2025 Volume 66 Issue 8 Pages 931-940
The microstructure changes in the precipitation hardening copper alloys during the thermal process after deformation are known to exhibit complex behavior due to the simultaneous progress of various phenomena such as precipitation, recovery, recrystallization, and grain growth. Understanding the mechanism is an important issue for designing the heat treatment processes.
In this study, the microstructure changes in the Cu-Co-P alloy during thermal process after a deformation was numerically simulated based on the N model coupled with the dislocation recovery model, where the interaction between dislocations and precipitation behavior was focused. As a result, the following peculiar phenomena were calculated:
The fcc-Co formation and the Co2P precipitation on dislocations take place simultaneously in the Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy. The Co particles are left inside the bulk Cu matrix in the recovery process of dislocations, and then the Co particles are re-dissolved into matrix phase. The following mechanism was proposed to explain this peculiar behavior. Since the Co particles inside a bulk matrix left by the dislocation recovery have high interfacial energy, they re-dissolve into the matrix phase to release this energy. Furthermore, the decrease in Co concentration in the matrix phase due to the Co2P precipitation on the dislocation accelerates the Co particle re-dissolution.
This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 87 (2023) 258–266.
Fig. 2 Time evolution of the number density of Co precipitates inside a balk matrix phase (Red), Co precipitates on dislocations (Gray), Co2P precipitates inside a balk matrix phase (Green) and Co2P precipitates on dislocations (Blue) of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy during thermal process after deformation. (online color)
Precipitation hardening copper alloys are manufactured through a process that begins with casting, followed by hot deformation, multiple cold deformation and thermal process. Among these processes, thermal process is responsible for the formation of precipitates, the reduction of crystal lattice defects such as dislocations and the coarsening of grain size, and is an important process to improve material properties [1–3].
Looking at the temporal changes of microstructures during thermal process after deformation, we obtain that the recovery, recrystallization, and grain growth occur, sequentially, with precipitation accompanying them simultaneously. In this case, precipitation progresses partially overlapping with recovery and recrystallization, and completely overlapping with grain growth. Note that precipitation and other phenomena do not proceed independently, but interact with each other (henceforth, referred to as “interaction”) [4]. In this study, we investigate the effect of the interaction between dislocations and precipitation on the time evolution of the number density of precipitates by coupling the dislocation recovery model [5, 6] with the N model of precipitation (Numerical model) [7].
First, as background to this study, the interaction between dislocations and precipitation in grains is summarized below. It is known that the collector plate mechanism (CPM) [8] accelerates the precipitate formation by the high speed diffusion of solute atoms through the dislocations (or grain boundaries). An important aspect of the CPM effect is that the number of precipitates depends on the dislocation density, since the nuclei form on the dislocations prior to the bulk matrix phase. On the other hand, if CPM is viewed as the kinetic aspect of the dislocation-precipitation interaction, there is also an energetic aspect. Around the dislocations, there exists elastic strain energy generated by the distorted atomic arrangement [8, 9]. The closer to the dislocation core, the greater this elastic strain energy [8, 9], and the nucleation rate on the dislocation is much faster than that in the bulk matrix phase due to the self-energy of the dislocation which is relaxed by nucleation [8]. Considering these phenomena as well as the suppression of dislocation motion due to precipitate formation such as Cottrell locking [6] and pinning [9], we realize the microstructure change during thermal process after deformation is considered to exhibit complex behavior due to the interaction between dislocations and precipitation. Therefore, Understanding the mechanism of this complex microstructure change is necessary for designing a rational thermal process.
Previously, Senuma et al. [10] constructed a computer system to simulate complex microstructure changes during the process for hot deformation of carbon steel and microalloyed steel. The system consists of several elemental models that predict recovery, recrystallization, grain growth, phase transformation, and precipitation, and each elemental model is linked to predict the overall evolution of microstructure changes [10–14]. Mean-field theories such as phenomenological constitutive equations for recovery, recrystallization, and grain growth, classical nucleation theory, and growth rate equation for precipitates were used to describe each microstructure change. In the field of copper, Fujiwata et al. [1, 2] studied simulation models based on mean-field theory for the thermal process of precipitation hardening copper alloys. Mean field theory was used in these studies because it is important, in process design, to predict the macroscopic microstructure conditions (volume fraction and size) that most affect material properties. However, these models do not explicitly incorporate interactions among individual phenomena.
