CN102156184B - Method for predicting space between aluminium-silicon alloy eutectic structure lamellas - Google Patents

Abstract

The invention relates to a method for predicting space between aluminium-silicon alloy eutectic structure lamellas; the method is characterized in that an experimental period can be greatly shortened by using a numerical simulation method to predict the space between aluminium-silicon alloy eutectic lamellas; the cost is low, and the prediction result is not limited by the experimental conditions; and the influences of a certain single factor to the space between the aluminium-silicon alloy eutectic structure lamellas can be inspected. The shortages that the cost of the existing experimental method for measuring the space between aluminium-silicon alloy eutectic structure lamellas is relatively high, the experimental period is long and the experiment results are influenced obviously by the experimental conditions are overcome; the computation is fast; the error is small; and the quantitative prediction to the space between eutectic lamellas can be realized.

Description

Method for predicting space between aluminium-silicon alloy eutectic structure lamellas

One, technical field:

The invention belongs to metallurgical technology field, relate generally to a kind of Forecasting Methodology of eutectic structure sheet interlayer spacing, particularly for the Forecasting Methodology of Alpax eutectic structure sheet interlayer spacing.

Two, background technology:

Alpax is one of most widely used aluminium alloy, and it is widely used in the fields such as automobile, electric power, aviation.Casting is a main method of Alpax moulding.But in casting process, al-si eutectic tissue is a kind of typical Solidification Microstructure Morphology.The sheet interlayer spacing of al-si eutectic tissue is a principal element that is directly connected to cast product mechanical property, Eutectic spacing for Alpax adopted experimental technique to measure more in the past, but the sheet interlayer spacing cost of determination of experimental method al-si eutectic tissue is higher, experimental period is long, and experimental result is affected significantly by experiment condition.

Three, summary of the invention:

1, goal of the invention:

The present invention is directed to the less present situation of Al-Si eutectic structure sheet interlayer spacing Quantitative study, a kind of method for predicting space between aluminium-silicon alloy eutectic structure lamellas is proposed, solve the existing problem of current determination of experimental method al-si eutectic tissue lamellar spacing, for Alpax eutectic structure performance prediction from now on provides theoretical direction, and can be applied directly to foundry method improvement, realize the sheet interlayer spacing of refinement al-si eutectic tissue.

2, technical scheme:

The present invention is achieved through the following technical solutions:

A method for predicting space between aluminium-silicon alloy eutectic structure lamellas, is characterized in that: the eutectic structure sheet interlayer spacing after doping Alpax under this cooling velocity and thermograde and solidify according to the cooling velocity of actual measurement and local thermograde, and step is as follows:

(1), set up Alpax eutectic structure sheet interlayer spacing forecast model:

First true origin QuαXiang center, determines the coordinate of y and z therefrom.Y axle is positioned in solid-liquid interface, and vertical with lamella.Problem reduction is become to two-dimensional state, and due to the symmetry of diffusion, its solution is in region, 0≤y≤λ/2.Solid-liquid interface is with speed stable growth in z direction of V, and diffusion equation is now:

$\frac{{\&PartialD;}^{2}C}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}+\frac{{\&PartialD;}^{2}C}{{\&PartialD;z}^2}+\frac{V}{D}\&CenterDot;\frac{\&PartialD;C}{\&PartialD;z}=0---\left(1\right)$

Wherein, C is silicon solute concentration, and D is the coefficient of diffusion of solute in liquid phase.

Due in process of setting, temperature can have a certain impact to the solute coefficient of diffusion of liquid phase, therefore will predict accurately Eutectic spacing, temperature must be taken into account the impact of solute coefficient of diffusion.

(2), calculate solute coefficient of diffusion:

$D={\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">D</figure-callout>}}_0\exp (-\frac{Q}{R_{g}T})---\left(2\right)$

D in formula ₀---equilibrium state solute coefficient of diffusion;

Q---atom diffusion activation energy;

R _g---air constant;

T---temperature.

