Method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion mass difference

Abstract

The application discloses a rare earth element selection method for reducing the thermal conductivity of ceramic materials based on ion mass difference, which comprises the following steps: step one, deducing a crystal phonon relaxation time expression after doping; step two, obtaining the crystal phonon relaxation time after doping each rare earth element according to the phonon relaxation time expression; and step three, sorting all the obtained phonon relaxation times to obtain the rare earth element corresponding to the lowest phonon relaxation time, which is the rare earth element that can most effectively reduce the thermal conductivity of the crystal. The application provides an executable rare earth element selection method for reducing the thermal conductivity of ceramic materials by doping rare earth elements based on strict theoretical derivation, is easy to operate, has low economic cost, can effectively reduce the research and development cost of the rare earth element doped ceramic materials, and shortens the research and development period.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of rare earth element doped ceramic materials. Specifically, it is a method for selecting rare earth elements for reducing the thermal conductivity of ceramic materials based on the mass difference of ions. BACKGROUND

[0002] Thermal barrier coating ceramic materials must have low thermal conductivity, high thermal expansion coefficient, high melting point, low sintering rate, good molten salt corrosion resistance, low modulus and high hardness, etc. Among them, the thermal conductivity is one of the important parameters, which determines the thermal protection effect on high temperature working parts. At present, one method for reducing the thermal conductivity of ceramic materials is to dope rare earth elements into the ceramic materials. However, for a certain specific ceramic material, which kind of rare earth element can greatly reduce its thermal conductivity, there is currently no executable theoretical method. If the one-by-one experimental method is used, due to the different types of rare earth elements and different doping concentrations, there will be a lot of possible situations to be tried, which will consume a lot of manpower and resources. If starting from the density functional theory, using the combination of commercial software such as VASP (Vienna Ab-initio Simulation Package, Vienna University Hafner Group developed for electronic structure calculation and quantum mechanics-molecular dynamics simulation software package) and software Phono3py to calculate each possible situation, it is still impossible to avoid the consumption of a large amount of material resources (server resources). SUMMARY

[0003] Therefore, the technical problem to be solved by the present application is to provide a method for selecting rare earth elements for reducing the thermal conductivity of ceramic materials based on the mass difference of ions, which has high reliability and is easy to operate.

[0004] To solve the above technical problems, the present application provides the following technical solutions:

[0005] The method for selecting rare earth elements for reducing the thermal conductivity of ceramic materials based on the mass difference of ions comprises the following steps:

[0006] Step one, based on lattice dynamics and Boltzmann transport equation, according to the principle that the mass difference between the doping ions and the host ions of the ceramic material after doping will cause phonon scattering, the expression of the phonon relaxation time of the crystal after doping is derived;

[0007] Step two, the mass of a plurality of rare earth element ions and the mass of the host ions of a certain specific ceramic material crystal are respectively substituted into the phonon relaxation time expression to obtain the phonon relaxation time of the crystal after doping each kind of rare earth element;

[0008] Step three, sort all the phonon relaxation times obtained, and the rare earth element corresponding to the lowest phonon relaxation time is the rare earth element that can most effectively reduce the thermal conductivity of the crystal.

[0009] Further, in step one, a physical model of the ceramic material is first established, and a perturbation term caused by the doping ions is calculated

[0010] Further, the step of establishing the physical model includes: assuming that the density of the undoped crystal is p, and it contains N0=N1N2N3 cells, each cell has a volume of Ω, and contains b0 atoms, and these b0 atoms are numbered as b=1, 2, …, b0.

