Materials Science and Non-Metallic Materials: Bonding, Crystal Structure, Polymers, Ceramics, Composites, and Electronic and Magnetic Properties

The theoretical density follows from the cell: ρ = nA/(V_c N_A), with n atoms per cell of atomic mass A in a cell of volume V_c = a³ (cubic) and N_A = 6.022 × 10²³ mol⁻¹. For copper, FCC with a = 0.3615 nm and A = 63.55 g/mol, ρ = 4 × 63.55/(6.022 × 10²³ × (0.3615 × 10⁻⁷ cm)³) = 8.94 g/cm³. Miller indices (hkl) of a plane are the reciprocals of its intercepts on the axes in units of the lattice parameters, cleared of fractions: a plane cutting x at 1, y at 2 and parallel to z has reciprocals 1, ½, 0 and indices (210). Directions [uvw] are the smallest integer components of a vector; in cubic crystals [hkl] is perpendicular to (hkl). Families are written {hkl} and ⟨uvw⟩. The spacing of (hkl) planes in a cubic crystal is d = a/√(h² + k² + l²). Linear and planar densities count atoms centred on a direction or plane per unit length or area; the densest planes are {111} in FCC and {110} in BCC, and these are the slip planes.

FCC contains four octahedral interstitial sites (at the cell centre and edge midpoints) and eight tetrahedral sites per cell; BCC has larger tetrahedral sites but its octahedral sites are distorted, which is why carbon causes tetragonal distortion in ferrite and why austenite dissolves far more carbon. Alloys form substitutional solid solutions when the Hume-Rothery rules are met — atomic sizes within about 15%, the same crystal structure, similar electronegativity and similar valence — and interstitial solid solutions when the solute is small (C, N, H, B). Where they are not met, intermediate phases and intermetallic compounds form. Ionic solids are built as a lattice of the larger anions with cations in the interstices: rock salt (NaCl, MgO, FeO; cations in all octahedral sites of FCC anions), cesium chloride, zinc blende (ZnS; half the tetrahedral sites), fluorite (CaF₂, UO₂, ZrO₂) and perovskite (CaTiO₃, BaTiO₃). Covalent solids have low-coordination networks: diamond cubic for C, Si and Ge; silicates built from SiO₄⁴⁻ tetrahedra sharing corners into chains, sheets and frameworks. Polymers are long chains whose structure — linear, branched, cross-linked, network — and conformation decide their behaviour.

A polymer is a distribution of chain lengths, so its molecular weight is an average. The number-average M̄_n = Σx_iM_i weights each fraction by the number (mole) fraction x_i of chains; the weight-average M̄_w = Σw_iM_i weights by the mass fraction w_i and is always at least as large, because long chains carry more mass. Their ratio M̄_w/M̄_n, the polydispersity index, is 1 for a uniform polymer and typically 1.5-3 or more for commercial ones. The degree of polymerisation is M̄_n divided by the mer (repeat-unit) molecular weight — 28 g/mol for polyethylene’s C₂H₄. Strength, melt viscosity and softening temperature rise with molecular weight.

Thermoplastics are linear or branched chains held together by secondary bonds; they soften on heating and harden on cooling reversibly, can be remoulded and recycled, and include polyethylene, polypropylene, PVC, polystyrene, nylon and PET. Thermosets are cross-linked into a three-dimensional network of primary bonds during curing; they do not melt, decompose on strong heating, are harder, stiffer and more dimensionally stable, and include epoxies, phenol-formaldehyde (Bakelite), polyesters and vulcanised rubber in the limit. Elastomers are lightly cross-linked polymers above T_g that extend enormously and recover. Copolymers contain two kinds of mer arranged randomly, alternately, in blocks or as grafts of one on a backbone of the other: styrene-butadiene rubber is random, ABS is a graft, and block copolymers form thermoplastic elastomers. Crystallinity requires regular chains — linear rather than branched, isotactic or syndiotactic rather than atactic, with no bulky irregular side groups — and slow cooling; crystalline regions are chain-folded lamellae, often grown into spherulites, and a polymer is never fully crystalline. The percent crystallinity is found from densities: %c = ρ_c(ρ_s − ρ_a)/(ρ_s(ρ_c − ρ_a)) × 100, with ρ_s the specimen density and ρ_c and ρ_a those of the fully crystalline and fully amorphous polymer. Higher crystallinity raises density, stiffness, strength and chemical resistance.

