Kinetic Model of the γ to α Phase Transformation at Grain Boundaries in Boron-bearing Low-alloy Steel

Abstract

The effect of boron on the nucleation and growth of ferrite at austenite grain boundaries is examined theoretically under the assumption that the junction of 4-austenite grain boundaries (i.e., the 4-grain junctions) are the dominant nucleation sites of ferrite. Boron segregates to the austenite grain boundaries and reduces the grain-boundary energy; it thereby retards ferrite nucleation at the grain boundary. The retardation is expressed as a decrease in nucleation density due to an increase in the critical activation energy for nucleation, and the calculated value of the fraction of active nucleation sites is in satisfactory agreement with the experimental results. The reduction of the austenite grain-boundary energy, which we obtained by applying the Gibbs isotherm for adsorption to the boron segregation, is of the same order of magnitude as the reduction is deduced from the results of calculations for a decrease in the nucleation density based on experimental results. The growth of ferrite was calculated using DICTRA, which yielded both the volume fraction and the grain size of transformed ferrite as functions of time; the results agreed with the experimental results. This agreement suggests that the influence of boron on the growth rate is negligible. However, the increase in the size of the diffusion cell due to the addition of boron is considered to be the main reason for the slightly larger grain size of ferrite compared with that in boron-free steel; this result is also in good agreement with experimental observations.

1. Introduction

Boron is well known to strongly affect the hardenability of steels by suppressing ferrite formation from austenite, even when added to steels in small amounts, e.g., at concentrations on the order of 0.002 mass% in low-alloy steels. This increased hardenability results from boron retarding ferrite nucleation at the austenite grain boundary, which has led researchers to hypothesize that a relationship exists between the segregation of boron at the austenite grain boundary and the retardation of the formation of ferrite.^1,2) The grain boundary energy of austenite is decreased by the segregation of boron and finally inhibits ferrite nucleation.^3,4,5,6,7)

The effect of boron on hardenability is utilized in the steel industry to strengthen steels;^9,10,11,12) however, some technical issues associated with the use of boron still remain, such as instability of the hardenability, which is caused by the formation of Fe₂₃(CB)₆,¹³⁾ and a substantial decrease in toughness, which is caused by an increase in the martensite–austenite (MA) constituent in the microstructure.¹⁴⁾ Furthermore, the simultaneous addition of Mo or Nb along with boron is known to result in a stronger effect on the hardenability.^13,14,15) Therefore, the establishment of the ability to reliably predict changes related to ferrite formation from austenite induced by the addition of boron is important for the effective and stable utilization of boron in steels.

The literature^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)} contains abundant reports related to the mechanism by which boron affects the formation of microstructures in steel; however, studies that consider the effects of both the thermodynamics and kinetics of boron segregation at the austenite grain boundary on the nucleation and the growth of ferrite from austenite are few. Shibata et al. studied the influence of boron segregated at the grain boundary on the hardenability of steels¹⁷⁾ using the alpha-particle track etching (ATE) method. Asahi et al. reported that hardenability increased according to the boron content to the extent that boron exists as a solid solution in steels.¹⁸⁾ Nevertheless, these studies did not include quantitative analyses based on thermodynamics, such as the change in free energy at the grain boundary induced by boron segregation or the extent of boron segregation. With respect to the nucleation and growth of ferrite in boron-added steels, one representative works is the experimental study by Tarui et al.,¹⁹⁾ who reported the increased incubation time for ferrite nucleation and the increased growth rate of ferrite during isothermal transformation by the addition of boron. They performed qualitative analyses, but they did not compare their results to those obtained from a thermodynamic point of view.

In contrast, the literature contains numerous studies related to the quantification of the boron segregation in steels. A pioneering study was reported by Hondros,²⁰⁾ who revealed the relationship between the degree of segregation and the amount of solid solution of alloy elements in steels. In addition, several other significant works have recently been published. Yoshitomi et al. analyzed thermodynamic data related to an extremely small quantity of boron in steels using a first-principles method,²¹⁾ which would be helpful in establishing a thermodynamic database for theoretical studies. Furthermore, Sawada studied the segregation of boron in the Σ9 grain boundary of austenite, which was thought to be close to a general grain boundary, using a first-principles method and concluded that the segregation energy derived from theory was sufficiently similar to the reported experimental values.²²⁾ However, these works were not referred to a specific effect of the change in thermodynamics on the hardenability of steel, as is the case with the γ–α transformation. This fact means that few studies on the specific effects of boron on the γ–α transformation from the viewpoint of the thermodynamics and the kinetics of ferrite nucleation and growth have been reported.

