Improved Thermodynamic Formula for Austenite/(Austenite+Cementite) Phase Boundary in Low Alloy Steels

Sutdy

The phase boundary between two different phases is a criterion that defines the start and finish of phase transformation processes. Phase transformation kinetics is increased by adding alloying elements that increase the stability of the product phase. For example, Cr addition increases the stability of ferrite (α) transformed from austenite decomposition. The addition of an alloying element that increases the stability of the parent phase can suppress the formation of the product phase from the parent phase, e.g., Mn addition increases the austenite (γ) stability, resulting in a retarded ferrite transformation. This is due to the fact that the addition of an alloying element decreases or increases the driving force required for the phase transformation.¹⁾ The behavior of alloying elements during phase transformation can be classified into two types from a thermodynamic point of view: ortho-equilibrium (OE) and para-equilibrium (PE).²⁾ OE is a full equilibrium situation, mainly observed at high temperature where the driving force for phase transformation is low and the diffusion of alloying elements is relatively fast. The alloying elements are fully partitioned between the parent phase and the product phase depending on the solubility in each phase. On the other hand, only the interstitial elements are partitioned, while no partitioning of an alloying element on substitutional sites occurs under PE conditions. The PE situation is generally observed in the low temperature region where the phase transformation kinetics is fast, and the diffusion of substitutional alloying elements is relatively slow.

Thermodynamic studies on the influence of alloying element partitioning on the phase boundaries of low alloy steels have been performed.^{3,4,5,6,7,8,9,10,11)} The partitioning effects of alloying elements have been studied with respect to the phase boundary between γ and (γ+α) regions,⁸⁾ known as the A_e3 temperature curve, and the eutectoid transformation temperature (A_e1 temperature), which is determined by two phase boundaries, γ/(γ+α) and γ/(γ+cementite(θ)).⁹⁾ Also, the influence of alloying element partitioning on the γ/(γ+θ) phase boundary, i.e., the A_cm temperature curve, has been studied and the following formula has been proposed.¹¹⁾   

$$\begin{array}{l}{A}_{cm}({}^{\circ }C)=224.4+992.4C-465.1C^2+46.7Cr\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+19.0CCr-6.1C{r}^2+7.6Mn\hspace{0.17em}+10.0Mo\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-6.8CrMo-6.9Ni+3.7CNi-2.7CrNi\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+0.8N{i}^2+16.7Si\end{array}$$(1)

However, this formula is limited to the hypo-eutectoid region with a carbon range from 0.2 to 0.7 mass%. Few or no formulae are available to predict the γ/(γ+θ) phase boundary above the A_e1 temperature for the hyper-eutectoid region. Therefore, in this study, a thermodynamic based formula considering the effect of alloying element partitioning was proposed to predict the γ/(γ+θ) phase boundary over a wide range of chemical compositions for low alloy steels.

The chemical composition range for the model proposed in this study is listed in Table 1. The maximum carbon content is 1.8 mass%, which can include both hypo- and hyper-eutectoid regions. Also, the effect of Cu addition was considered. Copper increases strengthening due to Cu precipitation and improves the hardenability and tempering resistance in HSLA steels.

Table 1. Range of chemical composition for the model (in mass%).

	C	Mn	Si	Ni	Cr	Mo	Cu
Min.	0.2	0.1	0.06	0.0	0.0	0.0	0.0
Max.	1.8	1.5	0.3	2.8	1.5	0.6	0.6

