Understanding the Rayleigh instability in humping phenomenon during laser powder bed fusion process

Figure 3 shows the wavelength and amplitude at various spatial offsets and laser powers. Values for power in the plot refer to the power of a single laser. The standard deviation is represented using shaded bands to enhance clarity.

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From figure 3(a), we notice that wavelength does not change significantly with spatial offset $l$ at a constant laser power. For example, wavelength remains constant between 600 and 700 μm for 40 W power inputs, even though the spatial offset increases from 0 to 300 μm. At the same time, wavelength increases from around 600 μm to around 1100 μm as laser power increases from 40 W to 100 W. This can be explained by approximating the molten track as a free-standing cylinder with radius $R$. The wavelength must exceed the circumference of the cylinder, i.e.

$$\begin{equation}\lambda \, > \,2\pi R,\end{equation} \tag{ 1 }$$

so that instability commences and perturbation begins to grow [17]. A higher-powered laser beam creates a molten track with larger radius and consequently greater circumference. Therefore, it is expected that wavelength increases with higher laser power.

More interestingly, the amplitudes vary significantly with varying spatial offsets, as shown in figure 3(b). The amplitude increases at specific spatial offsets $100 < l < 200\,\mu {\text{m}}$ for 40 and 60 W laser power, and between $120 < l < 270\,\mu {\text{m}}$ for 80 W laser power. At the same time, the amplitude is nearly constant for 100 W laser power. Here, we consider these molten tracks with low amplitudes (or more precisely, low ratio between amplitude and wavelength, $A/\lambda < 0.013$ as explained in section 3.2) as stable ones. Roughly speaking, unstable molten tracks exist because when the spatial offset increases from 0 to ∼150 μm, the energy density decreases as the effective shape of the combined beam becomes elongated. Consequently, a shorter contact length between the molten track and the substrate is created. The larger contact angle that results from this leads to larger undulations. The detailed explanation is provided in the next section. As the two spots are further separated ($l$ becomes larger than 200 μm), the two molten pools cease to remain connected. In such a case, the amplitude drops as the melting process is no longer different from using one laser beam to scan a single line twice, as the lagging beam can remelt and smoothen the molten track.

To further probe this instability, we turned toward the Rayleigh–Plateau instability of a liquid jet with infinite length. In specific, we consider the molten track as a liquid bead suspended on a flat surface, similar to some previous fluids models [18–20]. In such case, the original cylindrical system is modified with a quasi-1D model for a cylinder segmented on a flat surface. The equations for motion and mass conservation are:

$$\begin{equation}\rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}}} \right) = - \frac{{\partial p}}{{\partial x}}\end{equation} \tag{ 2 }$$

and

$$\begin{equation}\frac{{\partial S}}{{\partial t}} + u\frac{{\partial S}}{{\partial x}} = - S\frac{{\partial u}}{{\partial x}},\end{equation} \tag{ 3 }$$

where $S$ is the molten track's cross-sectional area as shown in figure 2, $u$ the flow velocity, $t$ time, $\rho$ the material density, and $p$ the pressure inside of the molten track.

To approximate the cross-sectional area of a molten track, we treat the cross-section as a circular segment. A geometrical analysis gives

$$\begin{equation}S = \frac{{{m^2}\left( {\theta - {\text{sin}}\theta {\text{cos}}\theta } \right)}}{{{{\left( {1 - {\text{cos}}\theta } \right)}^2}}}{\text{, }}\end{equation} \tag{ 4 }$$

where $m$ is the bead height from confocal measurement as sketched in figure 2, and $\theta$ is the contact angle calculated using the equation

$$\begin{equation}R = \frac{m}{{1 - {\text{cos}}\theta }} = \frac{b}{{2{\text{sin}}\theta }}{\text{, }}\end{equation} \tag{ 5 }$$

where $R$ is the radius of the circular segment modeled from the cross-section, and $b$ is the contact length. Illustrations for variables $R$, $\theta$, and $b$ are shown in figure 2. For each molten track, we assume the contact length $b$ is constant, as suggested in literature [21].

With $m\left( x \right)$ collected from confocal measurement, $S\left( x \right)$ can be calculated using $m\left( x \right)$ and $b\,.\,S\left( x \right)$ is then integrated and averaged over a period of length $\lambda$ to find the mean cross-sectional area $\bar S$ for each molten track, as shown below

$$\begin{equation}\bar S = \frac{1}{\lambda }\int\limits_{}^\lambda S \left( x \right)d\left( x \right).\end{equation} \tag{ 6 }$$

Therefore, we can estimate the original, unperturbed shape of the circular segment, which translates to the melting volume (when $\bar S$ is integrated over track length).

