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Anisotropy functions without missing orientations
Figure 3.8 shows the variation of θt as a function of the rotation angle θR, calculated by both BI and PF, for a simple two-fold anisotropy (m= 2) given by Eq. (3.3). Temperature gradient, growth velocity, and lamellar spacing were held constant. The results of both simulation methods follow quite closely the SP approximation, up to differences that do not exceed a degree. As expected, we find symmetric, non-tilted patterns when either the minimum or the maximum of γ is aligned with z. Somewhere in between these orientations, the value of θt passes thus through a maximum for an orientation that depends on the anisotropy function. We checked that this maximum increases when the anisotropy coefficient is increased. Overall, however, the tilt angle remains much smaller than θR over a full rotation range. In other words, a weak crystallographic anisotropy, though it has a clearly detectable effect on the dynamics of lamellar eutectic patterns, is not sufficient to induce a significant locking (θt ≈ θR). For convenience, we will call “unlocked patterns” such weakly anisotropic tilted lamellar patterns. Let us mention again that the values of d0/lD, lT /lD and Pe are actually different for BI and PF ; the results thus demonstrate once more that the influence of all of these parameters is weak in realistic conditions.Next, we have investigated a situation in which a mild lamellar locking takes place. We use an anisotropy function according to Eq. (3.5), with ǫg = 0.05, wg = 0.195, ǫ2 = 0.0854, ǫ4 = 0.0221 (Fig. 3.9). This function was chosen so as to smoothly reproduce the Wulff plot extracted (assuming the SP approximation) from the experimental data of Figure 7 of Ref. [41] (see discussion below). The corresponding Wulff shape is an oval with markedly flattened sides (but without straight facets, and without forbidden orientations). The variation of θsp with θR is continuous and univalued. The simulation results for θt nicely follow the SP-approximation curve.
Anisotropy with missing orientations
A strong lamellar-locking effect (Fig. 3.10a) can be reproduced by using an anisotropy function of the same form as that of Fig. 3.9, but with a deeper and sharper Gaussian (ǫg = 0.2 and wg = 0.1 ; see Eq. 3.4). This modification of the Wulff plot entails the appearance of two (quasi) facets, and four ”ears” with long metastable branches and sharp-edge junctions in the Wulff shape of the interphase boundary (see inset in Fig. 3.10a). Let θu and θl, where θu < θl, be the tilt angle values at which γ +γ′′ = 0 on the interval [0,π/2] (θu ≈25◦ and θl ≈ 70◦ in the example shown in Fig. 3.10). The SP approximation predicts three distinct parts in the θsp vs θR curve: (i) an essentially linear strongly locked branch with a slope close to 1, which runs from θR = 0 to θR = θl; (ii) an unlocked, although (weakly) anisotropic, branch for θR ranging from θu to π/2; (iii) an intermediate branch, which connects the end points of the locked and unlocked branches, and is, presumably, not observable given that the interface boundary is unstable (γ+γ′′ < 0) along its entire length. In the [θu,θl] interval, two (locked and unlocked) values of the lamellar tilt angle are possible for a given eutectic-grain orientation. In this bistable range, there is a value θe of the rotation angle (here, approximately 45.3◦) at which σ has the same value for both branches – this corresponds to a sharp edge in the convex equilibrium shape.
Both BI and PF simulations reproduce the two separate locked and unlocked branches (Fig. 3.10a). They demonstrate, in particular, the existence of a strong locking effect over a large orientation range, as predicted by the SP approximation. The two methods exhibit, however, a somewhat different behavior in the bistable interval. In the BI simulations, starting from the situation where the facet is aligned with the temperature gradient (θR = 0), the system closely follows the locked branch upon increasing θR, up to a limit angle at which it jumps abruptly to the unlocked branch. The jump occurs well before the turning point θl, at a value of θR which is close to 45◦. This point can be only attained if θR is successively increased along the locked branch, and the maximum value of θt depends on the step size in θR. It is thus clearly initial-condition dependent and has no obvious connection with the sharp-edge angle θe. This limit value of θt in the simulations is also compatible with previous in situ observations of tilted lamellar patterns. Reversely, when θR is decreased stepwise starting from θR = π/2, the system describes the whole unlocked branch, within numerical accuracy. This hysteresis is expected for a bistable system. It should be noted that the approach of the limit tilt angle along the locked branch also corresponds to a steep, apparently diverging increase of the δ angle (Fig. 3.10b). This indicates that, at large tilt angles, the SP approximation, while it still correctly predicts the values of the tilt angle, becomes inaccurate with regards to other aspects of the dynamics.
