Mechanical and thermal properties of Yb₂SiO₅: First-principles calculations and chemical bond theory investigations

Abstract

Ytterbium monosilicate (Yb₂SiO₅) is a promising candidate for environmental barrier coating. However, its mechanical and thermal properties are not well understood. In this work, the structural, mechanical, and thermal properties of Yb₂SiO₅ are studied by combining density functional theory and chemical bond theory calculations. Based on the calculated equilibrium crystal structure, heterogeneous bonding nature and distortion of the structure are revealed. Meanwhile, the full set of elastic constants, polycrystalline mechanical properties, and elastic anisotropy of Yb₂SiO₅ are presented. In addition, the minimum thermal conductivity of Yb₂SiO₅ was determined to be 0.74 W m⁻¹ K⁻¹. The theoretical results highlight the potential application of Yb₂SiO₅ in a thermal and environmental barrier coating.

APPENDIX: Chemical Bond Theory

Chemical bond theory, developed by Phillips and Van Vechten,53 Van Vechten,54 Levine,55 and Xue and Zhang,56 provides an efficient method to study the properties of the chemical bonds in complex crystals and to predict the physical properties of a crystal from a bonding standpoint. The implementation of the theory relies on the decomposition of complex crystal into a linear combination of subformula of various binary crystals.56

According to the chemical bond theory, for a giving chemical formula of a complex crystal, it is always possible to decompose it into the following simple subformula:56

$$A_{a1}^1A_{a2}^2 \cdots A_{ai}^iB_{b1}^1B_{b2}^2 \cdots B_{bj}^{\,j} = \sum\limits_{i,j} {A_{mi}^iB_{nj}^{\,j}} = \sum\limits_\mu {{{\left( {{A_m}{B_n}} \right)}^\mu }} \quad ,$$

(A1)

$$mi = {{N\left( {{B^{\,j}} - {A^i}} \right) \times ai} \over {{N_{\rm{C}}}\left( {{A^i}} \right)}}, nj = {{N\left( {{A^i} - {B^{\,j}}} \right) \times bj} \over {{N_{\rm{C}}}\left( {{B^{\,j}}} \right)}}\quad ,$$

(A2)

where N_C(Aⁱ) and N_C(B^j) are the coordination numbers of Aⁱ and B^j ions in the crystal. N(B^j − Aⁱ) is the nearest coordination fraction contributed by Aⁱ ion and vice versa. (A_mB_n)^μ is one of subformulas. Since the fictitious binary crystal contains only one species of chemical bond, its properties can be easily estimated.

For a fictitious binary crystal (A_mB_n)^μ, the total lattice energy U^μ can be separated into the ionic part U_i^μ and the covalent part U_c^μ, which can, respectively, be expressed as:57

$$U_{\rm{i}}^\mu = {{1270\left( {m + n} \right){Z^ + }{Z^ - }} \over {{l^\mu }}}\left( {1 - {{0.4} \over {{l^\mu }}}} \right)f_{\rm{i}}^\mu \left( {{\rm{kJ}} {\rm{mo}}{{\rm{l}}^{ - 1}}} \right)\quad ,$$

(A3)

$$U_{\rm{c}}^\mu = 2100m{{{{\left( {{Z^ + }} \right)}^{1.64}}} \over {{{\left( {{l^\mu }} \right)}^{0.75}}}}f_{\rm{c}}^\mu \left( {{\rm{kJ}} {\rm{mo}}{{\rm{l}}^{ - 1}}} \right)\quad ,$$

(A4)

where Z⁺ and Z⁻ are the valence of cation and anion in the binary crystal and l^μ is the length of ^μ type bond. f_i^μ and f_c^μ are the ionicity and covalency of A–B bond, and are defined as:55

$$f_{\rm{i}}^\mu = {{{{\left( {{C^\mu }} \right)}^2}} \over {{{\left( {E_{\rm{g}}^\mu } \right)}^2}}},f_{\rm{c}}^\mu = {{{{\left( {E_{\rm{h}}^\mu } \right)}^2}} \over {{{\left( {E_{\rm{g}}^\mu } \right)}^2}}}\quad ,$$