Due to D here ₀, Q and R _gbe constant, wherein D ₀=8 * 10 ^-9;

these two data substitution formula (2) are obtained to solute coefficient of diffusion is:

(3), calculate eutectic structure sheet interlayer spacing λ:

According to the periodic change condition of eutectic composition, can obtain:

$C=C_e+\mathrm{Aexp}(-\frac{V_Z}{D})+\mathrm{Bexp}(-\frac{2\mathrm{\&pi;Z}}{\mathrm{\&lambda;}})\cos \left(\frac{2\mathrm{\&pi;y}}{\mathrm{\&lambda;}}\right)---\left(3\right)$

Wherein, C _efor balance eutectic composition, V _zfor the eutectic growth speed in z direction, A, B are constants, and it is worth by formula (4), and (5) calculate:

A＝fC′-C ^β (4)

$B=\frac{f(1-f){\mathrm{V\&lambda;C}}^{\&prime;}}{2D\sin \left(\mathrm{\&pi;f}\right)}---\left(5\right)$

Wherein C ' is eutectic line length.

(4), the constitutional supercooling degree of α phase and β phase

with

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]---\left(6\right)$

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]---\left(7\right)$

Wherein, m _αfor the slope of α phase liquidus curve, m _βslope for β phase liquidus curve.In process of setting, the speed of growth at interface exerts an influence to solute distribution coefficient, adopts in the method its expression formula of solute redistribution coefficient of continuity model calculating α phase and β phase as follows:

$m_{\mathrm{\&alpha;}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">m</figure-callout>}}_{0\mathrm{\&alpha;}}(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+({\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+Y)\ln \left(\frac{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+\mathrm{<figure-callout\; id="0"\; label="Y\; k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">Y</figure-callout>}}{k_{}}\right))}{(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}})(1+Y)}---\left(8\right)$

$m_{\mathrm{\&beta;}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">m</figure-callout>}}_{0\mathrm{\&beta;}}(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}}+({\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}}+Y)\ln \left(\frac{k_{0\mathrm{\&beta;}}+\mathrm{<figure-callout\; id="0"\; label="Y\; k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">Y</figure-callout>}}{k_{}}\right))}{(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}})(1+Y)}---\left(9\right)$

Wherein $Y=\frac{\mathrm{R\&delta;}}{D_L},$

M _{0 α}---equilibrium state α phase liquidus curve slope;

M _{0 β}---equilibrium state β phase liquidus curve slope;

K _0a---nonequilibrium condition solute redistribution coefficient;

K _{0 β}---nonequilibrium condition solute redistribution coefficient;

D _l---non-equilibrium solute coefficient of diffusion.

(5), in eutectic growth process, the curvature of α phase and β phase is excessively cold

with

for:

${\mathrm{\&Delta;T}}_r^{\mathrm{\&alpha;}}=\frac{2{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(10\right)$

${\mathrm{\&Delta;T}}_r^{\mathrm{\&beta;}}=\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(11\right)$

Wherein, θ _αfor the contact angle of α phase solid-liquid phase, θ _βfor the contact angle of β phase solid-liquid phase, Γ _αfor α phase Gibbs-Thomson free energy coefficient Γ _βfor β phase Gibbs-Thomson free energy coefficient.

(6), the degree of supercooling of each phase is:

${\mathrm{\&Delta;T}}_{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(12\right)$

${\mathrm{\&Delta;T}}_{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(13\right)$

By Δ T _α| m _β|+Δ T _β| m _α|, Δ T _α=Δ T _β=Δ T obtains:

$\mathrm{\&Delta;T}=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\mathrm{V\&lambda;}\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}+\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\mathrm{\&lambda;}\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(14\right)$

In eutectic growth process, degree of supercooling can be write as:

$\mathrm{\&Delta;T}=K_c\mathrm{V\&lambda;}+\frac{K_r}{\mathrm{\&lambda;}}---\left(15\right)$

Wherein

$K_c=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\&CenterDot;\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}---\left(16\right)$

$K_r=\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(17\right)$

According to equation (15), λ is asked to local derviation, the λ value while obtaining minimum subcooled temperature:

$\frac{d\left(\mathrm{\&Delta;T}\right)}{\mathrm{d\&lambda;}}=K_{c}V-\frac{K_r}{{\mathrm{\&lambda;}}^2}=0---\left(18\right)$

