1 Introduction

Nanocrystalline alloys with grain sizes <100 nm are widely recognized for their excellent mechanical properties [1,2,3,4,5,6,7,8]. Weertman et al. observed that the yield strength and total elongation of high-density, high-purity nanocrystalline Cu and Pd surpassed those of their coarse-grained counterparts [9]. However, the presence of numerous high-energy interfaces limits the application of these materials at elevated temperatures [2,3,4, 10,11,12]. Upon exposure to high temperatures, these tiny grains spontaneously merge, which triggers the coarsening of the nanocrystalline structure [1, 4]. Consequently, enhancing the thermal stability of nanocrystalline alloys is essential for controlling their high-temperature mechanical properties [13, 14].

To prevent excessive grain growth in nanocrystalline alloys, various strategies, such as second-phase particle pinning [15,16,17] and grain boundary (GB) segregation [18,19,20,21], have been employed to hinder GB migration. Moreover, the interactions among the solutes accumulated at the GBs resulted in concentration-dependent activation energy, reducing the GB energy and driving force for GB migration. Wang et al. [22] prepared nanocrystalline (CoCrFeNi)93.5Mn6.5 high-entropy alloy films with an average grain size of 25 nm via inert gas condensation. Their results demonstrated that the dispersion of Cr-rich precipitates with body-centered cubic structures in the matrix effectively pinned GBs. Devaraj et al. [23] discovered that the coarsening resistance of nanocrystalline Al-Mg alloys during heating is attributed to the synergistic effect of the segregation of Mg at grain boundaries, which reduces the driving force for grain growth, and the formation of nanoscale Al3Mg2 precipitates pinning grain boundaries hinders grain boundary migration. Nevertheless, it is still challenging to comprehensively understand the competition between these multiple stabilization mechanisms and their synergistic effects on nanocrystalline materials by relying solely on experimental research.

Theoretical and computational methods are valuable for studying microstructural evolution, particularly those related to second-phase precipitation and solute segregation in nanocrystalline alloys [24,25,26,27]. Among these, the phase-field method, which is based on classical thermodynamics and kinetic theory, is frequently employed to simulate the phase transitions and evolution processes of material microstructures. This technique avoids the need to track the migration of phase interfaces during complex phase transformation processes [28,29,30,31] and is powerful for investigating the pinning effect of randomly distributed second-phase particles on the grain growth strain energy on solute segregation. As noted by Millett et al. [32], both the atomic size and solute concentration influence the GB energy. The results demonstrate that an increase in the atomic radius mismatch reduces the grain boundary energy. Furthermore, notably, the mismatch strain is inherently linked to atomic radius mismatch. However, the influence of the mismatch strain on the precipitation and segregation processes of nanocrystalline alloys has not been sufficiently considered by the phase-field model. Consequently, further research is required to explore the role of mismatched strains in these processes.

This study aims to develop a phase-field model of nanocrystalline alloys that incorporates mismatch strain to explore the influence of GB segregation and second-phase precipitation on grain growth. The competitive mechanism between these two events in terms of the average grain area distribution and resulting volume fraction was studied through statistical calculations and analysis. Based on these findings, an optimized strategy for tuning the grain stability of nanocrystalline materials was proposed. The improved model will be helpful in the development and refinement of nanocrystalline alloys with exceptionally high thermal stabilities for engineering applications.

2 Methods

2.1 Phase field model

In this study, the solute concentration field \(c = c\left( {\vec{r},t} \right)\) distinguishes the solvent from the solute, and the grains are denoted by the structural order parameter \(\left\{ {\eta_{i} \left( {\overrightarrow {r} ,t} \right),i = 1,...,N} \right\}\), where \(\vec{r}\), t and N denote the position vector, time and number of order parameters required to describe the microstructure of the nanocrystalline alloys, respectively. Each order parameter represents a grain in the microstructure, and order parameters belonging to different grains cannot overlap [33]. For ease of understanding, the phase field variables are defined as \(\overrightarrow {\eta } \equiv \left( {\eta_{1} ,\eta_{2} ,...,\eta_{N} } \right)\). Consequently, the grains in the nanocrystalline system are described by \(\overrightarrow {\eta } = \left( {0,0,...,\eta_{i} = 1,...,0,0} \right)\), where \(\overrightarrow {\eta } = \left( {0,0,...,\eta_{i} = 1,...,0,0} \right)\) represents the GBs, and \(0 < \eta_{i} ,\eta_{j} < 1\). In particular, we expand the phase-field model of GB segregation to immiscible nanocrystalline alloys developed by Abdeljawad et al. [34, 35]. The total free energy Ftot comprises chemical energy Fch, interface energy Fint, and elastic energy Fel, which are expressed as Eq. (1):

$$F_{{{\text{tot}}}} = F_{{{\text{ch}}}} + F_{{{\text{int}}}} + F_{{{\text{el}}}}$$
(1)

