Phase-field simulation method, device and storage medium for dielectric properties of relaxor ferroelectrics

Abstract

The present invention discloses a phase-field simulation method, device, and storage medium for the dielectric properties of relaxor ferroelectrics. The method includes the following steps: S1, introducing local compositional inhomogeneity for the relaxor ferroelectric to generate a concentration distribution; S2, setting a Landau energy based on the compositional inhomogeneity; S3, using a random electric field to describe other structural inhomogeneities resulting from compositional inhomogeneity and fluctuations; S4, obtaining the free energy of the relaxor ferroelectric dielectric properties based on the results of steps S1 to S3; and S5, searching for the order parameter distribution that minimizes the free energy to achieve steady-state phase-field simulation. Conventional phase-field simulations are unable to describe the compositional inhomogeneity of relaxor ferroelectrics and can only pre-set structures and parameters. The present invention directly introduces compositional inhomogeneity, associating ferroelectricity, structural inhomogeneity, and other related properties with the compositional distribution. This allows for the description and quantitative simulation of complex relaxor ferroelectrics through a concise and clear single path, with the results consistent with experimental observations.

Description

Technical Field

[0001] The present invention relates to relaxor ferroelectrics, and in particular to a phase field simulation method, device and storage medium for the dielectric properties of relaxor ferroelectrics. Background Art

[0002] Relaxor ferroelectrics are a class of materials of great importance in the fields of ferroelectrics, piezoelectrics, and dielectrics. However, compared to ferroelectrics, theoretical approaches to studying them are very limited. Phase-field simulation is an effective method for simulating ferroelectrics, but for relaxor ferroelectrics with complex compositions, constructing a relaxor ferroelectric phase-field model that does not rely on a large number of preset parameters and structures is extremely difficult. Many of the properties of relaxor ferroelectrics originate from local compositional fluctuations and inhomogeneities, which produce polar nanodomains or slush domain structures. For example, relaxor ferroelectrics are characterized by polar nanodomains, and are described as ferroelectric / paraelectric complexes with random spatial distribution (Science 365, 578-582 (2019)). In another example, a normally distributed random electric field is used to achieve the slush domain effect (Acta Materialia 225, 117558 (2022)). However, these models are highly dependent on a large number of preset structures and parameters.

[0003] Existing models are highly dependent on a large number of preset structures and parameters, and cannot reflect the impact of local composition fluctuations and inhomogeneities on the properties of relaxor ferroelectrics. Summary of the Invention

[0004] The purpose of the present invention is to provide a phase-field simulation method, device and storage medium for the dielectric properties of relaxor ferroelectrics, which can describe relaxor ferroelectrics by directly introducing component variables and realize phase-field simulation.

[0005] In phase field simulation, there is an order parameter and a free energy with the order parameter as the main variable. The present invention realizes steady-state phase field simulation by finding the order parameter distribution corresponding to the minimum free energy.

[0006] The technical solution of the present invention is:

[0007] A phase-field simulation method for dielectric properties of relaxor ferroelectrics comprises the following steps:

[0008] S1. For relaxor ferroelectrics, local inhomogeneity of composition is introduced to generate concentration distribution;

[0009] S2. Set the Landau energy according to the compositional heterogeneity;

[0010] S3. Use random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations;

[0011] S4, obtaining the free energy of the dielectric properties of the relaxor ferroelectric according to the results of steps S1 to S3;

[0012] S5. Find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation.

[0013] Preferably, the local heterogeneity of the component introduced in step S1 is:

[0014] The relaxor ferroelectric system is considered as a solid solution of multiple components A, B, C, etc., and their nominal concentrations are c A 0 , c B 0 , c C 0 , …, and satisfy .

[0015] Preferably, the method for generating the concentration distribution in step S1 is:

[0016] The true concentration distribution is generated by a Dirichlet distribution, i.e. , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated based on the nominal concentration, and k1 is a coefficient.

[0017] Preferably, the method for setting the Landau energy according to the composition heterogeneity in step S2 is:

[0018] Assume that the Landau energy at each position is a linear combination of the components, and the weight depends on the actual concentration of each component at this position, that is, ;in, is the Landau energy coefficient.

[0019] Preferably, the random electric field distribution in step S3 is related to the actual concentration gradient, i.e. , where E rand is the random field strength, is the configurational entropy calculated based on the nominal concentration, and k2 is another constant.

[0020] Preferably, the free energy is expressed as:

[0021] ;

[0022] Represent Landau energy, gradient energy, elastic energy and electric field energy respectively; specifically,

[0023] ;

[0024] ; ;

[0025] ;

[0026] in, , G, C, , , K, and E are the Landau energy coefficient, gradient energy coefficient, elastic stiffness coefficient, total strain, spontaneous strain, background dielectric constant, vacuum dielectric constant, and electric field, respectively.

