Effect of log-normal prior austenite grain size distributions on isothermal γ → α transformation in a low carbon low alloy steel

Abstract

Effect of log-normal prior austenite grain size (PAGS) distributions, with mode PAGS of 35, 65 and125 µm and skewness of 1.57, 1.44 and 1.13 respectively, on isothermal ferrite nucleation site density and initial growth behaviours was investigated. Both mode PAGS and skewness determined the PAG corner nucleation site density for ferrite formation. Faster ferrite transformation kinetics was observed for a higher PAG corner site density. Ferrite grain size refinement was achieved in the final microstructure with the encouragement of PAG corner nucleation, along with an Avrami exponent n of 1.2 for initial ferrite transformation (below 40 vol.%), through modifying mode PAGS / skewness of log-normal PAGS distributions in low carbon low alloy steels.

Keywords

log-normal prior austenite grain size distributions dilatometry isothermal austenite to ferrite transformation Avrami transformation analysis ferrite nucleation and growth

Introduction

Diffusional transformation from austenite to ferrite plays a significant role in determining the final microstructures (e.g. the final ferrite grain sizes and carbon content of second phases), and thereby the strength and toughness in a variety of low carbon low alloy steels. The drive towards leaner steel grades achieving a wider range of properties through processing rather than composition means that greater control of ferrite phase fraction and grain size is needed in chemically heterogeneous materials with a distribution of prior austenite grains (PAGs).

The Johnson–Mehl–Avrami–Kolmogorov (JMAK) analysis, equation (1), has been extensively used to describe the isothermal solid-state phase transformation mechanisms and kinetics that occur in the progression of nucleation, growth and soft / hard impingement in materials.^1–9

V α (t) = 1 - e^{- {kt}^{n}}

(1)

Here, n is termed as the Avrami exponent, which is extrapolated from the slope of a plot of ln [−ln(1−V_α(t))] versus ln(t). k is defined as the rate constant. V_α(t) is the volume fraction of ferrite formed at time t. The original JMAK model has limiting cases for given sets of assumptions, e.g. homogeneous nucleation, site saturation and isotropic growth, giving a range of n values between 1 and 4.⁶ Theoretical consideration of ferrite formation within mono-disperse (a single prior austenite grain size (PAGS)), normal and log-normal PAGS distributions gave n values varying from 3.5 (mono-disperse) to 2.8 (log-normal) under the three-dimensional site saturation nucleation condition.¹⁰ Experimental determination of n for the diffusional austenite to ferrite transformation in plain carbon and HSLA steel grades, considering a range of alloying compositions, falls into a lower range of 0.8–1.4.^11–14 The semi-empirically fitted n values are much lower than the theoretical value for a log-normal PAGS distribution,¹⁰ although the experimental PAGS distributions were not given in these studies. It can be assumed that they will be log-normal from the heat treatment used before transformation.^11–14

PAGS distributions, rather than an average PAGS, determine the PAG corner number density per unit area / volume (on a two- or three-dimensional basis) that have a low energy barrier for ferrite nucleation. A decrease in average / mode PAGS^15–18 and / or skewness within log-normal PAGS distributions can directly lead to an increase in PAG corner number density per unit area, which will accelerate ferrite nucleation, and thus, the overall transformation kinetics. Compared to a mono-disperse PAGS, normal and log-normal PAGS distributions have been predicted to broaden the γ → α transformation curves (the latter being greater), where the presence of larger PAGs in the distribution shifts the transformation curve to a longer time, and hence, a lower Avrami n value.¹⁰ Whilst the principles of how different PAGS distributions affect ferrite formation are recognised, quantitative analysis linking experimental log-normal PAGS distributions with ferrite nucleation and initial growth, and Avrami n values, has not been reported yet. Therefore, this paper focuses on the influence of log-normal PAGS distributions with different average / mode PAGS and skewness on ferrite nucleation and initial growth behaviours based on the JMAK analysis using dilatometric data and quantitative metallographic examination in a low carbon low alloy steel grade.

Material and methodology

Material

A commercial hot-rolled low carbon low alloy steel plate, Table 1, was used in this study. The steel had been continuously cast and so remnant micro-segregation (elongated as bands parallel to the rolling direction resulting in a banded ferrite and pearlite microstructure with the minimum band spacing of around 20 µm) was present.

Table 1.

Chemical composition (wt%) for the commercial low carbon low alloy steel plate used in this study.

	C	Mn	Si	Cu	Ni	Cr	Mo	P	S	Al	Ti	N
Nominal composition	0.127	0.806	0.317	0.310	0.275	0.122	0.055	0.014	0.018	0.021	0.036	0.0095

Furnace and dilatometric heat treatment routines

Prior to machining of dilatometry samples, 10 × 20 × 150 mm blocks were either normalised, air cooled to room temperature and then re-austenitised at 950–1150 °C for different times in a furnace followed by quenching into an ice + water mixture, Table 2, or just re-austerities to generate log-normal PAGS distributions with varied average / mode PAGS and skewness.

