc,av equals P_Laser for closed contours, i.e when t_Pos = 0. It is noted that for other process geometries such as ablation of areas with special hatching patterns, modified expressions might apply for Eqs. (14) and (15).

Two experimental examples to illustrate the validity of the above equations and assumptions are given in the following.

5. Materials and methods

For experimental verification of the model the kW-class ultrafast laser of the IFSW [16] was used to ablate grooves on CrNi-steel (1.4310) and cut 2 mm thick CFRP plates. The CFRP consisted of Toray T700S-12k carbon fibers with a RTM 6 matrix, which is a monocomponent resin. The carbon fibers were arranged in different layers with the orientation [0/90, −45/+45, 90/0, 0/90, −45/+45, 90/0]. The volume fraction of carbon fibers was 50%. The laser parameters used for the experiments are summarizes in Table 1.

Table 1. Laser parameters used for processing of CrNi steel and CFRP

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6. Surface ablation of CrNi-steel as an example for 1D HAP

Surface ablation of CrNi-steel as an example for heat accumulation between pulses (HAP) with one-dimensional (1D) heat flow was investigated experimentally using the laser parameters listed in Table 1. The beam was moved with a fast scanner over the surface with the velocity v_Feed in order to create shallow grooves in the surface of the samples. The large focus diameter allows to assume 1D heat flow as shown in Fig. 2 (left) as long as d_Focus is larger than the thermal diffusion length

ℓ_{D i f f} = \sqrt{\frac{4 \cdot κ \cdot N_{P u l e s}}{f_{l a s e r}}},

which holds for CrNi-steel over a duration covering more than 300 pulses at 300 kHz. For a given v_Feed with a number of pulses per spot according to N_Pulses = d_Focus·f_Laser / v_Feed the laser power was increased until a liquid surface in the grooves could clearly be identified by visual inspection. Figure 4 (left and center) shows two microscope images of typical results obtained with single scans with two different average laser powers and v_Feed = 1 m/s, which corresponded to 39 pulses per spot. The shiny surface of solidified liquid inside the groove that was produced with an average laser power of 610 W and the much deeper groove seen in the corresponding height-plot gives evidence that the process has significantly changed compared to the one observed with 306 W of laser power. Figure 4 (right) shows the minimum number of pulses at which a liquid surface is observed as a function of the average laser power. The number of pulses incident on one point in the groove was varied by changing the feed rate v_Feed. For a given number of pulses per spot on the grove, the data points show the mean value between the largest average power P_NL applied that did not lead to a noticeable solidified liquid and deep grooves and the smallest average power P_L that clearly produced a visible solidified liquid and a deep groove. The error bars extend from P_NL to P_L. The line shows the theoretically expected relation according to 1D-model given by Eq. (9a) using the material parameters ρ = 7900 kg/m³, λ_th = 25 W/mK (@800K), c_p = 559 J/kgK (@800K), and η_abs = 37% yielding C_Mat,1 = 7.2·10⁵ J/s^0.5/m²/K. The limiting increase of temperature was set to ΔT_Limit = 1500 K, and σ = 2 as the heat source is on the surface. The only fit parameter was η_Heat = 14%, which was chosen to yield the best fit to the data points.

Fig. 4 (Left and Center) Microscope images of the grooves created with 39 pulses/spot at two different average laser powers. Clear, shiny looking re-solidified liquid is visible inside the groove produced with 610 W of average power (Center). The corresponding height-plot below the image shows a deep groove and a solidified melt-wall on the right side. (Right) Number of pulses after which the surface is still liquid when the next pulse arrives. The squares are the experimental values, the line is a fit of Eq. (9a) to the experimental data using the parameters which are described in the text.

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The good agreement between the theoretical curve given by Eq. (9a) and the experimental findings confirm, firstly, the predominantly one-dimensional heat flow during the relevant time scale, secondly, the assumption of constant material properties, and thirdly, the assumption of a constant fraction η_Heat of the absorbed energy being converted to residual heat for at least the range of parameters considered in this experiment.