For instance, in contradiction with its basic assumption, the shape of the S-L interface becomes markedly asymmetric at large tilt angles, as can be seen in Fig. 3.10c.
Reconstructing the anisotropy function
A practical aim of the present study was to give a numerical support to a recent experimental work based on a rotating directional solidification (RDS) method [45, 41]. A brief overview of the method is given in Sec. 1.5. This method uses a standard thin-sample directional solidification setup, and permits, in addition, to rotate the sample at constant angular speed about an axis perpendicular to the (two-dimensional) sample. Under a few, not very restrictive, conditions (zero translation speed, center of rotation placed on the eutectic isotherm, quasi-two-dimensional and quasistatic nature of the front pattern dynamics), this is equivalent to continuously varying the rotation angle θR of a given eutectic grain with respect to the thermal gradient axis, as we have done in the above calculations. More precisely, the lamellar tilt angle θrds observed over time during a correctly set-up RDS experiment is equal to the steady-state tilt angle θt at the current value of θR. Moreover, in the SP approximation, this entails that the RDS trajectories of the trijunction points are centrosymmetric closed curves homothetic to the section of the Wulff shape of the interphase boundaries by the sample plane, from which a two-dimensional anisotropy function of the interphase interface can be derived (see Ref. [45]). The anisotropy function used in Fig. 3.9 was derived from the RDS pattern of a nearly locked grain by this method. As a test for the accuracy of this method, we have reconstructed the Wulff shape from the calculated θt(θR) points under the SP assumption. The two shapes are identical to within experimental error (a few percents of the position vector), as can be seen in Inset 2 of Fig. 3.9. In conclusion, the errors due to the SP approximation are generally not larger than the experimental uncertainties, which validates the use of the SP approximation in the exploitation of the RDS patterns.
Table of contents :
Acknowledgements
Abstract
1 Introduction
1.1 Directional solidification
1.2 Solidification of single phase alloys
1.2.1 Morphological instability
1.2.2 Constitutional supercooling (CS)
1.2.3 Mullins-Sekerka instability
1.3 Solidification of multiphase alloys: Eutectic solidification
1.4 Surface tension anisotropy: Origin and Effects
1.5 Rotating directional solidification (RDS)
1.6 Numerical methods
1.7 Thesis Outline and motivation
2 Simulation Methods
2.1 Sharp-interface problem
2.2 Basics of Phase Field Method
2.3 Grand-canonical Multi-phase-field Model
2.4 Thermodynamic description of the free energy
2.5 Incorporation of Anisotropy
2.6 Regularized Phase-field model
2.6.1 Willmore Regularization
2.6.2 Linear Regularization
2.7 Relation to sharp interface theory
3 Interphase Anisotropy Effects on Lamellar Eutectics
3.1 Introduction
3.2 Background
3.2.1 Anisotropic inter-phase boundaries
3.2.2 Theoretical predictions for the tilt angle
3.3 Boundary-integral method
3.3.1 Parameters
3.4 Results
3.4.1 General remarks
3.4.2 Anisotropy functions without missing orientations
3.4.3 Anisotropy with missing orientations
3.5 Discussion
3.5.1 Reconstructing the anisotropy function
3.5.2 Bistability in the numerical simulations
3.6 Conclusions and perspectives
4 Interphase Anisotropy Effects on Bulk Lamellar Eutectics
4.1 Introduction
4.2 Simulation Details
4.2.1 Parameters
4.2.2 Procedures
4.3 Results
4.3.1 Lamellar tilt
4.3.2 Structural selection in single eutectic grain
4.3.3 Multiple eutectic grains
4.4 Conclusions and Outlook
5 Influence of grain boundary anisotropy during directional solidification
5.1 Introduction
5.2 Background
5.2.1 Morphological instability
5.2.2 Grain Boundary (GB) and Sub-Boundary (SB)
5.2.3 Anisotropic interphase boundaries
5.2.4 Theoretical prediction of the tilt angle
5.3 Method
5.3.1 Parameters
5.4 Results
5.4.1 Behaviour of SB below CS
5.4.2 Above MS
5.4.3 Above CS, Below MS
5.4.4 Interaction between SBs
5.5 Conclusions and Outlook
6 Overall summary, Conclusions and Outlook
Appendix A Anisotropy Implementation Techniques
Appendix B Rotation Matrices in 2-D and 3-D
B.1 2-D
B.2 3-D
Appendix C Choice of Anisotropy γ Functions
C.1 4-fold
C.2 2-fold
C.3 In presence of a cusp
C.4 Composite Anisotropy Function
Bibliography
