On the Relationship between the Characteristic Ratio of a Finite Chain, C_n, and the Asymptotic Limit, C_∞

Abstract

Often experiments with chains containing a finite number of bonds, n, are interpreted with the assumption that the characteristic ratio, C n, is determined completely by C∞ and n. This assumption is supported by some, but not all, textbook models for simple flexible chains. The freely jointed chain, freely rotating chain with fixed bond angle, and simple wormlike chain predict C n ) f(C∞,n). These three models share the feature that the stiffness of the chain is specified by no more than one parameter. However, when more than one parameter affects the stiffness of the chain, as in the model with fixed bond angle and symmetric hindered rotation about independent bonds, C n is no longer determined by C∞ and n alone, Cn * f(C∞,n). Since virtually all real chains have hindered rotation, they cannot be expected to have the dimensions given by Cn ) f(C∞,n). This conclusion is supported by numerical calculations using previously published rotational isomeric state models for polyethylene, polyisobutylene, and poly-(dimethylsiloxane). Although these three polymers have similar values of C ∞, they may have quite different values of Cn. This conclusion from the calculations is consistent with the observed behavior of polyisobutylene and poly(dimethylsiloxane), as reported by Arbe et al. (Macromolecules 2001, 34, 1281 and by Sluch et al. (Macromolecules 2003, 36, 2721. The finite n effect in these polymers is three times stronger for the mean square unperturbed radius of gyration than for the mean square unperturbed end-to-end distance.

Atomistic models of three fluorinated polyimides in the amorphous state

“…This internal distance93 tends towards the mean square end‐to‐end distance, 〈 R 2 〉, as n approaches the degree of polymerization of the polymer chain. A plot of 〈 R 2 ( n )〉/ nl 2 versus n should thus be similar to that for a characteristic ratio 94. Such a plot is given in Figure 5, where the normalizing “length” of a monomer, l , has been defined to be the average distance between the central carbons of successive 6FDA fragments as determined from the last configurations of the PMC‐MD runs.…”

Ion mobility mass spectrometry as a powerful tool to analyze complex macromolecular systems

“…Mattice et al studied this relationship in depth and came to the conclusion that for relatively short chains (≈ n < 100) it is indeed expected that C n is significantly smaller than C ∞ . 156 The data obtained in their study can be extrapolated to determine a ratio C n /C ∞ of approximately 0.80 to 0.95 for the chain lengths evaluated in this study, which would bring the values of PAA, PMA and PBA for z = 1 determined in this work in very good agreement with the theoretical C ∞ value. For z = 2, PAA and PMA are also in agreement within the expected range.…”

“…Mattice et al had previously stated, that the ratio of C n /C ∞ depends on the length of the polymer chain. 156 Their results can be extrapolated to expect a ratio of approximately 0.80-0.95 for the chain lengths studied in this work. The findings once more underline that the z = 1 charge state yields more accurate results when evaluating C n .…”

Elucidating the Topology and Physical Properties of Triblock Copolymers Using Ion Mobility Mass Spectrometry

“…The values for PEG and PPG homopolymers are in good agreement with literature data. The literature value given for PMA represents a theoretically derived C ∞ value, which is expected to be slightly higher than the real C n. [ 23 ] The data for both systems clearly show a noticeable difference in C n for the different triblock copolymers based on their block sequence. While the copolymer data lines (see Figures 4 and 5) lie in the middle between the homopolymer lines and occupy a very similar part of the coordinate space, diligent evaluation of a sufficiently broad mass‐per‐charge region shows that the two species exhibit significantly different slopes, corresponding to significantly different C n .…”