4.2. Distribution of Cu-Rich Droplets
In the previous simulation work of immiscible alloy [9
], the droplet population was either calculated as one group or divided into various size classes, the visualization of plenty of droplets in a real, continuously-cooling process is not convenient. In this paper, we allow all thermophysical properties to vary with temperature and time during the cooling process. Meanwhile the model also memorizes the position and velocity of each droplet, which provide a way to visualize the spatial distribution of droplets during the whole coarsening process. Figure 3
shows the evolution of the spatial distribution of Cu-rich droplets with and without the HMF. There is a dramatic coarsening rate in the sample without the HMF, which leads to a significant macro segregation occurring in 0.3 s, whereas a similar segregation occurs in 6 s with the HMF. The HMF has the effect of retarding the coarsening of Cu-rich droplets. In two samples of similar segregation degree, such as 0.3 s in the 0 T case and 6 s in the 12 T case, the number of very small droplets in the former appear to be larger than that in the latter, which means the anti-coagulation effect of the HMF is more effective on larger droplets and less effective on very small droplets.
A droplet move in the matrix with temperature gradient will move to the region of higher temperature under Marangoni force. This force is relatively small compared with gravity, hence, Marangoni migration is hardly observed in the results.
shows the development of average radius and number of droplets in the samples with and without the HMF. In the sample without the HMF, the number of droplet decreases dramatically in the stage before 0.5 s, whereas the decreasing rate is slowed by the HMF. The velocities of droplets are retarded in the magnetic field and, thus, the collision frequency is also reduced. In the sample without the HMF, the average radius of droplets grows to a range of 35~43 μm in less than 0.5 s and then remains this size, whereas it keeps raising up under the HMF. In the early stage without the HMF, the fast decreasing rate in the droplet number implies that collision is a primary cause of coarsening. In the later stage, there are few droplets surviving after 0.5 s, and they have a low collision frequency due to the long distance between them, hence, their sizes grow slowly due to diffusional growth. On the contrary, the collision frequency is low under HMF, and coarsening is due to all the mechanisms including diffusional growth, Stokes coagulation, and Marangoni coagulation. The above inference could be confirmed by comparing the results with the previous simulation work of Guo and Liu [26
], in which the different coarsening mode condition in immiscible alloys are analyzed respectively. The development of droplet radius of 0 T case in Figure 4
follow a similar trend with the diffusional growth mode as in [26
]. The development of the droplet radius of the 12 T case shows a similar trend with the lines in [26
] containing all the effects of diffusional growth, Stokes coagulation, and Marangoni coagulation.
It is difficult to make a direct comparison between experiments and simulations because accurate values of the heating and cooling times are not available. Moreover, the magnitude of the temperature gradient varies with position and with time during an experiment. Nevertheless, a rough comparison is performed here between the experimentally-observed microstructure and the results predicted by the simulations. The melting and solidification experiments are performed with liquid immiscible Cu-55 at % Pb hypermonotectic alloys under a 12 T HMF. The experimental microstructure is shown in Figure 5
, in which the white round phase is the solidified droplets. The solidified droplets have different structure according to their size, and droplets with a radius larger than 40 μm show a net-shell type structure, with a net-like structure inside the spherical shell. Droplets smaller than 40 μm show an empty-shell type structure, which means the droplet consists of a Cu-rich spherical shell and the Pb-rich phase inside. It should be noted that the clusters of white-filled particles are not droplets, but Cu-rich dendrite. Without the HMF, the droplets will settle within the gravity field and the final state is an arrangement of two layers, with the lighter Cu liquid locating on the top. In the sample without the HMF, there are a large number of very small droplets, and most of them belong to the empty-shell type. The HMF appears to retard the coagulation of Cu-rich droplets, especially the larger ones. The amount of small empty-shell type droplets in a sample with the HMF is fewer than that without the HMF. This phenomenon is accordant with the simulation results in Figure 3
shows the evolutions of droplet radius distribution in number frequency form and mass density form, respectively. It is demonstrated that the collision-coagulation caused a broader distribution. In the initial distribution, droplets have an average radius of Rn
= 14 μm. Over the 1 s interval shown, there is an eight and four micron increase in the value of max radius frequency in the 0 T and 12 T case, respectively. The size distribution tends to spread to the region with larger values over time. The HMF affects the radius of the maximum frequency to rise slower. The measured drop-size distribution in the experiment is shown in Figure 6
a,b in number density form by a histogram with a bin size of five microns. It is seen that the numerical simulation results with the HMF has a distribution that more closely represents the experimental results. Without HMF, the distribution frequency concentrates in the range of 15–40 μm, which is due to the capturing of plenty of large droplets by the Cu-rich segregation layer, and only a few droplets survive and stay in the matrix, most of them having a radius less than 40 μm. In the simulation results, large droplets keep their spherical shape instead of forming a layer, as a simplification.