Consequently, the purpose of this study was to analyze the effects of boron on ferrite nucleation and growth, which is thermodynamically related to the segregation of boron at the austenite grain boundary, as well as to discuss an empirical thermodynamic based kinetic model for the γ–α transformation.

2. Theory and Calculations

2.1. Nucleation of Ferrite at the Austenite Grain Boundary

Evaluations of the change in the interfacial energy between ferrite and austenite per unit area and the change in the total Gibbs energy per unit volume associated with the formation of a ferrite nucleus are commonly used to perform thermodynamic analyses of ferrite nucleation at the austenite grain boundary.

Huang et al., who used an optical microscopy and a serial sectioning method, observed that a majority of the ferrite grains nucleated at the austenite 4-grain junctions (grain corners).²³⁾ However, they were able to observe an insufficiently low percentage (i.e., only 4.4 to 7.0%) of the ferrite nucleation at all grain corners. This fact implies a mechanism by which ferrite nucleation may not occur at all grain corners, though the presence of tiny nuclei in the early stage of ferrite nucleation could not be observed by an optical microscopy.

In a recent study, Takeuchi et al. observed ferrite nucleation at a grain corner in Co–Fe alloy.²⁴⁾ They reported that ferrite nucleation could be observed at approximately 60% of all grain corners, as well as at approximately 70% of the large-angle grain boundaries, by 3D-EBSD measurements.^24,25) They proposed that all grain corners do not inevitably become ferrite nucleation sites, although grain corners are certainly one of the primary nucleation sites.

In this study, focusing on the austenite 4-grain junctions, we estimated the volume of ferrite nuclei as well as the interfacial area between austenite and ferrite. In contrast, Clemm et al. derived the critical activation energy for the ferrite nucleation geometrically for cases of the 2-grain, 3-grain, and 4-grain junctions of austenite under the assumption that the grain-boundary energies of austenite–austenite and austenite–ferrite were constant, even if the curvature of interface changed.²⁶⁾ They finally found that the ratio between the critical activation energies of the 2-grain, the 3-grain, and the 4-grain junctions of austenite was 23:4.5:1, respectively. This conclusion is consistent with the experimental works of both Huang et al. and Takeuchui et al. in the sense that the 4-grain junctions are the main sites of ferrite nucleation.

For the purpose for evaluating the size and the volume fraction of ferrite as a function of time, we simulated the nucleation and growth of ferrite at the 4-grain junctions based on the geometrical analysis of Clemm et al. under the assumption that the grain-boundary energy of austenite–austenite changes with the degree of boron segregation.

The free energy change for the nucleation of ferrite can be formulated as a function of the curvature of the nuclei of ferrite, r, which is shown as Eq. (1):   

$$\mathrm{\Delta }G=\frac{\mathrm{\Delta }{G}_m}{V_m}c{r}^3+(b{\gamma }^{\gamma \alpha }-a{\gamma }^{\gamma \gamma })r^2$$(1)

where ΔG_m is the driving force for the phase transformation from austenite to ferrite according to the parallel tangent law, V_m is the molar volume, γ ^γγ is the grain boundary energy between two austenite grains, γ ^γ α is the grain boundary energy between a ferrite grain and a austenite grain, and a, b, and c are parameters derived from Clemm’s study of the geometry as functions of the contact angle defined as γ ^γγ/2γ ^γ α.

The critical size of ferrite nuclei, r^*, and the critical activation energy, ΔG^*, can be derived by minimizing ΔG in Eq. (1). These are expressed in Eqs. (2) and (3), respectively. In cases where ferrite nucleation occurs at the 4-grain junctions, the ferrite nuclei assumes the tetrahedral morphology shown in Fig. 1. Parameters a, b, and c can then be specified as functions of γ ^γγ/2γ ^γα in Eqs. (1), (2), and (3).   

$$r^{*}=-\frac{2(b{\gamma }^{\gamma \alpha }-a{\gamma }^{\gamma \gamma })}{3c\frac{\mathrm{\Delta }{G}_m}{V_m}}$$(2)

  

$$\mathrm{\Delta }{G}^{*}=\frac{4}{27}\frac{{(b{\gamma }^{\gamma \alpha }-a{\gamma }^{\gamma \gamma })}^3}{{\left(\frac{\mathrm{\Delta }{G}_m}{V_m}\right)}^2{c}^2}$$(3)

Fig. 1.

