Effect of Nb Addition on the Growth and Coarsening of Cu-particles in Ferritic Stainless Steel

Abstract

The growth and coarsening of Cu-particles in 18%Cr–1.5%Cu ferritic stainless steels as a function of Nb content was investigated quantitatively. The samples were solution-treated at 1250°C and then isothermally aged at 700°C for up to 86400 s. Fine, spherical Cu-particles nucleated and grew during aging. The radius of the Cu particles increased in proportion to the one-third power of the aging time in all the stainless steels. The radius of the Cu particles in the Nb-added stainless steels was always smaller than that in the Nb-free stainless steel. The normalized standard deviation of the size distribution of the Cu particles increased and reached saturation at a certain value during aging. The increase in the standard deviation during aging was delayed by the addition of Nb. The volume fraction of Cu particles increased and reached saturation at a certain value during aging. The addition of niobium delayed the saturation of the volume fraction of the Cu particles. The slow growth of Cu particles in Nb-added stainless steels is thought to be owing to the slow diffusion of Nb atoms from the Cu particles.

1. Introduction

The growth and coarsening of precipitates in the steel matrix greatly affects the strength of the steels. Growth of the precipitates occurs in which supersaturation of the solute atoms is still present in the matrix. The volume fraction and size of the precipitates increases during the growth process, resulting in a change in the strength of steel. The average size of precipitates continues increase even after the concentration of solute atoms reaches the equilibrium value, i.e., there is no remaining supersaturation. Such a process is called “coarsening,” and it begins at a later stage of the growth process. The driving force for coarsening is a decrease of the interfacial free energy of the precipitates. During the coarsening of precipitates, the average size of the precipitates increases, whereas the volume fraction of the precipitates remains almost constant. This results in a decrease in the number density of precipitates and an increase in the interparticle distance. Therefore, the coarsening of precipitates leads to a decrease in the strength of steels. The kinetics of the growth and coarsening of precipitates have been widely examined and reviewed in the literature.^{1,2,3,4,5,6,7,8,9,10,11)}

Because precipitation of Cu particles in steels improves the high-temperature and fatigue strength,^12,13,14,15) many studies on the precipitation of Cu in stainless steels have been conducted.^{16,17,18,19,20,21)} Bajguirani et al.^17,18,19) reported that precipitation of Cu particles in a 15-5 PH stainless steel occurs first by formation of coherent bcc solute clusters. These clusters then transform into twinned 9R particles, and then into 3R and fcc Cu particles. Although there have been many studies on Cu precipitation in terms of crystallography and morphology, reports on the kinetics of the growth and coarsening of Cu precipitates in stainless steel are limited.

The addition of Cu in stainless steels is intended to improve not only strength but also corrosion resistance.²²⁾ Niobium in ferritic stainless steels is also known as a beneficial element that increases both corrosion resistance and high-temperature strength.^23,24,25) The addition of Cu and Nb to stainless steel simultaneously is expected to greatly increase both strength and corrosion resistance. A previous study by the authors revealed that nucleation of Cu-rich solute zones was suppressed by the presence of solute Nb atoms in a 18Cr–1.5Cu–0.47Nb stainless steel.²⁶⁾ The effects of the addition of Nb on the growth and coarsening of Cu-rich zones in stainless steel, however, have not been well understood. The growth and coarsening of Cu-rich zones has a significant effect on the mechanical properties of the stainless steels such as hardness. The purpose of the present study, therefore, is to reveal the effect of the addition of Nb on the growth and coarsening of Cu-rich zones in an 18% Cr ferritic stainless steel.

2. Experimental Procedures

Ferritic stainless steel sheets of 18.4%Cr–1.53%Cu, 18.4%Cr–1.57%Cu–0.47%Nb, and 18%Cr–1.52%Cu–0.76%Nb (in mass%) manufactured in the laboratory were used. They are denoted as Nb-free, 0.47Nb, and 0.76Nb stainless steels, respectively. Specimens 10 × 10 mm were machined from the 2-mm-thick sheets. The specimens were solution-treated at 1250°C for 600 s and quenched in iced brine. They were then isothermally aged at 700°C for 10–86400 s. Thin foils for transmission electron microscopy (TEM) were prepared by a twin-jet polishing technique using an electrolyte containing 90 vol% acetic acid and 10 vol% perchloric acid. The samples were examined using JEM100CX and JEM2100 transmission electron microscopes operating at 100 and 200 kV, respectively. Bright-field TEM images were observed under the condition that the direction of the incident electron beam was nearly parallel to the <111>_α direction of the specimen. Particle radius of Cu-precipitates was measured from a bright-field TEM image. The normalized standard deviation of size distribution, s, of Cu-precipitates was evaluated by normalizing with the average radius of Cu-precipitates. Thickness of the TEM thin foil for the observed area was determined by using the thickness fringes. Hardness measurements were carried out on both solution-treated and aged specimens at room temperature under 500 gf load. Average hardness was calculated from five measuring points.

