Particle curvature effects on microstructural evolution during solid-state sintering: phenomenological insights from phase-field simulations

Abstract

Local curvatures have a profound influence on sintered microstructure. Here, using phase-field simulations, particle curvature effects were phenomenologically investigated by using geometrical configurations of two, three, and four particles, and by systematically varying particle curvatures. Some geometries, involving two, three and four particles, exhibited the expected smooth neck-length evolution, where the maximum neck length was determined by grain boundary (GB) energy ($\gamma _{GB}$) rather than surface energy ($\gamma _{S}$). In contrast, triangular arrangement of particles with unequal radii manifested a secondary necking event in form of a step during neck evolution. The secondary necking event coincided with internal pore collapse, and only specific range of particle radius ratios manifested such a mechanism. $\gamma _{S}$ played a dominant role in triggering the secondary necking event, while $\gamma _{GB}$ determined the remnant microstructure. Broadly, the geometries employed here allow us to computationally examine the sintering of particles that display wide variation in shapes and size distributions.

Introduction

Solid-state sintering is a processing technique that involves densifying green powder compacts by employing temperature, pressure or combination thereof [1,2,3]. This technique is used for processing high-temperature alloys, ceramics, nanocomposites [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and, even, ice compacts [4, 20]. Therefore, due to the wide application of sintering, a mechanistic understanding of the processes involved is mandated, particularly from the perspective of theory and simulation that can take into account coupling of multiple competing mechanisms [1, 21, 21,22,23,24,25,26,27, 27,28,29,30,31,32,33,34].

The “classical” two-particle model described sintering mechanism on the basis of surface, grain boundary (GB), lattice and vapor diffusion, and GB and surface energies [1, 21, 21,22,23,24, 35]. Such descriptions, involving kinetic and thermodynamic quantities, were used in the past to develop analytical models, or scaling laws, to quantify evolution of neck length between two particles of equal radius and their shrinkage kinetics. These scaling laws predict a smooth neck evolution, but do not allow us to view temporal change in microstructure near the neck region, i.e., they are “static” models. The description of such microstructural evolution is necessary, because technologically important powder materials increasingly contain extreme variations in size distribution, e.g., nanocomposites [14,15,16,17, 19] and shape (this category will be discussed later) [36,37,38].

Recent studies have shown that nano-sized powders exhibit rapid coarsening and grain boundary migration during sintering [14,15,16,17, 19, 31, 39], which is not accurately captured by the classical two-particle model [31]. Therefore, to describe such microstructural evolution, Lange et al. developed a thermodynamics model for two unequal sized particles, which subdivides the sintering process into three-stage processes [28, 29]. Broadly, they involve rapid neck growth (stage-I), competition between neck and grain growth (stage-II), and, finally, grain caused by rapid GB movement (stage-III). Note, Lange et al.’s model requires the movement of interfacial boundaries, which is not captured by the aforementioned classical model. The three-stage microstructure evolution mechanism was verified by Kumar et al. [31] and Ahmed et al. [34] by using phase-field modeling, a technique that simulates microstructure by capturing interfaces without explicitly tracking them [40, 41]. Later, Biswas et al. [33, 42] demonstrated that sintering kinetics can be enhanced by incorporating anisotropic microstructural properties, rigid body rotation and elastic strain energy to the phase-field models, compared to only thermodynamic considerations employed by Kumar et al. and Ahmed et al. [31, 34]. Notwithstanding, their phase-field simulations were consistent with Lange’s three-stage mechanism [31, 33, 34, 42], and showed smooth neck evolution across the three stages, at least in two-particle system. Importantly, these results demonstrate that microstructural evolution during sintering is overwhelmingly driven by the thermodynamic factors, i.e., GB and surface energies [28, 29, 31, 34], rather than specific microstructural and -mechanical features.

Beyond two-particle systems, several phase-field simulations of sintering have been carried out using “diverse” ensembles, where particles are arranged in differing multi-particle geometrical configurations, e.g., triangle, linear, square, two linear rows, etc., and as polycrystalline arrangements [43,44,45,46,47,48,49,50,51]. Most of these studies were limited to particles of equal radii, while very few have explicitly examined the role of particle curvature on microstructural evolution [31, 52]. It is well known that particle curvature determines the local chemical potential via Gibbs–Thompson effect, and gradient of that potential drives mass transport required for neck formation and grain growth [1, 28, 29, 31, 53, 54]. Consideration of curvature effects is now even more important because of two technical applications. First, powder particles of vastly different shapes are increasingly used for sintering applications, e.g., particles of spherical and plate-like shapes are mixed together to form a green-compact [36,37,38]. Second, in recent years, selective laser melting is extensively used to produce porous structures [55,56,57,58,59]. In this additive manufacturing technique, small particles are in contact with a flat base metal plate in addition to neighbors with spherical or near-spherical curvatures (Particles may be close to a GB while in contact with the flat base plate). In other words, these two cases show that local curvatures of individual particles, and their neighbors, can vary substantially.

Thus, taking these factors into account, we have investigated the phenomenological effect of particle curvature on the evolution of sintered microstructure, e.g., neck formation, change in pore shape, GB movement and the like. For this purpose, phase-field simulations were carried out using geometrical arrangements/configurations of two, three and four particles, where local curvatures were systematically varied from zero (i.e., a flat surface) to many times the radius of the smallest grain. Our phase-field model of the sintering process is based on thermodynamic description only, i.e., incorporating only effects from GB and surface energies, because the literature suggested that such considerations can reasonably capture evolution of the sintered microstructure [31, 34]. Through this approach, we have observed unique, particle curvature-mediated microstructural mechanisms that may participate during solid-state sintering. The remainder of the manuscript is organized as follows: "Computational details and initial microstructural geometries" section describes the phase-field model and geometrical configurations in detail, Results and analyses" section presents simulation results and corresponding analysis, and "Discussion: implications and limitations" section discusses our key findings.

Methodology

Phase-field model

Phase-field model for the sintering process was constructed using conserved ($\rho \{\vec {r},t\}$) and non-conserved ($\eta _i\{\vec {r},t\}$) variables (or order parameters). $\rho \{\vec {r},t\}$ is the solid density that acquires a value of 1 inside the solid phase and 0 with in pore regions. The non-conserved order parameter $\eta _i\{\vec {r},t\}$ (i=1,2,...,N) represents ensemble of “N” grains or particles, where $\eta _i$ is 1 inside the $i^{th}$ grain and 0 outside. Both $\rho \{\vec {r},t\}$ and $\eta _i\{\vec {r},t\}$ are related to the total free energy functional as

$$\begin{aligned} F = \int _V \left[ f(\rho ,\eta _i) + \frac{1}{2}\kappa _{\rho }\vert \nabla \rho \vert ^2 + \frac{1}{2}\sum _{i=1}^{i=N}\kappa _{\eta _i}\vert \nabla \eta _i\vert ^2 \right] dV, \end{aligned}$$

(1)

where $f(\rho ,\eta _i)$ is the bulk free energy, while the second and third gradient terms indicate excess energies due to the formation of interfaces, i.e., free surfaces and grain boundaries (GBs). In case of sintering, the gradient coefficients $\kappa _{\rho }$ and $\kappa _{\eta _i}$ have been shown to depend on surface and GB energies (discussed later) [31, 34, 60]. In Eq. (1), $f(\rho ,\eta _i)$ was described using a Landau-type polynomial of the form [32,33,34, 42, 60, 61]

$$\begin{aligned}&f(\rho ,\eta _i) = \alpha \left[ \rho ^2(1-\rho )^2 \right] + \beta \left[ \rho ^2 + 6(1-\rho )\sum _{i=1}^{i=N}\eta _i^2\right. \nonumber \\&\quad \left. -4(2-\rho )\sum _{i=1}^{i=N}\eta _i^3 + 3\bigg (\sum _{i=1}^{i=N}\eta _i^2\bigg )^2 \right], \end{aligned}$$

(2)

where $\alpha$ and $\beta$ are constants. The first term in Eq. (2) is a double-well potential, while the second term couples the conserved and non-conserved order parameters, and, taken together, both terms ensure that $f(\rho ,\eta _i)$ energy landscape acquires minima within solid phase/grains and pores.

