Intermolecular Forces — Why Drops Are Round and Insects Walk on Water 💧🐜

Intermolecular Forces — Why Drops Are Round and Insects Walk on Water 💧🐜

Early morning, dewdrops on leaves shimmer like pearls — and nearly all of them are round 🔮. A tiny insect strolls across the surface of a pond without sinking. None of this is coincidence. Behind all of it is an invisible force that binds molecules together, and — better yet — we can predict its behavior quantitatively with a formula.

The core idea in one paragraph 📌

Molecules exert two kinds of forces on each other: a short-range attraction (pulling them together when moderately close) and a very-short-range repulsion (keeping them from occupying the same space). Between molecules of the same substance this is called cohesion; between molecules of different substances, adhesion. Surface tension, wetting, and capillary action are all consequences of the balance between these two forces. The classic quantitative model is the Lennard-Jones potential.

Two flavors of force — definitions 🤝

Type Between molecules of… Intuitive symbol Example
Cohesion The same kind 💧 → 💧 Water ↔ water
Adhesion Different kinds 💧 → 🪟 Water ↔ glass

Intermolecular forces are short-range — beyond a few times the molecule’s own size, they drop off to nearly zero. That’s why you can’t glue shattered glass just by pressing the pieces together (the molecules have drifted too far apart), yet if you heat the glass until it softens, molecules come close again and rebond.

The Lennard-Jones potential — a quantitative model 📐

The most famous mathematical description of an intermolecular force is the Lennard-Jones potential:

\[ U(r) = 4\varepsilon \left[\left(\frac{\sigma}{r}\right)^{12} – \left(\frac{\sigma}{r}\right)^{6}\right] \]

Key insight: the minimum of the potential sits at \( r_{\min} = 2^{1/6}\sigma \) — meaning molecules prefer a specific equilibrium spacing, not zero.

Surface tension — the invisible skin 🛡️

Molecules deep inside a liquid are pulled from all sides. But molecules at the surface are only pulled from below and sideways (air or vacuum sits above). This imbalance makes the surface behave like a stretched skin.

Surface tension \( \gamma \) = energy per unit area, or force per unit length:

\[ \gamma = \frac{\text{work needed to grow the surface}}{\Delta A} \quad [\text{N/m or J/m}^2] \]

Reference values (at 20°C):

Liquid \( \gamma \) (N/m)
Pure water 0.072
Soapy water ~0.03 (half!)
Ethanol 0.022
Mercury 0.49 (very high)

A free droplet becomes spherical because a sphere has the smallest surface for a given volume → minimum surface energy 🔮.

Wetting — why water spreads but mercury balls up 🪟

When a liquid meets a solid, the contact angle \( \theta \) tells the story:

Capillary action — water defies gravity 🧪

Dip a very thin tube (radius \( r \sim 0.1 \) mm) into water and the water climbs up on its own. The rise height follows Jurin’s law:

\[ h = \frac{2\gamma \cos\theta}{\rho g r} \]

Python analysis 🐍

1) Plotting the Lennard-Jones potential

import numpy as np
import matplotlib.pyplot as plt

eps, sigma = 1.0, 1.0
r = np.linspace(0.9, 3.0, 400)
U = 4*eps*((sigma/r)**12 - (sigma/r)**6)

r_min = 2**(1/6) * sigma
plt.plot(r, U)
plt.axhline(0, color='gray', ls=':')
plt.axvline(r_min, color='red', ls='--', label=f'\( r_{{min}}={r_min:.3f} \)')
plt.xlabel('r/σ'); plt.ylabel('U/ε'); plt.ylim(-1.5, 3)
plt.title('Lennard-Jones potential'); plt.legend(); plt.grid()
plt.savefig('lj.png', dpi=120)

print(f"Energy minimum at r = {r_min:.4f} σ")

2) Jurin’s law — water rise in capillary tubes

import numpy as np

gamma = 0.072     # N/m (water at 20°C)
theta = 0         # water-glass contact angle
rho = 1000        # kg/m³
g = 9.81

radii_mm = np.array([1.0, 0.5, 0.1, 0.05, 0.01])
r_m = radii_mm * 1e-3

h_mm = (2*gamma*np.cos(theta)) / (rho*g*r_m) * 1000

for r_val, h_val in zip(radii_mm, h_mm):
    print(f"radius = {r_val:.2f} mm  →  rise = {h_val:.1f} mm")
# Output:
# radius = 1.00 mm  →  rise = 14.7 mm
# radius = 0.10 mm  →  rise = 147.0 mm
# radius = 0.01 mm  →  rise = 1470.0 mm !!

Takeaway: a capillary with a 10-µm radius can lift water ~1.5 m — the same mechanism trees use to move water from roots to the tips of their leaves 🌳.

Applications 🏗️

Take-home summary 🎁

Intermolecular force = attraction + repulsion, both short-range. Cohesion binds like-to-like; adhesion binds unlike-to-unlike. Surface tension (\( \gamma \)) and capillary action (Jurin’s law) are the macroscopic footprints of these microscopic forces. The standard quantitative model is the Lennard-Jones potential. And a few lines of Python are enough to predict how high water will climb 😎.

“Nice to know” box 💡

Liquid crystals are a curious in-between state: fluid like a liquid, but partially ordered like a solid. They were discovered by Austrian chemist Friedrich Reinitzer in 1888 and had no industrial use for decades. Today? The screen you’re reading this on is likely an LCD = Liquid Crystal Display. A seemingly useless discovery that changed the world 📱.

Test yourself 📝

References and further exploration 📚

Articles and reference

Videos (YouTube)

External simulators

On this site 🔗

In the next section: pressure in fluids — why your ears hurt at the bottom of a pool, how a submarine survives thousands of meters of ocean depth 🌊. See you there 👋

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