Then the capillary pressure at position $x$ can be written as

$$\begin{equation}p = \gamma \left( {1/R + 1/R^{^{\prime}}} \right){\text{, }}\end{equation} \tag{ 7 }$$

where $\gamma$ is the surface tension, $R$ computed from equation (5), and $1/R^{^{\prime}}$ is the average value of the longitudinal curvature of the molten track. The longitudinal curvature can be further simplified as shown below, according to Schiaffino [22]

$$\begin{equation}1/R^{^{\prime}} \approx - \beta {\partial ^2}h/\partial {x^2}{\text{, }}\end{equation} \tag{ 8 }$$

where $- {\partial ^2}h/\partial {x^2}$ is the curvature of the surface over the centerline. $\beta$ is a coefficient that accounts for the difference between the longitudinal curvature over the centerline and the average longitudinal curvature, as suggested from literature [20, 22]. Based on literature [20, 22], $\beta$ is between 0.65 and 0.56 when $0.42\;\pi < \theta < 0.86\;\pi$, which is the range of the contact angle in our work.

Now with the bead's height along the $x$-axis and over time $t$ approximated as

$$\begin{equation}h = {h_0} + \epsilon {{\text{e}}^{\omega t + ikx}}{\text{, }}\end{equation} \tag{ 9 }$$

where ${h_0}$ is the unperturbed height, $\epsilon$ the initial perturbation, $k = 2\pi /\lambda$ is the wave number, $\omega$ is the growth rate of the perturbation, and $h$ at $t =$ solidification time is equivalent to the measured height profile $m\left( x \right)$. The unperturbed height is the height of a homogeneous molten track before the instability grows (height of the uniform bead with cross-section area $\bar S$). The initial state of a molten track is shown in figures 1(b) and (e).

Then the dispersion relation is shown below, using equations (2), (3), (5) and (7) after substituting equations (8) and (9):

$$\begin{align}& \frac{{2\rho h_0^3{\omega ^2}}}{\gamma }\nonumber\\ & \quad = \frac{{{{\left( {k{h_0}} \right)}^2}\left[ {\bar\theta - {\text{sin}}\;\bar{\theta } \,{\text{cos}}\;\bar{\theta}}\, \right]\left[ {{\text{cos}}\;\bar {\theta } \left( {{\text{cos}}\,\bar \theta - 1} \right) - \beta {{\left( {k{h_0}} \right)}^2}} \right]}}{{{\text{sin}}\,\bar \theta - \bar \theta \,{\text{cos}}\,\bar \theta }}\end{align} \tag{ 10 }$$

as detailed derivation can be found in the previous literature [22]. Here, $\bar \theta$ describes the contact angle of a homogeneous molten track at the unperturbed state with a cross-section area of $\bar S$. Thus, $\bar \theta$ is constant for each molten track. This is in contrast to $\theta \left( x \right)$, the contact angle of the solidified molten track as shown in figure 2, which is a function of $x$ and can be calculated based on height profile $m\left( x \right)$ and contact length $b$. On the other hand, $\bar \theta$ is calculated from the geometry relations in equation (5) after $\bar S$ is determined.

For the molten track to be unstable, $\omega$ must be real. This condition, applied to equation (10), gives

$$\begin{equation}{\left( {k{h_0}} \right)^2} > {\text{cos}}\;\bar \theta \left( {{\text{cos}}\;\bar{\theta } - 1} \right)/\beta {\text{, }}\end{equation} \tag{ 11 }$$

which suggests that the instability can only arise at a large contact angle when $\bar \theta > \pi /2$.

From the dispersion relationship in equation (10), one can also find the most rapidly growing wave number ${k_{\text{*}}}$ with the corresponding growth rate ${\omega _{\text{*}}}$, as shown below [20]:

$$\begin{equation}{\left( {{k_{\text{*}}}{h_0}} \right)^2} = \frac{{{\text{cos}}\,\bar \theta \left( {{\text{cos}}\,\bar \theta - 1} \right)}}{{\left( {2\beta } \right)}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{equation} \tag{ 12 }$$

$$\begin{equation}\frac{{\rho h_0^3\omega _{\text{*}}^2}}{\gamma } = \frac{{{\text{co}}{{\text{s}}^2}\,\bar \theta {{\left( {{\text{cos}}\,\bar \theta - 1} \right)}^2}\left( {\bar \theta - {\text{sin}}\,\bar \theta {\text{cos}}\,\bar \theta } \right)}}{{8\beta \left( {{\text{sin}}\,\bar \theta - \,\bar \theta \,{\text{cos}}\,\bar \theta } \right)}}.\end{equation} \tag{ 13 }$$

