4.1 Experimental Results
As mentioned in Section 'Experimental Procedure', the determination of σsl is only possible if, at first, σvs and σlv are determined by another method. For a ternary alloy, it is possible to calculate σlv as a function of the system temperature and phases compositions by using the so-called Butler equation. For the calculation of the σvs-value, there is no such thermodynamics-based procedure available. Semi-empirical approaches seem to indicate that for pure metals there is an approximate direct proportionality between σvs and σlv. So let the quotient between these two values be defined as:
It was shown that, for many metals, kσ is approximately equal to 1.18. For other metals, this factor was found to be in the range [1.09, 1.33].
This proportionality was also assumed in the presented work for the case of the investigated Fe–Mn–C alloys. To define the kσ-value in the present case, measurements of the contact angle were performed at first for Armco iron. For the case of Armco iron, it was then assumed that σsl was equal to 0.204 J m−2, the classical value for pure iron obtained by Turnbull. Following the work by Kingery and Humenik the value σlv = 1.240 J m−2 was taken for Armco iron. By using Equation (1), it was then possible to calculate kσ as being equal to 1.098.
For the alloys S1, S2, and S3 (see Section 'Introduction'), it was possible to determine the corresponding values for σlv of each alloy by using the Butler equation. By applying the calculated proportionality factor of 1.098, the σlv-values were then calculated for each alloy. Through the measured values of the contact angle, one then finally determines the values for the liquid–solid interface energy, which are listed on Table 3 for each alloy.
Table 3. Measured values for the solid–liquid interface energy
|Alloy||σsl [J m−2]||Experimental error [J m−2]|
In the near future, new evaluations of the solid–liquid interface energy will be conducted at IEHK. These evaluations will also include the effect of additional alloy elements (e.g. aluminum) on this physical property.
As reported on the literature, the interface free energy for pure iron may be different from the original value determined by Turnbull (see, e.g. the theoretical investigations by Waseda and Miller or the approach by Granasy and Tegze whose results were checked against experiments using the grain boundary groove method). This would lead to a change of the present estimates for the solid–liquid interface energy in the case of Fe–Mn–C alloys.
4.2 Simulation Results
By observing the values of the solid–liquid interface energy from Section 3, it is concluded that the maximum value that was measured is σsl = 0.461 J m−2. It is important to notice that the compositions of the phases on the liquid and solid sides of the moving solidification interface are not equal to one another (as it was the case during the performed sessile-drop experiments), being instead of this related to one another by the local equilibrium condition. In order to investigate the effect of changing the interface energy on the simulation results, simulations were performed for both steel compositions C1 and C2, by varying, in each case, the interface energy from σsl = 0.204 J m−2 (value for pure iron) to σsl = 0.461 J m−2. The value of the anisotropy coefficient (γe), which controls the anisotropy of the interface energy was set equal to 0.05 for all simulations. The matrix for the different simulation set-ups depicted in Table 4 can then be constructed.
Table 4. Matrix describing the simulation set-ups
|σsl [J m−2]||Alloy composition|
On Figure 5a and b, simulations results are shown for the alloy Fe–23%Mn. All simulations shown in this contribution were performed for a domain, which corresponded to the upper right quadrant of the corresponding depicted box. The picture as a whole was then obtained by reflection. For the input parameters that were utilized in this case, dendritic sidebranches did not appear. Nevertheless, the interface shows already instabilities at the opening of the groove situated at a direction of 45° to the horizontal line. For a longer simulation time and a larger domain, sidebranches would then probably develop. Another important point is, although sidebranches did not appear, the dendrite main stem showed stable growth, in the sense that it grew with a stable velocity and a stable tip radius. This fact can be used to quantify and compare this result with the result of other simulations.
Figure 5. Simulation results for the iron-manganese alloy a) σsl = 0.204 J m−2, b) σsl = 0.461 J m−2. Both areas are 160 µm × 160 µm large. The figures show only a cut-out of the original simulation domain.
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When the main dendrite stems of Figure 5a and b (simulation C1E1) (simulation C1E2) are compared to one another, it is clear that dendrites are slenderer in simulation C1E1 (low interface energy) than in simulation C1E2 (high interface energy). This is an expected result, once a higher value for the interface energy tends to not only stabilize the interface against the growth of instabilities but also to increase the energetic cost of curving an interface, producing then larger curvature radii.
On Figure 6a and b, simulations results are shown for the alloy Fe–23%Mn–0.3%C. Results of simulation C2E1 (low interface energy) showed a much more ramified structure than simulation C2E2 (high interface energy). In addition to this, simulation C2E1 showed also a slenderer main stem (with apparently a smaller dendrite tip radius) than simulation C2E2. Notice that the breath of the region free from liquid on both sides of the symmetry axis of the vertical main stem is smaller in C2E1 than in C2E2.
