Distribution molecular weight, 7-10

The chain length distribution of free radical addition polymerization can also be derived from simple statistics. Thus, for polymer formed at any given instant, the distribution will be the most probable and will be governed by the ratio of the rates of chain growth to chain termination. 

From Eq. (2.21) it follows that the number- and weight-average chain lengths Xn and Xw are expressed by 

The value of X IXn for the cumulative polymermsy, of course, be much higher, depending on the changes in the value ofp with increasing conversion. It should be noted, however, that this is valid only where the growing chains terminate by disproportionation or transfer, not by combination. It can be shown in the latter case (Flory, 1953f) that the increment distribution is much narrower, i.e.. 

in summary, the kinetics of free radical polymerization are characterized by the following features  

Rate is directly proportional to the half-power of the initiator concentration. 

Two measurements of the bimolecular termination rate coefficient in the polymerization of ethylene by Ti complexes are available. For (7r-CsH5)2TiCl2/AlMe2Cl a value of 0.5 1 mole sec at 0°C has been reported [111] and for (7r-CsHs)2TiEtCl/AlEtCl2, 0.61 mole sec at0°C [202]. The rate coefficient of decomposition of the latter complex in the absence of ethylene was much lower being 5 x 10 1 mole sec at 20 C [202]. 

The diversity of initiation termination and transfer reactions and the changes in concentrations of types and activities of catalyst species during polymerization have so far precluded a theoretical treatment of molecular weight distributions. With some of the soluble catalysts, e.g. Zr(CH2Cg- 

Hs)4 for styrene [278], Ti(On-Bu)4/AlEtj for butadiene (125) or (7T-C4H7 1)2 or butadiene [61] molecular weight distributions are fairly narrow (Mw/M = 1—2), where the propagating species have long lifetimes and their concentrations remain reasonably constant throughout the polymerization. The soluble catalysts based on vanadium compounds likewise give relatively narrow distributions with ethylene or ethylene/ propene (M / n 2). Polymers prepared with heterogeneous catalysts 

Experimentally, linear log-normal distributions are observed for poly-ethylenes and polypropenes prepared using Ziegler—Natta catalysts (Fig. 14a—b). It has been shown that the main features of the distribution can either be explained by variable catalyst site activity or by termination of a propagating chain by a desorption process from the catalytic surface proportional to molecular weight. Such experimental evidence as is available favours the variable site hypothesis [302]. 

The broadness of the distribution is given by the parameter j, obtained from the slope (tan 0) of the log-normal plot where tan d = 7A/2. The polydispersity ratio (Af /Afp = exp(7 /2) no and x are constants in the distribution function not determinable from the log-normal line. (Special cases of the general distribution function have also been employed, e.g., the Lansing and Kraemer relationship where x = 0 [304] and the Wesslau relationship [305], where x = —1). Fractionation data can also be adequately represented by other exponential distributions such as that of Tung [306] W(n) dn = abn exp(—an ) dn. (W(n) = 1 — exp(—an )). Fig. 14a(c). 

The molecular weight distribution in this type of nonlinear polymerization will be much narrower than for a linear polymerization. Molecules of sizes vay much different from the average are less likely than in linear polymerization, since this would require having the statistically determined / branches making up a molecule aU very long or aU very short. The distribution functions for this polymerization have been derived statistically [Peebles, 1971 Schaefgen and Flory, 1948], and the results are given as 

The Type 1-condensed tannins consist of monomer units of molecular weight 272 (propelargonidin units 1, 4), or 288 (procyanidin units 2, 5), or 304 (pro-delphinidin units 3, 6), respectively (or 300 in rounded terms). The units, therefore, possess relatively large individual molecular weights compared with most natural or synthetic polymers. 

GPC has not so far been successfully applied directly to the phenolic polymers because of their highly polar and strongly hydrogen bonding character. Condensed tannins will dissolve only in very polar solvents (Sect. 1,123) and the best defined GPC supports are based on polystyrene or polystyrene bonded to silica. However, several groups have derivatized condensed tannins by methylation (66) or peracetylation (36, 124, 142, 148) and performed GPC analyses on the relatively less polar derivatives in chloroform or tetrahydrofuran. The usual acetylation method is with pyridine and acetic anhydride. 