(16) and (17) are brought in equation (18) and can be obtained:

${\mathrm{\&lambda;}}^{2}V=\frac{K_r}{K_c}=\frac{4\mathrm{\&pi;D}}{f(1-f)}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(19\right)$

${\mathrm{\&lambda;}}^{2}V=\frac{2D}{p^{\&prime;}}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(20\right)$

Wherein

$K_c=\frac{\stackrel{\&OverBar;}{m}{C}^{\&prime;}{P}^{\&prime;}}{f(1-f)D}---\left(21\right)$

$K_r=2\stackrel{\&OverBar;}{m}[\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{f\left|m_{\mathrm{\&alpha;}}\right|}+\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\left|m_{\mathrm{\&beta;}}\right|}]---\left(22\right)$

Wherein, P '=Σ (1/n ³π ³) sin ²(n π f),

By above-mentioned equation, can draw the al-si eutectic sheet interlayer spacing shown in Fig. 1 and Fig. 2 and degree of supercooling and speed of growth relation.

3, advantage and effect:

A kind of method for predicting space between aluminium-silicon alloy eutectic structure lamellas that the present invention proposes, the method tool has the following advantages:

Adopt method for numerical simulation prediction Alpax Eutectic spacing greatly to shorten experimental period, and cost is low, predicts the outcome and not limited by experiment condition, and can investigates the impact of a certain single technological factor on Alpax eutectic structure sheet interlayer spacing.Calculate fast, error is less, observes by experiment with result of calculation contrast and finds that the relative error of the method is less than 5%, can realize the quantification of Eutectic spacing is predicted.

Four, accompanying drawing explanation:

Fig. 1 is that Al-Si Eutectic spacing and degree of supercooling and speed of growth Δ T-λ-V are related to schematic diagram;

Fig. 2 is that the theoretical λ-V of Al-Si eutectic is related to schematic diagram.

Five, embodiment:

Eutectic structure is a kind of the most common Solidification Microstructure Morphology, and wherein the sheet interlayer spacing of eutectic structure pattern and eutectic structure is the principal element of the final eutectic cast properties of impact.Alpax is the most frequently used casting alloy, and it has good casting character, is therefore widely used in the fields such as automobile, electric power, machinery.According to classical Hall-Pitch formula, the size of crystal grain and the relation of mechanical property are known, and crystal grain is more tiny, and its mechanical property is higher.But in the eutectic structure of solidifying, Eutectic spacing is more tiny, the mechanical property of eutectic structure is better.Therefore refining eutectic sheet interlayer spacing is research and main contents controlling eutectic structure pattern.

Below in conjunction with specific embodiment, the present invention is described further:

A method for predicting space between aluminium-silicon alloy eutectic structure lamellas, is characterized in that: the eutectic structure sheet interlayer spacing after doping Alpax under this cooling velocity and thermograde and solidify according to the cooling velocity of actual measurement and local thermograde, and step is as follows:

(1), set up silicon eutectic structure sheet interlayer spacing forecast model:

First true origin QuαXiang center, determines the coordinate of y and z therefrom.Y axle is positioned in solid-liquid interface, and vertical with lamella.Problem reduction is become to two-dimensional state, and due to the symmetry of diffusion, its solution is in region, 0≤y≤λ/2.Solid-liquid interface is with speed stable growth in z direction of V, and diffusion equation is now:

$\frac{{\&PartialD;}^{2}C}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}+\frac{{\&PartialD;}^{2}C}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">z</figure-callout>}}^2}+\frac{V}{D}\&CenterDot;\frac{\&PartialD;C}{\&PartialD;z}=0---\left(1\right)$

Wherein, C is silicon solute concentration, and D is the coefficient of diffusion of solute in liquid phase.

Due in process of setting, temperature can have a certain impact to the solute coefficient of diffusion of liquid phase, therefore will predict accurately Eutectic spacing, temperature must be taken into account the impact of solute coefficient of diffusion.

