CN107315913A - A kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field - Google Patents
A kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field Download PDFInfo
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Abstract
The invention discloses a kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field, this method is on the basis of classical phase field model, with reference to micromagnetics magnetic energy calculation formula, the magnetic energy that ferromagnetism crystal grain is produced under external magnetic field introduces system free energy, constructs the universal model for simulating ferromagnetism crystal grain evolutionary process under external magnetic field.This model can be by introducing different magnetic energy（Zeeman energy, magnetocrystalline anisotropy energy, demagnetization field energy）, influence of each magnetic energy to ferromagnetic phase single domain Crystal Evolution under different external magnetic field strengths, and total magnetic energy are studied to its combined influence.The invention provides the phase field model for considering ferromagnetism Crystal Evolution under external magnetic field, and it is applied to Nd Fe B permanent-magnet alloys to verify its feasibility.
Description
Technical field
The present invention relates to Phase Field field, and in particular to a kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field.
Background technology
The fast development of computer technology has expedited the emergence of an emerging cross discipline --- Computer materials science, i.e., using computer
Many problems that the method for simulation is come in research material scientific and engineering field, can not only save experimental expenses and time, and
And by being combined with traditional theory and experiment research, can more be deep into the mechanistic aspect of material.In various meters
Calculate in analogy method, belong to the microcosmic Phase Field to meso-scale based on Material Thermodynamics and dynamics, material can be provided
The direction of interior microscopic microstructure Evolution and specific change path, reproduction material organize the differentiation with the time in various phase transition process
Process simultaneously provides its quantitative description, contributes to the inherent mechanism and rule of profound understanding microstructure evolution.It is other with meso-scale
Analogy method is compared, and Phase Field must arrive researcher without be tracked to the interface of material internal, therefore since being suggested
Favor, be widely used in materials science field, be more maturely used to simulating dendritic growth in subcooling films, flat
Look unfamiliar the processes such as long and solid-state phase changes.
In recent years, the Phase Field research for magnetic material has also obtained the concern of people.Due to the easy magnetic of ferrimagnet
Change and saturation magnetization is big, therefore great magnetic energy (including zeeman energy, magnetocrystalline anisotropy can be produced under external magnetic field
Energy, exchange energy and demagnetization energy etc.) so that the energy state of ferromagnetic phase changes.According to energy minimization principle, iron
Magnetic is met towards the minimum state evolution of each energy term (free energy of chemistry, interface energy, magnetic energy etc.) sum, therefore external magnetic field is to it
Tissue topography, which has, to be significantly affected.By theoretical calculation analog simulation, disclose what external magnetic field influenceed on specific system microstructure Evolution
Inherent law, to rationally controlling alloy microstructure morphology using external magnetic field in actual process, so that performance needed for obtaining has
Important directive significance.
At present, the Phase Field research of coupled magnetic field generally only considers single magnetic energy, and shortage considers to magnetic energy item,
It is therefore desirable to based on micromagnetics theoretical formula, build the phase field model for considering each magnetic energy comprehensively, under external magnetic field effect
Microstructure Evolution process and its rule realize quantitative study.
The content of the invention
It is an object of the present invention to based on classical phase field model, introduce the magnetic that ferromagnetism crystal grain is produced under external magnetic field
Energy (zeeman energy, magnetocrystalline anisotropy energy, demagnetization field energy), setting up can reflect each magnetic energy produced under external magnetic field to list
The phase field model of farmland ferromagnetism Crystal Evolution influence, so as to provide a kind of phase field analysis side of ferromagnetism Crystal Evolution under magnetic field
Method.
Technical scheme is as follows.
A kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field, this method on the basis of classical phase field model,
The magnetic energy that ferromagnetism crystal grain is produced under external magnetic field introduces system free energy, constructs and exists for simulating ferromagnetism crystal grain
The universal model of evolutionary process under external magnetic field, the universal model can be by introducing different magnetic energy, the different outer magnetic of research
The influence of each magnetic energy item respectively to ferromagnetism Crystal Evolution under field intensity, and total magnetic energy is to the synthesis shadow of ferromagnetism Crystal Evolution
Ring；The magnetic energy is zeeman energy, magnetocrystalline anisotropy energy and demagnetization field energy.