In this study, we performed microstructure simulations of thermal process after deformation of Cu-Co-P alloy using dislocation recovery model and N model to, quantitatively, understand the mechanism of microstructure change in this alloy, with a particular focus on the interaction between dislocations and precipitation. Cu-Co-P alloy is a typical precipitation hardening copper alloy with an excellent balance between strength and electrical conductivity [15]. For this alloy system, Shishido et al. [15] identified Co2P as the precipitate phase in Cu-0.4 mass%Co-0.1 mass%P after isothermal aging at 773 K based on the composition measurement by extracting residual analysis method and the diffraction pattern analysis by transmission electron microscopy (TEM). Kodan et al. [16] also calculated the phase diagram of the longitudinal section in Cu-0.1 mass%P of this alloy system using the CALPHAD [17, 18] method and showed that αCo2P is a stable phase at the composition and temperature mentioned above. On the other hand, due to a very large positive interaction parameter between Cu and Co drives fine precipitation in this alloy [19], the fcc-Co phase, coherent with matrix Cu phase, is expected to form early in the precipitation process as a metastable phase [20]. Fujiwara et al. [1, 2] performed a microstructure simulation using the recrystallization model and the Langer-Schwartz model of precipitation [7] for the thermal process after deformation of Cu-Cr alloy. It was shown that the actual precipitation behavior can be reproduced by considering the change in nucleation energy due to inhomogeneous nucleation, and the change in the interfacial atomic structure of the precipitate induced by the passage of recrystallized grain boundaries. However, the mechanism of microstructure change based on the CPM aforementioned, time evolution of dislocation density, and the existence of metastable phases remains unclear. Recently, Deschamps and Hutchinson have also written a review on precipitation phenomena in alloys, summarizing the progress of research on heterogeneous precipitation on crystal lattice defects [21]. However, a model that integrates the various interactions among dislocations and precipitation has not been established to date. As described above, although progress has been made in qualitative interpretation, the quantitative modeling and understanding of these interactions have not been fully developed. In this study, the importance of the interaction among dislocations and precipitation is discussed, based on the time evolution of the number density of various precipitates, considering the fcc-Co and Co2P phases, in addition to the competitive precipitation reaction [22] of the Co and Co2P phases.
The time evolution of the internal microstructure including dislocations and precipitates (Co and Co2P) is modeled based on a mean-field perspective. The dislocation recovery and precipitation phenomena (including precipitation on dislocations) during thermal process are considered only, and recrystallization and grain growth are excluded. This is because the objective of this study is to understand the interaction among dislocations and precipitation, and to exclude the effects of grain boundaries (such as dislocation pile-up and grain boundary precipitation). In the following, the recovery and precipitation models used in this analysis are described.
2.1 Recovery modelAs the evolution equation of dislocation density $\rho(t, T)$ during recovery, we adopted
| \begin{equation} \frac{\partial \rho (t,T)}{\partial t} = k_{0} + k_{1}(T)\rho (t,T) + k_{2}(T)\rho (t,T)^{2} \end{equation} | (1) |
(For simplicity, the temperature T is omitted in the following equations). Equation (1) expresses the time differential of the dislocation density in terms of the Taylor expansion for dislocation density, omitting the higher-order terms. k0, k1, and k2 are the expansion coefficients, and k0 = 0 was assumed because of $\partial \rho(t)/\partial t = 0$ at ρ(t) = 0. The second term on the right-hand side, which implies the decrease in dislocation density driven by the self-energy of dislocations, is expressed as [9, 23]
| \begin{equation} k_{1}\rho (t) = - M_{1} \cdot \Delta G_{1}(t) = - \frac{V_{\text{D}}}{2kT} \cdot \mu b^{2}\rho (t) \end{equation} | (2) |
As the annihilation mechanism of a single dislocation, annihilation at grain boundaries or climb motion are considered. Here, the symbol M1 is the relaxation coefficient of dislocation annihilation, ΔG1(t) is the self-energy of dislocation. The migration velocity of dislocation VD is expressed by $V_{\text{D}} = b\nu \cdot \exp(Q_{\text{s}}/R_{\text{g}}T)$, where b, ν, Qs, and Rg are the Burgers vector, thermal vibration frequency, activation energy of Cu self-diffusion, and the gas constant, respectively. In eq. (2), k, μ are the Boltzmann constant and the shear modulus of Cu, respectively. The third term on the right-hand side of eq. (1), which implies the dislocation dipole annihilation due to the interaction among dislocations, is expressed as [5]
| \begin{equation} k_{2}\rho (t)^{2} = - M_{2} \cdot \Delta G_{2}(t) = - 2\frac{D_{\text{s}}c_{\text{j}}b}{kT} \cdot \mu b^{2}\rho (t)^{2} \end{equation} | (3) |
The symbol M2 is the relaxation coefficient of dipole annihilation, ΔG2(t) is the interaction energy between dislocations, $D_{\text{s}} = D_{\text{s}}^{0} \cdot \exp(Q_{\text{s}}/R_{\text{g}}T)$ is the Cu self-diffusion coefficient, and cj is the concentration of dislocation jogs [5, 9] where $c_{j} = b\sqrt{\rho_{0}}$ was set using the initial dislocation density ρ0 [24]. Furthermore, when considering the effect of recovery due to precipitation on dislocations, eq. (1) was modified as follows:
| \begin{align} \frac{\partial \rho (t)}{\partial t} &= k_{0} + k_{1}\left\{ \rho (t) - 2N_{\text{d}}(t)\bar{R}_{\text{d}}(t) \right\} \\ &\quad + k_{2}\{ \rho (t) - 2N_{\text{d}}(t)\bar{R}_{\text{d}}(t)\}^{2} \end{align} | (4) |
Nd(t) and $\bar{R}_{\text{d}}(t)$ are obtained from eqs. (6) and (7) in Section 2.2.1, and are the number density and average radius of Co2P particles on dislocations, respectively. Equation (4) is a modified equation that reflects the reduction in dislocation length due to precipitation on dislocations. A study of the effect of recovery due to precipitation on dislocations was conducted by Zurob et al. [25]. In calculating the precipitation and recovery behavior of Nb steel during hot deformation, Zurob et al. used the recovery model assumed that the immobilization of dislocation segments due to pinning by precipitation on dislocations [25]. On the other hand, Takamiya et al. investigated the precipitation behavior of MnSe in thermal process after deformation that mimics hot deformation of 3%Si steel. And they found that recovery proceeds even in the dislocation microstructure where MnSe precipitates finely [26]. Among the precipitates formed in Cu-Co-P alloys, Co particles are expected to precipitate extremely finely due to the features (high driving force of precipitation and interfacial coherence with the Cu phase), previously mentioned. Therefore, in this study, we assume that dislocations with Co particles precipitated also contribute to recovery, based on the findings by Takamiya et al.