Table 2.

Furnace heat treatment routines to produce different log-normal PAGS distributions.

Specimen ID	Normalization		Re-austenitisation
	Temperature / °C	Time / mins	Temperature / °C	Time / mins
AQ950	-	-	950	15
NAQ1000	1050	30	1000	30
AQ1150	-	-	1150	60

Cylindrical samples, 10 mm in length and 4 mm in diameter, were machined from steel blocks listed in Table 2 (sample length perpendicular to the rolling direction) to carry out the isothermal ferrite transformation using a Bähr DIL 805A/D differential dilatometer. The dimensional variations were measured by a linear variable differential transformer (LVDT) in a gas-tight enclosure to perform all tests under vacuum or in an inert atmosphere. The resolution of the dilatometer is ±50 nm. Prior to loading between the silica pushrods of the dilatometer, the original sample length (L₀) was recorded using digital vernier callipers and an S-type thermocouple was spot welded to the specimen centre. After loading, the dilatometer chamber was pumped down to a pressure < 5 × 10⁻⁴ Pa prior to heat treatment. Isothermal ferrite formation after rapid cooling from the re-austenitisation treatment took place at 680–750 °C, Table 3, where final cooling after the isothermal hold was by helium to minimise any further diffusional transformation.

Table 3.

Dilatometry heat treatment routines for isothermal ferrite formation.

Specimen ID	Heating rate to T_A / °C/s	T_A / °C	t_A / mins	First cooling rate from T_A to T_iso / °C/s	T_iso / °C	t_iso / s	Second cooling rate from T_iso to T_f / °C/s
AQ950_T_iso_t_iso	10	950	5	50	680–750	3–5400	200
NAQ1000_T_iso_t_iso		1000
AQ1150_T_iso_t_iso		1150

T_A and t_A are the re-austenitisation temperature and time. T_iso and t_iso are isothermal austenite to ferrite transformation temperature and time. T_f is the final temperature of 100 °C.

Dilatometry curve analysis

The specimen length change (dilation, ΔL) was recorded during heating, isothermal holding and cooling cycles in the dilatometer and was converted into the fractional dilation change as ΔL/L₀, which was approximated to the volume fraction change using equation (2).

3 \frac{Δ L}{L_{0}} = \frac{Δ V}{V_{0}}

(2)

The time-dependent fractional dilation during isothermal holding was used to calculate the variation in ferrite volume fraction (V_α) with isothermal holding time by the method derived in,¹⁹ where the volume of one new austenite unit cell (V_γu) can be calculated either due to the volume change of the original austenite unit cell, or due to the formation of ferrite and the partitioning of carbon into the residual austenite with associated changes in the lattice parameters, equation (3).

V_{γ u} = 3 \frac{Δ L}{L_{0}} a_{γ}^{3} + a_{γ}^{3} = 2 V_{α} a_{α}^{3} + (1 - V_{α} {) a}_{γ}^{^{'} 3}

(3)

a_γ is the austenite lattice parameter for the nominal carbon content; a_γ’ is the austenite lattice parameter after the partitioning of carbon; a_α is the ferrite lattice parameter (all lattice parameters are calculated at the isothermal temperature).

Room temperature X-ray diffraction (XRD) patterns were obtained from an NAQ1000 specimen that had been tempered for 1 h at 600 °C using an Anton Paar XRDynamic 500 instrument with a Co target (wavelength λ = 1.79026 Å) and scanning time of 2 h for a 2θ range from 45° to 130°. Four peaks ((011)_α, (002)_α, (211)_α and (022)_α) were identified, from which individual lattice parameters were calculated and plotted against cos²θ so that an accurate lattice parameter (2.870 Å) could be obtained from extrapolation to the intercept with the vertical axis. The austenite lattice parameter for the nominal compositions at room temperature was calculated using equation (4)^20,21:

\begin{aligned} a_{γ} = 3.578 + 0.033 w t (C) + 0.00095 w t (Mn) \\ - 0 .0002 w t (Ni) + 0.0056 w t (Al) + 0.0006 w t (Cr) \\ + 0.0031 w t (Mo) + 0.0018 w t (V) \end{aligned}

(4)

Where wt(i) corresponds to the weight percent of elements “i” in austenite (bulk values for substitutional alloying elements). As interfacial equilibrium is assumed during ferrite formation then the volume change will be from carbon-partitioned austenite, i.e. the Thermo-Calc predicted equilibrium carbon content in austenite at the isothermal transformation temperature, which is the value used in equation (4) for calculating the austenite lattice parameter after carbon partitioning. Thermal expansion coefficients were determined from the linear portions of heating curves between 600 and 1000 °C before and after α → γ transformation as 1.49 × 10⁻⁵ / °C for ferrite and 2.41 × 10⁻⁵ / °C for austenite.

