Modeling Dissolution of Vanadium Carbide and Carbonitride in Fe–C–V(–N) Austenite

Abstract

A computer model is constructed to simulate the dissolution of V carbide and carbonitride with size distribution in steels. Assuming local equilibrium of carbon, nitrogen, and V at the particle/matrix interface, the dissolution rate is calculated using the mean-field and invariant field approximations. The fraction of particles and size distribution (PSD) of V carbide are in good agreement with those in an Fe–C–V alloy reported in the literature. The V mass fraction and PSD of carbonitride, measured by extraction replica in this study, were also reproduced well by simulation in an Fe–C–V–N alloy (N~20 ppm). Moreover, simulation with an equilibrium tie-line passing through the bulk alloy composition, as often done in the calculation of precipitate dissolution rate, yielded a large error.

1. Introduction

Grain refinement of ferrite and dislocation hardening caused by phase transformation from austenite, solid solution hardening due to alloying and precipitation hardening by alloy carbide are one of the leading means of strengthening of ferrite and bainite.¹⁾ V, Ti, and Nb, which are called micro-alloying elements, are effective for precipitation and grain refinement hardening, and they play a major role in hardening of HSLA (High Strength Low Alloy) steels.^2,3) In particular, V is an important element because it prevents martensitic steel from softening during tempering,⁴⁾ and it is used for alloy design and texture control of various high-strength steels.^5,6,7,8,9)

For V steels (steels to which V has been added as a hardening element), numerous studies have been made on the relationships of mechanical properties with heat treatment conditions and microstructure.^{10,11,12,13,14,15)} It is known that there are two modes of precipitation of V carbide. One is phase interface precipitation associated with proeutectoid ferrite transformation from austenite during continuous cooling and isothermal holding,^{16,17,18,19,20)} and the dependence of spacing between sheets of precipitates at the phase interface on temperature was studied.^21,22,23,24) The other is aging precipitation during tempering of martensite,^25,26) and the growth and coarsening kinetics of alloy carbides and carbonitrides were extensively studied in a V steel²⁷⁾ and an Fe–C–N–V alloy.²⁸⁾

Thermo-mechanical treatment is one of the main processes for microstructure control,²⁹⁾ and it is important to dissolve alloy carbides into a solid solution prior to thermo-mechanical treatment.^3,30) Thus, the dissolution kinetics of alloy carbide in austenite has been studied both experimentally and theoretically. In the past decades, Saito³¹⁾ analyzed the dissolution of Nb carbonitride during controlled rolling using Aaron and Kotler’s formula for binary alloys.³²⁾ Recently, Gong et al.³³⁾ studied experimentally the dissolution of Nb carbide in Nb steel during heating. They analyzed how the concentration of Nb in solution varied with time using the JMAK (Johnson-Mehl-Avrami-Kolmogorov) rate equation for a binary alloy. Moreover, the dissolution of NbC was simulated with commercially available software,³⁴⁾ where the rate controlling factors affecting the dissolution were not critically assessed in the model.

Compared with Nb carbide, studies of the dissolution of V carbide are relatively few. Reyes et al.³⁵⁾ studied the dissolution of V carbide in an Fe-0.5C-0.2V alloy through detailed microstructure observation, but they only compared the volume fractions of alloy carbide with thermodynamic calculation, and did not perform in-depth analysis of the dissolution rate of carbide. It is also known that the precipitation and dissolution of carbonitrides are affected greatly by a small amount of nitrogen.²⁸⁾ This report is thus intended to reveal experimentally the dissolution behavior of carbonitride in a quaternary Fe–C–N–V alloy and develop a simulation model for it. More specifically, we measured temporal changes in the volume fraction and size distribution of V carbonitride in an Fe-0.18C-0.3V-0.002N quaternary alloy during dissolution treatment, and performed a simulation under the assumption that local equilibrium of carbon, nitrogen and vanadium is maintained at the particle/matrix interface, and compared the results with experiment. Prior to the simulation in the quaternary alloy, the dissolution of V carbide in an Fe–C–V ternary alloy is simulated and results are compared with experiment by Reyes et al.³⁵⁾

