Macromolecular dimensions

In this section, we present statistical arguments for determining the size of polymer chains. Consider as an example a polyethylene chain 

Chains can rotate around dihedral angles. Inevitably then, the average length of polyethylene chains will be smaller than Lnax-The simplest model to determine the average length (L) of chains with M monomers is the freely jointed ideal chain. In a freely jointed chain, the dihedral angles can take on any value with equal probability. In an ideal chain there are no interactions between the monomers. In other words, the polymer is assumed to be a random walk in three dimensions (Fig. 11.2). Lord Rayleigh developed the model of unrestricted random walks in the early 19(X)s and Werner Kuhn first adopted this model in 1934 for polymer conformations. 

The polymer length (Fig. 11.3) is simply the vectorial sum of all bond length vectors projected on the dimension of the end-to-end vector of the chain  

If we denote with x, the position of monomer i, and with m, = — X, the bond vector between monomers, we can write 

Since the chain is freely jointed, we have (cos( )) = 0, assuming M 00. Then 

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F. T. Wall, F. Mandel. Macromolecular dimensions obtained by an efficient Monte Carlo method without sample attrition. J Chem Phys 63 4592, 1975. 

The silica nucleated only on the biomacromolecules. By decreasing the THEOS amount in solution from 3 to 0.03 wt.%, we observed a curtailment of the particle diameter to the macromolecular dimension. This constitutes evidence that most of the silica generated in situ was used to form a shell. 

MO 32 [16] and Mo2S6 [17]/ Mo368 [18] clusters. These clusters are compared (to scale) with C60 to demonstrate their macromolecular dimensions [4, 5]. 

As early as in 1958, Rich and Blow showed that NaDC forms a helical complex of macromolecular dimensions [72]. They observed that unusual changes occurred in NaDC (0.1 m) in the presence of glycylglycine (0.1 m). There was a large initial increase in the viscosity of the medium, and they were able to draw glassy and brittle fibers, which showed a regular structure by X-ray diffraction. With time the solution reached a gelatinous state and they could lift the cohesive mass, which was more like a gelly. 

According to the general definitions of the coil and the globule241, the macromolecule is in the coil state, if the fluctuations of the monomer concentration within the macromolecule are of order of the monomer concentration itself and the correlation radius of the fluctuations of concentration is of order of the macromolecular dimensions while in the globular state the concentration fluctuations are small compared with the concentration and the correlation radius is considerably smaller than the globular dimensions. 

In particular, it is well known that, if the macromolecule is supercooled below the 0 temperature, the phase transition isotropic coil-isotropic globule occurs. We emphasize that for the semiflexible macromolecule this is the peculiar phase transition between two metastable states. It should be recalled that the theory of the transition isotropic coil-isotropic globule for the model of beads is formulated in terms of the second and third virial coefficients of the interactions of beads , B and C24). This transition takes place slightly below the 0 point and its type depends on the value of the ratio C1/2/a3 if Cw/a3 I, the coil-globule transition is the first order phase transition with the bound of the macromolecular dimensions, and if C1/2/a3 1, it is a smooth second order phase transition (see24, 25)). 

By means of gel chromatography, structure of macromolecules can be elucidated (e.g., their branching [53]), and in some cases also their tacticity [54]. GPC was applied fairly often to the study of the interactions of the macromolecules such as association and aggregation [54-59], the association with ligands [60,61], or the preferential solvation of macromolecules in mixed solvents [62], and the interactions in the polyelectrolyte-salt-solvent [63] systems. GPC can provide information concerning the polymer-solvent interactions [64] as well as about the changes in the macromolecular dimensions with concentration in solution [65,66]. 

Miscible polymer blend Polymer blend, homogenous down to the molecular level, in which the domain size is comparable to macromolecular dimension, associated with the negative value of the free energy and heat of mixing,... 

Gomez, C. et al.. Physical and structural properties of barley (1-3), (l-4)-P-D-glucan. Part II. Viscosity, chain stiffness and macromolecular dimensions, Carbohydr. Polym., 32,17, 1997. 