[0011] Further, the perturbation term is calculated When the substitution atoms are assumed to only randomly replace the atoms of b=1, let P be the number of substitution atoms in the crystal, then the concentration of the substitution atoms in the crystal is According to the theory of lattice dynamics, the displacement operator of the bth atom in the lth cell is

[0012]

[0013] The Hamiltonian of the undoped lattice is

[0014]

[0015] In formula (2), the first term on the right side is the kinetic energy term, and v2 is the second-order term of the Taylor expansion of the potential energy. According to the assumption that only the atoms of b=1 are randomly replaced, the doping ions cause the mass of the atoms of b=1 in each cell to change, and in this case, the constant m1 is changed into a random variable m1(l), so the Hamiltonian of the doped crystal is written as

[0016]

[0017] The perturbation term caused by the doping ions is

[0018]

[0019] Further, according to the weak law of large numbers and the Born-von Karman boundary condition, the Hamiltonian is simplified, and finally formula (9) is obtained

[0020]

[0021] Further, according to formula (9) and the Fermi golden formula, the change rate of n qs per unit time is calculated, and the expression of the phonon relaxation time is obtained.

[0022] Further, according to the simplified Hamiltonian and the Fermi golden formula, the change rate of n qs ​The rate of change of the thermal conductivity of the ceramic material is obtained, and a phonon relaxation time expression is obtained:

[0023]

[0024] Further, a simplification step of formula (16) is included, and finally formula (19) is obtained:

[0025]

[0026] Formula (19) is the phonon relaxation time expression in step two, wherein ξ represents the molar concentration of doping, Δm represents the mass difference between the doping ion and the host ion, ρ represents the density of the undoped crystal in the physical model, Ω represents the volume of the cell in the undoped crystal, c represents the sound velocity, q represents the reciprocal lattice vector, e(1,qs) represents the polarization vector of the b=1 atom in the (qs) state, e * (1,qs') represents the transpose complex conjugate of e(1,qs').

[0027] Further, the method for selecting a rare earth element for reducing the thermal conductivity of a ceramic material based on the mass difference of ions comprises the following steps:

[0028] Obtaining the mass of the host ion;

[0029] Obtaining the mass of all rare earth element ions;

[0030] Bringing the mass of the host ion and the mass of the rare earth element ion into the phonon relaxation time expression for calculation;

[0031] Sorting the calculated phonon relaxation times from small to large, obtaining the rare earth element corresponding to the lowest phonon relaxation time, and the rare earth element is the rare earth element that most effectively reduces the thermal conductivity of the ceramic.

[0032] Further, the rare earth element has 17 kinds, and the 17 kinds of rare earth element ions are lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, lutetium, scandium and yttrium.

[0033] The technical scheme of the present application has the following beneficial technical effects:

[0034] The present application provides a simplified phonon relaxation time expression derived from strict theoretical derivation, which can be calculated after obtaining the mass of the doping ion and the host ion, the operation process is relatively simple, the operation speed can be greatly improved, the program based on the method can be executed on a general computer, and the consumption of manpower and material resources (server resources) is greatly reduced;

[0035] This invention provides an executable method for selecting rare earth elements to reduce the thermal conductivity of rare earth element-doped ceramic materials, based on rigorous theoretical derivation. It is easy to operate and has low economic cost. The rare earth elements screened by this method are among the preferred rare earth elements. Combined with the rare earth elements screened by other theoretical methods, the selection range of the optimal rare earth elements for doping can be greatly narrowed.

[0036] This invention uses theoretical derivation and can perform preliminary screening of rare earth elements without actual experimental operations. It can effectively reduce the R&D costs of rare earth element doped ceramic materials, such as manpower and materials, and shorten the R&D cycle. Attached Figure Description

[0037] Figure 1 This is a schematic diagram of the process of the present invention;

[0038] Figure 2 This is a schematic diagram of the microstructure of rare earth element-doped SrZrO3 according to an embodiment of the present invention.