It is preferred that, in classical phase kinetics equation
In system free energy GsysAdd magnetic energy E caused by external magnetic fieldmag
G′sys=Gsys+Emag (2)
Then the kinetics equation of gained universal model is
Wherein, η is phase S order parameter；T is time step；M is the intensity of magnetization；GsysFor system free energy；EmagFor outer magnetic
Magnetic energy caused by.
It is preferred that, the external magnetic field causes the magnetic energy E of ferromagnetism crystal grainmagBy zeeman energy Ezeem, magnetocrystalline anisotropy energy
EanisCan E with demagnetizationdComposition；By zeeman energy Ezeem, magnetocrystalline anisotropy energy EanisCan E with demagnetizationdOne by one instead of in (3) formula
EmagInfluence of the different magnetic energy to ferromagnetism Crystal Evolution can be drawn respectively, by zeeman energy Ezeem, magnetocrystalline anisotropy energy Eanis
Can E with demagnetizationdPlus and substitute into (3) formula can then draw combined influence of the external magnetic field to ferromagnetism crystal grain.
It is preferred that, the zeeman energy EzeemCan E with demagnetizationdIt is with the phase order parameter η modes connected
Therefore, the zeeman energy E under conditions of external magnetic field H is sufficiently strongzeemCan E with demagnetizationdIt can be expressed as
WhereinFor the saturation magnetization of ferromagnetic phase, ηferroFor ferromagnetic phase S order parameter；μ0For space permeability；H is
External magnetic field strength.
Wherein Ms is saturation magnetization；M (r) is unit magnetic moment；HdFor demagnetization field intensity, demagnetization field intensity expression formula is
Wherein φ (r) is magnetic potential,
Wherein ρ (r) is magnetic charge density,
Magnetocrystalline anisotropy energy EanisIntroducing rely on oriented S order parameter η=(ηx,ηy), oriented S order parameter η represents ferromagnetic phase
Magnetic moment direction, η=ηx+ηyThen represent the ferromagnetic volume fraction in the point, wherein ηxMagnetic moments parallel is represented in the ferromagnetic of x-axis
Phase component, ηyMagnetic moments parallel is represented in the ferromagnetic phase component of y-axis, thus magnetocrystalline anisotropy energy EanisIt is represented by
Wherein KuFor magnetocrystalline anisotropy constant；euan(r) it is the easy axis direction of ferromagnetic crystal grain.
It is preferred that, the phase field analysis method of ferromagnetism Crystal Evolution, comprises the following steps under a kind of magnetic field：
First, research system and simulation process are determined, corresponding thermodynamics and magnetics data is collected, determines initialization condition.
2nd, it is theoretical based on Ginzburg-Laudau, in classical phase kinetics equation
In system free energy GsysMagnetic energy E caused by middle addition external magnetic fieldmag：
G′sys=Gsys+Emag (2)
Then the phase field model kinetics equation for simulating ferromagnetism Crystal Evolution under external magnetic field that provides of the present invention is
3rd, magnetic energy computation model is set up.External magnetic field causes the magnetic energy E of single domain ferromagnetism crystal grainmagBy zeeman energy Ezeem, magnetic
Anisotropic crystalline energy EanisCan E with demagnetizationdConstitute, its magnetics calculation formula is respectively：
Wherein μ0For space permeability, H (r) is external magnetic field strength, and M (r) is the intensity of magnetization, euan(r, t) is ferromagnetic crystal grain
Easy axis direction, KuFor magnetocrystalline anisotropy constant, m (r, t) is unit magnetic moment, HdFor demagnetizing field, m (r) is unit magnetic moment.When
When external magnetic field is sufficiently large, the intensity of magnetization reaches that the intensity of magnetization of each point is directly proportional to each phase volume fraction of point in saturation, space：
WhereinFor i phase saturation magnetizations, fiFor i phase volume fractions, N is total number of phases.Because non-magnetic phase is in outer magnetic
Magnetization degree is much smaller than ferromagnetic phase under field action, and the non-magnetic phase intensity of magnetization is approximately 0 by this model, while by volume fraction fi
It is approximately ηi：
WhereinFor the saturation magnetization of ferromagnetic phase, ηferroFor ferromagnetic phase S order parameter.This model is considering external magnetic field
It is sufficiently strong so that in the case of magnetization direction and outer magnetic field direction identical, zeeman energy E in magnetic energyZeemExpression way be：
Demagnetization can EdCalculated by below equation：
Wherein IS(r)=μ0MS(r), φ (r) is magnetic potential, is calculated by following formula：
Wherein ρ (r) is magnetic charge density, is calculated by following formula：
ρ (r)=- μ0▽·(-MS(r)) (12)
With reference to (8) formula, magnetic charge density ρ (r) expression formulas in this model are：
Magnetocrystalline anisotropy energy EanisIntroducing rely on oriented S order parameter η=(ηx,ηy), it represents ferromagnetic phase magnetic moment direction,
η=ηx+ηyRepresent the ferromagnetic volume fraction in the point, ηxMagnetic moments parallel is represented in the ferromagnetic phase component of x-axis, ηyRepresent magnetic moment
Parallel to the ferromagnetic phase component of y-axis.Then magnetocrystalline anisotropy energy E in this modelanisIt is expressed as：
Wherein KuFor magnetocrystalline anisotropy constant, euan(r) it is ferromagnetism crystal grain easy axis direction.
