Crystal Structures and Atomic Bonding in Ceramics

Atomic Bonding in Ceramics

Ceramic materials are characterized by the types of chemical bonds that hold their atoms together, which significantly influence their properties. These bonds are primarily ionic and/or covalent.

• Bonding Types: Most ceramics contain a combination of ionic and covalent bonding.

• Percent Ionic Character: The degree of ionic character depends on the difference in electronegativity between the cations ($X_A$) and anions ($X_B$). It can be calculated using the formula:   $\% \text{ Ionic character} = \{ 1 - \exp[-(0.25)(X_A - X_B)^2]\} \times 100$

• Material Examples and Ionic Character:

  • $CaF_2$: $89 \%$ (Large ionic character)

  • $MgO$: $73 \%$

  • $NaCl$: $67 \%$

  • $Al_2O_3$: $63 \%$

  • $SiO_2$: $51 \%$

  • $Si_3N_4$: $30 \%$

  • $ZnS$: $18 \%$

  • $SiC$: $12 \%$ (Small ionic character)

Cations and Anions

Unlike metallic crystal structures, ceramic lattice sites are occupied by ions rather than neutral atoms. These ions are categorized into two types:

• Cations:

  • Positively charged ions.

  • They are the smaller of the two ions.

  • Formed when a more electropositive atom (usually a metal from the left side of the periodic table) loses electrons to a more electronegative atom.

• Anions:

  • Negatively charged ions.

  • They are the larger of the two ions.

  • Formed when a more electronegative atom (usually a non-metal from the right side of the periodic table) gains electrons from a more electropositive atom.

Factors Determining Crystal Structures

The arrangements of ions in a ceramic crystal are determined by two primary factors:

1. Charge Neutrality:

   • The bulk ceramic material must remain electrically neutral.

   • The net charge of the ions in the structure must sum to zero. This dictates the stoichiometry of the compound.

   • Examples of Charge Neutrality Failure:

     • $\text{MgO}_2$: Mg is divalent ($Mg^{2+}$) and O is divalent ($O^{2-}$). The net charge for one molecule would be $1(2+) + 2(2-) = -2$. This net negative charge is not allowed.

     • $\text{Cs}_2\text{Cl}$: Cs is monovalent ($Cs^{1+}$) and Cl is monovalent ($Cl^{1-}$). The net charge would be $2(1+) + 1(1-) = +1$. This net positive charge is not allowed.

   • Allowable Stoichiometries:

     • $Mn - O$: $Mn^{2+}O^{2-}$ (Net charge = $0$)

     • $Mn - F$: $Mn^{2+}F_2^{1-}$ (Net charge = $0$)

     • $Fe - O$: $Fe^{2+}O^{2-}$ (Net charge = $0$) or $Fe_2^{3+}O_3^{2-}$ ($2(3+) + 3(2-) = 0$)

     • $Ti - O$: $Ti^{4+}O_2^{2-}$ (Net charge = $0$)

     • $Si - O$: $Si^{4+}O_2^{2-}$ (Net charge = $0$)

     • $Al - O$: $Al_2^{3+}O_3^{2-}$ (Net charge = $0$)

2. Coordination Number (CN):

   • This is defined as the number of nearest-neighbor atoms or ions.

   • CN determines the possible crystal structures, which in turn determine the physical properties of the ceramic.

   • The stability of the crystal structure is influenced by ion contact; for a structure to be stable, the cations must be in contact with the surrounding anions.

   • As the radius ratio ($r_c/r_a$) increases, the coordination number also increases.

Coordination Numbers and Geometries

The coordination number is a function of the cation-to-anion radius ratio ($r_c/r_a$). Larger coordination numbers correspond to larger cation sizes relative to the anions.

• Radius Ratio Rules:

  • < 0.155: CN = $2$ (Linear geometry)

  • $0.155 - 0.225$: CN = $3$ (Equilateral triangle geometry)

  • $0.225 - 0.414$: CN = $4$ (Tetrahedral geometry)

  • $0.414 - 0.732$: CN = $6$ (Octahedral geometry)

  • $0.732 - 1.0$: CN = $8$ (Cubic geometry)

• Computation of Minimum Radius Ratio for Octahedral Site (CN = 6):

  • In an octahedral site, the geometry is defined by $a = 2r_{\text{anion}}$.

