Impermeable coil

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

It should be pointed out that in the Zimm, or impermeable coil, limit the specific value of R ff is uninfluential since the friction coefficient C( ) = C/v( ) is independent of R ff, see Eqs. (3.1.7 ) and (3.1.9), v( ) > 1. This is not true in the partial-draining case, in which v(q) is of order unity although larger than the Rouse free-draining limit v(q) = 1, and its full expression must be considered. 

A nondraining polymer molecule, also referred to as the impermeable coil, can be represented by an equivalent impermeable hydrodynamic sphere of radius R. The frictional coefficient of this sphere which represents the frictional coefficient of the non-draining polymer coil can thus be written,... 

The dimensions of a linear macromolecule in different states have been considered [51]. The following states are known [51, 52] to he the most typical 1 - compact globule, 2 - coil at the 0-point, 3 - impermeable coil in a good solvent, 4 - permeable coil (the state typical of rigid-chain macromolecules), 5 - completely uncoiled rod-like macromolecule. 

In the case of percolation clusters. Equation (11.10a) gives D = 2 lor d = 4/3 and d=3 [57], while for linear polymers, D = 5/3 for d = 1 and d = 3 [61], which corresponds to an impermeable coil in a good solvent (see Table 11.2, state 3). The resulting dimensions are typical of macromolecular coils in monomeric solvents. Using the concept of polymer fractals, one can answer the question of what would happen if the monomeric solvent is replaced by a polymeric one, i.e., whether the polymer clusters and the clusters of a high-molecular-mass solvent would be separated from one another or entangled . This question can be answered by the equation [61] ... 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

If a blob is considered as an non-draining (impermeable) coil of the radius where is the screening length of the excluded volume (subsection 3.1.1) and hydrodynamic ( ) interactions among segments, then... 

The extremely simple expressions for intrinsic viscosity at zero gradient and sedimentation coefficient containing in the denominator the square root of molecular weight agree very well with experimental data of theta solutions in the whole range from fully drained to impermeable coil. They turned out to be very good approximations of the more complicated expressions derived by Kirkwood-Riseman and Debye-Bueche ° and hence were used for the calculation of intrinsic viscosity of polydisperse samples. 

In view of these considerations, it is not surprising that experimental a values vary systematically with decreasing solvent goodness. As the solvent goodness decreases, the chain becomes more tightly coiled so that flow streamlines penetrate the coil to a lesser extent. In an extreme situation we can imagine the coil so impermeable to the solvent flow that it behaves as... 

Theoretical estimates of the quantity zv — 1 are in the range from 0.5 (non-draining Gaussian coil), to 1.11 (draining coil with excluded-volume interaction). A compilation of empirical values of K and of the power exponents for different polymers and different solvents may be found in the literature (Flory 1969, Tsvetkov et al. 1964). The empirical values of the exponent zv — 1 do not exceed 0.9, which indicates significant impermeability of the macro-molecular coil in a flow. We may note that once a relation of type (6.24)... 

In order to use Eq. (3.175) it is necessary to assume that the polymer coils are impermeable and noninteracting. Since the natural rubber is a high-molecular-weight flexible polymer and the solution is very dilute, these conditions may... 

PEG is a flexible molecule with a configuration in solution corresponding to that of a random coil (39). Assuming the coil is impermeable to the solvent, a radius of gyration of 18-20 A. was computed from viscosity measurements (38). With the further assumption, this is the actual... 

Equations (1.28) to (1.30) do not apply to flexible chain molecules, which are not rigid and can exhibit fluctuations in conformation. Here, one approach is to ignore shape anisometry and to make the assumption v = 2.5 in Eq. (1.30). This enables determination of an impermeable sphere-equivalent hydrodynamic volume for flexible chain macromolecules from [ ], provided that M is known. As noted above, the Mark-Houwink-Sakurada equation [Eq. (1.23)] is often used to relate [ j] to M when dealing with flexible coils. 

As noted above, for impermeable linear flexible coils, based on the assumption that [jj] Rg/M, the exponent in Eq. (1.23) is expected to vary between 0.5 for... 

Owing to the intrachain interaction the theoretical treatment of [r/] and Dq in non-0 solvents is far more difficult than that in 0 solvents. Usually, though still unjustified theoretically, it is assumed that the polymer coil in non-0 solvents is also impermeable to solvent molecules. Making this assumption and going to... 

The intrinsic viscosity ( /] is obtained by extrapolation of reduced viscosity of the dilute polymer solution (q-qs)lcqs> to zero polymer concentration, c—>0 (here rj is the viscosity of the polymer solution and the viscosity of pure solvent). In the nondraining limit of large N, the coils behave in a shear flow as impermeable for the solvent particles of effertive radius J ,. In dilute-solution limit, the Einstein equation i/ = i/s[l+ (5/2) ] applies, where is the volume fraction of particles in the solution. Hence, the intrinsic viscosity [i/] measures the (inverse) average intramolecular concenttation of the monomer units assuming that they are confined within a sphere of radius J ,. 

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