[0039] Figure 3 This is a table of relative atomic masses for some relevant elements. Detailed Implementation

[0040] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0041] like Figure 1 As shown, the method for selecting rare earth elements based on ion mass difference to reduce the thermal conductivity of ceramic materials includes the following steps:

[0042] Step one: Based on lattice dynamics and the Boltzmann transport equation, and according to the principle that the mass difference between the dopant ions and the host ions in a ceramic material causes phonon scattering, a phonon relaxation time expression for the doped crystal is derived; the derived phonon relaxation time expression is:

[0043]

[0044] In the formula, ξ represents the molar concentration of the dopant, Δm represents the mass difference between the dopant ion and the host ion, ρ represents the density of the undoped crystal in the physical model, Ω represents the volume of the cell in the undoped crystal in the physical model, c represents the speed of sound, q represents the reciprocal lattice vector, e(1,qs) represents the polarization vector of the b=1 atom in the (qs) state, and e * (1,qs′) denotes the transpose and complex conjugate of e(1,qs);

[0045] According to equation (19), under the same doping concentration, the greater the mass difference between the dopant ion and the host ion, the smaller the phonon delay time of each phonon state of the doped crystal, and therefore the smaller the thermal conductivity.

[0046] Step two, respectively, the mass of several rare earth element ions and a certain specific ceramic material crystal host ion into the phonon relaxation time expression, to obtain the phonon relaxation time of the crystal after doping each rare earth element; the commonly used rare earth element ions are 17, the 17 rare earth element ions are lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, lutetium, scandium and yttrium, the mass of each rare earth element is known, if there are other rare earth elements or other elements that may affect the thermal conductivity are screened out, this method can also be used;

[0047] Step three, sort all the phonon relaxation time, the lowest phonon relaxation time corresponds to the rare earth element that can most effectively reduce the thermal conductivity of the crystal; it should be noted that the most effective here refers to the most optimal option selected based on the mass difference of ions, according to other screening theories, the same or different rare earth elements may be obtained as the rare earth elements screened out in this application, these screened rare earth elements are the basis for further screening, and the optimal result theoretically appears in the range of these preliminary screened rare earth elements.

[0048] Further, the rare earth element selection method for reducing the thermal conductivity of the ceramic material based on the mass difference of ions comprises the following steps:

[0049] Obtain the mass of the host ion;

[0050] Obtain the mass of all rare earth element ions;

[0051] The mass of the host ion and the mass of the rare earth element ion are brought into the phonon relaxation time expression for calculation; the mass of the host ion and the mass of the rare earth element ion are calculated by using the relative atomic mass; based on the method of the present application, for a certain specific ceramic material, other values are the same, only the mass difference Δm is different;

[0052] Sort the calculated phonon relaxation time from small to large, and obtain the rare earth element corresponding to the lowest phonon relaxation time, which is the rare earth element that most effectively reduces the thermal conductivity of the ceramic.

[0053] Further, in practice, the method based on the present application needs to be programmed, the program can run on a common computer, in the program, the relative atomic mass of each atom is preset or the relative atomic mass required is directly input in the computer, then the program substitutes the data into formula (19) to perform operation, then the program finds the minimum result in the operation result, and marks the rare earth element corresponding to the minimum result as the rare earth element most effectively reducing the thermal conductivity of the doped ceramic material; after the rare earth element is obtained theoretically, an experimental method is used to verify, thereby the number of subsequent experiments required is reduced, and the consumption of manpower and material resources is greatly reduced.

[0054] Further, the derivation process of formula (19) is as follows:

[0055] Firstly, a physical model is established; it is assumed that the density of an undoped crystal is p, and the crystal contains N0=N1N2N3 cells, the volume of each cell is Omega, and contains b0 atoms, and the b0 atoms are numbered as b=1, 2, …, b0; it is assumed that the substitution atoms randomly replace the atoms of b=1, and P is the number of substitution atoms in the crystal, then the concentration of the substitution atoms in the crystal is According to the lattice dynamics theory, the displacement operator of the bth atom in the lth cell is

[0056]

[0057] In formula (1), , represents the reduced Planck constant, p represents the density of the undoped crystal in the physical model, Omega(l) is a random variable, representing the volume of the lth cell, omega(qs) represents the frequency corresponding to q in the s acoustic branch, e(b, qs) represents the polarization vector of the bth atom in the (qs) state, and a -qs respectively represent creation operators and annihilation operators, i represents an imaginary unit, q represents a reciprocal lattice vector, r(lb) represents the displacement operator of the bth atom in the lth cell in the crystal, omega represents a frequency, and t represents time;