3. Results

3.1. TEM Observations of Cu Precipitation

Bright-field images of Nb-free stainless steel aged at 700°C for 60, 100, 300, 600, 900, and 1800 s are shown in Figs. 1(a)–1(f), respectively. Enlarged images of Figs. 1(a), 1(b), and 1(c) are inserted in the upper right corner of their respective figures. Fine, spherical zones less than 10 nm in diameter were formed in the specimen aged for 60 s, as indicated by the arrows in the inset image of Fig. 1(a). These spherical zones were also detected in the specimen aged for 30 s, but they were not detected in the specimen aged for 10 s.²⁶⁾ The spherical zones were Cu-rich solute clusters described in a previous paper.²⁶⁾ Growth of the spherical zones is observed in Figs. 1(b)–1(f). Some of the spherical zones exhibited striped contrast after aging for 300 s, as shown in Fig. 1(c). The number of spherical zones with striped contrast increased upon further aging, as shown in Figs. 1(d)–1(f). These spherical zones with striped contrasts were Cu-rich particles with a 9R structure containing twins.²⁶⁾ Although Cu-rich solute zones with a bcc structure should be distinguished from Cu particles with a 9R structure from a crystallographic viewpoint, they are denoted together as “Cu particles” hereafter.

Fig. 1.

Bright-field images showing Cu-rich zones in the Nb-free stainless steel aged at 700°C for (a) 60, (b) 100, (c) 300, (d) 600, (e) 900, and (f) 1800 s.

Bright-field images of the 0.47Nb stainless steel aged at 700°C for 60, 100, 300, 600, 900, and 1800 s are shown in Figs. 2(a)–2(f). Enlarged images of Figs. 2(a) and 2(b) are inserted in the upper right corner of their respective figures. Fine, spherical Cu particles less than 10 nm in diameter were initially observed in the 0.47Nb stainless steel aged for 60 s [Fig. 2(a)], and growth of the spherical Cu particles is observed in Figs. 2(b)–2(f).

Fig. 2.

Bright-field images showing Cu particles in the 0.47Nb stainless steel aged at 700°C for (a) 60, (b) 100, (c) 300, (d) 600, (e) 900, and (f) 1800 s.

Bright-field images of the 0.76Nb stainless steel aged at 700°C for 60, 100, 300, 600, 900, and 1800 s are shown in Figs. 3(a)–3(f). Enlarged images of Figs. 3(a) and 3(b) are inserted in the upper right corner of their respective figures. Fine, spherical Cu particles less than 10 nm in diameter were initially observed in the 0.76Nb stainless steel aged for 60 s, as shown in Fig. 3(a), and growth of the spherical Cu particles is observed in Figs. 3(b)–3(f).

Fig. 3.

Bright-field images showing Cu particles in the 0.76Nb stainless steel aged at 700°C for (a) 60, (b) 100, (c) 300, (d) 600, (e) 900, and (f) 1800 s.

The increase of the average radius of the Cu particles as a function of aging time at 700°C is shown in Fig. 4. The average radii of the Cu particles were plotted in a log–log plot; fitted lines to the data are also drawn in the figure. The average radius of the Cu particles in all specimens increased monotonically with increased aging time. The average radius of Cu particles in the Nb-added stainless steels was smaller than that in the Nb-free stainless steel, suggesting that the addition of Nb suppressed the growth and coarsening of the Cu particles.

Fig. 4.

The increase of the Cu particle radius as a function of aging time in the (a) Nb-free, (b) 0.47Nb, and (c) 0.76Nb stainless steels.

