CN103678890A - Method for simulating influence of heating technology on premelting and melting of crystal boundaries by aid of crystal phase-field process - Google Patents

Abstract

The invention relates to a method for simulating the influence of a heating technology on premelting and melting of crystal boundaries by the aid of a crystal phase-field process, and belongs to the field of technologies for simulating crystals. The method includes steps of (1), building a crystal phase-field model, in other words, building the crystal phase-field model on the basis of an idea of the traditional phase-field process, introducing a periodic function form into the model, and setting periodic partial-time average atomic densities as order parameters to obtain a free energy function expression of crystals; (2), determining a numerical process and calculation parameters. The method has the advantages that premelting and melting microstructure morphology at different heating rates is researched by the aid of a simulated calculation process and the crystal phase-field process, the widths of liquid-phase films at the crystal boundaries are quantitatively calculated by an excess mass process when the crystal boundaries are premelted and melted at the different heating rates, accordingly, the widths of the liquid-phase films at the crystal boundaries are quantified when the crystal boundaries are premelted and melted, structural relations among numerical values of the widths of the liquid-phase films and the atomic crystal boundary structures are illustrated, and an effect is perfect.

Description

A kind of method of crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact that adopt

Technical field

The present invention relates to a kind of method of crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact that adopt, belong to crystal analogue technique field.

Background technology

While approaching melting temperature, grain boundaries is pre-formed liquid phase film, and this phenomenon is called crystal boundary fritting (Grain Boundary Premelting.The appearance of fritting has important impact for mechanical behaviors such as the intensity of polycrystalline material and fracture etc., and for example, it can change the macro property of polycrystalline material, reduces significantly the resistance of shear strain.Especially the fire check of the latter when alloy material high temperature process, can produce catastrophic material failure behavior.Therefore, crystal boundary fritting and fusing are the focuses of investigation of materials all the time, and heating rate is one of key factor affecting fritting and fusing.

Because crystal boundary only has the thickness of several atomic layers conventionally, be difficult to directly observe by experiment, the study of computer simulation crystal boundary problem has unique advantage.As MD, Lennard-Jones gesture is used in research, and potential energy between semiconductor and metallic atom, proves that it exists crystal boundary nonsequential bed when the different temperatures of fusing point above and below.The people such as Tang use basic Cahn critical point to soak model, the intercrystalline film morphology of subsolidus while obtaining premelt transformation.Kikuchi and Cahn are making significant research equally aspect crystal boundary premelt, they use LATTICE GAS MODEL WITH and the pre-melted thermokinetics performance of reunion-variation approximate evaluation crystal boundary, and its result is confirmed by MC simulation subsequently.

Although above method can confirm and simulate grain boundaries liquid phase layer film, due to the restriction of these models itself, the width of research premelt layer that cannot be quantitative and the relation between its influence factor.In recent years, the first method of approach that PFC method is come as being developed by DFT, has irreplaceable advantage in the atomic scale density wave configuration aspects that solves crystalline material.

Summary of the invention

The object of the present invention is to provide a kind of method of crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact that adopt, so that adopting crystal phase field method studies under different heating rates, the microstructure morphology of fritting and fusing, and the liquid phase thin-film width of grain boundaries while adopting superfluous mass method quantitatively to calculate fritting under different heating rates and fusing.

To achieve these goals, technical scheme of the present invention is as follows.

Adopt the method for crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact, specifically comprise the following steps:

(1) set up crystal phase field model:

The thought of crystal phase field model based on traditional phase field method builds, periodic function form is introduced in model, order parameter adopts has periodic local time average atomic density, is write density as Dimensionless Form here, and the simplest dimensionless free energy functional equation can be write as

$F=\∫d\stackrel{\→}{r}\left\{\frac{\mathrm{\ρ}}{2}\right[-\mathrm{\ϵ}+{({\▿}^2+1)}^2]\mathrm{\ρ}+\frac{1}{4}{\mathrm{\ρ}}^4\}---\left(1\right)$