2.2 Precipitation model 2.2.1 Basic formulaIn this study, the N model [7] describing the formation and growth of precipitates was used. the N model [7] is a phase transformation kinetic model that numerically calculates the evolution of the size distribution function (the distribution of the number density of precipitates in size space) over time and size space, from which the total number and mean radius of precipitates per unit volume and the solute concentration in the matrix can be obtained.
In this study, the fcc-Co and Co2P phases are the targets of calculations, and in order to consider the competitive precipitation reaction [22], simultaneously, among Co and Co2P phases with precipitation on dislocations and inside a bulk matrix, four different size distribution functions are employed: Co particles on dislocations, Co particles inside a bulk matrix, Co2P particles on dislocations, and Co2P particles inside a bulk matrix. For simplicity, the hcp-Co phase was not considered. The size distribution function for each precipitate is defined as a function of time t and size R as in $f_{i}(R,t)$, where the subscript i is a number indicating the type of precipitates, in the order of $i = 1,2,3,4$: Co particles on dislocations, Co particles inside a bulk matrix, Co2P particles on dislocations, and Co2P particles inside a bulk matrix. $f_{i}(R,t)$ satisfies the continuity equation in size space, therefore its evolution equation is given by [7]
| \begin{equation} \frac{\partial f_{i}(R,t)}{\partial t} = j_{i}(R,t) - \frac{\partial }{\partial R}\{ \nu_{i}(R,t)\cdot f_{i}(R,t) \} \end{equation} | (5) |
where $j_{i}(R,t)$ is the nucleation rate of precipitates i at size R and $\nu_{i}(R,t)$ is their growth rate. Using $f_{i}(R,t)$, we calculate the total number of precipitates per unit volume N(t) and the average radius $\bar{R}(t)$ as [7]
| \begin{equation} N(t) = \int_{0}^{\infty }\sum _{i = 1}^{4}f_{i}(R,t)\text{d}R \end{equation} | (6) |
| \begin{equation} \bar{R}(t) = \frac{1}{N(t)}\int_{0}^{\infty }\sum _{i = 1}^{4}Rf_{i}(R,t)\text{d}R \end{equation} | (7) |
Nd(t) and $\bar{R}_{\text{d}}(t)$ in eq. (4) are obtained by calculating eqs. (6) and (7) for the Co2P particles only on dislocations (i = 3), respectively. From the conservation conditions of solute atoms, the solute concentration in the matrix cj(t) is calculated as [7]
| \begin{equation} c_{j}(t) \cong c_{j}^{0} - \sum_{i = 1}^{4}(c_{j}^{\text{p}_{i}} - c_{j}^{\text{e}}) \int_{0}^{\infty }\frac{4\pi }{3} R^{3}f_{i}(R,t)\text{d}R \end{equation} | (8) |
(in the following equations, time t will be omitted and denoted as cj). The subscript j is a number indicating the atomic species, and $j = 1,2,3,4$ is Cu, Co, P, and Sn, respectively. Here, $c_{j}^{0}$, $c_{j}^{\text{p}_{i}}$, and $c_{j}^{\text{e}}$ are the average solute concentration, the solute concentration of the precipitate phase, and the equilibrium concentration of each solute, respectively.
