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High-temperature emissivity of silica, zirconia and samaria from ab initio simulations: role of defects and disorder

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Modelling Simul. Mater. Sci. Eng. 22 075004 (http://iopscience.iop.org/0965-0393/22/7/075004) View the table of contents for this issue, or go to the journal homepage for more

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Modelling and Simulation in Materials Science and Engineering Modelling Simul. Mater. Sci. Eng. 22 (2014) 075004 (15pp)

doi:10.1088/0965-0393/22/7/075004

High-temperature emissivity of silica, zirconia and samaria from ab initio simulations: role of defects and disorder Stanislav M Avdoshenko and Alejandro Strachan School of Materials Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, IN, USA E-mail: [email protected] Received 13 December 2013, revised 21 July 2014 Accepted for publication 15 August 2014 Published 19 September 2014 Abstract

Understanding and eventually controlling the high-temperature spectral emissivity of ceramic materials is important for a range of applications, including thermal barrier coatings. In this paper, we use ab initio density functional theory simulations to predict the emissivity of silica, zirconia and rare-earth oxides. High-temperature emissivity is dominated by processes with energies lower than the band gap of these materials and we focus on how dynamic and static features in the atomic structure of these materials (including defects, glasses and thermal fluctuations) enable transitions with desired energies. We find that neutral oxygen vacancies contribute significantly to the high emissivity of ZrO2 . On the other hand, neutral point defects in α and amorphous silica fail to provide transitions with energies significantly below the band gap, explaining the low emissivity of this material. In the case of Sm2 O3 , we find that transitions between localized f-electron states as well as point defects contribute to its high emissivity. Interestingly, dynamical changes in electronic structure in samples taken from molecular dynamics simulations of molten materials lead to a significant increase in their emissivity. Keywords: thermal barrier coatings, emissivity, rare-earth oxides S Online supplementary data available from stacks.iop.org/MSMSE/22/ 075004/mmedia (Some figures may appear in colour only in the online journal)

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Modelling Simul. Mater. Sci. Eng. 22 (2014) 075004

S M Avdoshenko and A Strachan

1. Introduction

The spectral emissivity of materials results from coupled processes involving ionic and electronic degrees of freedom, and its understanding at a fundamental level is important from a basic science perspective and can, at the same time, benefit applications that require engineering the electromagnetic response of materials [1]. This is particularly important in thermal barrier coatings (TBCs) where it is desirable to tune spectral emissivity to either avoid external electromagnetic radiation from heating the active structure or to use electromagnetic radiation to remove energy from the structure [2–4]. Examples of the first category are coatings used in a jet engine combustion chamber [5–9] and examples of the second are TBCs used to protect the leading edge of supersonic vehicles during re-entry. In the latter case, which motivates our work, fluid shockwaves are generated, resulting in ionized atmospheres and temperatures reaching up to 2000 K [3]. Our goal is to characterize the fundamental mechanisms that contribute to the high-temperature emissivity of oxides, focusing on the relationship between atomic structure, electronic properties and optical response. The key figure of merit for radiative cooling is spectral emissivity, the ratio between the emittance of the material and that of a blackbody at the temperature of interest. Both ionic and electronic processes lead to the absorption and emission of photons and, thus, contribute to emissivity [10, 11]. In the case of ceramics of interest in TBCs, ionic processes contribute significantly at low frequencies and energies, approximately up to 1500 cm−1 , corresponding to 0.186 eV and ∼6.6 µm or only slightly higher. Electronic processes, on the other hand, extend to significantly higher energy ranges. For example, excitations from the conduction band to the valence band involve energies higher than the band gap of the material (several eV). Interestingly, achieving high emissivities at the temperatures of interest (1000– 2000 K) requires processes involving intermediate energies, since the maximum in blackbody spectra at T = 2000 K occurs at ∼0.8 eV. Thus, atomic-level defects capable of creating electronic states within the band gap can play an important role in emissivity, as they can enable transitions at energies intermediate between those of ionic processes and band-to-band electronic transitions [12]. Ab initio electronic structure calculations using density functional theory (DFT) are becoming indispensable to establish relationships between atomic structure and electrooptical properties. These calculations are providing key information about the effects of composition [13, 14] and polymorphs [15] in optoelectronic properties and helping interpret optical and electronic experimental measurements [16]. This work has included materials of interest for thermal barrier coatings; see, for example, [15]. Despite such progress, little is known about processes with energies in the range that governs emissivity in the 1000 K to 2000 K temperature range of interest in many TBC applications and how they affect emissivity. In this paper, we use DFT, along with Planck’s blackbody radiation and Kirchhoff’s laws, to predict the electronic contribution to spectral emissivity of several ceramics: ZrO2 [17] and SiO2 , as well as rare earth oxide (REO) Sm2 O3 [18, 19], focusing on the spectral region that dominates emissivity at high temperatures. The paper is organized as follows. In section 2, we describe the theoretical methods and models used. As described in section 3, we find that neutral oxygen vacancies (Ov ) enhance the emissivity of ZrO2 but have relatively minor effects in silica and REOs. In section 4, we discuss the effects of localized fstates on the high-emissivity response of REOs, and in section 6, we quantify the effect of thermal fluctuations of atomic structure on emissivities. Finally, conclusions are drawn in section 7. 2