Figure 4 plots the stable and unstable regions based on equations (11) and (12). ${h_0}$, the unperturbed bead height can be calculated from the averaged cross-sectional area $\bar S$ and fixed contact length $b$ based on geometry correlations. Empty data points represent unstable molten tracks while filled ones for the stable tracks. In this work, a molten track is deemed stable if $A/\lambda < 0.013$. This threshold is chosen to best distinguish stable molten tracks upon visual inspection of all confocal images. Afterwards, data points are grouped based on their stability and the processing conditions in figure 3. Data points of Gaussian represent the experiments with a single Gaussian beam at various laser powers. An unstable region exists in figure 3 because for contact angle $\bar \theta > \pi /2$, the internal pressure moves liquid from lower $h\left( x \right)$ to higher $h\left( x \right)$, thus growing the perturbation.

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We notice that the results from molten tracks mostly follow the stability analysis in figure 4, suggesting a good alignment between the Rayleigh–Plateau instability theory and our experimental results. For example, many molten tracks produced by 100 W laser power remain in the stable regime, aligning with the observation from figure 3 in which 100 W laser produces tracks with long wavelength but small amplitude. One may find that the stable data points (filled points) in figure 4 still carry finite wavelengths, while unperturbed molten tracks should have infinite wavelength and ${h_0}/\lambda = 0$. This is because small ripples are difficult to avoid in PBF/LB-M and these molten tracks with small ripples are deemed stable based on our stability threshold. In summary, we think the trend between $h/{\lambda _0}$ and $\bar \theta$ follows the dispersion relation from linear stability analysis nicely.

From analysis above, we notice that the contact angle $\bar \theta$ is vital in determining the stability of the molten track: a contact angle greater than $\pi /2$ introduces growing perturbations. To correlate the contact angle to the processing parameters, we first normalize energy input with enthalpy at melting, ${h_{\text{s}}}$, similarly to previous literature [16, 23] to present all data from this work in the same context. In this case, ${h_{\text{s}}} = \rho c{T_{\text{m}}}$, where $c$ is the specific heat capacity and ${T_{\text{m}}}$ the melting temperature. We approximate the combined laser beam in this experiment as an elliptical beam, with a semi-major axis length $\left( {\sigma + l} \right)/2$ and a semi-minor axis length $\sigma /2$, similarly to our previous work [24]. With a dwell time $\tau \sim \left( {\sigma + l} \right)/\left( {2v} \right)$ proportional to the length of the major axis of the combined elliptical beam, the laser energy absorbed is ${A_{\text{b}}}P\tau$, where ${A_{\text{b}}}$ is the absorptivity, and $P$ is the total laser power input. This energy is distributed in a volume of $\pi \left( {\sigma /2} \right)\left[ {\left( {\sigma + l} \right)/2} \right]\sqrt {D\tau }$, where $D$ is thermal diffusivity. This gives an energy density of ${A_{\text{b}}}P\tau {\left\{ {\pi \left( {\sigma /2} \right)\left[ {\left( {\sigma + l} \right)/2} \right]\sqrt {D\tau } } \right\}^{ - 1}}$. Normalizing the energy density by the melting enthalpy and following the same modification as in previous literature [16, 25], we have the normalized enthalpy $\Delta H/{h_{\text{s}}}$ as

$$\begin{equation}\frac{{\Delta H}}{{{h_{\text{s}}}}} = \frac{{{A_{\text{b}}}P\tau }}{{{h_{\text{s}}}\left( {\sigma /2} \right)\left( {\sigma /2 + l/2} \right)\sqrt {\pi D\tau } }},\end{equation} \tag{ 14 }$$

where

$$\begin{equation}{h_{\text{s}}} = \rho c{T_{\text{m}}}{\text{,}}\end{equation} \tag{ 15 }$$

and

$$\begin{equation}\tau = \frac{{\sigma + l}}{{2v}}{\text{.}}\end{equation} \tag{ 16 }$$

.