2. Simulation Method

2.1. Dissolution Mode of Carbide Particle

With regard to the precipitate dissolution, several rate equations have been proposed for binary alloys.^36,37) Aaron and Kotler³²⁾ discussed the dissolution of V carbide assuming that the V diffusion in the matrix is rate-controlling the dissolution because it is slower than that of carbon. In this paper we intend to incorporate the effect of C and N diffusion on the dissolution of V carbonitride under the assumption that local equilibrium of both interstitial (C and N) and substitutional (V) elements is maintained at the carbide/matrix interface. In this section, we first describe the method of simulation in a ternary Fe–C–V alloy, and then, it is extended to a quaternary Fe–C–V–N alloy to evaluate the influence of N on the dissolution.

Figure 1 schematically shows the solubility curve of V carbide near the Fe-corner of the isothermal section of the Fe–C–V phase diagram. The cross (denoted P) and open circles (denoted A and A’) indicate the composition of V carbide and the austenite matrix at the start of dissolution. If the dissolution is rate-controlled by carbon diffusion, the u-fraction of V in the carbide, i.e. the occupancy fraction of V in the substitutional lattice, is constant during dissolution. This means that the carbon activity at the carbide/matrix interface is represented by point Q, intersection of the horizontal line from P with the solubility curve. The u-fraction of V is the occupancy of V in the substitutional lattice,and is related to the mole fraction of V and C as follows,   

$$u_V^{}=\frac{x_V}{1-x_C}$$(1)

Fig. 1.

Dissolution mode of V carbide under local equilibrium.

When the matrix composition is positioned at A’, the carbon activity at the interface, $a_C^1$, is higher than that in the matrix, $a_C^2\text{'}$. Thus, carbon is allowed to diffuse into the matrix and the carbide particle can dissolve. However, when the matrix composition is positioned at A, the right side of carbon isoactivity line $a_C^1$, carbon cannot diffuse into the matrix because the carbon activity in the matrix $a_C^2$ is higher than $a_C^1$, and the particle does not dissolve unless V diffuses to alter the carbon activity at the interface. As the temperature increases, the solubility curve moves toward the upper-right corner, and the isoactivity line $a_C^1$ also moves toward the right side and eventually reaches $a_C^2$. The temperature at which $a_C^1$=$a_C^2$ defines the boundary temperature between V diffusion controlled and carbon diffusion controlled dissolution. According to thermodynamic calculation, the carbide dissolution occurs under V diffusion control at both 950 and 1050°C (adopted in experiment) in this alloy. Indeed, the composition of the parent phase is in the A side even up to 1500°C, where steel melts, and it is considered that the dissolution of V carbide is controlled almost always by V diffusion.

2.2. Calculation of Particle/matrix Interfacial Tie Line

The particle/matrix interfacial tie line can be calculated as follows when the dissolution takes place by V diffusion. The compositions of the matrix and V carbide prior to dissolution are denoted A and P, as shown in Fig. 2. Carbon diffuses much faster than V and thus, the carbon activity is almost the same at the interface as that in the matrix far from the interface. The carbon isoactivity line passing through A intersects with the solubility curve of V carbide at point B, and BP is the interfacial tie-line during the dissolution under local equilibrium.

Fig. 2.

Interfacial tie-line for dissolution of V carbide (solid line). A dotted line is the bulk equilibrium tie-line at the end of dissolution. A dashed line indicates the critical tie-line at which the transition from V diffusion controlled to carbon diffusion controlled dissolution. (Online version in color.)

During dissolution carbon and V continue to be supplied to the matrix, so the matrix composition, point A, moves upward to the right. At the same time, the composition of B moves downward to the right along the solubility curve until it reaches B^∞. If the bulk alloy composition is in the γ+VC two-phase region like O₁, some carbide particles remain undissolved. If the bulk composition is in the γ single-phase region like O₂, all particles dissolve.

In the simulation, the relationship between compositions A and B are expressed by three regression formulae in advance, as shown below.

