Materials Science and Engineering. An Introduction

William D. Callister

Chapter 12

Structures and Properties of Ceramics - all with Video Answers

Educators

Chapter Questions

Problem 1

For a ceramic compound, what are the two characteristics of the component ions that determine the crystal structure?

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Problem 2

Show that the minimum cation-to-anion radius ratio for a coordination number of 4 is 0.225 .

Ameer Said

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Problem 3

Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is 0.414 . [Hint: Use the NaCl crystal structure (Figure 12.2), and assume that anions and cations are just touching along cube edges and across face diagonals.]

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Problem 4

Demonstrate that the minimum cation-toanion radius ratio for a coordination number of 8 is 0.732 .

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Problem 5

On the basis of ionic charge and ionic radii given in Table 12.3, predict crystal structures for the following materials: (a) CaO , (b) MnS , (c) KBr , and (d) CsBr. Justify your selections.

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Problem 6

Which of the cations in Table 12.3 would you predict to form fluorides having the cesium chloride crystal structure? Justify your choices

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Problem 7

Compute the atomic packing factor for the rock salt crystal structure in which $r_{\mathrm{C}} / r_{\mathrm{A}}=0.414$

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Problem 8

The zinc blende crystal structure is one that may be generated from close-packed planes of anions.
(a) Will the stacking sequence for this structure be FCC or HCP? Why?
(b) Will cations fill tetrahedral or octahedral positions? Why?
(c) What fraction of the positions will be occupied?

Ameer Said

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Problem 9

The corundum crystal structure, found for $\mathrm{Al}_2 \mathrm{O}_3$, consists of an HCP arrangement of $\mathrm{O}^{2-}$ ions; the $\mathrm{Al}^{3+}$ ions occupy octahedral positions.
(a) What fraction of the available octahedral positions are filled with $\mathrm{Al}^{3+}$ ions?
(b) Sketch two close-packed $\mathrm{O}^{2-}$ planes stacked in an $A B$ sequence, and note octahedral positions that will be filled with the $\mathrm{Al}^{3+}$ ions.

Ameer Said

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Problem 10

Beryllium oxide ( BeO ) may form a crystal structure that consists of an HCP arrangement of $\mathrm{O}^{2-}$ ions. If the ionic radius of $\mathrm{Be}^{2+}$ is 0.035 nm , then
(a) Which type of interstitial site will the $\mathrm{Be}^{2+}$ ions occupy?
(b) What fraction of these available interstitial sites will be occupied by $\mathrm{Be}^{2+}$ ions?

Narayan Hari

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Problem 11

Iron titanate, $\mathrm{FeTiO}_3$, forms in the ilmenite crystal structure that consists of an HCP arrangement of $\mathrm{O}^{2-}$ ions.
(a) Which type of interstitial site will the $\mathrm{Fe}^{2+}$ ions occupy? Why?
(b) Which type of interstitial site will the $\mathrm{Ti}^{4+}$ ions occupy? Why?
(c) What fraction of the total tetrahedral sites will be occupied?
(d) What fraction of the total octahedral sites will be occupied?

Narayan Hari

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Problem 12

Using the Molecule Definition Utility found in both "Metallic Crystal Structures and Crystallography" and "Ceramic Crystal Structures" modules of VMSE, located on the book's web site [www.wiley.com/college/callister (Student Companion Site)], generate (and print out) a three-dimensional unit cell for lead oxide, PbO , given the following: (1) The unit cell is tetragonal with $a=0.397 \mathrm{~nm}$ and $c=0.502 \mathrm{~nm}$, (2) oxygen atoms are located at the following point coordinates:
$$
\begin{array}{ll}
000 & 001 \\
100 & 101 \\
010 & 011 \\
110 & 111 \\
\frac{1}{2} \frac{1}{2} 0 & \frac{1}{2} \frac{1}{2} 1
\end{array}
$$
and (3) Pb atoms are located at the following point coordinates:

$$
\begin{array}{ll}
\frac{1}{2} 00.763 & 0 \frac{1}{2} 0.237 \\
\frac{1}{2} 10.763 & 1 \frac{1}{2} 0.237
\end{array}
$$

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Problem 13

Calculate the theoretical density of NiO , given that it has the rock salt crystal structure.

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Problem 14

Iron oxide ( FeO ) has the rock salt crystal structure and a density of $5.70 \mathrm{~g} / \mathrm{cm}^3$.
(a) Determine the unit cell edge length.
(b) How does this result compare with the edge length as determined from the radii in Table 12.3, assuming that the $\mathrm{Fe}^{2+}$ and $\mathrm{O}^{2-}$ ions just touch each other along the edges?