[0058] The Hamiltonian of the undoped lattice is

[0059]

[0060] In formula (2), m b represents the mass of the bth atom, represents the velocity of the bth atom in the lth cell, represents the transpose complex conjugate of , and v2 represents the second-order term of the Taylor expansion of potential energy;

[0061] where the first term on the right side of equation in formula (2) is a kinetic energy term, according to the aforementioned assumption that the atoms with b = 1 are randomly replaced, the doping ion causes the ion mass of b = 1 in each cell to change, in this case, the constant m1 is changed into a random variable m1(l), and therefore the Hamiltonian of the cell after doping is written as

[0062]

[0063] In formula (3), m1(l) represents a random variable changed from the constant m1, m1(l) represents a random variable changed from the constant m1, and m1(l) represents a random variable changed from the constant m1. b (l) represents a random variable changed from the constant m1, b (l) represents a random variable changed from the constant m1, vib(l, b) represents the velocity of the bth atom in the lth cell, represents represents the transpose complex conjugate of

[0064] Then the perturbation term caused by the doping ion is is

[0065]

[0066] In formula (4), ω(q's') represents the frequency corresponding to q' in the s'th acoustic branch, ω(qs) represents the frequency corresponding to q in the s'th acoustic branch, a q′s′ and a -qs represent annihilation operators, represent creation operators, e(1, qs) represents the polarization vector of the b = 1 atom in the (qs) state, e * (1, q's') represents the transpose complex conjugate of e(1, qs); represents the reduced Planck constant, m1 represents the mass of the b = 1 atom, m1(l) represents a random variable changed from the constant m1, Ω represents the volume of the cell in the undoped crystal in the physical model, ρ represents the density of the undoped crystal in the physical model, i represents the imaginary unit, q-q' represents the difference of the reciprocal lattice vector, and r(lb) represents the displacement operator of the bth atom in the lth cell in the crystal.

[0067] The formula (4) is simplified below to save and improve the calculation speed; according to the assumption of the physical model, when l takes 1,..., N0, are N0 independent and identically distributed random variables, and therefore they are not correlated; that is, for k ≠ l, the covariance is

[0068]

[0069] According to the weak law of large numbers, because N0 is very large,

[0070]

[0071] where Δm = m1(l) - m1, is the expectation value of

[0072] Further deformation gives

[0073]

[0074] According to the Born-von Karman boundary condition, then

[0075] ∑ l exp[i(q - q') · r(l1)] = N0Δ(q - q') (8)

[0076] where Δ(q - q') = 1 when q - q' is a reciprocal lattice vector; otherwise Δ(q - q') = 0; substituting equation (7) and equation (8) into equation (4), the perturbation term is

[0077]

[0078] Next, the transition probability is calculated which represents the change in the state (q, s) phonon concentration per unit time due to the transition of the initial state (phonon initial state) |i> = |n qs n q′s′ to the final state |f> = |n qs +1, n q′s′ -1>. According to Fermi's golden rule, the transition probability can be written as

[0079]

[0080] where only when E(f) = E(i), δ(E(f) - E(i) = 1, otherwise 0, so that the transition process obeys the law of conservation of energy, combined with equation (9), equation (10) and equation (12), we can get

[0081]

[0082] where equation (12) is

[0083]

[0084] Equation (12) is a formula in lattice dynamics, n qs and n q′s′ represent the concentrations of phonon states (q, s) and (q', s') respectively;

[0085] n qs and n q′s′The concentrations are respectively and Since the transition process satisfies the energy conservation, so According to the Bose statistics, we can get ψ qs is the measure of the deviation of the phonon state (q, s) from its equilibrium state, then

[0086]

[0087] And the change rate of n qs in unit time is According to the establishment of equations according to formula (11) and formula (14), we can get

[0088]