Figure 1. Systems of interest considered in this report. Top row: an atomistic representation of defect-free Sm2 O3 (free), a highlighted position of an oxygen vacancy (Ov) in Sm2 O3 structure and a superposition of time snapshots for Sm2 O3 molecular trajectory at 3000 K (D). Bottom row: schematic representation of essential structural changes, which might lead to the optical changes (structural rearmaments for defects (Ov) or time evaluation of local structural parameters).

2. Simulation details 2.1. Materials of interest

Figure 1 summarizes the systems of interest and possible structural features that could affect their optical response. In this report, we concentrate on the following cases: (i) crystalline, defect-free α-SiO2 [20], P 42/nmc-ZrO2 [21] and cubic Sm2 O3 [22]; (ii) SiO2 , ZrO2 and Sm2 O3 crystals containing a neutral oxygen vacancy; (iii) an ensemble of amorphous SiO2 structures [23–25] containing both defect-free and defective structures and (iv) high-temperature Sm2 O3 structures where atomic configurations are obtained from molecular dynamics simulations. Our goal is to correlate the atomic structure (including defects and thermal fluctuations) with the electronic structure (including donor/acceptor states) and ultimately with the high-temperature optical response of these ceramics. Additional details of the structures will be provided in the following sections. 2.2. Theory and methods

In the following subsections, we briefly review the definition of emissivity and its calculation from the spectral dielectric constant (real and imaginary components), which, in turn, is obtained directly from the ab initio calculations. We also provide details regarding the density functional calculations we use to predict atomic and electronic structures. 3



As mentioned above, spectral emissivity, E (ω, T), is defined as the ratio between the radiative energy emitted by a material and that of a blackbody at the same temperature; the total emissivity is computed from E (ω, T) and the blackbody (BB) radiation (uBB ) as E (ω, T )uBB (ω, T )dω . (1) E (T ) = uBB (ω, T ) dω

2.2.1. Calculation of emissivity.

From Kirchhoff’s law, we know that, in equilibrium, the amount of electromagnetic energy absorbed by a body at a given frequency should be emitted at the same frequency. Since spectral absorption (A(ω)), spectral reflectance (R(ω)) and spectral transmission (T(ω)) should add to unity, A(ω) + R(ω) + T(ω) = 1 [26, 27], emissivity can be computed from the material reflectance and absorption coefficient β(ω) as [28] E (ω) =

(1 − R(ω)) (1 − exp (−β(ω)l)) , 1 − R(ω) exp (−β(ω)l)

(2)

where l is the thickness of the specimen. The effect of thickness on emissivity will be discussed below and the results for the various materials of interest correspond to l = 10 and 100 µm. Reflectance and absorption coefficients are obtained from the real and imaginary parts of the refractive index (n and k) as R(ω) =

(n(ω) − 1)2 + k(ω)2 (n(ω) + 1)2 + k(ω)2

(3)

β(ω) = 4πωk(ω).