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Problem 15

Compute the theoretical density of diamond given that the $\mathrm{C}-\mathrm{C}$ distance and bond angle are 0.154 nm and $109.5^{\circ}$, respectively. How does this value compare with the measured density?

Ameer Said

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Problem 16

Compute the theoretical density of ZnS given that the $\mathrm{Zn}-\mathrm{S}$ distance and bond angle are 0.234 nm and $109.5^{\circ}$, respectively. How does this value compare with the measured density?

Ameer Said

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Problem 17

One crystalline form of silica $\left(\mathrm{SiO}_2\right)$ has a cubic unit cell, and from X-ray diffraction data it is known that the cell edge length is 0.700 nm . If the measured density is $2.32 \mathrm{~g} / \mathrm{cm}^3$, how many $\mathrm{Si}^{4+}$ and $\mathrm{O}^{2-}$ ions are there per unit cell?

Narayan Hari

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Problem 18

(a) Using the ionic radii in Table 12.3, compute the theoretical density of CsCl . (Hint: Use a modification of the result of Problem 3.3.)
(b) The measured density is $3.99 \mathrm{~g} / \mathrm{cm}^3$. How do you explain the slight discrepancy between your calculated value and the measured one?

Ameer Said

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Problem 19

From the data in Table 12.3, compute the theoretical density of $\mathrm{CaF}_2$, which has the fluorite structure.

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Problem 20

A hypothetical AX type of ceramic material is known to have a density of $2.10 \mathrm{~g} / \mathrm{cm}^3$ and a unit cell of cubic symmetry with a cell edge length of 0.57 nm . The atomic weights of the A and X elements are 28.5 and $30.0 \mathrm{~g} / \mathrm{mol}$, respectively. On the basis of this information, which of the following crystal structures is (are) possible for this material: sodium chloride, cesium chloride, or zinc blende? Justify your choice(s).

Narayan Hari

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Problem 21

The unit cell for $\mathrm{Fe}_3 \mathrm{O}_4\left(\mathrm{FeO}-\mathrm{Fe}_2 \mathrm{O}_3\right)$ has cubic symmetry with a unit cell edge length of 0.839 nm . If the density of this material is $5.24 \mathrm{~g} / \mathrm{cm}^3$, compute its atomic packing factor. For this computation, you will need to use ionic radii listed in Table 12.3.

Narayan Hari

Numerade Educator

Problem 22

The unit cell for $\mathrm{Al}_2 \mathrm{O}_3$ has hexagonal symmetry with lattice parameters $a=0.4759 \mathrm{~nm}$ and $c=1.2989 \mathrm{~nm}$. If the density of this material is $3.99 \mathrm{~g} / \mathrm{cm}^3$, calculate its atomic packing factor. For this computation use ionic radii listed in Table 12.3.

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Problem 23

Compute the atomic packing factor for the diamond cubic crystal structure (Figure 12.15). Assume that bonding atoms touch one another, that the angle between adjacent bonds is $109.5^{\circ}$, and that each atom internal to the unit cell is positioned $a / 4$ of the distance away from the two nearest cell faces ( $a$ is the unit cell edge length).

Ameer Said

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Problem 24

Compute the atomic packing factor for cesium chloride using the ionic radii in Table 12.3 and assuming that the ions touch along the cube diagonals.

Ameer Said

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Problem 25

For each of the following crystal structures, represent the indicated plane in the manner of Figures 3.10 and 3.11, showing both anions and cations: (a) (100) plane for the cesium chloride crystal structure, (b) (200) plane for the cesium chloride crystal structure, (c) (111) plane for the diamond cubic crystal structure, and (d) (110) plane for the fluorite crystal structure.

Ameer Said

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Problem 26

In terms of bonding, explain why silicate materials have relatively low densities.

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Problem 27

Determine the angle between covalent bonds in an $\mathrm{SiO}_4^{4-}$ tetrahedron.

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Problem 28

Would you expect Frenkel defects for anions to exist in ionic ceramics in relatively large concentrations? Why or why not?

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Problem 29

Calculate the fraction of lattice sites that are Schottky defects for cesium chloride at its melting temperature $\left(645^{\circ} \mathrm{C}\right)$. Assume an energy for defect formation of 1.86 eV .

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Problem 30

Calculate the number of Frenkel defects per cubic meter in silver chloride at $350^{\circ} \mathrm{C}$. The energy for defect formation is 1.1 eV , while the density for AgCl is $5.50 \mathrm{~g} / \mathrm{cm}^3$ at $\left(350^{\circ} \mathrm{C}\right)$.