7. Multi-scan cutting of thin sheets as an example for 1D HAS

As an example for heat accumulation caused by multiple scans (HAS), fast multi-scan ultra-short-pulsed laser cutting of a thin sheet is now considered as sketched in Fig. 5. This is an important application for generating cuts with very high quality in difficult materials such as thin copper sheets with plastic layers or carbon fiber reinforced plastics (CFRP). Multi-scan cutting is suffering in particular from HAS, when a large number of passes (>2000) over the same contour is required to cut through the whole material [10]. Provided that the following preconditions apply, it can again be assumed that the heat flow away from the cutting kerf into the thin sheet is predominantly 1D.

Fig. 5 Sketch of the cutting kerf during multi scan ablation. 1D-heat flow, represented by the red arrows, is assumed to be established after a large number of scans and sufficiently high scan speed.

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1D approximation precondition

One condition to be able to apply the approximation of 1-dimensional heat flow is that the temperature gradient along the contour $\vec{\nabla} T_{C o n t}$ is small due to the high scan speed, i.e. when the focus diameter $d_{F o c u s} \cdot v_{F e e d} > > κ$ [15]. Furthermore also the temperature gradient $\vec{\nabla} T_{z}$ from the top to the bottom of the material needs to be small, which can be assumed to be the case after a “large number” of scans, i.e. a “long” processing time t_Proc = N_Scans·t_Scan, and the bottom of the sheet is not in contact with a heat sink. More precisely this can be assumed to be the case when the thermal diffusion length ℓ_Diff significantly exceeds the material thickness d_Mat,

ℓ_{D i f f} = \sqrt{4 \cdot κ \cdot t_{P r o c}} > > d_{M a t},

i.e. when the processing time

t_{P r o c} > > d_{M a t}^{2} / (4 \cdot κ)

. For cutting, the input energy may in addition be assumed to be evenly distributed over the kerf walls which further reduces

\vec{\nabla} T_{z} .

In the geometry shown in Fig. 5, σ = 1 as the heat flows into both sides of the heated area. For metals, κ is in the range of about 10⁻⁴ m²/s to 10⁻⁵ m²/s, whereas for CFRP it is in the range of about 10⁻⁴ m²/s (along the fibers) to 10⁻⁶ m²/s (perpendicular to the fibers and through the matrix). In order to ensure 1D heat flow according to the above approximations, the processing time for e.g. a 2 mm thick sample must therefore be larger than about 10 ms to 100 ms for metals and about 1 s for CFRP. For comparison, the duration for 100 passes over a 100 mm long contour at the scanning speed of 10 m/s is 1 s. To ensure small temperature gradients along the contour, the scan speed $v_{F e e d} > > κ / d_{F o c u s}$ should amount to $v_{F e e d} > > 0.1 m/s$ for κ = 10⁻⁶ m²/s and to $v_{F e e d} > > 1 m/s$ for κ = 10⁻⁴ m²/s, assuming a focus diameter of 100 µm. In [11] it was shown that these assumptions hold and that calculating the 1D-heat flow using averaged material properties leads to results that are in excellent agreement with experimental data when cutting CFRP with a cw laser.

Limits of average power and maximum allowed number of scans

For the process geometry described in Fig. 5, the Eqs. (8a) and (9a) resulting from the model with 1D heat flow and Eqs. (14) and (15) for the multi-scan heat input apply. The area A over which the power is dissipated is given by A = ℓ_Contour · d_Mat and the process volume is about V_Proc ≅ d_Focus · ℓ_Contour · d_Mat. Inserting this and Eqs. (12), (14), (15) with P_Laser replaced by P_Inc,Limit,1 and σ = 1 into Eq. (8a) yields the limit

P_{I n c, L i m i t, 1} ≅ \frac{C_{M a t, 1} \cdot Δ T_{L i m i t} \cdot ℓ_{C o n t o u r} \cdot d_{M a t} \cdot \sqrt{v_{F e e d}}}{\sqrt{ℓ_{C o n t o u r} + v_{F e e d} \cdot t_{P o s}} (2 \sqrt{\frac{d_{F o c u s} \cdot d_{M a t} \cdot h_{P r o c} \cdot v_{F e e d}}{η_{P r o c} \cdot P_{I n c, L i m i t, 1}}} + C_{1})},

and uniquely defines the maximum average laser power that can be applied (in the considered situation with 1D heat flow) to prevent exceeding the temperature increase ∆T_Limit. The maximum allowed number of scans for a given laser power is found accordingly from Eq. (9a)

N_{S c a n s, L i m i t, 1} ≅ {(\frac{C_{M a t, 1} \cdot Δ T_{L i m i t}}{2} \frac{d_{M a t} \cdot \sqrt{v_{F e e d} (ℓ_{C o n t o u r} + v_{F e e d} \cdot t_{P o s})}}{P_{L a s e r}} - \frac{C_{1}}{2})}^{2} .