Here, the chemical free energy density fch is denoted as Eq. (2):

$$\begin{aligned} f_{{{\text{ch}}}} & = f\left( {c,\eta_{1} ,\eta_{2} ,...,\eta_{N} } \right) = h\left( {\overrightarrow {\eta } } \right)f_{{{\text{gb}}}} \left\{ {c,T} \right\} \\ & \quad + \left[ {1 - h\left( {\overrightarrow {\eta } } \right)} \right]f_{{\text{b}}} \left\{ {c,T} \right\} + W_{0} g\left( {\overrightarrow {\eta } } \right) \\ \end{aligned}$$
(2)

where the free energies of the GB fgb and bulk grain fb depend on the solute concentration at a given temperature, T. \(h\left( {\overrightarrow {\eta } } \right)\) and \(g\left( {\overrightarrow {\eta } } \right)\) are both functions interpolating the free energy between GBs and the grains. According to the thermodynamic model of subregular solvents [36], fgb and fb can be expressed as Eq. (3):

$$\begin{aligned} f_{\varphi } &= G_{\varphi }^{{\text{B}}} c + G_{\varphi }^{{\text{A}}} \left( {1 - c} \right) + \frac{RT}{{V_{{\text{m}}} }}\left[ {c\text{ln}c + \left( {1 - c} \right)\text{ln}\left( {1 - c} \right)} \right] \\ & \quad+{{\varOmega}}_{\varphi } c\left( {1 - c} \right),\,\varphi = {\text{gb,b}} \\ \end{aligned}$$
(3)

where the free energies of pure elements A and B are represented by GA and GB, respectively. The enthalpy of mixing is denoted as Ωφ. The ideal gas constant is represented by R, and the molar volume is denoted by Vm.

At the center of the GB region, \(h\left( {\overrightarrow {\eta } } \right)\) is equal to 1, while within the grains, it is equal to 0. Recently, this interpolation function form was used to study the dynamics of grain growth in the presence of resistance at the three-phase junction [37]. The variable W0 signifies the obstacle potential between the GB and bulk grain.

This study assumes that \(h\left( {\overrightarrow {\eta } } \right)\) share and \(g\left( {\overrightarrow {\eta } } \right)\) share the same functional form, represented by Eq. (4):

$$h\left( {\overrightarrow {\eta } } \right) = g\left( {\overrightarrow {\eta } } \right) = \frac{4}{3}\left[ {1 - 4\sum\nolimits_{i = 1}^{N} {\eta_{i}^{3} + 3\left( {\sum\nolimits_{i = 1}^{N} {\eta_{i}^{2} } } \right)^{2} } } \right]$$
(4)

The chemical free-energy density can be further simplified by combining Eq. (2) through Eq. (3), as Eq. (5):

$$f_{{{\text{ch}}}} = f_{{\text{b}}} \left\{ {c,T} \right\} + \left\{ {W - a_{1} c - a_{2} c^{2} } \right\}g\left( {\overrightarrow {\eta } } \right)$$
(5)

where W, a1 and a2 are expressed as Eq. (6):

$$\begin{aligned} W & = W_{0} + \left( {G_{{{\text{gb}}}}^{{\text{A}}} - G_{{\text{b}}}^{{\text{A}}} } \right) \\ a_{1} & = \left( {G_{{{\text{gb}}}}^{{\text{A}}} - G_{{\text{b}}}^{{\text{A}}} } \right) - \left( {G_{{{\text{gb}}}}^{{\text{B}}} - G_{{\text{b}}}^{{\text{B}}} } \right) - \left( {{{\varOmega}}_{{{\text{gb}}}} -{{\varOmega}}_{{\text{b}}} } \right) \\ a_{2} &= \left( {{{\varOmega}}_{{{\text{gb}}}} -{{\varOmega}}_{{\text{b}}} } \right) \\ \end{aligned}$$
(6)

Furthermore, the free energy density of the interface is defined as Eq. (7):

$$\begin{aligned} f_{{{\text{int}}}} & = \frac{{\kappa^{2} }}{2}\left( {\nabla c} \right)^{2} + \sum\nolimits_{i}^{N} {\frac{{\varepsilon^{2} }}{2}\left| {\nabla \eta_{i} } \right|^{2} } \\ \nabla c & = \frac{\partial c}{{\partial x}} + \frac{\partial c}{{\partial y}} \\ \end{aligned}$$
(7)

which represents the sum of the gradient energies between the phase boundaries and the GBs. In particular, κ and ε correspond to the gradient coefficients at the phase boundaries and GBs, respectively, and are correlated with their energies and interface widths.