4th, the initial pattern of tissue is determined, solute field diffusion equation is coupled, sets up the phase of ferromagnetism Crystal Evolution under external magnetic field
Field model.Program is write, cycle calculations obtain evolutionary process of the ferromagnetism crystal grain under external magnetic field, export and analyze knot
Really.
By being introduced separately into each magnetic energy, change external magnetic field strength, can study under different external magnetic field strengths, each magnetic energy is right
The influence that ferromagnetism grain structure develops；By introducing whole magnetic energy, the Evolution States that contrast is organized when not adding magnetic field, feasibility study
Study carefully combined influence of the external magnetic field to ferromagnetism Crystal Evolution.
Compared with prior art, the invention has the advantages that：
1st, it is theoretical based on Ginzburg-Laudau, each magnetic energy is introduced in classical phase field model, sets up and can be used for studying
The phase field model of ferromagnetism Crystal Evolution process under the influence of magnetic field；
2nd, each magnetic energy that magnetic field triggers ferromagnetic phase to produce is considered comprehensively, can individually study each magnetic energy brilliant to ferromagnetism
The Special Influence that grain develops, can consider combined influence of the external magnetic field to ferromagnetism Crystal Evolution again；
3rd, phase field model proposed by the present invention has versatility, can be under Coupling Thermal mechanics and magnetics data research external magnetic field not
With the microstructure Evolution of system differential responses process.
Brief description of the drawings
T under different magnetic field intensity of the Fig. 1 only to consider zeeman energy influence in simulation Nd-Fe-B system peritectic reactions1Xiang Ti
Fraction is with the change curve for simulating step number.
Magnetic moment under external magnetic fields of the Fig. 2 only to consider magnetocrystalline anisotropy energy influence in simulation Nd-Fe-B system peritectic reactions
The different T in direction1The volume fraction of phase component and its total amount is with the change curve for simulating step number.
Fig. 3 is only considers that demagnetization the presence or absence of can influence T under external magnetic field in simulation Nd-Fe-B system peritectic reactions1Phase crystal grain
Length-width ratio (a length of crystallite dimension parallel on outer magnetic field direction, a width of crystallite dimension on outer magnetic field direction) with
Simulate the change curve of step number.
Fig. 4 a consider T under the presence or absence of each magnetic energy influence external magnetic field to simulate in Nd-Fe-B system peritectic reactions1Phase
The volume fraction of crystal grain is with the change curve for simulating step number.
Fig. 4 b consider T under the presence or absence of each magnetic energy influence external magnetic field to simulate in Nd-Fe-B system peritectic reactions1Phase
Length-width ratio (a length of crystallite dimension parallel on outer magnetic field direction, a width of crystal grain chi on outer magnetic field direction of crystal grain
It is very little) with the change curve of simulation step number.
Embodiment
In order to more clearly describe technical scheme, the specific implementation below in conjunction with example and accompanying drawing to the present invention
It is further described, but the implementation of the present invention is not limited to this.
The invention provides a kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field, it is applied to Nd- below
Fe-B alloy systems.Using this method to generating Hard Magnetic phase T in rapid solidification under magnetic field1(Nd2Fe14B peritectic reaction)
(Liquid+γ-Fe(FCC)→T1) simulated, the present embodiment concrete operation step is as follows：
First, it is Nd-Fe-B to determine research system, and simulation process is peritectic reaction (Liquid+ γ-Fe (FCC) → T1), phase
Thermodynamic data is answered to be shown in Table 1, magnetics data and analog parameter are shown in Table 2, and each phase free energy of chemistry calculates free using regular solution
Can model：
Wherein GφFor φ phase free energy of chemistry, ciFor solute i solubility,It is free for solute i φ phases ground state chemistry
Can, R is gas constant, and T is temperature.
