Intermolecular forces and the liquid state

Lennard-Jones, hydrogen bonds, van der Waals, cohesive limit, spinodal.

A liquid is a continuum at macroscopic length scales, but its mechanical properties — its bulk modulus, its tensile strength, its surface tension — all derive from the forces between individual molecules. This chapter assembles the molecular-scale physics that the thermodynamics, free-energy, and surface-tension chapters take as given.

The Lennard-Jones pair potential

A neutral, closed-shell molecule like argon interacts with a neighbour by two competing effects:

Short-range repulsion. When electron clouds overlap, the Pauli principle forces them into higher-energy configurations. The energy cost rises steeply — empirically, like $1/r^{12}$ .
Long-range attraction. Fluctuating dipoles correlate; an instantaneous dipole on one molecule polarises its neighbour. The leading term scales as $-1/r^6$ . This is the London dispersion or van der Waals force.

A two-parameter potential that captures both is the Lennard-Jones (12-6) potential:

U(r) \;=\; 4\varepsilon\!\left[\left(\frac{\sigma}{r}\right)^{12} \;-\; \left(\frac{\sigma}{r}\right)^{6}\right].

$\sigma$ sets the length at which $U$ crosses zero; $\varepsilon$ is the depth of the well. For argon, $\sigma = 0.34\,\text{nm}$ and $\varepsilon/k_B = 120\,\text{K}$ .

r1.200 σ

U(r)-0.891 ε

F(r)-2.212 ε/σ

r_eq = 2^1/61.122 σ

r_inf = (26/7)^1/61.244 σ

max attraction |F|2.396 ε/σ

The repulsive (1/r¹²) term dominates at short range — pushing molecules apart when they overlap — and the attractive (−1/r⁶) term dominates at long range, pulling them together. The two balance at r_eq: the equilibrium spacing of two molecules at zero force. Pull a pair past r_inf and the restoring force decreases; beyond it, you've started breaking the bond.

Two important radii fall out:

Equilibrium separation $r_\text{eq} = 2^{1/6}\sigma \approx 1.122\sigma$ . Where $dU/dr = 0$ ; the molecules sit there at zero temperature.
Inflection point $r_\text{inf} = (26/7)^{1/6}\sigma \approx 1.244\sigma$ . Where $d^2 U/dr^2 = 0$ ; the attractive force is maximum. Pulling a pair beyond this radius gives less restoring force, not more — the bond is on its way to breaking.

From pair stiffness to lattice bulk modulus

Differentiating the LJ potential a second time at the minimum gives the pair stiffness:

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

For a 3-D lattice of LJ molecules at the equilibrium density, summing over $Z = 6$ nearest neighbours and per unit volume gives the bulk modulus

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

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).

For water this estimate gives $K \sim 2\,\text{GPa}$ — within a factor of two of the measured $K \approx 2.2\,\text{GPa}$ . This is one of the few first-principles predictions of macroscopic mechanics from molecular-scale parameters. Speed of sound in water then follows as $c = \sqrt{K/\rho} \approx 1480\,\text{m/s}$ .

Hydrogen bonds — the dominant interaction in water

Pure London dispersion does not capture water. Water molecules have a permanent electric dipole moment ( $\sim 1.85\,\text{D}$ ), and they form hydrogen bonds — directional, partially covalent attractions of energy $\varepsilon_\text{HB} \approx 0.25\,\text{eV} \approx 10\,k_B T$ at room temperature.

Misalignment angle θ = 15°

The hydrogen bond is *directional*: the O–H...O configuration is energetically favourable only when the three atoms are nearly collinear. Misalign by 30° and the bond strength drops by half; misalign by 60° and it nearly vanishes. Contrast with the isotropic Lennard-Jones potential (dashed grey), which has the same energy at every angle. The directionality is what gives water its tetrahedral structure, anomalous density behaviour, high surface tension, and high heat capacity.

The directionality is essential. A misalignment by 30° halves the bond strength; by 60° it nearly vanishes. The Lennard-Jones potential is isotropic — has no preferred direction — so it cannot capture this. The directional nature of hydrogen bonds is what gives water its tetrahedral arrangement, the density-maximum-at-4°C anomaly, the high heat capacity, and the high surface tension.

Each water molecule engages in up to four hydrogen bonds — two donating, two accepting. In liquid water, they form and break continuously, with an average lifetime of $\sim 10^{-12}\,\text{s}$ .