Figure 5 shows the changes in the normalized standard deviation of size distribution, s, of the Cu particles in the stainless steels during aging. In the initial stage of aging, the values of s for all the samples were almost the same, i.e., s = 0.15–0.17. The value of s in the Nb-free stainless steel increased significantly at 300 s aging and reached a constant value of approximately 0.28 after around 600 s aging. The value of s in the 0.47Nb stainless steel remained a constant value of approximately 0.17 until 600 s aging and then increased significantly to a constant value of 0.28 at 1800 s aging. The value of s in the 0.76Nb stainless steel was at a constant value of approximately 0.17 until 900 s aging and then increased significantly to a constant value of 0.28 at 3600 s aging. Although the initial and final values of s in all the samples were almost the same, the increase of s occurred at different aging times in each sample, i.e., the higher the Nb content in the sample, the slower the onset time for the increase of s.

Fig. 5.

Change in standard deviation of size distribution of Cu particle radius in the (a) Nb-free, (b) 0.47Nb, and (c) 0.76Nb stainless steels.

Figure 6 shows the change in the volume fraction of Cu particles in the Nb-free, 0.47Nb, and 0.76Nb stainless steels. The volume fraction of Cu particles in the Nb-free stainless steel increased up to approximately 300 s aging and reached a constant value of 0.0045, whereas those of the Nb-added stainless steels increased constantly throughout the aging period. These results imply that the growth stage of Cu particles in the Nb-free stainless steel was completed at approximately 300 s aging and the coarsening of the Cu particles followed. The higher the Nb content in the sample, the slower was the increase in the volume fraction of the Cu particles. It can thus be concluded that the addition of Nb suppressed the growth and coarsening of Cu particles in ferritic stainless steel.

Fig. 6.

Change in volume fraction of Cu particles in the (a) Nb-free, (b) 0.47Nb, and (c) 0.76Nb stainless steels.

3.2. Hardness Change during Aging

The increase of hardness during aging for the Nb-free, 0.47Nb, and 0.76Nb stainless steels is shown in Fig. 7. The increase of hardness was calculated on the basis of the hardness of the sample after solid-solution treatment (SST). The hardness of the Nb-free stainless steel initially increased at 10 s aging, and its peak hardness was obtained at 300 s aging. It should be noted that the volume fraction of Cu particles in the Nb-free stainless steel reached a saturated value after around 300 s aging, as shown in Fig. 6. This increase of hardness could be attributed to the nucleation and growth of the Cu particles. In the coarsening stage of the Cu particles, the hardness decreased. The age-hardening curve was shifted to a longer aging time by the addition of Nb. Furthermore, the peak hardness decreased with increasing Nb content in the sample, which indicates that the addition of Nb suppressed the age-hardening behavior in the stainless steels. It should be noted that a decrease in hardness occurred in the Nb-added stainless steels after 900 s aging, even though the volume fraction of Cu particles continued to increase after 900 s aging, as shown in Fig. 6.

Fig. 7.

Increase of Vickers hardness during aging of the (a) Nb-free, (b) 0.47Nb, and (c) 0.76Nb stainless steels.

4. Discussion

4.1. Growth and Coarsening of Cu Particles during Isothermal Aging

The normalized standard deviation of the size distribution of Cu particles, s, and the volume fraction of Cu particles, V_f, increased and then remained constant at certain values during aging. These two parameters, s and V_f, seem to be well correlated. Both values were saturated after 300 s aging in the Nb-free stainless steel. When V_f reaches an almost constant value, the supersaturation of the solute in the matrix should be almost zero, indicating the end of the nucleation and growth of particles.¹⁾ Subsequently, the coarsening stage of the particles started, and in this stage so-called self-similarity can be observed, e.g. one measures the same scaled size distribution by scaling the radii of the particles with the average particle radius.²⁾ In the present study, the value of s in the Nb-free stainless steel after 300 s aging was constant, indicating that self-similarity held in the microstructure. Therefore, one can say that the coarsening stage of Cu particles started at 300 s aging in the Nb-free stainless steel. According to the coarsening theories by Lifshitz and Slyozov,³⁾ Wagner,⁴⁾ Ardell,⁵⁾ and Greenwood,²⁷⁾ the particle radius increases in proportion to the one-third power of the aging time in the coarsening process. Such an increase of particle size during aging was confirmed in Fig. 4. It should be noted that the average radius of the Cu particles increased in proportion to the one-third power of the aging time even in the growth stage of the Cu particles, i.e., at less than 300 s aging in the Nb-free stainless steel. It is said that growth in proportion to the one-half power of the aging time, which is called parabolic growth, is expected to occur in particles where the supersaturation of the solute in the matrix is constant.²⁾