2.2.2 Nucleation rateAs integration of the nucleation rate $j_{i}(R,t)$ of a precipitate i at individual sizes R, the nucleation rate Ji(t) of a precipitate i is expressed as [7]
| \begin{equation} J_{i}(t) = \int_{0}^{\infty }j_{i} (R,t)\text{d}R \end{equation} | (9) |
Based on classical nucleation theory, and assuming that the nucleation rate at the critical nucleus size $R_{i}{}^{*}(c_{j})$ is dominant by the saddle-point method [27], $j_{i}(R,t)$ is given by [7]
| \begin{align} j_{i}(R,t) &= j_{i}(R_{i}{}^{*}(c_{j}),t) \\ &= S_{i}\beta_{i}^{*}(c_{j})Z_{i}(c_{j})\\ &\quad \times\exp \left(- \frac{\Delta G_{i}^{*}(c_{j})}{kT}\right)\left\{ 1 - \exp\left(- \frac{t}{\tau_{\text{inc}}(c_{j})}\right) \right\} \end{align} | (10) |
where Si, $\beta_{i}^{*}(c_{j})$, Zi(cj), $\Delta G_{i}^{*}(c_{j})$ and τinc(cj) are the precipitation sites per unit volume, the frequency of impingement of a solute atom on a critical nucleus, the Zeldovich factor, the energy barrier for the critical nucleus formation, and the incubation period, respectively. The precipitation sites Si is given as [28]
| \begin{align} &S_{i + 1}(t) = N_{\text{v}} - \rho (t)N_{\text{v}}{}^{\frac{1}{3}},\\ &S_{i}(t) = \rho (t)N_{\text{v}}{}^{\frac{1}{3}} ,\quad (i = 1,3) \end{align} | (11) |
for inside a bulk matrix and on dislocations, respectively. The subscripts i and i + 1 are used to distinguish precipitates on dislocations and inside a bulk matrix (In the following, this notation method is used to indicate the type of precipitates). Nv is the total number of atoms per unit volume of fcc-Cu. In the case of precipitation on dislocations, the number of sites is equal to the number of Cu atoms on dislocations [28], and in the case of precipitation inside a bulk matrix, the number of sites is equal to the total number of Cu atoms minus the number of Cu atoms on dislocations. Using the interfacial energy density σi and the driving force for precipitation ΔGi(cj), the critical nucleus size $R_{i}{}^{*}(c_{j})$ in eq. (10) is given as [7]
| \begin{equation} R_{i}{}^{*}(c_{j}) = - \frac{2\sigma_{i}}{\Delta G_{i}(c_{j})} \end{equation} | (12) |
by classical nucleation theory (note that all precipitates are assumed to be spherical shape). In this study, σi for Co particles is given as 0.15 J/m2 [20, 29], and for Co2P particles, it is given as 0.34 J/m2 from the growth rate of Co2P particles in Cu-Co-P alloy [15]. Using the chemical free energy $G_{\text{chem}}(c_{j}^{\text{p}_{i}})$, the chemical potential μj(cj) and the self-energy of dislocations annihilated by the critical nuclei formation $E_{\text{s}}(R_{i}{}^{*}(c_{j}))$, the driving force of precipitation $\Delta G_{i}(c_{j})$ is given as [7, 8]
| \begin{align} \Delta G_{i + 1}(c_{j})& = \frac{1}{V_{\text{m}}}\left(G_{\text{chem}}(c_{j}^{\text{p}_{i + 1}}) - \sum_{j = 1}^{4}c_{j}^{\text{p}_{i + 1}} \mu_{j}(c_{j})\right)\\ \Delta G_{i}(c_{j}) &= \frac{1}{V_{\text{m}}}\left(G_{\text{chem}}(c_{j}^{\text{p}_{i}}) - \sum_{j = 1}^{4}c_{j}^{\text{p}_{i}} \mu_{j}(c_{j})\right) \\ &\quad - E_{\text{s}}(R_{i}{}^{*}(c_{j})),\quad (i = 1,3) \end{align} | (13) |
for inside a bulk matrix and on dislocations, respectively. Here Vm is the molar volume. $G_{\text{chem}}(c_{j}^{\text{p}_{i}})$ is defined as
| \begin{align} G_{\text{chem}}(c_{j}^{\text{p}_{i}}) &= \sum_{j = 1}^{4}{ }^{0}G_{j}(T)c_{j}^{\text{p}_{i}} + R_{\text{g}}T\sum_{j = 1}^{4}c_{j}^{\text{p}_{i}}\ln (c_{j}^{\text{p}_{i}})\\ & \quad+ \sum_{j = 1}^{4}\sum_{\substack{k {=} 1\\(k{>} j)}}^{4}\left\{ \sum_{\nu = 0}^{n}L_{jk}^{(\nu)}(T)(c_{j}^{\text{p}_{i}} - c_{k}^{\text{p}_{i}})^{\nu } \right\} c_{j}^{\text{p}_{i}} c_{k}^{\text{p}_{i}} \end{align} | (14) |