The cohesive tensile limit and the spinodal

A liquid at rest is held together by attraction between molecules. To rupture it — to open a tiny vapour-filled void — requires doing work against this attraction.

If you stretch the liquid uniformly (lower its density below $\rho_\text{eq}$ ), the pressure becomes negative and proceeds along a $p(\rho)$ curve set by the equation of state. At some critical strain, $(\partial p/\partial \rho)_T = 0$ — the spinodal — and the homogeneous phase becomes mechanically unstable.

T / T_c0.950

p_spinodal (tension)0.846 p_c

Reduced T = 0.950 (T_c = 1)

Below T_c the isotherm develops a *van der Waals loop* with a region of (∂p/∂ρ) < 0 — mechanically unstable. The two extremes of this loop are the *spinodal* points: the minimum pressure (most negative) marks the in-principle tensile strength of the liquid. For water at room temperature this is ≈ −100 MPa. Above T_c the isotherm is monotonic — the critical point is the boundary between two-phase coexistence and a single supercritical phase.

Below the critical temperature $T_c$ , the van der Waals isotherm develops a loop with a region of negative slope. The two stationary points of this loop are the spinodal limits — the vapour spinodal at low density and the liquid spinodal at high density. The minimum pressure (most negative) on the liquid side marks the in-principle tensile strength: for water at room temperature, $p_\text{sp} \sim -100\,\text{MPa}$ .

The actual measured tensile strength of water is more like $-1\,\text{MPa}$ — three orders of magnitude weaker. The discrepancy is the central puzzle of the Cavitation book’s chapter 1.3. The resolution: real liquids contain pre-existing gas-filled crevices (heterogeneous nucleation sites) that fail long before the homogeneous spinodal is reached.

The van der Waals equation of state

The phenomenological van der Waals equation captures the LJ picture at coarse-grained level:

\left(p + \frac{a}{v^2}\right)(v - b) \;=\; R T.

The constant $a$ encodes the attractive part of the pair potential (the $1/r^6$ tail).
The constant $b$ encodes the excluded volume around each molecule (the $1/r^{12}$ wall).

a (attraction)3.00

b (excl. volume)0.333

T_c2.67

T / T_c1.000

Attraction a = 3.00 Excluded volume b = 0.333 Temperature T/T_c = 1.000

The vdW equation tweaks the ideal-gas law with two corrections: a/v² reduces pressure to account for inter-molecular attraction, and −b in the denominator reduces volume to account for excluded volume. Below T_c the isotherm develops the loop; above T_c it's monotonic. The critical point at (V_c, p_c, T_c) = (3b, a/(27b²), 8a/(27Rb)) is the only point where (∂p/∂V) and (∂²p/∂V²) both vanish.

Slide $a$ and $b$ ; watch the isotherm shape. The critical point lives at $(V_c, p_c, T_c) = (3b, a/(27b^2), 8a/(27Rb))$ . Below $T_c$ the system phase-separates into coexisting liquid and vapour; the Maxwell equal-area construction on this isotherm gives the coexistence pressure and the equilibrium densities of the two phases.

⏳ The history — London dispersion and the explanation of inert-gas cohesion

Until the 1930s the cohesion of inert gases — helium, neon, argon — was a puzzle. The molecules have no permanent dipole moments, no chemical bonds, no obvious mechanism for mutual attraction. Yet they condense to liquids and even solids at low temperatures.

Fritz London in 1930 derived the answer from quantum mechanics. Even non-polar molecules have fluctuating dipole moments due to zero-point motion of their electron clouds. An instantaneous dipole on one molecule induces a correlated dipole on its neighbour through the polarisability tensor; the resulting interaction averages to an attractive $-C_6/r^6$ tail.

The combination of London’s $1/r^6$ attraction with a phenomenological $1/r^{12}$ short-range repulsion — chosen by Lennard-Jones in 1924 mostly because $12 = 2 \times 6$ made the algebra clean — is the Lennard-Jones potential. The choice of exponent 12 is not derivable from first principles (the actual repulsion is closer to an exponential), but the form is so convenient that the LJ potential remains the workhorse pair potential for molecular simulations a century later.

For the cross-book applications — water’s bulk modulus, tensile-strength puzzle, hydrogen-bond effects on surface tension — see the key examples sub-page.