Guide of selecting substitutional elements for lower thermal conductive ceramics applied to thermal barrier coatings

Abstract

To decrease thermal conductivities of ceramic materials applied to thermal barrier coatings (TBCs), substitutional elements are introduced into ceramic materials through defect engineering. However, predicting what substitutional elements are more efficient for decreasing a given TBC ceramic material’s thermal conductivity among potential substitutional elements is still a challenge. In this work, a novel approach for predicting the most efficient substitutional elements is presented. It avoids the experimental trial-and-error method and optimizes computational processes, thus saving computational cost. The core part of the approach is the phonon relaxation time expression for a doped ceramic material, which was obtained through the introduction of random variables in the work. Besides, an application of the expression in combination with software Vienna Ab-initio Simulation Package (VASP) for doped SrZrO$_3$ with rare earth elements is given. It is found that La and Eu elements most efficiently decrease the thermal conductivity of SrZrO$_3$. To validate the correctness of the expression and the model, the thermal conductivities of all doped SrZrO$_3$ are calculated using Slack method. The doped SrZrO$_3$ with La and Eu has lower thermal conductivities of 1.2 W/(mK) and 1.3 W/(mK), respectively, compared to doped SrZrO$_3$ with other rare earth elements. Furthermore, it is found that variation trends of various doped materials’ reciprocal relative phonon relaxation time obtained by the expression and reciprocal thermal conductivities obtained by Slack method are the same, which suggests our model and expression are reasonable.

1 Introduction

TBCs are targeted at improving the life of the substrate. More specifically, TBCs act as an insulator and resist the heat transfer and also provide protection against high-temperature degradation of the underlying substrate [1,2,3]. The ceramics, as TBCs materials, are thermally sprayed and are popular due to their high thermal insulation, electrical insulation and oxidation resistance [1]. The most commonly used TBC material is 6–8 wt% Yttrium-stabilized zirconia $\text {Y}_2\text {O}_3$–$\text {ZrO}_2$ (YSZ), with a thermal conductivity of 2.3 W/(mK) at 1000 $^{\circ }$C [4,5,6,7]. However, when the temperature is higher than 1200 $^{\circ }$C, YSZ will not be used for long-term because of relatively high thermal conductivity and the metastable tetragonal phase degradation caused by low sintering resistance and poor phase stability [8,9,10].

New ceramic compositions with low thermal conductivity and excellent thermal stability, as TBC candidate materials, have been widely studied, such as rare-earth tantalates [11, 12], aluminates [13, 14], silicates [15], titanate [16], zirconates [17,18,19] and other oxides [20, 21]. Furthermore, continuous efforts have been done to improve existing ceramic materials [8]. The improved properties include the elastic plastic properties, cohesive strength, adhesive strength, fracture properties, thermo-mechanical fatigue properties, wear, erosion resistance and thermal resistance [4, 5, 8, 22].

To obtain lower thermal conductivity, defects are introduced into ceramic materials through defect engineering to accelerate the phonon scattering [23, 24], where one of the classical examples is YSZ. Besides, studies about $\text {SrZrO}_3$ doped with rare earth elements have been done by Ma and his groups [25,26,27,28,29,30,31,32]. It is found that properties of doped $\text {SrZrO}_3$ with rare earth elements indeed have been improved compared with pure $\text {SrZrO}_3$. In particular, doped $\text {SrZrO}_3$ has lower thermal conductivity than pure $\text {SrZrO}_3$. Thermal conductivity of doped SrZrO$_3$ with Gd (Sr(Zr$_{0.8}$Gd$_{0.2}$)O$_{2.9}$) is approximate 1.8 W/(mK) at 800$^{\circ }$C and thermal conductivity of doped SrZrO$_3$ with Yb (Sr(Zr$_{0.9}$Yb$_{0.1}$)O$_{2.95}$) is approximate 1.75 W/(mK) at 800$^{\circ }$C, which is lower than SrZrO$_3$ (approximate 2.2 W / (mK)) [28]; the thermal conductivity of Sr(Zr$_{0.9}$Ta$_{0.05}$Yb$_{0.05}$)O$_{3}$ is 1.65 W / (mK) at 800$^{\circ }$C [29] and that of Sr(Zr$_{0.9}$Y$_{0.05}$Yb$_{0.05}$)O$_{2.95}$ is 1.6 W / (mK) [32], which is also lower than SrZrO$_3$. Besides, behaviors of thermal conductivity increasing at very high temperatures can be explained by literature [33].

