LOOK AT THE UPLOADED PICTURE AND SOLVE ALL QUESTIONS $($a to d$)$ UNDERSTANDBLY PLEASE THANK YOU: 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$)(7$ pts$)$ Suppose this polymer network is exposed to a reservoir of cyclohexane at

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

when

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

$.$ Estimate ${}_e$ and find the resulting volumetric

expansion at swelling equilibrium $($defined in part $b).$

$($b$)(8$ 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 ${}_2=\frac{V_o}{V}.$ 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 resuOne 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$=150,000$ g$/$mol with the extent of vulcanization$=0\mathrm{.}003$

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

polyisoprene is $0\mathrm{.}91$ g$/$mole$.$

$($a$)(7$ pts$)$ Suppose this polymer network is exposed to a reservoir of cyclohexane at

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

$($i$)$

v

ln$(1$

$$

$$

$)$

$+$

$$

$2$

$=$

e

e

$+$

$$

ee v

$$

$1($

$$

$$

$1/3$

e

e

$)$

NV

Avg

$.$

o

$2$

$$ e

: equilibrium volume fraction of polymer in swollen gel $$

: polymer$-$solvent interaction parameter

v : number of elastically effective strands e

where

v : molar volume of the solvent

$1$

V

: volume of the dry polymer n

o

etwork

The molar volume of cyclohexane is $108$ cm$3.$ Estimate $$ and find the resulting volumetric

e

expansion at swelling equilibrium $($defined in part b$).$

$($b$)(8$ pts$)$ You will now estimate the shear modulus of the swollen network. Suppose a

network formed at dry volume Vo and then swollen with a solvent to a new volume V$.$ The volume fraction of the polymer in the resulting gel is $=$ Vo $/$V$.$ Assuming the

$2$

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

$$

kv

$,$ are multiplied

$$

S

$=$

e

$($

$$

$222+$

$+$

z

$$

$3)$

x

y

$2$

$-1/3.$ Starting from the resulting entropy change of the swollen network

$$

$2$

$)?$

GG

$($

$$

$)/$

$2$

dry

swelling equilibrium in cyclohexane, i$.$e$.2=$ e$$ and in dry form.

$4$

swollen by $($V$/$ Vo$)1/3=2-1/3.$ Therefore, the extension ratios in the definition of entropy of a polymer network, that is by a factor upon deformation and find the nominal stress$-$extension ratio relationship for a swollen network.

$($c$)(3$ pts$)$ What is the ratio of the moduli in swollen and dry state $($

$($d$)(2$ pts$)$ Find the shear modulus of the polyisoprene network described above at