(4)

The complex refractive index is computed from the complex dielectric constant (ε /ε ) tensor: ε (ω)2 + ε (ω)2 + ε (ω) (5) n(ω) = 2 ε (ω)2 + ε (ω)2 − ε (ω) . (6) k(ω) = 2 Finally, the frequency-dependent dielectric constants can be obtained directly from ab initio calculations. Within the projector-augmented wave (PAW) approach used here, the imaginary part is obtained from allowed, long wavelength transitions as [29] 1 4π 2 e2 limq→0 2 (ω) = 2wk δ(ck − vk − ω) × uck+eα q |uvk uck+eβ q |uvk ∗ , εαβ q c,v,k (7) where wk represent the weights of the k-points in the sum over the first Brillouin zone involving empty (uck ) and occupied (uvk ) states. The real part of the dielectric constant is obtained using the Kramers–Krönig relations [29]. To exemplify the complex non-linear relationships between dielectric response and emissivity, let us introduce a simple, single transition model system and discuss the various steps to compute emissivity (see figure 2). In figure 2(a), we show the spectral dielectric constants of the illustrative system; the imaginary component is described with a Gaussian function with broadening 10 cm−1 around a single resonance at ω = 500.0 cm−1 and oscillator strength = 5000.0 cm−2 . Using the relationship in equation (6), we obtained the refractive indices n and k shown in figure 2(b). The reflectance coefficient resulting from equation (3) 4



Figure 2. Applying the equations from section 2 to an artificial single transition system, the relationships between different characteristics can be better understood: (a) profile frequency-dependent dielectric constants ε /ε , (b) refractive indices n/k, spectral reflectance (red), (c) absorption coefficient R(ω)/β(ω) and (d) spectral emissivity with two different thicknesses, 10 µm (red) and 100 µm (light red).

and absorption coefficient from equation (4) are shown in figure 2(c). We see that the system can absorb and reflect light only for frequencies allowed by the transition. Finally, the spectral emissivity of the example is shown, together with Planck’s radiation law for various temperatures, in figure 2(d). We show emissivity for two sample thickness, 10 µm and 1 mm. As expected, spectral emissivity is large for frequencies around that of the absorption level. The bi-modal character is due to the fact that high emissivity requires high absorption and low reflectance, and the local minimum in emissivity at 500 cm−1 is due to high reflectance. The low emissivity for high and low frequencies is due to low absorption (the electromagnetic waves are transmitted through the sample); remember that we assume steady state conditions. The example also shows that increasing the film thickness leads to increased emissivity due to higher absorption. Significant total emissivity at a given temperature is achieved when the overlap between spectral emissivity and the corresponding blackbody radiation is high. The spectral emissivity of a blackbody at room temperature peaks at approximately 600 cm−1 and this peak moves toward higher values for the high temperatures of interest in this paper. We also note that the overall energy emitted by a blackbody increases with temperature as T 4 . 5



Density functional theory calculations. We use DFT to obtain the fundamental material properties needed to compute emissivity. This requires predicting atomic structures (both relaxed and from dynamic simulations) and electronic properties of the various materials of interest. For insulators with large band gaps, DFT tends to give reasonably accurate results [30–32]. The effect on emissivity of the well-known limitations of DFT’s single particle density of states (obtained from Kohn–Sham eigenvalues) will be discussed when results are presented. Equilibrium properties and geometrical optimizations were performed within the PBE flavor of the generalized gradient approximation (GGA) to describe exchange and correlation [33, 34] with the PAW method [35, 36] and the VASP code [37]. We used the soft oxygen pseudopotentials in VASP and the recommended kinetic energy cutoff of 400 eV. A harder pseudopotential would be appropriate for highly compressed structures for calculations that go beyond the independent-particle picture used here. The REOs were considered with an implicit description of f-electrons via pseudopotentials; the effect of electronic levels due to f-electrons in emissivity will be discussed in section 4. Structural relaxations were performed using Hellmann–Feynman forces [38] with a tight tolerance of 0.01 eV Å−1 . The role of thermal fluctuations on atomic structure and, consequently, emissivity is considered by analyzing the electronic structure of configurations obtained from an MD simulation. We performed Born–Oppenheimer MD under isochoric– isothermal conditions (NVT ensemble) with a Nosé–Hoover thermostat. We used a time step of 1 fs to integrate the equations of motion and thermostat coupling constant of ∼100 fs. We performed MD simulations at three different temperatures, 1000, 2000 and 3000 K, for about 15 ps [37]. All optimized structures discussed in this report are available online in the supplementary material (stacks.iop.org/MSMSE/22/075004/mmedia). We used a single point ( -point) to approximate integrals in reciprocal space for extended systems (supercells in vacancies, amorphous structures and supercells for MD) and the number of k-points for the crystal unit cells will be described in the following sections. The accuracy of our DFT-GGA calculations can be estimated by comparing the predicted high-frequency (clamped-ion or electronic) dielectric constants with experiments. The predicted value for α SiO2 is 1.8, slightly lower than the experimental value of 2.3 [39]; for ZrO2 , we predict 4.8 in close agreement with the experimental value of 4.9 [40]. This level of agreement is expected for GGA calculations, and higher levels of theory should be used for higher accuracy [29, 41]. Such calculations are significantly more computationally intensive and our choice of the level of theory reflects a balance between accuracy and our desire to study a variety of materials and structures, where we are more interested in trends than in the absolute accuracy of the individual predictions.