First, the γ/(γ+θ) phase boundary under OE conditions (hereafter called “ $A_{cm}^{OE}$ ”) and the γ/(γ+θ) phase boundary under PE conditions (hereafter called “ $A_{cm}^{PE}$ ”) are calculated for alloy compositions in Table 1. Equations related to the chemical potentials for each element and all thermodynamic parameters such as free energy change between phases used to calculate $A_{cm}^{OE}$ and $A_{cm}^{PE}$ are adopted in the literature.⁷⁾ Also detailed calculation procedures are described in the previous works.^7,11) Thermodynamic calculation is carried out using a personal program code including the equations and the thermodynamic parameters. The actual γ/(γ+θ) phase boundary (hereafter called “ $A_{cm}^{OE-PE}$ ”) is varied between two A_cm temperatures calculated at OE and PE conditions depending on the effect of alloying element partitioning. $A_{cm}^{OE-PE}$ at the eutectoid temperature can be determined by comparison with the $A_{e1}^{OE-PE}$ temperature. The $A_{e1}^{OE-PE}$ temperature calculated by considering partitioned alloying elements⁹⁾ indicates a tendency for alloy partitioning between PE and OE conditions, which directly relates to the alloy partitioning of $A_{cm}^{OE-PE}$ . The $A_{e1}^{OE-PE}$ temperature can be calculated by Eq. (8) in the reference.⁹⁾ Thus, the formularized $A_{cm}^{OE-PE}$ is given by:   

$$A_{cm}^{OE-PE}=A_{cm}^{OE}-\left(A_{cm}^{OE}-A_{cm}^{PE}\right)\left[\left(A_{e1}^{OE}-A_{e1}^{OE-PE}\right)/\left(A_{e1}^{OE}-A_{e1}^{PE}\right)\right]$$(2)

In the previous model,¹¹⁾ the partitioning effect of alloying elements linearly decreases below the A_e1 temperature, and no partitioning occurs at low temperature. Similarly, it is known that the alloying elements can be fully partitioned at high temperature. The partitioning of the alloying elements is strongly dependent on temperature, so the actual A_cm temperature with the temperature-dependent partitioning effect of alloying elements ( $A_{cm}^{OPT}$ ) is obtained as follows:   

$$\begin{array}{l}{A}_{cm}^{OPT}=\\ \{\begin{array}{l}{A}_{cm}^{OE}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\hspace{0.17em}\hspace{0.17em}for\mathrm{\hspace{0.17em} }{A}_{cm}^{OE-PE}\ge T_0^{high}\\ A_{cm}^{OE}-\left(A_{cm}^{OE}-A_{cm}^{OE-PE}\right)\left(\frac{T_0^{high}-A_{cm}^{OE-PE}}{T_0^{high}-A_{e1}^{OE-PE}}\right)\mathrm{\hspace{0.17em} }for\mathrm{\hspace{0.17em} }{A}_{e1}^{OE-PE}\le A_{cm}^{OE-PE}<T_0^{high}\\ A_{cm}^{PE}+\left(A_{cm}^{OE-PE}-A_{cm}^{PE}\right)\left(\frac{T_0^{high}-A_{e1}^{OE-PE}}{A_{e1}^{OE-PE}-T_0^{low}}\right)\mathrm{\hspace{0.17em} }for\mathrm{\hspace{0.17em} }{T}_0^{low}<A_{cm}^{OE-PE}<A_{e1}^{OE-PE}\\ A_{cm}^{PE}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } for\mathrm{\hspace{0.17em} }{A}_{cm}^{OE-PE}\le T_0^{low}\end{array}\end{array}$$(3)

where $T_0^{low}$ is the low standard temperature for full PE conditions (=25°C¹¹⁾), $T_0^{high}$ is the high standard temperature for full OE conditions, defined to be ( $A_{e1}^{OE-PE}$ +T_s)/2. T_s is the temperature for the maximum equilibrium carbon solubility on the γ/(γ+θ) phase boundary in a binary Fe–Fe₃C system (=1147°C). $T_0^{low}$ and $T_0^{high}$ are determined in order to separate the partitioning behavior with temperature.

