High-Temperature Specific Heat of Al–Cu–Ru Icosahedral Quasicrystals and 1/1 Crystal Approximants

Abstract

We report the results of specific heat measurements for Al–Cu–Ru icosahedral quasicrystals (i-QCs) and 1/1 crystal approximants (1/1-CAs) in the temperature range from 350 to 1250 K. The i-QCs and 1/1-CAs showed a marked upward deviation of the specific heat from the value of Dulong-Petit’s law above 600 K, reaching approximately 1.5 times the value at 1100 K. In addition, some i-QCs showed a large broad peak in the specific heat at approximately 1200 K. The origin of these excessive specific heat values above the Dulong-Petit value is discussed considering the high-dimensional nature of the structural order in the QCs and CAs.

Fig. 3 (a): Temperature dependence of specific heat per atom at constant pressure, c_P, for nine samples (four i-QCs, three 1/1-CAs, one MC, and one Cu sample). (b): Temperature dependence of specific heat per atom at constant volume, c_V, converted from c_P in (a).

1. Introduction

Quasicrystals (QCs) have a quasiperiodic translational order,^1,2) as opposed to the periodic order of conventional crystals. Mathematically, the quasiperiodic order has a high-dimensional nature; every quasiperiodic structure in three-dimensional (3D) physical space can be described as a 3D section of a hypothetical higher-dimensional periodic crystal (sometimes called a hyper-crystal), where the 3D space is incommensurate with the hyper-crystal lattice.²⁾ By changing the incommensurate orientation relationship to a commensurate one, the same hyper-crystal generates a series of periodic structures that are locally similar to the structure of the QCs. Such periodic structures are called crystal approximants (CAs) to the QCs.³⁾ Based on these structural features, both quasicrystals and their crystal approximants are termed “hyper-materials.”

One of the important objectives of research on QCs is to reveal the physical properties that truly reflect the peculiarity in the structural order of QCs. Unusual behavior in terms of the specific heat at high temperatures^4–6) is one of the candidates for such physical properties. High-temperature specific heat measurements have been reported for icosahedral Al–Pd–Mn^4,5) and Al–Cu–Fe,⁶⁾ and decagonal Al–Cu–Co^4,5) QCs, all of which show large upward deviations from the Dulong-Petit value; the specific heat per atom at constant volume c_V, is 4.5–5k_B (k_B: the Boltzman constant) at high temperatures, which is substantially larger than the Dulong-Petit value of 3k_B. Originating from the high-dimensional nature of quasiperiodicity, QCs have a special type of elastic degrees of freedom that are termed as phason degrees of freedom.^7–9) The generalized elasticity of quasicrystals is described in terms of the phason elastic field in addition to the phonon (conventional) elastic field. Such extra degrees of freedom in QCs may result in c_V values exceeding the Dulong-Petit value,^4,5) although it can also arise from an electronic origin.^6,10) In any case, to determine whether the unusual behavior of c_V described above is a characteristic feature commonly seen in QCs, we need to carry out more experiments with QCs of various alloy systems.

To clarify the origin of the excessive c_V in QCs, investigating the behavior of c_V in CAs could be helpful. High-temperature specific heat values have been measured for a 1/0 CA of an Al–Pd–Fe system, in which the upward deviation of c_V form the Dulong-Petit value is considerably small. This result supports the assumption that large upward deviations of c_V are characteristic features of QCs, possibly having a phasonic origin. However, it should be noted that local atomic jumps like phason flips and also longer-ranged structural fluctuations like phason fluctuations could also occur in high-order CAs closer to QCs. Considering this viewpoint, investigating the behavior of c_V in higher-order CAs (higher than 1/0) could be helpful and interesting.

Thus, in this study, we prepared icosahedral QCs (i-QCs) and 1/1 CAs of the Al–Cu–Ru system and performed high-temperature specific heat measurements using differential scanning calorimetry (DSC). Both phases showed considerably large deviations of c_V from the Dulong-Petit value. In this paper, we report the results of these measurements and discuss the possible origin of such excessive c_V values.