There are various methods to calculate the thermal conductivity of an insulator, including Molecular-Dynamics Simulation (MD) [24, 34, 35], the combination of lattice dynamics (LD) theory [36, 37] with the linearized Boltzmann transport equation (LBTE) [38,39,40,41,42], Clarke’s model [43], Cahill’s model [44], Slack’s model [45], density functional theory (DFT) [46, 47], and machine learning [48,49,50,51]. Usually, a combination of these methods is used to calculate thermal conductivity. Calculating thermal conductivity requires matrix of second- and third-order force constant generally, where main calculational cost is calculating third-order force constant matrix [39]. Considering that second- and third-order force constant matrix are related to interatomic potential, researchers introduce into machine learning to calculate interatomic potential, which reduces the calculational cost.

These methods calculate the thermal conductivity of a given crystal model well, but there are few special theories that can quantitatively predict what substitutional elements are more efficient for decreasing TBC ceramics thermal conductivity in terms of a given ceramic crystal.

The relation and model in the work give a method to select substitutional elements for decreasing thermal conductivity of TBC ceramics, where selected substitutional elements are more efficient to decrease the given ceramic’s thermal conductivity than other potential substitutional elements. It is worth noting that the relation cannot calculate values of thermal conductivity and it only calculate relative values of thermal conductivity. Thus, first select some substitutional elements through the relation, and then calculate thermal conductivity of doped ceramics through methods mentioned above.

Through lattice dynamics theory [36, 37], the expression for phonon relaxation time caused by substitutional elements is obtained in the work. First, in a unit cell, average interatomic distance, average atomic mass and average interatomic force are considered as random variable with respect to the unit cell position, where values of these random variable depend on whether the unit cell is doped. Second, the sum of random variables is changed into a constant through the weak law of large numbers. Finally, the phonon relaxation time is calculated by the single-mode relaxation time method [36].

This paper is organized as three parts. The first part introduces deductive reasonings for phonon relaxation time expressed by doped unit cell’s changes in second-order force constants, average neighbor interatomic distance and average atomic mass. The second part gives a application of the relation in combination with software VASP. Finally, we give conclusions of the work.

2 Expression for phonon relaxation time caused by substitutional atoms

The lattice thermal conductivity tensor $\kappa$ is expressed in a closed form [36]:

$$\begin{aligned} \kappa =\frac{1}{N_0\Omega }\sum _{\textbf{q}s}C_{\textbf{q}s}\textbf{v}_{\textbf{q}s}\otimes \textbf{v}_{\textbf{q}s}\tau _{\textbf{q}s}. \end{aligned}$$

(1)

Here, $N_0$, $\Omega$, $C_{\textbf{q}s}$, $\textbf{v}_{\textbf{q}s}$, and $\tau _{\textbf{q}s}$ denote the number of unit cells in the lattice, the volume of the unit cell, the mode-dependent heat capacity, group velocity, and phonon relaxation time of the phonon normal mode $\left( \textbf{q},s\right)$, respectively. The thermal conductivity tensor $\kappa$ is a symmetric $3\times 3$ matrix with respect to any three-dimensional spatial basis set. Moreover, there exists an orthonormal basis set, such that $\kappa$ is a diagonal $3\times 3$ matrix, where each diagonal element is proportional to the phonon relaxation time. According to the perturbation theory, substitutional elements only contribute to the phonon relaxation time. Thus, in the work, the phonon relaxation time of different materials is compared to compare their thermal conductivity.

The following is the model used to derive the expression for phonon relaxation time caused by substitutional ions: Consider a perfect crystal with density $\rho$ that contains $N_0=N_1N_2N_3$ unit cells, where each unit cell has a volume of $\Omega$ and contains $b_0$ atoms marked using $b=1, 2, \dots , b_{0}$. Assume that substitutional atoms only replace atoms labeled as $b=1$ randomly, and let $\xi =\frac{P}{N_0}$ denote the concentration of substitutional atoms, where P is the number of substitutional atoms in the crystal.

Based on the principles of lattice dynamics [36], we can describe the displacement operator $\textbf{u}\left( lb\right)$ of the bth atom in the lth unit cell using a Fourier expansion, as given by

$$\begin{aligned} \begin{aligned} \textbf{u}\left( lb\right) =&-i\sqrt{\frac{\hbar }{2\rho N_0\Omega }} \sum _{\textbf{q}s}\biggl \{\frac{1}{\sqrt{\omega \left( \textbf{q}s\right) }} \textbf{e}\left( b,\textbf{q}s\right) (a_{\textbf{q}s}^{\dag }-a_{-\textbf{q}s})\\&\times \exp \left[ i\left( \textbf{q}\cdot \textbf{r}(lb)-\omega t\right) \right] \biggr \}. \end{aligned} \end{aligned}$$

(2)

In Eq. (2), $\omega$ represents the harmonic frequency, while $a_{-\textbf{q}s}$ and $a_{\textbf{q}s}^{\dag }$ denote the annihilation and creation operators for the phonons, respectively. Furthermore, the polarization vector $\textbf{e}\left( b,\textbf{q}s\right)$ corresponds to the bth atom of the normal mode, which is characterized by the band index s and wave vector $\textbf{q}$. For simplicity in our derivation process, we replace the bth atomic mass with the mass density of the unit cell, which is denoted as $\rho$. Finally, $\hbar$ denotes the Planck constant.

