Mean chain length

Introducing the rate of thermal chain generation Wq, chain generation in the course of the reaction and the probability of path 1 referred to (according to [427]) as the chain propagation probability, co = a, the rate of the stationary reaction in terms of the scheme in Fig. 51 is 

It is seen from Eq. (35.1) that a stationary state can be achieved only for sa 1. As a 1, this condition always holds for unbranched reactions (s 1) and also for branched reactions (s 1) at sufficiently low a. The quantity v is called chain length. This notion 

According to [427], an important characteristic value of a branched chain reaction is 

The value 3vs can be considered as the branching probability over the whole length of a simple chain. It will be seen that even one branching within the whole chain length may cause an infinite number of reaction steps resulting in explosion. 

The concept of degenerate branching arose from the necessity of interpreting in terms of the kinetic chain theory the numerous slow reactions displaying characteristic features of branched chains. 

---

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

Note that the measured mean chain lengths of up to (L) some experimental systems of EP. 

It appears that the mean chain length (L) in a state of equilibrium polymerization depends essentially on the geometry constraints for sufficiently small T>. In contrast to recent theoretical predictions [47], which treat the... 

FIG. 14 Semi-log plot of mean chain length L vs width of the open slit D at various temperatures in 3d. Full symbols denote flexible chains and empty symbols semirigid chains with activation energy a = 0.5 [61]. 

FIG. 15 Mean chain length L vs. strip thickness D for semi-rigid (open symbols) and flexible (full symbols) polymer chains [61]. 

FIG. 19 Scaling plot for the relaxation of the mean chain length L t) after a T-jump from Tq = 0.35 to a series of final temperatures, given as a parameter along with the respective L o s. The same Monte Carlo results [64] as in Fig. 5 are used. Full line denotes the scaling function f x = = (0.215 + 8x) . In the inset the... 

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... 

Single-pulse proton nuclear magnetic resonance (NMR) spectroscopy has been used to demonstrate that the mean chain length of synovial fluid triglycerides is shorter than that in corresponding sera (Naughton etal.,... 

One of the main conclusions of Kenat et al., was that the largest changes in polymer, mean-chain length, occur from the effect of inlet temperature changes and that, therefore, controlling inlet temperature, rather than reactor temperature, is beneficial to reactor performance. 

LAS). The mean chain length of 12.1 in the applied sludge increased to 12.2 after a period of 23-31 days and was finally found to be 12.4 in the treated soil after 312-322 days. 

Fd h = 0) should increase as when the chain concentration increases. A very different picture is predicted in the case of adsorbing polymers [49]. The layer of adsorbed chains may be partially interpenetrated by free chains in the bulk and therefore the range and strength of the attraction are not determined by the solution concentration. Instead, they are rather sensitive to the coverage and thickness of the adsorbed chains which depend essentially on the solvent quality and on the mean chain length in the dilute regime. 

It seemed probable, from the early results of chromatography, that the hydrolyzate from Graham s salt contains metaphosphates with more than four phosphorus atoms in the anion ring 350). A little later 186) the metaphosphate content of a freshly prepared solution of Graham s salt with a mean chain length n of 100-125 was given as ... 

The end groups which can be determined titrimetrically (pH 4.5-9) provide a measure of the mean chain length n of the salt mixture Na -(HoPnOan+i)- The same conclusion was reached by van Wazer from theoretical considerations (888, 339). 

Chemical proof that the anions in Graham s salt arc linear chains of P04 tetrahedra follows from the observation that arsenato-phosphate glasses analogous to Graham s salt (see Section VI) are broken down on aqueous hydrolysis into ortho arsenate and mixture of low molecular polyphosphates with a mean chain length dependent only on the P As ratio in the glass (295, 308). 

Equilibrium conditions for the chain length distribution are set up in these reactions, the mean chain length n of the anion being greater the higher the temperature used in the preparation and the lower the water... 

Fig. 8. Mean chain lengths in the formation of Graham s salt. T = 650°C Pn,o = 55.3 Torr.

In these, As04 tetrahedra are built into the polyphosphate chains (309). Since the P—0—As bond has much the same sensitivity to hydrolysis as the As—0—As bond, they are rapidly hydrolyzed in aqueous solution to monoarsenate and mixtures of polyphosphates, the mean chain length of which depends on the As P ratio in the starting material (74)- Contrary to an idea based originally on gravimetric analysis (309), the As atoms are not distributed regularly in the chain, but statistically (308). The observation that, after careful hydrolysis, the phosphate content is exclusively in the form of polyphosphates provides chemical proof of catenation in the poly-arsenatophosphate anions. When arsenatophosphates with more than five P atoms per atom of As are hydrolyzed by hot water trimetaphosphates are formed, just as they are formed for all other high-molecular polyphosphates in solution (316). 

The question of chain entanglements has been studied more extensively in acrylonitrile-styrene copolymerization (17) (Table IV). At the first appearance of popcom particles in this system, the copolymer has a 1 1 molar ratio of acrylonitrile to styrene units. Its mean chain length is 20,000. 

Figure 1.6 Changes in the viscosities of solutions of polyphosphates of different chain lengths on keeping for 12 h, where the abscissa represents the mean chain length as determined by end-group titration (a) immediately after solution (b) after keeping for 12 h (Strauss and Treitler, 1955b).

Even when all prerequisites are fulfilled, solution of eqn. (68) only yields the degree of polymerization of the instantaneously produced polymer. In the stationary state, of course, the ratio of the growth and termination rates does not change. If the ratio [XT]/[M] also remains essentially unchanged, a polymer with equal mean chain length is formed in the course of the whole stationary period. When [XT]/ [M] increases, the degree of polymerization... 

Now the ratio A p(M)//c<(M-) = fcp(M)(M-)//c (M-) can be seen to be the ratio of the rate at which monomer is converted into polymer to the rate of termination of radical chains. If termination occurs by recombination, then this ratio is just one-half the average number of monomer units per final polymer chain, which we may represent by n, the mean chain length, or mean degree of polymerization. This permits us to write for the stationary radical concentrations... 

---