One of the distinguishing features of a lightly cross$-$linked polymer material is its ability to

imbibe and retain a large volume of solvent. Examples include hot melt adhesives, soft contact

lenses, diapers, and biological tissues. Clearly, swollen networks are of fundamental

importance in many areas of materials and biological sciences. In this question, I ask you to

work on two aspects of swelling phenomenon: equilibrium swelling, i$.$e$.$ how much solvent a

network can take up and the elastic modulus of a swollen network.

Consider polyisoprene chains with $M=150,000\frac{g}{m}ol$ with the extent of vulcanization $=0\mathrm{.}003$

which resulted in average of $6\mathrm{.}6$ crosslinks per chain and $M_x=22,000.$ The density of

polyisoprene is $0\mathrm{.}91\frac{g}{m}$ole.

$($a$)(5$ pts$)$ You will now estimate the shear modulus of the swollen network. Suppose a

network formed at dry volume $V_o$ and then swollen with a solvent to a new volume $V.$

The volume fraction of the polymer in the resulting gel is ${}_2=\frac{V_0}{V}.$ Assuming the

swelling is isotropic, the $x,y,z$ components of the end to end vector of each strand is

swollen by $(\frac{V}{V_0}{)}^{\frac{1}{3}}={}_2^{-\frac{1}{3}}.$ Therefore, the extension ratios in the definition of entropy

of a polymer network, that is $S=\frac{-k{v}_e}{2}({}_x^2+{}_y^2+{}_z^2-3),$ are multiplied by a

factor ${}_2^{-\frac{1}{3}}.$ Starting from the resulting entropy change of the swollen network upon

deformation, find the nominal $($engineering$)$ stress$-$extension ratio relationship for a

swollen network.

$($b$)(5$ pts$)$ What is the ratio of the moduli in swollen and dry state $(G\frac{{}_2}{G_{\text{d}\text{r}\text{y}}})?$

$($c$)(5$ pts$)$ Now, we will estimate the equilibrium swelling of the polymer network. Suppose

the polyisoprene network described above is exposed to a reservoir of cyclohexane at

room temperature. According to Flory$-$Rehner theory, equilibrium swelling is reached

when

$ln(1-{}_e)+{}_e+{}_e^2=\frac{v_e}{N_{\text{A}\text{v}\text{g}\text{.}}}\frac{\frac{?}{\text{b}\text{a}\text{r}}(v_1)}{V_o}(\frac{{}_e}{2}-{}_e^{\frac{1}{3}})$

${}_e$ : equilibrium volume fraction of polymer in swollen gel

$$ : polymer$-$solvent interaction parameter

$v_e$ : number of elastically effective strands

$\frac{?}{\text{b}\text{a}\text{r}}(v_1)$: molar volume of the solvent

$V_o$: volume of the dry polymer network

The molar volume of cyclohexane is $108c{m}^3.$ Estimate ${}_e$ and find the resulting volumetric

expansion $(\frac{V}{V_o})$ at swelling equilibrium of the polyisoprene network. Note that the interaction

parameter between poly $($isoprene$)$ and cyclohexane is best described by

$=0\mathrm{.}34+\frac{\frac{?}{\text{b}\text{a}\text{r}}(v{)}_1}{RT}({}_1-{}_2{)}^2.$

$($d$)(5$ pts$)$ Find the shear modulus of the polyisoprene network in cyclohexane at

equilibrium swelling, i$.$e$.{}_2={}_e,$ and in dry form.

$($e$)(5$ pts$)$ According to the Flory$-$Rehner theory, what value of $$ would be required to

get absolutely no uptake of solvent? What value of $$ would restrict the uptake to less

than $5\%?$