According to literature [37], in a perfect crystal, the potential of atoms that are labeled with $b=1$ is represented by

$$\begin{aligned} E_0 &= \frac{1}{2}m_1\sum _l \sum _\textbf{i}\biggl \{\frac{v^2}{a^2}\left[ \textbf{i}\cdot \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\\& \quad +\frac{v^2}{a^2}\left[ \textbf{i}\times \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\biggr \}, \end{aligned}$$

(3)

Here, we only consider the potential of atoms labeled as $b=1$ that is caused by their neighboring atoms, as described in literature [37]. The average neighboring interatomic distance is represented by a. To simplify the model and maintain its reasonability, we consider $m_1$ as the average atomic mass in a unit cell. The group velocity of longitudinal and transverse waves of long wavelength is represented by v. It is assumed that the velocities of longitudinal and transverse waves of long wavelength are uniform in all directions. The unit vector in the direction of the linkage from a neighbor atom to the $b=1$ atom is represented by $\textbf{i}$, and the sum is taken over all linkages. $k\left( \textbf{i}\right)$ represents the label of the unit cell of the neighbor atom in the $\textbf{i}$ direction. The first term in Eq. (3) is due to relative displacements parallel to the linkage, while the second term is due to relative displacements in the normal plane.

The interatomic forces between substitutional atoms and their neighboring atoms are distinct from those between normal atoms, leading to alterations in the second-order force constants as compared to the previous force constants. These modifications are expressed through changes $\Delta v$ in v.

The introduction of substitutional atoms into doped unit cells only affects the second-order force constants of those cells, while leaving the dispersion relation of the entire crystal unchanged, which is a self-consistent result. Specifically, several substitutional atoms alter a few elements of the second-order force constant matrix, leading to only minor differences compared to the matrix of the pure crystal. Thus, the substitutional atoms have little impact on the dispersion relation of the pure crystal, and consequently, the group velocities of the crystal remain unaltered. In other words, while substitutional atoms do affect the second-order force constants of a doped unit cell, they only act as a small perturbation on the entire crystal, resulting in negligible changes in the crystal’s group velocities.

In addition, in a doped unit cell, the substitutional atoms cause changes in the average atomic mass and the average neighboring interatomic distance from the undoped unit cell. Specifically, $\Delta m$ and $\Delta a$ represent the differences in these quantities caused by the presence of substitutional atoms. The perturbation Hamiltonian is

$$\begin{aligned} E&=\frac{1}{2}\sum _l (m_1 + \Delta m\left( l\right) )\\& \quad \times \sum _\textbf{i}\biggl \{\frac{\left( v+\Delta v(l)\right) ^2}{(a+\Delta a(l))^2}\left[ \textbf{i}\cdot \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\\& \quad +\frac{\left( v+\Delta v(l)\right) ^2}{(a+\Delta a(l))^2}\left[ \textbf{i}\times \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\biggr \}. \end{aligned}$$

(4)

Here, $\Delta m(l)$ and $\Delta a(l)$ denote the changes in the average atomic mass and interatomic distance of the lth unit cell, respectively. According to the previous assumption mentioned above, these quantities are random variables. The perturbation term $E_{1}$ due to the presence of substitutional atoms is given by

$$\begin{aligned} E_1&=E-E_0\\ &=\frac{1}{2}\sum _l \left( \frac{(m_1+\Delta m(l))(v+\Delta v(l))^{2}}{(a+\Delta a(l))^{2}}-\frac{m_1 v^2}{a^2}\right) \\& \quad \times \sum _\textbf{i}\biggl \{\left[ \textbf{i}\cdot \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\\& \quad +\left[ \textbf{i}\times \left( \textbf{u}\left( l1\right) - \textbf{u}\left( k\left( \textbf{i}\right) b\right) \right) \right] ^2\biggr \}. \end{aligned}$$

(5)

Substituting (2) into (5), we obtain

$$\begin{aligned} \begin{aligned} E_1=&\sum _{\textbf{q}^{'}s^{'},\textbf{q}s}\sum _\textbf{i} \frac{\hbar }{2\rho N_0\Omega }\frac{1}{\sqrt{\omega \left( \textbf{q}^{'}s^{'}\right) \omega \left( \textbf{q}s\right) }}\\&\times (a_{\textbf{q}^{'}s^{'}}-a_{-\textbf{q}^{'}s^{'}}^{\dag }) (a_{\textbf{q}s}^{\dag }-a_{-\textbf{q}s})\\&\times (A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})+B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\\&\times \sum _l c(l)\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] , \end{aligned} \end{aligned}$$