Narayan Hari

Numerade Educator

Problem 31

Using the data given below that relate to the formation of Schottky defects in some oxide ceramic (having the chemical formula MO), determine the following:
(a) The energy for defect formation (in eV ),
(b) the equilibrium number of Schottky defects per cubic meter at $1000^{\circ} \mathrm{C}$, and
(c) the identity of the oxide (i.e., what is the metal M?)
$$
\begin{array}{ccc}
\hline \boldsymbol{T}\left({ }^{\circ} \mathrm{C}\right) & \boldsymbol{\rho}\left(\mathrm{g} / \mathrm{cm}^{\mathbf{3}}\right) & \boldsymbol{N}_{\mathbf{s}}\left(\mathrm{m}^{-\mathbf{3}}\right) \\
\hline 750 & 3.50 & 5.7 \times 10^9 \\
1000 & 3.45 & ? \\
1500 & 3.40 & 5.8 \times 10^{17} \\
\hline
\end{array}
$$

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Problem 32

In your own words, briefly define the term "stoichiometric."

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Problem 33

If cupric oxide $(\mathrm{CuO})$ is exposed to reducing atmospheres at elevated temperatures, some of the $\mathrm{Cu}^{2+}$ ions will become $\mathrm{Cu}^{+}$.
(a) Under these conditions, name one crystalline defect that you would expect to form in order to maintain charge neutrality.
(b) How many $\mathrm{Cu}^{+}$ions are required for the creation of each defect?
(c) How would you express the chemical formula for this nonstoichiometric material?

Ameer Said

Numerade Educator

Problem 34

(a) Suppose that CaO is added as an impurity to $\mathrm{Li}_2 \mathrm{O}$. If the $\mathrm{Ca}^{2+}$ substitutes for $\mathrm{Li}^{+}$, what kind of vacancies would you expect to form? How many of these vacancies are created for every $\mathrm{Ca}^{2+}$ added?
(b) Suppose that CaO is added as an impurity to $\mathrm{CaCl}_2$. If the $\mathrm{O}^{2-}$ substitutes for $\mathrm{Cl}^{-}$, what kind of vacancies would you expect to form? How many of these vacancies are created for every $\mathrm{O}^{2-}$ added?

Ameer Said

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Problem 35

For the $\mathrm{ZrO}_2-\mathrm{CaO}$ system (Figure 12.26), write all eutectic and eutectoid reactions for cooling.

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Problem 36

From Figure 12.25, the phase diagram for the $\mathrm{MgO}-\mathrm{Al}_2 \mathrm{O}_3$ system, it may be noted that the spinel solid solution exists over a range of compositions, which means that it is nonstoichiometric at compositions other than $50 \mathrm{~mol} \% \mathrm{MgO}-50 \mathrm{~mol} \% \mathrm{Al}_2 \mathrm{O}_3$.
(a) The maximum nonstoichiometry on the $\mathrm{Al}_2 \mathrm{O}_3$-rich side of the spinel phase field exists at about $2000^{\circ} \mathrm{C}\left(3630^{\circ} \mathrm{F}\right)$ corresponding to approximately $82 \mathrm{~mol} \%\left(92 \mathrm{wt} \%\right.$ ) $\mathrm{Al}_2 \mathrm{O}_3$. Determine the type of vacancy defect that is produced and the percentage of vacancies that exist at this composition.
(b) The maximum nonstoichiometry on the MgO -rich side of the spinel phase field exists at about $2000^{\circ} \mathrm{C}\left(3630^{\circ} \mathrm{F}\right)$ corresponding to approximately $39 \mathrm{~mol} \%\left(62 \mathrm{wt} \%\right.$ ) $\mathrm{Al}_2 \mathrm{O}_3$. Determine the type of vacancy defect that is produced and the percentage of vacancies that exist at this composition.

Manik Pulyani

Numerade Educator

Problem 37

When kaolinite clay $\left[\mathrm{Al}_2\left(\mathrm{Si}_2 \mathrm{O}_5\right)(\mathrm{OH})_4\right]$ is heated to a sufficiently high temperature, chemical water is driven off.
(a) Under these circumstances, what is the composition of the remaining product (in weight percent $\mathrm{Al}_2 \mathrm{O}_3$ )?
(b) What are the liquidus and solidus temperatures of this material?

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Problem 38

Briefly explain (a) why there may be significant scatter in the fracture strength for some given ceramic material, and (b) why fracture strength increases with decreasing specimen size.

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Problem 39

The tensile strength of brittle materials may be determined using a variation of Equation 8.1. Compute the critical crack tip radius for a glass specimen that experiences tensile fracture at an applied stress of 70 MPa ( $10,000 \mathrm{psi}$ ). Assume a critical surface crack length of $10^{-2} \mathrm{~mm}$ and a theoretical fracture strength of $E / 10$, where $E$ is the modulus of elasticity.

Narayan Hari

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Problem 40

The fracture strength of glass may be increased by etching away a thin surface layer. It is believed that the etching may alter surface crack geometry (i.e., reduce crack length and increase the tip radius). Compute the ratio of the etched and original cracktip radii for a fourfold increase in fracture strength if half of the crack length is removed.