Schematic diagram of a ferrite precipitating at a junction of four grains assuming that its surface consists of four equivarent spherical morhology.

2.2. Growth of Ferrite Nuclei

The ferrite nuclei are not always spherical just immediately after the nucleation because of the facet nucleation of ferrite at the attached austenite grain boundary. However, the growth is thought to become gradually isotropic over time, in accordance with the domination of the partitioning of alloy elements at the interface between ferrite and austenite during the growth of ferrite. This morphological change of the ferrite nuclei will not significantly affect the calculation of the volume fraction of ferrite at the growth stage. In this study, we have therefore simulated the growth of ferrite by a 3-dimensional isotropic model; we expand a spherical ferrite particle at the center of an austenite sphere using the DICTRA software package.²⁷⁾ This package is based on a numerical solution of the multicomponent diffusion equations and assumes that the thermodynamic equilibrium is established locally at the migrating phase interface between ferrite and austenite. The local equilibrium is calculated by DICTRA using Thermo-Calc²⁸⁾ and its database. The growth rate is calculated from a set of mass balance equations.

3. Results and Discussion

3.1. Effect of Boron on the Grain-boundary Energy of Austenite

The grain boundary energy was evaluated in cases where the boron segregates at the austenite grain boundary. At first, the austenite grain boundary was assumed to be the independent phase with thickness δ and a specific Gibbs free energy that is ΔG_m^gb higher than that of the austenite phase. The austenite grain boundary energy, γ ^γγ, can be expressed as Eq. (4), which is derived from the parallel tangent theory shown in Fig. 2:²⁹⁾   

$${\gamma }^{\gamma \gamma }=\frac{\delta }{V_m^{gb}}\mathrm{\Delta }{G}_m^{gb}$$(4)

where V_m^gb is the molar volume of the austenite grain boundary.

Fig. 2.

Schematic diagram showing the change of free energy due to grain boundary segregation of boron.

In this study, the boron concentration is assumed to be sufficiently dilute (i.e., 0.0020 mass% in total) that we could adopt an approximation that the Gibbs free energies in both the austenite grain and the grain boundary are treated as a dilute solid solution because the concentration of boron at the austenite grain boundary will increase to only several mass percent at most, even if the segregated boron concentration at the grain boundary is approximately 1000 times larger than that in the grain. Therefore, the difference in the austenite grain boundary energy caused by the addition of boron, Δγ ^γγ, can be expressed as Eq. (5):   

$$\mathrm{\Delta }{\gamma }^{\gamma \gamma }\cong -\frac{\delta }{V_m^{gb}}RTk{x}^{\gamma }$$(5)

where k is the segregation coefficient (i.e., the ratio between the boron concentration at the austenite grain boundary and that in the grain), x^k is the boron concentration in the austenite grain boundary, R is the gas constant, and T is temperature. Notably, this equation coincides with the Gibbs isotherm equation.

According to the analysis of Shigesato et al., who used aberration-correlated TEM, the ratio between the average boron concentration and the segregated boron concentration at the austenite grain boundary is approximately 1500;³⁰⁾ therefore, k can be approximated as 1500. This value is similar to those derived from the segregation energy of boron (57–97 kJmol^–1) using various experimental methods, such as SIMS,³¹⁾ autoradiography,³²⁾ and the Auger-electron microscopy,³³⁾ and is also similar to the value reported by Hondros.²⁰⁾

If the molar volume of the austenite grain boundary, V_m^gb, is assumed to be 7.1 × 10^–6 m³mol^–1, then the thickness of the grain boundary, δ, is approximately three atomic layers (approximately 7.5 × 10^–10 m), and the boron concentration in the austenite grain, x^γ, is assumed to be 10^–4 (i.e., approximately 0.002 mass%), the difference in the austenite grain boundary energy from the grain, Δγ ^γ γ, can be calculated to be –0.12 Jm^–2. This value is slightly larger but still of the same order of magnitude as the grain boundary energy necessary for inhibiting the ferrite nucleation, as will be discussed later. Therefore, these results imply that the influence of the boron addition on the ferrite nucleation arises primarily from the reduction in the number of active nucleation sites caused by the decrease in the grain boundary energy of austenite.

