Figure 4.5 – Maximum irradiation of a powder grain by a top-hat laser beam.
From Tab. 4.2 it can be concluded, that for the selected parameters the melting of a typical 18Ni(300) Maraging Steel powder grain starts in a period of time less than 2µs, and the particle is completely molten after 3 − 5 µs.
In Tab. 4.2 the values of ts and tl are compared to the maximum time of grain irradiation tmaxi . To define this time we go back to the system ‘powder-laser’. Consider a top-hat laser beam of radiusω, travelling with a speed υ and irradiating a powder bed. We assume that the powder bed consists of equally spherical particles of radius r0. The maximum time while a powder grain can be located entirely within the border of the laser spot (see Fig. 4.5) is:
tmaxi = 2(ω − r0)/υ. (4.10)
According to (4.10), for laser scanning speeds 1 − 3 m/s, tmaxi is 24 − 71 µs (Tab. 4.2).
From Tab. 4.2 we can see, that the time of the particle complete melting is several times less than its irradiation time by the laser.
4.3 Evolution of powder bed properties
We have shown before, that the irradiation of a powder grain by a plane wave is not homoge-neous. The example of the temperature gradient inside the grain can be seen in Fig. 4.3. From this picture it is clear that when a part of the grain is already molten, a part of it still stays solid.
At the level of a powder bed, when a powder grain begins to melt, the molten material starts to form necks with the neighbouring particles. As it is shown in [1], at the first moment necks between particles are small and unstable, while during further laser irradiation they are merging and growing. Formation of stable interparticular necks between powder grains can be considered as a starting point of the evolution of the effective properties (absorptivity, thermal conductivity) of the powder bed.
Chapter 4. Single Grain Model
Figure 4.6 – Evolution of the proportion of the molten material during irradiation of two neighbouring grains: (i) t < t0; (ii) t ' t0, (iii) t > t0.
To connect the melting of a separate powder grain to the properties of a powder bed, i.e. to link the Single Grain Model to the homogeneous medium hypothesis, commonly used for SLS/SLM simulation (see Chapter 6 for details), we will estimate the evolution of the average temperature of the grain during its melting by a plane wave.
We need to define the moment when a neck starts to form between two neighbouring powder grains. For this purpose we will make some assumptions on the irradiated powder bed:
• Surface of the powder bed is perfectly ‘flat’, i.e. the centers of powder grains in the surface layer of the powder are situated in the same horizontal plane.
• Two neighbouring grains on the surface of the powder bed are irradiated in the same way by the laser.
• Back-scattering of laser irradiation by powder grains is not taken into account.
• Heat transfer between the grains during their melting is not taken into account.
The position of a contact point between two neighbouring grains is shown in Fig. 4.1. During laser irradiation of the grains, the time t0, when the melt front reaches the contact point, can be considered as the time of the neck-forming, i.e. when powder bed properties evolution starts.
We use the Single Grain Model to estimate the value of t0. We study the temperature evolution in a point on the surface of a spherical particle, irradiated by a plane wave. The particle is made of 18Ni(300) Maraging Steel (see properties in Tabs. 4.1 and 3.6), and the point we consider is a contact point to a neighbouring particle (see Fig. 4.1).
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4.3. Evolution of powder bed properties
Figure 4.7 – Temperature evolution in the contact point between two grains irradiated by a plane wave, approaching the laser of different powers.
The intensities of the plane are adapted to approach a top-hat laser beam with a radius 40 µm and powers 100 W, 200 W or 300 W (Tab. 4.3). The simulated temperature evolution is presented in Fig. 4.7. The time t0, when the melt front reaches the contact point, is also shown on this figure and its exact value is presented in Tab. 4.3.
Fig. 4.6 presents the stages of the evolution of the molten material proportion during laser irradiation of two neighbouring grains on the surface of a powder bed. As we can see, at the moment, when the melt front reaches the contact point between the particles, part of each particle still stays solid. To connect these results to the homogeneous medium model, we will estimate the average temperature of the grains Tav0 at the moment t0.
In Fig. 4.8 one can observe the evolution of the average temperature of the grains Tav0 during their irradiation with a plane wave. The exact values of Tav0 for the chosen intensities/powers are presented in Tab. 4.3.
Laser power P Wave intensity I0(4.4) C. p. melt time t0 Av. temp. Tav0
100 W 19894 W/mm2 3.92µs 1353◦C
200 W 39789 W/mm2 1.76µs 1213◦C
300 W 59683 W/mm2 1.04µs 1089◦C
Table 4.3 – Melting time t0of the contact point between two grains, irradiated by a plane wave of intensity I0, and the average temperature Tav0 of the grains at this moment.
Chapter 4. Single Grain Model
Figure 4.8 – Evolution of the average temperature of a powder grain (Tab. 4.1), irradiated by a plane wave, approaching the laser of different powers.
As it can be seen from Tab. 4.3, for the selected parameters, the melt front reaches the contact point, when the average temperature of the grain Tav0 is less than the melting point Tf = 1413◦C of the material.
It has to be observed on Fig. 4.8 that the time dependency of the average temperature of the grain Tav0, heated by a plane wave, is not linear. The non-linearity is mostly explained by the phase transition. For example, if we look at the curve, corresponding to a laser power of 100 W, we see that the non-linear part is enclosed between t1s= 1.84 µs and t1l = 6.88 µs. These values correspond to the moments when the grain starts to melt and when it is completely molten, respectively (see Tab. 4.2).
Outside the interval (ts, tl), Tavdepends almost linearly on time. Since we neglect losses for the surface of the grain, the slope of the graph is given by the ratio between the absorbed power and the (almost constant) thermal mass of the grain Mth
Mth=4
3πr03ρsCp. (4.11)
In this formula Cpis the heat capacity of 18Ni(300) Maraging Steel: Cp' 0.5 J/g/◦C for any temperature (see Fig. 3.3).
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