Selection method of rare earth elements to reduce thermal conductivity of ceramic materials based on ion volume difference

Abstract

The invention discloses a rare earth element selection method for reducing the thermal conductivity of a ceramic material based on ion volume difference, comprising the following steps: step one, based on lattice dynamics and the Boltzmann transport equation, according to the principle that the volume difference between doped ions and host ions after the ceramic material is doped will cause phonon scattering, deriving an expression for the phonon relaxation time of the crystal after doping; step two, according to the phonon relaxation time expression, obtaining the phonon relaxation time of the crystal after each rare earth element is doped; step three, the rare earth element corresponding to the lowest phonon relaxation time among all the obtained phonon relaxation times is the rare earth element that can most effectively reduce the thermal conductivity of the crystal; the invention provides an executable rare earth element selection method for reducing the thermal conductivity of a rare earth element doped ceramic material based on strict theoretical derivation, and is easy to operate, greatly reducing the consumption of manpower and material resources.

Description

Technical Field

[0001] The present invention relates to the technical field of rare earth element doped ceramic materials, and more specifically to a method for selecting rare earth elements based on ion volume difference to reduce the thermal conductivity of ceramic materials. Background Art

[0002] Thermal barrier coatings (TBC) are ceramic coatings that are deposited on the surface of high-temperature resistant metals or superalloys. They act as a heat insulator for the base material, lower the base temperature, and enable devices made of them (such as engine turbine blades) to operate at high temperatures. They can also increase the thermal efficiency of the device by more than 60%. In the application of TBC materials, thermal conductivity is a key feature for evaluating the performance of materials. Currently, one known method for reducing the thermal conductivity of ceramic materials is to dope ceramic materials with rare earth elements. However, for a certain ceramic material, which rare earth element is the most effective to add? Currently, most of them are tested one by one, which consumes a lot of manpower and material resources, making it difficult to improve the research and development cycle of rare earth element-doped ceramic materials. Currently, there is no executable theoretical method for screening rare earth elements. Summary of the invention

[0003] To this end, the technical problem to be solved by the present invention is to provide a rare earth element selection method for reducing the thermal conductivity of ceramic materials based on ion volume difference. A strict theoretical derivation is given to provide an executable rare earth element selection method for reducing the thermal conductivity of rare earth element-doped ceramic materials. The method is easy to operate and greatly reduces the consumption of manpower and material resources.

[0004] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0005] The method for selecting rare earth elements to reduce the thermal conductivity of ceramic materials based on ion volume difference comprises the following steps:

[0006] Step 1: Based on lattice dynamics and Boltzmann transport equation, according to the principle that the volume difference between doped ions and host ions after doping ceramic materials will cause phonon scattering, the expression of crystal phonon relaxation time after doping is derived;

[0007] Step 2, respectively substituting the volumes of multiple rare earth element ions and the host ion volume of a specific ceramic material crystal into the phonon relaxation time expression obtained in step 1 to obtain the phonon relaxation time of the crystal after each rare earth element is doped;

[0008] Step 3: The rare earth element corresponding to the lowest phonon relaxation time among all the phonon relaxation times obtained is the rare earth element that can most effectively reduce the thermal conductivity of the crystal.

[0009] Furthermore, the phonon relaxation time expression is:

[0010]

[0011] In the formula, ξ represents the molar concentration of doping, ΔV represents the volume difference between doped ions and host ions, ρ represents the density of undoped crystals in the physical model, Ω represents the volume of the unit cell in the physical model, c represents the speed of sound, q represents the reciprocal lattice vector, and m b represents the mass of the b-th atom, e(b,qs) represents the polarization vector of atom b in the qs state, and e * (b,qs′) represents the transposed complex conjugate of e(b,qs′).

[0012] Furthermore, in the derivation process of the phonon relaxation time expression, a physical model of the ceramic material is first established, and the perturbation term caused by the doped ions is calculated. According to the weak law of large numbers and the Born-Kármán boundary condition, the Hamiltonian Simplify.

[0013] Furthermore, according to the simplified Hamiltonian And Fermi's golden formula to calculate the unit time n qs The rate of change of is obtained, and the expression for the phonon relaxation time is obtained.

[0014] Furthermore, based on the Debye approximation, the expression of phonon relaxation time is further simplified under the condition that there are three acoustic branches.