2.2.2.

3. Spectral emissivity of perfect crystals and point defect contributions

High-emissivity ceramics of interest in TBC applications (such as ZrO2 ) [17] are mostly insulators with large band gaps (>4.5 eV) [42], but some can have relatively high electrical conductivity, like ZrB2 , for example. Thus, as discussed earlier, band-to-band electronic transitions are often too high in energy to explain the the reported experimental emissivities of up to 0.8 [42]. Our single-level example shows that a transition at the appropriate energy can have an important effect on total emissivity and point defects can provide the required states. In this section, we compute the spectral emissivity of perfect crystals and then investigate the role of oxygen vacancies in the optical response. 6



Figure 3. Emissivity responses (at 2000 K) for defect-free systems: (a) SiO2 , (b) ZrO2

and (c) Sm2 O3 , and defective oxides: (d) Ov -SiO2 , (f ) Ov -ZrO2 and (e) Ov -Sm2 O3 . Red lines show spectral emissivities together with Planck distribution profiles for two different temperatures (1000, 2000 K: blue and green, respectively). Insets show the dielectric function profiles (ε /ε ) as a function of frequency for each case. 3.1. Perfect crystals

We compute the optical responses for fully optimized ideal structures of α-SiO2 [20], P 42/nmcZrO2 [21] and C-type of Sm2 O3 [22]. The unit cell calculations for SiO2 and ZrO2 require sampling k-space beyond the gamma point and our calculations used a 4 × 4 × 4 mesh. Figures 3(a)–(c) show the calculated spectral emissivity of the perfect crystals (red lines) together with Planck’s distributions for T = 1000 K (blue) and 2000 K (green). As expected, all the ideal structures lead to emissivities dictated by their overall band gaps, as can be confirmed by the dielectric constants shown in the insets. SiO2 has the largest band gap (∼6.0 eV) of the present batch and even at a temperature of 2000 K (above its melting temperature) we predict a low emissivity of 0.03 for a thickness of l = 10 µm and 0.1 for l = 100 µm. Both ZrO2 and Sm2 O3 show slightly higher T = 2000 K emissivities; the values are 0.3 and 0.15 for l = 10 µm and 0.63 and 0.58 for l = 100 µm, respectively. The Kohn–Sham density of states, used for these calculations, is known to underestimate band gaps; thus, our predictions provide an upper bound for the emissivity of perfect crystals. We note that these calculations ignore the f-electrons in Sm2 O3 that are expected to contribute significantly to their emissivity; these effects will be discussed below. 3.2. Role of point defects

Defects can introduce localized electronic states within the band gap of semiconductors and insulators. These states play a central role in the performance and reliability of a wide range of electronic devices, and their prediction and understanding is a topic of significant current interest [23–25]. These electronic states also provide additional transitions and affect optical properties. Either donor or acceptor in nature, they can enhance the overall dielectric response and possibly emissivity. Controlling defects can thus provide an opportunity to tune spectral emissivity for various applications [2, 3]. 7



Figure 4. Density of states for defect (red) and defect-free (black) SiO2 (a) and ZrO2

(b). Fermi levels are highlighted. (c) The real space defect density (produced by oxygen vacancy) in ZrO2 integrated over energy range is captured in the gray box in (b).

