Advancing quantitative description of porosity in autogenous laser-welds of 304L stainless steel

Utilizing the reconstructions obtained and the known voxel resolutions for each weld sample, physical measures of pore size, population and frequency were calculated for pores constituting ninety-percent or more of the voided space within each sample. These values serve as a baseline and comparison for readily employed measures of pore presence.

Using tools developed by Voorhees et al.[19‐24], the interfacial morphology of porosity was also examined. The primary tools used to illustrate these quantitative descriptors of shape are the interfacial shape[20, 22] and interfacial normal[24] distributions. Given a triangulation representing the interface between two phases of a discretized microstructure, the mean (H) and Gaussian (K) curvatures can be calculated for each patch of interface according to the method of Meyer et al.[25]. These in turn, can be used to calculate the minimum and maximum principal curvatures, κ₁, κ₂ respectively, for each patch of interfacial area as follows.

This analysis allows for categorization of all patches into a specific (κ₁, κ₂) pairing. The method used to visualize this probability of principal curvature pairings is a two-dimensional color contour plot in which the horizontal and vertical axes are κ₁ and κ₂, respectively, and the color indicates the probability associated with each pair of principal curvatures. This visualization technique is called the Interfacial Shape Distribution (ISD)[20, 22, 23] and its descriptive legend is reproduced here for convenience, Figure 3. For the purposes of this study, the “L” phase, as illustrated in the figure, corresponds to the pores within welds (the figure was originally made for a solid–liquid system). To interpret the ISD, it is useful to arrange the curvature pairings into four major regions or categories. The uniqueness of these four categories is determined by the combination of positive or negative signs of H and K. Patches in which both H and K are positive correspond to Region 1, which are spherical or ellipsoidal with solid within. Patches in which the H is positive and K is negative correspond to region 2. Region 3 consists of patches that have negative H and K values. Regions 2 and 3 contain saddle shaped patches. Lastly, patches that have a negative H and positive K correspond to Region 4, which are spherical or ellipsoidal with the pore phase within. Based upon the population of each principal curvature within one complete three-dimensional reconstruction, which we will also refer to as a dataset, a probability can be assigned denoting the likelihood of encountering a particular pairing of principal curvatures for a given patch on the interface. It is important to note that the ISDs presented in this study have not been normalized by any characteristic length-scale and therefore illustrate a combined effect of both shape and size. As a result, the color bar associated with the ISDs presented later in the results section have units of μm² so that the integration of the probability function over the entire ISD is equal to 1.

The interfacial normaln ^ associated with each interfacial patch of a dataset is used to define a probability distribution for their orientation in three-dimensional space. The method used to visualize this probability distribution is the Interfacial Normal Distribution or IND[23, 24]. In this visualization technique, the two-dimensional projection of a sphere with respect to a given axis displays the probability of occurrence of a given normal orientation. In this study, all INDs are presented as projections along the positive z-axis, which is also the direction of travel for the work-piece beneath the welding laser. Thus, the upper and lower hemispheres correspond to the direction toward and away from the laser, respectively. The color values at each location in the IND indicate the probability of encountering a particular normal based on the population of normals within the dataset. The color bar associated with each IND presented later in the section on results represents non-dimensional probability.

To better understand the spatial distribution of pore interfaces, we develop a method to calculate interfacial-distance distributions (IDD) that provide the probability distribution of the inter-pore distances, as measured between the pore surfaces nearest to one another. In this paper, we refer to the IDD of the pores as “pore-interspacing distribution” (PID). The method is based on the scheme for calculating the channel-size distributions of complex three-dimensional microstructures[26]. Further details regarding this methodology will be available in a forthcoming publication. Briefly, to calculate the interface-distance distribution, the image data of the 3D microstructure of the weld is first converted to a signed distance function, the magnitude of which represents the distance from the nearest pore-solid interface. The sign of the value indicates phase within the dataset. In this study, points inside the pores are represented as positive distances and points in the solid are represented as negative distances. Each isodistance of a microstructure is an isosurface defined by thresholding the distance function associated with the microstructure at a particular distance value. We refer to the regions with distance function values greater than the threshold as “voids” to distinguish them from the original pores as the sizes and topological characteristics of the voids differ from those of the pores. (Note that the interface of the microstructure is described by the isosurface with a threshold value of zero.) Topological changes in the isodistance structure from one threshold to another denote the feature sizes of the structure. The differences between isodistance structures resulting from thresholding in the positive and negative directions are illustrated in the two-dimensional schematics of Figure 4, where the value of the distance function is represented by grayscale and the isocontour (analogous to isosurface in three dimensions) of each threshold value is marked by a black line.