2. Experimental Procedures

We produced samples of three phases: an i-QC, a 1/1-CA, and a monoclinic crystal (MC). First, we prepared alloys with the following compositions from the elemental constituents through arc melting under an argon atmosphere: Al₆₅Cu₂₀Ru₁₅, Al₆₆Cu₁₉Ru₁₅, and Al₆₇Cu₁₈Ru₁₅ for the i-QC; Al₅₈Cu₃₀Ru₁₂ for the 1/1-CA; and Al₁₃Ru₄ for the MC. The alloys were annealed first at 1073 K for 96 h, and subsequently at 823 K for 72 h, followed by water-quenching. Powder X-ray diffraction measurements were performed using Cu Kα radiation to identify the phases formed. From the annealed ingots, we cut out specimens with a circular column shape (ϕ5 mm × 1 mm) and performed specific heat measurements by DSC using a NETZSCH STA404F3 instrument in the temperature range from 350 to 1250 K with a heating rate of 20 K/min. For i-QC, we prepared two specimens from the ingot of Al₆₅Cu₂₀Ru₁₅, and one each from Al₆₆Cu₁₉Ru₁₅ and Al₆₇Cu₁₈Ru₁₅ ingots. For 1/1-CA, we prepared three specimens from the Al₅₈Cu₃₀Ru₁₂ ingot. We also prepared a reference Cu sample and performed DSC measurements under the same conditions. High-temperature X-ray diffraction experiments were conducted using Cu Kα radiation to evaluate the thermal expansion coefficients of the i-QC and MC in the temperature range from 300 to 1200 K. Using the evaluated thermal expansion coefficients and the bulk elastic moduli previously reported, the specific heats at constant pressure measured were converted to those at constant volume.

3. Results and Discussion

Figures 1(a)–(e) show the X-ray diffraction spectra of the prepared samples; their compositions are Al₆₅Cu₂₀Ru₁₅ (a), Al₆₆Cu₁₉Ru₁₅ (b), Al₆₇Cu₁₈Ru₁₅ (c), Al₅₈Cu₃₀Ru₁₂ (d), and Al₁₃Ru₄ (e). Here, the Cu Kα₂ component was removed from the raw data through data treatment. In Figs. 1(a)–(c), all peaks correspond to the i-QC. The peaks in Figs. 1(d) and (e) can be indexed as 1/1-CA and MC, respectively. Thus, we obtained single-phase i-QC (a)–(c), 1/1-CA (d), and MC (e) samples. All the X-ray diffraction peaks in Figs. 1(a)–(e) are sharp, and the peak widths are close to the instrumental resolution limit, indicating the high structural quality of the samples.

Fig. 1

X-ray diffraction spectra for the samples having the compositions of Al₆₅Cu₂₀Ru₁₅ (a), Al₆₆Cu₁₉Ru₁₅ (b), Al₆₇Cu₁₈Ru₁₅ (c), Al₅₈Cu₃₀Ru₁₂ (d), and Al₁₃Ru₄ (e). Diffraction peaks of i-QC, 1/1-CA, and MC phases are indicated.

Figure 2(a) presents the temperature dependences of d/d_373K (d = λ/(2 sin θ)) measured for the 664004 peak of the i-QC, and 620 and $62\bar{3}$ peaks of the MC. We performed curve fitting for the d/d_373K data, and the linear thermal expansion coefficients, α, given in Fig. 2(b) were calculated by differentiating the function fitted to the data. In Fig. 2(b), α values for Cu¹¹⁾ are also presented for comparison. The α values obtained for the i-QC and MC are slightly lower than those of Cu.

Fig. 2

(a) Temperature dependences of d/d_373K (d = λ/(2 sin θ)) measured for 664004 peak of i-QC, and 620 and $62\bar{3}$ peaks of MC. The results of curve fitting are also shown. (b): Linear thermal expansion coefficients α calculated by differentiating the fitted curves in (a). The α values for Cu¹¹⁾ are also presented for comparison.

Figures 3(a) presents the data of specific heat per atom at constant pressure, c_P, measured for nine samples (four i-QCs, three 1/1-CAs, one MC and one Cu sample). To compare the specific heat values with the Dulong-Petit value, we converted the measured c_P to c_V, namely, the specific heat per atom at constant volume, using the following thermodynamic relation:   

\begin{equation} c_{V} = c_{P} - 9VB \alpha^{2}T, \end{equation} (1)

where V is the atomic volume and B is the bulk modulus. We used the respective α values of i-QC, MC, and Cu given in Fig. 2(b). For the 1/1-CA, we used the α values of the i-QC. Tanaka et al.¹²⁾ have reported B measurements for an Al–Cu–Fe–Ru i-QC phase in the temperature range from 4 to 1073 K. We extrapolated these data up to 1250 K and used them for the i-QC and 1/1-CA. For the MC of Al₁₃Ru₄, B at only 0 K determined through first-principles calculations is reported.¹³⁾ We assumed the same temperature dependence as that of the B data for the Al–Cu–Fe–Ru system and used them for the MC. For Cu, the temperature dependence of B reported by Chang and Himmel¹⁴⁾ was used. The thus-obtained c_V data for the i-QC, 1/1-CA, MC, and Cu samples are presented in Fig. 3(b).

