The total amount of QC phases for ACF-II was about 15% greater than that in ACF-I, as seen in Table 1. However, the amount of ω-phase for the ACF-I was 13% higher. The particle size calculations indicate that the grain size of ACF-II was bigger than that of ACF-I. These calculations suggest that staying at 660°C for an extended duration can yield a greater concentration of quasicrystal. Furthermore, interesting differences were observed in the crystalline and quasicrystalline phases for ACF-I at RT and ACF-II at 660°C. These results indicate that the peak intensity at 660°C for ACF-II was sharper than that of the ACF-I sample at RT, which can be interpreted as evidence of random tiling. The random tiling of the plane by a set of objects, related by rotational symmetries, is a paradigm for the formation of quasiperiodic order due to entropy. The random tiling of the icosahedral phase is assumed to be attributed to the entropy term where the quasicrystalline state is selected by having a larger configurational entropy than the competitive crystal phases. Reversible structural transformation between a crystalline phase at a low temperature and a quasicrystalline structure at a high temperature via an intermediate modulated icosahedral phase, which seems to occur at 660°C.
Table 1. Results of Peak position(PP), intensity (Ir(%)), FWHM, crystallinity (IC(%)) and the particle size (PS) for the ψ-phases and ω-phase from integrated intensities graphs (Note: The intensity of Al2O3 was 100%.)
Figure 5 summarizes the phase evolutions that occurred throughout the heat treatment and cooling process and demonstrates that though Al, Cu and Fe each began from the same horizontal axis, they disappeared at different stages of heating. The θ-phase (Al2Cu) was the dominant phase at 350°C, at which point Al and Cu had completely mixed.  At 415°C, the θ–phase with [211] orientation started to change to a η-phase with [203, 403] orientation and then into a β-phase, suggesting that Fe had begun to interact with Al and Cu. At 465°C, the β-phase was the dominant phase. The ω-phase was dominant when the sample was annealed at 580°C for 20 minutes and disappeared when the temperature reached 710°C. The appearance of the ω-phase implies that the Al-Cu-Fe components had completely interacted. No phases appeared when the temperature increased from 710°C to 800°C, nor did the data show any phase changes after the sample was kept at 800°C (in-situ XRD, 40 minutes; in-situ TEM, 70 minutes). However, when the sample was cooled down to RT, a θ-phase, ω-phase and QC phase clearly emerged, as shown in the in-situ XRD.
Figure 5. Phase evolutions of Al-Cu-Fe from RT (initial) to RT (final), with the heating profile (left) and graph of the changing phases (right), shown as a time (Y- axis) and 2 theta (degree) (X-axis).
This formation was clearly observed in the in-situ TEM images. From these data, we can confirm that quasicrystalline structure forms and stabilizes at 660°C during the cooling process; these formations were stable from 660°C to RT. The in-situ TEM and in-situ XRD results show that to form a quasicrystalline structure, the sample needs to be in a liquid state (800°C). Then, as the sample cools to 660°C, the quasicrystal can be formed, and this formation remains unchanged with continued cooling. The additional in-situ XRD data provides evidence of random tiling, suggesting the quasicrystal is entropy-stabilized at 660°C.
Discussion
We used thin film samples to explore the growth mechanism involved in the formation of Al-Cu-Fe quasicrystals. The phase evolutions were captured by in-situ XRD. To confirm the in-situ XRD data, we further carried out secondary in-situ growth using in-situ TEM, in which we observed the real time formation of θ-Al2Cu, η-AlCu, β-Al(CuFe), ω-AlCuFe and ψ-phase (quasicrystal).
The overall growth mechanism was investigated and can be divided into the following stages: First, at the lower temperature, Cu began to move to the Al and voids were seen at the Al-Cu interface. With an increase in temperature, the Al and Cu reacted and formed θ-Al2Cu and η-AlCu binary phases, of which θ-Al2Cu was dominant. Second, as the temperature rose to 415°C, the cubic β-Al(CuFe) phase appeared and was dominate at around 470°C, indicating that the ternary phase is more stable than the binary phase at this stage. However, further increasing the temperature to 580°C showed that the ω-AlCuFe is more thermodynamically stable compared to the β-Al(CuFe) phase. Third, between 720°C to 800°C, a liquid state with a small amount of solid intermetallic compound was captured by TEM. This liquid state of Al-Cu-Fe facilitates better mixing, as homogenization has been reported to have a significant effect on boosting the quasicrystal phase \cite{Parsamehr_2018}. Finally, after reaching the peak temperature of 800°C, the sample was cooled to RT. As has been previously reported \cite{Tsai_1987}, it is during this cooling process that the QC phase forms. Specifically, we observed QC formation at 660°C. The structure was stable at this point, and remained stable throughout cooling to RT.
Whether a QC is energy or entropy stabilized, and whether its structure can be described by an ordered tiling or by a random tiling has been a topic of debate since stable QC were found. Our aim was to shed light on these issues. We found that both entropy and energy play important roles in stabilization. We observed that at high temperatures, entropy plays a key role in thermodynamic stability which is evidence of random tiling of the icosahedral phase in that it has a larger configurational entropy than the competing crystal phases. On the other hand, energy also contributes to stabilization during the cooling process. At lower temperatures, the structure becomes regularly ordered as ω-AlCuFe and θ-Al2Cu tetragonal structures with space groups P4/mnc and I4/mcm, respectively.
In summary, we observed the dynamic growth of Al-Cu-Fe QC using in-situ XRD and in-situ TEM. We microscopically identified the growth mechanism during the crystallization process and showed evidence of random tiling. Our study using in-situ techniques provides fundamental understanding on synthesis of the emerging QC thin film materials and paves the way to optimal design of functional nanostructures.