## 3 Results  
### Nanoparticle Size and Structure  
The particle diameters were ultrasmall (Figure 2a) and the average particle diameter (d\({}_{\rm avg}\)) was estimated to be 2.0 nm by fitting an observed particle diameter distribution to a log-normal function (Figure 2b and Table 1). The average hydrodynamic diameter (a\({}_{\rm avg}\)) of the nanoparticles dispersed in triple-distilled water (0.01 mM Gd) was estimated to be 12.7 nm by fitting an observed DLS pattern to a log-normal function (Figure 2c and Table 1). An aqueous nanoparticle solution sample is presented in Figure 3a. To confirm the colloidal dispersion of the nanoparticles in aqueous media, laser scattering (i.e., Tyndall effect) was conducted (Figure 3b). Laser scattering was observed only in the nanoparticle solution sample whereas no such scattering was observed in reference triple-distilled water, confirming the colloidal dispersion. In addition, the average zeta potential (\(\xi_{\rm avg}\)) was estimated to be \(-\)28.0 mV in aqueous media (Figure 3c). This high value supported the observed good colloidal stability.  
XRD patterns of the synthesized nanoparticles before and after TGA were recorded (Figure 4). The XRD pattern of the as-synthesized nanoparticles was broad and structureless, indicating that they were poorly crystallized (i.e., amorphous) due to their ultrasmall particle size [20]. However, sharp peaks were obtained after TGA, indicating crystallization and particle size growth of the nanoparticles after TGA, as observed previously [21]. The observed XRD pattern after TGA was in good agreement with that of cubic Gd\({}_{2}\)O\({}_{3}\) (JCPDS card no: 43-1014) and the estimated lattice constant (10.81 A) was comparable to the reported value (10.818 A) [22].  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{d_{avg}}\) (nm)} & \multirow{2}{*}{\(\mathbf{a_{avg}}\) (nm)} & \multicolumn{3}{c}{Surface-Coating Amount} & \multirow{2}{*}{\(\mathbf{\xi_{avg}}\) (mV)} & \multirow{2}{*}{\(\mathbf{\xi_{avg}}\) (mV)} & \multirow{2}{*}{\(\mathbf{\xi_{avg}}\) (mV)} & \multirow{2}{*}{\(\mathbf{\xi_{avg}}\) (mV)} \\  & & & & & & & & \multicolumn{1}{c}{} \\ \cline{3-4} \cline{6-7}  & **p 1** & **\(\sigma\) 2** & & & & **M at 2.0 T** & \\  & & **(wt.\%)** & **(1/nm\({}^{2}\))** & **N\({}_{\text{PAA}}\)3** & & & **Magnetism** & **and 300 K** & **(s\({}^{-1}\)mM\({}^{-1}\))** & **(s\({}^{-1}\)mM\({}^{-1}\))** \\ \hline
2.0 \(\pm\) 0.1 & 12.7 \(\pm\) 0.2 & 63.2 \(\pm\) 0.2 & 0.29 \(\pm\) 0.05 & 4 \(\pm\) 1 & \(-\)28.0 \(\pm\) 0.2 & Paramagnetic & 1.95 \(\pm\) 0.05 & 19.1 \(\pm\) 0.1 & 53.7 \(\pm\) 0.1 \\ \hline \hline \end{tabular} \({}^{1}\) Average surface-coating amount in wt.% of the PASA polymers per nanoparticle. 2 Average grafting density (i.e., average number of polymers coating a nanoparticle unit surface area). 3 Average number of PASA polymers coating a nanoparticle.  
\end{table}
Table 1: Average particle diameter (\(\mathbf{d_{avg}}\)), average hydrodynamic diameter (\(\mathbf{a_{avg}}\)), surface-coating amount (P, \(\mathbf{\sigma}\), \(\mathbf{N_{PAA}}\)), magnetic properties, and water-proton spin relaxivities (\(\mathbf{r_{1}}\), \(\mathbf{r_{2}}\)) of PASA-coated Gd\({}_{2}\)O\({}_{3}\) nanoparticles.  
Figure 3: **(a)** Photograph of an aqueous solution of PASA-coated Gd\({}_{2}\)O\({}_{3}\) nanoparticles. **(b)** Laser scattering (or Tyndall effect) (indicated using an arrow) due to collisions between laser light and nanoparticle colloids in solution whereas no such laser scattering was observed in triple-distilled water. **(c)** Zeta potential curve.  
Figure 2: **(a)** HRTEM image (dotted circles indicate PASA-coated Gd\({}_{2}\)O\({}_{3}\) nanoparticles). **(b)** Log-normal function fit to the particle size distribution to obtain \(\mathbf{d_{avg}}\). **(c)** Log-normal function fit to a DLS pattern to obtain \(\mathbf{a_{avg}}\).