(6)

where

$$\begin{aligned}{} & {} c(l)=\biggl [\frac{(m_1+\Delta m(l))(v+\Delta v(l))^{2}}{(a+\Delta a(l))^{2}}-\frac{m_1 v^2}{a^2}\biggr ], \end{aligned}$$

(7)

$$\begin{aligned} A(\textbf{q}s,\textbf{i}) & =[\textbf{i}\cdot \textbf{e}\left( 1,\textbf{q}s\right) ]-[\textbf{i}\cdot \textbf{e}\left( b,\textbf{q}s\right) ]\nonumber \\ & \quad \exp \left[ i\textbf{q}\cdot (\textbf{r}\left( k(\textbf{i})b\right) -\textbf{r}\left( l1)\right) \right] , \end{aligned}$$

(8)

and

$$\begin{aligned} B(\textbf{q}s,\textbf{i}) &=[\textbf{i}\times \textbf{e}\left( 1,\textbf{q}s\right) ]-[\textbf{i}\times \textbf{e}\left( b,\textbf{q}s\right) ]\nonumber \\{} & {} \quad \exp \left[ i\textbf{q}\cdot (\textbf{r}\left( k(\textbf{i})b\right) -\textbf{r}\left( l1)\right) \right] . \end{aligned}$$

(9)

$A^*$ and $B^*$ are conjugates of A and B, respectively.

To simplify $E_1$, consider that there are $N_0$ pairwise independent random variables c(l), which are identically distributed with respect to l. Thus, these variables are irrelevant to each other, meaning that for $k \ne l$, the covariance

$$\begin{aligned} \text {cov}(c(k),c(l))=0. \end{aligned}$$

(10)

Considering that $N_0$ is very large and the weak law of large numbers, there is

$$\begin{aligned}&\frac{1}{N_0}\sum _l\biggl \{c(l)\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] \nonumber \\&\quad -\xi \biggl [\frac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\frac{m_1 v^2}{a^2}\biggr ]\nonumber \\&\quad \times \exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] \biggr \}=0, \end{aligned}$$

(11)

where $\xi \biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right]$ is the expectation of $c(l)\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right]$.

After resolving the equation above, there is

$$\begin{aligned}&\frac{1}{N_0}\sum _l c(l)\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] \nonumber \\&\quad =\dfrac{\xi }{N_0}\biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\nonumber \\&\qquad \times \sum _l\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] . \end{aligned}$$

(12)

Applying Born–Karman boundary condition [52], we obtain

$$\begin{aligned} \begin{aligned} \sum _l\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] =N_0\Delta (\textbf{q}-\textbf{q}^{'}), \end{aligned} \end{aligned}$$

(13)

where $\Delta (\textbf{q}-\textbf{q}^{'})=1$ if and only if $\textbf{q}-\textbf{q}^{'}$ is a reciprocal lattice vector.

Thus, by substituting Eq. (13) into Eq. (12), there is

$$\begin{aligned}&\frac{1}{N_0}\sum _l c(l)\exp \left[ -i(\textbf{q}^{'}-\textbf{q})\cdot \textbf{r}\left( l1\right) \right] \nonumber \\&\quad =\xi \biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\Delta (\textbf{q}-\textbf{q}^{'}) . \end{aligned}$$

(14)

Through substituting Eq. (14) into Eq. (6), it can be obtained that

$$\begin{aligned} E_1=&\sum _{\textbf{q}^{'}s^{'},\textbf{q}s}\sum _\textbf{i} \frac{\hbar }{2\rho \Omega }\frac{1}{\sqrt{\omega \left( \textbf{q}^{'}s^{'}\right) \omega \left( \textbf{q}s\right) }}(a_{\textbf{q}^{'}s^{'}}-a_{-\textbf{q}^{'}s^{'}}^{\dag })\nonumber \\&\times (a_{\textbf{q}s}^{\dag }-a_{-\textbf{q}s})\nonumber \\&\times (A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})+B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\nonumber \\&\times \xi \biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\Delta (\textbf{q}-\textbf{q}^{'}) . \end{aligned}$$

(15)