(2), calculate solute coefficient of diffusion:

$D={\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">D</figure-callout>}}_0\exp (-\frac{Q}{R_{g}T})---\left(2\right)$

D in formula ₀---equilibrium state solute coefficient of diffusion;

Q---atom diffusion activation energy;

R _g---air constant;

T---temperature.

Due to D here ₀, Q and R _gbe constant, wherein D ₀=8 * 10 ^-9;

these two data substitution formula (2) are obtained to solute coefficient of diffusion is:

(3), calculate eutectic structure sheet interlayer spacing λ:

According to the periodic change condition of eutectic composition, can obtain:

$C=C_e+\mathrm{Aexp}(-\frac{V_Z}{D})+\mathrm{Bexp}(-\frac{2\mathrm{\&pi;Z}}{\mathrm{\&lambda;}})\cos \left(\frac{2\mathrm{\&pi;y}}{\mathrm{\&lambda;}}\right)---\left(3\right)$

Wherein, C _efor balance eutectic composition, V _zfor the eutectic growth speed in z direction, A, B are constants, and it is worth by formula (4), and (5) calculate:

A＝fC′-C ^β (4)

$B=\frac{f(1-f){\mathrm{V\&lambda;C}}^{\&prime;}}{2D\sin \left(\mathrm{\&pi;f}\right)}---\left(5\right)$

Wherein C ' is eutectic line length.

(4), the constitutional supercooling degree of α phase and β phase

with

for:

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]---\left(6\right)$

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]---\left(7\right)$

Wherein, m _αfor the slope of α phase liquidus curve, m _βslope for β phase liquidus curve.In process of setting, the speed of growth at interface exerts an influence to solute distribution coefficient, adopts in the method its expression formula of solute redistribution coefficient of continuity model calculating α phase and β phase as follows:

$m_{\mathrm{\&alpha;}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">m</figure-callout>}}_{0\mathrm{\&alpha;}}(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+({\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+Y)\ln \left(\frac{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}}+\mathrm{<figure-callout\; id="0"\; label="Y\; k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">Y</figure-callout>}}{k_{}}\right))}{(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&alpha;}})(1+Y)}---\left(8\right)$

$m_{\mathrm{\&beta;}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">m</figure-callout>}}_{0\mathrm{\&beta;}}(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}}+({\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}}+Y)\ln \left(\frac{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}}+\mathrm{<figure-callout\; id="0"\; label="Y\; k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">Y</figure-callout>}}{k_{}}\right))}{(1-{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DEST\_PATH\_101229162911.png"\; state="\{\{state\}\}">k</figure-callout>}}_{0\mathrm{\&beta;}})(1+Y)}---\left(9\right)$

Wherein $Y=\frac{\mathrm{R\&delta;}}{D_L}$

M _{0 α}---equilibrium state α phase liquidus curve slope;

M _{0 β}---equilibrium state β phase liquidus curve slope;

K _0a---nonequilibrium condition solute redistribution coefficient;

K _{0 β}---nonequilibrium condition solute redistribution coefficient;

D _l---non-equilibrium solute coefficient of diffusion.

(5), in eutectic growth process, the curvature of α phase and β phase is excessively cold

with

for:

${\mathrm{\&Delta;T}}_r^{\mathrm{\&alpha;}}=\frac{2{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(10\right)$

${\mathrm{\&Delta;T}}_r^{\mathrm{\&beta;}}=\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(11\right)$

Wherein, θ _αfor the contact angle of α phase solid-liquid phase, θ _βfor the contact angle of β phase solid-liquid phase, Γ _αfor α phase Gibbs-Thomson free energy coefficient Γ _βfor β phase Gibbs-Thomson free energy coefficient.