Fel represents the elastic strain energy over the entire volume of the system, as represented by V, which can be expressed as Eq. (8) [38,39,40]:

$$F_{{{\text{el}}}} = \frac{1}{2}\int_{V} {\sigma_{ij}^{{{\text{el}}}} } \varepsilon_{ij}^{{{\text{el}}}} {\text{d}}V$$
(8)

where \(\sigma_{ij}^{{{\text{el}}}}\) and \(\varepsilon_{ij}^{{{\text{el}}}}\) are expressed by:

$$\varepsilon_{ij}^{{{\text{el}}}} = \varepsilon_{ij} - \varepsilon_{ij}^{0} + \varepsilon_{ij}^{{{\text{ex}}}} ,$$
(8a)
$$\varepsilon_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right)$$
(8b)
$$\varepsilon_{ij}^{0} = \varepsilon_{00} \delta_{ij} f\left( c \right)$$
(8c)
$$\varepsilon_{00} = \left( {1/a} \right)\left( {da/dc} \right)$$
(8d)
$$f\left( c \right) = c^{3} \left( {10 - 15c + 6c^{2} } \right)$$
(8e)

where the elastic stress is represented by \(\sigma_{ij}^{{{\text{el}}}}\), and the elastic strain is represented by \(\varepsilon_{ij}^{{{\text{el}}}}\). In addition, \(\varepsilon_{ij}^{{{\text{el}}}}\) consists of the total strain \(\varepsilon_{ij}\), intrinsic elastic strain \(\varepsilon_{ij}^{0}\) and external strain \(\varepsilon_{ij}^{{{\text{ex}}}}\), as given by Eq. (8a). The expressions for \(\varepsilon_{ij}\) are given in Eq. (8b), where \(u_{i}\) and \(u_{j}\) are displacement field variables. \(\varepsilon_{ij}^{0}\), defined in Eq. (8c), is related to the lattice-misfit \(\varepsilon_{00}\), which is determined as the expansion coefficient calculated from the lattice parameter given by Eq. (8d). The Kronecker delta is represented by \(\delta_{ij}\) such that its value equals 1 when \(i = j\) and 0 when \(i \ne j\) [41].

Considering the difference in the alternate elastic modulus between the matrix and the precipitate:

$$\sigma_{ij}^{{{\text{el}}}} = C_{ijkl} \varepsilon_{ij}^{{{\text{el}}}}$$
(9)
$$C_{ijkl} = f\left( c \right)C_{ijkl}^{{\text{P}}} + \left( {1 - f\left( c \right)} \right)C_{ijkl}^{{\text{M}}}$$
(10)

where \(C_{ijkl}\) denotes the elastic modulus tensor. The elastic moduli of the precipitate and matrix are represented by \(C_{ijkl}^{{\text{P}}}\) and \(C_{ijkl}^{{\text{M}}}\), respectively. According to Hooker's law, the elastic stress applied to a system can be expressed as Eq. (11):

$$\sigma_{ij} = C_{ijkl} \varepsilon_{kl}^{{{\text{el}}}} = \left[ {f\left( c \right)C_{ijkl}^{{\text{P}}} + \left( {1 - f\left( c \right)C_{ijkl}^{{\text{M}}} } \right)\left( {\varepsilon_{kl} - \varepsilon_{kl}^{0} + \varepsilon_{kl}^{{{\text{ex}}}} } \right)} \right]$$
(11)

Substituting Eq. (11) into Eq. (8), the elastic strain Fel is determined as follows:

$$\begin{aligned} F_{{{\text{el}}}} &= \frac{1}{2}\int_{V} {C_{ijkl} \varepsilon_{ij}^{{{\text{el}}}} \varepsilon_{kl}^{{{\text{el}}}} } {\text{d}}V \\ &= \frac{1}{2}\int_{V} {\left[ {f\left( c \right)C_{ijkl}^{{\text{P}}} + \left( {1 - f\left( c \right)} \right)C_{ijkl}^{{\text{M}}} } \right]} \left( {\varepsilon_{kl} - \varepsilon_{kl}^{0} + \varepsilon_{kl}^{{{\text{ex}}}} } \right)\left( {\varepsilon_{ij} - \varepsilon_{ij}^{0} + \varepsilon_{ij}^{{{\text{ex}}}} } \right){\text{d}}V \\ \end{aligned}$$
(12)