Oxygen vacancies are common point defects in the materials of interest, and we introduce these defects in our simulation cells by removing an oxygen atom. In the cases of ZrO2 and SiO2 , we remove an oxygen atom in a 2 × 2 × 2 supercell due to the small size of the unit cells and use -point calculations in the resulting structures. After the removal of the atom, we relax the structures by energy minimization, maintaining the lattice parameters of the values of the perfect crystal. Figures 3(d)–(e) show spectral emissivities for the oxygen-deficient systems. Our results show that oxygen vacancies in SiO2 do not lead to a significant change in emissivity; the 2000 K emissivity remains essentially unchanged. In the case of Sm2 O3 , an oxygen vacancy leads to acceptor states that provide additional transitions and increase the emissivity at T = 2000 K and for a thickness of 10 µm to 0.23 (from 0.15 in the defect-free system). Interestingly, the neutral oxygen vacancy in ZrO2 induces drastic changes; its emissivity at T = 2000 K increases from 0.3 for the defect-free system to 0.6 for a sample with the vacancy (for l = 10 µm). The minor role of defects in SiO2 on emissivity is consistent with experimental spectral emissivities at high temperatures, which show little change in emissivity up to high temperatures without new transitions just above the phononic ones [43]; this is in contrast to Al2 O3 and MgO, where a significant increase in emissivity in the opaque region was observed. Our predictions for ZrO2 are consistent with its large high-temperature emissivity [44]. To understand the origin of such striking differences, we analyzed the electronic changes induced by the oxygen vacancies. Figure 4 compares the electronic density of states of silica (figure 4(a)) and zirconia (figure 4(b)); both perfect crystals (in black) and systems with a neutral oxygen vacancy (red lines) are shown. We can see that in the case of α-SiO2 , the neutral oxygen vacancy produces empty states near the conduction band that are unable to 8



provide transitions with the required energies in their neutral state. In contrast to SiO2 , the defect in ZrO2 leads to a mid-gap state with the donor nature, which allows vertical excitation with energy ∼1.0 eV. This excitation contributes significantly to the emissivity spectra and enhances the total emissivity to 0.6 at 2000 K. Real space visualization of the mid-gap state (see figure 4(c)) clearly indicates its localization around the oxygen vacancy. In this image we show the electron density iso-surface for the states in the energy range 4.5–5.5 eV; see the shaded area in figure 4(b).

4. Strongly localized f-shell states

Optically active, localized states can also result from f-shell states in lanthanides (Ln). Those localized states tend to be only weakly involved in chemistry, and consequently their optical response is rather independent of the details of the bonding; this also makes them easier to model. Fortunately, due to the shielding of 4f-electrons by the filled 5s and 5p shells, the optical spectra exhibit sharp features that can be explained from quantum selection rules [45] of the free ion [46]. The electronic structure of the Ln+3 ions is given by [Xe]4f n , with n = 0 for La and 14 in the case of Lu. The number of f-electrons in each Ln+3 determines their spectral properties. The series exhibits approximately symmetric properties with respect to the half populated system, namely, Gd+3 , which is naturally less reactive and has the largest band gap. On the contrary, Ln+3 ions with n in the ranges 1–6 and 9–12 exhibit excitations in the visible and near-infrared [47]. The absorption spectra of Sm+3 [Xe]4f 5 of interest here are very rich. While in the visible region all excitations are spin-forbidden and, thus, have a low intensity, the spin-allowed transitions (with S = 0) involve 6 H5/2 →6 FJ manifolds (where J = l/2; l = 1, 3, 5, 7, 9 and 11) and are quite efficient and play an important role in total emission [45]. Such transitions over the 6000–10 000 cm−1 range should be very effective emitters/absorbers within temperature ranges of interest, 1500–2000 K. One could expect Tm+3 to exhibit similar characteristics, where [[Xe]4f 7 ]4f 5 electronic structure maintains the similar 4f 5 behavior [48]. As discussed above, without considering f-electrons, the calculated emissivity of defectfree Sm2 O3 is very low. We now include the contribution of transitions between localized, environment-independent f-shell states by adding to the DFT spd dielectric response discussed above (that ignores the contributions of the f-shell) f-shell excitations taken from the free-ion Hamiltonian calculation reported by Jamalaiah et al [46]. We computed the f-shell contribution to εf (ω) from the excitation energies and strengths (transition moments) of the 6 H5/2 → 6 FJ resulting from solutions of the free-ion Hamiltonian. Keeping in mind that εspd (ω) and εf (ω) can be treated independently given the environment-independent nature of the f-states, we approximate the total dielectric response as a sum of the two contributions. As before, the Kramers–Krönig relationship gives the real part, εtotal (ω), and the approach previously described gives the total spectral emissivity. Figure 5 compares the predicted spectral emissivity of Sm2 O3 (red) with experimental values (black circles) at two different temperatures [49]. We see that the combination of DFT calculations for spd states plus the contribution of f states captures the main features of the spectra at energies relevant for high-temperature applications. The predicted total emissivity at T = 2000 K of 0.7 is in good agreement with the experimental value of 0.8 [50]. These calculations provide important insight into the emissivity of other rare earth oxides. One can expect that the high-temperature emissivity of La2 O3 , with its empty f-shell, and Gd2 O3 , with a large energy gap for transitions within the f-shell [48], to be described 9



