Dimensions, coils unperturbed

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

The exponent can vary from v=0.33 for hard spheres up to v=1.0 for rigid rods. For linear chains v=0.5 refers to unperturbed coil dimensions in -solvents and v=0.588 [6] to good solvent conditions. Equation (37) maybe re-writ-ten by expressing the molar mass as a function of the radius of gyration, i.e.. 

In some specific cases, dissolved macromolecules take up the shape predicted by the above theories of isolated chain molecules. In general, however, the interaction between solvent molecules and macromolecules has significant effects on the chain dimensions. In poor solvents, the interactions between polymer segments and solvent molecules are not that much different from those between different chain segments. Hence, the coil dimensions tend towards those of an unperturbed chain if the dimension of the unperturbed coil is identical to that in solution, the solution conditions are called conditions (ff solvent, temper-... 

For dilute solutions in good solvents the net excluded volume is positive, and coil dimensions are expanded beyond their unperturbed values. The expansion... 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

For the purpose of determination of the temperature coefficient for unperturbed dimensions of copolymer 3 (Table 1), [r ] values were measured in the same solvent (toluene) at different tempe-ratures. In accordance with the Shtockmayer-Fixman method, Kg=(/M)1/2xF0 values was deter-mined using the least-square technique. The temperature coefficient of unperturbed dimensions was calculated from the values obtained at different temperatures (Table 6) using the relation, suggested in the work [45] dinldT = 2/3x1 n A/ t//-. The coefficient of unperturbed coil dimension (dlnldl), determined for copolymer 3 (Table 1), equals 0.85xl0 3 deg"1 [44],... 

Direct Determination of Unperturbed Polyelectrolyte Coil Dimensions... 

It can be proved that these unperturbed dimensions approximate to the statistically determined dimensions of the isolated chain under 0 conditions the excluded volume effect is compensated by a positive, i. e. unfavorable, polymer-solvent interaction enogy. The rmperturbed dimensions of polymer chains are, therefore, subject in general to direct experimental determination in dilute solution in appropriate solvents in which the volume exclusion effect upon coil dimensions is nullified. 

Direct Determination of Unperturbed Polyel ti Iyte Coil Dimensions (under 0 Conditions)... 

Excluded volume effects (2) in polymers are defined as those effects which come about through the steric interaction of monomer units which are remotely positioned along the chain contour. Each Individual interaction has only a small effect, but, because there can be many such interactions in a long polymer, excluded volume effects become very large. One consequence of excluded volume is to expand the polymer coil dimensions over that predicted from simple random walk models. The "unperturbed values of the... 

It seems appropriate to discuss here the probabUity of interpenetration of polystyrene coils in the model networks. As already mentioned, according to the theoretical considerations of Flory [138] and De Gennes [139], polymeric coils in an amorphous solid state retain unperturbed dimensions. Since the volume fraction of the polymer in an unperturbed coil under -conditions is weU known to be very smaU, only about 2%, the transition from swoUen coils to solid state has to be accompanied by the replacement of aU solvent molecules with fragments of other polymeric molecules. In other words, theoretical notions predict extremely high mutual interpenetration of the polymeric chains in bulk state. Indeed, in order to maintain the coil dimension that is characteristic for a -solution, the coil must accommodate, on removing the solvent, a 50- to 100-fold amount of alien polymeric matter. In the 1970s this problem was discussed in fiiU [149-165], The authors of the tailor-made networks also took part in the discussion. 

The coil dimensions in the theta solvent are designated unperturbed . 

Unperturbed coil dimensions in solution are equal with those of macromol-ecirles in amorphous sohd state. 

End-to-end distance of a polymer coil with unperturbed dimensions... 

Some interesting results have been obtained for mixtures of glassy polymers which are similar in type and described in the literature as compatible. The coil dimensions of the copolymer of acrylonitrile and styrene (PSAN) dispersed in deuteropoly(methyl methacrylate) have been measured in those compositions in which the copolymer forms molecularly disperse mixtures. The coils are expanded as compared with the unperturbed coil dimensions. Measurements of the second viral coefficient at different temperatures show that the system is exothermic and has a negative excess entropy of mixing. Measurements of the coil dimensions in mixtures of poly(dimethyl siloxane) with the deuteriated polymer also show that the coil of the former expands compared to the 0-value as the... 

The term random coil is often used to describe the unperturbed shape of the polymer chains in both dilute solutions and in the bulk amorphous state. In dilute solutions the random coil dimensions are present under Flory 0-solvent conditions, where the polymer-solvent interactions and the excluded volume terms just cancel each other. In the bulk amorphous state the mers are surrounded entirely by identical mers, and the sum of all the interactions is zero. Considering mer-mer contacts, the interaction between two distant mers on the same chain is the same as the interaction between two mers on different chains. The same is true for longer chain segments. 

The mean dimensions of the macromolecular coils in the entangled system are found to approach their unperturbed values, i.e. values they would have in a 0-solvent. The coil dimensions in the concentrated system are the same as the dimensions of ideal coils. This is confirmed by direct measurements of the dimensions of macromolecular coils in concentrated solutions and melts by neutron scattering [30, 31]. 

Although these semiempirical treatments can be useful in predicting interfacial tensions, they are not successful from a fundamental standpoint and cannot be used to predict the interfacial composition profile. Furthermore, these theories neglect the entropy effects associated with the configurational constraints on polymer chains in the interfacial region. These effects are unique in polymers and arise because the typical thickness of the interfacial region between polymer phases is less than the unperturbed molecular coil dimensions of a high polymer. Major perturbations of... 

In earlier chapters an unperturbed coil referred to molecular dimensions as predicted by random flight statistics. We saw in the last chapter that this thermodynamic criterion is met under 0 conditions. 

Studies of the hydrodynamic properties and unperturbed dimensions of fractionated PCL have shown that it is a flexible coil (54,55). The following Mark-Houwink equations have been reported ... 

As explained earlier (Sect. 1.3.1), macromolecules in a low-molecular-weight solvent prefer a coiled chain conformation (random coil). Under special conditions (theta state) the macromolecule finds itself in a force-free state and its coil assumes the unpertubed dimensions. This is also exactly the case for polymers in an amorphous melt or in the glassy state their segments cannot decide whether neighboring chain segments (which replace all the solvent molecules in the bulk phase) belong to its own chain or to another macromolecule (having an identical constitution, of course). Therefore, here too, it assumes the unperturbed ) dimensions. 

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