(i) The activity of C in the matrix a_C is related to the C and V concentrations in the matrix, $x_C^A$ and $x_V^A$ as,   

$$a_C=p_1+p_2{x}_C^A+p_3{x}_V^A$$(2)

(ii) The relationship between the concentration $x_C^B$ and the activity of C at point B on the solubility curve is obtained as.   

$$x_C^B=q_1+q_2{a}_C+q_3{a}_C{}^2$$(3)

(iii) The concentration of V $x_V^B$ is related to the concentration of C on the solubility curve as,   

$$x_V^B=r_1+r_2{x}_C^B+r_3{\left(x_C^B\right)}^2$$(4)

where p₁.- - - - r₃ are the regression coefficients. See Tables 2 and 3 for the values of the coefficients calculated by Thermo-Calc with TCFE10 database. Incidentally, Eq. (4) is another expression of the solubility product, albeit it is a part of it.

Table 2. (a) Numerical data and (b) coefficients in regression formulae used in simulation in Fe-0.5mass%V-0.2mass%V alloy.

(a)
Phase		Temperature, °C
		800	870	920
austenite	xC	2.11×10−2	2.16×10−2	2.22×10−2
	xV	3.48×10−4	8.50×10−4	1.48×10−3
	Vm 10−6 m3	7.16	7.19	7.21
VC	xC	0.465	0.465	0.465
	xV	0.459	0.478	0.490
	Vm 10−6 m3	6.05	6.05	6.05

(b)
coefficient	Temperature, °C
	870	920
p1	−0.00843	0.00634
p2	3.25	2.48
p3	−1.46	0.927
q1	0.00232	0.00224
q2	0.349	0.463
q3	−0.506	−0.910
r1	0.00164	0.00284
r2	−0.0488	−0.0826
r3	0.585	0.968

Table 3. (a) Numerical data and (b) coefficients in regression formulae used in simulation in Fe-0.18mass%C-0.3mass%V-0.002mass%N alloy.

(a)
Phase	–	Temperature, °C
		870	950	1050
austenite	xC	7.10×10−3	8.17×10−3	8.30×10−3
	xN	8.17×10−7	8.76×10−6	7.19×10−5
	xV	1.83×10−3	3.03×10−3	3.25×10−3
	Vm 10−6 m3	7.25	7.28	7.33
V(CyN1-y)	xC	0.434	0.311	0.109
	xN	0.0277	0.156	0.366
	xV	0.509	0.516	0.514
	y	0.940	0.665	0.229
	Vm 10−6 m3	5.70	5.86	6.01

(b)
coefficient	Temperature, °C		coefficient	Temperature, °C
	950	1050		950	1050
p1	−1.79×10−4	−2.77×10−4	p1’	2.24×10−9	1.74×10−9
p2	1.69	1.09	p2’	−1.67×10−7	−1.25×10−7
p3	−0.274	−0.146	p3’	−3.94×10−7	−3.16×10−7
p4	−3.73	−1.77	p4’	0.00257	0.00257
q1	0.00140	0.00175	q1’	4.75×10−7	7.44×10−6
q2	0.539	0.874	q2’	2.82×10−5	−4.03×10−4
q3	0	0	q3’	0	0
q4	−6.53×103	−4.26×103	q4’	388	375
r1	0.00995	0.0350
r2	−0.991	−4.88
r3	34.9	194
r4	−129	−62.9

2.3. Interfacial Tie Line for Quaternary Alloy

For an Fe–C–V–N alloy, the activity of N is incorporated in the regression formulae as well as the activity of C.

(i) The carbon activity a_C and nitrogen activity a_N are expressed by the concentration (mole fraction) in the matrix $x_C^A$, $x_V^A$ and $x_N^A$.   

$$a_C=p_1+p_2{x}_C^A+p_3{x}_V^A+p_4{x}_N^A$$(5)

  

$$a_N=p_1\text{'}+p_2\text{'}{x}_C^A+p_3\text{'}{x}_V^A+p_4\text{'}{x}_N^A$$(6)

(ii) The relationship between the concentration and the activity of C and N at point B on the solubility surface is obtained as.   

$$x_C^B=q_1+q_2{a}_C+q_3{a}_C{}^2+q_4{a}_N$$(7)

  

$$x_N^B=q_1\text{'}+q_2\text{'}{a}_C+q_3\text{'}{a}_C{}^2+q_4\text{'}{a}_N$$(8)

(iii) The concentration of V $x_V^B$ is related to the concentration of C and N on the solubility surface as,   

$$x_V^B=r_1+r_2{x}_C^B+r_3{\left(x_C^B\right)}^2+r_4{x}_N^B$$(9)

The concentration of N is much smaller than that of C, and thus, the squared terms of x_N in Eqs. (8) and (9) are ignored.

