A solution of sodium tetradecyl sulfate ( ) in water at is prepared at

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Thermodynamic potentials and the formal structure of thermodynamics

Soft Matter at the Thermal Scale, Ch Amphiphiles and self-assembly

Problem

A solution of sodium tetradecyl sulfate (

C_{14}

) in water at

298\,\mathrm{K}

is prepared at

c = 1.0\,\mathrm{mM}

, well below its CMC of

2.1\,\mathrm{mM}

, so essentially all the surfactant is present as free monomers. ((a)) Each dissolved molecule dissociates into a free sodium ion and a free tetradecyl sulfate ion. Treating these ions as a dilute solution of point-like solutes, use the ideal dilute-solution limit for osmotic pressure,

\Pi = nk_BT

with

n

the number density of solute particles, to calculate

\Pi

for this solution. ((b)) A homologue, sodium hexadecyl sulfate (

C_{16}

, two more tail carbons), is then made. Using the empirical rule that

\log_{10}

CMC falls by about

0.3

per added carbon for sodium alkyl sulfates, estimate its CMC. ((c)) Above its CMC, this

C_{16}

surfactant forms short rod-like micelles of diameter

D = 4.5\,\mathrm{nm}

and length

L = 13.5\,\mathrm{nm}

. Using the Onsager excluded-volume estimate for rigid rods,

\phi_{\rm I} \approx 3.29\,D/L

, and given that these micelles pack into a hexagonal phase near

\phi \approx 0.34

, compare the formal estimate with the hexagonal onset, and say whether that comparison can decide if a nematic phase appears before hexagonal packing sets in.

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