3.2. Effect of Boron on the Nucleation Density of Ferrite

The number of ferrite nuclei, n_V , can be expressed as Eq. (6):   

$$n_V=N_{V}exp(-\mathrm{\Delta }{G}^{*}/k_{B}T)$$(6)

where N_V is the number of possible sites for the ferrite nucleation, n_v is the number of actual sites for the ferrite nucleation among N_V, and k_B is the Boltzmann constant.

According to Cahn’s theory,^34,35) the number of vertexes becomes 24 points if every austenite grain consists of a monotonous trisoctahedron (truncated octahedron). Therefore, the relationship between the number of possible sites for the ferrite nucleation and the equivalent radius of the austenite grain can be formulated as Eq. (7) under the assumption that ferrite nucleation is limited to only the 4-grain junctions of austenite grains:   

$$N_V=11.5{(d^{\gamma })}^{-3}$$(7)

This equation is valid under conditions where zero time ferrite nucleation holds and ferrites are distributed homogeneously with a monotonous spherical size. Furthermore, a relationship exists, as shown in Eq. (8), between n_v , which is defined in 3-dimensional space, and n_A, which is defined as the number of actual nucleation sites in 2-dimensional space by the observation of a cut surface of the specimen:   

$$n_A=2R{n}_V$$(8)

where R is defined as the radius of the ferrite grains.

Tarui et al. performed isothermal transformation experiments with the steels listed in Table 1 to elucidate the effects of boron addition on the γ–α transformation.¹⁹⁾ The experiments were conducted with 2-mm-thick specimens, which were isothermally transformed at 978 K for 2–1000 s in a salt bath. Specimens were initially heat-treated for 1.8 × 10³ s at 1323 K to give the austenite grains a uniform average size of 80 μm. They measured not only the number of ferrite grains but also the average size of ferrite grains as a function of the heat-treatment time. As shown in Table 2, the number of ferrite grains per unit volume of both the boron-free steel and the boron-bearing steel can be derived from their measurement using Eq. (8).

Table 1. Chemical composition of systems used for calculation and specimens used in the previous experimental work.

mass%
	C	Mn	Cr	B	Al	Ti	N	Fe
system for calculation	0.45	0.30	0.30	–	–	–	–	balance
B-bearing steel19)	0.43	0.35	0.31	0.0020	0.054	0.017	0.0058	balance
B-free steel19)	0.44	0.36	0.30	0	0.059	0.015	0.0053	balance

Table 2. Nucleation density evaluated from the metallographic observation.

	B-free steel	B-bearing steel
nV (1013m–3)	8.7	4.1

The average austenite grain size in the boron-free steel, was calculated to be 61.1 μm under the assumption that the number of actual nucleation sites, n_V , is almost equal to the number of possible nucleation sites, N_V , which is equal to the total number of 4-grain junctions in the case of the experimental result shown in Table 2. This value is very similar to the experimental result of 80 μm reported by Tarui et al. This similarity suggests that the ferrite nucleation occurred at all 4-grain junctions of austenite without incubation time in the boron-free steel. Hereafter, this mechanism is denoted as zero-time nucleation, which means that ΔG^* is substantially smaller than RT in Eq. (7). According to the analysis by Clemm et al., ΔG^* can be 0 or negligibly small in cases where the ratio between the austenite grain-boundary energy and the austenite–ferrite interfacial energy, γ ^γγ/2γ ^γ α, which is equivalent to the contact angle between austenite and ferrite, exceeds the critical limit because their geometrical balance is collapsed. They noted that the critical value was 0.817 for γ ^γγ/2γ ^γα in the case of the4-grain junctions of austenite. Furthermore, when γ ^γγ/2γ ^γα was greater than the critical value, the ferrite nucleation could occur without the thermal activation process, which suggests that zero-time nucleation could also occur.