This is the transformation of SH model, and the dimensionless functional is here relevant to classical density functional theory; Wherein F is free energy function, and ρ is atomic density, and ε is the degree of supercooling of system, the high higher temperature of ε correspondence; In PFC model, defining dimensionless chemistry gesture is

$\mathrm{\μ}=\frac{\mathrm{\δF}}{\mathrm{\δ\ρ}}=-\mathrm{\ϵ\ρ}+{({\▿}^2+1)}^2\mathrm{\ρ}+{\mathrm{\ρ}}^3---\left(2\right)$

In two-dimentional system, free energy function is obtained minimum by the solution of ρ (x, y), thus, can obtain the solution of the dimensionless local density of crystalline state in two-dimensional system (solid-state) hexagonal item

${\mathrm{\ρ}}_s(x,y)=\mathrm{\ρ}+A_t[\cos \left(\mathrm{qx}\right)\cos \left(\frac{\mathrm{qy}}{\sqrt{3}}\right)-\frac{1}{2}\cos \left(\frac{2\mathrm{qy}}{\sqrt{3}}\right)]---\left(3\right)$

$A_t=\frac{4}{5}(\mathrm{\ρ}\±\frac{1}{3}\sqrt{15\mathrm{\ϵ}-36{\mathrm{\ρ}}^2})---\left(4\right)$

Here ± represent respectively positive and negative ρ,

By (4) formula substitution (3) formula, (3) formula is brought (1) formula into, obtains the free energy function expression formula of crystal.

(2) determine numerical method and calculating parameter:

The canonical equation of motion of PFC model is

${\∂}_t\mathrm{\ρ}={\▿}^2\left(\frac{\mathrm{\δF}}{\mathrm{\δ\ρ}}\right)=(1-\mathrm{\ϵ}){\▿}^2\mathrm{\ρ}+2{\▿}^4\mathrm{\ρ}+{\▿}^6\mathrm{\ρ}+{\▿}^2{\mathrm{\ρ}}^3---\left(5\right)$

Formula (5) shows that density field ρ is local conservation law variable; Solid phase is different with density of liquid phase, and they reach equilibrium state separately by self-diffusion; For the Laplace operator in kinetics equation, adopt half hidden spectral method to solve, its discrete form is

$\frac{{\mathrm{\ρ}}_{\stackrel{\→}{k},t+\mathrm{\Δt}}-{\mathrm{\ρ}}_{\stackrel{\→}{k},t}}{\mathrm{\Δt}}=[(r-1)+2k^2-k^4]{\mathrm{\ρ}}_{\stackrel{\→}{k},t+\mathrm{\Δt}}+{\left({\mathrm{\ρ}}^3\right)}_{\stackrel{\→}{k},t}---\left(6\right)$

In formula, density is discrete turns to ${\mathrm{\ρ}}_{\stackrel{\→}{k},t}=\∫d\stackrel{\→}{x}{e}^{-i\stackrel{\→}{k}\·\stackrel{\→}{x}}{\mathrm{\ρ}}_t\left(\stackrel{\→}{x}\right),{\left({\mathrm{\ρ}}^3\right)}_{\stackrel{\→}{k},t}=\∫d\stackrel{\→}{x}{e}^{-i\stackrel{\→}{k}\·\stackrel{\→}{x}}{\mathrm{\ρ}}_t^3\left(\stackrel{\→}{x}\right),$ for Fourier space wave vector, and ${\stackrel{\→}{k}}^2={\left|\stackrel{\→}{k}\right|}^2.$

This beneficial effect of the invention is: the present invention adopts simulation method, adopt crystal phase field method to study under different heating rates, the microstructure morphology of fritting and fusing, and the liquid phase thin-film width of grain boundaries while adopting superfluous mass method quantitatively to calculate fritting under different heating rates and fusing, liquid phase thin-film width while having quantized to calculate crystal boundary premelt and fusing, and illustrate this numerical value and atom grain boundary structure relation, effect is ideal.

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the specific embodiment of the present invention is described, to better understand the present invention.