[0015] Furthermore, the method for selecting rare earth elements to reduce the thermal conductivity of ceramic materials based on ion volume difference comprises the following steps:

[0016] Obtain host ion volume;

[0017] Obtain the volumes of all rare earth element ions;

[0018] Substituting the volume of the host ion and the volume of the rare earth element ion into the phonon relaxation time expression for calculation;

[0019] All calculated phonon relaxation times are sorted from small to large to obtain the rare earth element corresponding to the lowest phonon relaxation time. This rare earth element is the rare earth element that most effectively reduces the thermal conductivity of the ceramic.

[0020] The technical solution of the present invention achieves the following beneficial technical effects:

[0021] The present invention provides an executable rare earth element selection method for reducing thermal conductivity of rare earth element-doped ceramic materials based on strict theoretical derivation. The method is based on the derived phonon relaxation time expression, which is relatively simple and can improve the calculation speed, so that the computer program based on the method can be executed on an ordinary computer, greatly reducing the consumption of manpower and material resources.

[0022] When making a selection according to the method of the present invention, the variables mainly required are the volumes of the doped ions and the host ions, and the volumes are known and only need to be brought into the expression of the phonon relaxation time. Then, the rare earth element corresponding to the lowest phonon relaxation time can be found from the calculation results. The method is easy to operate, convenient and fast, and one of the preferred rare earth elements can be preliminarily screened out without performing actual experimental operations, which reduces the room for selection for subsequent screening and can effectively shorten the research and development cycle. The method can quickly find the rare earth element that most effectively reduces thermal conductivity after being doped with a certain rare earth element, reduce the number of experiments for screening rare earth element types, and effectively reduce research and development costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 It is a schematic diagram of the process of the present invention;

[0024] Figure 2 A schematic diagram of the microstructure of rare earth element-doped SrZrO3 according to an embodiment of the present invention;

[0025] Figure 3 This is a table of relative atomic volumes of some related elements. DETAILED DESCRIPTION

[0026] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings in the embodiments of the present invention.

[0027] The method for selecting rare earth elements to reduce the thermal conductivity of ceramic materials based on ion volume difference comprises the following steps:

[0028] Step 1: Based on lattice dynamics and Boltzmann transport equation, according to the principle that the volume difference between doped ions and host ions after doping ceramic materials will cause phonon scattering, the expression of crystal phonon relaxation time after doping is derived;

[0029] Step 2, respectively substituting the volumes of multiple rare earth element ions and the host ion volume of a specific ceramic material crystal into the phonon relaxation time expression obtained in step 1 to obtain the phonon relaxation time of the crystal after each rare earth element is doped;

[0030] Step 3: sort all the phonon relaxation times obtained, and the rare earth element corresponding to the lowest phonon relaxation time among all the phonon relaxation times is 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 optimal option obtained by screening based on the ion volume difference, and other screening theories may also result in rare earth elements that are the same or different from the rare earth elements screened in this application. These screened rare earth elements serve as the basis for subsequent further screening, and the optimal result theoretically appears among these preliminarily screened rare earth elements;

[0031] There are currently 17 kinds of rare earth element ions commonly used, which 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 in the future or when screening other elements that may affect thermal conductivity, this method can also be used.

[0032] Specifically, Figure 1 As shown, the method for selecting rare earth elements to reduce the thermal conductivity of ceramic materials based on ion volume difference comprises the following steps:

[0033] Obtain host ion volume;

[0034] Obtain the volumes of all rare earth element ions;

[0035] The volume of the host ion and the volume of the rare earth element ion are substituted into the phonon relaxation time expression for calculation; the volume of the host ion and the volume of the rare earth element ion are both calculated using the corresponding ion radius in the Lang's Handbook of Chemistry;

[0036] All calculated phonon relaxation times are sorted from small to large to obtain the rare earth element corresponding to the lowest phonon relaxation time, which is the rare earth element that is most effective in reducing the thermal conductivity of the ceramic;

[0037] In practice, it is necessary to program the method based on the present invention. The program can run on an ordinary computer. The volume of each atom is preset in the program for retrieval or the required volume is directly input into the computer. The data is substituted into formula (19) for calculation by the program. Then the program finds the smallest value in the calculation result, and marks the rare earth element corresponding to the smallest value result as the rare earth element that is most effective in reducing the thermal conductivity of the doped ceramic material. That is, the rare earth element that can most effectively reduce the thermal conductivity of the ceramic material is selected. After the rare earth element is obtained theoretically, an experimental method can be used for verification. The present invention reduces the number of subsequent experiments required and greatly reduces the consumption of manpower and material resources.

