8.3 From pair stiffness to the bulk modulus

The curvature of the pair potential at its minimum is a spring constant, and a condensed phase is a lattice of such springs. Summing them gives a first-principles estimate of a macroscopic elastic modulus — one of the few places where a bulk mechanical property can be predicted from molecular parameters alone. This lesson carries the Lennard-Jones potential up to the bulk modulus of water and the speed of sound.

The pair as a spring

Near its minimum, any smooth potential is parabolic, and the molecule pair behaves as a Hookean spring whose stiffness is the curvature at the well bottom.

▶ Curvature of the Lennard-Jones well Derivation

Differentiate $U(r) = 4\varepsilon[(\sigma/r)^{12} - (\sigma/r)^6]$ twice and evaluate at $r_\text{eq} = 2^{1/6}\sigma$ :

k_\text{pair} \;=\; \left.\frac{d^2U}{dr^2}\right|_{r_\text{eq}} \;=\; \frac{72\,\varepsilon}{2^{1/3}\,\sigma^2}.

The stiffness is set by the well depth $\varepsilon$ divided by the square of the molecular size $\sigma$ — a stiffness scale $\varepsilon/\sigma^2$ times a pure number.

where

$k_\text{pair}$: pair spring constant N/m
$\varepsilon$: well depth J
$\sigma$: molecular size parameter m

Summing over the lattice

A condensed phase packs each molecule against $Z$ nearest neighbours (for a close-packed or simple-cubic estimate, $Z \approx 6$ – $12$ ). Compressing the material uniformly stretches and squeezes all those bonds at once. The bulk modulus — pressure per fractional volume change — is the lattice stiffness per unit volume:

K \;\sim\; \frac{Z}{2}\,\frac{k_\text{pair}}{r_\text{eq}} \;\sim\; \frac{\varepsilon}{\sigma^3}.

The result is clean: the bulk modulus is the well depth divided by the molecular volume, $K \sim \varepsilon/\sigma^3$ — an energy density built from the two Lennard-Jones parameters. Every factor has a meaning: deeper wells and more tightly packed molecules give stiffer matter.

ε1.00

σ1.00

k_pair = 72ε/σ²72.00

K (bulk)192.43 ε/σ³

Bond depth ε = 1.00 Bond length σ = 1.00

The bulk modulus of a solid (or strongly-bonded liquid) is essentially the *pair stiffness summed over nearest neighbours*. K ∼ ε/σ³ is dimensionally correct and within factor of 2 for water if ε is the hydrogen-bond energy and σ the molecular separation. This is how molecular-scale physics determines the macroscopic speed of sound √(K/ρ).

Testing it on water

Putting water’s molecular-scale numbers into $K \sim \varepsilon/\sigma^3$ gives $K \sim 2\,\text{GPa}$ , within a factor of two of the measured $K \approx 2.2\,\text{GPa}$ . For an estimate built from a single pair potential and a nearest-neighbour count, that is a genuine success — a macroscopic elastic constant predicted from the shape of a molecular potential. The speed of sound follows immediately from the bulk modulus and density,

c \;=\; \sqrt{\frac{K}{\rho}} \;\approx\; \sqrt{\frac{2.2\times10^9}{1000}} \;\approx\; 1480\,\text{m/s},

matching the measured sound speed in water. The chain molecular potential → curvature → bulk modulus → sound speed connects the microscopic force law of this chapter to a number one can measure with a stopwatch and a length of water.