1. Introduction
In recent years, the phase-field method has become a powerful and versatile technique to study microstructure evolutions in different material systems within simulations [
1,
2,
3]. By this technique, various patterns, evolved in the solidification processes, such as dendrites [
4,
5,
6], eutectics [
7,
8,
9], and peritectics [
10,
11] have been investigated. A general introduction to this method is given in [
12]. Focusing on the eutectics, in all mentioned simulation studies, the
directional solidification (DS) processes have been investigated by using a Bridgman furnace setup. In this setup, the involved solid phases grow into the liquid phase, guided by the applied temperature field. Depending on the system and the process parameters, different growth morphologies and instabilities, such as tilted growth [
13,
14,
15,
16] and oscillations in the solid phases boundaries [
17,
18], have also been reported in the literature. Following the theory of Jackson and Hunt [
19], for a constant velocity of the applied temperature gradient and benefiting domain sizes, a straight lamellar growth of isotropic solid phases is expected in two-dimensional simulations.
Next to the directional solidification process, a further experimental setup, called
rotating directional solidification (RDS), has been introduced by Oswald et al. [
20] and Akamatsu et al. [
21], in order to study the solidification of eutectic structures in thin samples. This rotation leads to the curved trajectories of the solidified phases and to a variation of the growth velocities at the solidification front. In these experimental setups, the sample is rotated with respect to an applied thermal gradient field with a predefined angular velocity. Hence, the growth velocities of the solids for example, vary proportionally to the distance from the center of rotation [
20,
21,
22].
Accordingly, due to the significant effect of the solidification velocity on the pattern formation [
23,
24,
25,
26], the existence of multiple velocities within a single experiment allows studying the interaction of neighboring lamellae with different growth velocities. This can give a better insight into the ongoing velocity-controlled mechanisms in the solidification process and is considered as one of the main advantages of the
RDS technique to the
DS process. A further advantage is the possibility to efficiently study the effects of anisotropic behaviors in the interfacial energies
on the pattern formation. For grains with a negligible solid/solid interface anisotropy
(floating grains), circular growth trajectories have been observed in
RDS experiments. On the contrary, a tilted growth of the solidified phases, with respect to the solid/liquid interface normals, occurs for grains with significant anisotropy amounts
(locked grains) [
21,
22]. Rátkai et al. [
27] have studied the effects of the interfacial energy anisotropies on the pattern formation in binary eutectic Ag-Cu system by means of the phase-field method. In order to mimic the process condition of the
RDS experiments, they have considered an angle between the anisotropy function long axis and the solidification direction. As in their study no rotating disk nor rotating temperature field is modeled, the obtained microstructures have waved lamellar patterns tending to semi-circles by increase in the anisotropy strength (see Figure 5 of [
27]). In addition, simulation studies with possibility of a full rotation can give new insights in role of the anisotropy in pattern formation, which have not so far been conducted in the literature.
Apart from the above mentioned advantages of the
RDS technique, it is worth mentioning that the method is limited to thin sample observations or bulks of materials with 2D growth patterns in steady states. The ternary eutectic system
Bi-In-Sn is an example of such a system with a 2D lamellar microstructure [
28]. Moreover, due to its low melting point (
K [
29]),
Bi-In-Sn has become a practical material system for an in situ observation of the ongoing mechanisms in solidification processes. In the works of [
23,
30,
31,
32], the evolved microstructures from
DS experiments are reported for different growth rates and experimental conditions. Based on the results, three-phase lamellar patterns with
stacking sequences are obtained in the stable growth regions of the floating grains, in which
denotes the
phases,
denotes the
phases, and
represents the
crystalline phases, respectively. In their investigated 2D samples, Witusiewicz et al. [
30] have shown that the resultant amounts of the lamellar spacings for different solidification velocities are in good agreement with the Jackson–Hunt relationship [
19]. They also show this agreement for previously reported results in bulk samples by Ruggiero et al. [
31]. Later, in the work of Bottin-Rousseau et al. [
23], which includes a very restricted velocity range, it is shown that the Jackson–Hunt relationship constant (
) agrees well with the results of [
30]. On the other hand, in case of locked grains, anisotropy can affect the microstructure, in which small domains of
or
superstructure are observed, wherein
a and
b are higher than unity integers [
23]. Mohagheghi et al. [
22] have conducted an
RDS study of the system
Bi-In-Sn in thin samples and have observed the same
stacking sequences as found in
DS experiments with circular trajectories. This structure is reported to be consistent for a wide variation range of the affecting parameters, such as the sample thickness and the growth velocity. However, a tilted growth of the solid phases has also been observed in the locked grains. It has been observed that the
interface anisotropy has a more affective role in the formation of the inclined lamellae compared with an anisotropy of the
interface [
22]. This investigated behavior of the interfacial anisotropies is another reason which makes the system
Bi-In-Sn favorable for the first investigations of the rotating directional solidification process within phase-field simulations.
