Chain-length dependence

Summarizing, one indeed observes a clear qualitative tendency of the reduction of the microcanonical effects for larger chains. The rapid decreases of latent heat and surface entropy qualitatively indicate that the adsorption transition of expanded polymers (DE to AET) crosses over from bimodal first-order-like behavior toward a second-order phase transition in the thermodynamic limit, 

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As with the rate of polymerization, we see from Eq. (6.37) that the kinetic chain length depends on the monomer and initiator concentrations and on the constants for the three different kinds of kinetic processes that constitute the mechanism. When the initial monomer and initiator concentrations are used, Eq. (6.37) describes the initial polymer formed. The initial degree of polymerization is a measurable quantity, so Eq. (6.37) provides a second functional relationship, different from Eq. (6.26), between experimentally available quantities-n, [M], and [1]-and theoretically important parameters—kp, k, and k. Note that the mode of termination which establishes the connection between u and hj, and the value of f are both accessible through end group characterization. Thus we have a second equation with three unknowns one more and the evaluation of the individual kinetic constants from experimental results will be feasible. 

The energy barrier between the two chemisorption modes on Au(lll) is very small, 10.5 kj/mol, (2.5 kcal/mol), (211), suggesting that the thiolate may easily cross from one of these minima to the other, enabling a facile annealing mechanism. This predicts that changing tilt direction may occur well below the melting point of the monolayer, and should be chain-length-dependent. 

Hamiltonian does not give rise to any crystalline order in the system. By employing models hke this, the quench-rate and chain-length dependence of the glass transition temperature, as well as time-temperature superposition, similar to experiments [23], were investigated in detail. 

The advantage of the simulations compared to the experiments is that the correspondence between the tracer diffusion coefficient and the internal states of the chains can be investigated without additional assumptions. In order to perform a more complete analysis of the data one has to look at the quench-rate and chain-length dependence of the glass transition temperature for a given density [43]. A detailed discussion of these effects is far beyond the scope of this review. Here we just want to discuss a characteristic quantity which one can analyze in this context. 

In Fig. 20 we show the MSQ of a system of GM [66] with different mean chain lengths (depending on 7, cf. Eq. (12)) for three values of LO=l, 0.1, 0. 01. Since the individual chains have only transient identity, it is meaningless to discuss their center of mass diffusion. It is evident from Fig. 20 that the MSQ of the segments, g t) = ([x( ) - x(O)j ), follows an intermediate sub-diffusive regime, g(t) oc which is later replaced by conventional diffusion at some characteristic crossover time which grows... 

Chain Length Dependence of Propagation Rate Constants... 

Most in depth studies of termination deal only with the low conversion regime. Logic dictates that simple center of mass diffusion and overall chain movement by reptation or many other mechanisms will be chain length dependent. At any instant, the overall rate coefficient for termination can be expressed as a weighted average of individual chain length dependent rate coefficients (eq. 20) 39... 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

Figure 5.4 Chain length dependence of A,I J predicted by (a) the geometric mean (eq, 25) or (b) the harmonic mean approximation (eq. 22) or the Smoluchowski mean (eq. 23) with a=1.0 and to=109 i and j are the lengths of the reacting chains.

Table 5.1 Parameters Characterizing Chain Length Dependence of Termination Rate Coefficients in Radical Polymerization of Common Monomers 1...

Bamford43,59 63 has proposed a general treatment for solving polymerization kinetics with chain length dependent kt and considered in some detail the ramifications with respect to molecular weight distributions and the kinetics of chain transfer, retardation, etc. 

Treatments (Smith-Ewart,79 pseudo-bulk77) have been devised which allow for the possibility of greater than one radical per particle and for the effects of chain length dependent termination. Further discussion on these is provided in the references mentioned above.77 751... 

This equation (eq. 5) is commonly known as the Mayo equation.1" The equation is applicable at low (zero) conversion and is invalidated if the rate constants are chain length dependent. 

Table 6.1 Chain Length Dependence of Transfer Constants (C )...

Some of the issues associated with RAFT emulsion polymerization have been attributed to an effect of chain length-dependent termination.528 In conventional emulsion polymerization, most termination is between a long radical and a short radical. For RAFT polymerization at low conversion most chains are short thus the rate of termination is enhanced. Conversely, at high conversion most chains are long and the rate of termination is reduced. 

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