Pervaded volume

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

The number of molecules per unit volume v is 6.02 x 1023 c/M. If the molecular centers are distributed randomly, the product vV is the average number of other molecules with centers lying within the pervaded volume of any one molecule. Accordingly, v V is a measure of the potential degree of coil overlap, and with = 2.68 xlO23 ... 

The pervaded volume V is the volume of solution spanned by the polymer chain... 

If the volume fraction cf) of the polymer solution is equal to the overlap volume fraction fill space and chains are just at overlap (0 = i i ) (see Fig. 1.15). ... 

In the definition of the overlap volume fraction 0, the pervaded volume V is taken at, since polymer size and hence its pervaded volume may change with concentration. 

However, polymer coils overlap and dominate most of the physical properties of semidilute solutions (such as viscosity). Thus, adding a very small amount of polymer to a solvent can create a liquid with drastically different properties than the solvent. This unique feature of polymer overlap is due to their open conformations. Linear polymers in solution are fractals with fractal dimension I) < 3. In semidilute solutions, both solvent and other chains are found in the pervaded volume of a given coil. The overlap parameter P is the average number of chains in a pervaded volume that is randomly placed in the solution ... 

At the overlap volume fraction (for 4> = < > ) P=, and as the concentration of linear chains is increased P steadily grows, reflecting the presence of additional chains inside the pervaded volume of each molecule. Notice that the overlap parameter counts the number of whole chains that share the pervaded volume. In reality, small parts of numerous chains are within... 

A macromolecule can adopt many conformations, defined by relative locations of its monomers in space. Polymer conformations are often self-similar (fractal) with pervaded volume Fmuch larger than their occupied volume Avmoti where Vmon is the monomer volume. The overlap volume fraction... 

A polymer with molar mass M = 10 g mol is at overlap in a solution with concentration c — 1.67 x 10 gcm , What is the pervaded volume V of each polymer chain ... 

A mean-field estimate of this probability can be made for the general case of an ideal chain in cf-dimensional space by replacing a chain with an ideal gas of N monomers in the pervaded volume of a coil R. The probability of a given monomer to contact any other monomer within this mean-field approximation is simply the overlap volume fraction (j>, of a chain inside its pervaded volume, determined as the product of the monomer volume and the number density of monomers in the pervaded volume of the coil NjR ... 

For good solvents (z > 1), chains repel each other strongly and do not interpenetrate. The volume excluded by a chain is of the order of its pervaded volume and the molar mass of the chain is M ... 

At very low concentrations, the polymers exist as isolated coils that are very far apart. The concentration increases moving from left to right in Fig. 5.1. At T=(9, there is a special concentration that equals the concentration inside the pervaded volume of the coil. This is the overlap concentration for -solvent [see Eq. (1.21)] ------------------------------... 

An important difference between randomly branched and linear polymers is that the fractal dimension of branched polymers is larger than the dimension of space (d—3). This severely limits the applicability of the mean-field theory to the crosslinking of long linear chains, called vulcanization. Long chains in the melt have a fractal dimension of P = 2, which leaves lots of room inside the pervaded volume of the chain (i.e., filled by other chains in a polymer melt). The extra room created by the linear sections between crosslinks allows the fractal dimension of P = 4 to exist in three-dimensional space on a certain range of length scales (see Section 6.5.4). 

In order for the fragments of molecules containing N monomers to be at overlap (for all A, the combined pervaded volume has to be independent of N (and equal to the total volume of the system) ... 

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