Van der Waals Force

In physical chemistry, condensed-matter physics, and structural biology, van der Waals forces (named after Dutch physicist Johannes Diderik van der Waals, who received the 1910 Nobel Prize in Physics for introducing finite molecular volume and intermolecular attraction into the equation of state of gases and liquids in 1873) encompass the distance-dependent electrodynamic interactions acting between electrically neutral atoms or molecules that do not involve covalent or ionic bond formation. In modern IUPAC nomenclature, the attractive van der Waals potential V_vdW(R) at center-to-center distance R is the sum of three distinct inverse-sixth-power (∝ −*R*⁻⁶) multipolar contributions—Keesom dipole–dipole orientation forces, Debye permanent-dipole–induced-dipole induction forces, and London dispersion forces (instantaneous fluctuating dipole–induced dipole forces)—balanced at contact distances by short-range Pauli exchange repulsion:

The classical ideal gas law (P · V_m = R · T, where V_m = V/n is the molar volume) assumes that gas molecules are point masses of zero volume that exert zero forces on one another between elastic collisions. Consequently, the ideal gas law cannot explain why real gases condense into liquids when cooled or compressed. In his 1873 doctoral thesis at Leiden University, van der Waals modified both pressure and volume terms to formulate the van der Waals equation of state:

• The Excluded Volume Parameter (b, L·mol⁻¹): Accounts for short-range Pauli exchange repulsion when the electron clouds of two molecules overlap at contact distance (equal to four times the hard-sphere volume of one mole of molecules, b = 4 N_A (4π*r*₀³/3)), which reduces the free volume available for translational motion to (V_m − b).
• The Attractive Cohesion Parameter (a, L²·bar·mol⁻²): Accounts for medium-range intermolecular attractions, which pull molecules near the container wall inward and reduce the wall impact pressure by an internal pressure term proportional to the square of molar concentration (Δ*P*_int = a / V_m²). Above the critical temperature T_c = 8*a* / (27*Rb*), thermal kinetic energy overwhelms intermolecular attraction (a), yielding a supercritical fluid; below T_c, the van der Waals isotherm predicts first-order gas–liquid phase condensation.

Between 1920 and 1930, Willem Hendrik Keesom, Peter Debye, and Fritz London derived the exact expressions for the three attractive R⁻⁶ mechanisms between neutral molecules A and B possessing permanent electric dipole moments (μ_A, μ_B), polarizability volumes (α′_A, α′_B, where α′ = α / 4π*ε*₀), and first ionization energies (I_A, I_B):

1. Keesom Orientation Force (Permanent Dipole – Permanent Dipole, 1921)

When two polar molecules with permanent dipole moments μ_A and μ_B (such as two HCl, SO₂, or acetone molecules) tumble thermally in a gas or liquid, the instantaneous electrostatic potential between two fixed dipoles scales as R⁻³ and averages to zero if all orientations are weighted equally. However, because attractive head-to-tail and antiparallel alignments have lower electrostatic energy than repulsive head-to-head alignments, the Boltzmann factor exp(−*V*_θ / k_BT) weights attractive orientations slightly more heavily than repulsive ones. Averaging over all rotational orientations for k_BT > V_dipole yields the thermally averaged Keesom orientation energy:

Notice that V_Keesom is inversely proportional to absolute temperature (T⁻¹): as temperature rises, thermal rotational tumbling scrambles dipolar alignment and weakens the Keesom attraction.

2. Debye Induction Force (Permanent Dipole – Induced Dipole, 1920)

When a polar molecule with permanent dipole moment μ_A approaches a second molecule B (whether polar or nonpolar) with polarizability α_B, the electric field of dipole A (E ∝ μ_A / R³) distorts the electron cloud of molecule B, inducing a dipole moment μ_ind,B = α_B · E ∝ α_Bμ_A / R³ that always points in an attractive orientation regardless of molecular tumbling. Summing the mutual induction of A on B and B on A gives the temperature-independent Debye induction energy:

3. London Dispersion Force (Instantaneous Fluctuating Dipole – Induced Dipole, 1930)

Derived via second-order quantum mechanical perturbation theory by Fritz London in 1930, the London dispersion force arises from correlated zero-point quantum fluctuations of electron clouds, whereby an instantaneous dipole on molecule A induces a synchronous, in-phase dipole on molecule B:

Unlike Keesom and Debye forces—which vanish completely when neither molecule possesses a permanent dipole moment (μ = 0)—London dispersion acts between all pairs of atoms and molecules, and in fact contributes 60% to 100% of the total van der Waals attraction in every non-hydrogen-bonded molecule (even in polar molecules such as CO, HCl, and CH₃Cl).

Equilibrium Contact Distance and Bondi Van der Waals Radii

When two non-bonded atoms approach closer than ~3–4 Å, their filled valence shells overlap. The Pauli exclusion principle forbids electrons of identical spin from occupying the same spatial region, forcing electron density into higher-energy nodes and generating an exponentially rising repulsive wall (V_rep(R) = A exp(−*bR*), or approximated algebraically as C₁₂ / R¹²). Combining short-range Pauli repulsion with the −*C*₆ / R⁶ van der Waals attraction gives the Lennard-Jones (12–6) potential:

The potential minimum (V = −ε) occurs at the equilibrium van der Waals contact distance R_min = 2^1/6 σ ≈ 1.122 σ. Half of this equilibrium non-bonded contact distance defines an element's van der Waals radius (r_w) (tabulated across the periodic table from X-ray crystal packing by Arnold Bondi in 1964: r_w(H) = 1.20 Å, r_w(C) = 1.70 Å, r_w(N) = 1.55 Å, r_w(O) = 1.52 Å, r_w(F) = 1.47 Å, r_w(Cl) = 1.75 Å). In structural biochemistry, the Ramachandran plot of allowed polypeptide backbone dihedral angles (φ, ψ) and the hydrophobic core packing of protein folding are governed directly by preventing interatomic distances from violating the sum of Bondi van der Waals radii.

