Figure 2 shows the inclusion composition in samples taken prior to calcium treatment in experiments A (left) and B (right). Compositions are plotted as mole fractions, with the size of the triangle proportional to the area fraction of inclusions with that composition. The thin grey line indicates the boundary of CaO-MgO-Al₂O₃ mixtures that would be 50% liquid at 1,550°C. As expected, experiment A formed only alumina inclusions whereas the MgO crucible of experiment B served as a source of magnesium, modifying the alumina deoxidation products to spinels.

Calcium treatment fully modified the inclusions, changing the composition to lie within the 50% liquid boundary (Figure 3). For brevity, only one ternary diagram is shown here for samples taken after calcium treatment, and Figure 4 presents a boxplot of the Ca/Al ratio of inclusions in all samples taken after calcium treatment. Each dashed line represents the stoichiometric ratio of a stable calcium aluminate in the CaO-Al₂O₃ phase diagram, including ( CaO ) 12 ( Al 2 O 3 ) 7 , denoted by C12A7, which is close to the eutectic composition in the CaO-Al₂O₃ binary. The calcium aluminates mainly lay between CA and C3A, within the desired composition range for full or partial liquefaction.

In both experiments, apparent inclusion diameters (as observed on the two-dimensional sample surface) roughly followed lognormal distributions (Figure 5). (The ridgeline plots in Figure 5 are partially overlapping histograms that create the impression of a mountain range; this type of visualization is useful to observe changes in size distributions over time.)

Figure 5A shows the inclusion size distributions of experiment A. Both samples taken before calcium treatment contained only solid alumina. However, the sample taken 2 minutes before calcium treatment (sample A2) showed almost twice the inclusion concentration (and had generally smaller inclusions) than that taken 4 minutes before calcium treatment (sample A1). The increase in inclusion concentration appears to have been caused by reoxidation of sample A2. Population density function (PDF) plots are useful for testing whether the difference in average inclusion size in samples A1 and A2 could have been caused by reoxidation of sample A2 (van Ende et al., 2013). The PDF plots in Figure 6 are a straight line for sample A1 (indicating size-independent inclusion growth), and more parabolic for sample A2, consistent with formation of a new population of alumina inclusion by reoxidation of sample A2 (van Ende et al., 2011).

Table 2 shows the chemical analysis (by optical emission spectrometry) of the steel puck that remained in the crucible after each experiment. The analyses are similar for the two experiments and indicate approximately 100% yield of Si (from calcium disilicide) and approximately 1% Ca yield.

For calcium-treated samples, the size distributions in Figure 5 are those of all inclusions that fell within the 50% liquid boundary shown in Figure 3: mainly solid inclusions are excluded from the plot, to be able to test whether liquid inclusions agglomerate. Although the lognormal fit was not good for some samples, the shapes of the distributions are all roughly lognormal, and all samples were considered to follow this distribution for comparison of average inclusion sizes.

For both experiments, a sharp decrease in inclusion size was observed upon calcium treatment, compared with inclusions in the samples taken before calcium treatment. This is difference is better visualized in Figure 7, which shows an interval plot of the change in average inclusion size over time. The average inclusion size was approximately 2   μm before calcium treatment, decreasing to sub-micron upon calcium treatment. Over time, the average inclusion size increased slowly, with a statistically significant difference in inclusion size between samples taken immediately after calcium treatment, and those taken later. Figure 7 also show the results of a Monte Carlo simulation of particle collision between liquid inclusions, discussed below.

To identify the main mechanism by which the liquid calcium aluminates collide and grow, the mechanisms described by Lei and He (2012) and Söder et al. (2004) were evaluated. Figure 8 shows the collision volumes for inclusions with diameters of 1, 5 or 10 μm.