JMAK-based transformation analysis

A modified JMAK-based transformation analysis,^22,23 taking the ratio between the experimental ferrite volume fraction at a certain time, V_α(t), and the equilibrium volume fraction, V_e, into consideration, is expressed as

\frac{V_{α} (t)}{V_{e}} = 1 - e^{- {kt}^{n}}

(5)

$\frac{V_{α} (t)}{V_{e}}$ approaches 1 when the isothermal holding time goes to infinity and the equilibrium condition is achieved. The experimental transformation plots reached a limiting ferrite volume fraction that was lower than the equilibrium value, and this limiting experimental value was used for V_e in equation (5). The predicted equilibrium ferrite volume fraction was used for V_e in the JMatPro predictions. The Avrami exponent n is the slope of a plot of ln [ln ( $\frac{V_{e}}{V_{e} - V_{α} (t)}$ )] vs ln(t) using linear regression, and ln k is the intercept with the vertical axis. JMAK analyses were carried out for both experimental (dilatometric) data and JMatPro predictions.

Microstructure characterisation

PAGS distributions

Furnace heat-treated steel blocks in Table 2 were sectioned from the RD (rolling direction) - TD (transverse direction) plane and polished to a 0.05 µm alumina polishing suspension finish, and dip etched in a mixture of aqueous picric acid, teepol (a wetting agent) and 37 wt% hydrochloric acid (Bechet and Beaujard's etchant²⁴) for 15–30 min. The etched samples were imaged under a Keyence VHX digital optical microscope and over 500 PAGs selected for each specimen to form an equivalent circle diameter (ECD) distribution using the image analysis software, ImageJ.

The PAGS distributions generated from the re-austenitisation treatment used in the dilatometer, Table 3, were also checked as above, and kept consistent with those developed using furnace heat treatments in Table 2 in terms of the mode PAGS and skewness for all log-normal PAGS distributions.

Ferrite grain development

The dilatometer heat-treated specimens were sectioned transversely at the central thermocouple position and polished as above for quantification of the development of allotriomorphic ferrite. Ferrite allotriomorphs were characterised using a JEOL 7800 SEM fitted with Oxford SYMMETRY electron backscatter diffraction (EBSD) and Keyence VHX digital optical microscope in terms of grain number density, length (along prior austenite grain boundaries (PAGBs)) and thickness (perpendicular to PAGBs). Over 300 ferrite grains were included for the ImageJ analysis of ferrite dimensions, where the fitting ellipse function was used to obtain the Feret max value (the major axis of the fitted ellipse) and Feret min value (the minor axis of the fitted ellipse) to represent the length along PAGBs and thickness normal to PAGBs of ferrite grains respectively.

Phase field simulations for determining the PAG corner number density per unit area

Synthetic PAG microstructures were generated using phase field simulations in MICRESS software. These microstructures were produced by distributing 1000 seed locations with various weighting factors to provide a range of final distributions. The Voronoi-based approach was utilised in MICRESS to generate synthetic PAGs to achieve a more realistic microstructure, and a homogenisation simulation was carried out at 1200 °C for 4 h to remove unstable features in the Voronoi grain distribution as the tessellation of Voronoi grains allows unrealistic grain shapes to be generated. The PAG morphology was taken at 1 h time intervals from 1 h to 4 h for a range of seed conditions providing > 200 microstructures, each of which contains > 2000 PAGs; an example is shown in Figure 1(a). The 3D PAG corners can be determined from the microstructure, as seen in Figure 1(b). The consistency between the PAGS distributions for the measured 2D experimental plane and a 2D section from the simulated 3D grain microstructure was also examined and discussed in Section 3.1.

Figure 1.

Synthetic PAG microstructures generated in MICRESS showing (a) the PAG morphology and (b) the 3D PAG corners.

The simulated microstructures were employed to predict the number density of intercepts between PAGBs per unit volume, N_{PAGB interc},. However, when observing a 2D surface during characterisation of ferrite allotriomorph number densities, out-of-plane nucleation events that occur can grow and project to the observation plane to augment the allotriomorphs from triple points that would be counted on a representative 2D section. For that reason a metric has been determined which relates the 3D intercept density to the effective grain corner-nucleated allotriomorphs on a random 2D section. This involves multiplying the 3D N_{PAGB interc} by half the mode 3D grain size to give N_{PAG corners}, with the unit of mm⁻², which allows characterisation to take the out-of-plane nucleation into consideration using equation (6).