The total free energy of the system was obtained by integrating the free energy density over the entire volume of the system, as represented by:

$$\begin{aligned} F_{{{\text{tot}}}} & = F_{{{\text{ch}}}} + F_{{{\text{int}}}} + F_{{{\text{el}}}} = \int_{V} {\left( {f_{{{\text{ch}}}} + f_{{{\text{int}}}} + f_{{{\text{el}}}} } \right)} {\text{ d}}V \\ & = \int_{V} {\left( {f_{{\text{b}}} \left\{ {c,T} \right\} + \left( {W - a_{1} c - a_{2} c^{2} } \right)g\left( {\overrightarrow {\eta } } \right) + \frac{{\kappa^{2} }}{2}\left( {\nabla c} \right)^{2} + \sum\nolimits_{i}^{N} {\left| {\nabla \eta_{i} } \right|^{2} } } \right){\text{d}}V} \\ & \quad + \frac{1}{2}\int_{V} {\left( {f\left( c \right)C_{ijkl}^{{\text{P}}} + \left( {1 - f\left( c \right)} \right)C_{ijkl}^{{\text{M}}} } \right)\left( {\varepsilon_{kl} - \varepsilon_{kl}^{0} + \varepsilon_{kl}^{{{\text{ex}}}} } \right)} \left( {\varepsilon_{ij} - \varepsilon_{ij}^{0} + \varepsilon_{ij}^{{{\text{ex}}}} } \right){\text{d}}V \\ \end{aligned}$$
(13)

The initial step of model exploration involves investigating how the enthalpy of mixing between the constituent elements and mismatch strain between the precipitate and matrix affect the GB free energy. For simplicity, we keep the free energy difference between the bulk grain and GB of both pure A and B equal, which implies that \(\left( {G_{{{\text{gb}}}}^{{\text{A}}} - G_{{\text{b}}}^{{\text{A}}} } \right) = \left( {G_{{{\text{gb}}}}^{{\text{B}}} - G_{{\text{b}}}^{{\text{B}}} } \right)\), thus leading to \(a_{1} = - a_{2} = - \left( {{{\varOmega}}_{{{\text{gb}}}} -{{\varOmega}}_{{\text{b}}} } \right)\).

In this study, we investigated four nanocrystalline systems with different lattice misfits, denoted by \(\varepsilon_{00}\) = 0, 0.02, 0.04 and 0.06 when \({{\varOmega}}_{{{\text{gb}}}}\) = +0.05. The derivatives of the GB free energy can be described as \(\frac{{\partial^{2} f_{{{\text{tot}}}} (c)}}{{\partial c^{2} }}\). According to the diffusion theory, when \(\frac{{\partial^{2} f_{{{\text{tot}}}} (c)}}{{\partial c^{2} }}\) is less than 0, the chemical driving force is sufficient to enable the nucleation process to occur. Within this compositional interval, small perturbations in the chemical composition cause an increase in the Gibbs free energy, thus the system becomes metastable, which is conducive to the generation of nucleation sites [42]. Referring to the work of Wang et al. [43], a simplified function of the elastic energy can be written as \(f_{{{\text{el}}}}{\prime} (c) = \frac{{V_{{\text{m}}} \varepsilon_{00}^{2} E}}{1 - \nu }(c - c_{0} )^{2}\) and \(\frac{{\partial^{2} f_{{{\text{el}}}}{\prime} (c)}}{{\partial c^{2} }}{ = }\frac{{2V_{{\text{m}}} \varepsilon_{00}^{2} E}}{{1{ - }\nu }}\). Based on the above definition, the simplified function of free energy is denoted as Eq. (14):

$$\begin{aligned} f_{{{\text{tot}}}}^{\prime} (c) & = f_{{\text{b}}} \left\{ {c,T} \right\} + \left\{ {W - a_{1} c - a_{2} c^{2} } \right\}g\left( {\overrightarrow {\eta } } \right) \\ & \quad + \frac{{V_{{\text{m}}} \varepsilon_{00}^{2} E}}{1 - \nu }(c - c_{0} )^{2} + \frac{{\kappa^{2} }}{2}\left( {\nabla c} \right)^{2} \\ \end{aligned}$$
(14)

where E is Young’s modulus, and v is Poisson ratio.