When the isodistance structures join together from thresholding with a negative threshold value, a change in the number of voids arises. This occurs at a distance value corresponding to half of the pore interspacing (the distance between interfaces at the narrowest point). Since we are examining systems that contain a variety of spatial distributions of pores, PIDs are calculated by measuring the rate at which pores are joining as a function of the distance threshold. Specifically, the PID is calculated by taking the negative derivative (−1 times the derivative) of the number of voids as a function of twice the distance threshold. Numerically, a central differencing method is used to calculate the derivatives. Each point in the PID represents the probability of finding a pair of pores with the pore interspacing at the corresponding distance threshold value. Furthermore, a characteristic pore interspacing is calculated by taking the weighted mean of the pore interspacing.

Similarly, when the isodistance structures are created for positive threshold values, a change in the number of voids also arises when the threshold becomes larger than the largest distance function value within a pore. The corresponding distance value is half the smallest dimension (the radius for a sphere) of the pore. Once again, measuring the rate at which pores disappear as a function of the distance threshold gives a pore-size distribution. Lastly, the characteristic pore interspacing is scaled to yield solid linear fraction (SLF):

where R is one half of the characteristic pore interspacing and r is the characteristic pore radius. The SLF provides a measure of local linear fraction of solid along the path connecting the center of the particles and passing through the narrowest matrix region. Unlike pore volume fraction, another commonly used measure of density, the SLF does not depend on the volume used for the calculation. This is of particular note for each weld schedule studied here, as laser weld porosity is generally a localized phenomena often occurring at the centerline of the weld and not distributed homogeneously throughout the weld. Furthermore, the SLF is useful as it yields a quantitative metric of solid material between regions of densely populated pores relative to the size of pores present. It is expected that this type of spacing sensitivity metric would have a strong influence on the mechanical properties of the weld.

As mentioned previously, a variety of population metrics were extracted from each weld schedule including but not limited to average and maximum pore volume, linear frequency, and total number of pores observed per weld case. Average pore volume for both focal lengths are shown in Figure 5 where each data series corresponds to a specific travel speed and data are presented as a function of delivered power. The trend is consistent in that for a given travel speed, increases in welding power produce larger pores. Slower travel speeds also generally produce larger pores with some exceptions to this trend. Linear frequency was obtained by taking the total number of observed pores for a given weld and dividing it by the weld length. Figure 6 illustrates a more complex dependence of linear frequency on power and travel speed. With the exception of the 2032 mm × min⁻¹ data series, an inflection point for the maximum frequency of pores was observed across all series of travel speeds, indicating that fewer pores can be obtained by increasing power delivered while maintaining the same travel speed. Furthermore, increases in travel speed shift the inflection point for diminishing pore presence to higher powers. An inflection point may be observed for the 2032 mm × min⁻¹ travel speed at higher powers as the data suggests an inflection point for this travel speed may lie just beyond the bounds of this study. While the aforementioned trends are consistent across both focal lenses, it is interesting to note that the 120 mm lens produces higher average porosity volume (Figure 5) but generally lower frequencies (Figure 6) at all parameter sets. In Table 1, maximum pore volume observed and total pore populations within each weld schedule are shown, with population counts denoted within parentheses. No porosity was observed in any weld produced at 200 W nor at higher travel speeds at 400 W under the 120 mm focal length.