N_{PAG corners} = 0.5 d_{mode} \times N_{PAGB interc}

(6)

Where d_mode is the mode PAGS for log-normal PAGS distributions.

The relationship between N_{PAG corners}, mode PAGS and skewness is shown in Figure 2. Qualitatively, Figure 2 clearly shows that N_{PAG corners} increases as the mode PAGS and / or skewness decreases. Quantitatively, a relatively high N_{PAG corners} value of around 3000 mm⁻² can be achieved in log-normal PAGS distributions with either a mode PAGS of 30 µm and a skewness of 1.13, or a mode PAGS of 45 µm and a skewness of 0.46. Therefore, both (theoretical) grain size distributions would be expected to lead to similar fine ferrite grain sizes in the final microstructure, assuming PAG corner nucleation dominates in both cases.

Figure 2.

The predicted relationship between N_{PAG corners}, mode PAGS (note this is the 3D mode grain size) and skewness for log-normal PAGS distributions.

The N_{PAG corners} values for the log-normal PAGS distributions were compared with those values (N’_{PAG corners}) for mono-disperse PAGS, considering average PAGS and using equation (6), where d_average was utilised to replace d_mode.

JMatPro simulations for the diffusional austenite to ferrite transformation

Ferrite volume fractions as a function of time were predicted using the chemical compositions, Table 1, and average PAGS for AQ950, NAQ1000 and AQ1150 samples at the three isothermal holding temperatures using JMatPro software (version 14) with the general steel database.

Results and discussion

Log-normal equiaxed PAGS distributions

The re-austenitisation treatments in Table 3 resulted in log-normal equiaxed PAGS distributions with different averages / modes / maxima and skewness values, as seen in Figures 3 and 4(a), and given in Table 4. In all cases, the maximum PAGS was < 3 times the average, so the distribution would not be considered to contain abnormal grains. The three experimental log-normal PAGS distributions show a gradual increase in the mode PAGS with a similar skewness between AQ950 and NAQ1000, and then a much more noticeable increase in the mode PAGS but reduced skewness for AQ1150. In addition, the experimental and predicted PAGS distributions show a good consistency in terms of mode PAGS and skewness values for AQ950 and NAQ1000, with an example shown in Figure 4(b). The predicted PAGS distribution was derived by taking a number of 2D sections from the simulated 3D grain structure, e.g. Figure 1(a), with a total number of PAGS as in the experimental measurement. It is obvious, however, that the experimental PAGS distribution for AQ1150 is tighter than the prediction, where the predicted skewness is 1.70, Figure 4(c).

Figure 3.

PAG morphologies for (a) AQ950, (b) NAQ1000 and (c) AQ1150 specimens.

Figure 4.

(a) The experimental log-normal PAGS distributions developed for the three samples in Figure 3, and the comparison of the experimental (2D grains) and corresponding 2D simulated (3D grains) log-normal PAGS distributions for (b) AQ950 and (c) AQ1150.

Table 4.

PAGS, skewness, and predicted N_{PAG corners} and N’_{PAG corners} for all three samples.

	AQ950	NAQ1000	AQ1150
Mode PAGS / μm	35	65	125
Average PAGS / µm	42	74	131
Maximum PAGS / μm	115	175	265
Skewness	1.57	1.44	1.13
Predicted N_{PAG corners} / mm⁻² (considering log-normal PAGS distributions)	1530	699	116
Predicted N’_{PAG corners} / mm⁻² (considering the average PAGS)	3474	1146	360

The changes in mode PAGS and skewness for all predicted log-normal PAGS distributions can translate into a gradual reduction in N_{PAG corners} in Table 4 which more than halved between AQ950 and NAQ1000, and decreased by an order of magnitude between AQ950 and AQ1150. The predicted N_{PAG corners} values for AQ950 and NAQ1000 are expected to provide an accurate determination of the balance of ferrite nucleation from PAG corners, edges and face sites (discussed in Section 3.4), due to the strong agreement between the simulated and experimental PAGS distributions in those samples. However, the relatively poor fit between simulated and experimental PAGS distributions in AQ1150 indicates that the predicted N_{PAG corners} value would be lower than experimentation due to a greater predicted skewness utilised. It is also noted that N’_{PAG corners} values for mono-disperse PAGS (using the average PAGS for each distribution) are 1.6–3.1 times greater than the predicted N_{PAG corners} values when log-normal PAGS distributions are considered for all three samples. But a lower ratio (<3.1) would be expected in AQ1150 if the smaller experimental skewness is used. Nevertheless, simulations^25,26 of ferrite nucleation and / or overall transformation kinetics, taking only average PAGS into account (i.e. JMatPro predictions, discussed in Section 3.2), are expected to be much quicker than experimental results.