Ameer Said

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Problem 41

A three-point bending test is performed on a spinel $\left(\mathrm{MgAl}_2 \mathrm{O}_4\right)$ specimen having a rectangular cross section of height $d 3.8 \mathrm{~mm}$ ( 0.15 in .) and width $b 9 \mathrm{~mm}$ ( 0.35 in.$)$; the distance between support points is 25 mm (1.0 in.).
(a) Compute the flexural strength if the load at fracture is $350 \mathrm{~N}\left(80 \mathrm{lb}_{\mathrm{f}}\right)$.
(b) The point of maximum deflection $\Delta y$ occurs at the center of the specimen and is described by

$$
\Delta y=\frac{F L^3}{48 E I}
$$

where $E$ is the modulus of elasticity and $I$ is the cross-sectional moment of inertia. Compute $\Delta y$ at a load of $310 \mathrm{~N}\left(70 \mathrm{lb}_{\mathrm{f}}\right)$.

Manik Pulyani

Numerade Educator

Problem 42

A circular specimen of MgO is loaded using a three-point bending mode. Compute the minimum possible radius of the specimen without fracture, given that the applied load is $5560 \mathrm{~N}\left(1250 \mathrm{lb}_{\mathrm{f}}\right)$, the flexural strength is $105 \mathrm{MPa}(15,000 \mathrm{psi})$, and the separation between load points is 45 mm ( 1.75 in .).

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Problem 43

A three-point bending test was performed on an aluminum oxide specimen having a circular cross section of radius 5.0 mm ( 0.20 in. ); the specimen fractured at a load of $3000 \mathrm{~N}\left(675 \mathrm{lb}_f\right)$ when the distance between the support points was 40 mm ( 1.6 in .). Another test is to be performed on a specimen of this same material, but one that has a square cross section of 15 mm ( 0.6 in .) length on each edge. At what load would you expect this specimen to fracture if the support point separation is maintained at 40 mm ( 1.6 in .)?

Narayan Hari

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Problem 44

(a) A three-point transverse bending test is conducted on a cylindrical specimen of aluminum oxide having a reported flexural strength of $300 \mathrm{MPa}(43,500 \mathrm{psi})$. If the specimen radius is $5.0 \mathrm{~mm}(0.20 \mathrm{in}$.) and the support point separation distance is 15.0 mm ( 0.61 in .), predict whether or not you would expect the specimen to fracture when a load of $7500 \mathrm{~N}\left(1690 \mathrm{lb}_f\right)$ is applied? Justify your prediction.
(b) Would you be $100 \%$ certain of the prediction in part (a)? Why or why not?

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Problem 45

Cite one reason why ceramic materials are, in general, harder yet more brittle than metals.

Ameer Said

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Problem 46

The modulus of elasticity for spinel ( $\mathrm{MgAl}_2 \mathrm{O}_4$ ) having $5 \mathrm{vol} \%$ porosity is 240 GPa $\left(35 \times 10^6 \mathrm{psi}\right)$.
(a) Compute the modulus of elasticity for the nonporous material.
(b) Compute the modulus of elasticity for $15 \mathrm{vol} \%$ porosity.

Narayan Hari

Numerade Educator

Problem 47

The modulus of elasticity for titanium carbide ( TiC ) having $5 \mathrm{vol} \%$ porosity is $310 \mathrm{GPa}\left(45 \times 10^6 \mathrm{psi}\right)$.
(a) Compute the modulus of elasticity for the nonporous material.
(b) At what volume percent porosity will the modulus of elasticity be 240 GPa $\left(35 \times 10^6 \mathrm{psi}\right) ?$

Narayan Hari

Numerade Educator

Problem 48

Using the data in Table 12.5, do the following:
(a) Determine the flexural strength for nonporous MgO assuming a value of 3.75 for $n$ in Equation 12.10.
(b) Compute the volume fraction porosity at which the flexural strength for MgO is $74 \mathrm{MPa}(10,700 \mathrm{psi})$.

Narayan Hari

Numerade Educator

Problem 49

The flexural strength and associated volume fraction porosity for two specimens of the $$
\begin{aligned}
&\text { same ceramic material are as follows: }\\
&\begin{array}{cc}
\hline \boldsymbol{\sigma}_{f s}(\boldsymbol{M P a}) & \boldsymbol{P} \\
\hline 70 & 0.10 \\
60 & 0.15 \\
\hline
\end{array}
\end{aligned}
$$
(a) Compute the flexural strength for a completely nonporous specimen of this material.
(b) Compute the flexural strength for a 0.20 volume fraction porosity.

Manik Pulyani

Numerade Educator