If the available average laser power exceeds P_Inc,Limit,1, i.e. if N_{Scans,Limit,1} < N_tot, the duration and number of the necessary processing pauses can be determined with Eqs. (10) and (11), respectively.

HAS scaling laws for cutting of thin sheets

As C₁/2 ≅ 1 and both, C₁ ≪ $2 \sqrt{N_{T o t}}$ and N_{Scans,Limit,1} ≫ 1 usually apply when the abovementioned preconditions that lead to 1D heat flow are fulfilled, simple HAS scaling laws can be derived from (18) and (19). For a constant process efficiency and closed contours, where t_Pos = 0, and neglecting C₁ one finds

P_{I n c, L i m i t, 1} \approx \frac{C_{M a t, 1}^{2} \cdot Δ T_{L i m i t}^{2} \cdot η_{P r o c}}{4 \cdot h_{P r o c}} \cdot \frac{d_{M a t} \cdot ℓ_{C o n t o u r}}{d_{F o c u s}},

and

N_{S c a n s, L i m i t, 1} \approx \frac{C_{M a t, 1}^{2} \cdot Δ T_{L i m i t}^{2}}{4} \cdot \frac{d_{M a t}^{2} \cdot v_{F e e d} \cdot ℓ_{C o n t o u r}}{P_{L a s e r}^{2}} .

Hence, as long as the abovementioned preconditions to justify the approximation by the 1D heat flow apply, the maximum applicable power increases linearly with increasing material thickness and contour length and decreases with increasing focus diameter and volume specific process energy. Under the same preconditions the maximum allowed number of consecutive scans increases linearly with increasing feed rate and the contour length. Unfavorably, however, it decreases with the square of the average laser power. The scaling with the square of the material thickness has to be interpreted with caution as it results from the assumption of the absorbing geometry, which is only allowed when the thermal diffusion length is much larger than the material thickness, which does not hold in the beginning of processing.

Multi-scan cutting of CFRP with a 1.1 kW ps-laser

The validity of the derived approximations for N_Limit,1 due to HAS were verified with multi-scan cutting of circles with a diameter of 50 mm in 2 mm thick CFRP [10] described in the section “materials and methods” using the processing parameters summarized in Table 1. The beam was moved with a fast scanner with a maximum scan speed of v_Feed = 30 m/s. To avoid HAP the minimum scan speed was set to 3 m/s corresponding to 10 pulses per spot.

The values of N_{Scans,Limit,1} were experimentally determined by increasing the number of scans at each processing parameter set, until a clear increase of the damage of the matrix material was visible in cross-sections inspected with an optical microscope. The measured N_{Scans,Limit,1} as a function of the feed rate and as a function of the average power are shown as squares in Fig. 6 left and right, respectively. The solid lines show the calculated relations as given by Eq. (19) assuming ∆T_Limit = 400 °C for the damage by evaporation of the matrix material and using the identical constant C_Mat,1 = 9.0·10³ J/s^0.5/m²/K for both graphs. This value for the material constant C_Mat,1 is motivated by assuming η_Abs = 90%, η_Heat = 90%, ρ = 1610 kg/m³ and c_p = 906 J/kg/K (the average of the material values according to the volume fraction), and λ_th = 2.9 W/m/K (the geometrical average of the thermal conductivity of 50 W/m/K, 5 W/m/K, and 0.1 W/m/K for Carbon parallel to the fibers, Carbon perpendicular to the fibers, and plastic, respectively (see also [11] for comparison).

Fig. 6 Limiting number of scans when cutting CFRP at which the matrix damage significantly increases as a function of the average laser power (left) and as a function of the feed rate (right). The squares are the experimental results. The lines are the results from the model using the identical material constant for both graphs as described in the text.

