QUESTION $2$: $(20$ pts$)$ Thermodynamics of polymer solutions

$($i$)$ Write down the Generalized Flory$-$Huggins Equation $($GFHE$)$ for binary mixtures,

A and B$,($per site$).$ Define each symbol you use and identify clearly the enthalpy

and the entropy terms.

$($ii$)$ Using GFHE above, find the critical point $({}_c,{}_c)$ in terms of $N_A$ and $N_B.$ Then,

determine the solubility criteria the mixture of two small molecules and for the

mixture of polymer and a solvent.

$($iii$)$ Now repeat part $($ii$)$ for a symmetric blend of two polymer $(N_A=N_B)$ in the limit of

infinite molecular weight. Compare your result with those from $($ii$)$ and discuss the

solubility criteria for this case.

$($iv$)$ Flory$-$Huggins theory predicts the phase diagrams with upper$-$critical$-$solution

temperature $($UCST$)$ only. This is mainly due to the assumptions in the theory, such

as no volume change upon mixing, no composition dependence of $$ and mean$-$field$-$

assumption. In reality, many other types of phase diagrams exist. The diagrams

below show some idealized symmetric binodal curves where $$ is not composition

dependent, i$.$e$.(),$ but $$ varies differently with temperature. Match the phase

diagrams, $T(),$ with corresponding $(T).$ Justify your choices $($Note that A and E

have essentially the same temperature trend$).$

$$

T

$$

T

Polystyrene$-$Polyvinyl

methyl ether

$$

T

Polyethylene

oxide$-$water

$$

Polystyrene$-$

acetone $($low $M_w)$

$$

Polystyrene$-$

acetone $($high $M_w)$Thermodynamics of polymer solutions

$($i$)$ Write down the Generalized Flory$-$Huggins Equation $($GFHE$)$ for binary mixtures,

A and B$,($per site$).$ Define each symbol you use and identify clearly the enthalpy and the entropy terms.

$($ii$)$ Using GFHE above, find the critical point $(\backslash$chi c$,\backslash$Phi c$)$ in terms of NA and NB$.$ Then,

determine the solubility criteria the mixture of two small molecules and for the mixture of polymer and a solvent. $($iii$)$ Now repeat part $($ii$)$ for a symmetric blend of two polymer $($NA$=$NB$)$ in the limit of

infinite molecular weight. Compare your result with those from $($ii$)$ and discuss the solubility criteria for this case. $($iv$)$ Flory$-$Huggins theory predicts the phase diagrams with upper$-$critical$-$solution

temperature $($UCST$)$ only. This is mainly due to the assumptions in the theory, such as no volume change upon mixing, no composition dependence of $\backslash$chi and mean$-$field$-$ assumption. In reality, many other types of phase diagrams exist. The diagrams below show some idealized symmetric binodal curves where $\backslash$chi is not composition dependent, i$.$e$.\backslash$chi $!=\backslash$chi $(\backslash$Phi $),$ but $\backslash$chi varies differently with temperature. Match the phase diagrams, T$(\backslash$Phi $),$ with corresponding $\backslash$chi $($T$).$ Justify your choices $($Note that A and E have essentially the same temperature trend$).$

$1->2->3->4->5->$