This reasoning is substantiated by the calculation of the growth curve of the particles, in which a change in the supersaturation of the solute in the matrix is taken into account. The calculated growth curve of spherical particles under decreasing supersaturation of the solute due to growth of particles can be expressed as²⁾ (see Appendix in detail)   

$$\begin{array}{l}x=\left\{ln\left[\left(1+y+y^2\right)/{\left(y-1\right)}^2\right]\right\}/6\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }-\left\{arctan\left[\left(2y+1\right)/\sqrt{3}\right]-arctan\left(1/\sqrt{3}\right)\right\}/\sqrt{3}\end{array}$$

where x and y represent the reduced time, t/τ, and the relative particle radius, R/R_m, respectively. The variable t is the aging time and τ is defined as τ = R_m²/(DS₀), where D and S₀ are the diffusion coefficient of the solute atoms and the supersaturation of the solute, respectively. The variables R and R_m are the average particle radius at t/τ and the time at which supersaturation reaches zero, respectively. The growth curve of spherical particles calculated by the above equation is shown in Fig. 8(a). The growth curves of spherical particles calculated by the equations y = $\alpha x^{\frac{1}{2}}$ , where α = 1.4, and y = $\beta x^{\frac{1}{3}}$ , where β = 1.06 are given in Figs. 8(b) and 8(c), respectively. The variables α and β are the fitting parameters. The growth curve of Fig. 8(a) is coincident with that of Fig. 8(b) in the initial stage of particle growth until t/τ = 0.125. This means that the particle growth under changing supersaturation conditions is parabolic because there is enough supersaturation in the matrix. After the transient region between t/τ = 0.125 and 0.25, the growth curve of Fig. 8(a) becomes coincidence with the curve of Fig. 8(c). This implies that the particle radius in the middle stage of growth between t/τ = 0.25 and 0.6 is expected to increase in proportion to the one-third power of the aging time. This probably corresponds to what we observed in Fig. 4(a) in the growth stage of Cu particles in the Nb-free stainless steel during aging between 30 and 300 s. Since the growth curve of Fig. 8(a) has a shallow slope after t/τ = 0.6, growth of the particles is thought to occur slowly due to some remaining supersaturation in the matrix. In an actual system, coarsening will occur in the later stage of growth, even though some supersaturation is still present in the matrix. Therefore, such slow growth could not be detected in Fig. 4(a).

Fig. 8.

Growth of spherical particles as a function of their relative particle radius, R/R_m, and the reduced time, t/τ.

We should discuss why the initial stage of growth, i.e., parabolic growth, could not be detected in the present experiments. In the parabolic growth stage at t/τ less than 0.125, the particle radius should be less than R/R_m = 0.5, as evaluated from Fig. 8(a). When the volume fraction reached a constant value in the Nb-free stainless steel, the average particle radius was 3.4 nm. This corresponds to the value of R_m in the Nb-free stainless steel. The parabolic growth should be observed when the average particle radius is less than 3.4 × 0.5 = 1.7 nm. In the present study, the minimum particle radius observed in the Nb-free stainless steel by using TEM was 2.3 nm. Therefore, we were not able to detect the parabolic growth stage of the Cu particles in the present experiments.

Such growth and coarsening stages of Cu particles in the Nb-free stainless steel were also found in the Nb-added stainless steels. However, the evolution of s and V_f in the Nb-added stainless steels occurred in a later stage of aging. The reason for this is discussed in the next section.

4.2. Delayed Cu Particle Growth on Nb Addition

The average radius of the Cu particles in the Nb-added stainless steels was always smaller than that of the Nb-free stainless steel. Furthermore, the higher the Nb concentration in the matrix, the slower the increase of both s and V_f during aging. These results indicate that the growth of Cu particles was suppressed by the addition of Nb. As the authors previously pointed out,²⁶⁾ the inclusion of Nb atoms in Cu particles is not preferable from a thermodynamic viewpoint. As a result, outward diffusion of Nb atoms from Cu particles is expected to occur during the growth of Cu particles. The diffusion coefficient of Nb in ferrite is substantially smaller than that of Cu in ferrite.²⁸⁾ The migration distance of Nb atoms for a random walk per second at 700°C is about one-fifth that of Cu atoms. The slow diffusion of Nb atoms must therefore delay the growth of Cu particles. Outward diffusion of Fe and Cr atoms from the Cu particles would also occur during growth. Since the diffusion rate of these elements in all the specimens is expected to be almost the same, they have little influence on the growth of the Cu particles.