[0038] Furthermore, the phonon relaxation time expression used in the calculation is:

[0039]

[0040] In the formula, ξ represents the molar concentration of doping, ΔV represents the volume difference between doped ions and host ions, ρ represents the density of undoped crystals in the physical model, Ω represents the volume of the unit cell in the physical model, c represents the speed of sound, q represents the reciprocal lattice vector, and m b represents the mass of the b-th atom, e(b,qs) represents the polarization vector of atom b in the qs state, and e * (b,qs′) represents the transposed complex conjugate of e(b,qs′).

[0041] Furthermore, the derivation process of the phonon relaxation time expression is as follows: firstly, a physical model of the ceramic material is established, and the perturbation term caused by the doping ions is calculated. According to the weak law of large numbers and the Born-Kármán boundary condition, the Hamiltonian Simplify; according to the simplified Hamiltonian And Fermi's golden formula to calculate the unit time n qs The rate of change of is obtained, and the expression for phonon relaxation time is obtained. At this time, the preliminary expression for phonon relaxation time is obtained. According to the Debye approximation, under the condition that there are 3 acoustic branches, the preliminary expression for phonon relaxation time is further simplified, and finally equation (19) is obtained.

[0042] Specifically, the derivation steps of the phonon relaxation time expression are:

[0043] First, a physical model is established: Assume that the density of the undoped crystal is ρ, and it contains N0 = N1N2N3 cells, each cell has a volume of Ω and contains b0 atoms; number these b0 atoms with b = 1, 2, ..., b0, respectively, and further assume that the substitution atoms only randomly replace the atoms with b = 1, let P be the number of substitution atoms in the crystal, then the concentration of the substitution atoms in the crystal is

[0044] According to the lattice dynamics theory, the displacement operator of the bth atom in the lth cell is

[0045]

[0046] In formula (1), represents the reduced Planck constant, ρ represents the density of the undoped crystal in the physical model, Ω represents the volume of the unit cell in the physical model, ω(qs) represents the frequency corresponding to q in the s-th acoustic branch, e(b,qs) represents the polarization vector of the b-th atom in the (qs) state, and a -qsdenote the generation operator and annihilation operator respectively, exp[i(q·r(lb)-ωt)] denotes e to the power of [i(q·r(lb)-ωt)], i denotes the imaginary unit, q denotes the reciprocal lattice vector, r(lb) denotes the displacement operator of the bth atom in the lth cell in the crystal, ω denotes the frequency, and t denotes the time;

[0047] The Hamiltonian for the undoped lattice is

[0048]

[0049] In formula (2), m b represents the mass of the b-th atom, represents the velocity of the b-th atom in the l-th cell, Indicates The transposed complex conjugate of , v2 represents the second-order term of the potential energy Taylor expansion;

[0050] The first term on the right side of equation (2) is the kinetic energy term. When only the volume change of each cell caused by doping ions is considered, the constant Ω is transformed into a random variable Ω(l). Therefore, the displacement operator of the bth atom in the lth cell after doping is written as

[0051]

[0052] In formula (3), represents the reduced Planck constant, ρ represents the density of the undoped crystal in the physical model, Ω(l) is a random variable representing the volume of the lth unit cell, ω(qs) represents the frequency corresponding to q in the sth acoustic branch, e(b,qs) is the polarization vector of the bth atom in the (qs) state, and a -qs denote the generation operator and the annihilation operator respectively, i denotes the imaginary unit, q denotes the reciprocal lattice vector, r(lb) denotes the displacement operator of the bth atom in the lth cell in the crystal, ω denotes the frequency, and t denotes the time;

[0053] Assume that the Hamiltonian of the doped crystal is The perturbation term caused by the doping ions is for

[0054]

[0055] 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 sth acoustic branch, and a q′s′ and a -qs represents the annihilation operator, represents the generation operator, e *(b,q′s′) represents the transposed complex conjugate of e(b,qs), e(b,qs) represents the polarization vector of atom b (qs) in the state, represents the reduced Planck constant, m b represents the mass of the b-th atom; Ω(l) is a random variable representing the volume of the l-th unit cell; Ω represents the volume of the unit cell in the undoped crystal, ρ represents the density of the undoped crystal in the physical model, i represents the imaginary unit, qq′ represents the difference of the reciprocal lattice vector, and r(lb) represents the displacement operator of the b-th atom in the l-th unit cell in the crystal.