Figure 3 shows the relationship between the austenite grain boundary energy, γ ^γγ, and the critical activation energy, ΔG^*, necessary for ferrite nucleation at the 4-grain junctions of austenite under conditions where the austenite–ferrite interfacial energy, γ ^γα, is 0.5 Jm^–2 and 0.6 Jm^–2. The interfacial energy has been estimated to be approximately 1 Jm^–2 in most of metals, as reported by Hillert.³⁶⁾ However, Gjostein et al. derived a value for γ ^γα of approximately 0.5 Jm^–2 in the case of 0.45 mass% carbon steel based on an analysis of their experimental data.³⁷⁾ Furthermore, γ ^γα has been estimated to range from 0.52 Jm^–2 to 0.63 Jm^–2 by Furuhara.³⁸⁾ Thus, the assumption that an appropriate value for γ ^γα lies between 0.5 Jm^–2 and 0.6 Jm^–2 appears to be valid for considering the relationship between γ ^γγ and ΔG^*. As evident in Fig. 3, ΔG^* tends to dramatically increase with a decrease in γ ^γγ, which means that the ferrite nucleation becomes difficult. In such a case of a smaller value of γ ^γ α where ferrite nucleation favors, a decrease in the value of γ ^γγ is also necessary to inhibit ferrite nucleation. Quantitatively, we estimated a 0.12 Jm^–2 decrease in the value of γ ^γγ for steel with 0.002 mass% boron is necessary to inhibit ferrite nucleation, as mentioned in the previous section.

Fig. 3.

Calculated activation energy for ferrite nucleation at 978 K as a function of austenite grain boundary energy, γ ^γγ, taking into account the influence of ferrite/austenite interfacial energy, γ ^γα, with two different 0.5 Jm^–2 and 0.6 Jm^–2, respectively.

We considered that the critical activation energy for the ferrite nucleation is either 0 or negligibly small under the assumption that ferrite nucleates at almost all the 4-grain junctions in boron-free steel. This low critical activation energy means that the ratio between the austenite grain-boundary energy and the austenite–ferrite interfacial energy, γ ^γγ/2γ ^γ α, should be similar to or greater than its critical value of 0.81 derived by Clemm et al. In addition, n_V in boron-free steel is considered to be almost equal to Nv i.e. n_V = Nv = 8.7 on the basis of the values in Table 2. However, the n_V of the boron-bearing steel is 4.1, as shown in Table 2, and n_V (for the boron-bearing steel)/Nv is calculated to be 4.1/8.7 = 0.47. This fact indicates that the ferrite nucleation density in the boron-bearing steel is almost half that in the boron-free steel. From Eqs. (6) and (7), in the case of n_V /Nv = 0.47, the critical activation energy that is necessary for the ferrite nucleation is estimated to be approximately 10^–20 J, which is finite but extremely small.

Enomoto noted that it is possible that ferrite nucleates at specific nucleation sites, thereby maintaining the specific crystallographic orientation relationship that has a coherent interface with low interface energy (much lower than the γ ^γ α value deduced from the balance of interfacial surface tension).³⁹⁾ Furthermore, the results of recent studies by Offerman et al. and van Dijk et al. indicated that the γ ^γ α value was extremely small based on their analyses using synchrotron X-ray diffraction.^40,41) In this case, the ferrite nucleation can occur without the thermal activation process, irrespective of the nucleation sites such as the 4-grain, 3-grain, or 2-grain junctions.

Figure 4 shows the influence of the activation energy for ferrite nucleation on n_V/N_V , which is the ratio of possible nucleation sites among the 4-grain junctions of austenite based on Eq. (6). The n_V /N_V ratio obviously varies drastically even if the change in the austenite grain boundary energy, γ ^γγ, is small, as shown in Fig. 3, because this ratio strongly affects the critical activation energy for ferrite nucleation, ΔG^*. This fact implies that the decrease in the austenite–austenite grain boundary energy, Δγ ^γγ, which is estimated to be 0.12 Jm^–2 for steel with 0.002 mass% boron, may inhibit the ferrite nucleation even when the critical activation energy for the ferrite nucleation is small because the ferrite nucleation occurs at an extremely small volume scale.

Fig. 4.

Effect of activation energy for ferrite nucleation on the fraction of active nucleation sites, n_V/N_V.

