Rigidity kinetic

The first term of the expression is proportional to the solvent viscosity t]s and the second to the internal viscosity (kinetic rigidity) of the macromolecule, so that measurement of the anisotropy of solutions in different solvents makes it possible to estimate the quantity... 

The problem of the relationship between the equilibrium and the kinetic rigidity of the chain is of paramount importance since both the behavior of a chain molecule in solution and the main properties of polymer materials are related to these molecular characteristics. 

The chains of polypeptides and polynucleotides in a helical conformation are even more cyclized by hydrogen bonds. They are characterized by very high equilibrium and kinetic rigidity ... 

In a laminar flow or in an external potential field a polymer molecule is subjected to forces that can both make it rotate as a whole and cause a relative shift of its parts leading to a deformation, i.e. changing its conformation. Which of these two mechanisms of motion predominates depends on the ratio of times required for the deformation and rotation of the molecule. If the time of the rotation of the molecule as a whole, tq, is shorter than the time required for its deformation, t r<, the deformation mechanism of motion will predominate and the molecule will be kinetically flexible. To characterize quantitatively the kinetic rigidity of chain molecules Kuhn has introduced the concept of internal viscosity - a quantity describing the resistance of the molecule to a rapid charge in its shape. Later, the theory of internal viscosity has been developed by Cerf ... 

In principle, the study of the motion of the macromolecule by the action of rotating forces can be used for characterizing its kinetic rigidity. 

The intrinsic viscosity of a solution of molecules described by a kinetically rigid chain necklace that can adopt any conformation (from the straight rod to the spherically symmetrical bead distribution) is given by > 2)... 

Rotational friction of a kinetically rigid worm-like chain has been considered by Hearst ). The position of the centers of hydrodynamic resistance (beads) in the worm-Uke model used by Hearst is determined in system of XYZ coordinates. Its origin coincides with the middle point of the chain and the direction of the Z axis coincides with the chain direction at this point (Fig. 11). It is assumed that the distribution of chain elements (beads) is cylindrically symmetrical with an axis of symmetry Z. In a molecular system of XY Z coordinates this distribution is given by... 

Comparisons of dynamo-optical properties of rigid- and flexible-chain polymers lead to the important conclusion that usually polymer molecules with high equilibrium chain r idity are also characterized by a high kinetic rigidity (Fig. 12). 

Hence, to understand correctly the mechanism of flow birefringence of r d-chain polymers it is necessary to take into account both the asymmetry of the pe of their molecules and their optical anisotropy. Both the properties are the main properties of the molecule and if one of them is absent, flow birefringence in solution of kinetically rigid molecules is impossible. 

As before, brackets < > in Eq. (55) refer to averaging over all chain conformations. This averaging is necessary because the conformations of various kinetically rigid molecules in solution, at a given moment, vary for different molecules and each of them is characterized by certain values of - 72. W and b, although they are virtually frozen. 

Curve I describes the dependence of y on x obtained for kinetically rigid worm-like chains disregarding the dependence of p on x. The asymptotic limit of Curve I is 0.833. .. (instead of y = 1 for Curve 4) because in Reference the anisotropy of the Gaussian chains is taken to be (3A/2 rather than 3/3A/5 used for plotting Curves 1 -4. 

For a solution of kinetically rigid particles (molecules) identical in size and shape, the characteristic orientation angle is uniquely related to coefficients of rotatory diffusion Dj and rotational friction W of a particle... 

G vs. X = 2 L/a for kinetically flexible chains with local rigidity according to the theory in Ref. A, B = values of G for kinetically rigid Gaussian coils at strong and weak hydrodynamic interactions, respectively... 

In contrast, if rigid chain molecules are of a rod-like shape (for a worm-like model x— 0), there is no conformational polydispersity in the assembly and comparison of Eqs. (61) and (62) gives G = 0.5. This is close to the value of G for a kinetically rigid (Jaussian coil both for weak (G = 0.667) and strong (G = 0.504) hydrodynamic interactions. 

Unfortunately, this procedure is not possible for various classes of polymers with moderate equilibrium chain rigidity shown in Fig. 23 since for all of them the dependence of [n]/[7j] on M cannot be expressed by a universal function A = A (x). The fact that for these polymers the experimental dependences A (x) differ greatly from Curve III in Fig. 23 implies that the dynamo-optical properties of their molecules cannot be described in terms of the theory of kinetical rigid chains. Presumably, flow birefringence in solutions of these polymers is related, to a certain extent, to the kinetic flexibility of their chains. 