Fig. 1 is heating rate k=4 * 10 in the embodiment of the present invention ^-6under, the atom shape appearance figure of low angle boundary θ=9.6, ε=-0.1 °.

Fig. 2 is heating rate k=4 * 10 in the embodiment of the present invention ^-8under, ε=-0.1, the atom shape appearance figure of low angle boundary θ=9.6 °.

Fig. 3 is heating rate k=4 * 10 in the embodiment of the present invention ^-6under, ε=-0.1, the atomic density distribution plan of low angle boundary θ=9.6 °.

Fig. 4 is heating rate k=4 * 10 in the embodiment of the present invention ^-8under, ε=-0.1, the atomic density distribution plan of low angle boundary θ=9.6 °.

Fig. 5 is different heating rates in the embodiment of the present invention, the liquid phase thin-film width-thetagram of low angle boundary θ=9.6 °.

Fig. 6 is low angle boundary θ=6.23 ° different heating rate later stage holding stage liquid phase thin-film width-thetagrams in the embodiment of the present invention.

Fig. 7 is low angle boundary θ=9.6 ° different heating rate later stage holding stage liquid phase thin-film width-thetagrams in the embodiment of the present invention.

Fig. 8 is heating rate k=4 * 10 in the embodiment of the present invention ^-6under, ε=-0.1, the atom shape appearance figure of crystal boundary θ=32.5, big angle °.

Fig. 9 is heating rate k=4 * 10 in the embodiment of the present invention ^-8under, ε=-0.1, the atom shape appearance figure of crystal boundary θ=32.5, big angle °.

Figure 10 is in the embodiment of the present invention under different heating rates, ε=-0.1, the atomic density distribution plan of crystal boundary θ=32.5, big angle °.

Figure 11 is in the embodiment of the present invention under different heating rates, the liquid phase thin-film width-thetagram of crystal boundary θ=32.5, big angle °.

Figure 12 is big angle crystal boundary θ=12.5 ° different heating rate later stage holding stage liquid phase thin-film width-thetagrams in the embodiment of the present invention.

Figure 13 is big angle crystal boundary θ=32.5 ° different heating rate later stage holding stage liquid phase thin-film width-thetagrams in the embodiment of the present invention.

Figure 14 is k=4 * 10 in the embodiment of the present invention ^-8, the liquid phase thin-film width-thetagram of each angle.

Figure 15 is the phase-field model Cu-Ag alloy crystal boundary fritting width curve map of group as a comparison of quoting in the embodiment of the present invention.

Embodiment

Embodiment

Adopt the method for crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact, specifically comprise the following steps:

(1) set up crystal phase field model:

The thought of crystal phase field model based on traditional phase field method builds, periodic function form is introduced in model, order parameter adopts has periodic local time average atomic density, is write density as Dimensionless Form here, and the simplest dimensionless free energy functional equation can be write as

$F=\∫d\stackrel{\→}{r}\left\{\frac{\mathrm{\ρ}}{2}\right[-\mathrm{\ϵ}+{({\▿}^2+1)}^2]\mathrm{\ρ}+\frac{1}{4}{\mathrm{\ρ}}^4\}---\left(1\right)$

This is the transformation of SH model, and the dimensionless functional is here relevant to classical density functional theory; Wherein F is free energy function, and ρ is atomic density, and ε is the degree of supercooling of system, the high higher temperature of ε correspondence; In PFC model, defining dimensionless chemistry gesture is

$\mathrm{\μ}=\frac{\mathrm{\δF}}{\mathrm{\δ\ρ}}=-\mathrm{\ϵ\ρ}+{({\▿}^2+1)}^2\mathrm{\ρ}+{\mathrm{\ρ}}^3---\left(2\right)$

In two-dimentional system, free energy function is obtained minimum by the solution of ρ (x, y), thus, can obtain the solution of the dimensionless local density of crystalline state in two-dimensional system (solid-state) hexagonal item