  • Using the Pythagorean theorem for the relationship across the diagonal:       $a^2 + a^2 = (2r_{\text{anion}} + 2r_{\text{cation}})^2$       $2a^2 = (2r_{\text{anion}} + 2r_{\text{cation}})^2$       $\sqrt{2}a = 2r_{\text{anion}} + 2r_{\text{cation}}$

  • Substituting $a = 2r_{\text{anion}}$:       $\sqrt{2}(2r_{\text{anion}}) = 2r_{\text{anion}} + 2r_{\text{cation}}$       $\sqrt{2}r_{\text{anion}} = r_{\text{anion}} + r_{\text{cation}}$       $r_{\text{cation}} = (\sqrt{2} - 1)r_{\text{anion}}$       $\frac{r_{\text{cation}}}{r_{\text{anion}}} = \sqrt{2} - 1 = 0.414$

AX-Type Crystal Structures

AX-type structures occur when there are equal numbers of cations and anions.

• Cesium Chloride (CsCl) Structure:

  • Coordination Number = $8$.

  • Radius ratio range: 0.732 < r_c/r_a < 1.0.

  • Though it resembles a Body-Centered Cubic (BCC) structure, it is not BCC because the center atom is a different species than the corner atoms.

• Rock Salt (NaCl) Structure:

  • Coordination Number = $6$.

  • Radius ratio range: 0.414 < r_c/r_a < 0.732.

  • Structure consists of two interpenetrating Face-Centered Cubic (FCC) lattices—one for cations and one for anions.

  • Examples: $NaCl, MgO, MnS, LiF, FeO$.

  • For NaCl specifically: $r_{Na} = 0.102\,nm$, $r_{Cl} = 0.181\,nm$, resulting in $r_{Na}/r_{Cl} = 0.564$.

• Zinc Blende (ZnS) Structure:

  • Coordination Number = $4$.

  • Radius ratio range: 0.225 < r_c/r_a < 0.414.

  • All ions are tetrahedrally coordinated; each atom is bonded to 4 atoms of the opposite type.

  • Anions typically occupy corner and face sites, while cations occupy interior tetrahedral sites.

  • Bonding is predominantly covalent.

  • Examples: $ZnS, ZnTe, SiC$.

AmXp and AmBnXp-Type Crystal Structures

These structures occur when the charges on cations and anions are not equal, requiring different numbers of each to maintain neutrality.

• $A_mX_p$ Structures (e.g., $AX_2$):

  • Fluorite ($CaF_2$): Radius ratio for $CaF_2 \approx 0.8$, leading to a CN of $8$.

  • $Ca^{2+}$ ions are positioned at the centers of cubes formed by $F^-$ ions at the corners.

  • A single unit cell is composed of eight such cubes.

  • Examples: $ZrO_2$ (cubic), $UO_2, PuO_2, ThO_2$.

• $A_mB_nX_p$ Structures:

  • These involve more than one type of cation ($A$ and $B$).

  • Barium Titanate ($BaTiO_3$): Contains $Ba^{2+}$ and $Ti^{4+}$ cations.

  • It exhibits the Perovskite crystal structure.

Predicting Crystal Structures

Crystal structures can be predicted using ionic radii and the radius ratio ($r_c/r_a$).

• Example: Predicting the structure of FeO

  • Ionic radius of $Fe^{2+} = 0.077\,nm$.

  • Ionic radius of $O^{2-} = 0.140\,nm$.

  • Ratio: $\frac{0.077}{0.140} = 0.550$

  • Since 0.414 < 0.550 < 0.732, the coordination number is $6$, and the predicted crystal structure is the Rock Salt (NaCl) structure.

• Reference Ionic Radii ($nm$):

  • Cations: $Al^{3+} (0.053)$, $Fe^{2+} (0.077)$, $Fe^{3+} (0.069)$, $Ca^{2+} (0.100)$.

  • Anions: $O^{2-} (0.140)$, $Cl^- (0.181)$, $F^- (0.133)$.