In an ionic crystal each cation surrounds itself with as many anions as can touch it without the anions overlapping, so the coordination number is fixed by the radius ratio r_c/r_a; and the arrangement must be electrically neutral overall, which fixes the stoichiometry and so the fraction of interstitial sites filled. The critical ratios come from geometry: a cation just touching three anions in a triangle has r_c/r_a = 0.155, four in a tetrahedron 0.225, six in an octahedron 0.414, eight at the corners of a cube 0.732. Na⁺ (0.102 nm) and Cl⁻ (0.181 nm) have a ratio of 0.56, between 0.414 and 0.732, so NaCl is six-coordinated: rock salt. Cs⁺ is larger, the ratio exceeds 0.732, and CsCl is eight-coordinated. Pauling’s further rules (the electrostatic bond strength reaching each anion balances its charge; polyhedra prefer to share corners rather than edges or faces) explain the silicates.

Point defects in ionic crystals must preserve charge neutrality. A Schottky defect is a pair (for a 1:1 compound) of a cation vacancy and an anion vacancy, as in NaCl and MgO; it lowers the density. A Frenkel defect is a cation that has left its site for an interstitial position, leaving a vacancy — favoured when the cation is small and the structure has room, as in AgCl and AgBr; it leaves the density unchanged. Aliovalent impurities are compensated by vacancies too: Ca²⁺ substituting for Na⁺ in NaCl creates one Na⁺ vacancy for each Ca²⁺, and Y³⁺ in ZrO₂ creates oxygen vacancies, which is what makes yttria-stabilised zirconia an oxygen-ion conductor. Non-stoichiometric compounds depart from their ideal ratio by changing the valence of some ions. Wüstite is Fe₁₋ₓO: some iron is Fe³⁺, and each pair of Fe³⁺ ions is balanced by one iron vacancy (metal deficiency); zinc oxide heated in zinc vapour takes up excess zinc as interstitials (metal excess), becoming an n-type semiconductor. Kröger-Vink notation writes these defects with their effective charges.

A composite combines a matrix — polymer (epoxy, polyester, thermoplastics), metal (Al, Ti, Mg) or ceramic (SiC, Al₂O₃, glass-ceramics) — with a reinforcement — particles, short (discontinuous) fibres, whiskers or continuous fibres of glass, carbon, aramid, boron, SiC or alumina — to get a property combination neither has alone: stiffness and strength with low density in polymer-matrix composites, stiffness and wear resistance in metal-matrix composites, and toughness in ceramic-matrix composites, where fibres bridge cracks. The matrix binds and protects the fibres and transfers load to them by shear at the interface; a fibre carries its full share of load only if it is longer than the critical length l_c = σ_f d/(2τ_c), where τ_c is the interfacial (or matrix) shear strength.

For continuous aligned fibres two limiting conditions bound the modulus. Loaded along the fibres, fibre and matrix strain equally (isostrain) and the loads add, so the rule of mixtures gives the upper bound E_c = V_fE_f + V_mE_m, and the fraction of the load carried by the fibres is F_f/F_c = V_fE_f/(V_fE_f + V_mE_m) — nearly all of it when E_f ≫ E_m. Loaded across the fibres, they carry equal stress (isostress) and the strains add, giving the lower bound 1/E_c = V_f/E_f + V_m/E_m, dominated by the compliant matrix. Particulate composites fall between the two bounds. The longitudinal strength of a continuous-fibre composite that fails when the fibres break is σ_c = σ_f V_f + σ′_m(1 − V_f), where σ′_m is the matrix stress at the fibre failure strain.