Cahn–Hilliard equation was used to describe the time evolution of the conserved order parameter $\rho$

$$\begin{aligned} \frac{\partial \rho }{\partial t} = \nabla \cdot D\nabla \frac{\delta F}{\delta \rho } = \nabla \cdot D\nabla \bigg (\frac{\partial f(\rho ,\eta _i)}{\partial \rho } - \kappa _{\rho }\nabla ^2\rho \bigg ), \end{aligned}$$

(3)

where $\frac{\partial f(\rho ,\eta _i)}{\partial \rho }=\mu$ is the chemical potential, and D is the diffusion/mobility coefficient that depends on the microstructure [31, 32]. Therefore, D is assumed to be a function of $\rho \{\vec {r},t\}$ and $\eta _i\{\vec {r},t\}$ and expressed as [32]

$$\begin{aligned}&D = D_{surf}\rho (1-\rho ) + D_{GB}\sum _i\sum _{j\ne i}\eta _i\eta _j \nonumber \\&\quad + D_{vol}\phi (\rho ) + D_{vap}\big [1-\phi (\rho )\big ]. \end{aligned}$$

(4)

Eq. (4) incorporates contributions to the “net” D from surface ($D_{surf}$), grain boundary ($D_{GB}$), volumetric ($D_{vol}$) and vapor ($D_{vap}$) phase diffusion by using the interpolation function $\phi (\rho )=\rho ^3(10-15\rho +6\rho ^2)$. Finally, time evolution of the non-conserved order parameter $\eta _i$ was computed using the Allen–Cahn equation

$$\begin{aligned} \frac{\partial \eta _i}{\partial t} = -L\frac{\delta F}{\delta \eta _i} = -L \bigg (\frac{\partial f(\rho ,\eta _i)}{\partial \eta _i} - \kappa _{\eta }\nabla ^2\eta _i \bigg ), \end{aligned}$$

(5)

where L is the grain boundary mobility.

Surface ($\gamma _{S}$) and grain boundary ($\gamma _{GB}$) interfacial energies are known well to drive mass transport during sintering, and consequently determine the evolution of microstructural features [1]. Such features include neck formation, densification and coarsening (i.e., merging small with larger grains) [31,32,33,34, 42]. Ahmed et al. have shown that $\gamma _{S}$ and $\gamma _{GB}$ are related to phase-field parameters $\alpha$, $\beta$, $\kappa _{\rho }$ and $\kappa _{\eta }$ as [34]

$$\begin{aligned} \gamma _{GB}=\frac{2}{\sqrt{3}}\sqrt{\beta \kappa _{\eta }}, \ \ \ \ \gamma _{S}=\frac{\sqrt{2}}{6}\sqrt{\kappa _{\rho }+\kappa _{\eta }}\sqrt{\alpha + 7\beta }. \end{aligned}$$

(6)

Thus, Eqs. (1)–(6) were utilized to systematically examine different types of microstructures (c.f. "Computational details and initial microstructural geometries" section).

Computational details and initial microstructural geometries

Numerical simulations were performed using non-dimensionalized form of Eqs. (3) and (5), as described by Ahmed et al. [34] and Wang [32] (In the manuscript, we will use "*" to indicate symbols corresponding to dimensionless time and length). Time integration was carried out using explicit Euler scheme with step size 0.001 ($=\Delta t^*$), and five-point stencil method for computing Laplacian (i.e., $\nabla ^2$) [34, 60, 62,63,64]. These numerical techniques were utilized to conduct 2D simulations involving $128 \times 128$ and $256 \times 256$ square grids of length 0.4 ($=\Delta x^*$). Other parameters/constants present in Eqs. (1)–(6) were adapted from the literature [34, 60, 64] and are listed in Table 1. The parameters in $1^{st}$ row of Table 1 (in bold) were employed in all the simulations, while 2$^{nd}$ and 3$^{rd}$ rows were used to perform sensitivity analysis for different ($\kappa _{\rho }$, $\kappa _{\eta }$) or ($\gamma _{S}$, $\gamma _{GB}$) combinations (using Eq. (6)).

Figure 1

Initial microstructures used in the simulations. Six different cases showing the small “probe” particle (marked as $\eta _1$) in contact with larger grains/particles with differing radii/curvatures. (a) case-I shows the smaller in contact with a larger grain $\eta _2$ that is 4 times the radius of $\eta _1$. (b) case-II depicts $\eta _1$ touching a flat surface (curvature=0). (c) case-III involves a $\eta _1$ lying at the grain boundary (GB) formed by two flat-surface grains. case-IVa in panel (d1) shows $\eta _1$ in contact with larger circular grains, $\eta _2$ and $\eta _3$, whose radii are 4 and 2 times that of $\eta _1$, respectively. While case-IVb in panel (d2) contains $\eta _1$ in contact with two equal-sized circular grains with 4 times larger radius. (e) $\eta _1$ touching a larger circular grain and a flat surface (case-V). (f) $\eta _1$ touching a larger circular grain and a flat surface with a GB between $\eta _1$ and $\eta _2$ (case-VI). The radius circular grain marked $\eta _2$ in (e) and (f) is 4 times that of $\eta _1$

Figure 2

Plots comparing total neck-length ($l^*_{neck}$) evolution in (a) cases I, II and III and (b) cases IVa-b, V and VI. Except for case-IV, a smooth three-stage neck evolution is exhibited in all cases, i.e., initial stage of neck formation and grow, concurrent grain (the larger ones) and neck growth at the intermediate stage, and the disappearance of small grain in the final stage. As an example, the three stages are marked as I, II and II in panel (a). Panels (c) and (d) plot the change in small grain area in all six cases as a function of normalized time. (e) compares the magnitude of rate of areal change ($\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$, left axis ), and time required for the small grain $\eta _1$ to disappear ($\mathbf{t }^{disappear}_{norm}$, right axis). Plots (a)–(e) were obtained using $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63 and $128 \times 128$ simulation box sizes

Table 1 List of non-dimensionalized parameters used in the simulations. Mobility parameters were fixed to $D_{surf}=10.0$, $D_{GB}=1.6$, $D_{vol}=0.04$, $D_{vap}=0.002$ and $L=1$. Computed values of $\gamma _{S}$ and $\gamma _{GB}$ for each $\{ \kappa _{\rho }$, $\kappa _{\eta } \}$ combinations are also indicated