In order to simulate the
RDS process, the utilized phase-field model is first introduced and the temperature formulation, which is to resemble the effects of the rotating temperature field, is presented. Then, the thermodynamic modeling of the system is shown, using the method from Noubary et al. [
33] and the
Calphad database from Witusiewicz et al. [
29]. Next, the simulation setup and the used parameters are introduced. To validate the generated material system,
DS simulations are performed in the desired growth velocity ranges of the experimental
DS and
RDS studies [
22,
30], respectively. The mentioned validation is performed by Jackson–Hunt analysis and by comparing the obtained results with the reported experimental data in the literature. Finally, the phase-field studies of the rotating directional solidification are performed, resulting in the circular trajectories of the solidified phases for the floating grains and inclined trajectories with respect to the solid–liquid interface normals, for the locked grains. The obtained results are compared with the existing experimental data of [
21,
22,
30] In a last step, the made observations are summarized and discussed and an outlook to upcoming simulation studies is given.
2. Methods
For the simulation studies in this work, the grand potential functional [
34,
35] serves as the basis of the utilized phase-field model. The model is explicitly described in [
36,
37,
38], while its utilization in the simulation of the directional solidification process has been reported for example in [
13,
38,
39]. The following descriptions of the phase-field model are implemented into the multi-physics phase-field-framework
Pace3D, version 2.4 [
40,
41]. By considering
N order parameters
for the involving phases, the local phase fractions are stored in the vector
. The phases are labeled by the Greek letters
. To avoid a confusion with the parameter of the interfacial energies
, the Greek letter
is not used to describe phases. The indices of the phases are marked with hat symbols
, in order to distinguish phase counters by a disparate labeling. The vector
indicates the corresponding
K amounts of the chemical potentials
. Based on an Allen–Cahn approach and Fick’s law, the time evolution equations
of the coupled phase fields and the chemical potentials are formulated as:
In Equation (
1), a diffuse interface is exploited to model the phase evolution in the simulation procedure. The gradient energy density
, the potential energy density w
, and the driving force
are the key parameters that define the shape of the interface [
38]. In order to resemble effects of anisotropy in the interfacial energies, a cubic positive anisotropy as given in [
42] is applied. In this formulation, the gradient energy density is expressed as
with
as the interfacial energy parameter,
as the generalized gradient vector and
as anisotropy function. The cubic positive anisotropy is defined as
with
as its strength [
42]. The potential energy density function is described in [
38] and includes the interfacial energies
and the higher-order term
. This higher order term is introduced, so as to suppress a third-phase appearance at the phase boundaries.
is adjusted to reflect the correct equilibrium angle conditions at the triple junctions [
38,
42]. The thickness of all interfaces is controlled by the parameter
and its kinetics is described by the relaxation coefficient
[
35]. In Equation (
1), the Lagrange multiplier
is exploited to accomplish the constraint
. In Equation (
2), the evolution of
is considered to characterize the diffusion processes. The information of the diffusion coefficient matrix
for the involved phases is included in the function
[
35] as the mobility, and the function
[
43] interpolates between the different phases. As is common in phase-field models, the widths of the interfaces are orders of magnitude larger than their physical values [
34], while the anti-trapping current
[
44,
45] helps to adjust the influences of these nonphysically enlarged interface widths. The concentrations of the
K chemical elements in the involved phases, are saved in the vector
including
K components. The driving force is defined by the grand potential deviations
of the evolved phases.
, for example, describes the grand potential of phase
. All grand potentials depend on the phase-field vector, the chemical potentials and the temperature
T and are stored in the vector
. Together with the concentrations
and the chemical potentials
, the grand potentials can be derived from
Calphad databases, by using the general method introduced in [
33]. In this method, the volume and the pressure are assumed to be constants, which ensures the thermodynamic consistency of the system. The numeric algorithm to solve the system of Equations (
1) and (
2), includes a spatial discretization scheme. Based on this scheme, the amounts of the phase-fields and chemical potentials at current time
t are considered as the inputs at each grid cell of the simulation domain. Initially, the phase-fields at time
are calculated from the inputs in which
stands for the considered time step. The outcome is utilized to calculate the chemical potentials at
. All these calculations are performed considering the amounts in the neighboring cells in forming the discretization scheme [
36]. The spatial derivations in the coupled set of the partial differential equations is discretized with finite differences and their time evolution is solved by an explicit Euler scheme [
46].
