Thermal expansion and why cyanide frameworks can shrink when heated

Most things grow when heated because a bond is easier to stretch than to squash

Intuition

Picture two atoms joined by a spring that is stiff when squeezed and soft when stretched. Heat makes them jiggle harder. Because the spring is soft on the stretch side, the jiggle spends more time stretched than squeezed. Their average gap gets bigger. That is thermal expansion.

Lavoisier (1789) wrote that every material, solid or fluid, increases in size in each dimension when heated. He thought heat was a real fluid (“calorique”) that got between the particles and pushed them apart. The modern picture is the potential energy curve from the bonding lectures. The potential $V(r)$ has a steep repulsive wall at short distance (the Pauli exclusion principle keeps electrons from sharing a state), an attractive tail at long distance and a minimum at the bonding distance.

As temperature rises, atoms have more thermal energy and the amplitude of the longitudinal vibrations, along the bond, increases. The well is asymmetric, so the average inter-atomic spacing, the bond length, increases.

The coefficient of thermal expansion, $\alpha$, is the relative change in size per unit change in temperature. For volume and for length:

$\alpha_V = \frac{1}{V}\left(\frac{dV}{dT}\right) \qquad \alpha_l = \frac{1}{l}\left(\frac{dl}{dT}\right)$

The lecture’s table gives $\alpha_l$ at 20 °C in units of $10^{-6}\ \text{K}^{-1}$: mercury 60, lead 29, aluminium 23, brass 19, stainless steel 17, copper 17, gold 14, nickel 13, concrete 12, steel 12, platinum 9, glass 8, GaAs 6, indium phosphide 5, Pyrex 3, silicon 3, diamond 1.

A sideways wobble can pull the ends of a link closer together

Intuition

Hold a skipping rope by its ends and swing it. The middle moves out sideways, and the ends are pulled closer together. The rope did not get shorter. It got more bent. A straight, thin link made of M–C≡N–M behaves like that rope when it wobbles from side to side.

Lengthening along a bond is not the only way atoms move. In a framework made of long thin links, there are also transverse vibrations, at 90° to the bond. These are low-energy modes. When the link bends sideways, the atoms at its two ends get closer together. If this effect outweighs the lengthening from longitudinal vibrations, the whole framework shrinks as it warms. That is negative thermal expansion (NTE).

Lecture 37 uses three related cyanide families, all shown on slides 23, 27 and 33: $Cd^{II}(CN)_2\cdot G$ (a single porous network, neutral framework, with guest molecules), $N(CH_3)_4Cu^IZn^{II}(CN)_4$ (a single network, anionic framework, with cations in the pores) and $M^{II}(CN)_2$ (two networks, neutral framework, no guests).

Four examples show NTE, normal expansion and zero expansion

Zn(CN)₂ contracts on warming. $\alpha = -16.5\times10^{-6}\ \text{K}^{-1}$. The lattice parameter falls from about +0.2% to −0.8% over 0 to 800 K in the plot (several diffraction methods agree). The explanation is transverse vibration.

Cd(CN)₂ with a CCl₄ guest. With the guest in the cavity, $\alpha = +10.0(2)\times10^{-6}\ \text{K}^{-1}$, so it expands. Around 300 to 380 K the lattice parameter starts to fall slightly as the solvent evaporates. At about 100 °C there is an abrupt collapse when all the solvent is gone. The desolvated clathrate, on cooling, has $\alpha = -33.4(5)\times10^{-6}\ \text{K}^{-1}$. At low temperatures, air molecules condense in the pores and the material expands again. So a guest in the pore can turn NTE off.

N(CH₃)₄CuZn(CN)₄. Between 200 and 400 K the lattice parameter is almost constant, called zero thermal expansion, because of two competing effects. As temperature rises, the framework’s transverse vibrations contract it. The cation, meanwhile, vibrates and rotates more and takes up more space, so the framework expands. The slide labels $\alpha$ as $-2.5$, $+0.2$ and $+6.0$ $\text{MK}^{-1}$ ($= 10^{-6}\ \text{K}^{-1}$) at about 250, 300 and 370 K. The space group changes from $P2_13$ below about 130 K to $F\bar{4}3m$ above.

Zn^II Pt^IV(CN)₆·xH₂O is a Prussian-blue type with cubic holes filled with water. Below room temperature it is near zero expansion, $\alpha = +2\times10^{-6}\ \text{K}^{-1}$. Above room temperature the water leaves and the lattice parameter decreases. The framework then shows NTE from transverse vibrations, $\alpha = -3\times10^{-6}\ \text{K}^{-1}$.

Compare

Longitudinal vibration

Along the bond. Larger amplitude on heating lengthens the average bond because the well is asymmetric. Causes expansion.

Transverse vibration

At 90° to the bond, a low-energy mode. The link bends sideways like a skipping rope and its ends move closer. Causes contraction in frameworks with long thin links.

The energies of these phonons, and so their amplitudes, depend on how strong the M(1)–C≡N–M(2) bonding is. Tightly bound cyanides give small amplitudes and small NTE. Loosely bound cyanides give large amplitudes and large NTE. The next topic shows how the choice of metal changes this.

Exam detail

Learn the five $\alpha$ values with their signs and what each example teaches: Zn(CN)₂ (NTE), Cd(CN)₂ with guest (positive) and desolvated (very negative), N(CH₃)₄CuZn(CN)₄ (zero, two competing effects), ZnPt(CN)₆ with and without water.

Pitfall

Do not say NTE means the bonds get shorter. The slides say longitudinal vibrations still lengthen each bond. It is the sideways motion that brings the atoms at the ends of the link closer together.

Recall

How would you explain to a friend that a framework can shrink when you heat it?

The links are thin and bendy. Heat makes them wobble sideways, like a skipping rope. When the rope bows out, its ends move closer. If the sideways effect is bigger than the small stretching of each bond, the whole framework gets smaller.