Figure 1 depicts the initial microstructures/configurations used in our simulations. These configurations examine the effect of different curvature environments and grain boundaries on the evolution of sintered microstructure by using a small circular particle or grain (of radius $R_{\eta _1}$) as a “probe” (marked $\eta _1$ in Fig. 1) (In the remainder, the terms particle and grains will used interchangeably). In total, six cases were examined. cases I and II depict $\eta _1$ touching a larger particle with $R_{\eta _2}$=4$\times R_{\eta _1}$(Fig. 1a), and a flat surface with 0 curvature or $R_{\eta _2}$ $\rightarrow \infty$ (Fig. 1b), respectively. The two-particle configuration, similar to case-I, has been investigated in several phase-field based studies [31,32,33, 42], and, therefore, case-I serves as baseline/control for this work. Case-III examines the effect of GB, when $\eta _1$ is in contact with flat surface (Fig. 1c). Case-IV probes the environment around two larger circular particles $\eta _2$ and $\eta _3$, when $R_{\eta _2}$ = 4$\times R_{\eta _1}$ and $R_{\eta _3}$ = 2$\times R_{\eta _1}$ (Case-IVa, Fig. 1d1), and $R_{\eta _2}$ = $R_{\eta _3}$ = 4$\times R_{\eta _1}$ (Case-IVb, Fig. 1d2). Fig. 1e depicts the geometry of Case-V, where $\eta _1$ is simultaneously in contact with a larger particle ($\eta _2$ and $R_{\eta _2}$ = 4$\times R_{\eta _1}$) and a flat surface ($\eta _3$). Finally, the configuration shown in Case-VI (Fig. 1f) was designed to examine the effect of GB on Case-V. It may be pointed out that presence of an initial internal pore further differentiates the microstructures in cases-IV, V and VI from I, II and III. Table  2 summarizes different features of the initial geometrical configurations/microstructures.

Table 2 Summary of particle radii and geometry of each initial microstructures. Presence or absence of grain boundary (GB) and prior internal pore is also indicated

Kinetics of microstructural evolution was quantified on the basis of total neck length ($l^*_{neck}$), i.e., summation of neck lengths between $\eta _1$ and its immediate neighboring particles, and $\eta _1$ area ($A_{\eta _1}$) as function of time . These two temporal quantities are commonly used in the literature to quantitatively characterize microstructural evolution during sintering [1, 31,32,33,34, 42, 60, 61]. Neck length was computed using $\sum _{i\ne 1}\eta _1 \eta _i ({\Delta x^*}^2 + {\Delta x^*}^2)^{\frac{1}{2}}$, and the area by $\sum _i\sum _j \eta _1 \Delta x_i^* \Delta x_j^*$ (i.e., $\int \eta _1dA$). Simulations were carried to till the smaller $\eta _1$ particle merges with its neighbors, meaning when it disappears during the evolution process [31, 34].

Results and analyses

Simulation results are presented in the following order. In Sect . 3.1, we will discuss kinetics of small particle evolution (c.f. $\eta _1$ in Fig. 1) for all the six cases (c.f. Table 1). Next, on the basis of these quantitative results, we will nominally categorize the six cases into fast, intermediate and slow kinetics, and in Sects. 3.2, 3.3 and 3.4, we examine their corresponding microstructures, respectively.

Kinetics of small particle evolution

The evolution of small particle, i.e., $\eta _1$ in Fig. 1, was quantified by measuring the temporal change in three quantities for each case: (i) its total neck length or $l^*_{neck}$ (Fig. 2a, b), (ii) area (Fig. 2c, d) and (iii) time taken by $\eta _1$ to disappear completely (Fig. 2e) . We emphasize that $l^*_{neck}$ is summation of the length of necks formed between $\eta _1$ and its surrounding grains/particles (e.g., necks between $\eta _2$–$\eta _1$ and $\eta _1$–$\eta _3$ in case-V of Fig. 1d), while $A_{\eta _1}$ is the area of $\eta _1$ alone.

The $l^*_{neck}$ v.s. normalized time $(\frac{t^*}{t^*_{max}})$ plot in Fig. 2a compares the neck evolution in cases I, II and III, while Fig. 2b compares the same feature in cases IVa-b, V and VI. Fig. 2a shows that, in case-I, the neck between the smaller and larger particle ($\eta _2$) evolves in three stages (indicate as I, II and II in Fig. 2a)—consistent with literature reports [28, 29, 31, 33, 34, 42]. As exemplified using case-I in Fig. 2a, stage-I involves rapid neck formation, followed by concurrent neck and grain growth in stage-II (i.e., growth of $\eta _2$ at the expense of $\eta _1$), and, finally, in stage-III the smaller particle is consumed by rapid GB movement [31, 34]. The last stage coincides with the disappearance of the $\eta _1$, which is seen in Fig. 2c as rapid reduction in $\eta _1$ area. Broadly, Fig. 2a, b demonstrates that cases I, II, III, V and VI follow the well-documented “three stage” particle evolution mechanism (also see Fig. 2c, d) (Differences between these cases will be presented later). In stark contrast, case-IVa-b prominently manifests an abrupt, step formation immediately after stage-I (Fig. 2b). Case-IVa exhibits that “step” slightly earlier than case-IVb, but both experience neck shrinkage and $\eta _1$ disappearance within the same time duration (shown using normalized time scale in Fig. 2b, d). In Sects. 3.3 and  3.4, we will provide a microstructural basis for the two-step mechanism. Notwithstanding, the two-step mechanism in case-IV distinctly highlights the effect of local curvature on the resulting microstructure. Other curvature effects can be seen in case-II (with a flat surface), which experiences a discernibly greater stage-II $l^*_{neck}$ than case-I (see Fig. 2a). In general, $l^*_{neck}$ at stage-II follows the trend case-IV > case-V $\approx$ VI > case-III > case-II > case-I.

While $l^*_{neck}$ was computed by taking into account all possible $\eta _1$ contact surfaces, it is rather cumbersome to examine neck evolution for each contact, specially in a multi-particle environment. Therefore, we have also monitored the change in $\eta _1$ area ($A_{\eta _1}$) for each case (Fig. 2c, d), because such a quantity directly probes the influence of local curvature/radii on $\eta _1$ evolution via Gibbs–Thompson effect, i.e., $\mu \propto \frac{\gamma }{R}$ [1, 54]. In this study, by construction (see Fig. 1 and Table 1), we have varied “R”, or the local curvature, for fixed values of $\gamma _{S / GB}$ (or $\kappa _{\eta }$ and $\kappa _{\rho }$ in Eq. 6). The $A_{\eta _1}$ v.s. $(\frac{t^*}{t^*_{max}})$ plots in Fig. 2c depict the areal change in cases I, II and III, while, separately, Fig. 2d compares cases IV, V and VI. A careful examination of these plots revealed two common features: first, an initial gradual decrease in $A_{\eta _1}$ (which is later approximated using a linear fit) and, second, its sudden drop due to the disappearance of $\eta _1$. The latter, near-vertical drop is very likely due to rapid $\eta _1$–$\eta _2$ GB movement, which swiftly consumes the smaller $\eta _1$ grain [28, 29, 31]. Crucially, we have extracted the magnitude of rate/slope of areal reduction or $\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$ and the time of $\eta _1$ disappearance $\mathbf{t }^{disappear}_{norm}$ (corresponding to the moment $A_{\eta _1}=0$) from the two distinct features outlined earlier in the plots presented in Fig. 2c, d. Values of the extracted quantities are listed in Table 3, while the plot presented in Fig. 2e compares them for all cases (Fig. 1). For $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 (see Table 1), we find that the kinetics of $\eta _1$ evolution becomes slower by increasing the number of grains/particles, or $\eta _i$, in its neighborhood, i.e., increase in $\eta _i$ decreases $\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$, while increasing $\mathbf{t }^{disappear}_{norm}$ (Discussion regarding the conditions that allows such a relationship between $\mathbf{t }^{disappear}_{norm}$ and number of particles will be delayed till the discussion section). A closer inspection of Fig. 2e and Table 3 revealed that $\eta _1$-kinetics can be nominally categorized into three distinct regimes, which, hereafter, will be referred to as fast, intermediate and slow kinetics regimes.