4.3. Hardness Change during Isothermal Aging

The hardness of all the samples increased during aging owing to the precipitation of Cu particles. The higher the Nb content in the stainless steel, the slower the increase of age hardening. Figure 4 shows that the growth of Cu particles was slowed by the addition of Nb. Furthermore, the increase in the volume fraction of Cu particles was delayed by the addition of Nb. Therefore, the slower age hardening in the Nb-added stainless steels is attributed to the slower nucleation and growth of Cu particles induced by the addition of Nb. A decrease of peak hardness was observed in the Nb-added stainless steels. TEM observations confirmed that the Laves phase was nucleated in the 0.47Nb and 0.76Nb stainless steels in the later stage of Cu particle precipitation. Our previous study showed that although precipitation of the Laves phase in 18%Cr–0.48%Nb stainless steel during aging at 700°C was observed, age-hardening by the precipitation of the Laves phase was minimal.²⁶⁾ Since precipitation of the Laves phase consumes solute Nb atoms, an increase in hardness by the precipitation of the Laves phase should accompany a decrease in hardness owing to the reduction of solution hardening by solute Nb atoms. For the 18%Cr–0.48%Nb steel, the effect of precipitation hardening was cancelled out by the solute depletion in the matrix. According to another experiment by the authors, an increase of hardness during aging at 700°C in a 18%Cr–0.87%Nb steel (27 Hv) was detected owing to the precipitation of the Laves phase. It can be said that the balance between solid-solution hardening by solute Nb atoms and precipitation hardening of the Laves phase will determine the age-hardening process of Nb-added stainless steels. The Laves phase in a ferrite matrix is expected to provide nucleation sites for Cu particles, which was clearly observed in the 0.76Nb stainless steel and shown in Fig. 9. Nucleation of Cu particles on the Laves phase results in a decrease in the number of Cu particles in the matrix capable of acting as obstacles for dislocation motion. Therefore, the formation of Laves phase changed the distribution of Cu particles in the matrix and decreased the effective number density of Cu particles which increases the hardness. The decrease of the maximum age hardening by the addition of Nb can probably be attributed to the heterogeneous nucleation of Cu particles on the Laves phase.

Fig. 9.

Bright-field image showing Cu particles nucleated on the Laves phase in the 0.76Nb stainless steel aged at 700°C for 3600 s.

5. Conclusion

The growth and coarsening of Cu particles in 18%Cr–1.5%Cu stainless steels as a function of Nb content are summarized as follows:

(1) Growth and coarsening of Cu particles in the ferrite matrix occurred during aging. The average radius of the Cu particles in all the samples increased in proportion to the one-third power of aging time. The average radius of Cu particles in the Nb-added stainless steels was smaller than that in the Nb-free stainless steel.

(2) The normalized standard deviation of the size distribution of the Cu particles, s, and the volume fraction of the Cu particles, V_f, increased and then remained constant at certain values during aging. The evolution of s and V_f in the Nb-added stainless steels occurred at a later stage of aging in comparison the Nb-free stainless steel.

(3) Solute Nb atoms delayed the growth and coarsening of Cu particles in the ferritic stainless steel. Since the inclusion of Nb atoms in the Cu particles is unfavorable, outward diffusion of Nb atoms from the Cu particles is needed for growth and coarsening of the Cu particles. Since the diffusion of Nb atoms in the ferrite matrix is substantially slower than that of Cu atoms, the growth and coarsening of the Cu particles is delayed by the slow diffusion of Nb atoms.

(4) The hardness of all the samples increased during aging owing to the precipitation of Cu particles. The higher the Nb content in the stainless steel, the slower the increase of age hardening. The delayed age hardening on addition of Nb was attributed to the slow growth and coarsening of Cu particles.

(5) The peak hardness attained during aging in the Nb-added stainless steels is smaller than that in the Nb-free stainless steel. Laves-phase particles formed in the Nb-added stainless steels during aging, which did not significantly affect their hardness. Nucleation of Cu particles on the Laves phase resulted in a decrease in the number of Cu particles in the matrix that were available to act as obstacles for dislocation motion. A decrease in the maximum age hardening by the addition of Nb can probably be attributed to the heterogeneous nucleation of Cu particles on the Laves phase.