Table 1
Table 2
2nd, three S order parameter η are introduced in the present embodimenti(i=1,2,3) Liquid, γ-Fe (FCC) and T are represented respectively1
Phase, sets up this model phase kinetics equation：
Wherein free energy G 'sysCalculated by below equation：
ωij=12 σij/δ (8)
Wherein, σijFor the interface energy between i phases and j phases, δ is interfacial thickness.
3rd, magnetic energy computation model is set up.T in the present embodiment1Phase zeeman energy can be calculated with demagnetization with below equation：
Demagnetization can EdCalculated by below equation：
Wherein IS(r)=μ0MS(r), φ (r) is magnetic potential, is calculated by following formula：
Wherein ρ (r) is magnetic charge density, is calculated by following formula：
Magnetocrystalline anisotropy energy EanisIntroducing rely on oriented S order parameter η=(η3,η4), it represents ferromagnetic phase T1Magnetic moment side
To η=η3+η4Represent ferromagnetic phase T1In the volume fraction of the point, η3Magnetic moments parallel is represented in the T of x-axis1Phase component, η4Represent magnetic
T of the square parallel to y-axis1Phase component.T in the present embodiment1Mutually easily axle is set to y-axis direction, i.e., (0,1), then T1Phase magnetocrystalline anisotropy
Can EanisIt is expressed as：
Wherein KuFor T1Phase magnetocrystalline anisotropy constant.
4th, the initial pattern of tissue is determined, the phase field model of ferromagnetism Crystal Evolution under magnetic field is set up.Program is write, is circulated
Calculate, obtain T1Evolutionary process of the crystal grain under magnetic field, is exported and analysis result.
Zeeman energy is to T as seen from Figure 11The facilitation of grain growth speed, and the bigger facilitation of external magnetic field is stronger
(T is the unit of external magnetic field strength).Magnetocrystalline anisotropy energy makes the magnetic moment T vertical with easy axis direction as seen from Figure 21Phase component
Volume fraction reduce, and make the magnetic moment T parallel with easy axis direction1Phase component increases, so as to cause magnetic moment inside ferromagnetic crystal grain
It is increasingly turned to easy axis direction, wherein η3Represent T of the magnetic moment direction perpendicular to easy axis direction1Phase component, η4Represent that magnetic moment direction is parallel
In the T of easy axis direction1Phase component, η3+η4For T1Phase total amount.Demagnetization can change T as can be seen from Figure 31Grain shape, makes its edge
Outer magnetic field direction is elongated.It can be seen that external magnetic field both promotes T from Fig. 4 a, Fig. 4 b1Grain growth, changing grain shape again makes it along magnetic
Field direction is elongated.
Claims (4)
1. a kind of phase field analysis method of ferromagnetism Crystal Evolution under magnetic field, it is characterised in that：This method is in classical phase field model
On the basis of, the magnetic energy that ferromagnetism crystal grain is produced under external magnetic field introduces system free energy, constructs for simulating iron
The universal model of magnetic crystal grain evolutionary process under external magnetic field, the universal model can be ground by introducing different magnetic energy
Study carefully the influence of each magnetic energy item respectively to ferromagnetism Crystal Evolution under different external magnetic field strengths, and total magnetic energy is drilled ferromagnetism crystal grain
The combined influence of change；The magnetic energy is zeeman energy, magnetocrystalline anisotropy energy and demagnetization field energy.
2. the phase field analysis method of ferromagnetism Crystal Evolution under a kind of magnetic field according to claim 1, it is characterised in that
Classical phase kinetics equation
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Wherein, η is S order parameter；T is time step；M is the intensity of magnetization；GsysFor system free energy；EmagCaused by external magnetic field
Magnetic energy.