When atoms come together in a solid their discrete energy levels broaden into bands. The occupancy of states is given by the Fermi-Dirac function f(E) = 1/(1 + exp((E − E_F)/kT)): at 0 K all states below the Fermi energy E_F are full and all above it empty, and at any temperature f(E_F) = ½. Whether a solid conducts depends on whether there are empty states immediately above the highest filled ones. In a metal the highest occupied band is partly filled, or a filled band overlaps an empty one, so electrons at E_F can be accelerated by an infinitesimal field: a conductor, with resistivity that rises with temperature because lattice vibrations scatter the electrons. In semiconductors and insulators the valence band is full and separated from the empty conduction band by a band gap E_g: small in semiconductors (Si about 1.1 eV, Ge about 0.67 eV, GaAs about 1.4 eV) and large in insulators (above roughly 4 eV; diamond about 5.5 eV).

In an intrinsic semiconductor, thermal excitation across the gap creates equal numbers of conduction electrons and holes, n = p = n_i ∝ T^(3/2) exp(−E_g/2kT), and E_F lies near mid-gap. Conductivity is σ = neμ_e + peμ_h, with the electronic charge e = 1.602 × 10⁻¹⁹ C and the mobilities μ; it rises steeply with temperature, the opposite of a metal. Doping makes it extrinsic: a group-V donor (P, As, Sb) in silicon gives an n-type semiconductor with electrons as majority carriers and E_F moved towards the conduction band; a group-III acceptor (B, Al, Ga) gives p-type, with holes and E_F moved towards the valence band. Over a wide range of temperature the extrinsic carrier concentration equals the dopant concentration (the saturation region), which is what makes doped devices stable; at high temperature intrinsic generation takes over again.

A field H (A/m) applied to a material produces a magnetisation M (magnetic moment per unit volume, A/m), and the flux density is B = μ₀(H + M) with μ₀ = 4π × 10⁻⁷ H/m. The susceptibility χ = M/H and relative permeability μ_r = 1 + χ classify the response. Diamagnetism is a weak opposing moment induced in every material by the field acting on orbital electrons: χ is small and negative, about −10⁻⁵ (Cu, Au, Zn, water, most polymers), and a superconductor is a perfect diamagnet, χ = −1. Paramagnetism arises in atoms with unpaired electrons whose permanent moments, randomly oriented, partly align with the field: χ is small and positive, about 10⁻⁵-10⁻³ (Al, Ti, Pt, O₂), and falls with temperature. In ferromagnets (Fe, Co, Ni, Gd) exchange interaction aligns neighbouring moments parallel within domains, giving very large, non-linear magnetisation that is lost above the Curie temperature. In antiferromagnets (MnO, NiO, Cr) exchange aligns neighbouring moments antiparallel and equal, so there is no net moment below the Néel temperature. In ferrimagnets (ferrites such as Fe₃O₄ and NiFe₂O₄, garnets) the antiparallel sublattices have unequal moments and a net magnetisation remains — ferromagnet-like behaviour in an insulating oxide, which is why ferrites are used at high frequencies where eddy currents would heat a metal core.

The B-H curve of a ferromagnet starts on the initial magnetisation curve, as domains aligned with the field grow by domain-wall motion and then rotate, up to saturation. On reducing the field the flux does not retrace its path: at H = 0 a remanence B_r remains, and a reverse field equal to the coercivity H_c is needed to bring B to zero. The area of the hysteresis loop is the energy lost per cycle per unit volume. Soft magnetic materials (Fe-Si electrical steel, permalloy Ni-Fe, soft ferrites, amorphous Fe-B-Si) have low coercivity, high permeability and a thin loop, for transformer and motor cores; they are made soft by removing whatever pins domain walls — impurities, inclusions, fine grains, internal stress. Hard magnetic materials (alnico, hard ferrites, Sm-Co, Nd-Fe-B) have high coercivity, high remanence and a large (BH)_max, for permanent magnets; they are made hard by pinning walls or by using single-domain particles. In Fe₃O₄ the Fe³⁺ ions on the two sublattices cancel and the net moment per formula unit is that of Fe²⁺, 4 Bohr magnetons (μ_B = 9.274 × 10⁻²⁴ A m²).