[0089] According to the single phonon relaxation time approximation, when qs≠q′s′, ψ q′s′ = 0; and This formula is the formula of the lattice thermal conductivity part in the lattice dynamics; by establishing equations, we have

[0090]

[0091] Divide the same terms on both sides of the equation, and transform the phonon relaxation time τ qs to one side of the equation alone, then we have

[0092]

[0093] Further simplification of equation (16): on the basis of the Debye approximation, only consider 3 acoustic branches, that is, the acoustic branch is 3, and believe that the frequency ω(q′s′) is only related to the size of q′, and has nothing to do with the direction of q′, and ω(q′s′) = c|q′|, then equation (16) can be simplified as

[0094]

[0095] If Δ(q-q′)≠0 in the first Brillouin zone, then q=q′, so

[0096]

[0097] After arrangement, we get formula (19)

[0098]

[0099] In equation (19), ξ represents the molar concentration of the dopant, Δm represents the mass difference between the dopant ion and the host ion, ρ represents the density of the undoped crystal in the physical model, Ω represents the volume of the cell in the undoped crystal, c represents the speed of sound, q represents the reciprocal lattice vector, e(1,qs) represents the polarization vector of the b=1 atom in the (qs) state, and e * (1,qs′) denotes the transpose and complex conjugate of e(1,qs′);

[0100] According to Equation (19), under the same doping concentration, the greater the mass difference between the dopant ion and the host ion, the greater the reciprocal of the phonon delay time, and the smaller the phonon delay time of each phonon state of the doped crystal, that is, the smaller the thermal conductivity. Thus, the rare earth element that most effectively reduces the thermal conductivity of ceramic materials can be obtained based on the calculation results.

[0101] Here is a specific example:

[0102] Rare earth doping is performed using SrZrO3-based ceramic materials, such as... Figure 2 The diagram shows the microstructure of rare earth element-doped SrZrO3. Figure 2 In the figure, (a), (b), and (c) represent the unit cell, supercell, and doped supercell of SrZrO3, respectively. Figure 2 In (c), X represents rare earth element ions; by replacing Zr ions with 17 rare earth elements, according to equation (19), the larger the mass difference Δm between the dopant ion and the host ion, the larger the reciprocal of the phonon delay time, that is, the smaller the phonon delay time, and the smaller the thermal conductivity; Figure 3 Chinese data ( Figure 3 The data (relative atomic masses from the Langevin Chemical Handbook) can be used to calculate that the element with the largest mass difference from Zr is Lu, with a difference of 83.78. Therefore, among all 17 rare earth elements, element Lu can most effectively reduce the thermal conductivity of SrZrO3 ceramic materials.

[0103] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of the claims of this patent application.

Claims

1. A method for selecting rare earth elements for reducing the thermal conductivity of ceramic materials based on the difference in ion mass, characterized in that, The method comprises the following steps: Step one, based on lattice dynamics and Boltzmann transport equation, according to the principle that the mass difference between the doping ions and the host ions of the ceramic material will cause phonon scattering, the expression of phonon relaxation time of the crystal after doping is derived; Step two, the mass of a plurality of rare earth element ions and the mass of the host ions of a certain ceramic material crystal are respectively substituted into the phonon relaxation time expression to obtain the phonon relaxation time of the crystal after doping each rare earth element; Equation (19) is the phonon relaxation time expression in Step Two, where ξ represents the molar concentration of the dopant, Δm represents the mass difference between the dopant ion and the host ion, p represents the density of the undoped crystal in the physical model, Ω represents the volume of the unit cell in the undoped crystal, c represents the speed of sound, q represents the reciprocal lattice vector, e(1,qs) represents the polarization vector of the b = 1 atom in the (qs) state, e * (1,qs') represents the transpose complex conjugate of e(1,qs'); Step three, sort all the obtained phonon relaxation times, and the rare earth element corresponding to the lowest phonon relaxation time is the rare earth element that most effectively reduces the thermal conductivity of the crystal.

