Coil molecules

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

This thermodynamic behaviour is consistent with stress-induced crystallisation of the rubber molecules on extension. Such crystallisation would account for the decrease in entropy, as the disorder of the randomly coiled molecules gave way to well-ordered crystalline regions within the specimen. X-Ray diffraction has confirmed that crystallisation does indeed take place, and that the crystallites formed have one axis in the direction of elongation of the rubber. Stressed natural rubbers do not crystallise completely, but instead consist of these crystallites embedded in a matrix of essentially amorphous rubber. Typical dimensions of crystallites in stressed rubber are of the order of 10 to 100 nm, and since the molecules of such materials are typically some 2000 nm in length, they must pass through several alternate crystalline and amorphous regions. 

The lowermost curve in Fig. 45 represents P(0) plotted against according to Eq. (31) for random coil molecules. The results of similar calculations for spherical and for rod-shaped particles of uniform density are shown also. The curve for the former of these is not very different from that for randomly coiled polymers at corresponding values of the abscissas the factor P(0) for rods differs appreciably, however. 

Michaeli (1960) opposed these views. He concluded that whatever the exact mechanism was, the binding of divalent cations caused contraction and coiling of the polyelectrolyte as was the case with adds. He disagreed with the concept of ionic crosslinking. The phenomenon of precipitation could be explained simply in terms of reduced solubility. From this he concluded that precipitation took place in an already coiled molecule and the matrix consisted of spherical macromolecules containing embedded cations. 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

For polymer melts or solutions, Graessley [40-42] has shown that for a random coil molecule with a Gaussian segment distribution and a uniform number of segments per unit volume, a shear rate dependent viscosity arises. This effect is attributed to shear-induced entanglement scission. 

Concentration of the polymeric adhesive. In general, the more concentrated the poly-merie adhesive, the lower its bioadhesive strength. The coiled molecules become solvent poor in a concentrated solution which, in turn, reduces the available chain length for interpenetration into the mueus layer. Therefore, a critical concentration of the polymeric adhesive is required for optimum bioadhesion [37]. 

We have used the uncharged polysaccharide dextran as a model describing the behaviour of water-soluble polymers. The dextrans used in this study have about 95 % oc-(l - 6) linkages within the main chain and side chains the 5 % non-a-(l -> 6) linkages are starting points of branched chains of which most are only stubs of about two glucose units 9). Therefore, while there is some branching in dextran, albeit low, its solution behaviour is that of a linear, random-coil molecule l0,ll). 

Since the polar groups repel each other, the expanded random coil molecules tend to become stiff rods. The nonpolar portions of the water-solubilized polymer face toward the organic phase at the organic-aqueous interfaces, and the polar portions preferentially point away from the organic phase. 

The essence of this model for the second virial coefficient is that an excluded volume is defined by surface contact between solute molecules. As such, the model is more appropriate for molecules with a rigid structure than for those with more diffuse structures. For example, protein molecules are held in compact forms by disulfide bridges and intramolecular hydrogen bonds by contrast, a randomly coiled molecule has a constantly changing outline and imbibes solvent into the domain of the coil to give it a very soft surface. The present model, therefore, is much more appropriate for the globular protein than for the latter. Example 3.3 applies the excluded-volume interpretation of B to an aqueous protein solution. 

Aharoni has stated that the observed rates of crystallization in polymers are inconsistent with the times required for random-coil molecules to separate themselves from the melt, and claims this as support for the collapsed coil model (43). No numerical comparisons are given, and it is difficult therefore to judge the basis for his assertion. 

Theories of k for random coil molecules are very difficult and still somewhat lacking in experimental confirmation (24,121). 

The derivation of this equation makes it seem more appropriate for small molecule liquids or suspensions of hard spheres than for interpenetrating random coil molecules however. Indeed, a somewhat modified version of the equation has approximately the form observed for rj(j) in concentrated suspensions of mono-disperse spheres. In this case I/r0 turns out to be of the order of the rotational... 

This expression is by the factor (rf/i/rf) smaller than the one originally given by Peterlin (76). However, since the coil expansion is a qualitative measure of the state of deformation of coil molecules in laminar shear flow and, moreover, the first relaxation time is, in general, by far the largest one, the original equation of Peterlin can be used unchanged, if desired. 

The peculiarity of this expression, however, is that it does not make sense for dilute solutions of Gaussian coil molecules. In fact, the free-draining case is characterized by the limit of infinetely smaE friction coefficient . For this case, the contributions of the chain molecules to the viscosity of the solution becomes zero. 

Perhaps the most striking feature of this equation is that the friction factor has disappeared. It cancels out in the product alv. Physically this means that only some overall dimensions of the coiled molecule are of importance for its hydrodynamic behaviour. These dimensions are proportional to / = < > / (3.58a)... 

In principle, it only remains to show that the non-draining case is actually prevailing in solutions of coil molecules. For this purpose, reference is made to the experiences gathered in the field of molecular weight determinations with the aid of intrinsic viscosity measurements. From eq. (3.37) one obtains, when eq. (3.58) is used for the relaxation... 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

The following observations seem to lead to a key for the understanding of the quasistatic treatment of the stress-optical behaviour (777). With the aid of third eq. (5.4) and eq. (5.8), the ratio of Maxwell constant and intrinsic viscosity, as valid for Gaussian coil molecules in a matching solvent, can be formulated as follows ... 

In any case, it can be demonstrated with the aid of the dumb-bell model that eq. (5.10) is a much better approximation for statistical coil molecules than eq. (5.11 a) for rigid rods. Two cases are considered for the purpose A rigid dumb-bell of fixed length hr and an elastic dumb-bell, according to the usual definition, possessing a root mean square length (Kyi. ... 

Thus, the observations made at the beginning of this section [see eqs. (5.10) and (5.11a)] with respect to coil molecules and rigid rods, are confirmed for the behaviour of the dumb-bell models. In particular, a comparison of eqs. (5.18) and (5.22) shows that, other than for the intrinsic viscosity, the Maxwell constant can only be calculated when, besides (hi) also (hi) is known. This remark will be of importance for the next section, where theories for short chain molecules will be discussed. 

Kuhn for statistically coiled molecules. The two dotted lines denoted by F and N stand for the free-draining and the non-draining case of Zimm s theoty for Gaussian coils. The hatched area indicates the area where the experimental points obtained on solutions of anionic polystyrenes are located (See Fig. 3.1). 

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