Length of the macromolecules

Multiple covalent bonds are formed in each macromolecule and, in general, statistical, polydispersed structures are obtained. In the case of controlled vinyl polymerizations, the average length of the macromolecule is determined by monomer to initiator ratios. If one views these polymerizations as extraordinarily long sequences of individual reaction steps, the average number of covalent bonds formed/chain may be visualized as shown in Scheme 2 ... 

The above developed assumptions may also be applied to the interactions of macromolecules with small particles, in particular with globulae, particles of sols, if the length of the macromolecule is sufficiently high to be capable of binding simultaneously with more than one particle. 

In any case, the mean square end-to-end distance of a long macromolecule (R2) is small compared to the length of the macromolecule. Whatever its chemical composition, a macromolecule which is long enough rolls up into a coil as a result of thermal motion, so that its mean square end-to-end distance becomes proportional to its molecular length... 

The picture considered in the previous section is idealised one the macromolecule does not exist in isolation but in a certain environment, for example, in a solution, which is dilute or concentrated in relation to the macromolecules (Des Cloizeaux and Jannink 1990). The important characteristic for the case is the number of macromolecules per unit of volume n which can be written down through the weight concentration of polymer in the system c and the molecular weight (or length) of the macromolecule M as... 

One can expect that the parameter of hydrodynamic interaction (2.11) behaves universally for subsequent division of the chain. One can reasonably guess that the quantity (2.11) does not depend on the length of the macromolecule and on the number of subchains. In this case, the hydrodynamic radius of the particle for the Gaussian chain... 

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

In the general case of arbitrary values of h, simple speculations appeared to be useful to determine the dependence of the resistance coefficient on the length of the macromolecule (Gennes 1979). The excluded-volume effects (see Section 1.5) can also be taken into account and one considers the resistance coefficient to be a function of two non-dimensional parameters h and v/b3. For a macromolecule, consisting of N smaller subchains, the friction coefficient can be written as... 

The dependence of the effective friction coefficient on the length of the macromolecule is of special interest. In a case when the hydrodynamic interaction of the particles of the macromolecule may be neglected, i.e. when the coil is, as it were, free-draining, the coefficient of resistance of the latter is proportional to the length of the macromolecule and the coefficient of friction of the particle associated with length M/N is proportional to this length... 

For strongly entangled systems (M > 10Me), the requirement of self-consistency is fulfilled identically, while the quantity x is connected with the intermediate length (see Section 5.1.2, formula (5.8)), or (as we shall see in Section 6.4.4, formula (6.55)) with the length of the macromolecule between adjacent entanglement Me, that is... 

In the last case, the parameter has the meaning of the ratio of a length of macromolecule between adjacent entanglements to the length of the macromolecule (see Section 5.1.2). The parameters t, x, , and Me appear to be equivalent for the strongly entangled systems. One of these parameters is used to describe polymer dynamics in either interpretation. 

One can see that the relaxation times at a > (ip/x) 2 are the Rouse relaxation times of the part of the macromolecule that correspond approximately to the length of the macromolecule between adjacent entanglements Me. There is an interval between slow and fast relaxation times, which is the bigger the longer the macromolecules. 

The self-diffusion coefficient for long chains is proportional to M 1 and is small. In this situation, however, a competing mobility mechanism gives a different dependence of the diffusion coefficient on the length of the macromolecule. This will be discussed further in this section. 

Formula (5.15) defines a certain intermediate scale, which can be compared to the intermediate scale revealed in the consideration of the diffusion of a macromolecule (see expression (5.7)). We ought to believe that the local displacement of any point of the macromolecule should depend neither on the number of subchains nor on the length of the macromolecule, so that we can identify the quantities (5.7) and (5.15) to find the relation between the parameters of the theory at B 1... 

The exponents in the above expressions can be estimated beforehand from the dependence of the limiting values of the characteristic viscosity at low and high frequencies on the length of the macromolecule. 

Investigation of viscoelastic behaviour of linear polymer solutions and melts shows that there are universal laws for dependencies of the terminal characteristics on the length of macromolecules, which allows to interpret these phenomena on the base of behaviour of a single macromolecule in the system of entangled macromolecules (Ferry 1980, Doi and Edwards 1986). The validity of the mesoscopic approach itself rests essentially on the fundamental experimental fact that quantities that characterise the behaviour of a polymer system have a well-defined unambiguous dependence on the length of the macromolecule. 

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... 

The dependence of the polarisability coefficient on the length of the macromolecule follows from equation (10.9) as... 

Of course, these relations are trivial consequences of the stress-optical law (equation (10.12)). However, it is important that these relations would be tested to confirm whether or not there is any deviations in the low-frequency region for a polymer system with different lengths of macromolecules and to estimate the dependence of the largest relaxation time on the length of the macromolecule. In fact, this is the most important thing to understand the details of the slow relaxation behaviour of macromolecules in concentrated solutions and melts. 

Values of Vp calculated on the basis ofx = L/d are presented in Table I. For PBG in the a-helical state, the contour length of the macromolecule may be calculated on the basis of a 1.5 A unit translation along the extended axis per amino acid residue (25). 

It should be stressed that such condensation of counterion by polyion is determined just by the structural parameter that defines charge density along the length of the macromolecule. It is not influenced by external condition, such as Cp or the addition of salt. The fact that the colligative properties of salt and polyelectrolyte are found to be additive [32,33] when salt is added to the polyelectrolyte provides insight with respect to the uniqueness of ( )p and y,. Such behavior is attributable to the inaccessibility of the polyion, the condensed Na" " ions and the solvent associated with the polyion domain, to the measurements being carried out. Their presence as a separate phase, however, is not detectable by the counterion activity measurement in the absence of simple salt. 

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