On thermal runaway and local endothermic/exothermic reactions during flash sintering of ceramic nanoparticles

Abstract

Numerical analysis of the heat balance at the flash event during flash sintering of granular ceramic nanoparticles was performed assuming continuum solid state as well as simultaneous surface softening/liquid formation and current percolation through the nanoparticle contacts. Assuming inter-particle radiations in the specimen volume, the electric Joule heat generated at the nanoparticle contacts partially lost by radiation from the specimen external surfaces. Considering the thermal effects due to rapid heating rate and free-molecular heat conduction regime, high-temperature gradients between the nanoparticle surfaces and the surrounding gas were developed. The attractive capillary forces, induced by the particle surface softening/liquid at the percolation threshold, lead to rapid rearrangement and densification of the nanoparticles. The excess Joule heat, already at the flash event, suffices the excess internal heat that is necessary for partial or full melting. Particle surface softening/liquid formation is a transient process, hence followed by crystallization immediate after the nanoparticle rearrangement. Thermal runaway is associated with local surface softening/melting and its solidification.

Appendix

Thermal heat

The internal heat was calculated using the following equation:

$$ \dot{Q}_{\text{int}} = \rho_{\text{g}} \rho_{0} \left( T \right)c_{\text{p}} \left( T \right)\frac{{{\text{d}}T}}{{{\text{d}}t}} $$

(9)

While neglecting the change in the green density up to the flash temperature, which fairly is correct [56]. The temperature-dependent specific heat of solid alumina was used, up to its melting point.

When liquid forms, most probably at the nanoparticle contacts, the change in the internal heat was calculated according to the following equation [47]:

$$ \dot{Q}_{\text{int}} = \left[ {\rho_{\text{s}} c_{\text{p}}^{\text{solid}} \left( {1 - x_{\text{melt}} } \right) + \rho_{\text{l}} c_{\text{p}}^{\text{liquid}} x_{\text{metl}} } \right]\frac{{{\text{d}}T}}{{{\text{d}}t}} $$

(10)

where x_melt is the volume fraction of the melt, and indices s and l refer to solid and liquid, respectively. Since current percolation is associated with the percolation phenomenon, the volume fraction of the melt is 0.247 at the invasive percolation threshold [57]. This value was used to calculate the heat capacity of the specimen when liquid forms.

For heat/energy balance, we added the enthalpy of fusion at the percolation threshold (i.e., for fusion of 0.247 volume fraction of the mass) to the internal heat versus temperature (see the shaded area in Fig. 2).

The internal heat in Eq. (9) was calculated using the finite differences approximation described elsewhere [18]:

$$ \dot{Q}_{\text{int}} = \rho_{\text{g}} \rho_{0} \left( {T_{i} } \right)c_{\text{p}} \left( {T_{i} } \right)\frac{{T_{\text{p}}^{i + 1} - T_{\text{p}}^{i} }}{\Delta t} $$

(11)

where the interval between two consecutive temperatures and their corresponding time interval Δt were determined from the heating rate during the experiment [2].

The following temperature-dependent functions were used:

• (a) Temperature dependencies of the density (g cm⁻³) of solid Al₂O₃ and its melt (liquid) [58]:

  $$ \rho_{\text{s}} = 3.9899 - 12 \times 10^{ - 5} \cdot T\left( {^\circ {\text{C}}} \right) $$

  (12)
  $$ \rho_{\text{l}} = 5.3243 - 11.27 \times 10^{ - 4} \cdot T\left( {^\circ {\text{C}}} \right) $$

  (13)

• (b) Temperature dependence of the specific heat of solid and liquid Al₂O₃ (J mol⁻¹ K⁻¹) [59]:

  $$ c_{\text{p}}^{\text{solid}} = 102.43 + 38.75 \cdot \theta - 15.91 \cdot \theta^{2} + 2.63 \cdot \theta^{3} - \frac{3.0075}{{\theta^{2} }} $$

  (14)

  in the temperature range 298–2327 K

  $$ c_{\text{p}}^{\text{liquid}} = 192.46 + 9.52 \times 10^{ - 8} \cdot \theta - 2.85 \times 10^{ - 8} \cdot \theta^{2} + 2.93 \times 10^{ - 9} \cdot \theta^{3} - \frac{{5.59 \times 10^{ - 8} }}{{\theta^{2} }} $$

  (15)

  in the temperature range 2327–4000 K

  $$ {\text{Where}}\quad \theta = \frac{{T\left( {\text{K}} \right)}}{1000} $$

Joule heat

The accumulated Joule heat was calculated using the following equation:

$$ \dot{Q}_{\text{Joul}} = \int\limits_{t = 0}^{{t\left( {T_{\text{onset}} } \right)}} {\frac{{V^{2} }}{{R_{\text{e}} \left( T \right)}}{\text{d}}t} $$

(16)

The temperature dependence of the electric resistivity of pure Al₂O₃ is strongly affected by the impurity content. Therefore, published data about flash sintering of alumina was used together with the specimen dimensions and the flash process parameters [2]. The average electric conductivity of the present alumina is therefore:

$$ \sigma_{\text{Solid}}^{\text{Alumina}} = 3.289\left( {\Omega \,{\text{cm}}} \right)^{ - 1} \cdot {\text{e}}^{{ - \left[ {\frac{21253}{{T\left( {\text{K}} \right)}}} \right]}} $$

(17)

and the electric conductivity of alumina melt is [25]:

$$ \sigma_{\text{melt}}^{\text{Alumina}} = 0.0032213\left( {\Omega \,{\text{cm}}} \right)^{ - 1} \cdot {\text{e}}^{{ - \left[ {0.0019314 \cdot T\left( {\text{K}} \right)} \right]}} $$

(18)

The Joule heat prior to any melt was calculated using the solid phase properties, both as dense, as well as a granular (percolative) system [34]. However, at the current percolation threshold, when continuous liquid forms, the specimen properties of a percolating media (with liquid volume fraction of 0.247) were used [34]:

$$ \sigma_{\text{specimen}}^{\text{percol}} = (\sigma_{\text{solid}} )^{0.72} \cdot (\sigma_{\text{melt}} )^{0.28} $$

(19))