Real polymer chains

The freely jointed chain is a simple model for predicting chain dimensions. It is, however, physically unrealistic. Since each carbon atom in a real polymer chain is tetrahedral with fixed valence bond angles of 109.5°, the links are subject to bond angle restrictions. Moreover, the links do not rotate freely because, as we have seen earlier, there are energy differences between different conformations (cf. Fig. 2.3). Both of these effects cause to be larger than that predicted 

The first modification to the freely jointed chain model is the introduction of bond angle restrictions while retaining the concept of free rotation about bonds. This is called the valence angle model. For a polymer chain with all backbone bond angles equal to 9, this leads to Eq. (2.5) for the mean square end-to-end distance 

Since 180° 6 90°, cosd is negative and (r ) is greater than nfi of the freely jointed chain model [Eq. (2.2)]. For polymers having C-C backbone bonds with 9 109.5°for which cos0 -j, the equation becomes 

Thus for polymers such as linear polyethylene, bond angle restrictions cause the RMS end-to-end distance to increase by a factor of V2 from that of the freely-jointed chain. 

Problem 2.4 For a linear molecule of polyethylene of molecular weight 1.4x10 what would be the RMS end-to-end distance according to the valence angle model as compared to that according to the freely-jointed chain model and the end-to-end distance of a fully extended molecule Comment on the values obtained, indicating Which one is a more realistic estimate of chain dimensions. 

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Flory P J 1949 The configuration of real polymer chains J. Chem. Phys. 17 303- 10... 

In a real polymer chain, rotation around backbone bonds is likely to be hindered by a potential energy barrier of height AEr. If AEr < RT, the population of the... 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The foregoing derivation may appear artificial in view of the assumptions involved. The contribution of a given bond to x is by no means restricted to the two unique values, + as has been assumed. On the contrary, one may show that all values of h from 0 to Z occur with equal probability for freely jointed connections between links. A more detailed study of the problem shows that the final result is unaffected by this assumption so long as n is large. The freely jointed chain model under consideration is an artifice also, but the form of the results obtained will be shown to apply also to real polymer chains. 

Id. Influence of Bond Angle Restrictions.—In all real polymer chains the direction assumed by a given bond is strongly influenced by the direction of its predecessor in the chain. The orientation of other nearby bonds (second, third, and possibly fourth neighbors) may also exert an appreciable influence, but the orientation of the immediate predecessor usually is of greatest importance. The exact nature of these restrictions on the direction assumed by a given bond... 

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

The study of dynamics of a real polymer chain of finite length and containing some conformational defects represents a very difficult task. Due to the lack of symmetry and selection mles, the number of vibrational modes is enormous. In this case, instead of calculating the frequency of each mode, it is more convenient to determine the density of vibrational modes, that is, the number of frequencies that occur in a given spectral interval. The density diagram matches, apart from an intensity factor, the experimental spectmm. Conformational defects can produce resonance frequencies when the proper frequency of the defect is resonating with those of the perfect lattice (the ideal chain), or quasi-localized frequencies when the vibrational mode of the defect cannot be transmitted by the lattice. The number and distribution of the defects may be such... 

By assuming the additivity of bond polarizabilities, a formal expression is derived for the difference Ay in principal polarizabilities of a real polymer chain having constant end-to-end distance r. An expression in a matrix form is derived for Ay of the PE chain. 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

The application of the lattice model to various systems requires a procedure to determine the lattice size and the chain bending energy, hence the number of segments and the chain stiffness in the lattice. Based on the equivalence between the contour length, volume, and gyration radius of a real polymer chain and that in the lattice model, the following equations were suggested 22... 

Since spectroscopic investigations of such a structure in real polymer chains present difficulties owing to the low concentration of the unit in the polymer (even for low molecular weight polymers), model compounds have been used widely in polymerization studies. 

The methods of conformational statistics, discussed so far, had as starting point the real polymer chain. The aim was to relate the dimensions of the coiled polymer molecule statistically to the mutual displaceability of the chain atoms. Nearly exact relationships are obtained for a large number of freely jointed or freely rotating elements. Under conditions of restricted movability, however, the statistical equations can generally not be solved and empirical factors like s, a and a are introduced. 

In the real polymer chains each monomer unit, essentially, does not remember the way of its introduction into the macromolecule. It is characterized only by its type and from this viewpoint is regarded as being uncolored. All the experimental characteristics of the copolymer microstructure are described undoubtedly by the sequences of the uncolored units. Hence, it is quite clear that each state of the sequence of the uncolored units Mx is the result of the unification of the corresponding pair of the colored units, i.e. M, = Sx + S2, M2 = S3 + S4. The rigorous kinetic consideration within the framework of the scheme (2,1) and (2.5) reveals [49, 60] that the sequences of the conditionally colored units in the macromolecules really form a certain Markov chain. The probabilities... 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

Real polymer chains of fixed bond angles and restricted rotation have... 

The negative deviations of the experimental initial slopes from the theoretical dependences I or II for polymer molecules with low equilibrium rigidity can be understood qualitatively by taking into account the finite character of d, the diameter of a real polymer chain, which was not included in the theories discussed in Chap. 3. 

The structural parameters of the freely jointed chain that can represent the real polymer chain and therefore fits a Gaussian function of end-to-end distances can be calculated. The first requirement that is going to be imposed is that the real chain and the model chain have the same value of mean square end-to-end distance therefore the product Nl is determined, but it does not permit N and / to be known independently. Consequently it will be necessary to add a further condition, which is that the two chains (the real one and the model) have the same length corresponding to that of the fully extended chain ... 

It is important to understand how the energy arising from these numerous contacts affects the conformations of a real polymer chain. The effective interaction between a pair of monomers depends on the difference... 

Although the extension of the infinite lattice model to the finite lattice corresponding to real polymer chains is straightforward in principle, the analytical expression for M is algebraically complex [5] and will not be reproduced here. It is of interest, however, to consider the predictions for the dependence of the efficiency of sampling of EPS, represented by the function (1 - M)/M, on the molecular weight of the aryl vinyl polymer. This is shown in Figure 2. [24]... 

The limitations of the random-flight model when applied to real polymer chains arise from... 

An idea of the stiffness of a polymer chain can be gained from the ratio (r )y /which is the square root of the characteristic ratio, Ooo = t )o / and indicates how much greater the RMS end-to-end distance of a real polymer chain is, compared to that of the freely jointed chain. From the v ues of Coo given in Table 2.1 it is seen that this ratio is 2-3. 

Thus, the combination of fixed bond angles and short-range steric interactions causes the end-to-end distances of real polymer chains to be greater than those of freely jointed chains by factors of 2-3. 

Problem 2.8 A real polymer chain consisting of n bonds each of length I may be usefully represented by an equivalent freely jointed chain of N links each of length b such that it will have the same end-to-end distance and the same contour length. Obtain N and b in terms of the characteristic ratio Coo of ths polymer chain. 

Thus it appears, as shown by Kuhn,84 that a real polymer chain is equivalent to a freely jointed chain in which monomer units are replaced by statistical units containing s monomer units. The number of statistical units in the molecule is... 

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