Lesson 7 · 30 min

Surfactants and micelles

Soap and detergent molecules have a water-loving head and a water-shunning tail: they cover the surface first, then huddle into micelles above the critical micelle concentration. With the measurements compiled by the US National Bureau of Standards in 1971, see the CMC halve with each CH₂ and fall with added salt.

In this lesson

Describe the two-faced structure of a surfactant, and explain why it first occupies the water surface and then, above the critical micelle concentration (CMC), gathers into micelles
Use the pseudo-phase model to compute how much surfactant is free and how much is in micelles above the CMC
Predict from measured data how the CMC changes with chain length (Klevens’ rule) and with added salt (Corrin–Harkins)

Shampoo, toothpaste and washing-up liquid all contain the same kind of molecule: at one end an ionic or polar group that likes water (the “head”), at the other a hydrocarbon chain that shuns it (the “tail”) — a surfactant. The sodium dodecyl sulfate (SDS) common in toothpaste is C₁₂H₂₅OSO₃⁻ Na⁺.

Add it to water little by little. At first the molecules crowd to the surface, tails poking into the air, and the surface tension falls. At a fairly sharp concentration the surface is full, and further molecules suddenly start huddling together in the water — micelles: dozens of molecules hide their tails in the middle and turn their heads out to the water. That concentration is the critical micelle concentration, about 8.2 mmol/L for SDS at 25 °C.

Predict first

Keep adding SDS to water past 8.2 mM. What happens to the concentration of free SDS molecules — those not in micelles?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

It keeps rising in proportion to the total It stays at about 8 mM It drops to zero It doubles

Why they huddle together

The water molecules around a hydrocarbon chain cannot hydrogen-bond freely as they do in pure water; they line up more “neatly” around the chain, at a cost in entropy. Pack dozens of chains into an oily core and the bound water is set free — at room temperature the driving force for micelles is mainly this gain in entropy, often called the hydrophobic effect.

On the other side, ionic heads carry like charges and repel one another when crowded. Part of the counter-ions (Na⁺ for SDS) cling to the micelle surface and cancel most of its charge, which lets the micelle hold together. So the CMC is sensitive to two things: how long the tail is, and how many counter-ions the solution holds.

c_1 = \min(c,\ \mathrm{CMC}), \qquad c_\mathrm{m} = \max(0,\ c - \mathrm{CMC})

The pseudo-phase model treats micelles as a separate “phase”: below the CMC everything is monomer; above it the monomer concentration is fixed at the CMC and the excess all goes into micelles.

$c$: total surfactant concentration
$c_1$: concentration of free monomers
$c_\mathrm{m}$: concentration of surfactant in micelles
$\mathrm{CMC}$: critical micelle concentration

Measured CMCs from the NBS compilation · the pseudo-phase model

How micelles form

Total 16.3 mM (2.00 × CMC) Molecules per micelle N = 60 (set by hand)

Surfactant in solution

Concentration as monomers and in micelles

monomers
in micelles

Monomers 8.16 mM, in micelles 8.16 mM, micelle concentration 136.0 μM.

Above the CMC: almost every further molecule joins a micelle, so the monomer concentration — and with it the surface tension and osmotic pressure — hardly changes.

At twice the CMC, half the SDS is free and half is in micelles. Switch to the shorter sodium octyl sulfate (C8) and micelles appear only at 130 mM; with sodium tetradecyl sulfate (C14), 2 mM is enough.

Measuring the CMC

Every property that depends on the monomer concentration breaks at the CMC: surface tension, conductivity, osmotic pressure, light scattering, the amount of dye the solution can dissolve… The US National Bureau of Standards’ 1971 compilation (NSRDS-NBS 36) lists 558 measurements for SDS alone and rates each one: the best-rated cluster at 8.1–8.4 mM at 25 °C, while lower-rated methods (surface tension read off a logarithmic plot, for example) scatter far more. The same quantity, measured by different methods in different laboratories, checked against itself — that is what data evaluation is for.

Chain length and salt

\lg \mathrm{CMC} = A - B\,n

Klevens’ rule: within a homologous series, lg CMC falls linearly with the number of carbons n in the chain. For sodium alkyl sulfates B = 0.30: each extra CH₂ halves the CMC.