By the golden rule formula, the transition probability $P_{\textbf{q}^{'}s^{'}}^{\textbf{q}s}$, as the rate at which the system makes a transition from an initial state $\vert i\rangle =\vert n_{\textbf{q}s},n_{\textbf{q}^{'}s^{'}}\rangle$ to a final state $\vert f\rangle =\vert n_{\textbf{q}s}+1,n_{\textbf{q}^{'}s^{'}}-1\rangle$, is given by

$$\begin{aligned} P_{\textbf{q}^{'}s^{'}}^{\textbf{q}s}= \frac{2\pi }{{\hbar }^2}\vert \langle f\vert E_1 \vert i\rangle \vert ^2 \delta \left( E\left( f\right) -E\left( i\right) \right) . \end{aligned}$$

(16)

The term $\delta \left( E\left( f\right) -E\left( i\right) \right)$ is a delta function that ensures the transition process conserves energy. It enforces the requirement that the final state energy is equal to the initial state energy, i.e., $E\left( f\right) -E\left( i\right) = 0$.

By combining Eq. (16) with expression (15), it can be obtained that

$$\begin{aligned} P_{\textbf{q}^{'}s^{'}}^{\textbf{q}s}&=\frac{2\pi }{{\hbar }^2}n_{\textbf{q}^{'}s^{'}}\left( n_{\textbf{q}s}+1\right) \delta \left( \omega \left( {\textbf{q}s}\right) -\omega ({\textbf{q}^{'}s^{'}})\right) \Delta (\textbf{q}-\textbf{q}^{'}) \nonumber \\& \quad \times \biggl \vert \sum _\textbf{i}\frac{\hbar \xi }{2\rho \Omega }\biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\nonumber \\&\quad \times \biggl [\sqrt{\frac{1}{\omega ({\textbf{q}^{'}s^{'}})\omega \left( {\textbf{q}s}\right) }}(A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})\nonumber \\&\quad +B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\nonumber \\&\quad +\sqrt{\frac{1}{\omega ({-\textbf{q}s})\omega \left( {-\textbf{q}^{'}s^{'}}\right) }}(A^*(-\textbf{q}s,\textbf{i})A(-\textbf{q}^{'}s^{'},\textbf{i})\nonumber \\&\quad +B^*(-\textbf{q}s,\textbf{i})B(-\textbf{q}^{'}s^{'},\textbf{i}))\biggr ]\biggr \vert ^2. \end{aligned}$$

(17)

The following relations have been used [36]:

$$\begin{aligned} a_{\textbf{q}s}^{\dag }a_{\textbf{q}^{'}s^{'}}\vert n_{\textbf{q}s},n_{\textbf{q}^{'}s^{'}}\rangle = \sqrt{n_{\textbf{q}^{'}s^{'}}\left( n_{\textbf{q}s}+1\right) }\vert n_{\textbf{q}s}+1,n_{\textbf{q}^{'}s^{'}}-1\rangle .\nonumber \\ \end{aligned}$$

(18)

In Eq. (18), $n_{\textbf{q}s}$ and $n_{\textbf{q}^{'}s^{'}}$ denote the phonon concentrations of normal modes $\left( \textbf{q},s\right)$ and $(\textbf{q}^{'},s^{'})$, respectively. The change rate of $n_{\textbf{q}s}$ in a unit time interval is

$$\begin{aligned} \frac{\partial n_{\textbf{q}s}}{\partial t}&=\sum _{\textbf{q}^{'}s^{'}\ne \textbf{q}s}P_{\textbf{q}^{'}s^{'}}^{\textbf{q}s}- P_{\textbf{q}s}^{\textbf{q}^{'}s^{'}}\nonumber \\ &=\frac{2\pi }{\hbar ^2}\sum _{\textbf{q}^{'}s^{'}\ne \textbf{q}s} \left( \psi _{\textbf{q}^{'}s^{'}}-\psi _{\textbf{q}s}\right) \left( \bar{n}_{\textbf{q}s}+1\right) \bar{n}_{\textbf{q}s}\nonumber \\& \quad \times \delta \left( \omega \left( {\textbf{q}s}\right) -\omega ({\textbf{q}^{'}s^{'}})\right) \Delta (\textbf{q}-\textbf{q}^{'})\nonumber \\& \quad \times \biggl \vert \sum _\textbf{i}\frac{\hbar \xi }{2\rho \Omega }\biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]\nonumber \\& \quad \times \biggl [\sqrt{\frac{1}{\omega ({\textbf{q}^{'}s^{'}})\omega \left( {\textbf{q}s}\right) }}\nonumber \\& \quad \times (A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})+B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\nonumber \\& \quad +\sqrt{\frac{1}{\omega ({-\textbf{q}s})\omega \left( {-\textbf{q}^{'}s^{'}}\right) }}\nonumber \\& \quad \times (A^*(-\textbf{q}s,\textbf{i})A(-\textbf{q}^{'}s^{'},\textbf{i})+B^*(-\textbf{q}s,\textbf{i})B(-\textbf{q}^{'}s^{'},\textbf{i}))\biggr ]\biggr \vert ^2. \end{aligned}$$