3. the phase field analysis method of ferromagnetism Crystal Evolution under a kind of magnetic field according to claim 2, it is characterised in that institute
State the magnetic energy E that external magnetic field causes ferromagnetism crystal grainmagBy zeeman energy Ezeem, magnetocrystalline anisotropy energy EanisCan E with demagnetizationdComposition；
By zeeman energy Ezeem, magnetocrystalline anisotropy energy EanisCan E with demagnetizationdOne by one instead of the E in (3) formulamagDifferent magnetic can be drawn respectively
Influence of the energy item to ferromagnetism Crystal Evolution, by zeeman energy Ezeem, magnetocrystalline anisotropy energy EanisCan E with demagnetizationdPlus and substitute into
(3) formula can then draw combined influence of the external magnetic field to ferromagnetism crystal grain.
4. the phase field analysis method of ferromagnetism Crystal Evolution under a kind of magnetic field according to claim 3, it is characterised in that institute
State zeeman energy EzeemCan E with demagnetizationdIt is with the phase order parameter η modes connected
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<mrow>
<mo>-</mo>
<msub>
<mi>I</mi>
<mi>S</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>V</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>-</mo>
<mo>&dtri;</mo>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein φ (r) is magnetic potential,
<mrow>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>4</mn>
<msub>
<mi>&pi;&mu;</mi>
<mn>0</mn>
</msub>
</mrow>
</mfrac>
<munder>
<mo>&Integral;</mo>
<mi>V</mi>
</munder>
<mo>&lsqb;</mo>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>r</mi>
<mo>,</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>/</mo>
<mo>|</mo>
<msup>
<mi>r</mi>
<mo>,</mo>
</msup>
<mo>-</mo>
<mi>r</mi>
<mo>|</mo>
<mo>&rsqb;</mo>
<msup>
<mi>dV</mi>
<mo>,</mo>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein ρ (r) is magnetic charge density,
<mrow>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&mu;</mi>
<mn>0</mn>
</msub>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<msub>
<mi>M</mi>
<mi>S</mi>
</msub>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&mu;</mi>
<mn>0</mn>
</msub>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<msubsup>
<mi>&Sigma;M</mi>
<mi>s</mi>
<mrow>
<mi>f</mi>
<mi>e</mi>
<mi>r</mi>
<mi>r</mi>
<mi>o</mi>
</mrow>
</msubsup>
<msub>
<mi>&eta;</mi>
<mrow>
<mi>f</mi>
<mi>e</mi>
<mi>r</mi>
<mi>r</mi>
<mi>o</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Magnetocrystalline anisotropy energy EanisIntroducing rely on oriented S order parameter η=(ηx,ηy), oriented S order parameter η represents the magnetic of ferromagnetic phase
Square direction, η=ηx+ηyThen represent the ferromagnetic volume fraction in the point, wherein ηxMagnetic moments parallel is represented in the ferromagnetic phase point of x-axis
Amount, ηyMagnetic moments parallel is represented in the ferromagnetic phase component of y-axis, thus magnetocrystalline anisotropy energy EanisIt is represented by
<mrow>
<msub>
<mi>E</mi>
<mrow>
<mi>a</mi>
<mi>n</mi>
<mi>i</mi>
<mi>s</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>K</mi>
<mi>u</mi>
</msub>
<munder>
<mo>&Integral;</mo>
<mi>V</mi>
</munder>
<mo>&lsqb;</mo>
<mn>1</mn>
<mo>-</mo>
<msup>
<mrow>
<mo>{</mo>
<msub>
<mi>e</mi>
<mrow>
<mi>u</mi>
<mi>a</mi>
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>&eta;</mi>
<mo>}</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>&rsqb;</mo>
<mi>d</mi>
<mi>V</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein KuFor magnetocrystalline anisotropy constant；euan(r) it is the easy axis direction of ferromagnetic crystal grain.
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| CN115090860A (en) * | 2022-06-20 | 2022-09-23 | 上海大学 | Method for regulating peritectic phase volume fraction in peritectic alloy |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109991387A (en) * | 2019-03-06 | 2019-07-09 | 华南理工大学 | A kind of phase Field Analyze Method for simulating gamma-TiAl alloy under non-isothermal condition |
| CN115090860A (en) * | 2022-06-20 | 2022-09-23 | 上海大学 | Method for regulating peritectic phase volume fraction in peritectic alloy |
| CN115090860B (en) * | 2022-06-20 | 2023-08-18 | 上海大学 | Method for regulating and controlling volume fraction of peritectic phase in peritectic alloy |
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