Acta Metallurgica Sinica, 2017, 53(1): 123-128
doi: 10.11900/0412.1961.2016.00274

A Model for Precipitation-Temperature-Time Curve Calculation

Abstract:

Nanometer precipitation is of great importance to the mechanical properties of the low carbon micro-alloyed steel. Precipitation process is controlled by the driving force for precipitation and the diffusion rate of atoms. Under the influence of these two factors, the fastest precipitation temperature for (Mx1Mv2M1-x-v3)(CyN1-y) phase is available, which is also known as nose temperature. The maximum number density of precipitates can be obtained through isothermal treatment at the nose temperature. The most effective tool for getting the value of nose temperature is the precipitation-temperature-time (PTT) curve. Due to that the diffusivity of substitutional atom is several orders of magnitude smaller than that of interstitial atom, the nucleation process and growth process of complex precipitation are controlled by the diffusion of substitutional atoms. So far no model has been established for calculating PTT curve of complex precipitation. All the existing models are established for simple precipitation. In this work, a kinetic model, based on the classical nucleation and growth theories and Johnson-Mehl-Avrami equation, employing Adrian thermodynamic model and L-J model, using average diffusivity to demonstrate the effects of forming elements on precipitation process, has been adapted to describe the precipitation kinetics following high temperature deformation in micro-alloy steels alloying with V, Nb and Ti. Using this model, the PTT curves for the kinetics of second phase were easily obtained. In the experiment, within the temperature range from 660 to 540 ℃, the nose temperature of carbonitride precipitation is equal to or slightly higher than 620 ℃. The value of nose temperature estimated from PTT curve is 628 ℃ which is consistent with the experimental observation. There are enough reasons to believe that the model proposed in this work can estimate accurately the nose temperature information in relatively small experiment case. This model has outstanding advantages in comparison with existing models: the mole fraction of precipitation and the driving force for precipitation per unit volume ∆Gv can be calculated directly without calculating the solubility formula of complex carbide in matrix; The proposed model can also be used to calculate the absolute solution temperature and the constituent of initial complex precipitation forming at critical temperature of precipitation, which can be used as the iterative initial values for calculating the equilibrium information between matrix and precipitation at relatively low temperatures.

Key words: micro-alloyed steel, ; classical nucleation　and growth theory, ; precipitation, ; PTT curve

1 析出动力学模型的建立
1.1 热力学计算

$y ln xy K M 1 C [ M 1 ] [ C ] + 1 - y ln x 1 - y K M 1 N [ M 1 ] [ N ] +$

$y 1 - y L CN R g T = 0$ (1)

$y ln vy K M 2 C [ M 2 ] [ C ] + 1 - y ln v 1 - y K M 2 N [ M 2 ] [ N ] +$

$y 1 - y L CN R g T = 0$ (2)

$y ln ( 1 - x - v ) y K M 3 C [ M 3 ] [ C ] + 1 - y ln ( 1 - x - v ) 1 - y K M 3 N [ M 3 ] [ N ] +$

$y 1 - y L CN R g T = 0$ (3)

$vy ln x [ M 2 ] K M 1 C v [ M 1 ] K M 2 C + 1 - x - v 1 - y ln 1 - x - v [ M 1 ] K M 3 N x [ M 3 ] K M 1 N + 1 - y ln x ( 1 - y ) K M 1 N [ M 1 ] [ N ] + y 2 1 - y L CN R g T = 0$ (4)

$M 0 1 = x 2 f + 1 - f [ M 1 ]$ (5)

$M 0 2 = v 2 f + 1 - f [ M 2 ]$ (6)

$M 0 3 = 1 - x - v 2 f + 1 - f [ M 3 ]$ (7)

$C 0 , C = y 2 f + 1 - f [ C ]$ (8)

$C 0 , N = 1 - y 2 f + 1 - f [ N ]$ (9)

$K MX = M X = A Fe 2 10 4 A M A X × 10 B - A / T$ (10)

1.2 临界形核功和临界半径

$Δ G = Δ G chem + Δ G int$ (11)

$Δ G chem = 4 3 π R 3 Δ G v$ (12)

$Δ G int = 4 π R 2 γ$ (13)

$Δ G v = - R g T V m [ ln [ M 1 ] 0 [ M 1 ] + ln [ M 2 ] 0 [ M 2 ] + ln [ M 3 ] 0 [ M 3 ] +$

$ln [ C ] 0 [ C ] + ln [ N ] 0 N ]$ (14)

$R c = - 2 γ Δ G v$ (15)

$R c$ 代入式(11)得到临界形核功为：

$Δ G c = 16 π γ 3 3 Δ G v 2$ (16)

1.3 形核率

$I = N n Zβ exp - Δ G c k B T$ (17)