[0027] Preferably, in the expression of the free energy:

[0028] Spontaneous strain comes from the electrostrictive effect of spontaneous polarization, that is, the electrostrictive coefficient , ;

[0029] The distribution of elastic strain is obtained by solving the mechanical equilibrium equation get;

[0030] The electric field is obtained by solving Poisson's equation, that is, ,in is the electric potential;

[0031] The overall solution process is as follows: first, a random polarization field is input, and the spontaneous strain and elastic equilibrium equations are calculated based on the polarization field to obtain the strain distribution; then, the solution of the Poisson equation is obtained based on the polarization field to obtain the electric field distribution; the next step of polarization structure evolution is carried out based on the polarization, strain and electric field distribution; and this cycle is repeated to obtain the final steady-state polarization field.

[0032] Preferably, the method for finding the order parameter distribution corresponding to the minimum free energy in step S5 is: selecting polarization is the order parameter, P1, P2, and P3 are the three components of electric polarization. According to the evolution equation , to obtain the minimum free energy, where L is a kinetic parameter, t is time, and F is free energy; methods for obtaining the minimum free energy include:

[0033] The evolution equation is written in discrete form:

[0034] ;

[0035] so ;

[0036] consider Small, with:

[0037] ;

[0038] Ensure that the direction of polarization evolution makes the free energy smaller and smaller, and finally reaches the convergent minimum.

[0039] The present invention also provides a phase-field simulation device for the dielectric properties of relaxor ferroelectrics, which performs the above-mentioned phase-field simulation method. The device comprises:

[0040] The first module introduces local inhomogeneity of composition and generates concentration distribution for relaxor ferroelectrics.

[0041] The second module sets the Landau energy according to the composition heterogeneity;

[0042] The third module uses random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations;

[0043] The fourth module obtains the free energy of the dielectric properties of the relaxor ferroelectric based on the results of the first, second and third modules;

[0044] The fifth module is to find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation.

[0045] The present invention also provides a computer-readable storage medium, wherein the storage medium stores a program, and the program is executed by a processor to implement the phase field simulation method.

[0046] The advantages of the present invention are:

[0047] Conventional phase-field simulations cannot describe the compositional inhomogeneity of relaxor ferroelectrics and can only preset structures and parameters. The present invention directly introduces compositional inhomogeneity, linking ferroelectricity, structural inhomogeneity, and other related properties to the compositional distribution. This allows for the description and quantitative simulation of complex relaxor ferroelectrics through a concise and clear single path, with the results consistent with experimental observations. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0049] Figure 1 Flowchart of the phase-field simulation method for the dielectric properties of relaxor ferroelectrics of the present invention;

[0050] Figure 2 Schematic diagram of the phase field simulation results of Example 1;

[0051] Figure 2 a is a schematic diagram of a compositionally heterogeneous structure;

[0052] Figure 2 b is the trend of domain structure from large to small;

[0053] Figure 2 c is the trend of the hysteresis loop becoming gradually thinner;

[0054] Figure 2 d is a schematic diagram of the change of domain size;

[0055] Figure 2 e is a schematic diagram of the change of dielectric energy storage density / efficiency;

[0056] Figure 3 Schematic diagram of the phase field simulation results of Example 2;

[0057] Figure 3 a, 3c are schematic diagrams of heterogeneous structures with more chaotic composition;

[0058] Figure 3 b is the trend of the hysteresis loop becoming gradually thinner;

[0059] Figure 3 d is a schematic diagram of the gradually decreasing and more chaotic spontaneous polarization;

[0060] Figure 3 e is a schematic diagram of the change in dielectric energy storage density / efficiency. DETAILED DESCRIPTION

[0061] To make the purpose, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts shall fall within the scope of protection of the present invention.

[0062] like Figure 1 As shown, a phase field simulation method for the dielectric properties of a relaxor ferroelectric comprises the steps of:

[0063] S1. For relaxor ferroelectrics, local inhomogeneity of composition is introduced to generate concentration distribution;

[0064] S2. Set the Landau energy according to the compositional heterogeneity;

[0065] S3. Use random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations;

[0066] S4, obtaining the free energy of the dielectric properties of the relaxor ferroelectric according to the results of steps S1 to S3;

[0067] S5. Find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation.

[0068] Specifically, the phase field simulation of the present invention is set up as follows for relaxor ferroelectrics:

[0069] (1) Step S1 introduces local inhomogeneity of composition: the relaxor ferroelectric system is considered as a solid solution of multiple components A, B, C, etc., and their nominal concentrations are c A 0 , c B 0 , c C 0 , …, and satisfy .

