Supplementary Data to "Simulation of dendritic-eutectic growth with the phase-field method" by Seiz et al. pt. 2

<p>Data from some additional simulations conducted during the review of the paper. The goal was to test whether the theory delineating the dendritic-eutectic regime from the eutectic regime would also be applicable at lower solidification speeds. I chose the gradient G = 99 K/mm for quicker convergence and a target crossover concentration of c_0 = 0.11 to roughly get a factor of 10 slower crossover velocity compared to the previous simulations. The crossover velocity, following the theory, ended up being V= 8.347 um/s. The grid spacing was increased by a factor of 3 motivated by the classical scaling laws VR^2 = const. (R= dendrite tip radius or lamellar spacing) and approximating sqrt(10) as 3. At first, three test concentrations (c_0 = 0.1, 0.11, 0.12) are considered. If the theory is right, only c_0 = 0.1 should yield dendritic-eutectic growth.</p> <p>&nbsp;</p> <p>Even after an increase of grid spacing, having more than 5 diffusion lengths between the solidification front and the boundary would have required in excess of 15k cells in the growth direction. Thus I opted to work with a smaller domain for which I later needed to account. I employed a domain of 1200 um (4k cells) in the growth direction, with the moving cutoff at 600 um, and a width of 270 um. This makes for about 2.5 diffusion lengths (l_d = 239.6 um) between the front and the boundary. This did influence the results, as the simulation with c_0 = 0.11 showed dendritic-eutectic growth. Fitting an exponential ansatz for the concentration c(x) = c_inf + dc exp(-x/l_d) (x starting from the dendrite tip) showed that the apparent far-field concentration c_inf=0.108243 was actually within the dendritic-eutectic regime.</p> <p>In order to approximate a simulation for which c_inf = 0.11, the ansatz was employed to solve for c(x_b = 600um) using the actual diffusion length for l_d and observed tip concentration to determine dc. This led to a boundary value of c(x_b) = 0.115, for which another simulation was run. This simulation did show eutectic growth dominating, though the apparent far-field concentration (c_inf = 0.112216) at simulation end did not match the target concentration, likely due to being not converged yet. However, this point does lie within the eutectic regime as predicted by the theory and thus the theory also works at lower velocities once the problem of finite domain sizes is accounted for.</p> <p>&nbsp;</p> <p>Videos showing the evolution of the four simulations with different boundary concentration are attached, though with somewhat variable time between frames. The videos show a 660 x 270 um view of the simulation, slightly beyond the moving window cutoff.</p> <p>Attached as well is the updated microstructure map (with the new simulations plotted over their apparent far-field concentration) and a plot of the height difference between the maximal observed position of the alpha and theta phase. The latter serves as an easy way to differentiate dominant eutectic from dominant dendritic-eutectic growth, as this difference goes to zero for dominant eutectic growth, but some significant non-zero value for dendritic-eutectic growth.</p>