Periodic vs random gels

We have a periodic gel structure $($like in $($b$)$ below$)$ made of long fibers of radius $r.$

The distance $2$a between the crosslinks is such that $a>r.$

In fact, $a$ is a good measure of the pore radius in these gels.

$($a$)$ Find the relation between $a()$ and the fractional volume $$ occupied by the gel fibers.

$($b$)$ The Ogston eq$.10\mathrm{.}5$ reads $f(,R)=$exp$(-\frac{}{4}[\frac{R+r}{a()}{]}^2),$ where $f(,R)$ is the fractional volume available to a sphere of radius $R$ in a random gel with a mean pore radius $a.$

$($b$1)$ Explain why $)>(a$ for a periodic gel.

$($b$2)$ If we use the function $a()$ found in $($a$)$ above in the Ogston equation, what is the probability $P(R=2a)$ as a function of $?$ Plot $P(R=2a)$ vs $$ for both types of gel.