The few data (105) available testing the validity of these assumptions are summarized in Table 7.7.3. While some values of P and P (weight-average degree of polymerization) obtained by GPC agree reasonably well with direct measurement of Pn (by NMR and vapour pressure osmometry, VPO) and P (by low- 

However, the extant data are certainly sufficient to establish the following facts about proarithocyanidin polymers  

1) They are polydisperse, the actual range of molecular sizes being dependent on the source of plant material. The observed range of sizes varies from the pro-cyanidins of Rubus idaeus leaves, which are dimeric, and those of Hordeum vulgare ears which are no larger than tetrameric, to other sources where there are polyflavanoid chains containing hundreds of units. 

Commercial polymers have broad distributions of molecular weights. Such molecular weight distributions are usually discussed in terms of different molecular weight averages. If one averages equally over all molecules, this leads to a number average molecular weight Af  

If one averages according to the mass of each molecular species, it leads to a weight average molecular weight  

Higher molecular weight averages have been defined. These are notably the z and z + 1 average molecular weights  

These average molecular weights in polydisperse systems order as 

Polymer scientists are veli aware of the importance of molecular weight distribution and have learned to precisely measure and control it. This enables us to control many important 

The number average molecular weight (M ) is calculated from Eq. 1.1. 

M = Molecular weight of chains in fraction i Nj = Number of chains in fraction i Wj = Weight of chains in fraction i 

We often use the ratio between the weight and number average molecular weights as a guide to summarize a polymer s overall molecular weight distribution. 

It is impossible to find a sample of a synthetic polymer in which all the chains have the same molecular weight. Instead, a distribution of molecular weights is reported. Some of the polymer chains will be much larger than the others, and some will be significantly smaller. The largest number of similar chains will be populated around a central point of the distribution, the highest point on the curve called the molecular weight distribution.  

Two other important average molecular weights are number average (Mn) and weight average (Mw). The first one is the weight calculated by average number of the molecules 

Weight average molecular weight of the polymer is usually measured by light scattering and is defined as 

The ratio of weight average to number average molecular weight is called the polydispersity, ra. The wider is the molecular weight distribution the higher the polydispersity value. The ra for unimolecular polymer is equal to 1. All natural polymers, such as peptides, DNA, and saccharides, have polydispersity equal to 1. 

Various experimental data suggest that the tube reorganization is important in linear polymers with molecular weight distribution. 

The discrepancy between the experimental results and eqn (7.265) is schematically explained in Fig. l.Tl shows two characteristic 

These results clearly indicate that the tube constraint for a polymer becomes weaker if it is made of shorter polymers. The weakening the tube can be expressed either by an increase in the step length, or by an increase in the constraint release process, or both. However, the interpretation seems to be still at a tentative level. 

The influence of catalyst supports is being widely studied at present. Metal complexes or clusters are applied un supports like silica, alumina or zeolites [42, I30[. Tliesc supports influence the molecular weight distribution, not only by their acidic or basic properties but also by their ability to stabilize metal dispersions and by their geometric properties. 

Blanchard and coworkers have shown the dependence of chain length of linear paraffins, formed over cobalt catalysts, from the pore size of the 

Support (alumina). SCS9 SCS9 SCS69 SCS350 

Shape selective FT synthesis catalyzed by zeolite entrapped ruthenium  

Chain limitation by zeolites with ruthenium catalysts 1132.1331  

Keii119 studied recently the effect of polymerization time, hydrogen, donor and aluminum alkyl on the MWD of propylene obtained with the MgCl2/TiCl4/EB— AlEtj/EB catalyst system. The isotactic and atactic fraction polydispersity indexes were found to be independent of the above parameters, and very close to each other (Mw/Mn = 3-5) as previously reported by Suzuki133) and Kashiwa,34). The Mw/Mn values of the overall polymer are lower than in the case of TiCl3 and are better represented by a Wesslau-type log-normal correlation than by a Tung-type correlation, as is the case of unsupported catalysts. 

Based on the independence of Mw/Mn on the hydrogen concentration, as well as kinetic evidence concerning an inhibition of the polymerization rate by CO, a surface heterogeneity dependence of the propagation rate constant has been proposed. As mentioned above, the electron donor has no effect on the polymer Mw/Mn. The polydispersity of the overall polymer only changes as a result of a variation of the relative quantity of atactic and isotactic polymer. Thus, the donor does not participate in the formation of active centers having different Mn and MWD. 

On the ground of these contrasting results it is felt that the role of the donor needs more in depth studies. With regard to the role of the support, the MgCl2-based catalysts permit to narrow the polypropylene MWD. In fact, the non-supported catalysts produce polymers having Klw/Mn in the range 6-10 53). 

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The ratio is one for a strictly unifonn molecular-weight distribution and larger than one for molecular-... 

We shall see that, as polymers go, this is a relatively narrow molecular weight distribution. 

