Influence of Particle Size on Apparent Diffusivity During Spark Plasma Sintering of Crystalline Powders


A theoretical framework is presented to determine an apparent diffusivity D during powder sintering and further elucidate the underlying interrelation between D and shrinkage behaviors of powders. Furthermore, to eliminate the influence of crystalline defects that are typically present in raw powders, complete crystallization of metallic glass powders is performed to allow the investigation of the influence of particle size solely on powder densification. Furthermore, to validate the framework developed, Ti40.6Zr9.4Cu37.5Ni9.4Sn3.1 crystalline alloy powders constituting particles with different sizes are examined to establish a correlation between D and the sintering behaviors of the powders. The findings show that the value of the apparent diffusivity D increases with the increasing particle size, which accelerates powder densification during spark plasma sintering. Furthermore, the results quantitatively support the argument that particle size can affect atomic diffusion during powder sintering.

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ρ :

Relative density

ρ 0 :

Initial relative density

H :

Powder height (mm)

H 0 :

Initial powder height (mm)

\( \dot{\rho } \) :

Densification rate (s−1)

T :

Time (s)

γ :

Surface energy (J/m2)

L :

Average particle size (μm)

ΔH/H 0 :

Powder shrinkage

B :

Geometry factor (constant)

η :

Viscosity, (pa s)

P :

The applied pressure (MPa)

a :

Average grain size (nm)

D :

Diffusivity (m2/s)

δ :

Atom diameter (Å)

k :

Boltzmann constant (J/K)

T :

Temperature (K)

D 0 :

Diffusion constant (m2/s)

\( Q \) :

Diffusion activation energy (kJ/mol)

C :

Heating rate (K/s)

\( D^{P} \) :

Pressure-related diffusivity (m2/s)

\( D_{0}^{P} \) :

Pressure-related diffusion constant (m2/s)

\( D_{0}^{T} \) :

Total diffusion constant (m2/s)

\( \Delta T \) :

The localized overheating (K)

X :

Distance from the surface of a powder (μm)

R :

The radius of powder particle, (μm)

I :

Current (A)

\( \rho_{r} \) :

Electrical resistivity (Ω cm)

\( \Delta t \) :

Pulse time (s)

\( C_{V} \) :

Heat capacity (J/K/mol)

\( \rho_{m} \) :

Density (g/cm3)

Φ :

Inner diameter of the die (mm)


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This work was supported by the National Natural Science Foundation of China (No. 51574128), the Guangdong Natural Science Foundation for Research Team (No. 2015A030312003), the Guangdong Application-oriented Special Funds for Science and Technology R&D (No. 2016B090931002), and the Fundamental Research Funds for the Central Universities (No. 2017PY014). We thank The Editing Team from Liwen Bianji, Edanz Editing China (, for editing the English text of a draft of this manuscript.

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Manuscript submitted April 2, 2019.

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Li, X.X., Yang, C., Liu, Z. et al. Influence of Particle Size on Apparent Diffusivity During Spark Plasma Sintering of Crystalline Powders. Metall Mater Trans B 50, 2843–2852 (2019).

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