In what follows, the impact of chain length, i, on kp and kt is discussed.
As termination is diffusion-controlled, the impact of chain length is much larger on kt than on kp, which is chemically controlled.
Macroradicals of different size coexist during polymerization and are subjected to propagation and termination. The propagation and termination rate coefficients mentioned so far were chain-length averaged quantities, which should be correctly denoted as <kp> and <kt>.
2.4.1 Chain-length Dependency of Propagation
As described in Section 2.3.1, propagation can be looked upon as a conversion-independent process up to very high conversion. The chain-length dependency of chemically controlled propagation may be interpreted in terms of the Transition State Theory (TST).84–88 A detailed discussion is given in Ref.87 Simulations suggest that, due to the increase in chain length, the internal mobility in the transition state structure is restricted which leads to a decrease in the entropy-driven pre-exponential, A(kp), thus lowering kp. The monomeric kp, kp(1), may exceed the limiting value for long chains, kp, as determined by PLP–SEC, by up to one order of magnitude. The decrease in kp(i) with increasing chain length is restricted to the very initial growth period up to about i = 10. As a consequence, propagation is adequately described by the long-chain value kp and the notation <kp> is omitted. On the basis of experimental studies,87 the following expression is proposed for chain-length dependent propagation (CLDP).
2.4 Chain-length Dependency of Rate Coefficients in CRP
2.4.2 Chain-length Dependency of Termination
Since termination is a diffusion controlled process, it is comes as no surprise that radicals diffuse and hence terminate slower upon increasing chain length. Consequently, the chain-length dependence (CLD) of kt reflects the impact of i on the self-diffusion coefficient, D, i.e., D ~ i−α, which is indeed true for very short radicals where center-of-mass-diffusion is be dominating.89 The exponent α, which is above zero, reflects the extent of chain growth on D. The situation is however more complex in that long-chain radicals exhibit another chain-length dependence, i.e., another power-law exponent.
The Composite Model by Smith, Russell and Heuts
According to Figure 2.4, the SD is the rate determining step in the initial period of the polymerization. The statement is however only true for large radicals. For two radicals at very small size, the TD is the dominating diffusion mode since the entanglement of two radicals which is necessary for SD, requires larger chain lengths. The CLDT of small radicals in the initial polymerization period is therefore expressed by a power-law expression (eq ( 2.28)) which is based on a TD approach with α being identified as the exponent in D ~ i−α The termination rate coefficient of two radicals of identical size is kt(i,i) with kt(1,1) referring to the termination of two monomeric radicals.
i
αAbove a critical chain length, the two growing radicals may entangle upon the formation of an encounter-pair and the SD becomes dominant.
The transition from TD (center-of-mass diffusion, short-chain regime) to CD (long-chain regime) was taken into account by the so-called Composite Model introduced by Smith, Russell and Heuts (eq ( 2.29)).90 According to this fundamental expression, the two regimes are separated by the crossover chain length, ic, and the CLDT is described by two exponents, αs
and αl, for the short-chain and long-chain regime, respectively. The composite-model behavior is widely accepted and so far no exception from
2 Theoretical Background
20
this general type of composite-model behavior has been reported. A comprehensive description of CLDT can be found in Ref.89
c hundred, i.e., for ethyl hexyl methacrylate, ic(EHMA) = (270± 30),91 methyl methacrylate, ic (MMA) = 100,92 and vinyl pivalate, ic (VPi) = 110 ± 30,42 macroradicals induced by the α-methyl group (see Appendices).91
In the short-chain regime, kt(1,1) may accurately be described by the Smoluchowski equation (eq ( 2.30)) with Rc being the chain-length independent capture radius and Pspin being the spin factor.
c holds for the self-diffusion coefficients of monomeric radicals and DA1 = DB1, the following expression is obtained with Pspin = 0.25.78,93 determining the theoretical maximum of kt(1,1) at given temperature and viscosity.
2.4 Chain-length Dependency of Rate Coefficients in CRP
21
1 10 100 1000
108 109
αl
αs
kt(1,1)
k0t
k t(i,i) / L⋅mol−1 ⋅s−1
i ic
Figure 2.5: Chain-length dependency of kt according to the Composite Model with kt(1,1) = 1.0·109 L·mol−1·s−1, αs = 0.60, ic = 30 and αl = 0.16.
η
⋅
= ⋅ ) 3 1 , 1
t(
T
k R ( 2.32)
This expression suggests a fundamental relationship between kt(1,1) and η, i.e, kt(1,1) ∝ η−1, meaning that kt(1,1) scales with fluidity. As a consequence, the activation energies for both quantities should be similar.
Depending on the structure of the macroradicals (random coil vs rod-like coil), αs values are expected to be in the range of 0.5 up to 1.0,41,90,94,95
whereas the theory of coil dynamics predicts an αl for two large macroradicals with the radical functionality at the chain-end to be 0.16.96–
98 In case that the radical functionalities of one or both species are located at a position along the backbone, numbers of αl are predicted to be 0.27 and 0.43, respectively.96 Thus, the CLDT in the long-chain regime will be different for SPR and MCR homo-termination. An increase in αl from 0.16 to 0.27 for ktst (SPR-MCR cross-termination) as well as to 0.43 for kttt (MCR homo-termination) is expected and was verified by investigations of Fröhlich et al.99 The change in kt(i,i) is depicted in Figure 2.5 for typical composite-model parameters.100
2 Theoretical Background
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Termination of Radicals Differing in Size
The Composite Model refers to termination between radicals of identical i. Under stationary conditions, however, radicals with different chain lengths occur. Three models have been proposed to estimate kt(i,j) for the termination of two radicals of sizes i and j: the geometric mean model (gm), the diffusion mean model (dm) and the harmonic mean model (hm).89,101,102 The models differ in the weighting of the chain lengths. For example, the dm model rests on the Smoluchowski equation in that the