(19)

Here, the definition of $\psi _{\textbf{q}s}$ is [36]

$$\begin{aligned} n_{\textbf{q}s}=\bar{n}_{\textbf{q}s}+\psi _{\textbf{q}s}\bar{n}_{\textbf{q}s}\left( \bar{n}_{\textbf{q}s}+1\right) . \end{aligned}$$

(20)

Besides, $\bar{n}_{\textbf{q}s}=\bar{n}_{\textbf{q}^{'}{s}^{'}}$ has been used [36]. $\bar{n}_{\textbf{q}s}$ represents the concentration of phonons in the normal mode $(\textbf{q},s)$ under equilibrium conditions, which is determined by the temperature of the system. The term $\psi _{\textbf{q}s}$ represents the deviation from this equilibrium concentration due to the non-equilibrium conditions. Then, due to the presence of $\delta (\omega \left( {\textbf{q}s}\right) -\omega ({\textbf{q}^{'}s^{'}}))$, we have $\hbar \omega \left( {\textbf{q}s}\right) =\hbar \omega ({\textbf{q}^{'}s^{'}})$, which means that the normal modes $\left( \textbf{q},s\right)$ and $(\textbf{q}^{'},s^{'})$ have the same energy. Thus, using Bose–Einstein statistics [52], we can conclude that $\bar{n}_{\textbf{q}s}=\bar{n}_{\textbf{q}^{'}s^{'}}$. The single-mode relaxation time expression can be derived by $\psi _{\textbf{q}^{'}s^{'}}=0$ for $\textbf{q}^{'}{s}^{'}\ne \textbf{q}s$ and $\dfrac{\partial n_{\textbf{q}s}}{\partial t}=-\dfrac{\left( \bar{n}_{\textbf{q}s}+1\right) \bar{n}_{\textbf{q}s}\psi _{\textbf{q}s}}{\tau _{\textbf{q}s}}$ [36] in Eq. (19):

$$\begin{aligned} \frac{\partial n_{\textbf{q}s}}{\partial t}&=\sum _{\textbf{q}^{'}s^{'}\ne \textbf{q}s}P_{\textbf{q}^{'}s^{'}}^{\textbf{q}s}- P_{\textbf{q}s}^{\textbf{q}^{'}s^{'}}\nonumber \\ &=\frac{2\pi }{\hbar ^2}\sum _{\textbf{q}^{'}s^{'}\ne \textbf{q}s} \left( -\psi _{\textbf{q}s}\right) \left( \bar{n}_{\textbf{q}s}+1\right) \bar{n}_{\textbf{q}s}\nonumber \\& \quad \times \delta \left( \omega \left( {\textbf{q}s}\right) -\omega ({\textbf{q}^{'}s^{'}})\right) \Delta (\textbf{q}-\textbf{q}^{'})\nonumber \\& \quad \times \biggl \vert \sum _\textbf{i}\frac{\hbar \xi }{2\rho \Omega }\biggl [\frac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\frac{m_1 v^2}{a^2}\biggr ]\nonumber \\& \quad \times \biggl [\sqrt{\frac{1}{\omega ({\textbf{q}^{'}s^{'}})\omega \left( {\textbf{q}s}\right) }}\nonumber \\& \quad \times (A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})+B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\nonumber \\& \quad +\sqrt{\frac{1}{\omega ({-\textbf{q}s})\omega \left( {-\textbf{q}^{'}s^{'}}\right) }}\nonumber \\& \quad \times (A^*(-\textbf{q}s,\textbf{i})A(-\textbf{q}^{'}s^{'},\textbf{i})+B^*(-\textbf{q}s,\textbf{i})B(-\textbf{q}^{'}s^{'},\textbf{i}))\biggr ]\biggr \vert ^2\nonumber \\ &= -\frac{\left( \bar{n}_{\textbf{q}s}+1\right) \bar{n}_{\textbf{q}s}\psi _{\textbf{q}s}}{\tau _{\textbf{q}s}}. \end{aligned}$$

(21)