3.3. Growth Simulation of Ferrite with DICTRA

DICTRA simulations of ferrite growth were performed on the Fe–C–Mn–Cr system for the compositions given in Table 1, which are simplified versions of the compositions used in the experiments related to the isothermal ferrite transformation in austenite performed by Tarui et al. All thermodynamic data for the calculation of thermodynamic equilibria were taken from the SGTE solution database.²⁸⁾ The boron, aluminum, titanium, and nitrogen concentrations are sufficiently dilute, as shown in Table 1, to justify the assumption that they affect almost no parameters related to the thermodynamic equilibrium; thus, the calculations were performed without consideration of the concentrations of these elements. To calculate the extent of diffusion necessary for the growth simulation of the ferrite grains in austenite, the mobilities of all elements except boron were adopted from the mobility database in DICTRA. The mobility of boron in austenite and ferrite was estimated from the diffusivities reported by Busby et al.,^42,43) and the boron can be treated as a dilute solute in steels. The temperature of 978 K used for the simulation was the same as that used in the isothermal transformation experiment performed by Tarui et al. As previously mentioned in section 2, the ferrite grains in this simulation were approximated as spheres growing in a spherical diffusion cell of austenite.

The equilibrium state at 978 K was calculated using Thermo-Calc, and the result is given in Table 3(a). The calculated equilibrium fraction of ferrite was 92%, whereas Tarui et al. reported that the amount of ferrite plateaus at 40% after approximately 3 × 10³ s at 978 K. The major contribution to this deviation is thought to be explained by the fact that the equilibrium calculation includes pearlite formation, i.e., the presence of ferrite in pearlite is taken into account. In contrast, the experimental observation was limited to primarily ferrite. The metastable equilibrium calculation shown in Table 3(b), where cementite was assumed to not form, i.e., the eutectoid reaction was assumed to not occur, yields 49% ferrite, which is closer to but still substantially greater than the 40% observed.

Table 3. Calculated results of fractions of ferrite and cementite for both equilibrium and metastable conditions by Thermo-Calc at 978 K.

(a) The equilibrium calculation included the presence of cementite.
	ferrite	cementite
molar fraction	0.920	0.080
(b) The metastable calculation excluded the presence of cementite.
	ferrite	cementite
molar fraction	0.492	0.508

Before the growth of ferrite is calculated in DICTRA, the initial size of the diffusion cell of austenite should first be determined. The diffusion cell size is defined as the volume in which a ferrite grain can grow without being influenced by other ferrite particles. In the DICTRA simulation in this study, we calculated the growth of a single ferrite grain, whereas, as a practical matter, several ferrite grains influence each other as they softly impinge during the latter stage of the transformation. Therefore, the DICTRA simulation is valid for the time range from the initiation of the transformation until the soft impingement by other ferrite grains occurs.

Figure 5 shows the change in the volume fraction of ferrite as a function of time, as calculated by the DICTRA simulation, when the diffusion cell sizes were 10^–6, 10^–5, and 10^–4 m. A logarithmic time scale was used for the x-axis in Fig. 5 because the time of rapid increase in the volume fraction of ferrite depended substantially on the diffusion cell size. All the curves show that the volume fraction of ferrite forms plateau at approximately 37% at a relatively rapid rate and that the growth continued at a much slower rate. For the simulation for the smallest cell size, 10^–6 m, the volume fraction of ferrite finally reached 49%, which is equal to the volume fraction at which the metastable equilibrium state is established.

Fig. 5.

DICTRA simulation for the volume fraction of ferrite as a function of time for 3 different cell sizes, 10^–6, 10^–5 and 10^–4 m.

However, the DICTRA simulations for cell sizes of 10^–5 and 10^–4 m did not reach the metastable equilibrium state because the calculation time, which was limited to 10⁶ seconds in this study, was apparently insufficient.

The relatively rapid growth at the early stage of the transformation is controlled only by rapid diffusion of an interstitial element (i.e., carbon) and allows no partitioning of substitutional elements because it is the non-partitioning local equilibrium (NPLE) mode. The growth of ferrite in this mode can persist until the gradient of the chemical potential of carbon becomes negligibly small at the γ/α interface as the transformation proceeds. The diffusion of substitutional elements is then required for further growth until the metastable equilibrium state is finally approached. This sequence means that the growth rate of the ferrite volume fraction substantially decreases because the transformation is controlled by extremely sluggish diffusion elements. This mode will persist until the metastable equilibrium condition (i.e., 49% of the ferrite volume fraction) is established. Therefore, the DICTRA simulation can be reasonably considered to agree with the experimental data of Tarui et al. provided that they determined their data as 40% of the ferrite volume fraction at a specific time condition within the substitute-diffusion-controlled growth.