Figure 5 plots the averaged contact angle $\bar \theta$ against normalized enthalpy, which describes the amount of energy density above that required to melt the material. It is clear that, as data points from various laser powers and beam shapes collapse into the same trend, contact angle decreases with increasing normalized enthalpy. Therefore, we observe more instability in lower laser power experiments as shown in figure 3, because lower laser power and consequently lower normalized enthalpy tend to create greater contact angles and introduce growing perturbations to a molten track, with the reasoning provided below.

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From figure 2 and geometry correlations, contact angle between the molten track and the substrate clearly depend on (a) the contact length between the liquid bead and the substrate, and (b) the volume of liquid accrued on the substrate (gray area in figure 2). A small contact length together with a large volume of liquid accrued is more likely to yield a large contact angle and therefore a strong instability and undulation with a higher amplitude. First, we discover that contact line length increases continuously with increasing normalized enthalpy (see supplementary materials figure S1 available online at stacks.iop.org/IJEM/4/015201/mmedia). At the same time, experimental data reveal that while changing beam shape can impact the liquid accrued when the laser power is constant, the volume of liquid accrued will reach a plateau with increasing power and normalized enthalpy (see supplementary materials figure S2) because only a finite volume of powder is deposited and available for melting. Therefore, a higher normalized enthalpy cannot continuously increase the melting volume of a molten track, but it can continuously increase the contact length between the molten track and the substrate. Thus, contact angle decreases with increasing energy input and introduces less amplitude growth.

With the stability criteria established in the previous sections, we now turn to the understanding of the correlation between the Rayleigh–Plateau instability and the resulting amplitude of the undulations. Theory only predicts a growth rate for the perturbations, and the molten track can ultimately break into small droplets to minimize its surface energy should the growth time be infinite. However, molten tracks solidify within a finite period of time in our PBF-LB/M experiment, preventing the perturbations from further growing. Thus, in additional to the perturbation growth rate from the dispersion relation, we need a time scale to represent the solidification time for a more comprehensive understanding towards the amplitude of the undulations.

Figure 6 plots the fastest growth rate for perturbations versus normalized enthalpy. Only tracks deemed unstable are plotted as stable tracks do not have ${\omega _{\text{*}}}$. We calculate the fastest growth rate ${\omega _{\text{*}}}$ from equations (5) and (13). Generally speaking, ${\omega _{\text{*}}}$ decreases with increasing normalized enthalpy. In particular, the growth rate drops rapidly for $\Delta H/{h_{\text{s}}} > 10$, a domain inhabited primarily by data points from the 100 W laser power experiments. Such domain corresponds to the small oscillations which generates small undulations in figure 3. As we discussed earlier, these molten tracks receive high energy input so their molten pools can extend deeply into the substrate, resulting in large contact length, small contact angle, and slower perturbation growth. Conversely, decreased normalized enthalpy leads to shortened contact length, enlarged contact angle, and accelerated perturbation growth.

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Figure 7 plots the ratio between amplitude $A$ and wavelength $\lambda$ versus the perturbation growth, a dimensionless number obtained by multiplying growth rate ${\omega _{\text{*}}}$ and a time scale, chosen to be the dwell time $\tau$ from equation (16). Figure 7 shows that there exists a clear threshold value for ${\omega _{\text{*}}}\tau$ near unity, below which the amplitude of the undulation remains low at around 0.015 of the wavelength $\lambda$. Once ${\omega _{\text{*}}}\tau$ exceeds the threshold, the ratio between amplitude and wavelength grows rapidly with increasing ${\omega _{\text{*}}}\tau$. The greatest $A/\lambda$ ratios are seen at lower laser powers, where the growth rate ${\omega _{\text{*}}}$ is very large compared to those at higher laser powers, allowing the undulation to grow to a greater amplitude before solidification. As $A \sim {{\text{e}}^{\omega t}}$ and $A \sim {\omega _{\text{*}}}\tau$ at small $\tau$, a faster growth rate together with a longer melting duration from a longer dwell time can lead to rapid growth of amplitude.

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Here, we recognize that dwell time cannot precisely represent the time scale, as dwell time is only affected by the spatial offset $l$, with speed $v$ being constant in our experiment. In reality, higher laser powers should create longer molten pools (the length of a liquid bead along the x-axis) for constant $l$ and $v$. A solidification time, calculated from observed molten pool length divided by laser scan speed, can be a more accurate time scale, but it is more difficult to measure. Nonetheless, scaling with ${\omega _{\text{*}}}\tau$ still does well to collapse datasets from various laser powers and different beam shapes into a single pattern in figure 7.