Furthermore, after performing a second-derivative operation on the function, this term can be approximated as Eq. (15):

$$\frac{{\partial^{2} f_{{{\text{tot}}}}^{\prime} (c)}}{{\partial c^{2} }} = \frac{RT}{{V_{{\text{m}}} }}({1 \mathord{\left/ {\vphantom {1 {c - {1 \mathord{\left/ {\vphantom {1 {(1 - c)}}} \right. \kern-0pt} {(1 - c)}}}}} \right. \kern-0pt} {c - {1 \mathord{\left/ {\vphantom {1 {(1 - c)}}} \right. \kern-0pt} {(1 - c)}}}}) + ( - 2a_{2} )g\left( {\overrightarrow {\eta } } \right) + \frac{{2V_{{\text{m}}} \varepsilon_{00}^{2} E}}{{1{ - }\nu }}$$
(15)

Figure 1a illustrates that the simplified free energy increases gradually as the mismatch strain increases. Figure 1b shows the derivative of the free energy shown in Fig. 1a. As the lattice misfit increases from 0 to 0.06, the bottoms of the curves shift towards the y = 0 guideline, resulting in a reduced region of spinodal decomposition. This indicates that increasing the mismatch strain makes the spontaneous nucleation process more challenging. Furthermore, we investigated three nanocrystalline systems with \({{\varOmega}}_{{{\text{gb}}}}\) =  −0.5, 0, +0.5 and \(\varepsilon_{00}\) = 0, as shown in Fig. 1c. As the value of \({{\varOmega}}_{{{\text{gb}}}}\) decreased from positive to negative, a gradual decline in the GB free energy curve was observed, with its minimum point residing below the maximum value of the bulk curve. Similarly, Fig. 1d shows the plots of the derivatives of the GB free energy curves shown in Fig. 1c. Owing to the reduction in the GB mixing enthalpy parameters, the region of spinodal decomposition steadily decreased, rendering spontaneous nucleation increasingly difficult.

Fig. 1
figure 1

Free-energy curves of bulk and GB systems a \(\varepsilon_{00}\) = 0, 0.02, 0.04, 0.06 and c Ωgb =  +0.5, 0 and −0.5; bulk thermodynamic parameters were assigned values of \(G_{{\text{b}}}^{{\text{A}}} = G_{{\text{b}}}^{{\text{B}}} = {{RT} \mathord{\left/ {\vphantom {{RT} {V_{{\text{m}}} }}} \right. \kern-0pt} {V_{{\text{m}}} }}\) = 0.6, \({{\varOmega}}_{{\text{b}}}\) = 3.0, and W = 0.8. Derivation of GB free energy curves with different lattice misfits and enthalpy of mixing. b \(\varepsilon_{00}\) = 0, 0.02, 0.04, 0.06 and d Ωgb =  +0.5, 0 and −0.5

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2.2 Kinetics and simulation conditions

The field variables evolve in time and space following the principle of energy dissipation in the system, as applied to the dynamic equations. The conserved field variable \(c\left( {\vec{r},t} \right)\) is usually described by the Cahn-Hilliard diffusion equation alongside a noise term [42, 44], which takes the form:

$$\frac{{\partial c\left( {\overrightarrow {r} ,t} \right)}}{\partial t} = \nabla M\nabla \frac{{\delta F_{{{\text{tot}}}} }}{{\delta c\left( {\overrightarrow {r} ,t} \right)}} + \xi \left( {\overrightarrow {r} ,t} \right)$$
(16)

For the nonconserved field variables, we used the relaxation equation, generally the Allen-Cahn type Eq. (17):

$$\frac{{\partial \eta_{N} \left( {\overrightarrow {r} ,t} \right)}}{\partial t} = L_{N} \frac{{\delta F_{{{\text{tot}}}} }}{{\delta \eta_{N} \left( {\overrightarrow {r} ,t} \right)}} + \zeta_{N} \left( {\overrightarrow {r} ,t} \right)$$
(17)

where M and \(L_{N}\) are the atomic and GB mobilities, respectively. \(\xi\) and \(\zeta_{N}\) are noise terms, which are random numbers conforming to a Gaussian distribution and must interact to satisfy the fluctuation and dissipation theorems [45].

$$\left\langle {\xi \left( {\overrightarrow {r} ,t} \right)\xi \left( {\overrightarrow {r}{\prime} ,t{\prime} } \right)} \right\rangle = - 2k_{{\text{B}}} TM\nabla^{2} \delta \left( {\overrightarrow {r} - \overrightarrow {r}{\prime} } \right)\delta \left( {t - t{\prime} } \right)$$
(16a)
$$\left\langle {\zeta_{N} \left( {\overrightarrow {r} ,t} \right)\zeta_{N} \left( {\overrightarrow {r}{\prime} ,t{\prime} } \right)} \right\rangle = - 2k_{{\text{B}}} TL_{N} \nabla^{2} \delta \left( {\overrightarrow {r} - \overrightarrow {r}{\prime} } \right)\delta \left( {t - t{\prime} } \right)$$
(17a)

where \(k_{{\text{B}}}\) is the Boltzmann constant. \(t{\prime}\) is a temporal variable. The noise term is closely related to the thermal fluctuation, which induces local fluctuations in the concentration and structural order parameter fields.

