The simulation for liquid phase sintering has been newly developed by Monte Carlo method. The basic three mechanisms in liquid phase sintering, namely (1) liquid wetting on solid surface, (2) rearrangement of solid particle by capillary force of liquid, (3) growth of solid particles by solution-reprecipitation through liquid phase (Ostwald ripening), have been introduced in the simulation. The liquid wetting on a solid plane or solid particle surface progressed more easily as decreasing the energy between solid and liquid (γ_SL). The introduction of rearrangement into the simulation brought the result that relative density increased with decreasing γ_SL and pores remained. The introduction of Ostwald ripening in addition to liquid wetting and rearrangement produced the result that relative density increased on the whole and the effect of γ_SL became clear. The simulation including Ostwald ripening was able to demonstrate clearly and continuously the influence of γ_SL on growth and contiguity of solid particles.

This Paper was Originally Published in Japanese in J. Jpn. Soc. Powder Powder Metallurgy 66 (2019) 259–265.

Fig. 11 Simulation by introducing Ostwald ripening in addition to liquid wetting and rearrangement as a function of γ_SL at 1000 MCS. (a) γ_SL = 0.3, (b) 0.8, (c) 1.0, (d) 1.3.

The sintering phenomenon in which the liquid phase partially exists in the solid phase particles (powder) is usually called liquid phase sintering. Liquid phase sintering is applied to many sintered materials such as electronic and tool materials because the densification of the powder compact is promoted compared to solid phase sintering. Many basic researches on liquid phase sintering have been done up to now. The mechanism and technology of liquid phase sintering have been systematized by prominent researchers of sintering such as Kingery,¹⁾ Petzow,^2,3) German.⁴⁾ Solid phase sintering can be understood as the sintering of particles and the movement of grain boundaries (grain growth) by mass transfer (mainly diffusion) in the solid state.^5,6) In contrast, the basic mechanisms of liquid phase sintering consist of ① wetting of the solid phase surface of the liquid phase, ② rearrangement of the solid phase particles by capillary force of the liquid phase, and ③ dissolution and reprecipitation of the solid phase particles into the liquid phase (Ostwald ripening).^1–6) Solid phase sintering and liquid phase sintering are common in powder sintering and driving force (surface energy), but the mass transfer mechanism may be considered fundamentally different.

Recently, computer simulations have attracted significant attention as methods for designing method novel materials, and their applications to sintered materials are also in progress.^5–8) Among them, the molecular dynamics (MD) method can handle the grain boundary, surface energy and diffusion,^8–11) but there are significant restrictions on the size (such as number of atoms) and time (sintering time) that can be handled. The finite element method (FEM) has no such limitation, but it is difficult to handle the microstructure of sintered materials (such as the µm order). The Monte Carlo (MC) method is positioned as an intermediate calculation method between the MD method and FEM.^5–8) Anderson et al. reported the simulation of grain growth by the MC method using the Potts model.¹²⁾ In the MC method, the assembly of atoms and molecules is moved as a unit cell based on the stochastic process of energy reduction. The use of MC method makes it possible to handle large mass transfer and extend time, which is difficult in the MD method, and to handle the material structure of µm order which is difficult in the FEM.^5–8) We have previously simulated the sintering and grain growth phenomenon using MC method and reported the results of the microstructure formation process, solid phase sintering, grain growth and Ostwald ripening.^5–19)

Tikare et al. also reported the results of the simulation research on solid phase sintering and Ostwald ripening using the MC method.^20–22) Although MC simulation of liquid phase sintering has been reported by Lee et al.,²³⁾ but sufficient study has not been conducted on the most basic mechanism of liquid phase sintering (especially ① and ②). Thus, it is believed that no study exists on liquid phase sintering in earnest in MC simulation research so far.

Therefore, in this study, a new MC simulation technology for liquid phase sintering is developed, and the basic mechanism of liquid phase sintering (① to ③) is analyzed in detail. The important parameters (factors) related to liquid phase sintering were analyzed, such as the effects of interfacial energy between the solid and liquid phases, solid phase particle rearrangement, Ostwald ripening and solid phase particle size. Furthermore, we examined and discussed the suitability and effectiveness of the MC simulation developed herein as a design tool for liquid phase sintering process.