based on the CALPHAD method [17, 18]. On the right-hand side of this equation, 0Gj is the Gibbs energy of a pure element, the second term is the energy due to the configurational entropy, and the third term is the excess enthalpy of mixing. In addition, the chemical potential μj(cj) is given as
| \begin{align} &\mu_{1}(c_{j}) = G_{\text{chem}}(c_{j}) - \sum _{j = 2}^{4}\left(\frac{\partial G_{\text{chem}}(c_{j})}{\partial c_{j}}\right)c_{j} ,\\ &\mu_{j}(c_{j}) = \mu_{1}(c_{j}) + \left(\frac{\partial G_{\text{chem}}(c_{j})}{\partial c_{j}}\right),\quad (j = 2,3,4) \end{align} | (15) |
using Gchem(cj). As the self-energy of dislocations annihilated by the critical nuclei formation $E_{\text{s}}(R_{i}{}^{*}(c_{j}))$, we adopted
| \begin{equation} E_{\text{s}}(R_{i}{}^{*}(c_{j})) \approx \frac{\mu b^{2}}{4\pi^{2}R_{i}{}^{*}(c_{j})^{2}}\ln \left(\frac{R_{i}{}^{*}(c_{j})}{R_{0}}\right) \end{equation} | (16) |
from the dislocation theory [9], where R0 is the radius of the dislocation core. Finally, for the energy barrier in the critical nucleus formation $\Delta G_{i}^{*}(c_{j})$, the frequency of impingement of a solute atom on a critical nucleus $\beta_{i}^{*}(c_{j})$, the Zeldovich factor Zi(cj), and the incubation period τinc(cj) in eq. (10), we used [7]
| \begin{equation} \Delta G_{i}^{*}(c_{j}) = \frac{16\pi \sigma_{i}{}^{3}}{3(\Delta G_{i}(c_{j}))^{2}} \end{equation} | (17) |
| \begin{equation} \beta_{i}^{*}(c_{j}) = \frac{4\pi R_{i}{}^{*}(c_{j})^{2}D_{i}c_{2}}{a^{4}} \end{equation} | (18) |
| \begin{equation} Z_{i}(c_{j}) = \frac{\nu_{\text{a}}}{4\pi^{3/2}}\left(\frac{3\Delta G_{i}^{*}(c_{j})}{kT}\right)^{1/2}\frac{1}{R_{i}{}^{*}(c_{j})^{3}} \end{equation} | (19) |
| \begin{equation} \tau_{\text{inc}}(c_{j}) = \frac{32\pi a^{4}R_{i}{}^{*}(c_{j})^{4}kT}{3\Delta G_{i}^{*}(c_{j})\nu_{\text{a}}{}^{2}D_{i}c_{2}} \end{equation} | (20) |
where a is the average lattice parameter of the precipitate and the matrix, νa is the volume per an atom which compose precipitate, and Di is the diffusion coefficient of Co atoms. Using the bulk-diffusion coefficient $D_{\text{b}} = D_{\text{b}}^{0} \cdot \exp(Q_{\text{b}}/R_{\text{g}}T)$ and the pipe-diffusion coefficient $D_{\text{p}} = D_{\text{p}}^{0} \cdot \exp(Q_{\text{p}}/R_{\text{g}}T)$, Di is set to [8, 30]
| \begin{align} &D_{i + 1} = D_{\text{b}},\\ &D_{i} \approx (D_{\text{b}}{}^{2}D_{\text{p}})^{\frac{1}{3}},\quad (i = 1,3) \end{align} | (21) |
for precipitates inside a bulk matrix and on dislocations, respectively.
2.2.3 Growth rate of precipitatesThe growth rate $\nu_{i}(R,t)$ of precipitates i of size R is given by [7]
| \begin{equation} \nu_{i}(R,t) = \frac{\text{d}R}{\text{d}t} = \frac{c_{2} - c_{2}^{\text{r}_{i}}(R)}{c_{2}^{\text{p}_{i}} - c_{2}^{\text{r}_{i}}(R)}\frac{D_{i}}{R} \end{equation} | (22) |
using the classical relation for particle growth, where $c_{2}^{\text{r}_{i}}(R)$ is the Co concentration of the matrix adjacent to the precipitates of radius R. From the Gibbs-Thomson equation [6], $c_{2}^{\text{r}_{i}}(R)$ is given by
| \begin{align} &c_{2}^{\text{r}_{i}}(R) = c_{2}^{\text{e}} \exp \left(\frac{2V_{\text{m}}\sigma_{i}}{R_{\text{g}}T}\frac{1}{R}\right),\\ &c_{2}^{\text{r}_{i + 2}}(R)(c_{3}^{\text{r}_{i + 2}})^{c_{3}^{\text{p}_{i + 2}}/c_{2}^{\text{p}_{i + 2}}}(c_{4}^{\text{r}_{i + 2}})^{c_{4}^{\text{p}_{i + 2}}/c_{2}^{\text{p}_{i + 2}}}\\ &\quad = c_{2}^{\text{e}}(c_{3}^{\text{e}})^{c_{3}^{\text{p}_{i + 2}}/c_{2}^{\text{p}_{i + 2}}}(c_{4}^{\text{e}})^{c_{4}^{\text{p}_{i + 2}}/c_{2}^{\text{p}_{i + 2}}}\exp \left(\frac{2V_{\text{m}}\sigma_{i + 2}}{R_{\text{g}}T}\frac{1}{R}\right),\\ &\qquad (i = 1,2) \end{align} | (23) |