Interfacial shape distributions for porosity observed at 600 W across all five travel speeds are shown for a focal length of 120 mm in Figure 7. All ISDs are plotted on a uniform color scale in units of square microns to aid in comparison. In the discussion in this section and those following, welds are grouped by travel speed into three categories: low (254 and 508 mm × min⁻¹), moderate (1016 mm × min⁻¹ only), and high (1524 and 2032 mm × min⁻¹). At low travel speeds, the majority of interfacial patches lie in Region 4 of the ISD, which largely corresponds to elliptical shapes with the edge of the shape distributions reaching the κ₁=κ₂ line, which corresponds to completely spherical shapes. For low travel speeds, the shape distributions also extend into Region 3 due to the presence of some saddle-shaped patches, as can be seen in Figure 2b in the “kidney bean”-shaped void at the upper right corner of the 3D reconstruction. At moderate travel speed, curvature pair maxima tend to lie upon the κ₁=0 axis with a broad spread into regions 3 and 4. At this moderate speed, the distribution does not approach the κ₁=κ₂ line closely, providing indication that voids are less spherical and generally more complex in nature. At high travel speeds, shape distributions are more diffuse than at low speeds and the overwhelming majority of the curvature distributions reside in Region 4. Additionally, the peaks of these shape distributions are again very close to the κ₁=κ₂ line, indicating voids are rather spherical at these higher travel speeds with very few saddle-shaped patches. The ISD peaks shift toward lower magnitudes of curvatures with increasing travel speed as observed in Figure 7, which is consistent with the fact that average pore volume decreases at higher travel speeds as shown in Figure 5.

The curvature distributions for a given travel speed are rather consistent across all power levels. The primary difference in ISDs relating to power variation, see Additional file3,4 is that the peak of the curvature distributions exist at increasingly negative values of κ₂ with decreases in power. This change corresponds to more spherical pore morphologies being formed with decreases in weld power. These trends were observed consistently across both focal length welds. A full set of calculated ISDs for all weld cases in this study having more than twenty pores each are included in the Additional file3,4 to this article.

Interfacial normal distributions with respect to the positive z-axis are shown in Figure 8 for porosity in weld samples discussed in the previous section, again at 600 W and under a focal length of 120 mm. Each IND is plotted with a uniform color scale with a maximum of 3.0 × 10⁻³ (units of dimensionless probability) to enable direct comparison. At moderate and high travel speeds, local anisotropy in directionality appears such that pronounced clusters of normal orientations populate the lower and upper hemispheres of each projection in Figure 8c – e. Taken together with the ISDs, the net change in pore morphology with travel speed can be summarized as follows. At low travel speeds, pores are on average larger and are mainly near spherical or ellipsoidal (Figure 7a – b). At the same time, the orientation of the normals are more isotropic, as observed from the more uniform INDs (Figure 8a – b). Thus, we can conclude that ellipsoidal pores are randomly oriented. At the moderate travel speed, pore sizes decrease, and pore morphologies become less ellipsoidal and more irregular (Figure 7c), and the IND shows preferential normal orientation toward and away from the incident laser (see the enhanced probability in the upper and lower hemispheres in Figure 8c). At high travel speeds, pore morphologies once again become closer to spherical, with some ellipticity still present, as evidenced by the fact that the peaks of the ISDs do not lie directly on the κ₁=κ₂ lines (Figure 8d – e). Also, as in the moderate-speed case, interfacial normals are aligned with the direction of laser incidence, resulting in the peaks of the INDs occurring in the upper and lower hemispheres at these high speeds. The same trends were observed in the negative z-axis projections as well. Succinctly stated, increasing weld speed results in decreased pore sizes and in interfacial normals preferentially orienting in the direction of laser incidence (both toward and away from). These trends in ISDs and INDs were observed across all powers and across both focal lengths. For completeness, calculated INDs for all weld cases in this study are also included in the Additional file5,6,7 and8 to more fully demonstrate the trend.

The pore-interspacing and pore-size distributions were calculated for a total of 18 weld schedules to more closely examine the interspacing of pores in three dimensions as a function of weld power and weld speed. These cases span both focal lenses and include welds for which power was varied while travel speed was held constant at 1016 mm × min⁻¹ and for which the travel speed was varied for a constant power of 600 W; see Table 2. These selections traverse the central column and row of both sample matrices where porosity was observed, shown in Table 1, encompassing median values for mm-scale laser-welds. The probability distributions in Figures 9 and10 were calculated as described in the section on spatial analysis. To alleviate redundancy, PIDs for only the 120 mm focal length welds are plotted in Figure 9, as the trends illustrated here are consistent for the 80 mm focal length results as well. In Figure 9, each data series represents a speed of 1016 mm × min⁻¹ under a different power. Overall, the pore-interspacing probability distributions shift to higher values of distances as the weld power increases. Stated simply, pores are spaced farther apart at higher power.

