Ethylene homo- and copolymerization chain-transfers: A perspective from supported (ⁿBuCp) ₂ ZrCl ₂ catalyst active centre distribution

Abstract

Polymerization chain termination reactions and unsaturation of the polymer backbone end are related. Therefore, in this study, the parameters resulting from the modelling of the active centre distribution of the supported catalyst—silica/MAO/(ⁿBuCp)₂ZrCl₂—were applied to evaluate the active-centre-dependent ethylene homo- and copolymerization rates, as well as the corresponding chain termination rates. This approach, from a microkinetic mechanistic viewpoint, elucidates better the 1-hexene-induced positive comonomer effect and chain transfer phenomenon. The kinetic expressions, developed on the basis of the proposed polymerization mechanisms, illustrate how the active site type-dependent chain transfer phenomenon is influenced by the different apparent termination rate constants and momoner concentrations. The active centre-specific molecular weight M _ni (for the above homo- and copolymer), as a function of chain transfer probability, $p_{CT_{i}}$, varied as follows: $log\left ({p_{CT_{i}} } \right )=log\left ({mw_{ru}} \right )-log\left ({M_{ni}} \right )$, where mw _ru is the molecular weight of the repeat unit. The physical significance of this finding has been explained. The homo- and copolymer backbones showed all the three chain end unsaturations (vinyl, vinylidene, and trans-vinylene). The postulated polymerization mechanisms reveal the underlying polymer chemistry. The results of the present study will contribute to develop in future supported metallocene catalysts that will be useful to synthesize polyethylene precursors having varying chain end unsaturations, which can be eventually used to prepare functional polyethylenes.

Appendix A: Kinetic Derivation

We begin the kinetic derivation by considering the fact that the MAO cocatalyst first monomethylates one of the chloride ligands attached to transition metal (Zr). Next, it abstracts the chloride ligand to generate the active metallocenium cation (Zr⁺).[17,19,30–37]See Equation A1. Reversible complex formation[11,38–41]with the transition metal active site, as per the trigger mechanism of Ystenes,[42] is a pre-requisite to propagation. See Equations A2 and A3. As per Ystenes, the coordination site is never free; it is always occupied by a monomer. This complexed monomer gets inserted into the growing polymer chain as soon as another incoming monomer is ready to complex through expansion of the coordination sphere. This associates two monomers with the active centre in the form of a transition state.[37] Equations A4 to A7 represent the post-complexation four-step propagation reactions.Activation:

$$ Zr+MAO\xrightarrow{k_{i}} Zr^{+}\left({metallocenium\;cation} \right)+MAO^{-} $$

(A1)

Reversible complex formation:

$$ Zr^{+}+M_{1}\;{{\rightleftharpoons}_{k_{r1}}^{k_{f1}}}\; Zr^{+}\mathellipsis M_{1} $$

(A2)

$$ Zr^{+}+M_{2} \;{{\rightleftharpoons}_{k_{r2}}^{k_{f2}}}\;Zr^{+}\mathellipsis M_{2} $$

(A3)

Propagation:

$$ P_{n}^{M_{1}} -Zr^{+}+M_{1} \xrightarrow{k_{p11}} P_{n+1}^{M_{1}} -Zr^{+} $$

(A4)

$$ P_{n}^{M_{1}} -Zr^{+}+M_{2} \xrightarrow {k_{p12}} P_{n}^{M_{1}} -M_{2} -Zr^{+} $$

(A5)

$$ P_{n}^{M_{2}} -Zr^{+}+M_{2} \xrightarrow {k_{p22}} P_{n+1}^{M_{2}} -Zr^{+} $$

(A6)

$$ P_{n}^{M2} -Zr^{+}+M_{1} \xrightarrow {k_{p21}} P_{n}^{M_{2}} -M_{1} -Zr^{+} $$

(A7)

The effective ethylene homopolymerization chain transfer reactions, according to Schemes I to III (developed as per the backbone end saturations, determined in this study, by FTIR spectroscopy), and citation in the literature,[2,4–10]can be written as follows:

$$\begin{array}{@{}rcl@{}} P_{n}^{M_{1}}{}&-&{}Zr^{+}+M_{1} \left({CH_{2} =CH_{2}} \right)\;^{{\underrightarrow {k_{tr,\; \beta -H\to M_{1}}^{vinyl}}}}\; \overset{\equiv}{P}_{n,vinyl} \\ &&+Zr^{+}-CH_{2} -CH_{3} \end{array} $$