In practice, ferrite nucleation can occur from PAG corners and, edge and face sites along PAGBs, with the balance between the number density of these nucleation sites depending on the full log-normal PAGS distribution, rather than the average / mode PAGS. A larger mode PAGS and / or skewness for log-normal PAGS distributions results in a lower N_{PAG corners} value, as shown in Figure 2; hence, PAG edge / face nucleation, which has a higher energy barrier^27,28 compared to PAG corner nucleation, is more likely to occur in AQ1150 compared to AQ950 and NAQ1000 (more analysis in Section 3.4), as the PAG corner nucleation and initial growth might not consume these PAG edge / face sites rapidly.

Dilatometric measurements on the isothermal austenite to ferrite transformation

An example of the measured relative length change for the NAQ1000_T680_1800s specimen is shown in Figure 5(a). The segment AB corresponds to thermal expansion of the initial martensitic microstructure during continuous heating without any occurrence of phase transformation. The segment BC represents the formation of austenite on heating with its associated length contraction. The segments CD and DE are the thermal expansion and contraction of austenite on heating and initial cooling after the re-austenitisation hold respectively. The segment EF is the dilation associated with isothermal austenite to ferrite transformation, where the relative length increases with increasing isothermal holding time. FG represents the dilation associated with the remaining austenite to bainite / martensite transformation coupled with thermal contraction during post-isothermal hold helium gas quenching.

Figure 5.

(a) An example showing the measured relative length change for the NAQ1000_T680_1800s specimen during the dilatometric heating, isothermal holding and cooling cycle; (b) the comparison between predicted and experimental isothermal ferrite formation for NAQ1000 specimens at different undercoolings; and (c) the comparison between predicted and experimental isothermal ferrite formation at 720 °C for the three log-normal PAGS distributions. The experimental error for deriving the ferrite volume fraction using dilatometric data is within ± 5%.

Dilation changes for the isothermal austenite to ferrite transformation, i.e. EF in Figure 5(a), have been replotted as ferrite volume fraction against time, where the transformation traces yield a sigmodal time dependence of ferrite volume fraction, illustrating the progression of nucleation, growth and impingement during isothermal ferrite formation, as shown in Figure 5(b) and (c). It is noted that the experimental isothermal austenite to ferrite transformation kinetics and limiting ferrite volume fractions for all NAQ1000_T680 / 720 / 750 specimens in Figure 5(b) are significantly lower than the predictions where an average PAGS and uniform compositions have been taken into account using JMatPro. The limiting ferrite volume fractions increase with undercooling, i.e. from 45.0 vol.% at 750 °C to 77.2 vol.% at 680 °C, even though the continued austenite to ferrite transformation is still occurring sluggishly after an extended isothermal holding at 750 °C. This trend is in line with the predicted equilibrium ferrite volume fractions at different undercoolings, although the difference in limiting ferrite volume fractions between 680 and 750 °C is over twice that of predictions. This might be attributed to a synergistic effect from carbon partitioning upon transformation, which causes a variation in the carbon content in the untransformed austenite (hence, a driving force variation), and Mn segregation across bands, leading to a deceleration and / or even stopping of transformation, constraining ferrite within bands.¹⁹

When different log-normal PAGS distributions are considered, the fastest initial transformation is seen in AQ950_T720 with the smallest mode PAGS compared to NAQ1000 / AQ1150_T720 in Figure 5(c). In principle, the initial transformation rate depends on (i) the rate of nucleation, determined by the balance between nucleation site density and potency of nucleation at specific sites,^16–18 and (ii) the competition between nucleation and initial growth.²⁹ A decrease in N_{PAG corners} from AQ950 through NAQ1000 to AQ1150 in Table 4 theoretically leads to a retardation in nucleation at the same undercooling, hence, delaying the overall transformation kinetics. Consequently, the quickest and slowest experimental transformation kinetics are observed in AQ950_T720 and AQ1150_T720 respectively, aligning well with the predicted ferrite transformation kinetics for these samples, although to different extents quantitatively. The lower apparent limiting ferrite volume fraction with increasing mode PAGS is also expected. The lower amount of grain boundary per unit volume as mode PAGS increases means that these sites are fully decorated at lower ferrite volume fractions leaving more of the total ferrite volume fraction to be realised by 1D growth into the PAG interior normal to the PAGBs.

