A phase diagram of polystyrene$-$block$-$polybutadiene copolymer is shown in the following Figure.

$($a$)$ Assume that volume fraction of polystyrene is $f_{PS}0\mathrm{.}45$ and the copolymer shows a lamellar morphology. If $d$ spacing of the plane $($for the first order reflection, $n=1)$ is determined to be $19\mathrm{.}54nm$ from small$-$angle X$-$ray scattering experiment, determine scattering angle $(22)$ of this experiment.

$0\mathrm{.}4526$

$($b$)$ If volume fraction of polystyrene is increased from $0\mathrm{.}45$ to $0\mathrm{.}55,$ determine the associated thickness of the polybutadiene layers.

$8\mathrm{.}79nm$

$($c$)$ Considering copolymers A $($f $fS0\mathrm{.}25,$ red dot$)$ and B $(f_{PS}0\mathrm{.}75,$ blue dot$)$ both with hexagonal morphology, explain how nano$-$morphology of the copolymer $A$ is different from that of the polymer $B.$

Assumption: $(1)$ The small$-$angle X$-$ray scattering experiment is performed by using CuK$$ radiation with a wavelength of $0\mathrm{.}154nm$; $(2)$ If $y=sin(x),$ then $x=si{n}^{-1}(y).$