[0070] The method for generating the concentration distribution in step S1 is as follows: the true concentration distribution is generated by a Dirichlet distribution, i.e. , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated based on the nominal concentration, and k1 is a coefficient.

[0071] (2) Step S2 sets the Landau energy according to the component heterogeneity: Assume that the Landau energy at each position is a linear combination of the components, and the weight depends on the actual concentration of each component at this position, that is, . is the Landau energy coefficient, and all subscripts can be 1, 2, or 3, representing the x, y, and z components, respectively.

[0072] (3) Step S3 uses a random electric field to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations; the random electric field is a distribution related to the actual concentration gradient, i.e. , where E rand is the random field strength, is the configurational entropy calculated based on the nominal concentration, and k2 is another coefficient.

[0073] Apart from this, all other parameters are constants with no distribution.

[0074] In step S4, the free energy of the dielectric properties of the relaxor ferroelectric is obtained; the free energy is expressed as:

[0075] ;

[0076] They represent Landau energy (i.e., the intrinsic properties of relaxor ferroelectrics), gradient energy (measure of domain wall energy), elastic energy (measure of the energy of relaxor ferroelectric elasticity and electrostriction), and electric field energy. Specifically,

[0077] ;

[0078] ; ;

[0079] ;

[0080] in, , G, C, , , K, and E are the Landau energy coefficient, gradient energy coefficient, elastic stiffness coefficient, total strain, spontaneous strain, background dielectric constant, vacuum dielectric constant, and electric field, respectively.

[0081] In the expression of the free energy, the spontaneous strain comes from the electrostrictive effect of spontaneous polarization, that is, , ; The distribution of elastic strain is obtained by solving the mechanical equilibrium equation The electric field is obtained by solving Poisson's equation, that is, ;

[0082] The overall solution process is as follows: first, a random polarization field is input, and the spontaneous strain and elastic equilibrium equations are calculated based on the polarization field to obtain the strain distribution; then, the solution of the Poisson equation is obtained based on the polarization field to obtain the electric field distribution; the next step of polarization structure evolution is carried out based on the polarization, strain and electric field distribution; and this cycle is repeated to obtain the final steady-state polarization field.

[0083] The method for finding the order parameter distribution corresponding to the minimum free energy in step S5 is: select polarization is the order parameter, according to the evolution equation , to get the minimum free energy, where L is a kinetic parameter, t is time, and F is free energy.

[0084] Methods for obtaining the minimum free energy include:

[0085] The evolution equation is written in discrete form as follows:

[0086] ;

[0087] so ;

[0088] consider Small, with:

[0089] ;

[0090] Ensure that the direction of polarization evolution makes the free energy smaller and smaller, and finally reaches the convergent minimum.

[0091] Example 1

[0092] This example takes xBiFeO3-(1-x)SrTiO3 solid solution as an example, and the simulation grid is , grid spacing The parameter settings are shown in Table 1.

[0093] Table 1 BiFeO3-SrTiO3 parameter setting table

[0094]

[0095] like Figure 2 The results shown in the figure show that the method of the present invention can effectively simulate the composition heterogeneity ( Figure 2 a) The trend of domain structure from large to small ( Figure 2 b) and the trend of the hysteresis loop becoming gradually thinner ( Figure 2 c), the change of domain size ( Figure 2 d) and changes in dielectric energy storage density / efficiency ( Figure 2 e) corresponds well with existing experimental values.

[0096] Example 2

[0097] In this embodiment, Bi4(Ti,A,B,C,D)3O 12 For example, a solid solution is considered to be obtained by equimolar doping of elements A, B, C, and D. Therefore, when there are n components A, B, C, etc., the nominal concentration of each component is 1 / (n+1) (because there is also Bi4Ti3O 12 ), the entropy of the system is The simulation grid is , grid spacing The parameter settings are shown in Table 2.

[0098] Table 2 Bi4(Ti,A,B,C,D)3O 12 Parameter setting table

[0099]

[0100] like Figure 3 The results shown show that as the entropy of the material increases, the material composition becomes more disordered ( Figure 3 a, 3c), the hysteresis loop changes from wide to thin ( Figure 3 b), the spontaneous polarization gradually decreases and becomes more chaotic ( Figure 3 d), the energy storage performance is significantly improved and the optimal point is located in the medium-high entropy range, which is consistent with the reported experimental results.

[0101] Example 3

[0102] The present invention also provides a phase-field simulation device for the dielectric properties of relaxor ferroelectrics, which performs the above-mentioned phase-field simulation method. The device comprises:

[0103] The first module introduces local inhomogeneity of composition and generates concentration distribution for relaxor ferroelectrics.