Fig. 3

(a): Temperature dependence of specific heat per atom at constant pressure, c_P, for nine samples (four i-QCs, three 1/1-CAs, one MC, and one Cu sample). (b): Temperature dependence of specific heat per atom at constant volume, c_V, converted from c_P in (a).

First, the c_V values for the MC and Cu samples obey the Dulong-Petit law quite well in the entire temperature range, as shown in Fig. 3(b). However, their c_P values are not constantly equal to 3k_B; they deviate upward from 3k_B with increasing temperatures. This finding indicates that converting c_P to c_V is indispensable for precisely evaluating the amount of deviation from the Dulong-Petit value. In contrast to the c_V values for the MC and Cu samples, those for the i-QC and 1/1-CA samples show a marked deviation from the Dulong-Petit value; they exhibit deviations at approximately 600 K, reaching 4–4.5k_B at 1100 K. Here, while the melting temperature of the i-QCs is 1325 K, that of the 1/1-CA is 1113 K. Therefore, we could perform measurements only up to 1100 K for the 1/1-CA samples. One of the i-QC samples shows two peaks at 850 and 990 K, which correspond to the melting of the Al₂Cu and Al₆Ru phases, respectively; the peaks indicate a slight contamination of these phases, which was undetected by X-ray diffraction. Above 1150 K, the c_V values of the i-QC samples show more rapid increases, and one of the samples exhibits a peak at 1220 K.

The elasticity of conventional crystals is described in terms of the phonon (conventional) elastic field, u(r). Within linear elasticity, the elastic energy, E_el, can be expressed as   

\begin{equation} E_{\text{el}} = \frac{1}{2}\int K_{ijkl}u_{ij}u_{kl}d\mathbf{r}, \end{equation} (2)

where K_ijkl is the elastic constant tensor and $u_{ij} = \frac{1}{2}(\frac{\partial u_{i}}{\partial r_{j}} + \frac{\partial u_{j}}{\partial r_{i}})$ is the elastic strain. Using the Fourier transform of u(r), i.e.,   

\begin{equation} \tilde{\mathbf{u}}(\mathbf{q}) = \int \mathbf{u}(\mathbf{r})e^{-i\mathbf{q}\cdot \mathbf{r}}d\mathbf{r}, \end{equation} (3)

eq. (2) can be rewritten as   

\begin{align} E_{\text{el}} &= \frac{1}{2}\int \frac{1}{(2\pi)^{3}}K_{ij}(\mathbf{q})\tilde{u}_{i}(\mathbf{q})\tilde{u}_{j}(\mathbf{q})d\mathbf{q} \\ &= \sum_{k=1}^{N} \frac{\Delta \mathbf{q}}{(2\pi)^{3}}K_{ij}(\mathbf{q}_{k})\tilde{u}_{i}(\mathbf{q}_{k})\tilde{u}_{j}(\mathbf{q}_{k}), \end{align} (4)

where   

\begin{equation} K_{ij}(\mathbf{q}) = K_{ikjl}q_{k}q_{l}, \end{equation} (5)

and N is the number of allowed q, which is equal to the total number of atoms. The expected value of E_el at temperature T should be   

\begin{equation} \langle E_{\text{el}} \rangle = Z^{-1}\int E_{el} e^{-E_{\text{el}}/k_{\text{B}}T}d\Gamma, \end{equation} (6)

where $d\Gamma = d\tilde{\mathbf{u}}(\textbf{q}_{1})d\tilde{\mathbf{u}}(\textbf{q}_{2}) \ldots d\tilde{\mathbf{u}}(\textbf{q}_{N})$, and   

\begin{equation} Z = \int e^{-E_{el}/k_{\text{B}}T}d\Gamma \end{equation} (7)

is the partition function. From eqs. (4), (6), and (7), we obtain   

\begin{equation} \langle E_{\text{el}} \rangle = Nc_{V}T \quad \left( c_{V} = \frac{3}{2}k_{\text{B}} \right). \end{equation} (8)

Here, eq. (8) can be interpreted in terms of the low of equipartition of energy. That is, we have 3N degrees of freedom in atomic displacements, each of them contributing $\frac{1}{2}k_{\text{B}}T$ to the total elastic energy. We have another 3N degrees of freedom in atomic momenta, which give the kinetic energy:   

\begin{equation} \langle E_{\text{k}} \rangle = Nc_{V}T \quad \left( c_{V} = \frac{3}{2} k_{\text{B}} \right). \end{equation} (9)