${\mathrm{\ρ}}_s(x,y)=\mathrm{\ρ}+A_t[\cos \left(\mathrm{qx}\right)\cos \left(\frac{\mathrm{qy}}{\sqrt{3}}\right)-\frac{1}{2}\cos \left(\frac{2\mathrm{qy}}{\sqrt{3}}\right)]---\left(3\right)$

$A_t=\frac{4}{5}(\mathrm{\ρ}\±\frac{1}{3}\sqrt{15\mathrm{\ϵ}-36{\mathrm{\ρ}}^2})---\left(4\right)$

Here ± represent respectively positive and negative ρ,

By (4) formula substitution (3) formula, (3) formula is brought (1) formula into, obtains the free energy function expression formula of crystal.

(2) determine numerical method and calculating parameter:

The canonical equation of motion of PFC model is

${\∂}_t\mathrm{\ρ}={\▿}^2\left(\frac{\mathrm{\δF}}{\mathrm{\δ\ρ}}\right)=(1-\mathrm{\ϵ}){\▿}^2\mathrm{\ρ}+2{\▿}^4\mathrm{\ρ}+{\▿}^6\mathrm{\ρ}+{\▿}^2{\mathrm{\ρ}}^3---\left(5\right)$

Formula (5) shows that density field ρ is local conservation law variable; Solid phase is different with density of liquid phase, and they reach equilibrium state separately by self-diffusion; For the Laplace operator in kinetics equation, adopt half hidden spectral method to solve, its discrete form is

$\frac{{\mathrm{\ρ}}_{\stackrel{\→}{k},t+\mathrm{\Δt}}-{\mathrm{\ρ}}_{\stackrel{\→}{k},t}}{\mathrm{\Δt}}=[(r-1)+2k^2-k^4]{\mathrm{\ρ}}_{\stackrel{\→}{k},t+\mathrm{\Δt}}+{\left({\mathrm{\ρ}}^3\right)}_{\stackrel{\→}{k},t}---\left(6\right)$

In formula, density is discrete turns to ${\mathrm{\ρ}}_{\stackrel{\→}{k},t}=\∫d\stackrel{\→}{x}{e}^{-i\stackrel{\→}{k}\stackrel{\→}{x}}{\mathrm{\ρ}}_t\left(\stackrel{\→}{x}\right),{\left({\mathrm{\ρ}}^3\right)}_{\stackrel{\→}{k},t}=\∫d\stackrel{\→}{x}{e}^{-i\stackrel{\→}{k}\·\stackrel{\→}{x}}{\mathrm{\ρ}}_t^3\left(\stackrel{\→}{x}\right),$ for Fourier space wave vector, and ${\stackrel{\→}{k}}^2={\left|\stackrel{\→}{k}\right|}^2.$

In the present embodiment, time and space step-length that simulation adopts are Δ x=Δ y=0.5, Δ t=0.0075, and boundary condition is periodic boundary condition, zoning is 512 Δ x * 512 Δ y, the 0<x<L in zoning _x/ 4 and 3L _x/ 4<x<L _xplace adopts the ρ that is oriented to θ/2, at L _x/ 4<x<3L _x/ 4 places adopt the ρ of be oriented to-θ/2, have obtained so the poor symmetrical canting crystal boundary of different orientation.

In the present embodiment, mainly study the relation of the rate of heat addition, orientation declinate and liquid phase film width, choose 6.2 ° of many group orientation declinates, 8.23 °, 9.6 °, 12.5 °, 17.5 °, and 32.5 ° carried out analog computation.Choose ρ ₀=-0.196; At the simulation initial stage, definition ε=-0.3, system, in solid-state, defines ε=-0.1 subsequently, and system is the increase along with the time of evolution close to fusing point (fusing point of pure material is ε=-0.09), occurs fritting or melting phenomenon in system.In embodiment, select different heating rate (k=4 * 10 ^-6, k=4 * 10 ^-7, k=4 * 10 ^-8) make temperature be elevated to ε=-0.1 from ε=-0.3, then allow system at this temperature, be incubated 150000 steps, the impact of research heating rate on fritting and fusion process.