(6), the degree of supercooling of each phase is:

${\mathrm{\&Delta;T}}_{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(12\right)$

${\mathrm{\&Delta;T}}_{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(13\right)$

By Δ T _α| m _β|+Δ T _β| m _α|, Δ T _α=Δ T _β=Δ T obtains:

$\mathrm{\&Delta;T}=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\mathrm{V\&lambda;}\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}+\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\mathrm{\&lambda;}\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(14\right)$

In eutectic growth process, degree of supercooling can be write as:

$\mathrm{\&Delta;T}=K_c\mathrm{V\&lambda;}+\frac{K_r}{\mathrm{\&lambda;}}---\left(15\right)$

Wherein

$K_c=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\&CenterDot;\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}---\left(16\right)$

$K_r=\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(17\right)$

According to equation (15), λ is asked to local derviation, the λ value while obtaining minimum subcooled temperature:

$\frac{d\left(\mathrm{\&Delta;T}\right)}{\mathrm{d\&lambda;}}=K_{c}V-\frac{K_r}{{\mathrm{\&lambda;}}^2}=0---\left(18\right)$

(16) and (17) are brought in equation (18) and can be obtained:

${\mathrm{\&lambda;}}^{2}V=\frac{K_r}{K_c}=\frac{4\mathrm{\&pi;D}}{f(1-f)}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(19\right)$

${\mathrm{\&lambda;}}^{2}V=\frac{2D}{p^{\&prime;}}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(20\right)$

Wherein

$K_c=\frac{\stackrel{\&OverBar;}{m}{C}^{\&prime;}{P}^{\&prime;}}{f(1-f)D}---\left(21\right)$

$K_r=2\stackrel{\&OverBar;}{m}[\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{f\left|m_{\mathrm{\&alpha;}}\right|}+\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\left|m_{\mathrm{\&beta;}}\right|}]---\left(22\right)$

Wherein, P '=Σ (1/n ³π ³) sin ²(n π f),

By above-mentioned equation, calculate the al-si eutectic sheet interlayer spacing shown in Fig. 1 and Fig. 2 and degree of supercooling and speed of growth relation.

Embodiment:

Said method material therefor parameter is as follows: Al-12.6mass%Si, and liquidus curve is 577 ℃, eutectic composition is 12.6%, m _{0 α}=-7.5, m _{0 β}=17.5, k _0a=0.13, k _{0 β}=2 * 10 ^-4, P '=8.9 * 10 ^-3, Γ _α=1.96 * 10 ^-7, Γ _β=1.7 * 10 ^-7, θ _α=30 °, θ _β=65 °,

the pouring temperature of alloy is 700 ℃, and mold temperature is 300 ℃.

To test β phase size and Eutectic spacing on the metallographic structure photo obtaining according to above-mentioned steps, measure.

That adopt is λ || metering system, λ || be that line is measured in the vertical direction of lamella, repeated measurement is averaged for 10 times.

Table 1 analog result and Comparison of experiment results

The sheet interlayer spacing of analog computation is 13.9 μ m as can be seen from Table 1, and the sheet interlayer spacing of the al-si eutectic tissue of practical measurement is 14.6 μ m, and error is 0.7 μ m, and relative error is only 5%, illustrates that model calculates reliable.

Eutectic structure sheet interlayer spacing after can doping Alpax under this cooling velocity and thermograde and solidify according to the cooling velocity of actual measurement and local thermograde; And temperature and the impact of cooling velocity on material thermal physical property parameter have been added in prediction al-si eutectic tissue lamellar spacing model, be mainly the impact of temperature and the cooling velocity liquidus curve slope of the coefficient of diffusion in aluminium liquid, aluminium silicon phasor on silicon, thereby guaranteed that model can realize the accuracy that realizes prediction in wider temperature range and larger cooling velocity interval.

The inventive method is equally applicable to the sheet interlayer spacing prediction of the eutectic freezing tissue of other alloys, the thermograde that can obtain by actual monitoring and cooling velocity, the sheet interlayer spacing of prediction eutectic.

This method for predicting space between aluminium-silicon alloy eutectic structure lamellas provided by the invention, adopt method for numerical simulation prediction Alpax Eutectic spacing greatly to shorten experimental period, and cost is low, predict the outcome and not limited by experiment condition, and can investigate the impact of a certain single factors on Alpax eutectic structure sheet interlayer spacing.Calculate fast, error is less, is suitable for commercial Application.

