Question 4 of 8: Liquid Phase Sintering of Tungsten Carbide — Mass-Transport Pathways, No-Shrinkage Driving Force, Cobalt Requirements, and Capillary Densification Stress

Question 4: Liquid Phase Sintering of Tungsten Carbide — Mass-Transport Pathways, No-Shrinkage Driving Force, Cobalt Requirements, and Capillary Densification Stress (20 marks: a–5, b–5, c–5, d–5)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

4.1 — (a) The five mass-transport pathways in initial-stage solid-state sintering

All five pathways carry matter from a SOURCE region to the neck (SINK), but they differ in where the source material is drawn from:

1. Surface diffusion (1). Atoms migrate along the particle's own free surface directly into the neck.
2. Lattice diffusion from the surface (2). Atoms diffuse through the particle's bulk (volume), starting from a surface source, and terminate at the neck.
3. Vapour transport (3). Atoms evaporate from the (higher vapour-pressure, convex) particle surface, cross the pore space, and condense at the (lower vapour-pressure, concave) neck surface.
4. Grain-boundary diffusion (4). Atoms diffuse along the grain boundary that forms at the particle–particle interface itself, directly into the neck.
5. Lattice diffusion from the grain boundary (5). Atoms diffuse through the bulk, starting from the grain boundary as the source, and terminate at the neck.

(Plastic/viscous flow, sometimes listed as a sixth mechanism in a general two-sphere sintering model, is omitted here because WC is a hard, brittle covalent carbide with negligible dislocation mobility at sintering temperature — it contributes essentially nothing to neck growth in this system, consistent with the question specifying exactly five pathways.)

4.2 — (b) The no-shrinkage driving force

Pathways 1–3 (surface diffusion, lattice diffusion FROM the surface, and vapour transport) all draw their source material from the particle's own free SURFACE and deposit it at the neck. Because the material removed never comes from the region between the two particle CENTRES, the centre-to-centre distance is unchanged as the neck grows by these routes — the neck fills in, but the compact does not shrink. (By contrast, pathways 4–5 draw material from the grain boundary itself, i.e. from between the centres, so the centres approach each other and the compact densifies — this is exactly why WC, which sinters almost entirely by surface diffusion and vapour transport in the solid state, cannot reach high density without a liquid phase.)

The driving force for these non-densifying (surface-to-neck) pathways is purely geometric: the difference in curvature, and hence in chemical potential per atom, between the convex particle surface (both principal radii $D/2$, curvature sum $4/D$) and the saddle-shaped neck surface at point $P$ (concave radius $\rho$ in the plane of the drawing, convex radius $x$ around the neck). With $\rho$ and $x$ taken as positive magnitudes, the Laplace–Kelvin relations give the source-to-sink driving force:

for surface energy $\gamma_{sv}$ and atomic volume $\Omega$. Because the neck's concave radius is much smaller than both $x$ and $D$ in the initial stage ($\rho\approx x^2/D$), the bracket is positive and dominated by $1/\rho$: the particle surface has the higher chemical potential (and higher equilibrium vapour pressure) and the neck the lower, so atoms flow from surface to neck — entirely a surface-curvature effect, with no dependence on, or coupling to, the particle centre-to-centre spacing.

4.3 — (c) Requirements on the Co addition for effective liquid phase sintering

1. Must be liquid at the sintering temperature. Co (or the Co-rich WC–Co eutectic, melting near 1320 °C) must actually melt at the chosen hold temperature (typically 1400–1500 °C for WC–Co), so a genuine liquid phase is present, not just a solid-state sintering aid.
2. Must wet the WC solid. The contact angle must be low (Young: $\cos\theta=(\gamma_{SV}-\gamma_{SL})/\gamma_{LV}$, ideally $\theta\to0$, i.e. spreading when $\gamma_{SV}\ge\gamma_{SL}+\gamma_{LV}$), and the dihedral angle $\phi$ at WC/WC boundaries must also be low ($\gamma_{SS}=2\gamma_{SL}\cos(\phi/2)$), so the liquid spreads over the grains and penetrates between them rather than beading up in isolated pockets.
3. Must have appreciable solubility for WC (W and C). Some solubility of the solid in the liquid is required to enable solution–reprecipitation densification — WC dissolves preferentially at highly stressed, high-curvature contact points and reprecipitates at lower-energy sites, which is the dominant LPS densification mechanism once the liquid has wetted and rearranged the grains.
4. Must be present in a suitable volume fraction. Enough liquid to fill the pore network by capillary action and permit particle rearrangement, but not so much that the compact slumps or loses shape (commercial WC–Co grades use roughly 6–25 wt% Co).
5. Must have low enough viscosity and fast enough solute diffusivity. Rapid particle rearrangement and solution–reprecipitation both need adequately fast transport within the liquid itself, so full densification is achieved within a practical sintering hold time.

4.4 — (d) Compressive contact stress from liquid capillary pressure

Step 1 — Laplace pressure in the liquid. With the gas pressure in the pore taken as zero and the liquid meniscus curvature at the pore surface approximated by the single radius $r$ (the stated pore radius), the Laplace equation gives the liquid pressure DEFICIT relative to the (zero) gas pressure:

i.e. the liquid is in tension (below the surrounding gas pressure) by $\Delta P = 2\gamma_{LV}/r$.

Step 2 — net attractive force on a particle. This pressure deficit acts hydrostatically over the liquid film wetting each particle, pulling the two particles of a contacting pair together with a net force equal to $\Delta P$ acting over the particle's own cross-sectional area, $\pi R^2$:

Step 3 — stress across the solid–solid contact. This entire attractive force is carried through the small solid–solid contact area at the neck, $\pi x^2$ (with $x\ll r$, the contact is a small fraction of the particle's own cross-section). Equating the force transmitted through the contact to the force above:

Solving for the contact stress:

Because $x\ll r\le R$, the geometric factor $(R/x)^2$ is large, so the liquid capillary pressure — itself a modest quantity, of order $10^5$–$10^6$ Pa for a typical metallic liquid and micron-scale pore — is amplified by orders of magnitude into a very large LOCAL compressive stress at each solid–solid contact. This amplified contact stress is what drives particle rearrangement and the solution–reprecipitation densification of Question 4(c): it locally increases the solubility/chemical potential of the solid at the stressed contact (by the same curvature-driving-force logic as Question 4(b)), promoting preferential dissolution there and reprecipitation at unstressed, lower-energy surfaces.