(A5)

$$f_{\rm{i}}^\mu + f_{\rm{c}}^\mu = 1\quad ,$$

(A6)

where E_g^μ is the average energy bond gap and is composed of homopolar E_h^μ and heteropolar C^μ parts. These two parts of energy can be estimated by:55

$$E_{\rm{h}}^\mu = {{39.74} \over {{{\left( {{l^\mu }} \right)}^{2.48}}}} ({\rm{eV}})\quad ,$$

(A7)

$$\eqalign{& {C^\mu } = 14.4{b^\mu } {\rm{exp}}\left( { - k_{\rm{s}}^\mu {r^\mu }} \right)\left( {{{{{\left( {Z_{\rm{A}}^\mu } \right)}^*}} \over {{r^\mu }}} - \left( {n/m} \right){{{{\left( {Z_{\rm{B}}^\mu } \right)}^*}} \over {{r^\mu }}}} \right) \cr & \quad \quad ({\rm{eV}})({\rm{if}} n m)\quad , \cr} $$

(A8a)

$$\eqalign{& {C^\mu } = 14.4{b^\mu } {\rm{exp}}\left( { - k_{\rm{s}}^\mu {r^\mu }} \right)\left( {\left( {m/n} \right){{{{\left( {Z_{\rm{A}}^\mu } \right)}^*}} \over {{r^\mu }}} - {{{{\left( {Z_{\rm{B}}^\mu } \right)}^*}} \over {{r^\mu }}}} \right) \cr & \quad \quad ({\rm{eV}}) ({\rm{if }}m n)\quad , \cr} $$

(A8b)

where b^μ is a structural correction factor, r^μ = l^μ/2 is the average ion radius expressed in angstroms, exp(−k_s^μr^μ) is the Thomas–Fermi screening factor, and (Z_A^μ)* and (Z_B^μ)* are the effective valence electron numbers of A and B ions, respectively.

The lattice energy density u^μ of a binary crystal, which is an important parameter in estimating the bulk modulus and linear TEC of a binary crystal, is defined as:58

$${u^\mu } = {{{U^\mu }} \over {{N_{{\rm{Av}}}}{n^\mu }v_{\rm{b}}^\mu }}\quad ,$$

(A9)

where N_Av is the Avogadro constant, n^μ is the number of the chemical bond in one formula unit, and v_b^μ is the volume of ^μ type chemical bond. Then, the bulk modulus B^μ of a certain type of binary crystals can be calculated on the basis of lattice energy density:58

$${B^\mu } = {\delta ^\mu } + {{{u^\mu }} \over {{\beta ^\mu }}}\quad ,$$

(A10)

where δ^μ is a constant and β^μ is a proportion factor depending on the average valence and average coordination number of subformula A_mB_n. The bulk modulus of the complex crystals is expressed as:59

$${B_{\rm{m}}} = {1 \over {{k_{\rm{m}}}}}\quad ,$$

(A11)

$${k_{\rm{m}}} = {{\Delta V} \over {{V_{\rm{m}}}\Delta P}} = {1 \over {{V_{\rm{m}}}}}\sum\limits_\mu {{{\Delta {V_\mu }} \over {\Delta {P_\mu }}}} = {1 \over {{V_{\rm{m}}}}}\sum\limits_\mu {{V^\mu }{k^\mu }} = {1 \over {{V_{\rm{m}}}}}\sum\limits_\mu {{{{V^\mu }} \over {{B^\mu }}}} \quad ,$$

(A12)

where B^μ is the bulk modulus of μ type chemical bond, and V^μ is the volume of μ type bond in one molecule.

The linear TEC of a complex crystal is closely related to the lattice energy and can be estimated by the following equations:49

$$\alpha = \sum\limits_\mu {{F^\mu }{\alpha ^\mu }} \quad ,$$

(A13)

$${\alpha ^\mu } = - 31685 + 08376{\chi ^\mu }\quad ,$$

(A14)

$${\chi ^\mu } = {{{k_{\rm{B}}}Z_{\rm{A}}^\mu N_{{\rm{CA}}}^\mu } \over {{U^\mu }{\Delta ^\mu }}}{\beta ^\mu }, {\beta ^\mu } = {{m\left( {m + n} \right)} \over {2n}}\quad ,$$

(A15)

where parameter Δ^μ is a correction parameter from the analysis of experimental results, which depends on the position of cation in the periodic table. F^μ is the fraction of μ type bond in the complex crystal.