Figure 5. Experimental (black dots) and theoretical spectral emissivity functions for Sm2 O3 systems. Experimental profiles were adapted from [49] and were measured at T = 1823 K and T = 1513 K; the theoretical spectral emissivity function was calculated at T = 2000 K, where spectral emissivity is computed based on total ε /ε given by the /εspd and free-ion Hamiltonian based εf /εf (see the text direct sum of DFT-based εspd for details).

accurately by the spd–DFT calculations. However, the theoretical emissivity at T = 2000 K of approximately 0.25 is small compared to experiments (0.5–0.6) [44]. Complex defects could be partially responsible for the discrepancy, but thermal fluctuations on the atomic structure, including the motion of defects, could also affect the optical properties at elevated temperatures; these will be explored in section 6 below. 5. Amorphous silica

Amorphous dielectrics are an important class of material with a wide range of applications. Due to the lack of long-range order, these materials exhibit significant atomic-level variability and higher densities of defects than their crystalline counterparts. The atomic structure of amorphous dielectrics, including intrinsic defects, can now be predicted theoretically; see [23–25] and references therein. Here, we analyzed previously reported stoichiometric amorphous SiO2 structures (with 72 atoms in each simulation cell) and non-stoichiometric, oxygen-deficient cases (71 atoms) from [23–25]. Two sets of 60 statistically independent samples for each kind (see the online supplementary material) were re-optimized using VASP, and the spectral emissivity was calculated as described above (stacks.iop.org/MSMSE/22/075004/mmedia). These structures were obtained by annealing liquid samples to room temperature using molecular dynamics with the ReaxFF force field followed by a DFT relaxation. While this approach is state of the art and among the best methods to produce amorphous samples, uncertainties regarding the accuracy of the resulting structures remain. Assessing such uncertainties is beyond the scope of this paper. The distribution of total emissivities at various temperatures and the associated dielectric response functions are shown in figure 6 for both classes of structures. Consistent with our findings for the crystalline case, amorphous SiO2 has a low emissivity at the high temperatures 10



Figure 6. (a)–(b) Distribution of emissivity for amorphous structures [23–25] estimated at different temperatures. (c)–(d) Superposed dielectric function for all systems within manifolds 72 (c) and 71 (d).

and the presence of point defects does little to increase this value. The variability in atomic structure leads to approximately 10% variation in the total emissivity, and the presence of defects in the oxygen-deficient cells increases the emissivity calculated at T = 2000 K to about 0.1; see figure 6. 6. Thermal fluctuations and dynamics

The thermal motion of atoms can also be expected to have an effect on the electronic and optical response of oxides, especially at temperatures near or above melting, where rapid bond formation and breaking occurs. Since in many applications the materials of interest are used at high temperatures, we now turn our attention to such dynamical effects. It is widely accepted that structural transformations and fluctuations have a strong effect on the electronic and optical properties of soft materials and systems with soft degrees of freedom. For example, the efficiency of charge transfer and transport at nano- and sub-nanometer scales is strongly affected by structural processes in single molecules [51, 52], polymers at ambient conditions [53, 54], bio-molecular assemblies [55], stacks of organic molecules [54], etc. In such applications, electronic and ionic processes exhibit different timescales and the Born– Oppenheimer approximation holds, i.e. the electrons see the ions as static. In section 5, the ensemble of SiO2 glasses was obtained by relaxing (energy minimization) structures from annealing simulations; this approach explores the minima in the configurational energy landscape. In this section, we focus on Sm2 O3 structures obtained from 11