Table 3 List of $\eta _1$-kinetic parameters extracted from Fig. 2c, d. The slope $\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$ was obtained from a fit to the initial linear segment, while $\mathbf{t }^{disappear}_{norm}$ corresponds to the moment when $A_{\eta _1}=0$. Numerical simulations were performed using parameters $\alpha$=16, $\beta$=1, $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63, and $128 \times 128$ simulation box sizes. Additional calculations were also conducted using $256 \times 256$ grids to confirm the trends

Fast kinetics was manifested in cases I and II, because of their highest $\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$ and lowest $\mathbf{t }^{disappear}_{norm}$ values among all the cases (see Fig. 2e and Table 3). Slow kinetics was observed in cases V and VI due to their lowest $\left| \frac{\mathbf{dA }_{\eta _1}}{\mathbf{dt }_{norm}}\right|$ and highest $\mathbf{t }^{disappear}_{norm}$ values. Accordingly, cases III and IV lay in the intermediate regime. It may be noted that, even though microstructures of cases III, IVa-b and V contained the same number of particles/grains (Fig. 1), case-V has slower $\eta _1$-kinetics than the former two microstructures (see Fig. 2e). On the other hand, the initial microstructures of cases III (with flat surfaces) and IVa-b (without flat surfaces) differed significantly in terms of local curvatures, still they manifest comparable $\eta _1$-kinetics (although neck formation mechanism is substantially different). Such contrasting kinetics within a similar category of microstructures, i.e., with same number of grains/particles, underscores the effect of local curvatures in a multi-particle environment on their microstructural evolution during sintering. In the next three subsections, we will systematically examine mechanisms contributing to the evolution of each configuration, lying within their respective each regime (Fig. 2c), in detail.

Fast kinetics regime in two-particle environments: Cases I and II

From Fig. 2e and Table 3, we find that cases I and II (with a flat surface) have comparable the rate of area reduction and disappearance time of the smaller $\eta _1$ particle. These results demonstrate that, in the two-particle configuration/geometry, flat surface has minimal impact on the $\eta _1$ evolution kinetics, at least for the chosen set of $\kappa _{\rho }$ and $\kappa _{\eta }$ (see Tables 2 and  3). However, such a surface have a noticeable influence on the neck length, where case-II has a longer stage-II neck than in case-I (Fig. 2a). To better understand this difference, we have compared and contrasted the microstructural evolution in both cases.

Microstructural evolution in cases I and II is presented in panels  3(a1)–(a3) and  3(b1)–(b3) of Fig. 3, respectively, and shows changes from the middle of stage-II till $\eta _1$ disappears (compare Fig. 2a with Fig. 3a, b). Furthermore, the images in both cases are positioned in such a way that the movement of $\eta _1$–$\eta _2$ GB interface could be discerned. For example, in case-I, $\eta _1$ is observed to shrink due to GB movement (compare panels  3(a1) and 3(a2)), which is consistent with the growth of larger $\eta _2$ grain [28, 29, 31, 34]. Such GB-movement-mediated grain growth is also noted in case-II, although there was one key difference between the two cases. A careful comparison between panels 3(a1)–(b1) and 3(a2)–(b2) revealed that the GB in case-II, or the neck , had a noticeable curvature, while that neck in case-I is relatively flatter. Our results are also consistent with the theoretical studies by Colbeck [26], who showed that as the ratio of grain size, i.e., $\frac{R_{\eta _2}}{R_{eta_1}}$, increases the GB becomes more curved. Furthermore, our simulations reveal that the enhanced GB neck-curvature in case-II may have been facilitated by the formation of, diffusion-induced, protrusion on the prior flat surface (compare Fig. 3b1 and Fig. 1b). Regardless, it appears that curvature in the neck serves to increase $l^*_{neck}$ in case-II. After the disappearance of smaller particle, i.e., at the end of grain growth stage, the second grain/particle in each case (represented as $\eta _2$ in Fig. 1a, b) develops a dome-shaped protrusion on their surfaces (Fig. 3a3 and Fig. 3b3), consistent with sintered microstructures seen in many materials [4,5,6,7,8,9,10,11,12,13].

We have also systematically examined the effect of surface and GB energies on neck formation by varying the gradient coefficients $\kappa _{\eta }$ and $\kappa _{\rho }$ (see their corresponding $\gamma _{S},\gamma _{GB}, \frac{\gamma _{S}}{\gamma _{GB}}$ in Table 1), and the maximum total neck length $l^{*,max.}_{neck}$ was numerically evaluated for each ($\gamma _{S},\gamma _{GB}$) combinations (see bar chart in Fig. 3c ) (Here, $l^{*,max.}_{neck}$ corresponds to the neck length during stage-II of sintering). From Fig. 3c, broadly, we find that the $l^{*,max.}_{neck}$ increases with decreasing $\frac{\gamma _{S}}{\gamma _{GB}}$ ratio–consistent with other two-particle sintering studies [33, 34, 42]. Note that increasing $\gamma _{GB}$ (with fixed $\gamma _{S}$) tends to increase $l^{*,max.}_{neck}$ for both cases, while increasing $\gamma _{S}$ (while keeping $\gamma _{GB}$ fixed) reduces neck length. Importantly, Fig. 3c demonstrates that $l^{*,max.}_{neck}$ in case-II is consistently longer than case-I for all the ($\gamma _{S},\gamma _{GB}, \frac{\gamma _{S}}{\gamma _{GB}}$) combinations, and that $\gamma _{GB}$ has more influence the neck length when the smaller $\eta _1$ grain/particle is in contact with a flat surface. Therefore, our simulations show that, in a two grain/particle system, presence of a flat surface, with $R_{\eta _2}$ $\rightarrow \infty$, enhances neck formation during the sintering process.

Intermediate kinetics regime in three-particle environments: Cases III and IV

Geometrically, by virtue of the large particle curvatures (i.e., $\eta _2$ and $\eta _3$ in Fig. 1d1–d2), case-IV microstructure contains an internal triangle-shaped pore, while, in case-III, contact between the flat $\eta _2$ and $\eta _3$ grains produced a GB case-III(Fig. 1c) (In both cases, $\eta _1$ is in contact with $\eta _2$ and $\eta _3$). Such starkly different microstructures caused little difference between their $\eta _1$-kinetics (Fig. 2e and Table 3). However, the two cases differed substantially during the stage-II of neck formation: case-IVa-b manifested two-step neck formation mechanism (Fig. 2b), while neck evolution in case-III followed a “classically” smooth stage-II transition (Fig. 2a). Therefore, to better understand the neck evolution in both cases we have carefully monitored changes in their microstructure.

