crystalline amorphous and anisotropic solids

A solid keeps its shape and volume, and the difference between crystalline and amorphous bodies is set by how their particles are arranged relative to one another. In a crystal the particles (atoms, ions, molecules) form long-range order: a repeating motif — the unit cell — is reproduced over distances far larger than the interatomic spacing, so melting occurs at a sharply defined temperature $T_f$ and the $T(t)$ graph shows a horizontal plateau. An amorphous body has only short-range order, as a liquid does: on heating it softens gradually over a range of temperatures. Anisotropy is the dependence of physical properties (thermal conductivity $\lambda$, refractive index $n$, Young's modulus $E$, expansion coefficient $\alpha$) on direction inside the body. In a single crystal anisotropy follows directly from the ordered lattice: interparticle spacings and bond types differ along different crystallographic directions. An amorphous body has no preferred directions and is therefore isotropic; a polycrystal also behaves as an isotropic body while its grains are small compared with the sample, because grain orientations average out.

A sample of a solid is heated slowly. Its temperature-versus-time graph shows a horizontal plateau at $273\ \text{K}$, and X-ray diffraction gives a set of sharp narrow peaks. Is this body crystalline or amorphous? Justify your answer.

Find the density $\rho$ of a copper single crystal with an FCC lattice if the lattice parameter is $a = 3{,}61\times10^{-10}\ \text{m}$, the molar mass is $M = 63{,}55\ \text{g/mol}$ and Avogadro's constant is $N_A = 6{,}02\times10^{23}\ \text{mol}^{-1}$.

Graphite has a layered structure: inside a layer the atoms are joined by strong covalent bonds, while weak van der Waals forces act between the layers. The thermal conductivity along the layers is $\lambda_{\parallel} \approx 400\ \text{W/(m}\cdot\text{K)}$ and across the layers $\lambda_{\perp} \approx 4\ \text{W/(m}\cdot\text{K)}$. Explain the origin of the anisotropy, estimate the ratio $\lambda_{\parallel}/\lambda_{\perp}$ and predict whether a pressed graphite powder will be anisotropic. Justify your answer.

Work with models only, without heating and without hazardous substances.

1. Open a browser-based crystal-lattice simulation or a molecular-dynamics visualisation of a solid. Build SC, BCC and FCC cells, record the number of particles per cell ($1$, $2$, $4$) and compute the packing fraction

$$k = \frac{n \cdot \frac{4}{3}\pi r^3}{a^3},$$

assuming the particles touch one another. Compare your result with the known values $0{,}52$, $0{,}68$, $0{,}74$.
2. In a birefringence simulation, place a calcite crystal in the beam path and rotate it slowly, noting whether the beam splits into two. Then replace the calcite with isotropic glass and repeat.
3. In a molecular-dynamics model of an amorphous body, raise the temperature and record how particle mobility changes: does a sharp jump appear at one temperature, or a gradual softening?
4. Write the summary as three lines: type of body — observed feature — explanation in terms of particle arrangement.

Show the link between particle arrangement, melting behaviour and the direction dependence of properties, so that one figure explains the crystal, the amorphous body and anisotropy at once.

Three parts on one sheet.

(a) Particle-arrangement sketch: on the left, a flat single-crystal lattice with the unit cell outlined by a dashed contour; on the right, a chaotic cluster of particles of an amorphous body with short-range order.

(b) $T(t)$ graphs: for the crystal, a rising slope, a horizontal plateau at $T_f$, then a rise again; for the amorphous body, a smooth softening curve with no plateau.

(c) Polar diagram of a property (for example $\lambda$ or $E$): for the single crystal, a closed curve elongated along the direction of strong bonding; for the amorphous body, a circle showing the same value of the property in all directions.

A racing frame of mass $1{,}1\,\mathrm{kg}$ carries a cyclist of mass $70\,\mathrm{kg}$; in a sprint the rider produces $500\,\mathrm{W}$ and travels at $45\,\mathrm{km/h}$. Carbon fibres are laid along the frame tubes: the elastic modulus along a fibre is about $230\,\mathrm{GPa}$, while across it is about $15\,\mathrm{GPa}$.

This exercises anisotropy: the elastic properties of crystalline graphite and of the fibre depend on direction. In a graphite layer strong covalent bonds act in the basal plane, while weak van der Waals forces act between planes; therefore stiffness along the tube and across it are unequal, which is described by the elasticity tensor of an anisotropic solid.

A smartphone screen is a layer of amorphous glass $0{,}7\,\mathrm{mm}$ thick; the camera cover is crystalline sapphire $0{,}3\,\mathrm{mm}$ thick. Glass hardness is $6$–$7$ on the Mohs scale, sapphire hardness is $9$; in a fall from $1{,}5\,\mathrm{m}$, glass more often gets scratches, while sapphire chips along certain planes.

Here the distinction between amorphous and crystalline states is at work: glass has no long-range order, so its hardness and refractive index are the same in all directions; sapphire has a periodic lattice, and its hardness, thermal conductivity and refractive index are anisotropic, while cleavage runs along crystallographic planes.

A chocolate bar of mass $100\,\mathrm{g}$ is heated to $45\,^{\circ}\mathrm{C}$, then cooled to $27\,^{\circ}\mathrm{C}$ and warmed again to $31\,^{\circ}\mathrm{C}$. The stable crystalline form V of cocoa butter melts at $34\,^{\circ}\mathrm{C}$, while rapidly cooled amorphous and unstable forms melt at $23$–$28\,^{\circ}\mathrm{C}$; at room temperature $20\,^{\circ}\mathrm{C}$ a tempered bar snaps firmly, while an untempered one is soft and dull.

The example shows crystalline and amorphous states of one substance: crystalline form V has long-range order and a sharp melting point at $34\,^{\circ}\mathrm{C}$, whereas the amorphous/glassy phase has no long-range order and melts/softens over a broad interval. Different crystalline polymorphs are different lattices made of the same molecules, which changes strength and gloss.