Concrete outcome is as follows:

(1) impact of heating rate on low-angle boundary fritting and fusing: when Fig. 1, Fig. 2, Fig. 3, Fig. 4 are respectively low-angle boundary θ=9.6 °, adopt different heating rates, while being warmed up to uniform temp, atomic density crystal boundary shape appearance figure and corresponding atomic density distribution plan.As can be seen from the figure, on low angle boundary, occur by independent liquid phase region around independent dislocation, the distance of some grating constant sizes of being separated by between them, heating rate is less, the size of the droplet of formation is larger, as shown in Figure 2.Fig. 3 and Fig. 4 are atomic density distribution plan, and its figure presents vibration mode, and the density peaks of crystal boundary area, lower than internal system atomic density, illustrates and now necessarily occur liquid phase region, and disordering appears in atom.Along with atom present position is gradually near crystal boundary, density function oscillation amplitude increases.This is because the Burgers direction vector of the adjacent independent dislocation of grain boundaries is different, and is evenly distributed in the both sides of crystal boundary, so that occurs above-mentioned phenomenon.Because dislocation exists the lattice distortion causing, cause system capacity in body to increase, so dislocation atom dislocation around moves along crystal boundary, until the strain energy of dislocation reaches minimum value, this just shows as adjacent dislocation merging, volumetric stress has relaxed.Heating rate is larger, and smectic battery limit (BL) atom oscillation amplitude is larger.

The present embodiment adopts the theoretical crystal boundary liquid phase width that calculates of the superfluous quality of Mellenthin, this theoretical advantage is: it can calculate the liquid phase film width under various complicated Grain-boundary structures accurately, has solved the shortcoming that bright and sharp interface theory can only calculate solid-liquid interface liquid phase film width when completely separated.In the present embodiment, the system gross mass of simulation is definite, and the chemical potential that superfluous quality theory can be converted into grain boundaries changes to define film thickness, and the computing formula of liquid phase film width is:

$w=\frac{L_y[\mathrm{\&mu;}-{\mathrm{\&mu;}}_s\left(\mathrm{\&rho;}\right)]}{{\mathrm{\&mu;}}_l\left(\mathrm{\&rho;}\right)-{\mathrm{\&mu;}}_s\left(\mathrm{\&rho;}\right)}---\left(7\right)$

μ _l(ρ) and μ _s(ρ) be respectively that system approaches equilibrium state, the chemical potential of liquid phase and solid phase during equal densities.μ is the total chemical potential of system, L _yfor the boundary length of vertical grain boundaries in simulated domain, grating constant

Fig. 5 is under different heating rates, the liquid phase thin-film width variation with temperature curve of low-angle boundary, and along with temperature raises, grain boundaries liquid phase thin-film width also increases thereupon, and the trend comparison that starts increase is mild, increases fast afterwards.When temperature is lower, heating rate does not almost affect the width of liquid phase film, because low angle boundary consists of the network of dislocation, and complex structure, during low temperature, atom diffusion ratio is slower, so heating rate is less on the impact of grain boundaries liquid phase region.When temperature approaches fusing point, liquid phase thin-film width increases rapidly, under uniform temp, heating rate is less, and the width of liquid phase film is larger, when this is because approaches fusing point, atomic thermal motion aggravation, heating rate is less, and atom can spread more fully, while being therefore warming up to uniform temp, its liquid phase thin-film width is larger.

Fig. 6 and Fig. 7 are holding stage, and under different heating rates, over time, in initial insulating process, the system liquid phase thin-film width that heating rate is large is less than the system that heating rate is little to liquid phase film width, and this changes consistent with temperature rise period liquid phase thin-film width.Prolongation along with temperature retention time, the system liquid phase thin-film width that heating rate is large surmounts the system that heating rate is little, this is because larger at temperature rise period heating rate, atom diffusion is insufficient, the existence of grain boundaries dislocation causes lattice distortion, quite high distortional strain energy is stored, and the interior of system can be raise.System is in thermodynamic instability state, there is the trend that changes to reduce free energy, the strain energy of dislocation becomes the driving force of holding stage diffusion, therefore system temperature rise period heating rate is larger, holding stage atom driving force is larger, the driving force of droplet coalescence is larger, so liquid phase film width is just larger.