for Co and Co2P particles, respectively [30]. Here $c_{3}^{\text{r}_{i + 2}}$ and $c_{4}^{\text{r}_{i + 2}}$ are the concentrations of P and Sn in the matrix adjacent to the Co2P particle. In a ternary alloy, the solute concentration of the matrix adjacent to the precipitate depends on the diffusion coefficients of the two solute atoms (Co and P in this alloy). If these coefficients are close and both diffuse sufficiently, ortho-equilibrium is achieved by the distribution of solute atoms. In this case, the solute concentration of the matrix adjacent to the precipitate can be determined from the tie line passing through the average solute concentration. On the other hand, in steels containing interstitial atoms such as C and N, the difference in diffusion coefficients between interstitial and substitutional atoms is extremely large, and in some cases substitutional atoms hardly diffuse at all. In such situations, it is necessary to assume No-partitional local equilibrium or para-equilibrium instead of ortho-equilibrium [8]. In the case of Cu-Co-P alloy, the diffusion coefficients of Co and P in Cu are 9 × 10−22 m2/s and 5 × 10−18 m2/s, respectively, at 659 K (the conditions of calculation in this study), with a difference of about 103 [31]. Despite this large difference in diffusion coefficients, the calculations in this study were performed assuming ortho-equilibrium. Because the present study targets the microstructure changes in which Co atoms (substitutional) diffuse sufficiently (precipitation on dislocations and inside a bulk matrix, and competitive precipitation reaction of Co and Co2P particles).
Finally, we discuss the effect of Sn on the growth of the Co2P phase. In order to understand the effect, we show the calculation results of the driving force for precipitation with and without Sn. When Co and P concentrations are set to c2 = 0.0032 and c3 = 0.0021 from the calculation conditions of this study, the driving force in c4 = 0 (without Sn) and c3 = 0.0011 (with Sn) are calculated as −22,196 J/mol and −22,171 J/mol, respectively, which are almost the same values. From these results, it is suggested that the Sn content hardly contributes to the growth rate of the Co2P phase.
2.2.4 Change in size distribution function due to recoveryThe precipitates that exist on dislocations are left behind in the bulk matrix outside the dislocation due to the dislocation movement (migration or annihilation). To express this behavior, using the dislocation length Δρ = (dρ/dt)Δt that annihilated during the minute time Δt, the number of precipitates of size R that left behind in the bulk matrix was calculated as
| \begin{align} &f_{i + 1}(R,t) = f_{i + 1}(R,t) + f_{i}(R,t)\Delta \rho/\rho ,\\ &f_{i}(R,t) = f_{i}(R,t) - f_{i}(R,t)\Delta \rho/\rho ,\quad (i = 1,3) \end{align} | (24) |
at each time step of numerical calculation.
2.3 Calculation conditionsThe numerical calculation conditions are explained here. We conducted the simulation of recovery and precipitation by coupling calculation of the evolution equations for dislocation density and precipitate size distribution function (eqs. (4) and (5)). As initial conditions, the supersaturated solid solution of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn, which contains dislocations introduced by deformation, was set. The fourth-order Runge-Kutta method [32] was employed as the numerical calculation method. The time step in eqs. (4) and (5) was set to 0.1 sec, and the minimum value of size step in eq. (5) was set to 0.25 nm. The profile of thermal process (time evolution of temperature) was assumed to be a constant heating rate of temperature increase at 0.1 K/s followed by isothermal holding at 659 K. The numerical values used in the models described in Sections 2.1 and 2.2 are shown in Table 1–Table 3.