The collision volume gives the rate at which an inclusion with a given diameter collides with inclusions of other sizes, depending on the concentration of the inclusions. If the efficiency of collisions is assumed to be 100% (all collisions lead to agglomeration) the resulting change in the concentration of particles ( n k ) of a given particle size ( r i j ) can be discretized as shown by Smoluchowski (1917): d n k dt = 1 2 ∑ i + j = k W ij n i n j − n k ∑ i = 1 ∞ W ik n i

The first term in this equation gives the rate at which inclusions consisting of " i " particles collide with clusters containing “j” particles to form a “k” cluster of particles. The second term is the depletion term (Lei and He, 2012)—removal of k-sized inclusions by collisions with other inclusions. The expressions and parameters used to calculate W _ij are summarized in Table 3.

For a gas-stirred industrial ladle, turbulence is the main inclusion collision mechanism (shown in red in Figure 8). In the case of the induction furnace, the red curve is not applicable, since no external agitation is present. Rather, Stokes collision and laminar-flow collisions (indicated by the line labeled “Laminar Flow”) are the main mechanisms. For 5–10 µm inclusions, the rates of Brownian collision are generally much lower than the other mechanisms. For a smaller (1 µm) inclusion, the collision volume with other small inclusions ( ≤   1  μm ) is dominated by Brownian collision, when no external agitation is present. For collision with larger inclusions ( ≥   1  μm ) , first laminar collision and then Stokes collision are the dominant mechanisms.

Using the literature expressions for the collision volumes (Lei and He, 2012; Söder et al., 2004), the change in average inclusion volume was calculated for induction-furnace conditions, using Monte Carlo simulations. The input parameters to the simulation are the parameters of the lognormal size distribution, the initial number of inclusions per unit volume of steel (concentration), and the time constant for inclusion flotation. The time constant for flotation is taken to be equal to the depth of steel (in the crucible), divided by the mass transfer coefficient in the steel.

An initial number of inclusions is chosen, with sizes randomly following the lognormal distribution. The simulated sub-volume of steel is calculated based on the initial number of inclusions, and the initial inclusion concentration. In each time step, the following are evaluated:

For each inclusion, its collision rate with every other inclusion (by the Stokes, Brownian and turbulent mechanisms) is calculated. Each pair of inclusions is allowed to agglomerate randomly with a probability that is proportional to the length of the time step and the calculated collision rate for that pair, taking the collision efficiency to be unity. If a pair of inclusions collides and hence agglomerates, the new inclusion size is calculated based on the assumption of spherical shape. In addition, each inclusion is also allowed to float out randomly, with a probability of flotation equal to the length of the time step divided by the time constant for flotation.

The simulation is very sensitive to the initial inclusion size distribution and the inclusion concentration (number of inclusions per unit volume). Ten simulations were performed, considering an initial number of 2,500 inclusions, with a lognormal size distribution based on the measured sizes in the sample taken 1 minute after calcium treatment. The parameters of the lognormal distribution were as follows: location ( μ ( ln ⁡ d i n c l u s i o n / μm ) = − 0.937 and scale ( σ ( ln ⁡ d i n c l u s i o n / μm ) = 0.39 . The initial inclusion concentration was first estimated using the software CSDCorrections (Higgins, 2000; Higgins, 2002). However, because of resolution limitations the majority of small inclusions present immediately following calcium treatment are not detected (discussed in more detail in the next section). To fit the experimental data presented in this work, an initial particle density of 1.0 × 10 7 particels mm 3 was used, which is one order of magnitude larger than calculated from the measured 2D distribution using CSDCorrections. The collision rate due to laminar flow depends on the velocity gradient, which was taken to be 2 s⁻¹, based on modeling of natural convection in the liquid metal in the induction furnace (Piva, 2018).

It was assumed that inclusions remained spherical throughout. Following each time step, the sizes of all inclusions were recalculated, to serve as input to the calculated collision rate in the next time step. Piva et al. (2019) used a similar approach and concluded that Monte Carlo simulations for inclusion growth by collision validated their experimental work. Results of simulated inclusions sizes are shown in Figure 7. Each black line in Figure 7 is the result of an individual simulation based on Smoluchowski’s approach; the trends can be seen to similar to those of the experimental data.