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Strictly speaking and in view of the many uncertainties concerning the actual material parameters, C_Mat_,1 here has the role of a fitting parameter. Nonetheless, the good agreement between the theoretical model and the experimental results shown in Fig. 6 confirm the basically inverse quadratic scaling of the maximum allowed number of scans with respect to the average power (left) and the predominantly linear scaling of the maximum allowed number of scans with the feed rate (right) as explicitly seen from the approximation given in Eq. (21). It should be mentioned, that C_Mat_,1 is identical for the calculated limits in both, Fig. 6 (a) and (b). Furthermore, the large error bars have to be taken into account to not be misled by the apparently constant experimental values between about 8 m/s and 20 m/s in Fig. 6 (b).

Using the above material constants, h_Proc = 45 J/mm³, η_Proc = 10%, and the contour length of 157 mm, the upper limit of the average laser power given by Eq. (20) yields 24.8 W, which applies when the whole sample is to be cut through without pauses. By inserting this average power into Eq. (11) and Eq. (10) one finds 38 pauses with a duration of 15 are required, if the available laser power of 1.1 kW shall be applied instead. In the experiments that were performed before developing the present theory, very high quality cuts were achieved by inserting pauses with a duration of 60 s after every 200 passes [10].

8. Conclusion

The approximation formulas derived in this paper for the three different dimensionalities nD ∈ {1,2,3} of the heat flow allow to predict the maximum allowed average power P_Limit,nD and the corresponding maximum number of heat inputs N_Limit,nD that can be applied for a given process before the local temperature increase due to heat accumulation reaches a specified limit ∆T_Limit.

The approximation formulas show that the maximum number N_Limit,1 of consecutively applicable heat inputs in the important case of 1D heat flow unfavorably decreases with the inverse of the square of the average laser power, but increases linearly with the feed rate and the contour length. If more than N_Limit,1 scans are required to produce a certain structure, pauses have to be introduced into the process to avoid thermal damage. The duration and the number of these pauses can be calculated with Eqs. (10) and (11), respectively.

These findings agree very well with experimental results: Two examples of practical relevance, surface ablation of CrNi-steel as an example for heat accumulation due to pulses (HAP) and cutting of thin CFRP-sheets as an example for heat accumulation due to multiple scans (HAS) were experimentally investigated, and an excellent agreement of the scaling of the limiting number of heat inputs with respect to average power and feed rate was found.

A major and important task for future work will be to model and determine C_Mat,nD, and in particular η_Heat, in order to specify the basic physical limits for the maximum possible number of heat inputs and the maximum average power.

Funding

German Research Foundation (DFG) (grant GR 3172/17-1); Russian Foundation of Basic Research (grant 15-02-91347); and the BMBF (grant 13N13931).

References and links

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10. C. Freitag, M. Wiedenmann, J.-P. Negel, A. Loescher, V. Onuseit, R. Weber, M. Abdou Ahmed, and T. Graf, “High-quality processing of CFRP with a 1.1-kW picosecond laser,” Appl. Phys., A Mater. Sci. Process. 119(4), 1237–1243 (2015). [CrossRef]

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16. J.-P. Negel, A. Voss, M. Abdou Ahmed, D. Bauer, D. Sutter, A. Killi, and T. Graf, “1.1 kW average output power from a thin-disk multipass amplifier for ultrashort laser pulses,” Opt. Lett. 38(24), 5442–5445 (2013). [CrossRef] [PubMed]

Laser property	CrNi-Steel	CFRP
Wavelength, λ_L	1030 nm	1030 nm
Pulse duration, t_P	8 ps	0.0627
M²	1.3	1.3
Pulse frequency, f_Laser	300 kHz	300 kHz
Average power, P_Laser,max	≤600 W	≤ 1100 W
Pulse energy, E_P,max	≤2 mJ	≤3.7 mJ
Focus diameter, d_Focus	130 µm	125 µm

Processing constraints resulting from heat accumulation during pulsed and repetitive laser materials processing

Abstract

1. Introduction

Heat accumulation scenarios

Heat accumulation equation

2. Approximations for large numbers of heat inputs

3. Heat accumulation limits

4. Frequency of heat inputs and average incident power

5. Materials and methods

6. Surface ablation of CrNi-steel as an example for 1D HAP

7. Multi-scan cutting of thin sheets as an example for 1D HAS

1D approximation precondition

Limits of average power and maximum allowed number of scans

HAS scaling laws for cutting of thin sheets

Multi-scan cutting of CFRP with a 1.1 kW ps-laser

8. Conclusion

Funding

References and links

Cited By

Figures (6)

Tables (1)

Equations (31)

Optics Express