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Figure 1 shows the time evolution of the number density and mean radius of precipitates (Fig. 1(a)), and volume fraction of precipitates and dislocation density (Fig. 1(b)) during thermal process after deformation of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy (henceforth, Cu-Co-P-Sn alloy). First, in order to understand the overall precipitation behavior, the results are not distinguished by the type of precipitates, but rather are summed for all. The horizontal axis is the time plotted logarithmically. The solid and dotted lines indicate the number density and the mean radius of precipitates (Fig. 1(a)), and volume fraction of precipitates and dislocation density (Fig. 1(b)), respectively. In Fig. 1(a), we confirm the peak in the number density of precipitates and the slowing down of the particle growth rate at that point, which shows that this calculation clearly describe the nucleation, growth, and coarsening processes. On the other hand, Fig. 1(b) shows that recovery and precipitation overlap in the early stage of precipitation. This indicates that nucleation and growth proceed, accompanied by a decrease in the preferential precipitation sites (dislocations). Figure 1(a) and (b) also show that the changes in the number density and volume fraction of precipitates during the coarsening process (after the peak of the number density) are steep in the early stage and then slow down. In order to understand these behaviors in detail, Fig. 2 shows the time evolution of the number density of Co and Co2P particles inside a bulk matrix and on dislocations, respectively. The number density of precipitates is taken into account here, because in the theory of alloy strengthening mechanisms, the number density (i.e., precipitate spacing) is a major factor in determining the amount of strengthening [33]. The color of each solid line corresponds to the type of precipitates in the graph: Co particles inside a bulk matrix (red), Co particles on dislocations (gray), Co2P particles inside a bulk matrix (green), and Co2P particles on dislocations (blue). Other notations are the same as that in Fig. 1. As illustrated in Fig. 2, the initially formed precipitates are Co particles on dislocations, and the physical explanation is as follows: the Co atoms, which are encountered on the dislocation by CPM [8], are grouped together by a very large positive interaction parameter between Cu and Co, and form the precipitate with same fcc structure as the matrix phase. Although the fcc-Co particles are chemically unstable, they are more stable than the Co2P particles at the critical nucleus size due to the interfacial coherency with the matrix phase, and thus are presumed to be the first to nucleate. Here, we confirm from Fig. 2 that three types of precipitates (Co particles inside a bulk matrix, Co2P particles on dislocations, and Co2P particles inside a bulk matrix) are formed following the Co particles on dislocations. There are two reasons for this: first, some Co particles formed on dislocations are left inside the bulk matrix due to dislocation movement (migration or annihilation); second, Co and Co2P particles precipitate in the competitive precipitation reaction. In Fig. 1(b), the dislocation density did not decrease to zero with time, but converged to a constant value, which indicates that some Co2P particles (and some Co particles) on dislocations continue to lock the dislocations. In this calculation, we confirmed that Co particles nucleate inside a bulk matrix and on dislocations in parallel. However, the majority of the Co particles inside a bulk matrix are that nucleated on the dislocations and then left behind in the bulk matrix by the dislocation migration. Finally, we confirm from Fig. 2 that the change in the number density during thermal process after deformation (Fig. 1(a)) mainly depends on the behaviors of Co2P particles on dislocations, Co particles on dislocations and Co particles inside a bulk matrix. It is also observed that a small amount of Co2P particles inside a bulk matrix are formed by the annihilation of immobilized dislocations, which decreases with time. The mechanism of the precipitation kinetics of these particles is discussed next.
Time evolution of (a) the number density and mean radius of precipitates, (b) the dislocation density and volume fraction of precipitates of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy during thermal process after deformation.
Time evolution of the number density of Co precipitates inside a balk matrix phase (Red), Co precipitates on dislocations (Gray), Co2P precipitates inside a balk matrix phase (Green) and Co2P precipitates on dislocations (Blue) of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy during thermal process after deformation. (online color)
In Fig. 2, the precipitation kinetics of Co and Co2P particles on dislocations is understood to be the disappearance of Co particles by the competitive precipitation reaction [22] on locked dislocations and the subsequent coarsening of Co2P particles, which is promoted by the high speed diffusion of Co atoms through the dislocation (CPM [8]). Therefore, we will focus on the precipitation kinetics of Co particles inside a bulk matrix in the following. The characteristic feature of Co particles inside a bulk matrix is that their number density decreases gradually after reaching a peak. This cannot be explained by the decrease in number density of Co particles associated with Ostwald ripening, since long-range diffusion of Co within the bulk matrix hardly occurs at 659 K, based on the diffusion coefficient of Co atoms in Cu matrix [34]. Therefore, it is likely that re-dissolution of the Co particles is occurring. For re-dissolution to occur, we considered that the following two conditions are necessary.
Figure 3 shows the time evolution of energy (integrated values of chemical free energy and interfacial energy in unit volume of microstructure) due to precipitation for Co particles inside a bulk matrix and Co2P particles on dislocations, during thermal process after deformation. Here, the integrated values of chemical free energy Gi(t) and interfacial energy Ei(t) was calculated, respectively, as
| \begin{align} &G_{i}(t) = \int\nolimits_{0}^{\infty }\frac{4\pi }{3}R^{3}f_{i}(R,t)\frac{1}{V_{\text{m}}}(G_{\text{chem}}(c_{j}^{\text{p}_{i}}) - G_{\text{chem}}(c_{j}^{0}))\text{d}R ,\\ &E_{i}(t) = \int\nolimits_{0}^{\infty }4\pi R^{2}f_{i}(R,t)\sigma_{i}\text{d}R \end{align} | (25) |
Figure 3(a) and (b) show the calculation results for Co particle inside a bulk matrix and Co2P particle on dislocations, respectively. The horizontal axis is the logarithmic plot of time, and the solid and dotted lines are the chemical free energy and interfacial energy, respectively. First, Fig. 3(a) shows that the interfacial energy of Co particles inside a bulk matrix is higher than the chemical free energy. We understood the reason for this as follows: After nucleation on the dislocation, the Co particles are left behind in the bulk matrix by dislocation movement. Since only few solute atoms diffuse inside a bulk matrix, the Co particles are frozen in place immediately after being left behind. Originally, the self-energy relaxation of dislocations acted as a driving force for precipitation, but because this quantity disappears, the Co particles left behind in the bulk matrix will contain high energy (mainly interface energy). As a consequence, we understood that condition (1) is satisfied.