Pore interspacing was also calculated for various weld speeds, as shown in Figure 10. Again, to reduce redundancy and to make the trend clear, only select results are shown for welds made at multiple speeds in conjunction with the 120 mm focal length at 600 W. While the probability of finding pores at interspacing distances below 250 microns is relatively high across all cases, the distributions appear to be broader for low and high travel speeds, with the high travel-speed case potentially exhibiting a bimodal distribution. However, the statistics are insufficient to conclusively determine whether a bimodal distribution exists; further examination of larger weld samples or a larger number of samples under the same processing parameters are required to do so.

The characteristic pore interspacing, characteristic pore radius, and SLF are listed in Table 2 for all pore structures considered in Figures 9 and10. To better illustrate the variation with process parameters, calculated SLF are plotted as functions of delivered power and weld speed across all process pairs examined in Figure 11. The method for calculating pore interspacing and radius is accurate up to half a voxel[26]. Therefore, the accuracy of each calculation is dependent on the resolution of the measurements; the uncertainty is listed within parenthesis in Table 2 and shown by error bars in Figure 11.

As described earlier, pore interspacing is a measure of the proximity of pores in the weld structure, while the SLF measures the proximity of pores relative to the distance between their centers and the characteristic pore size. For the samples where weld power is varied, the smallest pore interspacing was found at 600 W for a speed of 1016 mm × min⁻¹ for both 80 and 120 mm lens welds (see Table 2), while the minimum SLF occurs at weld powers of 800 – 1000 W for the same travel speed (Figure 11a). This is consistent with the results of Figure 6, where the structure with the highest pore frequency per unit length arises at a weld power of 800 W for the 1016 mm × min⁻¹ speed weld series. While these results are consistent, SLF provides a more insightful detail of the pore structures present; for example, in the case of 800 W welds formed at 1016 mm × min⁻¹ with a 120 mm focus lens, the pore interfaces are separated by a distance that is 0.39 times the center-to-center distance between neighboring pores on average. Additionally, it is valuable to point out that the SLF is in the range of 0.4 to 0.6 for welds with a broad range of process parameters, which indicates that characteristic pore interspacing is approximately the same as the characteristic pore diameter in these cases. This suggests that for many weld cases, the characteristic pore interspacing can be approximated by the average pore diameter, which is generally easier to measure. However, high SLF values (> 0.6) are observed at the lowest power and the highest speed, indicating that pores may be spaced farther apart relative to their size at low delivered energy (Table 2 and Figure 11).

The measures presented above are useful in quantifying the influence of specific weld parameters on resultant pore microstructures. These measures revealed various degree of variability within each weld. For example, ISDs are not sharply peaked, indicating some pore surfaces have high curvatures while others have low curvatures. Here, we examine the width of the distribution for pore sizes to better elucidate the variability in this measure. To this end, the coefficient of variation (c.v.) associated with equivalent pore radii is calculated. The coefficient of variation provides a quantitative measure of the variability of any population by taking the ratio of the standard deviation (σ) to the average (μ), Equation 4.

The coefficient of variation is useful as it allows for comparison across populations having very different means, varying populations and dissimilar distributions. For distributions having a standard deviation less than the average, a c.v. less than one is returned, indicating very low variability in the population. Alternatively, c.v. values greater than unity indicate notable amounts of variability where the value of c.v. indicates how many times greater the variability in the distribution is than the mean. To illustrate the trend is consistent across both focal lenses the c.v. values will be presented for both the 80 mm and 120 mm focal lens series, Figure 12. While no trend in pore size variability is apparent for changes in weld speed a minor increase in variability can be observed with an increase in delivered power. Additionally, the values of c.v. found for the pore size distributions, which are all well below unity, indicate that the spread in radii for all weld schedules are relatively small with respect to their mean value. Variability measures such as the one presented above will likely aid the understanding of microstructure’s influence on mechanical response.