All the parameters used in this study were normalized [35] and listed in Table 1. The simulation area is a square with an area of \(N_{x} = N_{y}\) = 100 nm, with each grid point being \(\Delta x = \Delta y\) = 1.0 nm distance apart. In Sects. 3.1 and 3.2, \(c_{0}\) = 0.25 was used as the uniform initial concentration, whereas in Sect. 3.3, four sets of concentrations, \(c_{0}\) = 0.05, 0.10, 0.15 and 0.20, were selected to study the effects of the different solute concentrations. These values were chosen to represent the actual uniform initial distribution, thereby facilitating a more detailed analysis of the system behavior under different initial conditions.

Table 1 Normalized parameters used in present study
Full size table

3 Results and discussion

3.1 Effect of mismatch strain on precipitation in immiscible nanocrystalline alloys

Despite numerous experimental findings suggesting that the mismatch strain significantly impacts the precipitation behavior, tracing the effects of different lattice misfits on the precipitation process in real-time experimental states remains challenging. To investigate the role of the mismatch strain between the precipitate and matrix in the microstructural evolution and growth dynamics of the second phase during the phase transformation, four sets of lattice misfits were selected for analysis. Furthermore, the mixing enthalpy was set as positive with \({{\varOmega}}_{{{\text{gb}}}}\) =  +0.5, and \(c_{0}\) = 0.25. As shown in Fig. 2a–d, under specific mismatch conditions, the solute concentration at the GBs gradually increased with aging time. The precipitates are shown in red, and those from the blue matrix are shown. Precipitates that reached the critical nucleus size (indicated in red) continued to expand, whereas smaller precipitates were either engulfed by larger surrounding particles or dissolved into the matrix.

Fig. 2
figure 2

Effect of lattice misfits a \(\varepsilon_{00}\) = 0, b \(\varepsilon_{00}\) = 0.02, c \(\varepsilon_{00}\) = 0.04 and d \(\varepsilon_{00}\) = 0.06 on microstructure evolution of second-phase precipitation in immiscible nanocrystalline alloys

Full size image

However, as the lattice misfit increases from 0 to 0.06, the second-phase precipitation process is suppressed at an equivalent aging time, resulting in a reduced volume fraction and density of the precipitates. Figure 2a, b illustrates the microstructures of systems with lattice misfits of 0 and 0.02, respectively. As can be seen, the model with a lattice misfit of 0.02 contains nearly no secondary phase within its grain structure. Solute enrichment and precipitation at GBs effectively impede GB migration, hindering grain coarsening [15, 20]. As the lattice misfit increases to 0.04, the mismatch strain exerts a stronger inhibitory effect on the precipitation process over time. However, when the lattice misfit becomes excessively significant, the resulting increase in the elastic mismatch strain may increase the nucleation resistance of second-phase precipitation, leading to solute enrichment occurring solely at the GBs (Fig. 2d).

Statistical calculations based on ImageJ were conducted to analyze the distribution of the grain area in nanocrystalline alloys with varied lattice misfit strains at 200-time intervals. Precipitation was observed in areas where the concentration exceeds 0.8. In addition, the volume fraction of the precipitated phase (Vp) was measured over time. As shown in Fig. 3, the corresponding system exhibits a declining trend in the volume fraction of the precipitates as the lattice misfit increases. Particularly, in nanocrystalline alloys with a 0 lattice misfit, the grain area distribution ranged from 10 to 1375 nm2. However, as the lattice misfit escalates to 0.02, the maximum grain size in the alloy significantly reduces from 1375 to 960 nm2, resulting in a decrease in the fast-grown grains. This decrease primarily results from the substantial elastic effect that inhibits the precipitation of the secondary phase. The solutes in the alloy predominantly accumulated at the GBs, consequently impeding GB migration. Moreover, the grain size gradually increased from 960 to 1355 nm2 as the lattice misfit increased to 0.06, resulting from the mismatch strain exerted on the precipitation at the GBs.

Fig. 3
figure 3

Effect of lattice misfits a \(\varepsilon_{00}\) = 0, b \(\varepsilon_{00}\) = 0.02, c \(\varepsilon_{00}\) = 0.04 and d \(\varepsilon_{00}\) = 0.06 on grain area distribution in immiscible nanocrystalline alloys

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These findings indicate that as the lattice misfit increases, the particle volume fraction decreases. When the coupling mismatch strain is small, precipitation within the grains is effectively suppressed. However, when the mismatch strain becomes excessively large, the precipitation process of the second phase at the GBs is inhibited, resulting in a weakened pinning effect on grain boundary migration, thereby forming rapidly growing grains.