Claims (2)

1. a method for predicting space between aluminium-silicon alloy eutectic structure lamellas, it is characterized in that: the eutectic structure sheet interlayer spacing after doping Alpax under this cooling velocity and thermograde and solidify according to the cooling velocity of actual measurement and local thermograde, step is as follows:

(1), set up Alpax eutectic structure sheet interlayer spacing forecast model:

First true origin QuαXiang center, determines the coordinate of y and z thus; Y axle is positioned in solid-liquid interface, and vertical with lamella; Problem reduction is become to two-dimensional state; Due to the symmetry of diffusion, its solution is in region, 0≤y≤λ/2; Solid-liquid interface is with speed V stable growth in z direction, and diffusion equation is now:

$\frac{{\&PartialD;}^{2}C}{{\&PartialD;y}^2}+\frac{{\&PartialD;}^{2}C}{{\&PartialD;z}^2}+\frac{V}{D}\&CenterDot;\frac{\&PartialD;C}{\&PartialD;z}=0---\left(1\right)$

In formula (1), C is silicon solute concentration, and D is the coefficient of diffusion of solute in liquid phase;

(2), calculate solute coefficient of diffusion:

$D=D_0\exp (-\frac{Q}{R_{g}T})---\left(2\right)$

In formula (2): D ₀---equilibrium state solute coefficient of diffusion;

Q---atom diffusion activation energy;

R _g---air constant;

T---temperature;

Due to D ₀, Q and R _gbe constant, wherein D ₀=8 * 10 ^-9;

these two data substitution formula (2) are obtained to solute coefficient of diffusion is:

(3), calculate eutectic structure sheet interlayer spacing λ:

According to the periodic change condition of eutectic composition, obtain:

$C=C_e+\mathrm{Aexp}(-\frac{V_Z}{D})+\mathrm{Bexp}(-\frac{2\mathrm{\&pi;Z}}{\mathrm{\&lambda;}})\cos \left(\frac{2\mathrm{\&pi;y}}{\mathrm{\&lambda;}}\right)---\left(3\right)$

Wherein, C _efor balance eutectic composition, V _zfor the eutectic growth speed in z direction, A, B are constants, and its value is calculated by formula (4), (5):

A＝fC′-C ^β (4)

$B=\frac{f(1-f){\mathrm{V\&lambda;C}}^{\&prime;}}{2D\sin \left(\mathrm{\&pi;f}\right)}---\left(5\right)$

Wherein C ' is eutectic line length;

(4), the constitutional supercooling degree of α phase and β phase with

for:

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]---\left(6\right)$

$\mathrm{\&Delta;}{T}_c^{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]---\left(7\right)$

Wherein, m _αfor the slope of α phase liquidus curve, m _βslope for β phase liquidus curve;

(5), in eutectic growth process, the curvature of α phase and β phase is excessively cold

with

for:

${\mathrm{\&Delta;T}}_r^{\mathrm{\&alpha;}}=\frac{2{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(10\right)$

${\mathrm{\&Delta;T}}_r^{\mathrm{\&beta;}}=\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(11\right)$

Wherein, θ _αfor the contact angle of α phase solid-liquid phase, θ _βfor the contact angle of β phase solid-liquid phase, Γ _αfor α phase Gibbs-Thomson free energy coefficient Γ _βfor β phase Gibbs-Thomson free energy coefficient;

(6), the degree of supercooling of each phase is:

${\mathrm{\&Delta;T}}_{\mathrm{\&alpha;}}=\left|m_{\mathrm{\&alpha;}}\right|[A+B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;f}}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\mathrm{f\&lambda;}}---\left(12\right)$

${\mathrm{\&Delta;T}}_{\mathrm{\&beta;}}=-\left|m_{\mathrm{\&beta;}}\right|[A-B\frac{\sin \left(\mathrm{\&pi;f}\right)}{\mathrm{\&pi;}(1-f)}]+\frac{{2\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\mathrm{\&lambda;}}---\left(13\right)$

By Δ T _α| m _β|+Δ T _β| m _α|, Δ T _α=Δ T _β=Δ T obtains:

$\mathrm{\&Delta;T}=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\mathrm{V\&lambda;}\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}+\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\mathrm{\&lambda;}\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(14\right)$