2.4. Rate Equation of Dissolution and Mass Balance Equations

Assuming that carbide particles have a spherical shape, the dissolution rate is calculated using the invariant-field approximation for a spherical particle.³⁶⁾ The diffusion field of V is given by the equation,   

$$x_V=x_V^A+\left(x_V^B-x_V^A\right)\frac{R}{r}$$(10)

where R is the radius of carbide particles and r is the distance from the center of the particle. According to the flux balance equation, the dissolution rate can be expressed as:³⁷⁾   

$$\frac{dR}{dt}=v=D^{\mathrm{\gamma }}\frac{\mathrm{\Omega }}{R},\mathrm{\hspace{0.17em} }\mathrm{\Omega }=\frac{x_V^A-x_V^B\prime }{x_V^p-x_V^B\prime }$$(11)

where $x_V^p$ is the V concentration in the carbide. D^γ (= 8.2 × 10^–5·exp(–264200/R_gT)m²/s) is the V diffusion coefficient in austenite, R_g (=8.314 J/ mol·K) is the gas constant, and T is temperature.³⁸⁾ According to the Gibbs-Thomson (G-T) effect, the relationship of $x_V^B\prime$ with $x_V^B$ is expressed by the equation,   

$$x_V^B\text{'}=x_V^B\left(1+\frac{2\sigma V_m^p}{{\text{R}}_{\text{g}}T}\frac{k_V}{R}\right),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{k}_V=\frac{1-x_V^B\prime }{x_V^p-x_V^B\prime }$$(12)

where V_m^P is the molar volume of V carbide (or carbonitride), and σ is the particle/matrix interfacial energy.

Next, the temporal variation of the austenite matrix composition is calculated by means of mass balance equations of solute. In this model, if the initial size of individual particles is known, the temporal change of the particle sizes can be calculated, but here the distribution of initial particle size is given. Thus, the volume fraction of all particles $f_V^P$ can be represented as,   

$$f_V^P=\frac{4\pi N_p}{3}\sum _j^{}{f}_{N,j}{R}_j^3$$(13)

where N_p is the total number of particles per unit volume, and f_N,j is the number fraction of particles that belong to the j-th size class. Assuming that $x_V^0$ is the bulk V concentration, $x_V^A$ can be calculated from the mass balance as follows:   

$$\frac{x_V^0}{V_m^{\gamma }}=\frac{f_V^P}{V_m^P}\cdot x_V^P+\frac{1-f_V^P}{V_m^{\gamma }}\cdot x_V^A$$(14)

where $V_m^P$ and $V_m^{\gamma }$ are the molar volumes of the particles and austenite matrix. The mass balance equations of C and N can be expressed as,   

$$\frac{x_C^0}{V_m^{\gamma }}=\frac{f_V^P}{V_m^P}\cdot x_C^P+\frac{1-f_V^P}{V_m^{\gamma }}\cdot x_C^A$$(15)

  

$$\frac{x_N^0}{V_m^{\gamma }}=\frac{f_V^P}{V_m^P}\cdot x_N^P+\frac{1-f_V^P}{V_m^{\gamma }}\cdot x_N^A$$(16)

The relationship between the volume fraction of precipitates $f_V^P$ and the mass fraction of V in the extraction residue³⁵⁾ $f_m^V$ can be expressed for carbonitride as,   

$$f_V^P=f_m^V\frac{{\rho }_{Fe}}{{\rho }_p}\cdot \left\{\frac{y{M}_C+\left(1-y\right)M_N}{M_V}+1\right\}$$(17)

where y is the site occupancy of carbon in the carbonitride, ρ_p is the density of V carbonitride, ρ_Fe is the density of austenite, and M_i is the atomic weight of element i.