Thus,

$$\begin{aligned} \tau _{\textbf{q}s}^{-1}=&\frac{\pi \xi ^2}{2\rho ^{2}\Omega ^2}\biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]^2\nonumber \\&\times \sum _{\textbf{q}^{'}s^{'}\ne \textbf{q}s} \delta \left( \omega \left( {\textbf{q}s}\right) -\omega ({\textbf{q}^{'}s^{'}})\right) \Delta (\textbf{q}-\textbf{q}^{'})\nonumber \\&\times \biggl \vert \sum _\textbf{i}\biggl [\sqrt{\frac{1}{\omega ({\textbf{q}^{'}s^{'}})\omega \left( {\textbf{q}s}\right) }}\nonumber \\&\times (A^*(\textbf{q}^{'}s^{'},\textbf{i})A(\textbf{q}s,\textbf{i})+B^*(\textbf{q}^{'}s^{'},\textbf{i})B(\textbf{q}s,\textbf{i}))\nonumber \\&+\sqrt{\frac{1}{\omega ({-\textbf{q}s})\omega \left( {-\textbf{q}^{'}s^{'}}\right) }}\nonumber \\&\times (A^*(-\textbf{q}s,\textbf{i})A(-\textbf{q}^{'}s^{'},\textbf{i})+B^*(-\textbf{q}s,\textbf{i})B(-\textbf{q}^{'}s^{'},\textbf{i}))\biggr ]\biggr \vert ^2, \end{aligned}$$

(22)

and we let

$$\begin{aligned} c=\biggl [\dfrac{(m_1+\Delta m)(v+\Delta v)^{2}}{(a+\Delta a)^{2}}-\dfrac{m_1 v^2}{a^2}\biggr ]^2. \end{aligned}$$

(23)

According Eq. (22), for different species substitutional ions, cs are different while others are the same. Therefore, we just calculate c to compare their phonon relaxation time.

3 Application of the expression in combination with software VASP about $\text {SrZrO}_3$ doped with rare earth elements

Given a crystal and a set of substitutional ions, VASP can calculate perfect crystal’s and every doped crystal’s velocities of longitudinal and transverse waves of long wavelength. Then, the relation in this paper can be used as a sifter to “mark” every substitutional ion in the set. These marks are relative values of reciprocal phonon relaxation time, c, caused by substitutional ions respectively. The higher the mark it is, the more efficient the substitutional ion is for decreasing thermal conductivity. With its application in doped SrZrO$_3$ in combination with software VASP, the schematic view of the process is illustrated in Fig. 1.

Fig. 1

Schematic view about usage of the relation in combination with software VASP for doped SrZrO$_3$ with rare earth elements

The relation is applied to doped $\text {SrZrO}_3$ with rare earth elements in combination with software VASP. We do not consider element Pm because of its radioactivity. The crystal structures of SrZrO$_3$ and doped SrZrO$_3$ are shown in Fig. 2.

Fig. 2

Pure $\text {SrZrO}_3$ unit cell and doped $\text {SrZrO}_3$ are shown as (a) and (b), respectively. The blue, purple, red and yellow spheres represent Zr, Sr, O and rare earth ions, respectively

The first-principles calculations in this study were performed using the VASP code [46, 47]. The calculations utilized the plane-wave basis projector augmented wave method [53] within the framework of density functional theory (DFT). The exchange correlation potential was described by the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof revised for solids (PBEsol) [54]. A plane-wave energy cutoff of 500 eV was used. The reciprocal spaces of crystals used for calculating elastic constants were sampled by an $8\times 8\times 8$ mesh with half grid shifts along all three directions from the $\Gamma$-point centered mesh. The total energies were considered as minimum when the energy change became less than $10^{-8}$ eV, and the elastic constants were then obtained.

The acoustic velocities were calculated using the elastic constants and the method reported in [55], and the resulting data is listed in Table 1. For undoped SrZrO$_3$, the average atomic distance a, average acoustic velocity v, and average atomic mass $m_1$ were calculated to be 2.4 Å, 4449.656 m/s, and 45.37 g/mol, respectively. The data in Table 1 is also presented in Fig. 3, with $(\Delta a, \Delta v, \Delta m)$ characterizing the rare earth element as a spatial vector. By combining the data in Table 1 and relation (23), all c values were calculated and are shown in Fig. 4.

Table 1 Required data about corresponding doped crystals of different substitutional ions [56]

Fig. 3

Spatial distribution of data in Table 1. $\vert \Delta a\vert$, $\vert \Delta v\vert$, and $\vert \Delta m\vert$ together character the rare earth element’s capacity of decreasing thermal conductivity, and thus $(\vert \Delta a\vert , \vert \Delta v\vert , \vert \Delta m\vert )$ can describe the rare earth element as a vector

To verify the correctness of the relation and the model, we calculate all doped crystals’ thermal conductivities $\kappa$ using Slack’s method reported in [45] and results are shown at Fig. 4 and Table 2. According Fig. 4, we find that variation trends of c and $\dfrac{1}{\kappa }$ are the same, which means two methods get consistent results. Thus, our model and relation are reasonable. For doped SrZrO$_3$, we got element Eu and La can more efficiently decrease thermal conductivity of SrZrO$_3$ than others.