At 25°C, the Mark-Houwink exponent for poly(methyl methacrylate) has the value 0.69 in acetone and 0.83 in chloroform. Calculate (retaining more significant figures than strictly warranted) the value of that would be obtained for a sample with the following molecular weight distribution if the sample were studied by viscometry in each of these solvents ... 

We saw in Chap. 1 that the ratio M /M is widely used in polymer chemistry as a measure of the width of a molecular weight distribution. If the effect of chain ends is disregarded, this ratio is the same as the corresponding ratio of n values ... 

The preceding discussions of the kinetics and molecular weight distributions in the step-growth polymerization of AB monomers are clearly exemplified by the esterification reactions of such monomers as glycolic acid or co-hydroxydecanoic acid. Therefore one method for polyester synthesis is the following ... 

The diacid-diamine amidation described in reaction 2 in Table 5.4 has been widely studied in the melt, in solution, and in the solid state. When equal amounts of two functional groups are present, both the rate laws and the molecular weight distributions are given by the treatment of the preceding sections. The stoichiometric balance between reactive groups is readily obtained by precipitating the 1 1 ammonium salt from ethanol ... 

Howardt describes a model system used to test the molecular weight distribution of a condensation polymer The polymer sample was an acetic acid-stabilized equilibrium nylon-6,6. Analysis showed it to have the following end group composition (in equivalents per 10 g) acetyl = 28.9,... 

The molecular weight distribution for a polymer like that described above is remarkably narrow compared to free-radical polymerization or even to ionic polymerization in which transfer or termination occurs. The sharpness arises from the nearly simultaneous initiation of all chains and the fact that all active centers grow as long as monomer is present. The following steps outline a quantitative treatment of this effect ... 

The phenomena we discuss, phase separation and osmotic pressure, are developed with particular attention to their applications in polymer characterization. Phase separation can be used to fractionate poly disperse polymer specimens into samples in which the molecular weight distribution is more narrow. Osmostic pressure experiments can be used to provide absolute values for the number average molecular weight of a polymer. Alternative methods for both fractionation and molecular weight determination exist, but the methods discussed in this chapter occupy a place of prominence among the alternatives, both historically and in contemporary practice. 

Polydisperse polymers do not yield sharp peaks in the detector output as indicated in Fig. 9.14. Instead, broad bands are produced which reflect the polydispersity of synthetic polymers. Assuming that suitable calibration data are available, we can construct molecular weight distributions from this kind of experimental data. An indication of how this is done is provided in the following example. 

The basic premise of this method is that the magnitude of the detector output, as measured by hj for a particular fraction, is proportional to the weight of that component in the sample. In this sense the chromatogram itself presents a kind of picture of the molecular weight distribution. The following column entries provide additional quantification of this distribution, however. 

Three polystyrene samples of narrow molecular weight distribution were investigatedf for their retention in GPC columns in which the average particle size of the packing was varied. In all instances the peaks were well resolved. The following results were obtained ... 

Fig. 14. Molecular weight characteristics of novolac resins. Shown is the size-exclusion chromatogram for a typical commercial novolac polymer. The unsymmetrical peak shape reflects the multimodal molecular weight distribution of the polymer.

These normal stresses are more pronounced for polymers with a very broad molecular weight distribution. Viscosities and viscoelastic behavior decrease with increasing temperature. In some cases a marked viscosity decrease with time is observed in solutions stored at constant temperature and 2ero shear. The decrease may be due to changes in polymer conformation. The rheological behavior of pure polyacrylamides over wide concentration ranges has been reviewed (5). 

A brief review has appeared covering the use of metal-free initiators in living anionic polymerizations of acrylates and a comparison with Du Font s group-transfer polymerization method (149). Tetrabutylammonium thiolates mn room temperature polymerizations to quantitative conversions yielding polymers of narrow molecular weight distributions in dipolar aprotic solvents. Block copolymers are accessible through sequential monomer additions (149—151) and interfacial polymerizations (152,153). 

Narrow, regular, and broad refer to molecular weight distribution. 

Tetiafluoioethylene—peifluoiopiopyl vinyl ethei copolymeis [26655-00-5] aie made in aqueous (1,2) oi nonaqueous media (3). In aqueous copolymerizations water-soluble initiators and a perfluorinated emulsifying agent are used. Molecular weight and molecular weight distribution are controlled by a chain-transfer agent. Sometimes a second phase is added to the reaction medium to improve the distribution of the vinyl ether in the poljmier (11) a buffer is also added. 

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