# Porous Organic Polymers-Supported Metallocene Catalysts for Ethylene/1-Hexene Copolymerization
## 2 Results and Discussion
### Theoretical Explanation  
The distribution of chain length and chemical composition for ethylene/1-hexene copolymers can be interpreted based on the Stockmayer bivariate distribution [44]. The Stockmayer distribution is given by the following Equation (2):  
\[w_{r,F_{\mathrm{A}}}=r\tau^{2}e^{-r\tau}\sqrt{\frac{r}{2\pi\beta}}e^{-\frac{r( F_{\mathrm{A}}-\tau_{\mathrm{A}})^{2}}{2\beta}} \tag{2}\]  
where the parameters r and \(\tau\) have the same meanings as in the Flory distribution; F\({}_{\mathrm{A}}\) represents the molar fraction of monomer A in a copolymer chain; \(\overline{\mathrm{F_{A}}}\) is the average molar fraction of monomer A in  
Figure 6: Scanning electron micrographs of PE copolymers particles. (**a–c**) PE-1 from POP-1-supported metallocene catalyst; (**d–f**) PE-2 from POP-2-supported metallocene catalyst; (**g–i**) PE-4 from 2408 silica-supported metallocene catalyst.  
the whole copolymer, and its value can also be measured by \({}^{13}\)C NMR and Fourier transform infrared spectroscopy (FTIR).The parameter \(\beta\) is described as following Equation (3):  
\[\beta=\overline{F_{A}}\big{(}1-\overline{F_{A}}\big{)}\sqrt{1-4\overline{F_{A}} \big{(}1-\overline{F_{A}}\big{)}\big{(}1-r_{A}r_{B}\big{)}}. \tag{3}\]  
By integrating Equation (2), the CCD component of the Stockmayer distribution can be changed into Equation (4):  
\[w_{F_{A}}=\frac{3\tau^{2}\beta^{2}}{\Big{[}2\beta\tau+\big{(}F-\overline{F} \big{)}^{2}\Big{]}^{5/2}}=\frac{3}{4\sqrt{2\beta\tau}\bigg{[}1+\frac{\big{(}F- \overline{F}\big{)}^{2}}{2\beta\tau}\bigg{]}^{5/2}}. \tag{4}\]  
[PERSON] and [PERSON] [44] illustrated how the Stockmayer distribution was used to describe the CCD of ethylene/1-hexene copolymers prepared from a metallocene catalyst. The CRYSTAF peaks moved to lower crystallization temperatures and the profiles became broader as 1-hexene content increased. [PERSON]'s distribution can be used successfully to explain this behavior. The Stockmayer parameters \(\tau\) and \(\beta\) were calculated, assuming that the copolymers are random (\(r_{\rm A}r_{\rm B}\approx 1\)). The product \(\beta\tau\) increases with increasing 1-hexene content in the polymer, so the CRYSTAF peaks were expected to become broader. As shown in Figure 7, that results are in good agreement with the experimental data. From Equation (4), we can infer that the CCD becomes broader as the product \(\beta\tau\) increases. As expected from our previous analysis ofthe Stockmayer distribution,\(\tau\) increases when \(r_{\rm n}\) decreases and \(\beta\) increases when \(r_{\rm A}r_{\rm B}\) increases, and both factors broaden the CCD.  
As in this case, through different diffusion rates of ethylene/\(\alpha\)-olefin and TIBA to the active sites controlled by the different pore structure (especially the micro- and the narrow meso-pores) of the POP supports, different reaction micro-environments of the same active species were established in the nanoreactors, which caused the variations of concentration of 1-hexene and of \(\beta\tau\) value, thus leading to the broadening of CCDs as seen from the TREF results in Table 3. Additionally, different CCDs and polydispersity indexes (PDIs) of these PE samples were observed from POP supports with different pore structure. From this perspective, the design and synthesis of well-defined POP-supported pore structure would be practical for controlling the end-use properties (e.g., the CCD and the molecular weight distribution) of the polyolefin.  
Figure 7: (a) CRYSTAF profiles of ethylene/1-hexene copolymers prepared from a single-site catalyst; (b) chemical composition distributions predicted with Stockmayer distribution [44]. CRYSTAF: is a fully automated instrument intended for the fast measurement of the Chemical Composition Distribution (CCD) in Polyolefins, and performs the Crystallization Analysis Fractionation technique to separate the polymer by its comonomer content. HDPE: high density polyethylene.