Time evolution of the chemical free energy and the interfacial energy of (a) Co precipitates inside a bulk matrix phase and (b) Co2P precipitates on the dislocations of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy during thermal process after deformation.
Next, a comparison of Fig. 3(a) and (b) shows that the time, at which the decreasing rate of chemical free energy of the Co2P particles on dislocations reaches its maximum, corresponds approximately to the peak of the interfacial energy of the Co particles inside a bulk matrix. To clarify this relationship, Fig. 4 shows the time evolution of energy due to precipitation of Co particles inside a bulk matrix and Co2P particles on dislocations, during thermal process after deformation of Cu-0.32 at%Co-X at%P-0.11 at%Sn alloy. Here, X is the concentration of P, and simulations were performed at X = 0.003, 0.01, and 0.1. Figures 4(a)–(c) and Figs. 4(d)–(f) show the results for X = 0.003, 0.01, and 0.1 for Co particles inside a bulk matrix and Co2P particles on dislocations, respectively. Other notations are the same as that in Fig. 3. The results for X = 0.003 (Fig. 4(a) and (d)) show that the interfacial energy of the Co particles inside a bulk matrix gradually decreases even when only a few Co2P particles on dislocations exist. As mentioned earlier, since the Co particles inside a bulk matrix hardly grow by Ostwald ripening at 659 K based on the diffusion coefficient of Co atoms in Cu matrix, this mechanism is inferred to be the re-dissolution of the Co particles into the matrix (Mechanism 1). This mechanism 1 was also confirmed for X = 0.01 and 0.1. From the comparison of the calculation results at X = 0.003, 0.01, and 0.1 (Fig. 4(a), (d), (b), (e), (c) and (f)), we confirm that the peak of the interfacial energy of the Co particles inside a bulk matrix shifts to the shorter time side in response to the decrease in chemical free energy of Co2P particles on dislocations. This is understood to be due to the decrease in Co concentration of the matrix by the formation of Co2P particles on dislocations, and the resulting accelerated positive diffusion (re-dissolution) from the Co particles inside a bulk matrix to the matrix phase (Mechanism 2). The important point here is that Mechanism 2 accelerates Mechanism 1, there is a simultaneous release of the interfacial energy of the Co particles inside a bulk matrix and the chemical driving force of the Co2P particles on dislocations, that is, the energy of the entire system is efficiently released. This suggests the existence of a mechanism that satisfies condition (2).
Time evolution of the chemical free energy and the interfacial energy of Co precipitates inside a bulk matrix phase [(a), (b), and (c)], and Co2P precipitates on the dislocations [(d), (e), and (f)] of Cu-0.32 at%Co-X at%P-0.11 at%Sn alloy during thermal process after deformation, where X = 0.003 in (a) and (d), X = 0.01 in (b) and (e), and X = 0.1 in (c) and (f).
From the above, we understood that the characteristic of the precipitation kinetics in this alloy is the re-dissolution of Co particles inside a bulk matrix. And the decrease in Co concentration of the matrix, accompanied by the transfer of Co atoms from the matrix to Co2P particles on dislocations, accelerates this re-dissolution. Finally, Fig. 5 shows the precipitate size distribution at t = 4000 and 40000 sec in the simulation of the Cu-Co-P-Sn alloy (Fig. 1–Fig. 3). While the fine Co particles inside a bulk matrix disappear, the formation of Co2P particles on dislocations occurs. In addition, the number of fine Co2P particles inside a bulk matrix is decreasing, and we infer that the same mechanism as the Co particles inside a bulk matrix works at the size close to the critical nucleus.
Particle size distribution of the precipitates at (a) t = 4000 s and (b) t = 40000 s after deformation. The colors: red, gray, green, and blue, correspond to the Co precipitates inside a bulk matrix phase, Co precipitates on the dislocations, Co2P precipitates inside a bulk matrix phase, and Co2P precipitates on the dislocations of Cu-0.32 at%Co-0.21 at%P-0.11 at%Sn alloy, respectively. (online color)
Microstructure simulation of thermal process after deformation of Cu-Co-P alloy was performed by coupling the dislocation recovery model with N model to investigate the mechanism of microstructure change in this alloy, focusing on the interaction among dislocations and precipitation. And the following conclusions were obtained.
Part of this research was based on discussions in JST SIP (“Materials Integration” for Revolutionary Design System of Structural Materials) and in a Grant-in-Aid for Scientific Research on Innovative Areas on High Entropy Alloys (Grant Number JP18H05454) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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