Acknowledgments

The authors wish to express their sincere thanks to the Integrated Center for Sciences (INCS), Ehime University for use of their TEM (JEM2100).

Appendix

a. Particle Growth with Constant Supersaturation

We suppose that the concentrations of B solute atoms in a binary system exhibiting a miscibility gap whose tie-line ends for the matrix (α phase) and precipitate (β phase) are $c_B^{\alpha }$ and $c_B^{\beta }$ . We now consider the situation in which a spherical particle with radius R will grow in a supersaturated matrix. The initial concentration of the supersaturated matrix is $c_B^0$ . The concentration outside the particle at r, c_B, is calculated by solving the Laplace equation in spherical coordinates:   

$${\nabla }^2{c}_B=\frac{{\partial }^2{c}_B}{\partial r^2}+\frac{2}{r}\frac{\partial c_B}{\partial r}=0$$(1)

with the boundary conditions   

$$c_B=c_B^0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(\text{at}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r\to \infty )$$(2)

  

$$c_B=c_B^{\alpha }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(\text{at}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r\to R)$$(3)

We assume that at the interface local equilibrium exists, i.e., the concentration of B atoms in the matrix is already down to the value given by the phase diagram, i.e., c_B = $c_B^{\alpha }$ (at r → R). The solution to Eq. (1) is c_B (r) = $a_0+\frac{a_1}{r}$ . The boundary conditions determine the constants to be a₀ = $c_B^0$ and a₁ = $\left(c_B^{\alpha }-c_B^0\right)R$ . Thus, the concentration outside the particle, c_B, is   

$$c_B\left(r\right)=c_B^0+\left(c_B^{\alpha }-c_B^0\right)\frac{R}{r}$$(4)

The solute flux balance at the interface of the particle is given as follows:   

$$D{\left(\frac{\partial c_B}{\partial r}\right)}_{r=R(t)}=\left(c_B^{\beta }-c_B^{\alpha }\right)\frac{dR}{dt}$$(5)

By derivation of Eq. (4), we obtain   

$${\left(\frac{\partial c_B}{\partial r}\right)}_{r=R}=-\left(c_B^{\alpha }-c_B^0\right){\left(\frac{R}{r^2}\right)}_{r=R}=\left(c_B^0-c_B^{\alpha }\right)\frac{1}{R}$$(6)

By substitution of (6) into (5), we get   

$$D\left(c_B^0-c_B^{\alpha }\right)\frac{1}{R}=\left(c_B^{\beta }-c_B^{\alpha }\right)\frac{dR}{dt}$$(7)

  

$$\frac{dR}{dt}=\frac{D}{R}\frac{\left(c_B^0-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}=\frac{D}{R}{S}_0$$(7)′

If the supersaturation, given by $S_0=\frac{\left(c_B^0-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}$ , is constant in the growth process, Eq. (7)′ can be integrated by separation of the variables:   

$$\int RdR=D\int S_{0}dt$$(8)

  

$$\frac{1}{2}\left(R^2-R_0^2\right)=D{S}_0\left(t-t_0\right)$$(8)′

By choosing R(t = 0) = 0, we obtain   

$$R^2=2D{S}_{0}t$$(9)

Equation (9) indicates that shortly after nucleation the particles followed a parabolic growth pattern.

b. Particle Growth with Varying Degrees of Supersaturation

In most metallic alloys, the supersaturation decreases with time until it is zero and growth is finished. However, the coarsening process usually becomes important at low levels of supersaturation. In this section, by extending the treatment described in the previous section, we will solve the time-dependent particle growth with varying degrees of supersaturation. The volume fraction of β-phase particles after complete decomposition, $V_f^e$ , is   

$$V_f^e=\frac{\left(c_B^0-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(=S_0)$$(10)

This is the same as the supersaturation, S₀. During decomposition the average matrix composition, c_B, varies with time from $c_B^0$ to $c_B^{\alpha }$ . The time-dependent volume fraction can be written as   

$$V_f=\frac{\left(c_B^0-c_B\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}$$(11)

If the particles all have the same size, the volume fraction can be expressed by the radius of the particles, R, and their number density, n_v:   

$$V_f=\frac{4}{3}\pi R^3{n}_v$$(12)