Macroscopic Bodies: The Hamaker Constant (A_H) and DLVO Colloid Stability

Whereas the van der Waals interaction energy between two isolated molecules decays rapidly as −*C*₆ / R⁶, integrating the pairwise interaction across all atoms inside two macroscopic bodies (such as two colloidal nanoparticles, two lipid bilayers, or a gecko spatula and a glass surface) at surface-to-surface separation D (performed by Hendrik C. Hamaker in 1937 and generalized via macroscopic dielectric continuum theory by Evgeny Lifshitz in 1956) yields a much longer-ranged attraction:

• Two Spheres of Radius R_s at Small Gap D ≪ R_s: V_vdW(D) = −(A_H · R_s) / (12 · D) (decays only as 1 / D!).
• Two Parallel Planar Surfaces per Unit Area: W_vdW(D) = −*A*_H / (12π · D²) (decays as 1 / D²).

Here A_H = π² · C₆ · ρ₁ · ρ₂ is the Hamaker constant (typically 0.4 × 10⁻¹⁹ to 4 × 10⁻¹⁹ J across condensed media). In DLVO theory (Derjaguin–Landau–Verwey–Overbeek), the stability of colloidal suspensions, clays, milk, blood cells, and latex paints is governed by the competition between this universal 1/D Hamaker van der Waals attraction (which drives coagulation/flocculation) and electrostatic double-layer repulsion between charged particle surfaces.

How does a hydrogen bond differ from an ordinary Keesom van der Waals dipole–dipole interaction?

Although a hydrogen bond (D–H···A, where D and A are strongly electronegative N, O, or F atoms) involves a strong electrostatic dipole–dipole attraction, it differs from non-specific van der Waals forces in three ways:

1. Bond Energy: Hydrogen bonds are typically 15 to 40 kJ·mol⁻¹ (10 times stronger than a typical van der Waals pair contact of 1–4 kJ·mol⁻¹).
2. Partial Covalent Charge Transfer (n → *σ*\*): In a hydrogen bond, the non-bonding lone pair (n) of the acceptor atom donates electron density directly into the empty antibonding *σ*\*_D–H orbital of the donor bond (lengthening the D–H covalent bond and shifting its IR stretching frequency downward by hundreds of cm⁻¹), enforcing strict linear directionality (D–H···A angle ≈ 170°–180°).
3. Sub-van der Waals Penetration: Because the positively polarized H atom lacks any inner core electron shells, the H···A distance in a hydrogen bond (~1.7–2.0 Å in water or ice) is 0.7 to 1.0 Å shorter than the sum of the van der Waals radii of H and O (1.20 + 1.52 = 2.72 Å).

Why is the London dispersion term almost always larger than the Keesom dipole–dipole term, even in polar molecules like HCl and CH₃Cl?

Students are often taught that dipole–dipole (Keesom) forces dominate in polar molecules while London dispersion forces only matter in nonpolar molecules. Quantitative evaluation of the C₆ coefficients (see Table above) proves the exact opposite: in HCl (μ = 1.08 D), London dispersion provides 81.4% of the total attraction while Keesom orientation provides only 14.4% (and in HBr and HI, London dispersion provides >95% and >99%!). The physical reason is that the Keesom orientation force requires aligning entire molecular dipoles against thermal rotational scrambling (k_BT in the denominator), whereas London dispersion correlates high-frequency electronic fluctuations across all valence electrons simultaneously without being degraded by thermal rotation. Only in small molecules with very large dipole moments and low polarizability (such as H₂O and NH₃, which also form hydrogen bonds) does the Keesom term exceed the London term.

How does atomic force microscopy (AFM) use van der Waals forces to image individual atoms and molecules on non-conductive surfaces?

Invented by Gerd Binnig, Calvin Quate, and Christoph Gerber in 1986, atomic force microscopy (AFM) scans an atomically sharp silicon or silicon nitride tip attached to a microcantilever across a sample surface at sub-nanometer distances. Unlike scanning tunneling microscopy (STM), which requires an electrical tunneling current and therefore works only on conductive metals and semiconductors, AFM measures the van der Waals and Pauli force gradient (F = −d*V*_LJ/d*z*) acting between the tip apex atom and surface atoms: at distances of 0.5–2 nm (non-contact attractive regime), long-range van der Waals forces pull the cantilever downward, whereas at < 0.3 nm (contact repulsive regime), short-range Pauli R⁻¹² repulsion deflects the cantilever upward, enabling sub-angstrom imaging of insulating minerals, polymers, and biological membranes in water.

1. van der Waals, J. D. (1873). Over de Continuiteit van den Gas- en Vloeistoftoestand [On the Continuity of the Gaseous and Liquid States]. Doctoral dissertation, Leiden University.
2. Keesom, W. H. (1921). "Die van der Waalsschen Kohäsionskräfte." Physikalische Zeitschrift, 22, 129–141.
3. London, F. (1930). "Zur Theorie und Systematik der Molekularkräfte." Zeitschrift für Physik, 63(3–4), 245–279. https://doi.org/10.1007/BF01421741
4. Bondi, A. (1964). "van der Waals volumes and radii." The Journal of Physical Chemistry, 68(3), 441–451. https://doi.org/10.1021/j100785a001
5. Israelachvili, J. N. (2011). Intermolecular and Surface Forces (3rd ed.). Academic Press. ISBN 978-0-12-391927-4