A large number of combinations of alloying elements in Table 1 were used to calculate the $A_{cm}^{OPT}$ temperature according to Eq. (3). A simple formula for $A_{cm}^{OPT}$ temperature prediction has been generated by regression analysis as   

$$\begin{array}{l}{A}_{cm}^{OPT}({}^{o}C)=297.5+655C\left(1-0.205C\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}+13.3Mn-13.3Ni\left(1-0.06Ni\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}+6.5Mo-16.6Cu+79.8Cr\left(1+0.055Cr\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}-C\left(4.7Mn-25.6Si-9.6Ni+36.7Cr-8.7Cu\right)\end{array}$$(4)

where the amount of each alloy is in mass percent. The R² value for the accuracy of the regression was 0.9981 when the $A_{cm}^{OPT}$ temperatures were compared to those calculated using Eqs. (3) and (4).

Figure 1 compares the γ/(γ+θ) phase boundaries calculated by the previous formula (Eq. (1)) and the proposed formula (Eq. (4)) for the binary Fe–Fe₃C alloy system. The experimental data points of the γ/(γ+θ) phase boundary were measured from Fe–Fe₃C alloys with different carbon contents higher than 0.8 mass%.¹²⁾ The γ/(γ+θ) boundary curve calculated by Eq. (1) is underestimated compared with the experimental data because the thermodynamic data used to derive Eq. (1) were limited to the hypo-eutectoid composition range of the low alloy steels. The results calculated using Eq. (4) are in good agreement with experimental data, which indicates that Eq. (4) can overcome the inaccuracies of the previous formula for hyper-eutectoid region.

Fig. 1.

Calculated γ/(γ+θ) phase boundary by Eq. (1) (red dash line) and by Eq. (4) (black solid line). Experimentally measured A_cm temperatures are plotted as data points.

Twelve Ac_cm temperatures of various Fe–C–X alloys measured experimentally were collected from the literature.^13,14) Those Ac_cm temperatures are higher than the equilibrium A_cm temperatures since the Ac_cm temperature is measured during continuous heating.¹⁵⁾ Overheating beyond the equilibrium temperature is required to complete the dissolution of all cementite particles. The calculated equilibrium A_cm temperatures are compared with the measured Ac_cm temperatures for various Fe–C–X alloys,^13,14) as shown in Fig. 2. The equilibrium A_cm temperatures obtained using the newly proposed formula are in an acceptable range relative to the experimental data considering the non-equilibrium nature of the experiment. The compositions of the alloys used for comparison are within the composition range in Table 1 except for one alloy (0.81C-3.69Ni-0.78Cr). The A_cm temperatures calculated using the previous formula are too low for the same reason illustrated in Fig. 1. There is no doubt that the prediction accuracy could be lowered when exceeding the composition range of data used to derive the equations.

Fig. 2.

Effect of substitutional alloying element partitioning on calculated A_cm temperature. “Ac_cm” is the A_cm temperature experimentally measured during heating.

The equilibrium volume of proeutectoid cementite can be obtained from the γ/(γ+θ) phase boundary. The volume of the proeutectoid cementite in a hyper-eutectoid AISI 52100 (Fe-0.92C-0.397Mn-0.27Si-0.025Ni-1.471Cr-0.009Mo-0.065Cu in mass%) was experimentally measured to confirm the accuracy of the proposed formula. Small coupon samples of AISI 52100 were isothermally held at different temperatures higher than the A_e1 temperature for 1 h in a heat-treating furnace with an Ar atmosphere to prevent surface oxidation or decarburization. The samples were directly quenched into water and were etched by nital. An image analyzer was used to measure the volume of proeutectoid cementite. Figure 3(a) compares the volume of proeutectoid cementite calculated using Eqs. (1) and (4) with the measured results. The lever rule was used to calculate the volume of proeutectoid cementite from the γ/(γ+θ) phase boundary, as was used for the proeutectoid ferrite calculation.¹¹⁾ First, the mass fractions for proeutectoid cementite and austenite (W_θ and W_γ) were calculated.   