As already mentioned, for polymers with high equilibrium chain rigidity the experimental dependence of [n]/[t ] on molecular we t agrees with the theoretical relation (59) for kinetically rigid chains. Thhi equation can be utilized for the determination of the parameters A and 0 from experimental data on flow birefringence of polymer or fraction of varying molecular we t. 

These G values are slightly higher than the theoretical values for kinetically rigid drained coils (denoted on the ordinate by B) and increase with decreasing chain lei h for all the polymers investigated. This may be due to some polydlspersity of samples the effect of which should probably increase with decreasing characteristic angles [x/g]. ... 

Values of x/gj plotted according to data in Ref. 43) and molecular weights according to data in Ref.43) (a) 4 poly(amide benzimidazole) in sulfuric acid 44) ( ) 5 poly(p-phenylene-1,3,4-oxadiazole) in sulfuric acid43) (v). The curve corresponds to theory as in Ref. 1). A and B are theoretical values for kinetically rigid Gaussian coils with strong (A) and weak (B) hydro-dynamic interactions 4 3)... 

In contrast, the motion in the electric field of kinetically rigid molecules for which tq is less than should be governed by the mechanism of the orientation of the molecule as a whole and the rate of this process should be determined by the rotational friction coefficient W or rotatory diffusion coefficient Dj (Eq. (53)). Hence, the study of the orientation kinetics of kinetically rigid macromolecules in the electric field permits the determination of their rotational mobility (i.e. W and Dj). 

Fig. 58a, b. Possible polarization mechanisms of a chain molecule in the electric field, a Kinetically flexible chain (deformational mechanism) b kinetically rigid chain (orientational mechanism)... 

For this purpose, the Kerr effect in the alternatii (sinusoidal) field can be used. The theory of this effect for solutions of kinetically rigid molecules has been formulated by Peterlin and Stuart According to this theory, the character of the dependence of the observed birefringence An on frequency v = co/2 n of the applied field clearly differs for the two cases ... 

In the same frequency range, the dispersion of the dielectric increment Ae of solution can be observed. The curves of dielectric dispersion almost coincide with the dispersion curves for EB (Fig. 59) so that both mechanisms of molecular motion are identical and are represented by dispersion Eq. (81) for kinetically rigid mol ules. 

The orientational mechanism of EB in solutions of r id-chain polymers and the possibility of determining rotatory diffusion constants of their molecules from dispersion curves may be utilized for the characterization of equilibrium conformational properties of their drains. The theory of rotational friction of kinetically rigid molecules developed by Hearst makii% use of the statistics of worm-like chains can be employed for this purposes. The results of this theory for the two limiting cases of molecular conformation refering to the slightly bent rod and the worm-like coil are expressed by Eqs. (27) and (28) (Sect. 2.3). 

This discussion demonstrates that, in contrast to the equilibrium (static) rigidity, the concept of the kinetic rigidity of a chain molecule is not universal. The kinetic flexibility of the chain depends on the character of the process in which this flexibility is manifested. 

Equation (85) represents a general relationship between the Kerr constant K and the dipolar and optical properties of a kinetically rigid particle. To establish the quantitative dependence of K on the conformation and structure of a rigid-chain polymer molecule, the molecular model describing its electro-optical properties should be specified. For this purpose, we use a kinetically rigid worm-like chain, just as for the study of the FB problem. 

Fig. 73. Graph of relative Kerr constant K/Ko=, = q vs. x = 2L/A for a kinetically rigid worm-like chain. Figures on curves denote angle d formed by the dipole of the monomer unit and the chain direction...

The quantitative measure of kinetic rigidity of the molecule is the time required for a change in its conformation. It is the kinetic rigidity that determines whether the molecule is oriented as a whole by the orientational mechanism or by the deformational mechanism. This is reflected experimentally in the value of relaxation time t. 

The experimentally determined relaxation times of dipole orientation make it possible to obtain an important hydrodynamic characteristic of the molecule its rotational diffusion coefficient. For kinetically rigid mole-... 

An important property of relaxation spectra of chain models exhibiting limited thermodynamic bending flexibility has been establidied It was found that the initial slope of = and CP2(cos 6)) = 3/2 [ — 1/3] does not depend on the change in the parameter of thermodynamic flexibility. Hence, the initial slope of F(r/ij) and the characteristic time Tjnit = r i = < 1 /t> are determined only by the (mrameters of a kinetically rigid segment in the continuous model (by its length and effective frictional coefficient) or by the size and micro-struaure of a set of kinetic units in discrete lattice models. On the other hand, the greater the thermodynamic chain r idity, the more the plot of F(r/tj) deviates from the initial slope. 

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