# Explaining differences in rate of propagation based on monomer structure

The values given below have been taken from my lecture hand out, and they weren't cited so I don't know where my instructor got them from

The $k_\mathrm{p}$ (rate constant for propagation in radical polymerisation ) step for the following monomers is given below:

$$\begin{array}{cc} \hline \text{Monomer} & k_\mathrm{p}~/~\mathrm{mol^{-1}~s^{-1}} \\ \hline \text{Styrene} & 106 \\ \text{Vinyl ester} & 3000\mathrm{-}4000 \\ \text{Ethylene} & 16 \\ \hline \end{array}$$

And the instructor explained the difference between the first two as follows:

the benzyl radical is stabilised by conjugation, however the radical produced in a vinyl ester lacks stabilisation by conjugation and thus reacts more rapidly.

Seems like a fair explanation, however in the same table the $k_\mathrm{p}$ for ethylene is given as $16~\mathrm{mol^{-1} s^{-1}}$, the slowest propagation step of the three. Thus his explanation is not consistent with this observation I believe, because similar to the vinyl ester radical this one too is not stabilised by conjugation and should be similarly eager to react.

My instructor did not offer a satisfactory explanation and I was wondering if people could weigh in on this here.

• Perhaps its due to the C-H bond energy in ethylene being quite a bit larger than a C-O bond and possibly a C-H in the styrene? – porphyrin Oct 20 '16 at 14:11
• I feel like I remember a radical $\mathrm{\alpha}$ to a carbonyl as being moderately stabilized via conjugation even though it is destabilized due to the electron withdrawing effect from the carbonyl... – Zhe Oct 20 '16 at 21:49

The argument presented by your instructor is based on one factor: the stability of the radical terminus of the polymer. But for bimolecular polymerization reactions we need to consider 4 things:

2. Stability of the monomer
3. Stability of the transition state
4. The probability of the radical and the monomers being correctly oriented

The first three items affect the activation energy ($E_\mathrm{a}$), while the final item affects the frequency factor ($A$), or pre-exponential factor in the Arrhenius equation:

$$k = A \exp{\left(\frac{-E_\mathrm{a}}{RT}\right)}$$

For styrene, vinyl acetate (a vinyl ester), and ethylene, I have found the following values for radical polymerization processes.[1,2] These have been calculated to provide an apples-to-apples comparison:

$$\begin{array}{cc} \hline \text{Monomer} & E_\mathrm{a}~/~\mathrm{kJ~mol^{-1}} & A~/~\mathrm{L~mol^{-1}~s^{-1}} & T~/~^\circ\mathrm{C} & k_\mathrm{p}~/~\mathrm{L ~mol^{-1}~s^{-1}} \\ \hline \text{Styrene} & 32.5 & 4.3 \times 10^7 & 60 & 341 \\ \text{Vinyl acetate} & 20.4 & 1.5 \times 10^7 & 60 & 9460 \\ \text{Ethylene} & 27.7\mathrm{-}32.8 & 2.0 \times 10^3 \mathrm{-} 3.0 \times 10^4 & 60 & 0.1\mathrm{-}0.2 \\ \hline \end{array}$$

Note:

"the propagation rate coefficient for most monomers is strongly pressure-dependent with large negative activation volumes; thus an increase in pressure leads to an increased $k_\mathrm{p}$ value."[1]

This is particularly relevant for the ethylene example, which is done at $100\ \mathrm{bar}$. At $60\ ^\circ\mathrm{C}$, this should be about $0.15\ \mathrm{g~mL^{-1}}$,[3] which is a factor of 7 less dense than the styrene and vinyl acetate systems.

With those caveats noted, the argument of the vinyl ester having a lower activation energy ($E_\mathrm{a}$) due to the reactants being less stable relative to the transition state seems valid. However, the data for ethylene shows that the pre-exponential factor ($A$) is likely responsible for the lesser rate of propagation displayed by ethylene.

Sources:

1. Krzysztof Matyjaszewski & Thomas P. Davis, Handbook of Radical Polymerization, 2003, p 199.
2. Etienne Grau, Jean-Pierre Broyer, Christophe Boisson, Roger Spitz, and Vincent Monteil, Macromolecules 2009, 42, 7279–7281. DOI: 10.1021/ma901622u
3. Thomas, W. & Zander, M. Int J Thermophys (1980) 1: 383. DOI: 10.1007/BF00516565
• Thanks for the great answer, and the references were especially helpful :) – getafix Oct 26 '16 at 6:48