Figure 6. Simulation results for the iron-manganese carbon alloy a) σsl = 0.204 J m−2, b) σsl = 0.461 J m−2. Both areas are 360 µm × 360 µm large. Figures show only a cut-out of the original simulation domain.
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All these facts can be related to the higher energetic cost for creating or bending interfaces in the case of higher interface energy.
In order to compare simulation results, it is important to define a common parameter that can be calculated for all simulations. One important characteristic of the dendrite is its radius at the tip (Rtip). This value was calculated from the simulations for each one of the simulation set-ups. The calculation procedure involved approximating the tip region by a parabolic function, y(x) = ax2 + bx + c, by employing a non-linear regression with the least-squares method. In all simulations, only the region of the tip was included in the polynomial regression. A correlation coefficient for the regression of over 0.99 was obtained in all four cases. Once the polynomial coefficients are known, the determination of the tip curvature is trivial. It is important to point out that it was shown in the literature that a functional relationship exists between the Rtip-value and the value of the secondary dendrite arm spacing established during growth in a region near to the dendrite tip (initial secondary dendrite spacing). Table 5 shows the calculated values for Rtip in a highly undercooled regime, which expectedly generates radii values much smaller than the ones that would be observed in a conventional solidification experiment.
Table 5. Results of dendrite tip radius for the various simulation set-ups
| ||Simulation set-up|
It is noticeable that, for the case of the Fe–Mn–C alloy, an increase of the interface energy from 0.204 to 0.460 J m−2 caused an increase of the tip radius of only 2.5%. Nevertheless, there was a considerable change on the microstructure (see Figure 6). In the case of the Fe–Mn alloy, there was, as expected, also an increase of the radius as a consequence of the increase of the interface energy value. But in this last case, the radius variation was much larger, of about 20%.
For the same value of the interface energy, the alloy with carbon showed a larger dendrite radius than the binary one, but the difference between radii was larger for the lower value of the interface energy (compare data for C1E1 and C2E1 in Table 5). Carbon diffuses in the melt much quicker than manganese and once the dendrite radius increases with the diffusion length, it is reasonable that by alloying with carbon, the dendritic radius should become larger.
It is expected that the simulation set-up C1E1 will show much slenderer dendrites than all other alloys/set-ups, which were studied in this work, because its dendrite tip radius is much smaller than the other ones. A simulation for alloy C1E1 and for a larger domain will probably show this and it will be reported on this aspect in a near future. The relative absence of sidebranches in the simulations C1E1 and C1E2 can be explained by the synergy of three important factors: (i) the high supersaturation that was imposed in this case, (ii) the low diffusion rate of manganese in the liquid compared to the one of carbon for the composition C2, (iii) the narrower solidification interval for the iron-manganese alloy (ΔT = 21 K) compared to the one for the iron-manganese-carbon alloy (ΔT = 66 K).
Another question that will be answered in future works is the one related to the roughening of the structure when carbon is added to the alloy: alloying the Fe–23%Mn steel with 0.3% carbon leads to a larger dendrite radius. From this fact, it does not necessarily follow that the initial secondary dendrite spacing will get larger through carbon addition, once the stability of the interface is related not only to the diffusion rate in the liquid but also to the details of the phase diagram. Those details are much different for a Fe–Mn and for a Fe–Mn–C alloy (see chapter 3 of the book by Kurz and Fisher for a discussion of the binary case).
Although simulations results were checked by changing the interface thickness by a factor of 0.5, new simulations will be performed by including the corrections which are necessary to make the results concerning the kinetics and the growth morphology really independent of the interface thickness.[29, 30] The present results can nevertheless be considered to supply a reasonable estimate of the dendrite tip radii and its dependence on the value for the interface energy. In addition to this, the effect of a even higher value for the solid–liquid interface energy on simulation results will also be investigated in detail in future works (see Section 'Experimental Results').
These simulations did not include the effect of the change in the interface energy on the dynamics of the Ostwald-ripening. According to Kurz and Fisher, the interface energy influences the Ostwald ripening of secondary dendrite arms like following:
where M is defined as
and where ΔT is the solidification interval for the alloy, dT/dt is the cooling rate, Δsf is the fusion entropy, Dl is the liquid diffusion coefficient, is the concentration of the liquid at the end of solidification, C0 is the nominal composition of the alloy, m is the liquidus slope and k is the equilibrium distribution coefficient.
These equations show that, for the binary case, the exact value of the interface energy will have an important effect on the secondary dendrite spacing after the Ostwald ripening. Quite recently, the Ostwald ripening of dendrites in a high-alloyed steel was investigated by utilizing both the PFM and an analytical model.