3. Experimental Procedure

The alloy (Fe-0.18mass%C-0.3mass%V) was vacuum induction melted using high-purity materials. Table 1 shows the chemical composition of the alloy. A 30 kg ingot was heated to 1200°C, held for 1800 s, and forged at 900°C to a slab, 40 mm in thickness, 160 mm in width and 100 mm in length. After cooling, it was solution-treated at 1250°C for 86400 s to reduce the micro-segregation of V. Then, the slab was heated at 1250°C for 3600 s, hot-rolled at a temperature above 1000°C, and water-cooled to form a plate, 3.5 mm in thickness. The hot-rolled plate was machined to remove scale and cold-rolled to a plate, 1.4 mm in thickness. The cold-rolled plate was heated to 870°C to have VCN precipitated, and isothermally held at 950 and 1050°C for different periods of time to observe the dissolution behavior. Hereafter, the precipitation temperature and time are denoted T_p (=870°C) and t_p, and the temperature and time for isothermal dissolution are denoted T_d and t_d, respectively.

Table 1. Chemical composition of the steel studied (mass%).

Fe	C	N	V	Si	Mn	Al	P	S
Bal.	0.18	0.002	0.30	0.01	<0.01	0.02	<0.002	<0.001

The melted alloy contained approximately 20 ppmN. Figure 3(a) shows the vertical section of the Fe–C–V–N quaternary phase diagram at 0.3mass%V and 0.002mass%N calculated by Thermo-calc with TCFE10 database. The figure indicates that the alloy is in the γ+VCN two-phase region at both 950 and 1050°C, and it is considered that the VCN does not completely dissolve even if it was held for a prolonged period of time. For comparison, Fig. 3(b) shows the vertical section of the Fe–C–V ternary phase diagram. The ternary alloy without N is in the γ single-phase region at the holding temperature of dissolution and thus, if the alloy did not contain N, V carbide is expected to dissolve completely.

Fig. 3.

(a) Vertical section of the Fe–C–V–N phase diagram calculated by Thermo-calc at 0.3mass%V and 0.002mass%N. (x) and (+)indicates, respectively, the precipitation and dissolution temperatures used in this study. (b) Vertical section of the Fe–C–V phase diagram at 0.3mass%V.

To measure the fraction and size distribution of V carbonitride, the extraction replica method³⁹⁾ was adopted. After holding, the alloy specimen was electrolytically extracted with 10% acetylacetone-1% tetramethylammonium chloride-methanol solution, and the residue was decomposed with acid. Then, the V concentration in the residue was measured by means of inductively coupled plasma optical emission spectrometry (ICP-OES). To analyze the particle size distribution (PSD) of VCN, specimens were isothermally held for different periods of time and observed under transmission electron microscope. The precipitate phase was identified by means of selected area electron diffraction (SAED), and the precipitate composition was analyzed by means of energy dispersive X-ray spectroscopy (EDS).

4. Results of Experiment

Figures 4(a) and 4(b) show the TEM bright field image and SAED pattern of an extraction replica specimen after precipitation treatment for t_p = 3600 s. From the SAED pattern, black particles are an NaCl-type (B1) carbide (carbonitride). The composition analysis of the particles by EDS indicate that the concentration ratio of Fe and V is 3.2 : 96.8 in mole fraction, and the particles are identified to be V carbide (or carbonitride).

Fig. 4.

(a) Bright field image of extraction replica of a specimen held at 870°C for 3600 s. (b) Indexed SAED pattern of a precipitate particle.

Figure 5(a) displays the extraction replica TEM image of the same sample as Fig. 4(a) in lower magnification and Fig. 5(b) is the particle size distribution (PSD) in this sample. Precipitation treatment was carried out at t_p = 600 and 3600 s. There was no significant difference in the V concentration in the residue of them.

Fig. 5.

(a) Extraction replica micrograph, and (b) particle size distribution (PSD) of Fe–C–N–V specimen held at 870°C for 3600 s (total particle number is 101).

Figures 6(a) through 6(c) show the TEM images of the extraction replica samples isothermally held at T_d = 950°C for t_d = 30, 100, and 1000 s. It is seen that dissolution proceeded significantly in t_d = 30 s and thereby, the particle size decreased. In Figs. 6(b) and 6(c), small particles are relatively few, and larger particles with the diameter 30 to 50 nm or even greater are observed. Figure 7 shows the evolution of V concentration in the residue with dissolution time at T_d = 950 and 1050°C. At these temperatures, the particles did not dissolve completely.