JMAK-based transformation analysis

The double ln - ln(t) Avrami plots of experimental data for the range of log-normal PAGS distributions / undercoolings studied show very similar, non-linear behaviour, Figure 6. Possible reasons have been previously proposed for the deviations of experimental kinetics from purely linear Avrami plots^30,31 including: (i) the non-random distribution of heterogeneous nucleation sites; (ii) the overlapping of two or more simultaneous reactions; and (iii) the effects of the experimental uncertainties in calculating product volume fractions. By contrast, the predicted Avrami plots (dashed lines in Figure 6 based on the JMatPro predicted ferrite transformation) display a linear shape over a shorter time scale to achieve the equilibrium ferrite volume fraction, and a single linear fit over the entire range of the predicted data results in a high Avrami exponent n value of around 3.5. This n value keeps constant with undercoolings and average PAGS, indicating that the austenite to ferrite transformation is treated more as a three-dimensional ferrite growth with a reducing nucleation rate time dependency in the JMatPro prediction, similar to theoretical calculations of n for ferrite formation considering mono-disperse PAGS in.¹⁰ In addition, the rate constant k differs with undercooling and average PAGS, i.e. changing from 0.003 at 680 °C to 2.3 × 10⁻⁴ at 750 °C for NAQ1000, or varying from 0.010 for AQ950 to 1.3 × 10⁻⁴ for AQ1150 at 720 °C. Hence, the difference in Avrami plots between JMatPro predictions and experimental data clearly shows the importance of considering the full log-normal PAGS distribution.

Figure 6.

(a) Comparisons of Avrami plots using the experimental and predicted ferrite volume fractions for NA1000_T680 / 720 / 750 specimens, and (b) comparisons of Avrami plots using the experimental and predicted ferrite volume fractions for AQ950 / NAQ1000 / AQ1150_T720 specimens within three log-normal PAGS distributions. A lower slope at the very early transformation stage for the NAQ100_T750 sample in Figure 6(a) has been identified, probably due to the retardation of ferrite nucleation at a lower undercooling, hence, a smaller driving force.

Differentiation of the non-linear experimental Avrami plots in Figure 6 results in a linear regime when the ferrite volume fraction is less than 40 vol.% of the overall phase constitutions, where Avrami exponents, n, of 1.0–1.2 are determined, Table 5. The identified n value range is slightly narrower than the semi-empirically fitted n of 0.8–1.4 for ferrite formation in plain carbon and HSLA steel grades.^11–14 The linear regime plots for different undercoolings and PAGS distributions show a reduction in n as undercooling decreases or mode PAGS increases. For different undercoolings studied in the NAQ1000 samples, Figure 6(a), the n value decreases slightly with decreasing undercooling, which is consistent with lower driving forces reducing the nucleation rate, whilst higher temperature gives more rapid diffusional growth and so an overall decrease in number density of nucleation sites activated. For different PAGS distributions studied at the same undercooling, Figure 6(b), the slight reduction in n is attributed to a decrease in PAG corner site density, related with an increase in mode PAGS and / or skewness, to retard the overall nucleation rate but encourage growth. The rate constant, k, which depends on both nucleation and growth rates,^22,23 varies with undercooling and PAGS distributions, where an increase in undercooling and / or mode PAGS leads to a higher k. As n and k are linked with the time dependence / dimensionality of ferrite nucleation and growth behaviour within this linear regime, then these values should be reflected in the dimensions and number densities of ferrite allotriomorphs with isothermal holding time.

Table 5.

Avrami exponent n, rate constant k, R² and ferrite volume fraction range for the linear regime of the experimental data in Figure 6.

	Avrami exponent n	Rate constant k	R²	Ferrite volume fraction range (ln(t) range) / %
NAQ1000_T680	1.2₆	0.042	0.994	0–39.4 (0–2.4 s)
NAQ1000_T720	1.2₀	0.028	0.997	0–30.8 (0–2.7 s)
NAQ1000_T750	1.0₇	0.009	0.994	0–25.7 (0–4.2 s)
AQ950_T720	1.2₄	0.033	0.999	0–35.6 (0–2.5 s)
AQ1150_T720	1.0₂	0.014	0.994	0–30.6 (0–4.0 s)

The development of ferrite allotriomorphs

The variation in number density of allotriomorphs and their dimensions along (half-length) and normal (half-thickness) to the PAGBs for the NAQ1000 specimens at all three isothermal transformation temperatures are shown in Figure 7. Figure 7(a) shows that the number densities of ferrite allotriomorphs in the NAQ1000_T680 and NAQ1000_T720 samples already exceed the simulated N_{PAG corners} for the first data point (ln(t) = 1.6 and t = 5 s), which suggests that these most favourable nucleation sites saturate very quickly. The continued increase in ferrite allotriomorph number density is consistent with nucleation continuing at PAG edges and faces up to the end of the linear region (ln(t) = 3 and t = 20 s). Nucleation is accompanied by growth along PAGBs from already formed nuclei and more limited growth normal to PAGBs in both samples, as seen in Figure 7(b) and (c). The faster growth along PAGBs at the higher isothermal transformation temperature of 720 °C removes more potential nucleation sites so that the ferrite allotriomorph number density at the end of the linear region for NAQ1000_T720 is lower than for NAQ1000_T680. The allotriomorphs formed at 720 °C have larger dimensions than those formed at 680 °C, but the reduced number density and lenticular shape mean that a smaller volume fraction of ferrite is present at the end of the linear region, Figure 5(a). The apparent reduction in number density at times beyond the linear region and reduced growth rates, Figure 7(b) and (c), suggest that the linear region corresponds to grain boundary nucleation (at PAG corners, edges and faces) along with 2D diffusional growth along the PAGB plane and 1D diffusional growth normal to the PAGBs until complete decoration of the PAGBs at around ln(t) = 3. Metallographic images in Figure 8 support the development of PAGB ferrite allotriomorphs with complete decoration of the PAGBs at around 20 s at 720 °C, Figure 8(b). The balance of nucleation and growth give an Avrami n value of around 1.2, Table 5.