Combining eqs. (8) and (9), we obtain the total energy:   

\begin{align} \langle E_{\text{tot}} \rangle = \langle E_{\text{el}} \rangle + \langle E_{\text{k}} \rangle &= Nc_{V}T \quad (c_{V} = 3k_{\text{B}})\\ &\quad (\text{Dulong-Petit's law}). \end{align} (10)

As described in section 1, QCs have additional elastic degrees of freedom, termed phason degrees of freedom; the generalized elasticity of QCs is described in terms of the phason elastic field, w(r), in addition to u(r).^7–9) Within linear elasticity, the elastic energy, E_el, of QCs comprises three quadratic terms, i.e., pure phonon, pure phason, and phonon-phason coupling terms:   

\begin{align} E_{el} &= E_{u\unicode{x2013}u} + E_{w\unicode{x2013}w} + E_{u\unicode{x2013}w} = \frac{1}{2} \int K_{ijkl}^{u\unicode{x2013}u}u_{ij}u_{kl}d\mathbf{r} \\ &\quad + \frac{1}{2}\int K_{ijkl}^{w\unicode{x2013}w}w_{ij}w_{kl}d\mathbf{r} + \int K_{ijkl}^{u\unicode{x2013}w}u_{ij}w_{kl}d\mathbf{r}, \end{align} (11)

where $w_{ij} = \frac{\partial w_{i}}{\partial r_{j}}$ is the phason strain, and $K_{ijkl}^{u\text{–}u}$, $K_{ijkl}^{w\text{–}w}$ and $K_{ijkl}^{u\text{–}w}$ are elastic constant tensors. Here, the argument of eqs. (2)–(8) can be applied independently to E_u–u and E_w–w, resulting in the additional c_V = (3/2)k_B originating from the latter. Neglecting the phonon-phason coupling effect, which is usually small, we could expect c_V = 3k_B + (3/2)k_B = (9/2)k_B for QCs. Note that this additional c_V should be observable only at high temperatures where phasons are activated.

According to Fig. 3(b), c_V of our i-QC samples increases from approximately 3k_B at 400 K to approximately 4–4.5k_B at 1100 K, which may be attributable to the phason effects described above. The 1/1-CA samples also show a similar increase in c_V, as observed in Fig. 3(b); this behavior is different from that of the 1/0 CA of the Al–Pd–Fe system, which showed much lesser increase in c_V.⁵⁾ In general, the higher the degree of approximation of an CA, the closer its structure is to that of a QC. Then, local atomic jumps like phason flips and also longer-ranged structural fluctuations like phason fluctuations could occur in high-order CAs, where phason flips and phason fluctuations are defined as the local and long-ranged structural variations, respectively, in QCs, which are brought about by the variation in w(r).¹⁵⁾ However, the order 1/1 is still a relatively low order, thus making it difficult for us to interpret the c_V increase observed for the 1/1-CAs in Fig. 3(b) as due to the phason-like effects. In conventional crystals, the c_V increases at high temperatures are generally caused by the formation of vacancies in thermodynamic equilibrium and/or anharmonic lattice vibrations. Further experiments are needed to clarify the cause of the c_V increase in the 1/1-CAs.

4. Conclusions

We produced high-quality single-phase samples of Al–Cu–Ru i-QCs and 1/1 CAs, and an MC of Al₁₃Ru₄, and measured their specific heats in the temperature range from 350 to 1250 K. We also measured the specific heat of Cu for comparison. The measured specific heats per atom at constant pressure, c_P, were converted to those at constant volume, c_V, using the thermal expansion coefficients and bulk elastic moduli. While the c_V values of the MC and Cu samples obeyed the Dulong-Petit law (c_V = 3k_B) quite well in the entire temperature range, those of the i-QCs and 1/1-CAs showed a marked upward deviation from the Dulong-Petit value above 600 K, reaching c_V = 4–4.5k_B at 1100 K. In addition, some i-QCs showed a broad specific heat peak at approximately 1200 K. We showed theoretically that phason fluctuations could contribute an additional specific heat of Δc_V ≈ 1.5k_B to the Dulong-Petit value of c_V = 3k_B. The excessive specific heat observed for i-QCs can partly be attributed to this phason effect.

Acknowledgments

We thank Y. Kamimura of our group for his help in manuscript preparation. This work was supported by KAKENHI Grant-in-Aid (No. JP19H05821) from the Japan Society for the Promotion of Science (JSPS).

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