Length of chain

This form is obeyed fairly well above x values of 5-10 dyn/cm in Fig. Ill-15c. Limiting areas or a values of about 22 per molecule result, nearly independent of chain length, as would be expected if the molecules assume a final orientation that is perpendicular to the surface. Larger A values are found for longer-chain surfactants, such as sodium dodecyl sulfate, and this has been attributed to the hydrocarbon tails having a variety of conformations [127]. 

Grassberger P 1997 Pruned-enriched Rosenbluth method simulations of theta polymers of chain length up to 1,000,000 Phys. Rev. E 56 3682... 

For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

After polymeri2ation is carried out by blending mono- and difunctional chlorosilanes ia excess water, the siloxanes are separated from the water and neutraH2ed. Ratio of the mono-chain stopper to di-chain extender controls the length of the polymer. Once an equiHbrium mixture of chain lengths is catalyticaHy formed, volatile light ends are removed and the desired product results. 

Consequently, the development of chain length of the polymer is restricted and the average molecular weight tends to be low. PhthaUc anhydride has a tendency to sublime. Its heat of sublimation is 598 J /g (143 cal/g), and the heat of vaporization is 365 J/g (87.2 cal/g). Therefore, care must be taken to prevent its loss (see later discussions on Production Processes). 

Using both condensation-cured and addition-cured model systems, it has been shown that the modulus depends on the molecular weight of the polymer and that the modulus at mpture increases with increased junction functionahty (259). However, if a bimodal distribution of chain lengths is employed, an anomalously high modulus at high extensions is observed. Finite extensibihty of the short chains has been proposed as the origin of this upturn in the stress—strain curve. 

If the initiation reaction is much faster than the propagation reaction, then all chains start to grow at the same time. Because there is no inherent termination step, the statistical distribution of chain lengths is very narrow. The average molecular weight is calculated from the mole ratio of monomer-to-initiator sites. Chain termination is usually accompHshed by adding proton donors, eg, water or alcohols, or electrophiles such as carbon dioxide. 

Equation 20 is the rate-controlling step. The reaction rate of the hydrophobes decreases in the order primary alcohols > phenols > carboxylic acids (84). With alkylphenols and carboxylates, buildup of polyadducts begins after the starting material has been completely converted to the monoadduct, reflecting the increased acid strengths of these hydrophobes over the alcohols. Polymerization continues until all ethylene oxide has reacted. Beyond formation of the monoadduct, reactivity is essentially independent of chain length. The effectiveness of ethoxylation catalysts increases with base strength. In practice, ratios of 0.005—0.05 1 mol of NaOH, KOH, or NaOCH to alcohol are frequendy used. 

FIG. 3 Probability distributions of chain lengths at four temperatures (given as parameter) [28]. In the inset, oscillations of the MWD due to the formation of rings are shown for T = 0.2. 

We use the off-lattice MC model described in Sec. IIB 2 with a square-well attractive potential at the wall, Eq. (10), and try to clarify the dynamic properties of the chains in this regime as a function of chain length and the strength of wall-monomer interaction. 

S. K. Kumar, M. Vacatello, D. Y. Yoon. Off-lattice Monte Carlo simulations of polymer melts confined between two plates. 2. Effects of chain length and plate separation. Macromolecules 25 2189-2197, 1990. 

Table 14.1 Sulfur-Nitrogen Chains as a Function of Chain Length...

As Eq. (3) sh vs, the critical composition (ticn can be controlled by the asymmetry of chain lengths. Particularly interesting is the limit Na = N, Nb = I (which physically is realized by polymer solutions, B representing a solvent of variable quality). Checking the deviations from the mean field predictions, Eq. (3), further contributes to the understanding of the statistical mechanics of mixtures. 

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