In the closed-association model

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

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

Problem

In the closed-association model, the mole fraction of aggregates of size

N

obeys

X_N \propto X_1^N

at fixed

\alpha

, where

X_1

is the free-monomer mole fraction. Suppose

X_1

increases by exactly

1\%

(so

\Delta X_1/X_1 = 0.01

) at fixed

\alpha

. ((a)) Find the resulting fractional increase

\Delta X_N/X_N

for aggregation numbers

N = 5

N = 20

and

N = 60

. ((b)) Explain in one or two sentences why this makes the onset of micellisation appear sharper for surfactants with larger aggregation numbers.