Fig. 6.

Extraction replica micrographs of Fe–C–N–V specimen, (a) held at 950°C for 30 s, (b) for 100 s, and (c) for 1000 s.

Fig. 7.

Evolution of V mass fraction in residue plotted against holding time at T_d = 950 and 1050°C.

5. Simulation Results

5.1. Dissolution of V Carbide in Fe–C–V Alloy

As seen above, some particles remained undissolved after holding at T_d = 950 and 1050°C for a prolonged period of time. This is probably due to the ~20 ppmN in the alloy presently studied (see Fig. 3) and thus, the effect of a small amount of N cannot be ignored. Under this circumstance, the simulation is first compared with experiment in an Fe-0.5mass%C-0.2mass%V alloy reported by Reyes et al.³⁵⁾ more than a decade ago. The concentration of impurities in this alloy was 5 ppm or less. The fraction of V carbide vs holding time plots are available at two temperatures: the precipitation treatment was carried out at T_p = 800°C and the dissolution holding was done at T_d = 870 and 920°C.

Table 2 shows the values of regression coefficients and numerical data for the other thermodynamic parameters evaluated from Thermo-calc with TCFE10 database.⁴⁰⁾ Figures 8(a) and 8(b) display the temporal evolution of volume fraction $f_V^P$ simulated by the present model. The initial size distribution f_{N, j} was taken from Fig. 5(a) in ref. 35). The value of interfacial energy σ(= 0.3 J/m²) was obtained from the coarsening experiment of V carbide in Fe–C–V austenite.⁴¹⁾ The initial value of N_P(=2.3 × 10¹⁹ m⁻³) was fitted to $f_V^P$ before dissolution holding. There are no other fitted parameters. Since the particles disappear in order from the smallest ones, N_P decreases with time. At both temperatures, the alloy was in the γ+VC two-phase region. It is seen that dissolution occurred only partially and the evolution of carbide fraction is reproduced quite well.

Fig. 8.

Calculated volume fraction of VC as a function of holding time for comparison with measurements reported by Reyes et al.³⁵⁾ The temperature of dissolution is (a) 870°C, and (b) 920°C.

Figure 9 displays the temporal variation of particle size. It indicates that only particles of smaller size groups dissolved completely, and the particle size of larger particles varies slowly. Figures 10(a) and 10(b) compare the calculated and measured PSD.

Fig. 9.

Evolution of particle size during holding at 870°C. R_av is the mean radius of remaining particles.

Fig. 10.

Comparison of PSD calculated for the alloy of Reyes et al. with measurements.³⁵⁾ The dissolution temperature is 870°C. (a) t_d=2 min, and (b) t_d=20 min.

5.2. Dissolution of V Carbonitride in Fe–C–V–N Alloy

Table 3 shows the values of regression coefficients and parameters used in simulation in the quaternary alloy. As the σ value, the value obtained from the coarsening experiment (σ = 0.15 J/m²) was used.⁴²⁾ According to the table, the occupancy fraction of carbon y and that of nitrogen 1-y in the carbonitride V(C_yN_1-y) vary considerably with temperature. Maugis and Gouné²⁸⁾ simulated the evolution of the occupancy fraction of C and N during isothermal holding. It is assumed in this simulation that the change in occupancy fraction occurs in a short time during heating from T_p to T_d. They also reported that the V concentration in the carbide does not change significantly with temperature and thus, it is assumed that the V concentration changes instantly upon heating to the dissolution temperature. Figures 11(a) and 11(b) compare the calculated and measured V mass fraction in the residue of replica at T_d = 950 and 1050°C. The figures indicate that dissolution occurred only partially, as expected from the vertical section of the phase diagram (Fig. 3).

Fig. 11.

Calculated V mass fraction (or volume fraction) of V carbonitride as a function of holding time for comparison with measurements. The temperature of dissolution is (a) 950°C, and (b) 1050°C. N_p = 8.87×10¹⁹ m⁻³.