A cell has six nearest cells, assigned solid, liquid or gas phase (pore) cells. The solid phase contains the number of crystal orientation (Q), which is 64 in this study. The excess energy (γ) is set as the interface of the different phases. We give γ_GB at different Q solid phase interfaces, γ_SL between the solid and liquid phases, γ_SV between the solid and pore phases, and γ_LV between the liquid and pore phases.

Figure 1 shows three types of initial structures. The black and gray parts represent the liquid and solid phases respectively. The white parts around them represent pores. Figure 1(a) shows the structure that evaluates the wetting of the liquid phase on the solid surface. The diameter of the liquid phase is 24 cells. Figure 1(b) shows the structure for evaluating the liquid phase wetting on the surface of solid phase particles and densification by particle rearrangement. The size of the whole calculation cell is 200 × 200. As in the previous report,¹⁸⁾ among these cells, the five outer cells are the outer space (the same as the pores), and the solid phase, liquid phase and pore cells are located at the inner 190 × 190 cells. The average particle sizes of the solid and liquid phases are 12 and 5 cells respectively, the liquid phase ratio is 20%, and the porosity is 30%. Figure 1(c) shows the structure for evaluating Ostwald ripening, and the size of the whole calculation cell is 200 × 200. The handling of the five outer cells is the same in Fig. 1(b). The average particle sizes of the solid and liquid phases are 3 cells and a cell respectively, the liquid phase ratio is 20%, and the porosity is 30%.

Fig. 1

Initial structures for simulations. (a) wetting of a liquid particle (phase) on a solid plane, (b) solid particles (gray) with the diameter of 12 cells and liquid particles (black). (c) solid particles with the diameter of 3 cells and liquid particles.

In MC simulation including the solid and liquid phases particles and pore, the liquid phase wetting to the solid surface is reproduced by an algorithm that exchanges the liquid phase and pore cells. If a randomly selected cell is the liquid phase, and a liquid phase surface exists in a randomly selected direction starting from that cell, then a trial of liquid phase diffusion is entered. The liquid phase moves on the surface of the liquid or solid phase and decides whether to stop or continue moving with a probability of 1/2. When it stops, it exchanges itself with the pore cell. Then, the success or failure of the trial is judged from the energy change of the whole system. If the selected cell is a pore and any of the six surrounding cells is the liquid phase, the pore will randomly walk in the liquid phase, and if it reaches another pore, it will contact itself with the liquid cell adjacent to that pore. The success or failure of the trial is determined from the energy change of the whole system.

If the combination of a randomly selected cell and its neighboring cells is a pore and liquid phase, a straight line that reaches the outermost surface of the entire calculation cell in the direction connecting the liquid phase from the pore is assumed. All particles passing the straight line move by one cell in the pore direction. If a particle cannot be moved by a solid phase particle, it does not move only for that particle. When solid phase particles move, the liquid phase cell moves to the opposite side of the solid phase particles. Furthermore, the success or failure of the trial is judged from the energy change caused by the exchange of the selected pores and the outermost liquid phase.

The algorithm of this process (mechanism) is almost like that of previous reports.^7,8,13–16) The main points are described as follows. When a combination of randomly selected cells and their adjacent cells is in the solid and liquid phases, the solid phase is exchanged with the liquid phase. The solid phase randomly walks in the liquid phase, and when it reaches another solid phase, it exchanges itself with the liquid phase cell adjacent to that solid phase. Simultaneously, Q of the solid-phase cell randomly walking is set to the same Q of the solid-phase cell that was encountered. The success or failure of the trial is judged from the energy change due to the exchange. The frequency factors (F) of the liquid phase wetting, rearrangement of solid phase particles and solid phase particle growth trial in the presence of a liquid phase represent F_phd, F_lhd, F_ra, and F_ost, respectively. All these values lie within the range of 0∼1. The value of F_ra is 0 or 0.05, those of F_phd and F_lhd are 0.5 and that of F_ost is 0 or 0.5. The number of calculation steps is represented by Monte Carlo Step (MCS). One MCS means that the number of trials of mass transfer is the total number of calculation cells regardless of success or failure. MCS can be thought of as a measure corresponding to time. Table 1 summarizes the calculation cell conditions, excess energy, frequency factor, calculation MCS and other conditions.