2. The method for selecting rare earth elements based on reducing the thermal conductivity of ceramic materials according to claim 1, characterized in that, In step one, a physical model of the ceramic material is first established and the perturbation term caused by the doping ions is calculated 3. The method of claim 2, wherein the rare earth element is selected from the group consisting of Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, and combinations thereof. The establishment of the physical model comprises the following steps: assuming that the density of the undoped crystal is ρ, and the undoped crystal contains N0=N1N2N3 cells, the volume of each cell is Ω, and the cell contains b0 atoms, and the b0 atoms are numbered as b=1, 2, …, b0.

4. The method of claim 3, wherein the rare earth element is selected from the group consisting of Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, and combinations thereof. Computing the perturbation term When the concentration of the substitutional atom in the crystal is low, it is assumed that the substitutional atom is only randomly substituted for the atom of b = 1. Assuming that P is the number of substitutional atoms in the crystal, the concentration of the substitutional atom in the crystal is According to the theory of lattice dynamics, the displacement operator of the bth atom in the lth cell is The Hamiltonian of the undoped lattice is In formula (2), the first term on the right side is the kinetic energy term, ν2 is the second-order term of the Taylor expansion of the potential energy, according to the assumption that only the atom of b=1 is randomly replaced, the mass of the atom of b=1 in each cell is changed due to the doping ions, in this case, the constant m1 is changed into a random variable m1(l), and therefore the Hamiltonian of the doped crystal is then the perturbation term due to the dopant ions is 5. The method of claim 4, wherein the rare earth element is selected from the group consisting of Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, and combinations thereof. According to the weak law of large numbers, the Hamiltonian is simplified under the Born-von Karman boundary conditions: : According to the model assumption, when l takes 1,..., N0, are N0independent and identically distributed random variables, so they are uncorrelated; that is, for k ≠ l, the covariance is According to the weak law of large numbers, because N0 is very large, the following formula is obtained where Δm = m1(l) - m1, is the expected value of Further, the following formula is obtained According to the Born-van Karman boundary condition, the following formula is obtained ∑ l exp[i(q-q') · r(l1)] = N0Δ(q - q') (8) When q-q' is a reciprocal lattice vector, Δ(q-q')=1; otherwise, Δ(q-q')=0; by substituting formula (7) and formula (8) into formula (4), the perturbation term is 6. The method of claim 5, wherein the ceramic material is selected from the group consisting of YAG, YSZ, and Gd203. The rate of change of n qs in unit time is calculated according to formula (9) and the Fermi Golden Rule, and a phonon relaxation time expression is obtained.

7. The method for selecting rare earth elements based on reducing the thermal conductivity of ceramic materials according to claim 6, characterized in that, According to the Fermi Golden Rule is written as Where δ(E(f)-E(i)) makes the transition process comply with the law of conservation of energy, in combination with formula (9), formula (10) and formula (12), the following formula is obtained Where formula (12) is In formula (12), n qs and n q′s′ represent the concentrations of phonon states (q, s) and (q', s'), respectively. n at equilibrium qs and n q′s′ respectively and then ψ qs is a measure of the deviation of the phonon state (q, s) from its equilibrium state, then while the rate of change of n qs over time is Combining equations (11) and (14) gives​ According to the single phonon relaxation time approximation, when qs≠ q's', ψ q′s′ = 0; and then there is By simultaneously canceling the same terms on both sides of the equation, the following formula is obtained 8. The method for selecting rare earth elements based on reducing the thermal conductivity of ceramic materials according to claim 7, characterized in that, The method comprises the following steps: On the basis of the Debye approximation, it is assumed that there are three acoustic branches, the frequency ω(q's') is only related to the size of q', and is independent of the direction of q', and ω(q's')=c|q'|, and formula (16) is simplified to If Δ(q-q')≠0 in the first Brillouin zone, then q=q', so the following formula is obtained 9. The method for selecting rare earth elements based on reducing the thermal conductivity of ceramic materials according to claim 1, characterized in that, The 17 rare earth elements are lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, lutetium, scandium and yttrium.

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