[0056] The simplification steps of equation (4) include: according to the physical model assumption, when l takes 1, ..., N0, are N0 independent and identically distributed random variables, so they are uncorrelated; that is, for k≠l, the covariance is

[0057]

[0058] According to the weak law of large numbers, since N0 is very large,

[0059]

[0060] Where ΔV = Ω(l)-Ω, and yes The expected value of

[0061]

[0062] According to the Born-von Karman boundary condition,

[0063] ∑ l exp[i(qq′)·r(l1)]=N0Δ(qq′) (8)

[0064] Where, when qq′ is a reciprocal lattice vector, Δ(qq′) = 1; otherwise Δ(qq′) = 0. Substituting (7) and (8) into (4), the perturbation term

[0065]

[0066] Subsequent conversion probability Operation, transformation probability Indicates the initial state (the initial state of the phonon) in unit time |i>=|n qs n q′s′ >Transformed to the final state |f>=|n qs +1,n q′s′ -1> caused by the change in state (q,s) phonon concentration, according to Fermi's golden formula, It can be written as

[0067]

[0068] Among them, only when E(f)=E(i), δ(E(f)-E(i)=1, otherwise it is 0, so that the transformation process obeys the law of conservation of energy; the initial state per unit time |i>=|n qs n q′s′ >Transformed to the final state |f>=|n qs +1,n q′s′ -1> caused by the state (q, s), according to the lattice dynamics formula, the change of phonon concentration has the following relationship

[0069]

[0070] In formula (12), n qs and n q′s′ represent the concentrations of phonon states (q,s) and (q′,s′), respectively;

[0071] Therefore, combining equations (9), (10) and (12), we can get

[0072]

[0073] Then in unit time n qs The rate of change of

[0074]

[0075] where ψ qs The definition of

[0076]

[0077] Among them, ψ qs It is a measure of the deviation of the phonon state (q, s) from its equilibrium state; is the concentration of the phonon state (q, s) when it is in equilibrium;

[0078] The condition Since the transition process of phonons from the initial state to the final state satisfies the law of energy conservation, So according to Bose statistics, we get According to the single-phonon relaxation time approximation, when qs≠q′s′, we have ψ q′s′ = 0; and according to the formula of thermal conductivity in lattice dynamics, By establishing an equation with equation (13), we have

[0079]

[0080] The phonon relaxation time τqs By transforming one side of the equation separately and canceling out the same terms on both sides of the equation, we can get the preliminary expression of phonon relaxation time:

[0081]

[0082] Next, we further simplify formula (16):

[0083] According to the Debye approximation, only three acoustic branches are considered, and it is assumed 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 to

[0084]

[0085] In the first Brillouin zone, if Δ(qq′)≠0, then q′=q, so

[0086]

[0087] After sorting

[0088]

[0089] In formula (19), ξ represents the molar concentration of doping, ΔV represents the volume difference between doped ions and host ions, ρ 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, and m b represents the mass of the b-th atom, e(b,qs) represents the polarization vector of atom b in the qs state, and e * (b,qs′) represents the transposed complex conjugate of e(b,qs′).

[0090] According to the actual situation, |ΔV|<<Ω. In this value range, at the same doping concentration, the larger the volume difference ΔV between the doped ions and the host ions, the larger the inverse of the phonon delay time, and the smaller the phonon delay time of each phonon state of the doped crystal, so the thermal conductivity is smaller. Therefore, the corresponding rare earth element ions can most effectively reduce the thermal conductivity of the ceramic material, and the rare earth element ions are selected accordingly.

[0091] Here is a specific example:

[0092] SrZrO3-based ceramic materials are used for rare earth doping, such as Figure 2 The figure shows the microstructure of rare earth element doped SrZrO3. Figure 2 (a), (b), and (c) represent the unit cell, super cell, and doped super cell of SrZrO3, respectively. Figure 2In (c), X represents a rare earth element ion; 17 rare earth elements are used to replace Zr ions. According to formula (19), the larger the volume difference ΔV between the doped ions and the host ions, the larger the inverse of the phonon delay time, that is, the smaller the phonon delay time, that is, the smaller the thermal conductivity; Figure 3 Chinese data( Figure 3 The data comes from the relative atomic mass in the "Lan's Handbook of Chemistry". When the volume difference between La and Zr ions is the largest, the difference is 26.1. Therefore, among all the 17 rare earth elements, La can most effectively reduce the thermal conductivity of SrZrO3 ceramic materials.