(2) impact of heating rate on high-angle boundary fritting and fusing:

When Fig. 8, Fig. 9, Figure 10 are respectively high-angle boundary θ=32.5 °, adopt different heating rates, while being warmed up to uniform temp, atomic density crystal boundary shape appearance figure and corresponding atomic density distribution plan.As can be seen from the figure, the liquid phase film being comprised of fuzzy lattice structure in atom shape appearance figure, forms " pipeline " that be communicated with.Heating rate is less, and the atom randomness peak width of grain boundaries is larger.Atomic density distribution plan does not present the vibration mode (shown in Figure 10) as low-angle boundary, but occurs obvious trough.Heating rate is larger, and the atom average density in intracrystalline district is less, and the atom average density in crystal boundary area is larger.Heating rate is larger, and under uniform temp, grain boundaries liquid phase layer width is less.

Figure 11 is under different heating rates, the liquid phase thin-film width variation with temperature curve of high-angle boundary, and figure is similar with low angle boundary width, and the trend comparison that starts increase is mild, increases fast afterwards.When temperature is lower, heating rate is on the almost not impact of the width of liquid phase film, and when temperature approaches fusing point, liquid phase thin-film width increases rapidly, and under uniform temp, heating rate is larger, and the width of liquid phase film is less.

Figure 12 and Figure 13 are holding stage, and under different heating rates, wide-angle liquid phase film width over time.Figure is different with low angle boundary holding stage liquid phase width, and in whole insulating process, heating rate is less, and grain boundaries liquid phase thin-film width is larger.Because big angle crystal boundary is in the temperature rise period, liquid phase is evenly communicated with distribution along crystal boundary, and this motion for hot activation atom provides passage easily, and therefore diffusion is more abundant, and heating rate is less, and liquid phase width is larger.At holding stage subsequently, by means of wider diffusion admittance previous stage, the system that heating rate is less, the liquid phase width of holding stage is still greater than the system that heating rate is larger.

Figure 14 is k=4 * 10 in the embodiment of the present invention ^-8, the liquid phase thin-film width-thetagram of each angle.As shown in Figure 14, rising with temperature, place, a plurality of crystal boundaries angle liquid phase thin-film width that the present invention simulates all constantly increases, and at the same temperature, orientation declinate is larger, and liquid phase thin-film width is larger, and it is theoretical that this curve meets Landau shot-range interaction, along with the rising of temperature, liquid phase thin-film width is dispersed when approaching fusing point.Figure 15 is the phase-field model Cu-Ag alloy crystal boundary fritting width curve map of group as a comparison of quoting in the embodiment of the present invention.The present invention adopt crystal boundary liquid phase thin-film width that superfluous quality method calculates and said method curve obtained quite similar, verified accuracy of the present invention.

The above is the preferred embodiment of the present invention; it should be pointed out that for those skilled in the art, under the premise without departing from the principles of the invention; can also make some improvements and modifications, these improvements and modifications are also considered as protection scope of the present invention.

Claims (1)

1. adopt the method for crystal phase field method simulation heating technique on crystal boundary fritting and fusing impact, it is characterized in that: described method specifically comprises the following steps:

(1) set up crystal phase field model:

The thought of crystal phase field model based on traditional phase field method builds, and periodic function form is introduced in model, and order parameter adopts has periodic local time average atomic density, and dimensionless free energy functional equation can be write as

$F=\&Integral;d\stackrel{\&RightArrow;}{r}\left\{\frac{\mathrm{\&rho;}}{2}\right[-\mathrm{\&epsiv;}+{({\&dtri;}^2+1)}^2]\mathrm{\&rho;}+\frac{1}{4}{\mathrm{\&rho;}}^4\}---\left(1\right)$

