QUESTION $3$: $(30$ pts$)$ Conformation in an Ionic Liquid

About the polymer: PEO is hydrophilic, biocompatible and easy to crosslink to form gels with

tunable mechanical properties suitable for many biomedical applications such as drug release.

In addition, PEO is one of the most widely studied polymers for ion conducting membranes in

lithium$-$ion batteries due its ability to form complexes with a variety of salts. The glass

transition temperature of PEO is $T_g$~~$213K.$ Due to the lack of bulky groups on the chain

backbone, PEO crystallizes below $T_m$~~$340K$ to adopt a semi$-$crystalline phase $($coexisting

mixture of amorphous and crystalline phases$).$ The molecular structure of PEO $(C_2{H}_{4}O{)}_n$ is

given below:

PEO in ionic liquid: In their work published in Macromolecules, $2017,50(21),$ pp $8739-$

$8744,$ Kharel and Lodge investigate the conformation of PEO in an ionic liquid as the chain

conformation is directly related to the polymer's conducting properties. Please read the

background information below.

'Ionic liquids exhibit negligible vapor pressure, high thermal and chemical stability, and high

ionic conductivity, which make them attractive for use in a variety of applications including

fuel cells and electrochemical devices. One essential property that ionic liquids lack is

mechanical integrity, which can instead be provided by the addition of suitable polymer

architectures. However, optimization of these composite materials requires a thorough

understanding of the behavior of polymers in ILs.

Among numerous combinations of polymers and ILs, the use of poly$($ethylene oxide$)($PEO$)$ is

particularly attractive from both fundamental and technological perspectives. For example,

polymer gel electrolytes based on PEO and ILs are being investigated as replacements for

current battery technologies that otherwise employ flammable and volatile organic solvents."

From Kharel and Lodge, Macromolecules, $2017,50(21),$ pp $8739-8744$

They investigated the chain dimensions of PEO for various molecular weights in in the ionic

liquid $1-$butyl$-3-$methylimidazolium tetrafluoroborate $([$BMIM$][$BF$4])$ at $80C$ using small

angle neutron scattering $($SANS$)$ which simply measures the radius of gyration of the polymer

$($we will learn this technique later$).$

$($a$)(7$ pts$)$ Estimate the radius gyration of PEO in a theta solvent and fill in the table above.

$($b$)(7$ pts$)$ The experiments are to be performed in the dilute polymer regime. Authors

specifically chose this concentration to be the half of the overlapping concentration $(c^{**})$

for each molecular weight. Calculate $c^{**}$ values $($in $m\frac{g}{m}l)$ and fill in the table.

$($c$)(8$ pts$)$ The last column of the table displays the measured $R_g$ values of PEO in the ionic

liquid at $80C.$ Find the corresponding scaling exponent for $R_g{N}^v$ for PEO in ionic

liquid. How does it compare with the scaling in theta solvent. Comment on the solvent

quality of ionic liquid for PEO.

$($d$)(8$ pts$)$ Define chain expansion coefficient. Based on your finding in $($c$),$ what is the

scaling relationship for the chain expansion coefficient and the degree of polymerization

for PEO in the ionic liquid $($i$.$e$.$ what is $p$ in $N^p)$