Figure 12 displays the evolution of particle sizes with holding time at T_d = 950°C which indicates that small particles disappear in a short time while larger particles decrease in size very slowly. The average value repeatedly increases and decreases due to disappearance of particles in group units, it tends to increase gradually as a whole. According to Fig. 11(a), $f_m^V$ approached the equilibrium fraction at t_d~ 100 s, and continued to decrease slowly and turned to increase at prolonged holding. This can be interpreted as a result of the G-T effect: as dissolution proceeds, $x_V^B\prime$ becomes large especially for small particles. This accelerates dissolution, and as a result, $f_V^P$ (and $f_m^V$) drops below the equilibrium fraction. On the other hand, the V concentration at the interface of large particles $x_V^B\prime$ does not increase significantly, and $x_V^B\prime$ becomes even smaller than $x_V^A$, causing the particle size to increase (dR/dt > 0). Figure 12(b) indicates that small particles (class 1 and 2) decrease in size monotonously. Medium-sized particles (class 6) have a temporary decrease in dissolution rate in the middle period, i.e. t_d=100–1000 s, but eventually disappear completely. In contrast, dR/dt of the largest particles (class 9) becomes slightly positive from about 1000 s, and as a result, the particle size increases, albeit extremely slowly. The same can be observed in Fig. 8(d). Indeed, when the simulation was carried out excluding G-T effect (σ = 0 J/m²), the curve of $f_m^V$ monotonously approached the equilibrium fraction.

Fig. 12.

(a) Evolution of particle size simulated at 950°C. R_av is the mean radius of remaining particles. (b) Evolution of interface migration rate (dissolve if negative, grow if positive) at the same temperature. The number indicates the size class number.

Figures 13(a) and 13(b) compare the simulated PSD at t_d = 30 and 100 s with measured ones. In the simulation of a multi-particle system, a simulation using a diffusion cell that depends on the particle size has been proposed.⁴³⁾ In the present simulation, the mean field approximation is used where large particles and small particles are dissolved in a matrix having the same V concentration. Hence, the difference in dissolution rate due to the difference in particle size may be amplified. Moreover, the size distributions with different holding time were measured in different areas (or specimens). Considering these, the agreement between simulation and experiment is satisfactory.

Fig. 13.

Comparison of measured and calculated PSD of VCN in an Fe–C–V–N quaternary alloy held at 950°C (a) for 30 s (total particle number 100) , and (b) for 100 s (16).

5.3. Simulation Using a Fixed Tie Line

In the calculation of dissolution rate, the equation for a binary alloy is often used assuming that the slow-diffusing solute controls the dissolution.^31,32) In a binary alloy, the tie-line does not move during isothermal holding. In the present case, however, the concentrations of carbon and nitrogen in the matrix increase and thereby, the V concentration at the interface decreases (see Fig. 2) as dissolution proceeds. In order to see the effect of tie-line motion, the simulation was carried out with the identical initial PSD using tie line PB^∞, which passes through the bulk composition. As shown in Fig. 14 (solid curve), it took ~1500 s, 7.5 times longer to reach the final V mass fraction, than that calculated using the tie-line which moves maintaining local equilibrium (~200 s). This indicates clearly that the effects of C and N diffusion cannot be ignored in the dissolution of alloy carbide.

Fig. 14.

Evolution of V mass fraction in extraction replica calculated using the bulk equilibrium tie-line (solid curve) for comparison with that obtained by the present method (dashed curve). The dissolution temperature is 950°C.

6. Conclusions

A computer program for the dissolution of V carbide and carbonitride was developed assuming that local equilibrium of interstitial (carbon and nitrogen) and substitutional solutes (V) at particle/matrix interfaces were maintained. The dissolution occurred under the V diffusion-controlled mode. The dissolution rate was calculated using the mean field approximation and invariant field approximation.³⁶⁾ The particle size at the start of simulation was given by the measured size distribution. For the Fe–C–V ternary alloy, the results were in good agreement with experiment reported in the literature.³⁵⁾ For the Fe–C–V–N quaternary alloy containing 20 ppm N, the model reproduced well the temporal variation of V mass fraction in the residue of replica measured in this study. Calculations using a fixed tie-line produced a large error, as is often done in the calculation of dissolution of alloy carbides.

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