3.2 Effect of mismatch strain on precipitation in immiscible nanocrystalline alloys

Nanocrystalline materials possess structures characterized by loose GBs, which render them favorable for rapid atom (ion) diffusion and susceptibility to impurity atom (ion) segregation. The disordered alignment of atoms along the GBs results in several defects, such as vacancies and dislocations. These imperfections increase the boundary energy, causing the Gibbs free energy of GBs to exceed that of a single crystal with the same atomic component. The Gibbs free energy serves as an indicator of the thermodynamic potential of a system, and various mechanisms may be activated to minimize the total Gibbs free energy of a system. Among these mechanisms is the aggregation of solute atoms at GBs. Furthermore, precipitated phases can exhibit preferential nucleation at the GBs because these high-energy regions provide favorable sites for precipitate nucleation [46].

GB segregation is likely to occur when \({{\varOmega}}_{{{\text{gb}}}}\) < 0. For the concentration \(c_{0}\) = 0.25, four sets of lattice misfits with values of \(\varepsilon_{00}\) = 0, 0.02, 0.04 and 0.06 were evaluated. In addition, the mixing enthalpy was set to a negative value of \({{\varOmega}}_{{{\text{gb}}}}\) =  −0.5. As shown in Fig. 4a–d, the red-colored areas in the figures correspond to precipitates that emerge from the blue matrix, while the yellow regions signify segregation. Furthermore, the segregation process undergoes gradual suppression as the lattice misfit rises from 0 to 0.06. This suppression is attributed to an inhibitory effect of the segregation process induced by elastic mismatch strain. Similarly, the microstructures of the systems with the lattice misfit of 0 and 0.02 are shown in Fig. 4a, b, respectively. Clearly, precipitation within the grains is significantly reduced when the lattice misfit increases to 0.02, with the precipitation primarily confined to GBs. However, when the lattice misfit further increases to 0.06, precipitation at GBs nearly disappears, resulting in the progressive accumulation of the solute at GBs.

Fig. 4
figure 4

Effect of lattice misfits a \(\varepsilon_{00}\) = 0, b \(\varepsilon_{00}\) = 0.02, c \(\varepsilon_{00}\) = 0.04 and d \(\varepsilon_{00}\) = 0.06 on microstructure evolution of GB segregation in miscible nanocrystalline alloys

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Statistical analyses were performed to determine the grain area distribution of the nanocrystalline alloys at 200-time intervals and the time-dependent volume fraction of solute segregation (Vs) for different lattice misfits. As shown in Fig. 5, when the lattice misfit \(\varepsilon_{00}\) = 0, the volume fraction of solute segregation at the GBs in the system gradually decreased during the later stages of aging. This trend was mainly due to the enhanced precipitation of the second phase in the studied system, as the growth of the precipitated phase throughout the late aging period consumed some solutes from the matrix, resulting in a reduction in solute segregation at the GBs. When the lattice misfit increased to 0.02, the precipitation process was hindered, resulting in an increased solute volume fraction at the GBs. However, as the lattice misfit increased to 0.06, it restricted the extent of solute segregation at the GBs. In comparison with the alloy system with a mismatch of 0.02, the volume fraction of solute segregation at the GBs in the system with a mismatch of 0.06 is substantially lower.

Fig. 5
figure 5

Effect of lattice misfits a \(\varepsilon_{00}\) = 0, b \(\varepsilon_{00}\) = 0.02, c \(\varepsilon_{00}\) = 0.04 and d \(\varepsilon_{00}\) = 0.06 on grain area distribution of GB segregation in miscible nanocrystalline alloys

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Moreover, for the nanocrystalline alloy system with a lattice misfit of 0, the grain area distribution ranged from 5 to 1162 nm2. As the mismatch strain increased to 0.02, the maximum grain size of the alloy decreased significantly from 1162 to 766 nm2. This reduction is attributed to the suppression of second-phase precipitation in the grains, primarily because of the relatively large elastic mismatch strain, which causes most solutes in the alloy to accumulate at the GBs, thereby impeding GB migration. As the lattice misfit continued to increase to 0.04, the grain size progressively expanded from 766 to 825 nm2, which is due to the onset of elastic strain caused by the lattice mismatch. Ultimately, this strain completely halts the precipitation in miscible nanocrystalline alloys. At \(\varepsilon_{00}\) = 0.06, the solute segregation process within the system is also influenced by the elastic resistance. The inhibitory effect of solute segregation at the GBs on GB migration is weakened, resulting in a further increase in the maximum grain size to 1059 nm2.