Table 1 Simulation conditions concerning calculation size, the initial configuration, energy values, frequency factors.

Using the simulation results, the relative density was mainly calculated from the ratio of the solid and liquid phases to the pores in the internal 114 × 114 cell of the 200 × 200 calculation cell. The contiguity (C_SS) between the solid phases was calculated from the simulation results based on the following equation.   

Figure 2 shows the simulation results of the liquid phase wetting to the solid phase when γ_SL is 0.3, indicating that the liquid phase wets and spreads to the solid phase as MCS increases. Evidently, the liquid phase loses its shape, and one cell is first wetted on the solid surface, and then the liquid phase for one cell spreads on the solid surface. Then, another liquid phase moves (diffuses) on the expanded liquid phase cell. Figure 3 compares the difference in the spread of wetting due to the difference in γ_SL. When γ_SL is 0.3∼0.8, wetting spreads, the contact angle can be regarded as 0°. However, the rate of wetting and spreading changes as γ_SL increases. When γ_SL is 1.0∼1.3, the contact angle is 60°, and when γ_SL is 1.6 or more, the contact angle is 120°.

Fig. 2

Simulation by using Fig. 1(a) for wetting of a liquid on a solid plane as a function of MCS. The value of γ_SL is 0.3. (a) 50MCS, (b) 500MCS, (c) 1000MCS, (d) 5000MCS, (e) 10000MCS, (f) 15000MCS.

Fig. 3

Simulation for wetting of a liquid phase on a solid plane as a function of γ_SL at the constant step (20000 MCS). (a) γ_SL = 0.3, (b) 0.5, (c) 0.8, (d) 1.0, (e) 1.3, (f) 1.6. MCS.

Figure 4 shows the simulation results of the liquid phase with γ_SL = 0.5 spreading on solid particles (position is fixed) with an average particle size of 12 cells. First, when compared with the initial structure (Fig. 1), the shape of the liquid phase particles collapses and spreads by wetting on the surface of the solid phase particles. As the MCS increases, the places where the liquid phase is connected are observed, but other places where the liquid phase does not wet on solid particles are founded. Figure 5 shows the effect of γ_SL on how the liquid phase wets and spreads to the solid particles. Here, MCS is fixed at 20,000. As γ_SL decreases, the easier it is for the liquid phase to wet spread. In the case of γ_SL = 0.3, the entire solid particle surface is wet in the liquid phase. As γ_SL increases, the liquid phase is more integrated than covering the surface. Using the results of Fig. 4 and Fig. 5, Fig. 6 shows the number of interfaces between the solid phase and the pores (the solid surface) in relation to MCS. The smaller the interface, the more the solid surface is covered with the liquid phase. Evidently, the number of interfaces decreases greatly at the initial stage of MCS, and the decrement rate increases as γ_SL decreases, and hardly changes at MCS > 500.

Fig. 4

Simulation by using Fig. 1(b) for wetting of liquid phase on solid particles as a function of MCS. The value of γ_SL is 0.5. (a) 10MCS, (b) 50MCS, (c) 100MCS, (d) 1000MCS.

Fig. 5

Simulation for wetting of liquid phase on solid particles as a function of γ_SL at 1000 MCS. (a) γ_SL = 0.3, (b) 0.8, (c) 1.0, (d) 1.3.

Fig. 6

Number of solid/pore interfaces – MCS curves for the simulations of Figs. 4 and 5.

Figure 7 shows the calculation results for γ_SL = 0.5 by introducing particle rearrangement with liquid phase wetting. In this simulation, the solid phase particles do not change their shape, indicating no mass transfer on the solid phase. The introduction of particle rearrangement reduces the distance between solid particles as the particles get wet in the liquid phase. Consequently, the pores decrease; that is, densification progresses. In Fig. 7(a), small liquid phase particles can be seen in the pores, but in the initial stage of the simulation, the rearrangement of particles occurs actively, so that the liquid phase cell may leave the solid phase. Figure 8 shows the γ_SL effect on the calculation results by introducing liquid phase wetting and rearrangement. Here, the MCS is set as 1000. The size and number of pores decrease as the γ_SL decreases. Concerning the results of Fig. 7 and Fig. 8, Fig. 9 shows the relative density (area ratio) in relation to MCS. The relative density increases as the γ_SL decreases. However, even if the MCS is lengthened, the relative density does not reach 100% and some pores remain.