[0093] Obviously, the above embodiments are merely examples for the purpose of clear explanation, and are not intended to limit the implementation methods. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the implementation methods here. The obvious changes 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 to reduce thermal conductivity of ceramic materials based on ion volume difference, characterized in that: The steps include: Step 1: Based on lattice dynamics and Boltzmann transport equation, according to the principle that the volume difference between doped ions and host ions after doping ceramic materials will cause phonon scattering, the expression of crystal phonon relaxation time after doping is derived; Step 2, respectively substituting the volumes of multiple rare earth element ions and the host ion volume of a specific ceramic material crystal into the phonon relaxation time expression obtained in step 1 to obtain the phonon relaxation time of the crystal after each rare earth element is doped; Step 3, the rare earth element corresponding to the lowest phonon relaxation time among all the phonon relaxation times obtained is the rare earth element that can most effectively reduce the thermal conductivity of the crystal; The phonon relaxation time expression is: In the formula, ξ represents the molar concentration of doping, ΔV represents the volume difference between doped ions and host ions, ρ represents the density of undoped crystals in the physical model, Ω represents the volume of the unit cell in the physical model, c represents the speed of sound, q represents the reciprocal lattice vector, and m b represents the mass of the b-th atom, e(b,qs) represents the polarization vector of atom b in the qs state, and e*(b,qs′) represents the transposed complex conjugate of e(b,qs′).

2. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 1, characterized in that: In the derivation of the phonon relaxation time expression, the physical model of the ceramic material is first established, and the perturbation term caused by the doping ions is calculated. According to the weak law of large numbers and the Born-Kármán boundary condition, the Hamiltonian Simplify.

3. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 2, characterized in that: Assuming that the density of the undoped crystal is ρ, and it contains N0=N1N2N3 cells, the volume of each cell is Ω, and contains b0 atoms, a physical model is established; Number these b0 atoms with b=1, 2, ..., b0 respectively, and assume that the substitution atoms only randomly replace the atoms with b=1. Let P be the number of substitution atoms in the crystal, then the concentration of substitution atoms in the crystal is According to the lattice dynamics theory, the displacement operator of the bth atom in the lth cell is The Hamiltonian for the undoped lattice is Among them, the first term on the right side of equation (2) is the kinetic energy term, and ν2 is the second-order term of the potential energy Taylor expansion; under the condition that only the volume change of each cell affected by the doping ions is considered, the constant Ω is transformed into a random variable Ω(l), and the displacement operator of the doped crystal is written as Assume that the Hamiltonian of the doped crystal is The perturbation term caused by the doping ions is for The simplification steps of equation (4) include: according to the physical model assumption, when l takes 1, ..., N0, are N0 independent 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, since N0 is very large, Since ΔV=Ω(l)-Ω, and yes The expected value of According to the Born-von Karman boundary condition, ∑ l exp[i(q-q′)·r(l1)]=N0Δ(q-q′) (8) Where, when qq′ is a reciprocal lattice vector, Δ(qq′) = 1; otherwise Δ(qq′) = 0. Substituting (7) and (8) into (4), the perturbation term 4. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 3, characterized in that: According to the simplified Hamiltonian And Fermi's golden formula to calculate the unit time n qs The rate of change of is obtained, and the expression for the phonon relaxation time is obtained.

5. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 4, characterized in that: According to Fermi's golden formula, It can be written as Among them, δ(E(f)-E(i)) makes the transformation process obey the law of conservation of energy; Initial state per unit time |i>=|n qs n q′s′ >Transformed to the final state |f>=|n qs +1,n q′s′ -1> The change in state (q, s) phonon concentration caused by In formula (12), n qs and n q′s′ represent the concentrations of phonon states (q,s) and (q′,s′), respectively; Combining equations (9), (10) and (12), we can get Then in unit time n qs The rate of change of where ψ qs The definition of According to the single-phonon relaxation time approximation, when qs≠q′s′, we have ψ q′s′ =0; and By establishing an equation with equation (13), we have The phonon relaxation time τ qs Transforming only one side of the equation, we have 6. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 5, characterized in that: According to the Debye approximation, the expression of phonon relaxation time is further simplified under the condition that there are three acoustic branches.

7. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 6, characterized in that: The set 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 to In the first Brillouin zone, if Δ(qq′)≠0, then q′=q, so After sorting 8. The method for selecting rare earth elements for reducing thermal conductivity of ceramic materials based on ion volume difference according to claim 1, characterized in that: The following steps are involved: Obtain host ion volume; Obtain the volumes of all rare earth element ions; Substituting the volume of the host ion and the volume of the rare earth element ion into the phonon relaxation time expression for calculation; All calculated phonon relaxation times are sorted from small to large to obtain the rare earth element corresponding to the lowest phonon relaxation time. This rare earth element is the rare earth element that most effectively reduces the thermal conductivity of the ceramic.

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