Figure 3

Time snapshots showing microstructural evolution of (a1)–(a3) case-I and (b1)–(b3) case-II at $1 \times 10^5$, $2\times 10^5$ and $3\times 10^5\Delta t^*$. Corresponding normalized times, i.e., $\frac{\Delta t^*}{\Delta t^*_{max}}$ , are also indicated. The microstructures in panels (a) and (b) correspond to $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63. (c) bar chart comparing the sensitivity of maximum, stage-II, neck length ($l^{*,max}_{neck}$) to surface and grain boundary interfacial energies. Case-II had noticeable neck curvature and longer $l^{*,max}_{neck}$ than case-I. This longer $l^{*,max}_{neck}$ in case-II remains largely unaffected by variation in $\frac{\gamma _{S}}{\gamma _{GB}}$

Figure 4

Snapshots showing microstructural evolution of case-III at (a1) $2 \times 10^5$, (a2) $3\times 10^5$ and (a3)$4\times 10^5 \Delta t^*$, and case-IVb at (b1) $0.8 \times 10^5$, (b2) $1.2\times 10^5$, (b3) $1.4\times 10^5$ and (b4) $4\times 10^5\Delta t^*$. Corresponding normalized times, i.e., $\frac{\Delta t^*}{\Delta t^*_{max}}$, are also indicated. (c) Plots comparing the development of necking between $\eta _2$–$\eta _3$ and $\eta _2$–$\eta _1$–$\eta _3$ particles with $\frac{\Delta t^*}{\Delta t^*_{max}}$. Inset in (c) shows the onset pf secondary necking event. Microstructures shown in (a) and (b) were simulated using $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63. The smaller particle in case-III enhances grain boundary grooving. The collapse of the internal pore in case-IV causes an abrupt increase in neck length

Figure 5

Plots comparing the effect of ( $\gamma _{S}$ and $\gamma _{GB}$ on (a)$l^{*,max}_{neck}$ in cases III (olive color) and VI (blue color) and (b) secondary neck formation in case-IV. The onset of secondary event is marked with arrows in panel (b). Case-IVb had consistently longer neck lengths than case-III, irrespective of $\frac{\gamma _{S}}{\gamma _{GB}}$ ratio. In case-IVb, $\gamma _{S}$ drives the two-step neck evolution mechanism, while the subsequent $l^{*,max}_{neck}$ was determined by $\gamma _{GB}$

Figure 6

Snapshots showing microstructural evolution of (a1)–(a3) case-V and (b1)–(b3) case-VI at 1000, $1\times 10^5$, and $5\times 10^5\Delta t^*$. (a3) and (b3) indicates loss of axial symmetry, while (b3) additionally depicts grain boundary rotation. Microstructures were obtained using $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63. The presence of GB in case-VI changes the evolution pathway of the internal pore by changing its shape compared to case-V

Evolution of case-III is presented in panels (a1), (a2) and (a3) of Fig. 4, which depict microstructures near the middle of stage-II ($\frac{\Delta t^*}{\Delta t^*_{max}}$=0.4 in Fig. 2a), partial consumption of the smaller $\eta _1$ grain due to the growth of larger $\eta _2$ and $\eta _3$ grains ($\frac{\Delta t^*}{\Delta t^*_{max}}$=0.6), and the complete disappearance of $\eta _1$ at $\frac{\Delta t^*}{\Delta t^*_{max}}$ = 0.8, respectively. Here, comparable to case-II (Fig. 3b), neck formation involved protrusions on the flat surfaces of $\eta _2$ and $\eta _3$ (Fig. 4a1). Such protrusions grew in size, with $\eta _1$–$\eta _2$ and $\eta _1$–$\eta _3$ GBs partially consuming the smaller $\eta _1$ particle (Fig. 4a2). Fig. 4a3 shows the microstructure after stage-III (see Sect. 3.1), where $\eta _1$ has been fully absorbed by $\eta _2$ and $\eta _3$. The microstructure at $\frac{\Delta t^*}{\Delta t^*_{max}}$ = 0.8 also contained a GB groove, a feature that has been experimentally observed in many materials [65,66,67,68,69,70]. Although phase-field simulations by other groups have shown such GB grooving [52, 71,72,73,74], the current work shows that the presence of small particles at grain boundary interfaces will substantially enhance GB grooving. Focusing on neck formation, due to the presence of smaller particle at the $\eta _2-\eta _3$-GB, the individual $\eta _1-\eta _2$ and $\eta _1-\eta _3$ neck lengths were half of the total neck length $l^*_{neck}$ in Fig. 2a. (Note that the such individual neck lengths in case-III are smaller than in case-II). Sensitivity of neck formation at $\eta _2-\eta _3$-GB was further examined by performing simulations with differing $\gamma _{GB}$ and $\gamma _{S}$ (see Table 1), and comparing the corresponding $l^{*,max.}_{neck}$ for case-III in Fig. 5a. Comparable to other cases (see Sect. 3.2), increase in $\gamma _{GB}$ enhances neck formation, while $\gamma _{S}$ does not. Combined, these results demonstrate that effect of GB on sintering of case-III-like configurations.

In contrast to case-III, the presence of an internal pore in case-IV, due to larger $\eta _2$ and $\eta _3$ curvatures (see Fig. 1 and Table 2), resulted in a very different neck and microstructure evolution (Fig. 4b1–b3). To illustrate this difference, for simplicity, we will focus on case-IVb only, while noting that both IVa and IVb configurations exhibited the two-step neck formation mechanism (Fig. 2b). In Fig. 4b1–b3, we show the microstructures resulting in the secondary (or 2$^{nd}$) necking event, and correlated them with the evolution of $\eta _2$–$\eta _1$–$\eta _3$ and $\eta _2$–$\eta _3$ neck formation using $l^*_{neck}$ v.s. $\frac{\Delta t^*}{\Delta t^*_{max}}$ plots (Fig. 4c). Panel  4b1 shows the microstructure prior to the 2$^{nd}$ neck formation event at $\frac{\Delta t^*}{\Delta t^*_{max}}$=0.15, where a triangular-shaped internal pore is trapped between the three particles (marked with an arrow). The onset of secondary event is shown in panel  4b2, at $\frac{\Delta t^*}{\Delta t^*_{max}}$=0.25 (also see the plot in Fig. 4c), where the internal pore becomes smaller. In a short span of time, at $\frac{\Delta t^*}{\Delta t^*_{max}}$=0.3, the internal pore nearly disappears or collapses (panel 4b3), which corresponded to the “rise” in secondary necking event in Fig. 4c (marked with a dotted line). Fig. 4c also shows that the secondary necking event also occurs for both $\eta _2$–$\eta _3$ between 0.25$\le \frac{\Delta t^*}{\Delta t^*_{max}}\le$0.3; in addition to $\eta _2$–$\eta _1$–$\eta _3$. However, beyond $\frac{\Delta t^*}{\Delta t^*_{max}}>$ 0.3, $\eta _2$–$\eta _3$ neck continues to increase, while $\eta _2$–$\eta _1$–$\eta _3$ neck length reduced monotonically after reaching its maximum value. Microstructure presented in panel  4b4 shows a snapshot within stage-III at $\frac{\Delta t^*}{\Delta t^*_{max}}$=0.6, where the $\eta _1$ particle has substantially reduced in size. We also find from Fig. 4c that $\eta _2$–$\eta _3$ neck also manifest a tertiary event at $\frac{\Delta t^*}{\Delta t^*_{max}} \sim$0.78, when the $\eta _1$ particle disappears completely. Therefore, we learn that three-particle (circular/spherical) system, with unequal radii, will experience “abrupt” changes in the neck dimensions, due to the disappearance of smaller microstructural features during sintering. Our simulations show that such features are either small internally trapped porosity or particles themselves, and their disappearance results in a two-step formation during neck evolution (Fig. 2b and Fig. 4c).