It should be noted that even a small number of large droplets, which might appear negligible in number density form, can dramatically alter the evolution of a drop-size distribution. Smaller droplets have a much higher number density than larger ones and, hence, the figure of the number density frequency can hardly describe the size and number of large droplets. The relative mass fraction of the dispersed phase in the different drop-size categories is assessed and shown in Figure 6
c,d. It is seen that the shift to larger droplets over time is more evident in terms of mass density rather than number density. The figures of mass density form also show the effect of the HMF to reduce the growth rate of droplets. The experimental result without the HMF is approximately accordant with the simulated distribution curve of 0.1 s, whereas the experimental curve with the HMF is accordant with the simulated curve of 2 s. As shown in Figure 4
, the growth rate without the HMF tends to be slow after 0.3 s, the reason being that large droplets have floated up to the top, and the remaining small droplets have a small number and, hence, have a low collision frequency. In the experiment, the large droplets formed a segregation layer on the top, and the remaining small droplets grow slowly.
The whole time for liquid-liquid separation is 10.81 s from 963 °C to 955 °C. Below 955 °C the moving speed of Cu-rich solid particle is much slower than the liquid droplets, and the distribution of droplets in the end of the liquid-liquid separation is similar to the results of a solidified ingot [28
]. A rough comparison between experiments and simulation shows that experimentally-observed drop-size distributions are not accordant to the final distribution at 10.81 s in the end of liquid-liquid separation, but accordant to the radius distribution at an early time predicted by the simulation, that is 0.3 s without the HMF and 2 s with the HMF. The discrepancy between the experiment and simulation is first caused by the existence of a segregation layer in the experiment, which captures some large droplets. A second explanation for this discrepancy is that the accuracy values for the quenching temperature and cooling rate in the experiment are not available. A third explanation for the discrepancy between simulation and experiment is that the forces calculated in this model do not reflect all the forces acting on a droplet. There are interactions among droplets which are complicated and need to be investigated in future research.
In the simulation work of Rogers and Davis [4
], which used a population balance model to simulate a Zn-Bi system, the time evolution of the drop-size show a bimodal distribution. It is discussed that the second distribution of larger droplets developed perhaps due to droplet collisions. In this work, collisions are also considered, but the results only contain unimodal distribution. This discrepancy is perhaps due to that the coarsening stage being different. In the work of Rogers and Davis [4
], the first peak of the bimodal distribution is less than 5 μm, the very small droplets have lower collision frequency; meanwhile, the larger droplets have faster collision frequencies and form a second peak of distribution. This work focus on the later stage of coarsening process, most droplets have larger size and they all take part in the collision coarsening process, which lead to a unimodal distribution of droplet radius.
We assume that the gravity melt flow due to temperature difference has an effect on the motion of small droplets, whereas this effect is difficult to depict in the figure of spatial distribution. Therefore, we divide the sample into two parts, the left part and the right part. The left part means the center part of the sample, and the right part means the outer part of the sample. The melt flow is mainly upward in the left part and mainly downward in the right part (Figure 2
). Figure 7
shows the distribution of droplet radius with and without the HMF, and the time is chosen to be 0.2 s in the 0 T case and 2 s in the 12 T case. The distribution tends to be different for these two parts, especially when the droplets are small. In the sample without the HMF, the amount of small droplets (smaller than 40 μm) in the left part tends to be higher than that in the right part. In the sample with the HMF, the amount is also higher in the left part for droplets smaller than 60 μm. The large droplets have larger velocities and higher collision frequency, and the influence of melt flow on large droplets is small and negligible.
shows the velocities of the droplets in the y direction (vertical upward) in different size categories. By comparing the distributions in the left and right parts in Figure 8
, it is seen that the right part has a shift of the droplet velocity to higher values. Driven by the Stokes force, droplets move in the same direction with the melt flow in the left part, and in the opposite direction of the melt flow in the right part. A larger spread in the distribution in the right part implies that the relative velocities of the droplets are greater and, therefore, a higher rate of collision will be observed. When the radius is smaller than 40 μm, this shift of velocity to higher values is significant in the sample without the HMF. By comparing the distributions in the left and right parts in Figure 8
, it is seen that the application of the HMF leads to a much slower velocity for the droplets. The velocity of the melt flow in the matrix is reduced under the HMF and, hence, it has a smaller influence on the velocity of the droplets.
The moving velocity of droplets in this work is much slower compared with the result of Jingjie Guo’s work [26
], which is about the coarsening simulation of an Al-In system. The possible reason of this discrepancy is due to the different alloy system, and the Al-In system has a much larger density difference between droplets and matrix compared with the Cu-Pb system. The moving velocity in Figure 8
is also higher than in previous results by mechanical calculation of a single droplet in a Cu-Pb system [28
]. The slow velocity in this work is partly due to the method to calculate the phase density. In the temperature region of miscibility gap, the minority Cu-rich phase and matrix Pb-rich phase (corresponding to the liquid L1
in the Cu-Pb phase diagram) have a much smaller density difference than pure Cu and pure Pb. In this paper, we try a new method to allow the density and all thermophysical properties to vary with temperature and time during the cooling process, which decreases the density difference and, hence, leads to the decrease of the droplet velocity. Compared with the experimental results, this method of calculation is more accordant to the real physical process. Melt flow in this model also changes the relative velocities of droplets, which leads to a different collision frequency. Melt flow has a significant influence especially at the first stage of coagulation when all droplets are small and their velocities are slower than the melt flow, and droplets tend to move along the direction of melt flow. Later, as droplets grow, Stokes motion takes on a key role. Large droplets move up, but fluid flow still pushes some small droplets, and changes the velocity and direction of them.