Hydrodynamic chain dimension

It has been established that Ko normally is independent of the solvent and the molecular weight of the polymer, though often dependent to some extent on the temperature. It is therefore possible to deduce values for the expansion factor in good solvents from intrinsic viscosities measured in them. From Eqs. (3.181) and (3.184) the linear expansion factor a.jj, which is a measure of long range interactions and pertains to hydrodynamic chain dimensions, is thus given by... 

For small expansions of Gaussian polymer chains the expansion parameter, Oif), for the hydrodynamic chain dimensions is given by a closed expression of the form... 

Little is known about the chain dimensions of PPC in solution. Recently, a comparison of the hydrodynamic volume of polystyrene (PS) and PPC has been reported for tetrahydrofuran (THF) as solvent in connection with a size exclusion chromatography (SEC) analysis [78, 79]. The basis for the calculation was the assumption of an immortal PO/CO2 alternating copolymerization, and thus that absolute values of Mn relate to starter and PO/CO2 ratios. Narrow molecular weight distributed PPCs with various molecular weights were prepared from adipic acid as starter. The absolute molecular weight has a relationship of K = K(ps)... 

Usually, the values of the hydrodynamic diameter, d,calculated from experimental data on sedimentation and diffusion correspond in their order of magnitude to transverse chain dimensions obtained from structural analysis. However, the precision of the determinations of d, using hydrodynamic data, is not high. 

From the above considerations it is clear that measurements of chain dimensions and hydrodynamic properties of an isolated, unperturbed ring polymer... 

The intrinsic viscosity (IV) is physically related to the "hydrodynamic volume" of the chain (its volume during flow), so that when the chain dimensions (as defined by end-to-end distance) increase in the solution, the IV will increase en suite. In practice, a large deviation is found between IV in a "bad" solvent (worst conditions at the so-called theta (0) state, where a = 0.5), and in an "excellent" one (where high values of IV are fotmd and a = 0.8). 

It is obvious that both intra-molecular and inter-polymer phenomena in polyelectrolyte solutions are dominated by coulomb forces. Repulsive interactions would be expected to diminish aggregation. At the same time, extended chain dimensions and the long-range character of electrostatic effects can promote forms of ordering unique to polyelectrolytes. Hydrodynamic, spectroscopic and thermodynamic methods have all been brought to bear on the coupled problems of conformation, counterion distribution and inter-polymer ordering in polyion solutions. These approaches are well represented in Part III by the works on Paoletti, Berry and Jamieson. 

With regard to differences in polymer behavior in solution versus the bulk state, several points must be made. Clearly, it is now well-established that the choice of theta solvent can affect chain dimensions to some extent [42-44, 46, 47]. Hence, only the chain in an amorphous melt of identical neighbors can be considered to be in the unperturbed state. Particularly striking are some of the differences noted in temperature coefficients measured by different techniques. Is it possible that the thermal expansion of a polymer molecule is fundamentally different in the bulk and in solution Can specific solvent effects exist and vary in a systematic way within a series of chemically similar theta solvents Does the different range of temperatures usually employed in bulk versus solution studies affect K Are chains in the bulk (during SANS and thermoelastic experiments) allowed adequate time to completely relax to equilibrium All of these issues need further attention. Other topics perhaps worthy of consideration include the study of the impact of deuterium labelling on chain conformation (H has lower vibrational energy than does H ) and the potential temperature dependence of the Flory hydrodynamic parameter . 

Photon correlation spectroscopy (PCS) is particularly suitable for studies of the hydrodynamic behaviour of polymers, and for studies of the effects of solvency conditions (e.g. solvent, temperature) and skeletal structure (e.g. linear, branched, cyclic) upon chain dimensions. The development of PCS has been greatly assisted by developments in multi-bit processors (which have enabled g r) to be determined with ever greater accuracy) and in the analytical methods by which information is extracted from... 

Hydrodynamic properties of dilute solutions of flexible-chain polymers are of interest in the context of this chapter because they are closely associated with chain dimensions. The concentration... 

Equation [156] underestimates the ratio of Rh/Rg by about 15% ° when compared with the experimental one at the 6 temperature. Similar calculations can be made for Gaussian chains with other architectures. In real chains, the expansion of chain dimension by exduded-volume effects brings about an expansion of the hydrodynamic radius. ... 

It is known that polymer dynamics is strongly influenced by hydrodynamic interactions. When viewed on a microscopic level, a polymer is made from molecular groups with dimensions in the angstrom range. Many of these monomer units are in close proximity both because of the connectivity of the chain and the fact that the polymer may adopt complicated conformations in solution. Polymers are solvated by a large number of solvent molecules whose molecular dimensions are comparable to those of the monomer units. These features make the full treatment of hydrodynamic interactions for polymer solutions very difficult. 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

All factors related to the arrangement of the polymer chain in space are classified as tertiary structure. Parameters measurable directly (the radius of gyration RG, the end-to end distance h, the hydrodynamical radius RH, and the asymmetry in light scattering intensity) or indirectly (interaction parameters, the second virial coefficient A2) are related to the dimensions, such as size and shape of the polymer chain in a specific solvent under given conditions of temperature and pressure. For the exact determination of the coil size of macromolecules, it is necessary to ensure that measure-... 

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