We assume that during growth, the value of n_v is constant. By substituting Eq. (12) into Eq. (11), one can express c_B as   

$$c_B=c_B^0-V_f\left(c_B^{\beta }-c_B^{\alpha }\right)=c_B^0-\frac{4}{3}\pi R^3{n}_v\left(c_B^{\beta }-c_B^{\alpha }\right)$$(12)′

The equation for the growth rate given above in Eq. (7)′ can be written as   

$$\frac{dR}{dt}=\frac{D}{R}\frac{\left(c_B-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}$$(7)″

Then,   

$$\frac{dR}{dt}=\frac{D}{R}\frac{\left(c_B^0-\frac{4}{3}\pi R^3{n}_v\left(c_B^{\beta }-c_B^{\alpha }\right)-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}$$(13)

  

$$\frac{dR}{dt}=\frac{D}{R}\frac{\left(c_B^0-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}-\frac{4}{3}\pi R^2{n}_{v}D$$(13)′

Here, supersaturation S₀ and $\tilde{D}$ are expressed as   

$$S_0=\frac{\left(c_B^0-c_B^{\alpha }\right)}{\left(c_B^{\beta }-c_B^{\alpha }\right)}$$(14)

  

$$\tilde{D}=\frac{4}{3}\pi n_{v}D$$(15)

We can rewrite Eq. (13)′ as   

$$\frac{dR}{dt}=\frac{D}{R}{S}_0-R^2\tilde{D}$$(16)

This ordinary differential equation for the radius can be solved by separation of the variables   

$$\frac{RdR}{D{S}_0-R^3\tilde{D}}=dt$$(17)

and integration from zero to R,   

$${\int }_0^R\frac{RdR}{D{S}_0-R^3\tilde{D}}={\int }_0^{t}dt$$(18)

The left-hand side of Eq. (18) can be written as   

$${\int }_0^R\frac{RdR}{D{S}_0-R^3\tilde{D}}=\frac{1}{D{S}_0}{\int }_0^R\frac{RdR}{1-\frac{\tilde{D}}{D{S}_0}{R}^3}=t$$(19)

When the decomposition is complete and V_f is $V_f^e$ , R reaches R_m. By substituting R_m in Eq. (12),   

$$V_f^e=S_0=\frac{4}{3}\pi R_m^3{n}_v$$(20)

we obtain   

$$\frac{\tilde{D}}{D{S}_0}=\frac{\frac{4}{3}\pi n_{v}D}{D\frac{4}{3}\pi R_m^3{n}_v}=\frac{1}{R_m^3}$$(21)

By substituting Eq. (21) into Eq. (19), we obtain   

$$\frac{1}{D{S}_0}{\int }_0^R\frac{RdR}{1-\frac{R^3}{R_m^3}}=t$$(22)

  

$${\int }_0^R\frac{\frac{R}{R_m}{R}_{m}dR}{1-\frac{R^3}{R_m^3}}=D{S}_{0}t$$(23)

By defining   

$$\hat{r}=\frac{R}{R_m}$$(24)

we can rewrite Eq. (23) as   

$${\int }_0^{\hat{r}}\frac{\hat{r}{R}_m{ }^{2}d\hat{r}}{1-{\hat{r}}^3}=D{S}_{0}t$$(25)

  

$${\int }_0^{\hat{r}}\frac{\hat{r}d\hat{r}}{1-{\hat{r}}^3}=\frac{D{S}_0}{R_m{ }^2}t$$(25)′

  

$${\int }_0^{\hat{r}}\frac{\hat{r}d\hat{r}}{1-{\hat{r}}^3}=\frac{t}{\tau }$$(25)″

where τ is defined as   

$$\tau =\frac{R_m{ }^2}{D{S}_0}$$(26)

This integration of Eq. (25)″ can then be performed, yielding   

$$\frac{1}{6}\frac{ln\left(1+\hat{r}+{\hat{r}}^2\right)}{{\left(\hat{r}-1\right)}^2}-\frac{1}{\sqrt{3}}\left(arctan\left(\frac{2\hat{r}+1}{\sqrt{3}}\right)-arctan\left(\frac{1}{\sqrt{3}}\right)\right)=\frac{t}{\tau }$$(27)

Equation (27) expresses the relation between the particle radius and aging time during the growth process of particles with varying degrees of supersaturation, which is depicted in Fig. 8.

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