(A8)

$$ P_{n}^{M_{1}} -Zr^{+}\;{^{{\underrightarrow {k_{tr,\; \beta -H\to Zr^{+}}^{vinyl}}}}}\; \overset{\equiv}{P}_{n,vinyl} +Zr^{+}-H $$

(A9)

$$ P_{n}^{M_{1}} -Zr^{+}\;{^{{\underrightarrow {k_{tr1,\; \beta -H\to Zr^{+}}^{t-vinylene}}}}}\; \overset{\equiv}{P}_{n,t-vinylene} +Zr^{+}-H $$

(A10)

$$\begin{array}{@{}rcl@{}} &&P_{n}^{M_{1}} -Zr^{+}\,+\,M_{3} \left({macromer} \right)\,{^{{\underrightarrow {k_{tr,\, \beta -H\to M_{3}~}^{vinyledene}}}}}\; \overset{\equiv}{P}_{n,vinyledene}\\ && +Zr^{+}-H \end{array} $$

(A11)

Similarly, the effective ethylene-1-hexene copolymerization chain transfer reactions, according to Scheme IV (developed as per the backbone end saturations, determined in this study, by FTIR spectroscopy), and citation in the literature,[43,44]can be listed as:

$$\begin{array}{@{}rcl@{}} &&P_{n}^{M_{1}} -Zr^{+}+M_{2} (CH_{2} =CH-Bu)\\ &&{~}^{\underrightarrow{k_{tr,\; \beta -H\to M_{2}}^{vinyl} +({Route\;A} )}}\; \overset{\equiv}{{P}_{n,vinyl}} +Zr^{+}-H \end{array} $$

(A12)

$$\begin{array}{@{}rcl@{}} &&P_{n}^{M_{1}} -Zr^{+}+M_{2} ({CH_{2} =CH-Bu})\\ &&{~}^{\underrightarrow{k_{tr2,\; \beta -H\to Zr^{+}}^{t-vinylene}({Route\;B} )}}\; \overset{\equiv}{{P}_{n,t-vinylene}} +Zr^{+}-H \end{array} $$

(A13)

$$\begin{array}{@{}rcl@{}} &&P_{n}^{M_{1}} -Zr^{+}+M_{2} \left({CH_{2} =CH-Bu} \right)\\ &&{~}^{{\underrightarrow{k_{tr,\; \beta -H\to Zr{^{+}}}^{vinyledene} ({Route\;C} )}}} \overset{\equiv}{{P}_{n,vinyledene}} +Zr^{+}-H\\ \end{array} $$

(A14)

From Equations A2 and A3, as per steady state assumption, we can write the following:

$$ \frac{d\left[ {M_{1} -Zr^{+}} \right]}{dt}=k_{f_{1}} \left[ {Zr^{+}} \right]\left[ {M_{1}} \right]-k_{r_{1}} \left[ {M_{1} -Zr^{+}} \right]=0 $$

(A15)

$$ \frac{d\left[ {M_{2} -Zr^{+}} \right]}{dt}=k_{f_{2}} \left[ {Zr^{+}} \right]\left[ {M_{2}} \right]-k_{r_{2}} \left[ {M_{2} -Zr^{+}} \right]=0 $$

(A16)

According to long chain hypothesis, we next derive the following:

$$\begin{array}{@{}rcl@{}} &&\left[ {M_{1} -Zr^{+}} \right]=\left[ {P_{n}^{M_{1}} -Zr^{+}} \right]=\frac{k_{f_{1}} } {k_{r_{1}} } \left[ {Zr^{+}} \right]\left[ {M_{1}} \right]\\ &&\hspace*{0.8pc}=k_{1} \left[ {Zr^{+}} \right]\left[ {M_{1}} \right];\quad k_{1} =\frac{k_{f_{1}} } {k_{r_{1}} } \end{array} $$