Figure 7.

(a) The time-dependent number densities of ferrite allotriomorphs, (b) ferrite lengthening and (c) thickening kinetics for NAQ1000_T680 / 720 / 750 specimens.

Figure 8.

The development of allotriomorphic ferrite for (a) NAQ1000_T720_5s, (b) NAQ1000_T720_20s, and (c) NAQ1000_T750_120s samples at the early transformation stage.

Transformation at an even higher isothermal holding temperature (720 vs 750 °C) results in a reduced extent and rate of nucleation compared with growth and so a slight reduction in n value, Table 5. This shift from nucleation to growth and reduction in n value (to around1.0) is exaggerated as the isothermal holding temperature is raised to 750 °C, Table 5. Under this reduced undercooling and driving force, nucleation is slowed sufficiently that the ferrite allotriomorph number density is still around the value for PAG corners at ln(t) = 3. Nucleation then proceeds until around ln(t) = 4.8 (t = 120 s) with the peak ferrite number density gained, as the metallographic image is shown in Figure 8(c), after which nucleation stops, Figure 7(a), and growth changes to a much slower 1D diffusional rate similar to those seen for the other isothermal holding temperatures. Overall, the development of ferrite allotriomorphs is the same at all three holding temperatures.

When taking the three log-normal PAGS distributions into consideration, a consistent Avrami exponent n of 1.0–1.2 is also derived for the linear regime, as listed in Table 5, following the same 3D nucleation coupled with 2D diffusional growth along the PAGB plane and limited 1D diffusional thickening normal to PAGBs behaviour discussed above. Ferrite nucleation occurs very rapidly at PAG corners for AQ950_T720 (ln(t) = 1.6, similar to NAQ1000_T720 as above), Figure 9(a). The increase in ferrite allotriomorph number density aligns well with the continued PAG edge and face nucleation up to the end of the linear region (ln(t) = 3 and t = 20 s). However, the ratio between the peak allotriomorph number density and N_{PAG corners} for AQ950_T720 (over 3.0) is lower than that for NAQ1000_T720 (over 5.0). This indicates that as N_{PAG corners} increases, due to a decrease in the skewness for a constant mode PAGS, or vice versa, as seen in Figure 2, the PAGB length between corners decreases reducing the number density of potential PAG edge / face nucleation sites. At the same undercooling growth along the PAGBs will be similar (as shown in Figure 9(b)) so that a greater portion of potential sites are consumed and a lower ratio on edge / face to corner nucleation.

Figure 9.

(a) The time-dependent number densities of ferrite allotriomorphs, (b) ferrite lengthening and (c) thickening kinetics for AQ950 / NAQ1000 / AQ1150_T720 specimens.

The log-normal PAGS distribution developed for AQ1150 has a very large mode PAGS but a reduced skewness, which results in much slower ferrite formation due to the retardation in nucleation with a reduced availability of PAG corners; correspondingly, a reduced Avrami n of 1.0 is gained in this sample, as listed in Table 5. PAG corners are saturated first (<ln(t) = 2.3), and then nucleation moves to PAG edge and face sites which have a higher energy barrier.^27,28 The peak ferrite allotriomorph number density for AQ1150_T720 is considerably lower than that for AQ950 / NAQ1000_T720 in Figure 9(a), as N_{PAG corners} is very low in the former case, Table 4; but the ratio between the peak value and N_{PAG corners} (slightly above 7.0) is higher than the later samples. This suggests an even stronger involvement of PAG edges and faces to accommodate ferrite nuclei, in addition to PAG corner nucleation, in AQ1150_T720. The balance between corner and edge / face nucleation brought about by the large mode PAGS sample (AQ1150_T720) is comparable to that reduced isothermal holding temperature in a finer mode PAGS sample (NAQ1000_T680) with the ratio between the peak allotriomorph number density and N_{PAG corners} slightly exceeding 7.0, where reduced growth along PAGBs has reduced the consumption of edge / face nucleation sites by grain-corner nucleated allotriomorphs.

