Simulation study on size distribution evolution of powder particles in liquid phase sintering
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摘要: 液相烧结过程中, 粉末颗粒的熟化长大和聚并同时发生。利用群体平衡模型定量预测了相邻颗粒聚并效应作用下的粒径分布演化规律, 提出了一种基于欧式范数的方法以确定液相烧结过程是否达到稳态, 研究了从瞬态到稳态的转变过程中的粒径上限及其变化率, 发现颗粒瞬态粗化之后将得到稳态粒径分布。模型计算得到的粒径分布和实验数据之间吻合良好, 表明本数值模型具备定量预测能力。通过引入布朗粗化频率描述液相烧结过程中的聚并现象, 模拟结果表明颗粒的聚并行为显著延缓了瞬态向稳态的转变过程, 甚至可能导致最终得到非稳态粒径分布。Abstract: The Ostwald ripening and aggregation of neighboring particles occur simultaneously during liquid phase sintering (LPS). A population balance equation was applied to predict the evolution of particle size distribution in aggregation effect during LPS process, a method based on the Euclidean norm was proposed to determine the steady state of particle size distribution in LPS, and the proposed model was applied to predict the upper limit and change rate of particle size during the transition from transient to steady-state. It is found that the particle size distribution reaches a steady-state after the transient coarsening. The modeling and the experimental data reported in literature show good agreement, demonstrating the reliable quantitative prediction by the present numerical model. The present model was adopted to study the particle aggregation by taking Brownian motion into account during LPS. It is also found that the aggregation obviously retards the transition from transient to steady-state, and may even lead to an unsteady-state particle size distribution.
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图 2 液相烧结过程中颗粒成分分布和二元合金相图
Figure 2. Composition distribution in solid particle during LPS and the phase diagram for a binary alloy
图 3 瞬态到稳态的液相烧结过程: (a) 不同时间粒径分布; (b) ║f (ρi, τ+ (35) τ) -f (ρi, τ) ║ (i=1, …, Nξ) 欧式范数随时间变化关系曲线
Figure 3. LPS from the transient to the steady state: (a) PSD at different time; (b) Euclidean norm of║f (ρi, τ+ (35) τ) -f (ρi, τ) ║ (i=1, …, Nξ) as a function of time
图 4 颗粒体积分数ϕ分别为0.13和0.40条件下的欧式范数║f (ρi, τ+ (35) τ) -f (ρi, τ) ║ (a), 最大粒径ρm1 (b) 和最大粒径变化速率(dρm1/dτ) 随时间变化的函数关系(c)
Figure 4. Euclidean norm of║f (ρi, τ+ (35) τ) -f (ρi, τ) ║ (a), the maximum particle size (ρm1) (b), and the changing rate of the maximum particle size (dρm1/dτ) as a function of time (c) at ϕ=0.13 and 0.40
图 5 非零颗粒体积分数下计算得到的稳态粒径分布与实验结果: (a) ϕ=0.13; (b) ϕ=0.40
Figure 5. Experiment and calculation results of particle size distribution in steady state at nonzero particle volume fraction: (a) ϕ=0.13; (b) ϕ=0.40
图 6 瞬态到稳态的液相烧结过程中粒径分布(f (ρ, τ)) 和负聚并速率(-Qagg (ρ, τ)) 的演化: (a) τ=0.90; (b) τ=1.34; (c) τ=1.57; (d) τ=1.79; (e) τ=4.48; (f) τ=22.40, 89.59, 170.22和188.14
Figure 6. Evolutions of particle size distribution (f (ρ, τ)) and the negative of aggregation term (-Qagg (ρ, τ)) during LPS from the transient to the steady state: (a) τ=0.90; (b) τ=1.34; (c) τ=1.57; (d) τ=1.79; (e) τ=4.48; (f) τ=22.40, 89.59, 170.22, and 188.14
表 1 不同颗粒体积分数下τcr1, τcr2, ρm1 (τcr2) 及f (ρm1, τcr2)
Table 1. Values ofτcr1, τcr2, ρm1 (τcr2), and f (ρm1, τcr2) for the different particle volume fractions
ϕ τcr1 τcr2 ρm1(τcr2) f(ρm1, τcr2) 0.13 4.9 6.0 1.624 1.56×10-3 0.40 5.4 6.7 1.695 5.22×10-4 
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