(A17)

$$\begin{array}{@{}rcl@{}} &&\left[ {M_{2} -Zr^{+}} \right]=\left[ {P_{n}^{M_{2}} -Zr^{+}} \right]=\frac{k_{f_{2}} } {k_{r_{2}} } \left[ {Zr^{+}} \right]\left[ {M_{2}} \right]\\ &&\hspace*{0.8pc}=k_{2} \left[ {Zr^{+}} \right]\left[ {M_{2}} \right];\quad k_{2} =\frac{k_{f_{2}} } {k_{r_{2}} } \end{array} $$

(A18)

At small time scale, the total concentration of active sites is constant. Therefore, we can write:

$$ C_{t} =\left[ {Zr^{+}} \right]+\left[ {P_{n}^{M_{1}} -Zr^{+}} \right]+\left[ {P_{n}^{M_{2}} -Zr^{+}} \right] $$

(A19)

Using Equations A17 to A19, we can write the following:

$$ \left[ {Zr^{+}} \right]\left({uncomplexed} \right)=\frac{1}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2} } \right]}\times C_{t} $$

(A20)

$$\begin{array}{@{}rcl@{}} \left[ {P_{n}^{M_{1}} -Zr^{+}} \right]\left({complexed} \right)&=&\frac{k_{1} \left[ {M_{1}} \right]}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\\ &&\times C_{t} \end{array} $$

(A21)

$$\begin{array}{@{}rcl@{}} \left[ {P_{n}^{M_{2}} -Zr^{+}} \right]\left({complexed} \right)&=&\frac{k_{2} \left[ {M_{2}} \right]}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\\ &&\times C_{t} \end{array} $$

(A22)

Now, we deduce the desired kinetic rate expressions for ethylene homopolymerization. Using Equations A4 and A22, we can write the following expressions:

$$ C_{t,homopolym}=\left[ {Zr^{+}} \right]+\left[ {P_{n}^{M_{1}} -Zr^{+}} \right];\;\left[ {M_{2}} \right]=0 $$

(A23)

$$\begin{array}{@{}rcl@{}} &&R_{p,homopolym} =-\frac{d\left[ {M_{1}} \right]}{dt}=k_{p11} \left[ {P_{n}^{M_{1}} -Zr^{+}} \right]\left[ {M_{1}}\right]\\ &&\hspace*{3.7pc}=\frac{k_{p11} k_{1} \left[ {M_{1}} \right]^{2}}{1+k_{1} \left[ {M_{1}} \right]}\times C_{t,homopolym} \end{array} $$

(A24)

Considering Equations A8 to A11, and A21, we write ethylene homopolymerization termination rate as follows:

$$\begin{array}{@{}rcl@{}} R_{tr,homopolym}&=&\left(k_{tr,\; \beta -H\to M_{1}}^{vinyl} \left[ {M_{1}} \right]+k_{tr,\; \beta -H\to Zr^{+}}^{vinyl}\right.\\ &&\left.~+k_{tr1,\; \beta -H\to Zr^{+}}^{t-vinylene} +k_{tr,\; \beta -H\to M_{3}}^{vinyledene}\left[ {M_{3}} \right] \right)\\ &&\times \frac{k_{1} \left[ {M_{1}} \right]}{1+k_{1} \left[ {M_{1}} \right]}\times C_{t,homopolym} \end{array} $$

(A25)

Using $\frac {mw_{ru}} {M_{n,homopolym}} =\frac {R_{tr,homopolym}} {R_{p,homopolym}} =\frac {1}{l_{n,homopolym}}$, and Equations A23 and A24, we write the following final expression:

$$\begin{array}{@{}rcl@{}} \frac{1}{l_{n,homopolym}}&=&\frac{k_{tr,\;\; \beta -H\to M_{1}}^{vinyl}} {k_{p11}} \\ &&+\frac{k_{tr,\; \beta -H\to Zr^{+}}^{vinyl} +k_{tr1,\; \beta -H\to Zr^{+}}^{t-vinylene}} {k_{p11}}\\ &&\times \frac{1}{\left[ {M_{1}}\right]}+ \frac{k_{tr,\; \beta -H\to M_{3}}^{vinyledene}} {k_{p11}} \times \frac{\left[ {M_{3}} \right]}{\left[{M_{1}} \right]}\\ \end{array} $$

