Volume of the polymer coil

Relative methods measure properties that depend clearly on molecular weight, for example, the hydrodynamic volume of the polymer coils (GPC, vis-cosimetry) or their solubility as a function of chain length. However, these measurements can only be evaluated with respect to the molecular weight of the mac-... 

What is the size of the overlapping volume The complete independence of the constant in Equation 1 from the degree of polymerization shows that the overlapping volume always consists of the same portion of the volume of the polymer coil (12). This can be easily understood by assuming that two polymer coils are able to migrate nearly unhindered through each other. Then the mean depth of permeation and, therefore, the time of overlapping is determined only by the statistics of the free Brownian motion. Equation 1 is based on this assumption. 

For determining the molar mass of branched polymers gel permeation chromatography can be used. An important quantity in this connection is the hydrodynamic volume of the polymer coil, which, as shown before in Eq. (9.27), is proportional to the product [rj M. According to Benoit and co-workers (1966) the hydrodynamic volume is the key size parameter in the establishment of a universal calibration curve for gel permeation chromatography columns (see Chap. 2) if log (h/]M) is plotted versus the elution volume for a variety of polymers, the data fit a single curve. 

The volume of the polymer coils l poiymer can be described via the ratio of the mass polymer of t e polymer to its density Pequ- This density p qu does not correspond with the density in the dry state, but with the density of the polymer in solution, where solvent molecules surround the polymer chain ... 

Therefore, we expect the scaling law for the correlation time t as being proportional to the molecular volume of the polymer coil, and in view of Rg N, ... 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

According to his interpretation, both the degradation characteristics and the extent of drag reduction are determined by the relationship between compact and flexible chain segments. As a consequence of this, it is not the absolute volume of the (gel) coil which is the decisive value, but the behavior of the polymer chain. 

The calculation proceeds as follows (13). First, we determine the frequency of encounter of two polymer chains following Smoluchowskis treatment. After two chains have come into contact with each other, we allow them to move freely in all directions so we obtain a mean volume and a mean lifetime of the overlapping. This mean volume is given by about 1/10 of the diameter of the polymer coil. This treatment is illustrated in Figure 4. 

Garvey et al.85) made a similar sedimentation study on poly(vinyl alcohol) adsorbed on polystyrene latex particles. Adsorbance of the polymer was also measured. Both the thickness of the adsorbed layer and the adsorbance increased linearly with the square root of the molecular weight. The volume occupied by a polymer molecule in the adsorbed layer was approximately equal to that of the effective hydrodynamic sphere in bulk solution. However, the measured values of LH were greater than the hydrodynamic diameters of the polymer coils in solution. Thus, it may be concluded that adsorbed poly(vinyl alcohol) assumes a conformation elongated in the direction normal to the surface. 

Due to the chain architecture and the large size of the macromolecules, the wetting behaviour of polymer liquids can be different from that of simple liquids. The effect becomes particularly strong when the dimension of the liquid phase, e.g. film thickness and droplet diameter, approaches the dimension of the polymer coil. In addition to the spreading coefficient and the surface pressure effects, entropic elasticity of the polymer chain provides a strong contribution to the free energy for a constant volume V0=Ad ... 

If a polymer molecule in solution behaves as a random coil, its average end-to-end distance is proportional to the square root of its extended chain length (see page 25) - i.e. proportional to Ai 5, where Mr is the relative molecular mass. The average solvated volume of the polymer molecule is, therefore, proportional to M 5 and, since the unsolvated volume is proportional to A/r, the average solvation factor is proportional to (i.e. Af 5). The intrinsic viscosity of... 

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... 

M and v are the molecular weight and the partial specific volume of the polymer, jjo p are the viscosity and the density of the solvent, respectively, and P and O are functions of relative chain length L/A and of the parameter of hydrodynamic interaction, d/A, respectively. These functions have been represented in an analytical form and tabulated over a wide range of changes in the L/A and d/A parameters At extremely high molecular weights (at IVA -> ), functions P and ap oach an asymptotic limit P— Po = 5.11 — 4>, = 2.862 x 10 (the Flory constant). This corresponds to the conformation of a hydrodynamically undrained Gaussian coil. 

The overlap concentration can be estimated as the concentration at which the number of coils per unit volume, v, times the volume pervaded by a single coil, R, is roughly unity, where Rg = R )q /Vb is the radius of gyration of the polymer coil. Now v = cNa/M, where c is the mass per unit volume of polymer in solution and Na is Avogadro s number. Therefore, in a theta solvent (a solvent at a temperature where polymer excluded-volume interactions are negligible see Section 2.3.1.2), the overlap concentration, c, is given by... 

Depletion flocculation is produced by addition of a free nonadsorbing polymer [7]. In this case, the polymer coils caimot approach the particles to a distance A (this is determined by the radius of gyration of free polymer, Rq), as the reduction in entropy on close approach of the polymer coils is not compensated by an adsorption energy. The suspcakesension particles or emulsion droplets will be surrounded by a depletion zone with thickness A. Above a critical volume fraction of the free polymer, the polymer coils wiU be squeezed out from between the particles... 

The shape and size of macromolecules together with the segment distribution within these forms determine the excluded volume of the polymer. Compact molecular shapes such as helices, ellipsoids, and spheres have only an external (intermolecular) excluded volume the space occupied by a given volume in space cannot be occupied by others, and so is an excluded volume for other molecules. Coils, on the other hand, with their loose internal structure, also have, additionally, an internal (intramolecular) excluded volume, since the space occupied by one segment is not available to another segment of the same molecule. 

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