Figure 7. (a)–(b) Comparison of radial distribution functions obtained from the MD simulations of Sm2 O3 at temperatures of 2000 and 3000 K for Sm–O (black), Sm–Sm (red) and O–O (green) pairs. (c)–(d) Emissivity distribution at 1000 K (c) and 2000 K (d), obtained based on 100 instant dielectric functions along MD trajectories. Insets illustrate the dielectric function profiles (ε /ε (spd)) as a function of frequency (energy range 0–10 eV).

high-temperature dynamical simulations (i.e. without quenching) to characterize the role of thermal fluctuations on emissivity. We performed ab initio MD simulations at isothermal, isochoric conditions at temperatures T = 2000 K and T = 3000 K. All calculations are performed using lattice parameters that correspond to the optimized system at T = 0 K, i.e. 10.915 Å. As before, the DFT calculations are performed with the localized f described in the pseudopotential. As was discussed in section 4, the significant contribution of environment-independent f-electrons can be added to the dielectric response from the DFT calculations. In figures 7(a) and (b), we show the radial distribution functions obtained from the MD simulations of Sm2 O3 at temperatures of 2000 and 3000 K, and the distribution of total emissivities obtained at the corresponding temperatures from 100 samples extracted from the simulations are shown in figures 7(c) and (d). The radial distribution functions show that at T = 2000 K our samples remained crystalline, but at T = 3000 K the system melted. This is reasonably consistent with the experimental melting temperature of Sm2 O3 (T∼2100 K) [56], although our simulations correspond to the T = 0 K lattice parameter and are, thus, pressurized. The emissivity calculations show that even at 2000 K, thermal fluctuations do not affect the optical response significantly; the resulting distribution of instantaneous values has a mean value of 0.19 with a standard deviation of 0.02; this can be compared to ∼0.15 for the minimized 12



Table 1. Total emissivity predictions for all materials at 2000 K for defect-free and

defected materials, with thicknesses (l-parameter in equation (2)) 10 µm and l = 100 µm in parenthesis. System

Defect free

Oxygen vacancy

f-shell

SiO2 ZrO2 Sm2 O3

0.03(0.10) 0.30(0.63) 0.15(0.58)

0.04(0.08) 0.60(0.78) 0.23(0.67)

— — 0.72(0.88)

structure at T = 2000 K (see section 3) without considering f-shell electrons. We note that while these are high-temperature simulations, there are no defects in the simulation cells. Thus, we are ignoring the processes of bond breaking and formation as defects move at high temperatures, and we expect our calculations to underestimate emissivity. Interestingly, the results from the T = 3000 K snapshots show that the dynamic breaking and formation of bonds in the melt affect emissivity significantly, increasing its mean value to 0.6; see 7(d). This expectation number is rather close to the high temperature limit of f-shell independent oxides in the REO series, like (Y,La,Gd)2 O3 [44]. These results are consistent with experiments showing the importance of defects and their motion on spectral emissivity [12, 43] but, interestingly, no significant change in emissivity was observed during the melting of several rare earth oxides [44]. We attribute this observation to the high emissivity caused by the large density of defects and their rapid bond breaking/forming processes in the high-temperature solids. 7. Conclusions

High-emissivity materials are important for a broad spectrum of applications where electromagnetic radiative heat transfer may reduce the temperature of a working body. Here, we explored possible reasons for the high emissivity of ZrO2 and rare-earth oxides of interest for TBCs and studied silica, which is known to have a low high-temperature emissivity. Through the use of density functional theory electronic structure calculations, we characterize the optical properties of defect-free and defective crystals, as well as glassy materials, and for samples with thermal fluctuations, we quantified the contribution of the various factors to the hightemperature emissivity. We found that static and dynamic disorder play important roles in their emissivity. We focused on the high-temperature emissivity of silica, zirconia and samaria, important in thermal barrier applications. We demonstrate that density functional theory calculations combined with Kirchhoff’s law and Planck’s blackbody radiation law can be used to quantitatively predict the emissivity of these materials. Table 1 summarized the emissivity predictions at T = 2000 K. Results for 10 and 100 µm highlight the importance of absorption in achieving high emissivity. We find that point defects can, in some cases, lead to localized electronic states within the material band gap that significantly affect their emissivity at the temperatures of interest. In particular, we found that the neutral oxygen vacancy in zirconia increases the emissivity significantly; we believe that neutral oxygen vacancies contribute significantly to the high emittance of this material [44]. On the other hand, the variety of defects studied in silica (crystalline and amorphous) fail to provide transitions with appropriate energies and do not increase emissivity appreciably; this explains high-temperature spectral emissivity measurements [43]. In the case of samaria, both localized f-electron states and point defects 13