Fig. 4

Reciprocal relative phonon relaxation time c and reciprocal thermal conductivity $\dfrac{1}{\kappa }$ for every doped crystal. For their orders of magnitudes are the same, c has been divided by $10^{15}$. It can be found that two curves’ variation trends are the same

Table 2 Thermal conductivity of doped SrZrO$_3$ with rare earth elements at 1200 K

Data combination of Table 1 with 2 can character the rare earth element as a four-dimension vector and these are shown at Fig. 5.

Fig. 5

As the four-dimension vector, $(\Delta a,\Delta v,\Delta m,\kappa )$ character the rare earth element. To make dimensionless, every vector is divide by the vector of La element

Substitutional elements introduce into three changes: $\Delta a$, $\Delta v$ and $\Delta m$. To get which factor is the most influential, Only one or two factors are considered through variable-controlling approach. All cases are shown in Fig. 3, where $c(\Delta a)$, $c(\Delta v)$, $c(\Delta m)$, $c(\Delta a,\Delta v)$, $c(\Delta a,\Delta m)$, $c(\Delta v,\Delta m)$ only consider $\Delta a$, $\Delta v$, $\Delta m$, $(\Delta a,\Delta v)$, $(\Delta a,\Delta m)$ and $(\Delta v,\Delta m)$, respectively. It can be found that curves of $c(\Delta v)$, $c(\Delta a,\Delta v)$ and $c(\Delta v,\Delta m)$ reveal the consistency with thermal conductivity and the results are shown at Fig. 6.

Fig. 6

Comparisons with the variation trend of reciprocal thermal conductivity. When acoustic velocity $\Delta v$ is considered, curves’ variation are almost same. Thus, the factor of second-order force constant’s change have the largest influence on thermal conductivity

To more accurately certify that second-order force constant factor $\Delta v$ is the most influential in terms of decreasing thermal conductivity, the matrix of correlation coefficients is calculated. With $(\vert \Delta a\vert , \vert \Delta v\vert , \vert \Delta m\vert ,\kappa )$ as the random vector, the matrix of correlation coefficients is

$$\begin{aligned} \begin{gathered} \Sigma = \begin{bmatrix} 1 &{} 0.72 &{} -0.23 &{} -0.87 \\ 0.72 &{} 1 &{} 0.25 &{} -0.93\\ -0.23 &{} 0.25 &{} 1 &{} 0.06\\ -0.87 &{} -0.93 &{} 0.06 &{} 1 \end{bmatrix}. \end{gathered} \end{aligned}$$

(24)

The last column of matrix (24) is shown at Fig. 7, and they are the correlation coefficients of thermal conductivity $\kappa$ with $\vert \Delta a\vert$, $\vert \Delta v\vert$, $\vert \Delta m\vert$ and $\kappa$, respectively. $\kappa$ is negative correlated with $\vert \Delta v\vert$ and $\vert \Delta a\vert$, and further second-order force constant’s changes have the largest correlation with thermal conductivity.

Fig. 7

Correlation coefficients of thermal conductivity $\kappa$ with $\vert \Delta a\vert$, $\vert \Delta v\vert$, $\vert \Delta m\vert$ and $\kappa$, respectively. $\kappa$ is negative correlated with $\vert \Delta v\vert$ and $\vert \Delta a\vert$, and further second-order force constant’s changes have the largest correlation with thermal conductivity. The correlation coefficient of $\kappa$ with itself is 1

4 Conclusions and discussion

In this work, we propose an economic approach to select efficient substitutional elements for decreasing the thermal conductivity of TBC ceramics. This approach uses the expression (23) and software such as VASP to predict which substitutional elements have lower thermal conductivities when added into a given TBC ceramic material. In addition to VASP, other software related to the first-principles calculations can also be combined with the expression (23). This approach saves computational cost and avoids the need for experimental enumeration. Furthermore, this approach can be applied to other insulator materials to decrease their thermal conductivity.

In deductive reasoning, we deal with the complicated sum of random variables by the weak law of large numbers. We also investigate the effect of substitutional elements on the thermal conductivity of TBC ceramics, changes in the average neighboring interatomic distance, second-order force constant, and average atomic mass. We find that the second-order force constant change has the most significant influence on decreasing the thermal conductivity, as shown in Fig. 7. We confirm the validity of our approach by comparing our results with those obtained using Slack’s method.

Finally, our method is applied to SrZrO3 doped with rare earth elements using software VASP, and we find that La and Eu elements are the most effective in decreasing the thermal conductivity of SrZrO3 among the rare earth elements.

Appendix

Mathematical symbols in the work and their explanation as shown in Table 3.

Table 3 Mathematical symbols in the work and their explanation