$$\{\begin{array}{l}{W}_{\theta }=\left(C_{\gamma /\theta }-C_0\right)/\left(6.69-C_0\right)\\ W_{\gamma }=1-W_{\theta }\end{array}$$(5)

where C₀ is the mass percent carbon in the AISI 52100 (=0.92) and C_γ/θ is the mass percent carbon on the the γ/(γ+θ) phase boundary at a given temperature. Therefore, C_γ/θ is influenced by the partitioning of alloying elements since the alloy partitioning influences the γ/(γ+θ) phase boundary. C_γ/θ can be obtained by solving Eq. (4) for C as the temperature and amounts of alloying elements except for C are given. The exact volume fraction of proeutectoid cementite (V_θ) is expressed as follows:   

$$V_{\theta }=\left(W_{\theta }/{\rho }_{\theta }\right)/\left(W_{\theta }/{\rho }_{\theta }+W_{\gamma }/{\rho }_{\gamma }\right)$$(6)

where ρ_θ and ρ_γ are the densities of cementite and austenite. The alloy partitioning varied from PE to mixed (PE+OE), and finally reached to OE conditions as raising temperature influences the γ/(γ+θ) phase boundary (A_cm temperature) as seen in Eq. (3) and brings about the change of the cementite volume fraction. The calculated volume curve, obtained using Eq. (4), matches well with the measured data points. Figure 3(b) shows the microstructure of the quenched sample after holding at 800°C, where the small spherical white particles are the proeutectoid cementite.

Fig. 3.

(a) Comparison of proeutectoid cementite volume calculated by Eqs. (1) and (4) for hyper-eutectoid AISI 52100. Experimental data were obtained by quantitative image analysis. (b) Microstructure of the sample held at 800°C for 1 h, followed by water-quenching.

As found in Figs. 1, 2, 3, the formula proposed in this study accurately predicts the γ/(γ+θ) phase boundary in the high temperature hyper-eutectoid region. Thus, the use of Eq. (4) in the low temperature hypo-eutectoid austenite region was also verified. Figure 4 shows the calculated volumes of proeutectoid ferrite below the A_e1 temperature compared with the experimental results. The calculated volume curve using Eq. (4) is similar to that using Eq. (1), showing good accuracy. Similar to the calculation in Fig. 3, the mass fractions for proeutectoid ferrite and austenite (W_α and W_γ) were calculated from the Fe–C phase diagram of the alloy as   

$$\{\begin{array}{l}{W}_{\alpha }=\left(C_{\gamma /\theta }-C_0\right)/C_{\gamma /\theta }\\ W_{\gamma }=1-W_{\alpha }\end{array}$$(7)

The carbon solubility in ferrite was ignored. The exact volume fraction of proeutectoid ferrite (V_α) is given by   

$$V_{\alpha }=\left(W_{\alpha }/{\rho }_{\alpha }\right)/\left(W_{\alpha }/{\rho }_{\alpha }+W_{\gamma }/{\rho }_{\gamma }\right)$$(8)

where ρ_α is the density of ferrite. In this case, the alloy partitioning also influences C_γ/θ and finally V_α.

Fig. 4.

Comparison of proeutectoid ferrite volume calculated by Eqs. (1) and (4) for hypo-eutectoid AISI 5140 (Fe-0.36C-0.72Mn-0.27Si-0.06Ni-1.11Cr in mass%). Experimental data were obtained by quantitative image analysis.¹¹⁾

In conclusion, OE and PE thermodynamic calculations were carried out to predict the γ/(γ+θ) phase boundary for hypo- and hyper-eutectoid low alloy steels. The temperature variation of the partitioning effect of substitutional alloying elements on the actual γ/(γ+θ) phase boundary was considered. A new A_cm temperature formula was proposed as a function of chemical composition based on the thermodynamic calculations, which could predict the accurate γ/(γ+θ) phase boundary of various low alloy steels not only in the high temperature hyper-eutectoid region, but also in the low temperature hypo-eutectoid region. It is anticipated that the proposed formula can be useful for the calculation of various phase transformations of multi-component austenite with wide ranges of temperature and composition.

Acknowledgement

This research was partially supported by Industry, University and Research Institute Core Technology Development & Industrialized Supporting Business conducted by Jeonbuk province in 2013.

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