In eutectic growth process, degree of supercooling is:

$\mathrm{\&Delta;T}=K_c\mathrm{V\&lambda;}+\frac{K_r}{\mathrm{\&lambda;}}---\left(15\right)$

Wherein

$K_c=\frac{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|}{\left|m_{\mathrm{\&alpha;}}\right|+\left|m_{\mathrm{\&beta;}}\right|}\&CenterDot;\frac{C^{\&prime;}}{2\mathrm{\&pi;D}}---\left(16\right)$

$K_r=\frac{2(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)+2f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{f(1-f)\left(\right|m_{\mathrm{\&alpha;}}|+|m_{\mathrm{\&beta;}}\left|\right)}---\left(17\right)$

According to equation (15), λ is asked to local derviation, the λ value while drawing minimum subcooled temperature:

$\frac{d\left(\mathrm{\&Delta;T}\right)}{\mathrm{d\&lambda;}}=K_{c}V-\frac{K_r}{{\mathrm{\&lambda;}}^2}=0---\left(18\right)$

Formula (16) and formula (17) are brought in equation (18) and are obtained:

${\mathrm{\&lambda;}}^{2}V=\frac{K_r}{K_c}=\frac{4\mathrm{\&pi;D}}{f(1-f)}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(19\right)$

${\mathrm{\&lambda;}}^{2}V=\frac{2D}{p^{\&prime;}}\frac{f\left|m_{\mathrm{\&alpha;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)+(1-f)\left|m_{\mathrm{\&beta;}}\right|{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{\left|m_{\mathrm{\&alpha;}}\right|\left|m_{\mathrm{\&beta;}}\right|C^{\&prime;}}---\left(20\right)$

Wherein

$K_c=\frac{\stackrel{\&OverBar;}{m}{C}^{\&prime;}{P}^{\&prime;}}{f(1-f)D}---\left(21\right)$

$K_r=2\stackrel{\&OverBar;}{m}[\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&alpha;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}\right)}{f\left|m_{\mathrm{\&alpha;}}\right|}+\frac{{\mathrm{\&Gamma;}}_{\mathrm{\&beta;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{\&beta;}}\right)}{(1-f)\left|m_{\mathrm{\&beta;}}\right|}]---\left(22\right)$

Wherein, P '=Σ (1/n ³π ³) sin ²(n π f),

By above-mentioned equation, can calculate al-si eutectic sheet interlayer spacing and degree of supercooling and speed of growth relation.

2. method for predicting space between aluminium-silicon alloy eutectic structure lamellas according to claim 1, it is characterized in that: in process of setting, the speed of growth at interface exerts an influence to solute distribution coefficient, adopts continuity model to calculate the solute redistribution coefficient of α phase and β phase, and its expression formula is as follows:

$m_{\mathrm{\&alpha;}}=\frac{m_{0\mathrm{\&alpha;}}(1-k_{0\mathrm{\&alpha;}}+(k_{0\mathrm{\&alpha;}}+Y)\ln \left(\frac{k_{0\mathrm{\&alpha;}}+Y}{k_{0\mathrm{\&alpha;}}(1+Y)}\right))}{(1-k_{0\mathrm{\&alpha;}})(1+Y)}---\left(8\right)$

$m_{\mathrm{\&beta;}}=\frac{m_{0\mathrm{\&beta;}}(1-k_{0\mathrm{\&beta;}}+(k_{0\mathrm{\&beta;}}+Y)\ln \left(\frac{k_{0\mathrm{\&beta;}}+Y}{k_{0\mathrm{\&beta;}}(1+Y)}\right))}{(1-k_{0\mathrm{\&beta;}})(1+Y)}---\left(9\right)$

Wherein $Y=\frac{\mathrm{R\&delta;}}{D_L},$

M _{0 α}---equilibrium state α phase liquidus curve slope;

M _{0 β}---equilibrium state β phase liquidus curve slope;

K _0a---nonequilibrium condition solute redistribution coefficient;

K _{0 β}---nonequilibrium condition solute redistribution coefficient;

D _l---non-equilibrium solute coefficient of diffusion.

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