Ferrite growth behaviours are consistent in AQ950 and NAQ1000, taking the experimental errors into consideration, as shown in Figure 9(b) and (c), but in AQ1150, ferrite allotriomorphs grow to a greater size with a reduced number density. This indicates the retardation of soft impingement occurring in the largest mode PAGS sample along with the continued transformation observed in Figure 5(c).

Control of diffusional austenite to ferrite transformation is a key metallurgical tool to tailor ferrite grain sizes in the final microstructure during steel processing. Log-normal PAGS distributions in the starting microstructure (i.e. after austenite recrystallization and austenite grain growth during hot rolling³² and / or during re-austenitisation followed by controlled cooling³³) determine the nucleation site density for ferrite, especially the N_{PAG corners} value, which is influenced by both the mode PAGS and skewness, as seen in Figure 2. A narrower range of Avrami exponent n values of 1.0–1.2 is identified for ferrite nucleation and initial growth, compared to the reported experimental n range of 0.8−1.4 in plain carbon and HSLA steels,^11–14 in the studied log-normal PAGS distributions with N_{PAG corners} increasing from around 120 mm⁻² to >1530 mm⁻². The minor decrease of n from 1.2 to 1.0 is related to a retardation in nucleation rate, which is attributed to either a 70 °C reduction in undercooling (thus, driving force) for the same log-normal PAGS distribution with N_{PAG corners} of around 700 mm⁻², or a 1400 mm⁻² decrease in N_{PAG corners} for varied log-normal PAGS distributions at a constant undercooling of 145 °C.

It is noted that the amount of PAG edge / face nucleation for the log-normal PAGS distribution with N_{PAG corners} of around 120 mm⁻² is twice that for the log-normal PAGS distribution with N_{PAG corners} > 1530 mm⁻². However, the PAG corner nucleation site density, which also determines the balance of PAG corner, edge and face nucleation, is the dominant factor in controlling the n values and ferrite grain size. Consequently, a higher N_{PAG corners}, along with the Avrami exponent n of 1.2, gives rise to a finer ferrite grain size in the final microstructure, as shown in Figures 7 and 9.

Conclusions

The effects of log-normal PAGS distributions on the isothermal ferrite nucleation and growth behaviour with respect to different undercoolings have been investigated in a low carbon low alloy steel based on dilatometric measurements, JMAK theory and microstructural characterisation. The main conclusions are:

Log-normal PAGS distributions determine the nucleation site density for ferrite formation, and the PAG corner number density per unit area, N_{PAG corners}, depends on the mode PAGS and skewness.

Log-normal PAGS distributions influence the transformation kinetics and limiting ferrite volume fractions during isothermal austenite to ferrite transformation. The fastest ferrite transformation kinetics and highest limiting volume fraction were observed in the smallest mode PAGS sample at the same undercooling.

Linear Avrami transformation analysis was carried out for ferrite volume fraction < 40 vol.%, where the determined Avrami exponent, n, of 1.0–1.2 represents the transformation mechanism of continued 3D nucleation, 2D diffusional growth along the PAGB plane and limited 1D thickening normal to PAGBs. A retardation in nucleation, due to either a lower undercooling or a reduced N_{PAG corners}, resulted in a slight decrease in n from 1.2 to 1.0.

The balance of ferrite nucleation from PAG corner, edge and face sites along PAGBs has been established for different log-normal PAGS distributions. The amount of PAG edge / face nucleation for the log-normal PAGS distribution with N_{PAG corners} of around 120 mm⁻² is twice that of the log-normal PAGS distribution with N_{PAG corners} > 1530 mm⁻², although PAG corner site saturation is significantly delayed in the former case.

Refined ferrite grain sizes in the final microstructure were observed when there was a greater amount of PAG corner nucleation, seen for log-normal distributions with a higher N_{PAG corners}, which could be gained from either a smaller mode PAGS or lower skewness.

Footnotes

Acknowledgements

This work was supported by Engineering and Physical Sciences Research Council (EPSRC) (grants No. EP/S005218/1 and EP/S018107/1).

Author contribution(s)

Yulin Ju: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Validation; Writing – original draft.

Carl Slater: Conceptualization; Software.

Martin Strangwood: Conceptualization; Supervision; Validation; Writing – review & editing.

Nathan Dixon: Methodology.

Claire Davis: Formal analysis; Project administration; Supervision; Writing – review & editing.

Data availability

Data will be made available on request.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Engineering and Physical Sciences Research Council, (grant number EP/S005218/1, EP/S018107/1).

ORCID iDs

Yulin Ju

Carl Slater

Nathan Dixon

Claire Davis

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