contribute to their high emissivity, making the Sm ion attractive for high-temperature thermal barrier coatings [50]. In addition, we found that the dynamic process of bond breaking and formation at high-temperatures (liquids and solids near their melting temperature) play an important role in emissivity, and the interplay between atomic structure and electronic properties at high temperatures should be accounted for explicitly in theoretical predictions. Acknowledgments

This work was supported by the US Air Force Office of Scientific Research (AFOSR) via grant# FA9550-11-1-0079. Program manager: Dr Ali Sayir. Computational resources of nanoHUB are gratefully acknowledged. References [1] Torsello G, Lomascolo M, Licciulli A, Diso D, Tundo S and Mazzer M 2004 Nature Mater. 3 632–7 [2] Jenkins D R 2007 Space Shuttle: The History of the National Space Transportation System (Stillwater: Voyageur Press) [3] Hale W 2010 Wings in Orbit NASA/SP pp 182–99 [4] Zhao X, He X D, Sun Y and Wang l D 2011 Mater. Lett. 65 2592–4 [5] Biamino S, Liedtke V, Badini C, Euchberger G, Olivares I H, Pavese M and Fino 2008 J. Eur. Ceram. Soc. 28 2791–800 [6] Bartuli C, Valente T and Tului M 2002 Surf. Coat. Technol. 155 260 [7] Ordine A, Achete C A, Mattosa O R, Margarit I C P, Camargo S S Jr and Hirsch T 2000 Surf. Coat. Technol. 133 583–8 [8] Banks B A et al 1988 Arc-Textured Metal Surfaces for High Thermal Emittance Space Radiators, NASA-TM-100894 [9] Cockeram B V, Measures D P and Mueller A J 1999 Thin Solid Films 355–6 17–25 [10] He X, Li Y, Wang L, Sun Y and Zhang S 2009 Thin Solid Films 17 5120–9 [11] Lim G and Kar A 2009 J. Phys. D: Appl. Phys. 42 155421 [12] Brecher C, Wei G C and Rhodes W H 1990 J. Am. Ceram. Soc. 73 1473–88 [13] Bondi R J, Lee S and Hwang G S 2010 Phys. Rev. B 81 195207 [14] Tripathi M N, Shida K, Sahara R, Mizusekl H and Kawazoe Y 2012 J. Appl. Phys. 111 103110 [15] Jiang H, Gomez-Abal R I, Rinke and Scheffler M 2010 Phys. Rev. B 81 085119 [16] Chiodo L, Garc´ıa-Lastra J, Iacomino A, Ossicini S, Zhao J, Petek H and Rubio A 2010 Phys. Rev. B. 82 045207 [17] Ng D and Fralick G 2001 Rev. Sci. Instrum. 72 1523 [18] Krishna M G, Rajendran M, Pyke D R and Bhattacharya A K 1999 Sol. Energy Mater. Sol. Cells 59 337 [19] Lowe A, Chubb D L, Farmer S C and Good B S 1994 Appl. Phys. Lett. 64 3551 [20] 1963 Crystal Data, Determinative Tables, ACA Monograph No 5, American Crystallographic Association [21] Bondars B, Heidemane G, Grabis J, Laschke K, Boysen H, Schneider J and Frey F 1995 J. Mater. Sci. 30 1621 [22] Hanic F et al 1984 Acta Cryst. B 40 76 [23] Anderson N, Vedula R, Schultz P A, Ginhoven R and Strachan A 2012 Appl. Phys. Lett. 100 172908 [24] Vedula R, Anderson N and Strachan A 2012 Phys. Rev. B 85 205209 [25] Vedula R, Anderson N, Schultz P A, Van Ginhoven R M and Strachan A 2011 Phys. Rev. Lett. 106 206402 [26] Auslender M and Hava S 1995 Infrared Phys. Technol. 36 1077 [27] Pigeat P, Rouxel D and Weber B 1998 Phys. Rev. B 57 9293 [28] Lim G and Kar A 2009 J. Phys. D: Appl. Phys. 42 155412 14


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