Fig. 7

Simulation by using Fig. 1(c) and by introducing rearrangement of the solid particle as well as liquid wetting as a function of MCS. The value of γ_SL is 0.5. (a) 10MCS, (b) 50MCS, (c) 100MCS, (d) 1000MCS.

Fig. 8

Simulation by introducing rearrangement of the solid particle as well as liquid wetting as a function of γ_SL at 1000 MCS. (a) γ_SL = 0.3, (b) 0.8, (c) 1.0, (d) 1.3.

Fig. 9

Relative density (area ratio) – MCS curves for the simulations of Figs. 7 and 8.

Figure 10 shows the calculation results by introducing the Ostwald ripening process with liquid phase wetting and particle rearrangement when γ_SL = 0.5, in relation to MCS. Notably, the particles grow with increasing MCS. When Ostwald ripening is introduced, the number of pores is reduced, and densification progresses more than when it is absent (Figs. 7 and 8). Figure 11 shows the effect of γ_SL on the calculation results by introducing Ostwald ripening. The MCS is fixed at 1000. Evidently, as the γ_SL decreases, the number of pores decrease and densification increases, and the particle size increases. It can be seen that when γ_SL is small or large, the liquid phase becomes thin or clumped, respectively. Concerning the results of Fig. 10 and Fig. 11, Fig. 12 shows the relationship between the relative density and MCS. As in Fig. 9, the relative density increases rapidly in the initial stage, and the relative density increases as γ_SL decreases. However, comparing Fig. 9 and Fig. 12, it is noted that by introducing Ostwald ripening, the relative density increases entirely, and also that the relative density increase is remarkable, as γ_SL decreases.

Fig. 10

Simulation by using Fig. 1(c) and by introducing Ostwald ripening in addition to liquid wetting and rearrangement as a function of MCS. The value of γ_SL is 0.5. (a) 10MCS, (b) 50MCS, (c) 100MCS, (d) 1000MCS.

Fig. 11

Simulation by introducing Ostwald ripening in addition to liquid wetting and rearrangement as a function of γ_SL at 1000 MCS. (a) γ_SL = 0.3, (b) 0.8, (c) 1.0, (d) 1.3.

Fig. 12

Relative density – MCS curves for the simulations of Figs. 10 and 11.

Concerning the results of Fig. 11 and Fig. 12, Fig. 13 shows the relationship between the average particle size of solid phase particles and MCS. The particle size increases with MCS. Also, the particle size increases as γ_SL decreases.

Fig. 13

Particle size (diameter) – MCS curves for the simulations of Figs. 10 and 11.

Figure 14 shows the C_SS of solid phase particles in relation to MCS using the results of Figs. 7, 8, 10, and 11. Figure 14(a) shows the calculation results when liquid phase wetting and rearrangement are introduced (when Ostwald ripening is not introduced), whereas Fig. 14(b) shows the results when liquid phase wetting, rearrangement, and Ostwald ripening are introduced. In Fig. 14(a) without Ostwald ripening, C_SS increases as MCS increases, and increases slightly as γ_SL increases. In Fig. 14(b) with Ostwald ripening, C_SS decreases as MCS increases which is contrary to Fig. 14(a). C_SS in Fig. 14(b) increases as γ_SL increases, and its tendency in Fig. 14(b) is considerably larger than that in Fig. 14(a). The initial C_SS is close to 10% in Fig. 14(a), whereas it is close to 55% in Fig. 14(b), depending on how the initial structure was created. Thus, it is not meaningful to compare Figs. 14(a) and (b) for early C_SS.

Fig. 14

Contiguity (C_SS) – MCS curves. (a) simulation by introducing rearrangement of the solid particle and liquid wetting in Figs. 7 and 8, (b) simulation by introducing Ostwald ripening in addition to liquid wetting and rearrangement in Figs. 10 and 11.