To better understand the neck formation mechanism in case-IV, simulations were carried out by varying $\kappa _{\rho }$ and $\kappa _{\eta }$ (see Table 1), which allowed us to systematically vary $\gamma _{S}$ and $\gamma _{GB}$, i.e., fix $\gamma _{S}$ and increase/decrease $\gamma _{GB}$ and vice versa, similar to other cases (Sect. 3.2). The simulation results are presented in Fig. 5a and Fig. 5b, which compares the effect of $\gamma _{S}$ and $\gamma _{GB}$ on maximum neck length $l^{*,max}_{neck}$ and secondary necking event or two-step neck formation mechanism, respectively. Fig. 5a demonstrates that $l^{*,max}_{neck}$ increases upon increasing $\gamma _{GB}$ (from 1.63 to 2.45 with fixed $\gamma _{S}$=2.99), but decreases marginally upon reducing $\gamma _{GB}$ (from 2.99 to 4.49 with fixed $\gamma _{GB}$=1.63). In other words, the maximum neck length in case-IV is highly dependent on $\gamma _{GB}$ than $\gamma _{S}$—similar to cases I, II and III. We also find that, irrespective of $\gamma _{S}$ and $\gamma _{GB}$ values, the $l^{*,max}_{neck}$ in case-IV is consistently greater than case-III (containing flat surfaces, Fig. 4a), which suggests that, in a three-particle system, neck formation is enhanced by curved rather than flat surfaces with a GB. Interestingly, in contrast to $l^{*,max}_{neck}$, Fig. 5b shows that the two-step neck formation mechanism is strongly influenced by $\gamma _{S}$ instead of $\gamma _{GB}$. Here, the secondary necking event is retained upon increasing $\gamma _{S}$ (from 2.99 to 4.49 with fixed $\gamma _{GB}$=1.63), while such an event is barely visible upon increasing $\gamma _{GB}$. Thus, our simulations show that the competition between surface and GB energies are responsible for the two-step neck formation mechanism in a three-particle system. $\gamma _{S}$ dominates till the disappearance of internal porosity (Fig. 4b1–b3), and, subsequently, $\gamma _{GB}$ drives $\eta _2$–$\eta _1$–$\eta _3$ neck to acquire peak length $l^{*,max}_{neck}$ and, then, facilitates the disappearance of smaller $\eta _1$ particle via $\eta _2$–$\eta _1$ and $\eta _3$–$\eta _1$ GB movement (Fig. 4b3 and 4b4).

Slow kinetics regime in three- and four-particle environments: Cases V and VI

Cases V and VI exhibited largely similar neck formation (Fig. 2a) and $\mathbf{t }^{disappear}_{norm}$(Fig. 2e) tendencies, in contrast to the cases presented Sect. 3.2 and  3.3. It may be emphasized that case-V is a three-particle system like case-IV, except that the larger $\eta _3$ grain is replaced with a flat surfaced grain, which results in a slightly elongated internal pore case-V than case-IV (see Fig. 1). Finally, in case-VI, we have examined the effect of grain boundary interfaces within the sintered microstructure of case-V by placing a GB in contact with the internal pore (Fig. 1f).

Figure 7

Snapshots showing magnified view of the microstructure near the small grain region of case-V at (a) $3.3 \times 10^5$, (b) $3.4\times 10^5$, (c) $3.6\times 10^5$ and (d) $3.7\times 10^5\Delta t^*$. Microstructures were obtained using $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63. An interfacial neck-contact is noted in (c), which separates the smaller $\eta _1$ grain from the lenticular shaped pore. (d) also shows that the internal pore disappears before $\eta _1$

Figure 8

Snapshots showing magnified view of the microstructure near the small grain region of case-VI at (a) $3.5 \times 10^5$, (b) $4\times 10^5$, (c) $4.3\times 10^5$ and (d) $4.4\times 10^5\Delta t^*$. Microstructures were obtained using $\kappa _{\rho }$=5 and $\kappa _{\eta }$=1 or, $\gamma _{S}$=2.77 and $\gamma _{GB}$=1.15. An interfacial neck-contact is noted in (b), which separates the smaller $\eta _1$ grain from the lenticular shaped pore. (c) and (d) also show that $\eta _1$ disappears before the internal pore

Figure 9

Plots comparing neck evolution in (a) case-V and (b) case-VI for different ($\kappa _{\rho },\kappa _{\eta }$) or ($\gamma _{S}$, $\gamma _{GB}$) combinations. Both configurations experience similar influences from varying $\gamma _{S}$ and $\gamma _{GB}$ . Only minor variations in the maximum neck lengths are barely discernible. Also not the absence of secondary necking events in panel (a), even though case-V have the same number of grains as case-IV

Figure 10

Plots comparing the effect of $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$ on the secondary neck formation in case-IVb. (a) shows neck evolution for $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$=1,2,4,6 and 8. Inset marks the radii of each grain in the three-particle system of case-IVb, while the double arrow in panel (a) exemplifies the secondary peak height. (b) plots the onset time for secondary necking and height the secondary peak as a function of $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$. Crucially, these plot demonstrate that secondary neck formation is most pronounced within a specific range, i.e., 2$\le \frac{R_{\eta _2,\eta _3}}{R_{\eta _1}} \le$6 (highlighted within a box). Simulations parameters used for generating the plots were $\kappa _{\rho }$=5 and $\kappa _{\eta }$=2 or, $\gamma _{S}$=2.99 and $\gamma _{GB}$=1.63

Figure 11

Bar chart comparing the effects of no. of grains, their curvature and $\frac{\gamma _{S}}{\gamma _{GB}}$ ratio on the $\eta _1$ grain $t^{disappear}_{norm}$. Broadly, $t^{disappear}_{norm}$ increases with the no. of grains when $\gamma _{S}$ dominant; that trend is marked with arrows for $\frac{\gamma _{S}}{\gamma _{GB}}$=1.83 and 2.75. The plot also reveals noticeable differences in $t^{disappear}_{norm}$ in the configurations comprising three grains, i.e., case-III, IVb and V, which points the effect of differing particle/grain curvatures

Microstructural evolution of cases V and VI is presented in Fig. 6a and  6b, respectively, which show snapshots at 1000, 100000 and 500000$\Delta t^*$. These time intervals depict microstructures at the onset of neck formation in stage-I (6a1 and 6b1), near peak neck length at stage-II (6a2 and 6b2), and the disappearance of $\eta _1$ and internal pore (6a3 and 6b3). Nominally, Fig. 6 shows that, irrespective of the GB, both cases manifest two common features: (i) similar $\eta _1$ shape evolution (unlike the other cases—see Figs. 3–5) and (ii) loss of axial symmetry in the large circular grain due to the absorption of $\eta _1$ by the surrounding grains. The latter feature is depicted by a 24$^{\circ }$ angular separation between dotted (initial axis of symmetry) and solid (axis of symmetry after $\eta _1$ disappearance) lines in Fig. 6a3 and 6b3. In case-VI, we also noted a discernible rotation/bending of the GB by 7$^{\circ }$ with respect to its initial location (see the dotted line in Fig. 6b2 and Fig. 6b3 and inset in Fig. 6b3), which was likely caused by the disappearance of $\eta _1$ (The presence of a GB in case-VI also results in a triple junction in Fig. 6b3). The GBs also affected the shape of internal pores (insets in Fig. 6a2 and Fig. 6b2). In case-V, the triangle-shaped pore lies at a triple junction, while the additional GB in case-VI forces the pore to acquire a quadrilateral shape (lodged at the intersection of four GBs). Note that, in both cases, the pore is in contact with the smaller $\eta _1$ grain (marked with arrows). Next, we examine the evolution of internal pore in detail, using Fig. 7 (case-V) and Fig. 8(case-VI), which show magnified views of the region near $\eta _1$ in the same scale.