As the evolved microstructures of the bulk samples are experimentally reported to be 2D in the steady states [
28], the investigated samples in this work are considered to be two-dimensional, and a 2D heat distribution is assumed in the temperature field formulation. However, the derived formulation can still be used in the 3D cases of such material systems. Considering the solution of the heat equation in steady-state
, the temperature formulation can be expressed as following in form of a linear function with
space and
time as its variables:
As utilized in the works of [
13,
47,
48] for
DS simulations, in this formulation
indicates the base temperature,
denotes the temperature gradient,
v is the temperature gradient velocity,
t represents the simulation time, and
x refers to the growth direction, respectively. Hence, a linear temperature increase occurs in the solidification direction, with an overall decrease over time. In case of
RDS simulations, despite rotating the sample as described in the depicted experimental setup in
Figure 1a, a rotation of the temperature profile is realized within the
Pace3D framework. By considering a rotating temperature field, the same physical effects can be reproduced with a reduced computational effort. In order to derive such a temperature formulation, moreover to satisfaction of the heat distribution equation, the following constraints should be considered:
- (i)
The sought formulation for the temperature has to represent the effects of the hot and cold isothermal blocks, the effects of the temperature profile in between, and the caused variations of the temperature, due to the rotation. To resemble the near uniform distribution of the heat in the vicinity of the blocks, a low temperature gradient is required in these segments, whereas in the segments nearer to the disk center, containing the solidification front, a sharper temperature change is necessary. Hence, a linear function for the whole simulation domain with a constant temperature gradient amount, as described in Equation (
5), is not favored. By using such a function, the system temperature can rise dramatically with increasing distance from the rotation center, which can lead to a destabilization of the modeled material system, specially in large domain simulations.
- (ii)
To ensure a correct calculation of the evolution equations (Equations (
1) and (
2)), a continuously differentiable function is needed for the temperature, with respect to the space, in which differing amounts of the derivatives can exist in different space points. Based on this constraint, a piecewise function, composed of three linear functions with different temperature gradients can not be considered due to non-continuity of the first derivatives in the connection points.
One option to fulfill these constraints is the usage of a
function, as schematically illustrated in
Figure 1b. In this case, the upper and lower asymptotes of the function can resemble the hot and cold temperature blocks or the desired maximum and minimum temperatures in the simulation domain. The higher temperature gradient in the middle, physically represents the sharper temperature change at the disk center in comparison with the outer side. With this, well-defined solidification and melting fronts can establish themselves within the simulations. In order to formulate the discussed temperature profile, the start setting without disk rotation is considered first. By using the applied coordinate system of
Figure 1a, the temperature is formulated as a function of
y, in the form:
in which
is the temperature and
is its gradient in the rotation center (
).
is a constant coefficient to determine the amounts of the mentioned asymptotes. In this formulation,
will be equal to
, as expected. Next, the rotational matrix is used to model the angle of rotation
, depending on the time
t, with an angular velocity of
. The new coordinate system (
) is obtained as:
The combination of Equations (
6) and (
7) results in the final space- and time-dependent formulation for the rotating temperature profile:
The used parameters for
and
are given in
Table A3 in the
Appendix A. With these parameters and utilization of Equation (
8), the average deviation form the exact solution of the steady-state heat equation
in the simulation points is calculated as
. This deviation is negligible in authors’ opinion. The introduced description of a rotating temperature field is used to perform the subsequently shown
RDS phase-field simulation studies. It is worth mentioning that the implemented temperature formulation is independent from the investigated material system.
4. Summary and Outlook
In this work, phase-field simulations of the rotating directional solidification are performed, which are mentioned in the literature for the first time. This enables a precise study of the most important affecting parameters of the pattern formation, such as the solidification velocity, which can be studied as a process parameter, or the anisotropy of the interfacial energies, which can be regarded as a system parameter. In order to simulate this process, the effect of a rotating sample on the microstructure evolution, caused during the solidification, is rebuilt by an imprinted rotating temperature field. The modeled
Bi-In-Sn system is validated in two-dimensional simulations of the
DS process and is subsequently used to perform the simulation of the
RDS process. By using isotropic interfacial energies, circular trajectories with the expected
repeat units are obtained, which represent the expected microstructure of the floating grains in
RDS simulations. Furthermore, by introducing anisotropy of the interfacial energies into the material system, the growth of tilted lamellae is observed. In this case, the experimentally reported dominance of the
interfaces in the formation of the locked grains is recovered in the phase-field simulations. It has to be mentioned that, the effect of the applied thin
phase layer on the resulting microstructures can not be ascertained in its full extent. As the adjustment of the solid phases is influenced by several parameters, such as the growth velocities, the surrounding phases and the boundary conditions, an independent investigation of the different influences is required in forthcoming studies. For this, the implementation of a new barrier formulation without the need of an additional thin solid phase layer is planned, to improve the calculations of the interactions between the phase fields, concentration fields and a curved barrier section for upcoming investigations in future works. However, the presented results in this work show the general applicability of the presented method to investigate different growth morphologies within
RDS simulations, which lays the foundation for further investigations. The effect of angular velocity variations on the microstructure evolution, for example, can be studied within the presented simulation setup. Initial simulations have already shown that an increase in the angular velocity
can lead to instabilities, such as oscillations of the phase boundaries and the elimination of lamellae. Such behavior has also been observed in experimental works [
23,
28]. In addition, the systematic variation of anisotropic interfacial energies strength, is a further possible topic for upcoming research, which can be investigated with the presented
RDS setup. The presented simulation setup is suitable for any eutectic material system which has 2D pattern in its stable growth, independent from the number of evolving phases.