The surface ratio (
\(R_{\text {sur}}\)) was defined as the ratio between the number of surface cellulose chains,
\(n_{\text {sur}}\), and the total number of cellulose chains of the CNCs. The
\(R_{\text {sur}}\) was calculated using WAXS data and Eq.
5, where
\(d_{\left( 110\right) }\, {\text {and}}\, d_{\left( 1\bar{1}0\right) }\) are the plane spacings,
\(L_\text {2}\) and
\(L_\text {1}\) are the height and the width of the crystal, respectively (Habibi et al.
2006; Eyley and Thielemans
2014).
$$\begin{aligned} R_{\text {sur}}=\frac{n_{\text {sur}}}{\sum n}=\ \frac{2\left( \frac{L_1}{d_{\left( \text {110}\right) }}\right) +\ 2\left( \frac{L_2}{d_{\left( 1\bar{1}0\right) }}\right) }{\frac{L_1L_2}{d_{\left( 110\right) }d_{\left( 1\bar{1}0\right) }}} \end{aligned}$$
(5)
The degree of surface substitution (DSS) of DO, DSS
\(_{\text {DO}}\) (Eq.
6), and the sulfate half-ester groups, DSS
\(_{\text {S}}\) (Eq.
7), were determined as being the amount of 2,3-dialdehyde-AGUs,
\(n_{\text {ox}}\), and AGUs with the sulfate half-ester,
\(n_{\text {S}}\), per surface cellulose chains. The degree of surface sites used, DSS
\(_\text {U}\), was defined as the percentage of the surface modified AGUs and the actual surface of the AGUs that was modifiable (Eq.
8). Due to the 180
\(^{\circ }\) angle of the
\(\beta \)-1,4-glycosidic linked glucose units, every other superficial AGU was considered modifiable by oxidation. Thus, the DSS
\(_\text {U}\) was calculated as the number of molecules substituted by the surface molecules modifiable by periodate oxidation. In Eq.
7,
\(\text{c}_{\text S}\) is the sulfur molar concentration from elemental analysis.
$$\begin{aligned} \text {DSS}_{\text {DO}} = \frac{n_{\text {ox}}}{n_{\text {sur}}} = \frac{\text {DO}}{R_\text {sur}} \end{aligned}$$
(6)
$$\begin{aligned} \text {DSS}_{\text {S}} = \frac{n_{\text {S}}}{n_{\text {sur}}} = \frac{c_{\text {S}}}{R_\text {sur}} \end{aligned}$$
(7)
$$\begin{aligned} \text {DSS}_{\text {U}} = \frac{n_{\text {ox}}}{\frac{n_{\text {sur}}}{2}} \end{aligned}$$
(8)