$β = 4 π R c 2 D ev X 0 a 4$ (18)

$D ev = x D M 1 + v D M 2 + ( 1 - x - v ) D M 3$ (19)

$D M i = D M io exp ( - Q i R g T )$ (20)

$X 0 = x M 1 0 + v M 2 0 + ( 1 - x - v ) M 3 0$ (21)

$I t = I exp ( - t τ e )$ (22)

$N s = ∫ 0 ∞ I t d t = I τ e$ (23)

1.4 析出相的长大

$v = d R d t = D ev R X 0 - X i X p - X i$ (24)

$R 2 = 2 D ev X 0 - X i X p - X i t$ (25)

1.5 析出曲线

$f = 1 - exp - W t n$ (26)

$f = 1 - exp ( - 4 3 π Ι τ e λ 3 D ev 3 2 t 3 2 )$ (27)

$lg t 0.05 = 1 1.5 ( - 1.28994 - lg 4 π 2 τ e λ 3 N n 15 a 4 -$

$lg X 0 - 5 2 lg D ev - 2 lg R c + 1 ln 10 Δ G c k B T )$ (28)

$lg t 0.05 t 0 = 1 1.5 ( - 1.28994 - 5 2 lg D ev - 2 lg R c + 1 ln 10 Δ G c k B T )$ (29)

$lg t 0.95 t 0.05 = 1 1.5 lg ( ln 0.05 ln 0.95 )$ (30)

2 模型验证与讨论

2.1 全固溶温度的计算

2.2 平衡固溶量的计算

Fig.1 Thermodynamic equilibrium information of TixVvNb1-x-vCyN1-y for steel above 1473 K(a) solute concentrations in matrix(b) the occupation ratio of atoms in precipi- tate

2.3 终轧结束时实际固溶量的计算

1200 ℃保温3 min后,将钢以10 ℃/s的冷却速率冷却到900 ℃,再施加60%的变形,在此过程中会有一定量的第二相析出。由于时间较短,假定实际析出量仅为平衡析出量的20%,则900 ℃下的实际固溶量为1200 ℃下平衡固溶量减去900 ℃下平衡析出量的20%。

2.4 复合相在铁素体中的析出行为

Fig.2 Kinetic behaviors of complex precipitates precipitated in ferrite (t0 is the absolute time when second phase begins to precipitate, t0.05 and t0.95 are the times of volume fraction of second phase reaching 5% and 95% equilibrium precipitation amount at given temperatures, respectively) (a) critical nucleus radius(b) critical nucleation energy(c) PTT diagram

3 结论

(1) 基于经典形核长大理论和Johnson-Mehl-Avrami方程,采用平均扩散率表征成核原子对第二相形核长大过程影响的思想,建立了计算第二相PTT曲线的模型。计算结果与实验结果吻合良好。

(2) 该模型适用于不同基体中各类第二相析出动力学的计算,也可用于计算全固溶温度及不同温度下第二相平衡析出量。

The authors have declared that no competing interests exist.

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And the time dependence of volume fraction and the particle radius of strain induced precipitation are also predicted on the basis of classical nucleation theory. In order to estimate the effect of deformation on nucleation, the change in elastic energy of dislocation with nucleation is calculated. Experimental results showed that combination of Nb and Ti addition, decreased the solubility of carbonitrides and accelerated the precipitation rate from supersaturated austenite because of the formation of complex precipitates. Such experimental results are in good agreement with the prediction by the present model. And both the acceleration of precipitation rate and the refinement of precipitates particles due to hot deformation are also quantitatively explained.      URL     [本文引用:3] [23] Zener C.Theory of growth of spherical precipitates from solid solution[J]. J. Appl. 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[25] Quispe A, Medina S F, Gómez M, et al.Influence of austenite grain size on recrystallisation-precipitation interaction in a V-microalloyed steel[J]. Mater. Sci. Eng., 2007, A447: 11 By means of torsion tests using small specimens, the influence of austenite grain size on strain induced precipitation kinetics has been determined in a vanadium microalloyed steel. Determination of recrystallisation–precipitation–time–temperature (RPTT) diagrams for two austenite grain sizes allows values of the aforementioned magnitudes to be determined. An ample discussion is made of the quantitative influence found and its relation with nucleation and growth mechanisms of precipitates. The results are compared with the quantitative influence exerted by the other variables, reaching the conclusion that the austenite grain size has a notable influence on strain induced precipitation kinetics which should not be underestimated. Finally, the influence of austenite grain size is included in a strain induced precipitation model constructed by the authors of this work and which also takes into account the other aforementioned variables.      URL     [本文引用:1]

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micro-alloyed steel,
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YANG Yong
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LI Tianrui
JIA Tao
LI Xiaolin
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