The calculation results in Fig. 5 also imply that any change in the ferrite volume fraction also substantially depends on the diffusion cell size. For the case where the diffusion cell size is set to be large, the increase in the ferrite volume fraction is remarkably slowed. If all ferrite nuclei are assumed to initially be formed simultaneously and to begin to grow at t = 0, i.e., zero-time nucleation, the cell represents the average distance between the ferrite nuclei. The number of ferrite nuclei per unit volume, n_v, can be approximately expressed by Eq. (9) as a function of the cell radius, R_cell:   

$$n_V\cong \frac{1}{\frac{4}{3}\pi R_{Cell}{}^3}$$(9)

In their experiments, Tarui et al. measured the sizes and the number of ferrite grains per unit volume as functions of time. Table 4 shows the cell radius, R_cell in both the boron-free steel and the boron-bearing steel; these radii were derived from Eq. (9) using the experimental data. The values of R_cell were determined to be 14 μm in the boron-free steel and 18 μm in the boron-bearing steel. These results indicate that R_cell increases by approximately 20% upon the addition of 0.0020 mass% boron.

Table 4. Diffusion cell sizes for boron-free and boron-bearing steels evaluated from metallographic observation.

	B-free steel	B-bearing steel
Rcell (10–6 m)	14	18

Using these R_cell values, we performed a DICTRA simulation to quantitatively study the growth of ferrite. Figure 6 shows the change in the radius of ferrite grains as a function of time during the early stage, i.e., 0–100 s, of ferrite growth in both boron-free steel and boron-bearing steel; the x-axis is indicated as the square root of time. The results of the DICTRA simulation are shown as two lines in Fig. 6. The dotted line represents the results of the calculations with R_cell equal to 14 μm, which is assumed to be the case for boron-free steel. The solid line represents the results of the calculations with R_cell equal to 18 μm, which is assumed to be the case for boron-bearing steel. In addition, the plotted points show the experimental data of Tarui et al., where open marks indicate the results for boron-free steel and solid marks represent those for boron-bearing steel. As evident in Fig. 6, the results of the calculations satisfactorily agree with the experimental results despite the calculations not including an adjustment parameter.

Fig. 6.

Change of ferrite particle radius as a function of square root of time for boron-free steel and boron-bearing steel during relatively early stage of transformation.

After 5 s^–1/2, i.e., approximately 25 s, of transformation time, the deviation between the calculated and the experimental results for the boron-bearing steel increases. The experimental results for the ferrite volume fraction of the boron-bearing steel increase, whereas the experimental results for the boron-free steel appear to become saturated, as do the calculation results for both steels. These results imply that the growth of ferrite is facilitated by the depletion of the carbon concentration at the γ-side of the γ/α interface because of the precipitation of Fe₂₃(CB)₆ in the boron-bearing steel; this hypothesis has been advocated by Tarui et al. In our simulation, certain mechanistic aspects of ferrite growth, such the precipitation of Fe₂₃(CB)₆ at the γ/α interface and the depletion of the carbon, are not taken into account. This omission may explain the deviation between the calculated and experimental results for the boron-bearing steel.

With respect to a comparison between the calculated and the experimental results for the ferrite radius in boron-free steel, the agreement is satisfactory given the lack of any adjusting parameters, even though the calculated value is larger than the experimental value in the time range of 0–100 s. In particular, the gradients of the curves for both the boron-free steel and the boron-bearing steel, i.e., the change in ferrite grain radius as a function of the square root of time in Fig. 6, appears to support the validity of the calculation model for ferrite growth.

In the DICTRA simulation, the influence of the added boron, which increases the diffusion cell size, R_cell, from 14 μm to 18 μm by reducing the total number of ferrite nuclei nucleated by boron, on ferrite growth is small, although a slight enlargement of the ferrite radius with prolonged heat-treatment times was observed.