Figs. 7a, b (330000 and 340000 $\Delta t^*$ for case-V) and  8a (330000 $\Delta t^*$ for case-VI) qualitatively show that continued sintering have reduced the contact area between the internal pore and the smaller $\eta _1$ in both cases (compared to the microstructures at 100000$\Delta t^*$ (see insets in Fig. 6a2 and  6b2)). Subsequently, V and VI develop new interfacial “neck-contacts” between the larger grains $\eta _2$ and $\eta _3$ that were in prior contact with $\eta _1$. This neck-contact is indicated using yellow colored arrows in Figs. 7c and 8b. It is worth noting that such interfacial neck-contacts separate the internal pore from $\eta _1$ in both case-V and VI, which contrasts with case-IVb (compared Fig. 4b1–b3 with Figs. 7c and 8b). However, the main difference between the two $\eta _2$–$\eta _3$ interfacial neck-contacts is that, in case-V, it lasts for very short period of time ($\approx 10000\Delta t^*$), while such a contact persists in case-VI for a longer duration, i.e., $\approx 10000\Delta t^*$, an order of magnitude greater than V. It may be pointed out that the $\eta _2$–$\eta _3$ interfacial neck-contact is essentially a grain boundary between the two grains. Subsequently, the initial $\eta _2$–$\eta _3$ interfacial neck-contact in case-VI evolves by elongating into an extended $\eta _2$/$\eta _3$ grain boundary (Fig. 8b-c). In the following, we will discuss how the evolution of neck-contacts changes their local environments.

The development of $\eta _2$–$\eta _3$ interfacial GB neck-contacts also alters the pore shape in both cases, in comparison their shapes at 100000$\Delta t^*$(see insets in Fig. 6a2 and 6b2). In case-V, the pore acquires an ellipsoidal/lenticular shape, while being trapped between $\eta _2$ / $\eta _3$ bicrystal interface (Fig. 7c). On the other hand, the pore in case-VI had a triangle shape, and subsequent sintering time “locates” it at the $\eta _2$–$\eta _3$–$\eta _4$ triple junction (Fig. 7b). The differing shapes also determine the constrains imposed on them by the surrounding GBs, i.e., bicrystal v.s. triple junction interfaces [75,76,77], which, in turn influences their stability with sintering time. Evans et al. [75, 76], and Riedel and Svoboda [77] have shown the three GBs, at triple junction, exert a greater drag force than bicrystal interface. Consequently, smaller drag forces on the ellipsoidal/lenticular allow it collapse sooner than the triangular-shaped pore (compare Figs. 7b and 8d), and extend the $\eta _2$–$\eta _3$ interfacial neck-contact duration in case-VI. The differing GB drag forces also influences the relative stabilities of the pore and smaller $\eta _1$ grain. Constrains imposed by the triple junction on the pore allows it last longer than $\eta _1$ in case-VI (Fig. 7), while drag forces on the bicrystal pore were insufficient to prevent it from collapsing before $\eta _1$ disappearance in case-V (Fig. 8). Therefore, the examination of case-V and VI demonstrate that the presence (or absence) of an additional GB interface in microstructures with comparable geometrical configurations produces very different grain and pore evolution pathways, e.g., compare Figs. 7d and 8d.

Pore evolution in cases V and VI can also be examined from the perspective of coordination number ($N_{coord}$), i.e., number of particles in contact with the pore. From Fig. 1, we note that the initial $N_{coord}$ in cases V and VI was three and four, respectively. Kingery [21] introduced coordination number to study pore shrinkage during sintering. Later, Wakai et al. [78, 79] applied this notion to examine the pore shrinkage in a multi-particle environment using 3D Surface Evolver program [80], and demonstrated that pore shrinkage proceeds with progressive reduction in $N_{coord}$. Our 2D phase-field simulations show that, during sintering, $N_{coord}$ in case-V reduces from three to two (before collapsing), and four to three in case-VI (Figs. 1, 6,  7 and  8). Therefore, our simulations have captured $N_{coord}$ reduction with pore shrinkage, along with the movement of corresponding interfaces.

The case-V simulations further allowed us to compare and contrast with those obtained from case-IVb, and gather insight into their differing neck evolution tendencies (Fig. 2b): case-IVb exhibited two-step neck formation, while case-V showed a smooth stage-II transition (Both cases comprised three grains, but had different $\eta _3$ curvatures—see Fig. 1). Recall that, in case-IVb, the collapse of internal pore coincided with a secondary neck formation event (Fig. 4b-c), while that triangular pore remained in contact with $\eta _1$; till its final collapse prior to $\eta _1$ disappearance (Fig. 4b1–b3). In contrast, case-V develops an interfacial neck-contact that discernibly separates the lenticular internal pore from $\eta _1$ (Fig. 7c). Such pore-$\eta _1$ spatial interaction, i.e., contact or separation, may have influenced $\eta _1$ neck evolution. It appears that, because of the direct pore-$\eta _1$ contact in case-IV, the space occupied by the pore is completely consumed toward forming the $\eta _2$–$\eta _1$–$\eta _3$ interfaces, which contributes to the secondary necking event. On the other hand, in case-V, the space occupied by the lenticular pore is consumed to form $\eta _2$–$\eta _3$ neck, since the pore was trapped between the two grains, which have minimal impact on $\eta _2$–$\eta _1$–$\eta _3$ neck formation (A similar mechanism may be argued for case-VI). Consequently, $\eta _1$ grain in case-V (and case-VI) experienced a smoother stage-II neck evolution. Studies are in progress to gain a better mechanistic understanding of the two-step neck formation mechanism.

Furthermore, our simulations have shown that the secondary necking event is highly susceptible to $\gamma _{S}$ (Fig. 7b). Since both cases IV and V have the same number of grains, we have conducted simulations to probe secondary neck formation in case-V, by varying $\gamma _{S}$ and $\gamma _{GB}$ values (Table 1) and using the approach described in Sects. 3.2 and  3.3. The results from those simulations are plotted in Fig. 9a, which did not reveal discernible secondary necking events in case-V. However, Fig. 9a shows that the maximum neck length depends on $\gamma _{GB}$, and is comparable to cases I, II and III. For sake of completion, we performed similar calculations for case-VI (Fig. 9b), and the results exhibited similar trends as seen in case-V (Comparison of Fig. 9a and 9b revealed only minor variations in the maximum neck lengths).