This is the transformation of SH model, and the dimensionless functional is here relevant to classical density functional theory; Wherein F is free energy function, and ρ is atomic density, and ε is the degree of supercooling of system; In PFC model, defining dimensionless chemistry gesture is

$\mathrm{\&mu;}=\frac{\mathrm{\&delta;F}}{\mathrm{\&delta;\&rho;}}=-\mathrm{\&epsiv;\&rho;}+{({\&dtri;}^2+1)}^2\mathrm{\&rho;}+{\mathrm{\&rho;}}^3---\left(2\right)$

In two-dimentional system, free energy function is obtained minimum by the solution of ρ (x, y), thus, can obtain the solution of the dimensionless local density of crystalline state in two-dimensional system (solid-state) hexagonal item

${\mathrm{\&rho;}}_s(x,y)=\mathrm{\&rho;}+A_t[\cos \left(\mathrm{qx}\right)\cos \left(\frac{\mathrm{qy}}{\sqrt{3}}\right)-\frac{1}{2}\cos \left(\frac{2\mathrm{qy}}{\sqrt{3}}\right)]---\left(3\right)$

$A_t=\frac{4}{5}(\mathrm{\&rho;}\&PlusMinus;\frac{1}{3}\sqrt{15\mathrm{\&epsiv;}-36{\mathrm{\&rho;}}^2})---\left(4\right)$

Here ± represent respectively positive and negative ρ,

by (4) formula substitution (3) formula, (3) formula is brought (1) formula into, obtains the free energy function expression formula of crystal;

(2) determine numerical method and calculating parameter:

The canonical equation of motion of PFC model is

${\&PartialD;}_t\mathrm{\&rho;}={\&dtri;}^2\left(\frac{\mathrm{\&delta;F}}{\mathrm{\&delta;\&rho;}}\right)=(1-\mathrm{\&epsiv;}){\&dtri;}^2\mathrm{\&rho;}+2{\&dtri;}^4\mathrm{\&rho;}+{\&dtri;}^6\mathrm{\&rho;}+{\&dtri;}^2{\mathrm{\&rho;}}^3---\left(5\right)$

Formula (5) shows that density field ρ is local conservation law variable; Solid phase is different with density of liquid phase, and they reach equilibrium state separately by self-diffusion; For the Laplace operator in kinetics equation, adopt half hidden spectral method to solve, its discrete form is

$\frac{{\mathrm{\&rho;}}_{\stackrel{\&RightArrow;}{k},t+\mathrm{\&Delta;t}}-{\mathrm{\&rho;}}_{\stackrel{\&RightArrow;}{k},t}}{\mathrm{\&Delta;t}}=[(r-1)+2k^2-k^4]{\mathrm{\&rho;}}_{\stackrel{\&RightArrow;}{k},t+\mathrm{\&Delta;t}}+{\left({\mathrm{\&rho;}}^3\right)}_{\stackrel{\&RightArrow;}{k},t}---\left(6\right)$

In formula, density is discrete turns to ${\mathrm{\&rho;}}_{\stackrel{\&RightArrow;}{k},t}=\&Integral;d\stackrel{\&RightArrow;}{x}{e}^{-i\stackrel{\&RightArrow;}{k}\stackrel{\&RightArrow;}{x}}{\mathrm{\&rho;}}_t\left(\stackrel{\&RightArrow;}{x}\right),{\left({\mathrm{\&rho;}}^3\right)}_{\stackrel{\&RightArrow;}{k},t}=\&Integral;d\stackrel{\&RightArrow;}{x}{e}^{-i\stackrel{\&RightArrow;}{k}\&CenterDot;\stackrel{\&RightArrow;}{x}}{\mathrm{\&rho;}}_t^3\left(\stackrel{\&RightArrow;}{x}\right),$

for Fourier space wave vector, and ${\stackrel{\&RightArrow;}{k}}^2={\left|\stackrel{\&RightArrow;}{k}\right|}^2.$

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