Question 5 of 7: CaO Coordination Number; Porosity/Grain Size in Ceramics; Weibull Statistics

Question 5: CaO Coordination Number; Porosity/Grain Size in Ceramics; Weibull Statistics (20 marks)

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Given. (a) $r_{Ca^{2+}}=0.100$ nm, $r_{O^{2-}}=0.140$ nm. (b)–(c) General ceramic microstructure (porosity, grain size) and brittle-fracture behaviour.

Find. (a) Predicted coordination number of Ca$^{2+}$ in CaO. (b) How porosity and grain size affect tensile strength. (c) Why ceramic failure is treated statistically, and how the Weibull distribution is used.

Approach

(a) uses the standard radius-ratio rule: the cation-to-anion radius ratio is compared against the geometric thresholds for each coordination number. (b)–(c) are answered from brittle fracture-mechanics reasoning: ceramics fail from the propagation of their most severe pre-existing flaw, so anything that changes the size/severity of that flaw population changes strength.

1. (a) Radius ratio. $$\frac{r_{Ca^{2+}}}{r_{O^{2-}}}=\frac{0.100}{0.140}$$ $$\boxed{\text{ratio}\approx0.714}$$ This falls in the range $0.414\le\text{ratio}<0.732$, which predicts coordination number 6 (octahedral coordination). This matches the observed structure of CaO, which crystallizes in the rock-salt (NaCl-type) arrangement with CN$=6$ for both Ca$^{2+}$ and O$^{2-}$.
2. (b) Effect of porosity. Pores act simultaneously as (i) stress concentrators — geometrically similar to internal cracks — and (ii) a reduction of the effective load-bearing cross-section. Increasing the volume fraction of porosity $P$ reduces strength roughly exponentially, $\sigma\approx\sigma_0\exp(-nP)$ (the Ryshkewitch–Duckworth-type relation), because pores are typically the largest, most severe flaws present and directly control the Griffith fracture stress $\sigma_f\propto1/\sqrt{a}$ (flaw size $a$).
3. (b) Effect of grain size. Finer grain size increases strength, both through a Hall–Petch-type mechanism ($\sigma_y=\sigma_0+Kd^{-1/2}$, grain boundaries impede whatever elementary process initiates fracture) and, more importantly for ceramics, because the size of the largest intrinsic flaw (a grain-boundary or triple-point microcrack) tends to scale with grain size $d$ — a finer grain structure has a smaller maximum flaw size, which by the Griffith relation directly raises the fracture strength. Fine-grained ceramics are therefore both stronger and more consistent (narrower strength scatter).
4. (c) Why statistics are needed. Ceramics have essentially no plasticity to blunt or redistribute stress around a crack tip, so failure is governed entirely by the single most severe pre-existing flaw (pore, inclusion, machining damage) in the loaded volume — a "weakest-link" process, analogous to a chain failing at its weakest link, rather than a deterministic bulk-average property like a metal's yield strength. Because flaw size, location and orientation vary randomly from specimen to specimen, and larger specimens/components sample a larger flaw population (and so, statistically, contain a more severe worst flaw), measured strength is inherently scattered and size-dependent — a single strength number is not physically meaningful for design.
5. (c) The Weibull distribution. The standard statistical model is the survival probability $$P_{surv}(\sigma)=\exp\!\left[-\left(\frac{V}{V_0}\right)\left(\frac{\sigma}{\sigma_0}\right)^m\right]$$ where $m$ is the Weibull modulus (a large $m$ means a narrow, consistent strength distribution — a reliable material with a tightly controlled flaw population; a small $m$ means wide scatter, dominated by a few severe outlier flaws), $\sigma_0$ is a characteristic (scale) strength, and the $V/V_0$ ratio captures the size effect. Designers select an allowable design stress corresponding to an acceptably low probability of failure — not a simple safety factor on a mean strength — and $m$ itself is reported as a key quality/reliability metric for structural ceramics.