As shown in Figure 7, the measured diameter of calcium-treated inclusions was 3.5 times smaller than prior to calcium treatment. Moreover, the reaction with calcium was rapid and even shortly after calcium addition, almost no solid inclusions could be observed. The sharp decrease in inclusion size is consistent with the formation of a new population of calcium aluminate inclusions upon addition of calcium, and dissolution of the pre-existing alumina or spinel inclusions (Piva and Pistorius, 2021). The activity of dissolved oxygen in the vicinity of the calcium aluminates would be much lower than adjacent to alumina or spinel inclusions (because of the much lower activity of alumina in calcium aluminates). The decrease in oxygen activity following calcium treatment destabilizes alumina and spinel inclusions, causing their dissolution. This mechanism implies that calcium treatment of alumina or spinel inclusions involves dissolution of alumina and spinel, following formation of a new population of calcium aluminates.

The decrease in inclusion size was even larger than indicated by Figure 7, because many inclusions (in the samples taken immediate after calcium treatment) were too small to be detected by backscattered electron imaging. This is indicated by the observation that in the samples taken immediately following calcium treatment, the measured area fraction of inclusions was much lower than expected; see Figure 9. This figure shows the measured area fractions of inclusions, with expected trends in inclusions concentration based on first-order kinetics of inclusion flotation. When using an instrument with improved spatial resolution (Quanta 600) the measured area fraction was larger (and within the expected range for samples taken 3 minutes or more after calcium addition). However, the low measured area fractions in the samples taken immediately after calcium addition indicate that many inclusions were too small to be detected.

The lines in Figure 9 were calculated based on the assumption of first-order kinetics of inclusion flotation (Schwerdtfeger, 1983), with the rate-limiting step being mass transfer of steel to the steel-argon interface. The lines were calculated based on a mass transfer coefficient of m =   4.5 × 10 − 5 m / s (Piva et al., 2017), giving a rate constant of m A / V = 6.5 × 10 − 4 s − 1 (time constant of approximately 26 min), where A is the projected are of the steel-gas interface, and V is the volume of liquid steel. The initial oxygen concentration ( [ O ] initial ) is taken as the area fraction of inclusions before calcium treatment in each experiment.

Figure 9 shows that the measured decrease in inclusion area fraction broadly agrees with the expected area fraction. A notable exception is the much smaller than expected area fraction following calcium treatment (as already mentioned). Another is that the sample taken 2 min before calcium treatment in Experiment A had a much larger inclusion area fraction than expected, supporting the conclusion that this sample suffered reoxidation.

To test the idea that shortly after calcium treatment, a large number of small inclusions—below the instrument detection limit—are created, the sample taken 1 minute after Ca in experiment B was deep-etched. Electrolytic etching used a solution of acetylacetone (10%) and tetramethylammonium chloride (1%) in anhydrous methanol.

A representative SEM image in Figure 10 illustrates the profusion of small inclusions in the sample taken immediately after calcium treatment. Point analyses of the inclusions confirmed that these are calcium aluminates. Since the smaller inclusions are below the resolution limit of backscattered electron (BSE) images of polished cross-sections, these would not have been detected during automated inclusion analysis. Figure 10 includes (at right) simulated BSE images of inclusion cross-sections, based on the measured spatial resolution of the two SEMs (Tang et al., 2017). The simulated images are consistent with the suggestion that many smaller inclusions would not be detected during automated inclusion analyses.

In contrast, a sample taken 13 min after calcium treatment (Figure 11) did not contain sub-micron calcium aluminates, and the area fraction of inclusions fitted the expected trend (Figure 9). Agreement of the measured inclusion fraction with the fitted trend indicates that the calcium aluminates in this sample had grown sufficiently large to detected during automated inclusion analysis. The sub-micron inclusions were no longer present because the inclusions had agglomerated by collision, in line with the simulations shown in Figure 7.