$n$: carbon atoms in the alkyl chain
$A,\ B$: constants fitted to measurements

Measurements compiled in NSRDS-NBS 36 · lines are least-squares fits

Two rules for the CMC

Carbons in the chain n = 12

lg CMC of sodium alkyl sulfates against n (25 °C)

lg CMC = 1.520 − 0.3006 n (r² = 0.999999): each CH₂ divides the CMC by 2.00. For n = 12 the predicted CMC is 8.18 mM.

Moving one CH₂ from water into the micelle core: ΔG ≈ −(1 + β)RT ln10 · 0.301 = −2.93 kJ/mol (β = 0.70 from the salt data).

The four measurements for C8, C10, C12 and C14 fall almost exactly on a line (r² > 0.99999). Turned into energy, the slope says that moving one CH₂ from water into the micelle core releases about 2.9 kJ/mol — the same size as moving it from water into a liquid hydrocarbon.

Worked example

Predicting the CMC of SDS from its neighbours

At 25 °C sodium decyl sulfate (C10) has a CMC of 32.7 mM. Use “each CH₂ halves the CMC” to estimate the CMC of sodium dodecyl sulfate (C12), and compare it with the measured 8.16 mM.

Try the calculation

Sodium hexadecyl sulfate crystallises out at 25 °C before reaching its CMC. If it stayed dissolved, what would Klevens’ rule (lg CMC = 1.520 − 0.3006 n, CMC in mol/L) give for its CMC, in mM?

Add NaCl and the extra Na⁺ screens the negative charge of the micelle surface better, weakening the repulsion between heads and making micelles easier to form: in 1 mol/L NaCl the CMC of sodium decyl sulfate is less than a tenth of its value in pure water. Corrin and Harkins (1947) found that lg CMC against lg(CMC + c_salt) is a straight line whose slope is the fraction β of counter-ions bound to the micelle.

Measurements compiled in NSRDS-NBS 36 · lines are least-squares fits

Two rules for the CMC

Added NaCl 0.10 mol/L

Sodium decyl sulfate: lg CMC against lg(CMC + c_NaCl)

lg CMC = −2.504 − 0.705 lg(CMC + c_NaCl) (r² = 0.9941). With 0.10 mol/L NaCl the predicted CMC is 14.4 mM (32.7 mM measured without salt).

The seven measurements give β ≈ 0.70: bound Na⁺ cancels about seven tenths of the micelle surface charge. Drag the salt concentration and watch the CMC predicted by the line fall.

Concept check

Why does salt lower the CMC of an ionic surfactant such as SDS markedly, while hardly affecting a nonionic surfactant?

Salt screens the electrostatic repulsion between ionic heads; nonionic heads have no such repulsion to screen Salt reacts chemically with the hydrocarbon chains Salt raises the boiling point of water Salt turns the micelles into solids

Key takeaways

Surfactants have a water-loving head and a water-shunning tail: they first gather at the surface and, at the CMC, assemble into micelles; SDS has a CMC of about 8.2 mM at 25 °C.
Pseudo-phase model: above the CMC the monomer concentration stays at the CMC and the excess enters micelles, so surface tension, conductivity and other properties break there.
Klevens’ rule: each extra CH₂ halves the CMC of sodium alkyl sulfates (about −2.9 kJ/mol); salt lowers the CMC, and the Corrin–Harkins slope shows about 70 % of the counter-ions bound.

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Data and references

P. Mukerjee & K. J. Mysels, Critical Micelle Concentrations of Aqueous Surfactant Systems, NSRDS-NBS 36 (National Bureau of Standards, 1971).
B. D. Flockhart, J. Colloid Sci. 16, 484 (1961); E. D. Goddard & G. C. Benson, Can. J. Chem. 35, 986 (1957); K. J. Mysels & P. Kapauan, J. Colloid Sci. 16, 481 (1961); E. K. Mysels & K. J. Mysels, J. Colloid Sci. 20, 315 (1965).
H. B. Klevens, “Structure and aggregation in dilute solution of surface active agents”, J. Am. Oil Chem. Soc. 30, 74–80 (1953).
M. L. Corrin & W. D. Harkins, “The effect of salts on the critical concentration for the formation of micelles in colloidal electrolytes”, J. Am. Chem. Soc. 69, 683–688 (1947).