Discussion: implications and limitations

The systematic examination of multiple sintering geometries has revealed a unique pore-mediated, two-step neck evolution mechanism in three-particle system of unequal radii, which were organized in a triangular arrangement (Fig. 1d1–d2, Fig. 2b and Fig. 4b-c). Such a mechanism may be present in powder compacts with a wide particle size distribution [12, 13, 38, 81,82,83], where local arrangements may produce triangular required geometry (Fig. 1d). Furthermore, since this neck formation mechanism is determined by surface energies (see Sect. 3.3 and Fig. 5b), granular ensembles comprising particles with high surface to volume ratios may also manifest the two-step neck evolution mechanism, e.g., during the sintering of nano-sized particles [14,15,16,17,18,19]. However, the two-step neck evolution will not be exhibited for all particle sizes. To elucidate this matter, using case-IVb, we have examined neck evolution of geometries with different particle radius ratios (i.e., $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$=1,2,4,6,8) , and extracted the onset time for the secondary necking event and height of the 2$^{nd}$ peak (exemplified using a double arrow in Fig. 10a) for each $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$. Fig. 10b plots the extracted quantities as a function of $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$. We find that the two-step neck evolution is most prominent within 2$\le \frac{R_{\eta _2,\eta _3}}{R_{\eta _1}} \le$6, i.e., a regime containing clearly identifiable secondary neck onset and height (highlighted with a box in Fig. 10b). In contrast, this mechanism is completely absent when the particles are of equal size ($\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$=1) and when $\eta _2$ and $\eta _3$ are substantially larger than $\eta _1$, i.e., $\frac{R_{\eta _2,\eta _3}}{R_{\eta _1}}$=8 (see inset of Fig. 10a to see the geometry). In this context, notionally, the neck evolution in case-V (where $\eta _3$ is a flat surface—Fig. 1e ) will lie on the extreme right-hand side of Fig. 10b, since that mechanism was not seen in this configuration (Fig. 9a).

We also note that our simulations involving two-particle systems (cases I and II) did not exhibit the secondary necking events. However, Biswas et al. have demonstrated that two-particle system may show such events, when rigid body rotation and GB anisotropy are incorporated into Cahn–HIlliard and Allen–Cahn equations of motion (see Fig.10a in [42]). Importantly, compared to such results [42], the absence and/or presence of secondary necking events in case-I and II, and IV, respectively, point to a more fundamental, underlying influence of curvature and thermodynamic forces ($\gamma _{S}$ and $\gamma _{GB}$). These influences will couple with the microstructural elements and micromechanics of sintering (as indicated in [42]), and result in a rather complex process. Such complexity behavior will arise for powders with wide size distribution and shape variations, i.e., spherical and flat surfaces, [37, 38], where the inter-particle contacts can be locally represented using a combination of case-I, II, III, IV V and V (Fig. 1).

Few comments are required regarding the effect of no. of particles/grains, and their local curvatures on the disappearance time of the smaller $\eta _1$ grain, i.e., $t^{disappear}_{norm}$. Fig. 11 examines such effects for different $\gamma _{S}$ and $\gamma _{GB}$ or, $\frac{\gamma _{S}}{\gamma _{GB}}$ ratios ($\gamma _{S}$ and $\gamma _{GB}$ values are listed in Table 1). Broadly, we find that $t^{disappear}_{norm}$ increases with no. of grains for $\frac{\gamma _{S}}{\gamma _{GB}}$=1.83 and 2.75 (marked with arrows in Fig. 11), while that effect is rather subdued for $\frac{\gamma _{S}}{\gamma _{GB}}$=1.22. In the latter, $\eta _1$ for case-III takes a longer time to disappear than the other cases, presumably because of the presence of an extended prior $\eta _2$–$\eta _3$ grain boundary (Fig 1c and Fig. 4a). Regardless, the effect of local curvature can be seen in all cases. Particularly in the configurations of three grains, i.e., cases III, IVb and V, where they exhibit very different $t^{disappear}_{norm}$ for a given $\frac{\gamma _{S}}{\gamma _{GB}}$, especially when $\gamma _{S}$ is “significantly” greater than $\gamma _{GB}$. Furthermore, it should be noted that we have examined sintering of multi-particle systems using a phenomenological approach using $\gamma _{S}$ > $\gamma _{GB}$ (Table 1); a relationship exhibited in many materials [54, 84, 85]. Therefore, the demarcation of sintering kinetics into different regimes will hold for many materials systems (Sect. 3). For example, in case of powder compacts with varying particle shape and size distributions, our results imply that sintering kinetics (leading to the final densification) will vary between locations depending on the local particle curvatures.

Finally, it may be pointed out that our phase-field simulations, while having clearly identified several particle curvature effects on neck formation, are currently constrained in two ways. First, these simulations were carried out using fixed values of diffusivities (see eq. 4 and Table 1). Since diffusivities are a function of temperature [53, 54], the current phase-field and geometrical models only simulate isothermal sintering. Second, the role of rigid body motion has been neglected [32, 33, 42]. Studies are underway to examine two-step mechanism in metallic alloys by taking into account both temperature variation and rigid body rotation.

Summary

In this work, we have phenomenologically investigated the effect of particle curvatures on the neck formation, and corresponding microstructural evolution, during a sintering process. Phase-field simulations were carried out using closed-packed geometrical arrangements/configurations of two, three and four particles/grains, where their local curvatures were systematically varied from zero (i.e., a flat surface) to eight times the radius of the smallest grain. Curvature effects within such geometries were characterized by the tracking neck evolution, surface area and local microstructural evolution around a “probe” grain. The influence of grain boundaries was also examined. Key findings from our simulations are as follows:

• 1. Some geometries, involving two, three and four particles, manifested the “classical” three-stage neck evolution mechanism, where the neck length smoothly varied from neck initiation to the final grain growth stage. The presence of flat surfaces and grain boundaries did not alter this behavior. In these geometries, local curvature affected only the maximum neck lengths. Grain boundary energy influenced the maximum neck lengths more than surface energy.

• 2. In contrast, neck evolution of ensembles with three circular particles of unequal radii in a closed-packed triangular arrangement, and with an internal pore trapped between the three particles, significantly departed from the classical behavior. Such a geometry consistently manifested a secondary necking event in form of a step during the neck evolution. Surface energy played a dominant role in triggering the secondary necking event, while grain boundary energy determines the remnant microstructure after the neck acquires its maximum length. Microstructurally, such a two-step neck formation mechanism is heavily influenced by the evolution of the internal pore, whose collapse (before the disappearance of the smallest grain) coincided with the secondary necking event. The secondary necking event in such three-particle ensembles was not observed for all values of particle radii. Only a range of particle radii displayed the two-step neck formation mechanism. Secondary necking event was not observed in configurations with particles of same radius, particle sizes exceed a particular ratio, or, when, one of the particle/grain had a flat surface.

• 3. The effect of grain boundaries on the evolution of internal pores was also evaluated using three- and four-particle configurations, while one of them was a flat surface. Our simulations showed that such boundaries can alter the pore shape and delay their collapse before the disappearance of the smallest grain. Grain boundary rotation associated with pore collapse, and the subsequent absorption of the smallest grain by the larger particles, was also captured.

• 4. Broadly, phase-field modeling of geometries employed here establishes the preliminary building blocks of a computational platform to investigate the sintering of powder compact, where particles display wide variation in shapes and size distributions. Studies are underway to examine secondary necking events in metallic alloys.