[0104] The second module sets the Landau energy according to the composition heterogeneity;

[0105] The third module uses random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations;

[0106] The fourth module obtains the free energy of the dielectric properties of the relaxor ferroelectric based on the results of the first, second and third modules;

[0107] The fifth module is to find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation.

[0108] Example 4

[0109] The present invention also provides a computer-readable storage medium, wherein the storage medium stores a program, and the program is executed by a processor to implement the phase field simulation method.

[0110] The above embodiments are intended only to illustrate the technical concepts and features of the present invention. Their purpose is to enable those skilled in the art to understand the contents of the present invention and implement them accordingly. They are not intended to limit the scope of protection of the present invention. Any modifications made based on the spirit of the main technical solution of the present invention shall be included in the scope of protection of the present invention.

Claims

1. A phase-field simulation method for the dielectric properties of relaxor ferroelectrics, characterized in that: Including steps: S1. For relaxor ferroelectrics, local inhomogeneity of composition is introduced to generate concentration distribution; S2. Set the Landau energy according to the compositional heterogeneity; S3. Use random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations; S4, obtaining the free energy of the dielectric properties of the relaxor ferroelectric according to the results of steps S1 to S3; S5. Find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation; The free energy is expressed as: ； Represent Landau energy, gradient energy, elastic energy and electric field energy respectively; specifically, ； ； ； ； in, , G, C, , , K, and E are the Landau energy coefficient, gradient energy coefficient, elastic stiffness coefficient, total strain, spontaneous strain, background dielectric constant, vacuum dielectric constant, and electric field, respectively; The method for finding the order parameter distribution corresponding to the minimum free energy in step S5 is: select polarization is the order parameter, P1, P2, and P3 are the three components of electric polarization, according to the evolution equation , to obtain the minimum free energy, where L is a kinetic parameter, t is time, and F is free energy; methods for obtaining the minimum free energy include: The evolution equation is written in discrete form as follows: ； so ; consider Very small, with: ； Ensure that the direction of polarization evolution makes the free energy smaller and smaller, and finally reaches the convergent minimum.

2. The phase-field simulation method for the dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that: The local inhomogeneity of the components introduced in step S1 is: The relaxor ferroelectric system is considered as a solid solution of multiple components A, B, C, etc., and their nominal concentrations are c A 0 ,c B 0 , c C 0 , …, and satisfy .

3. The phase-field simulation method for the dielectric properties of relaxor ferroelectrics according to claim 2, characterized in that: The method for generating the concentration distribution in step S1 is: The true concentration distribution is generated by a Dirichlet distribution, i.e. , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated based on the nominal concentration, and k1 is a coefficient.

4. The phase-field simulation method for the dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that: The method for setting the Landau energy according to the composition heterogeneity in step S2 is: Assume that the Landau energy at each position is a linear combination of the components, and the weight depends on the actual concentration of each component at this position, that is, , mnl represents the tensor index; among them, is the Landau energy coefficient.

5. The phase-field simulation method for the dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that: The random electric field distribution in step S3 is related to the actual concentration gradient, that is, , where E rand is the random field strength, is the configurational entropy calculated based on the nominal concentration, and k2 is another coefficient.

6. The phase-field simulation method for the dielectric properties of relaxor ferroelectrics according to claim 5, characterized in that: In the expression of the free energy: Spontaneous strain comes from the electrostrictive effect of spontaneous polarization, that is, the electrostrictive coefficient , ; The distribution of elastic strain is obtained by solving the mechanical equilibrium equation get; The electric field is obtained by solving Poisson's equation, that is, ,in is the electric potential; The overall solution process is as follows: first, a random polarization field is input, and the spontaneous strain and elastic equilibrium equations are calculated based on the polarization field to obtain the strain distribution; then, the solution of the Poisson equation is obtained based on the polarization field to obtain the electric field distribution; the next step of polarization structure evolution is carried out based on the polarization, strain and electric field distribution; and this cycle is repeated to obtain the final steady-state polarization field.

7. A phase-field simulation device for the dielectric properties of relaxor ferroelectrics, characterized in that: The phase field simulation method according to any one of claims 1 to 6 is performed, wherein the apparatus comprises: The first module introduces local inhomogeneity of composition and generates concentration distribution for relaxor ferroelectrics. The second module sets the Landau energy according to the composition heterogeneity; The third module uses random electric fields to describe other structural inhomogeneities arising from compositional inhomogeneities and fluctuations; The fourth module obtains the free energy of the dielectric properties of the relaxor ferroelectric based on the results of the first, second and third modules; The fifth module is to find the order parameter distribution that minimizes the free energy to achieve steady-state phase field simulation.

8. A computer-readable storage medium, characterized in that The storage medium stores a program, and the program is executed by a processor to implement the phase field simulation method according to any one of claims 1 to 6.

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