The results and effects of liquid phase wetting, rearrangement and Ostwald ripening introduced in simulating liquid phase sintering by the MC method are discussed in order. The wetting of the liquid phase on the solid surface is expressed by Young’s formula representing the contact angle, θ.⁴⁾   

For rearrangement, it is effective to consider the effect of attracting the solid phases to the liquid phase between solid phase particles, the so-called capillary force ΔP.⁴⁾   

The effect of introducing Ostwald ripening is deemed the most important outcome in this study. First, comparing Fig. 9 and Fig. 12 regarding the relative density, by introducing Ostwald ripening, the relative density increases and densification proceeds when γ_SL is small. The trend indicates that although rearrangement is the main process of liquid phase sintering, rearrangement alone does not provide sufficient densification. Also, it indicates that the densification needs the occurrence of Ostwald ripening which changes the position of the solid phase particles and furthermore needs the shape which is adapted (grain shape accommodation).⁴⁾ When Ostwald ripening was introduced (Fig. 13), γ_SL had the effect that grain growth was more likely to occur as γ_SL was smaller. To enter the trial of Ostwald ripening, the solid surface needs to be wet with the liquid phase. As apparent from the simulation of the liquid phase wetting in Figs. 2∼6, the smaller γ_SL is, the easier the liquid phase wets the solid phase, explaining the possibility of Ostwald ripening.

Due to the introduction of Ostwald ripening, the C_SS results in Fig. 14 are considered to have significant meaning. In the simulation with Ostwald ripening, when MCS increased to some extent (>100), C_SS showed a significant effect of γ_SL. As the γ_SL increases, it becomes more disadvantageous for the energy to form a solid-liquid interface. In other words, it is easier to form an interface between solid phases (a grain boundary) than to form a solid-liquid interface. Comparing Figs. 14(a) and (b) in detail, the effect of γ_SL on C_SS in (b) with the Ostwald ripening is stronger than that in (a) without the Ostwald ripening; C_SS becomes further larger or smaller when γ_SL value is larger or smaller. Also, the C_SS value affects the opportunity that the solid phase dissolves in the liquid phase, which is required for Ostwald ripening. Besides, consequently, it can be explained that the possibility of grain growth is affected by the C_SS value. Thus, the simulation in which three processes of liquid phase wetting, solid phase particle rearrangement and Ostwald ripening are introduced can be expected to be a powerful tool that can design the influence of various factors of liquid phase sintering materials.

Finally, a brief discussion is given on the relationship with the experiment (real materials). In the actual liquid phase sintering material, sintering progresses when the solid and liquid phases are well wetted. Yamaguchi et al. conducted experiments by changing the wettability and solubility of glass, which is a liquid phase, for silver sintering, and found that they contributed to the sintering of this material.²⁴⁾ For other liquid phase sintering materials, such as cermet (WC–Co and TiC–Ni), ceramics and glass mixed phase materials (Al₂O₃ and Si₃N₄), the liquid phase sintering is the most basic liquid phase sintering.^4,25) The simulation results also show that reducing γ_SL improves the wettability to the solid phase, facilitates rearrangement, and increases the relative density, which is consistent with the real materials. It is difficult to experimentally change only γ_SL independently, and it is difficult to analyze the influence of only one factor expected, so it is advantageous to analyze this point by simulation. Many physical properties (values) related to liquid phase sintering are conceivable, not limited to γ_SL, and how these relate to the simulation conditions (Table 1) is briefly described. The values of γ_SL, γ_SV and γ_LV can be obtained by experimental data such as wetting and dihedral angles, and MD calculation and can be directly introduced into this simulation. Besides, data, such as the diffusion coefficient and solubility in the liquid phase, can be reflected in the frequency factor (F_ost) of Ostwald ripening, and the viscosity coefficient of the liquid phase can be reflected in the frequency factors of wetting and rearrangement (F_lhd and F_ra). In any case, how much this simulation can reproduce the sintering and grain growth of various real materials obtained by liquid phase sintering is the most important research topic to determine whether it can be applied as a material design tool in the industry for research and development.

The Monte Carlo (MC) simulation of liquid phase sintering was developed by introducing the basic mechanisms of liquid phase wetting to solid phase surfaces, rearrangement of solid phase particles by the capillary force of the liquid phase, and dissolution and reprecipitation of solid phase particles into the liquid phase (Ostwald ripening). Then, the influence of important parameters on liquid phase sintering was analyzed, and the following results were obtained.

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