(A26)

Next, we develop the final expression for $\frac {1}{l_{n,copolym}}$. In this regard, considering Equations A4 to A7, we write the copolymerization rates for M₁ and M₂ as follows:

$$\begin{array}{@{}rcl@{}} R_{p_{1}}&=&-\frac{d\left[ {M_{1}} \right]}{dt}=k_{p11} \left[ {P_{n}^{M_{1}} -Zr^{+}} \right]\left[ {M_{1}} \right]\\ &&+k_{p21} \left[ {P_{n}^{M_{2}} -Zr^{+}} \right]\left[ {M_{1}} \right] \end{array} $$

(A27)

$$\begin{array}{@{}rcl@{}} R_{p2}&=&-\frac{d\left[ {M_{2}} \right]}{dt}=k_{p22} \left[ {P_{n}^{M_{2}} -Zr^{+}} \right]\left[ {M_{2}} \right]\\ &&+k_{p12} \left[ {P_{n}^{M_{1}} -Zr^{+}} \right]\left[ {M_{2}} \right] \end{array} $$

(A28)

Using Equations A21 and A22, we write the above equations, respectively, as follows:

$$ R_{p_{1}} =\frac{k_{p11} k_{1} \left[ {M_{1}} \right]^{2}+k_{p21} k_{2} \left[ {M_{1}} \right]\left[ {M_{2}} \right]}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\times C_{t} $$

(A29)

$$ R_{p_{2}} =\frac{k_{p22}\,k_{2} \left[ {M_{2}} \right]^{2}+k_{p12}\, k_{1} \left[ {M_{1}} \right]\left[ {M_{2}} \right]}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\times C_{t} $$

(A30)

Eqs. A29 and A30 lead to the following:

$$ R_{p,copolym} =\frac{k_{p11} \,k_{1} \left[ {M_{1}} \right]^{2}+\left({k_{p21} \,k_{2} +k_{p12} \,k_{1}} \right)\left[ {M_{1}} \right]\left[ {M_{2}} \right]+k_{p22} \,k_{2} \left[ {M_{2}} \right]^{2}\;}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\times C_{t} $$

(A31)

Applying Equations A12 to A14, and A21, we can write the overall copolymerization termination rate as follows:

$$\begin{array}{@{}rcl@{}} R_{tr,copolym}&=&\left(k_{tr,\; \beta -H\to M_{2}}^{vinyl} +k_{tr2,\; \beta -H\to Zr^{+}}^{t-vinylene} \right.\\ &&\left.+k_{tr,\; \beta -H\to Zr^{+}}^{vinyledene} \right)\\ &&\times\frac{k_{1} \left[ {M_{1}} \right]\left[ {M_{2}} \right]}{1+k_{1} \left[ {M_{1}} \right]+k_{2} \left[ {M_{2}} \right]}\times C_{t} \end{array} $$

(A32)

Using Equations A29 and A30, we finally write:

$$ \frac{1}{l_{n,copolym}} =\frac{\left({k_{tr,\; \beta -H\to M_{2}}^{vinyl} +k_{tr2,\; \beta -H\to Zr^{+}}^{t-vinylene} +k_{tr,\; \beta -H\to Zr^{+}}^{vinyledene}} \right)k_{1}} {\left({k_{p12} \,k_{1}} \right)r_{1} \frac{\left[ {M_{1}} \right]}{\left[ {M_{2}} \right]}+\left({\frac{k_{p11} \,k_{1}} {r_{1}} +\frac{k_{p22} \,k_{2}} {r_{2}} } \right)+\left({k_{p21} \,k_{2}} \right)r_{2} \frac{\left[ {M_{2}} \right]}{\left[ {M_{1}} \right]}} $$

(A33)

where $r_{1} =\frac {k_{p11}}{k_{p12}}$ and $r_{2} =\frac {k_{p22}}{k_{p21}}$ are the reactivity ratios of M₁ and M₂, respectively.

Note that under the present situation, the concentration of a given active site type is not known. Therefore, all the aforesaid rate constants should be considered to be apparent rate constants.