Figure 7 shows the change in the volume fraction of ferrite as a function of time. The results of the DICTRA simulations are shown as two lines in Fig. 7. One line represents the as-calculated result for the condition that R_cell is equal to 14 μm. The other line represents the result for the condition that R_cell is equal to 18 μm. In addition, the plotted points show the experimental data of Tarui et al., where open marks represent the results for boron-free steel, and solid marks represent the results for boron-bearing steel. At the early stage of transformation, the agreement between the calculated and experimental results is satisfactory. This agreement means that the difference in the size of the diffusion cell determined by the calculation is influenced by the addition of boron. More specifically, the boron added to steels inhibits the nucleation of ferrite and does not affect the growth rate of ferrite. As previously discussed, Tarui et al. considered that ferrite growth was facilitated by carbon depletion via the precipitation of Fe₂₃(CB)₆ at the later stage of phase transformation when boron was added. However, the simulation is performed without consideration of such effects; the simulation considers only the settings for the appropriate diffusion cell sizes. Thus, this fact implies that the simulated and experimental results can lead to different interpretations of the same phenomena.

Fig. 7.

DICTRA simulation showing the change in volume fraction of ferrite as a function of time.

Furthermore, we note that one of the causes of deviations between the calculated and experimental results is that only nucleation at the 4-grain junctions of austenite are treated in the calculations. In this study, we assumed that ferrite nucleates at every 4-grain junction. Given that ferrite nucleation does not actually occur at every 4-grain junction, as mentioned in section 2, the diffusion cell size may be larger than 14–18 μm under the conditions used in this study. Therefore, the nucleation sites may be overestimated as well as the diffusion cell size may be underestimated to the point that it results in deviations between the calculated and experimental results to some extent.

The deviation in the ferrite volume fraction becomes larger in both the boron-bearing steel and the boron-free steel when the transformation time exceeds approximately 100 s. The experimental results are smaller than the calculated results for transformation times of 50 s to 200 s. This deviation may be caused by the emergence of soft impingement by neighboring ferrite particles during the experiment, whereas this effect is not taken into account in the calculations. Furthermore, at transformation times longer than 200 seconds, the experimental results became greater than the calculated results. This deviation is thought to be caused by error in the prediction of the volume fraction of ferrite because the ferrite transformation is controlled by sluggish diffusion of substitutional alloy elements in steel as well as by soft impingement as a negative effect.

From the results shown in both Figs. 6 and 7, the addition of boron decreases the nucleation density and enlarges the diffusion cell size for the growth of ferrite and finally slows the increase in the ferrite volume fraction. However, these results also imply that boron may not directly affect the growth of ferrite.

4. Conclusion

We performed a nucleation-and-growth simulation to evaluate the retardation of isothermal ferrite transformation by the addition of boron to low-alloy steel. The simulation estimated the contribution of boron segregation in the austenite grain boundary to the austenite–ferrite interfacial energy according to Clemm’s geometrical analysis based on the classic nucleation model, and the results were compared with those estimated from experimental observations reported by Tarui et al. Furthermore, we used DICTRA to calculate the volume fraction of ferrite as a function of the transformation time under the condition of three-dimensional growth of spherical ferrite grains. We thereby confirmed the following points.

(1) The effect of boron segregation at the austenite grain boundary was evaluated, and the estimated austenite grain boundary energy was found to decrease by 0.12 Jm^–2 with the addition of 0.002 mass% of boron. This value is slightly larger than that derived from the estimations based on the experimental observations of Tarui et al. but is, however, still of the same order of magnitude.

(2) The result shown above implies that the mechanism of the decrease in the nucleation density of ferrite at the austenite grain boundary, where segregated boron decreases the austenite grain-boundary energy and where the activation energy for the ferrite nucleation simultaneously increases, can be predicted from the kinetic model.

(3) If the ferrite nucleation is assumed to occur at every 4-grain junction of austenite, the calculated austenite grain radius of the boron-free steel agrees with the radius derived from the experimental observations of Tarui et al. In principle, this result supports the model of zero-time nucleation and growth. However, the model also predicts that the nucleation density also substantially decreases at the 4-grain junctions with the addition of boron.

(4) The cell size required for the growth simulation of a ferrite grain was estimated from the experimental information and was verified to increase with the addition of boron. With these evaluated values, the agreement between the DICTRA simulation and the experimental information for the ferrite volume fraction as a function of time is satisfactory, despite the lack of any adjustable parameters in the calculation.

(5) The results imply that boron does not affect ferrite growth during its early stages because the ferrite radius is proportional to the square root of the transformation time for both the boron-free steel and the boron-bearing steel in the DICTRA simulation.

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