A model surfactant forms aggregates that are all of one size

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

A model surfactant forms aggregates that are all of one size,

N=40

molecules, in the closed-association model. Each molecule gains

\alpha=10

(that is,

10k_BT

) in standard chemical potential by joining an aggregate rather than remaining a free monomer. The free-monomer mole fraction

X_1

and the mole fraction of amphiphile held in aggregates,

X_N

, satisfy

X_N = N\left(X_1e^{\alpha}\right)^N, \qquad X_{\rm tot} = X_1 + X_N,

where

X_{\rm tot}

is the total amphiphile mole fraction (solute plus aggregated amphiphile, water as solvent). ((a)) Writing

c \equiv X_1e^{\alpha}

and

X^\ast \equiv e^{-\alpha}

, show that the conservation law reduces to a single equation for

c

\;c^N = \dfrac{X^\ast}{N}\left(\dfrac{X_{\rm tot}}{X^\ast} - c\right)

. ((b)) Solve this equation for

c

by bisection, to two significant figures, for

X_{\rm tot} = X^\ast

X_{\rm tot}=2X^\ast

and

X_{\rm tot}=3X^\ast

, and hence give the free-monomer mole fraction

X_1

at each of these three total concentrations. ((c)) State in one sentence what the size of the change in

X_1

across this threefold increase in

X_{\rm tot}

shows about the free monomer once aggregates have begun to form.