The simulation results suggest that introducing an elastic mismatch strain increases the nucleation barrier, thereby slowing the precipitation of the secondary phase inside the grains. Consequently, solute enrichment occurs in the GBs. Furthermore, precipitation at the GBs was suppressed as the mismatch strain increased. Ultimately, segregation in miscible nanocrystalline alloys is mitigated when the elastic mismatch strain reaches an excessively high value. This phenomenon not only weakens the pinning effect of the second-phase precipitation at the GBs, but also inhibits solute segregation at the GBs. Consequently, accelerated grain growth occurred within the system.

3.3 Effect of mismatch strain on precipitation in immiscible nanocrystalline alloys

Referring to the free-energy derivation curves in Fig. 1, we studied four groups of solute concentrations to explore the interplay between the mismatch strain and composition, both within and beyond the spinodal decomposition domain. According to the aforementioned simulation results, the stabilization of grain sizes via solute segregation and second-phase pinning can be enhanced when \(\varepsilon_{00}\) = 0.02. Figure 6 shows the microstructure evolution of immiscible nanocrystalline alloys with \(\varepsilon_{00}\) = 0.02, \({{\varOmega}}_{{{\text{gb}}}}\) =  +0.5, and \(c_{0}\) = 0.05, 0.10, 0.15 and 0.20, where the red-colored areas show the precipitates emerging from the blue matrix. No precipitation occurred in the system when \(c_{0}\) = 0.05, and solute enrichment occurred at the three-phase junctions between several grains. When the solute composition increased to \(c_{0}\) = 0.10, which lies within the spinodal decomposition region, a gradual emergence of second-phase precipitation was observed at the GBs.

Fig. 6
figure 6

Microstructure evolution of immiscible nanocrystalline alloys with a \(c_{0}\) = 0.05, b \(c_{0}\) = 0.10, c \(c_{0}\) = 0.15, d \(c_{0}\) = 0.20 and \(\varepsilon_{00}\) = 0.02

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The interfaces between the precipitates and grains consisted of incoherent boundaries without coherent strain, which subsequently induced a higher interfacial energy. Consequently, the propensity for second-phase precipitation along the GBs was notably higher than that within the bulk grains [15]. In such circumstances, second-phase precipitation exerted a substantial pinning effect on the GBs, thereby posing substantial challenges to GB migration and effectively preventing fast grain growth. As shown in Fig. 6b–d, the increase in solute concentration not only enhanced the solute enrichment process at the GBs, but also escalated the precipitation process.

Figure 7a–d displays the statistically calculated grain area distribution in immiscible nanocrystalline alloys, accounting for different compositions at 200-time steps. At a low solute composition of \(c_{0}\) = 0.05, the grain area ranges from 4 to 1587 nm2. Increasing the concentration from 0.05 to 0.20 significantly decreases the maximum grain size from 1587 to 813 nm2.

Fig. 7
figure 7

Grain area distribution of immiscible nanocrystalline alloys with a \(c_{0}\) = 0.05, b \(c_{0}\) = 0.10, c \(c_{0}\) = 0.15, d \(c_{0}\) = 0.20 and \(\varepsilon_{00}\) = 0.02

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At time-step t = 200, the volume fraction of precipitation is ~4.6%, a value significantly lower than 10%. This finding suggests that the imposed elastic misfit strain inhibits the precipitation process within the system. For the solute composition of \(c_{0}\) = 0.15 and 0.20, the volume fraction of precipitation increases from 11.2% to 16.9%, respectively. These fractions are significantly lower than the initial solute concentration because the elastic strain inhibits the precipitation of the second phase. This observation further demonstrates a decrease in grain area as the volume fraction of precipitate presents at GBs increases.

4 Conclusion

In this study, a phase-field model was constructed by incorporating the elastic mismatch strain and factors such as solute concentration and mixing enthalpy. The stabilization of the grain size via solute segregation and second-phase pinning can be enhanced by the appropriate coupling of the elastic misfit strain, thereby highlighting the existence of a critical threshold for the elastic mismatch. Furthermore, GB segregation is favored in nanocrystalline alloys with solute concentrations exceeding the spinodal decomposition domain, whereas precipitation at the GBs is preferred under other circumstances. Moreover, in immiscible nanocrystalline alloys with positive mixing enthalpies, most solutes tend to precipitate at the GBs, in contrast to miscible nanocrystalline alloys with negative mixing enthalpies, where solutes exhibit a preference for GB segregation. The findings of this study provide valuable insights for the design of nanocrystalline alloys with improved stability.