The intrinsic viscosity is a measure of the macromolecular dimensions. Thus, for flexible macromolecules, chain skeleton parameters (bond lengths, valence angles, degree of polymerization, mass of the monomeric unit), the steric hindrance parameter a, or a measure of the hindrance to rotation, and the expansion factor a as a measure of the interaction with the solvent, determine the magnitude of the intrinsic viscosity. In theta solvents, a = 1. Thus, from Equations (9-148) and (9-152)... 

Among the early examples of the successful use of electric fields to probe ionic structures and electrical and optical anisotropies are the linear polyelectrolytes. Basic information about macromolecular dimensions, size, and shape have been derived from the relaxation of field-induced changes in optical properties and in electrical parameters of the electrically and optically anisotropic systems. The analysis of electric conductivity measurements has demonstrated that linear polyelectrolytes are electrically anisotropic. It was established that the extremely large dipole moments, which the electric field produces by displacement of the counterion atmosphere parallel to the long axis of the polyions, are responsible for their orientations in the direction of the external field. 

Despite being an absolute method that provides very important information on the fundamental properties of the polymer in solution, including its conformation, chain stiffness and its interaction with the environment, the use of SLS is limited as it requires expensive equipment and strict experimental conditions including very pure and fully soluble polymers and dust-free. In addition, Zimm plots of experimental data can be irregularly shaped and difficult to interpret when polymer aggregation or association occurs [41]. Several examples of further studies addressing the macromolecular dimensions and conformation of chitin [42] and chitosan [40,43-48] can be found in the hterature. 

The Staudinger index is a measure of the macromolecular dimensions. Thus, for flexible macromolecules, chain skeleton parameters (bond lengths, valence angles, degree of polymerization, mass of the monomeric unit),... 

The effect of dye on the macromolecular dimensions differs from a usual screening effect due to the presence of salts as indicated by the viscosity values given in Table III. 

The chromatographic behavior of low MW salts could be expected to be less complex than that of synthetic polyelectrolytes with variable macromolecular dimensions, and also simpler than that of proteins, for which hydrophobic interactions need to be considered. For this reason, and also on historical grounds, we will first review the literature on the elution of salts. Next we will consider the behavior of synthetic polyelectrolytes, treating polycations separately, because these compounds pose special difficulties. The role of electrostatic effects in the bEC of proteins will be the third topic. Lastly, we discuss models put forward to semi-quantitatively treat electrostatic effects. 

A number of fundamental questions related to chromatographic equilibria and dynamics have been explicated. The fundamental issue of the nature of the macromolecular dimensions that determine Ksfc s the subject of Chapter 1. The resolving power of aqueous SEC columns is considered from the perspective of column pore size distributions in Chapter 6. These and other aspects of efficiency are explored in detail in Chapter 7. Interactions between macromolecules and stationary phases are covered in both Chapter 2 (hydrophobic effects) and Chapter 3 (electrostatic effects). 

Note the presence of two screening lengths and is of macromolecular dimension and is simply related to the radius of gyration = Kg/y/l. p, on the other hand, is short range at high density and is determined from the core conditioa Note also that at high densities there is a wide intermediate region of intersite separation, r where power law correlations occur. 

Upon the physical properties both in the solid state and in solution [2,3]. Furthermore, the macromolecular dimensions can be controlled since the synthesis involves a repetitive sequence of steps. Dendrimers are synthesised from monomers of the AB type (where X 3= 2) through a step-growth polymerisation process. 

Miscible polymer blend—Blends, homogenous to the molecular level, or having the domain size comparable to the macromolecular dimension, negative value of the free energy of mixing, AGm AHm 0, and AGJd(j> > 0. 

Figure 1 The coil in the center of the panel is intended to represent the length scales of a single coil in an amorphous polymer melt, nominally the radius of gyration (ffg). The other cartoons depict the different confinement geometries discussed, where the length scales of confinement begin to approach the macromolecular dimensions.

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