
\sect{Adsorbsion and fractal dimension}

%Baruch's  A41

Surfactant Adsorption: A dilute solution of surfactants
can be regarded as an ideal three dimensional gas. As surfactant
molecules can reduce their energy by contact with air, a fraction
of them migrate to the surface where they can be treated as a two
dimensional ideal gas. Surfactants are similarly adsorbed by other
porous media such as polymers and gels with an affinity for them.
\begin{itemize}
\item[(a)] Consider an ideal gas of classical particles of mass $m$ in
$d$ dimensions, moving in a uniform potential of
strength $\epsilon_d$. Show that the chemical potential at a
temperature $T$ and particle density $n_d$, is given by
\[\mu_d=\epsilon_d+k_BT\ln[n_d\lambda^d] \qquad \mbox{where}
\qquad \lambda=\frac{h}{\sqrt{2\pi mk_BT}}\]
\item[(b)] If a surfactant lowers its energy by $\epsilon_0$ in
moving from the solution to the surface, calculate the
concentration of coating surfactants as a function of the solution
concentration $n$ (at $d=3$).
\item[(c)] Gels are formed by
cross-linking linear polymers. It has been suggested that the
porous gel should be regarded as fractal, and the surfactants
adsorbed on its surface treated as a gas in $d_f$ dimensional
space, with a non-integer $d_f$ . Can this assertion be tested by
comparing the relative adsorption of surfactants to a gel, and to
the individual polymers (assuming it is one dimensional) before
cross-linking, as a function of temperature?\\

\end{itemize}

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