Excluded volume determination

Excluded volume determines space occupancy in biopolymer solutions. Competition between macromolecules for space in a mixed solution determines the phase separation threshold. In a dilute solution of biopolymers, macromolecules hardly interact with one another, individual macromolecules are independent of one another, and biopolymers are cosoluble. The effects of spatial limitations are enhanced by the transition from a dilute mixed solution, to a semi-dilute biopolymer solution where molecules come into contact with one another, interact, compete for the same space, and do not mix in all proportions. 

Thus, in salt solutions, conformations of a polyelectrolyte chain are those of a self-avoiding chain with the excluded volume determined by the electrostatic repulsion between charged monomers. With deaeasing salt concentration, the chain size increases. It becomes comparable with the size of a polyelectrolyte chain in a salt-free solution (see eqn [10]) at salt concentration for which the value of the Debye screening length is equal to ro b up)... 

Solvent-excluded surfaces correlate with the molecular or Connolly surfaces (there is some confusion in the literature). The definition simply proceeds from another point of view. In this c ase, one assumes to be inside a molecaile and examines how the molecule secs the surrounding solvent molecules. The surface where the probe sphere does not intersect the molecular volume is determined. Thus, the SES embodies the solvent-excluded volume, which is the sum of the van der Waals volume and the interstitial (re-entrant) volume (Figures 2-119. 2-120). 

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. 

Figure 8.10 Excluded volume for two spheres (dotted surface) as determined by the distance of closest approach.

The reduced chromatogram contains four peaks the first will be the dead volume peak (which, as has been shown previously, must be the fully excluded peak determined from the retention volume of a salt or solute of large molecular weight). 

Where the slope s determined by the plot is the extension coefficient. The I-1/2 dependence viewed in terms of an increased Debye length can be explained as the electrostatic excluded volume contribution. 

A theoretical investigation of the use of NMR lineshape second moments in determining elastomer chain configurations has been undertaken. Monte Carlo chains have been generated by computer using a modified rotational isomeric state (RIS) theory in which parameters have been included which simulate bulk uniaxial deformation. The behavior of the model for a hypothetical poly(methylene) system and for a real poly(p-fluorostyrene) system has been examined. Excluded volume effects are described. Initial experimental approaches are discussed. 

The volume of the solid phase Vp is usually measured by a pycnometric technique, which measures the excluded volume of a pycnometric fluid, whose molecules cannot penetrate the solid phase of PS. A simple example of a pycnometric fluid is helium [55], The pycnometric fluid fill in all void space (pores) accessible to it, and presumably do not adsorb on the surface of PS. In the case of microporous PSs, measurement of the volume accessible for guests with various sizes allows the determination of a distribution of micropores volume vs. the characteristic size of guest molecules. This approach lays the basis of the method of molecular probes. The essence of this method is in the following we have a series of probe molecules with different mean sizes (dl>d2>d3>---). The pycnometric measurements of the excluded volume will give a series The difference A V=Vpi-Vpi(i>j) corresponds to the volume of micropores with pycnometric sizes of d in a range dt[Pg.283]

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

Before a volume determination can be estimated, a corrected water table contour map is prepared that depicts the configuration of the water table excluding the... 

Ky is the Flory-Huggins interaction parameter between the i and j monomers. In Eq. 6.6, the matrices have a dimension (m) (m). We note that the s-depen-dence of the excluded volume matrix is solely determined by the contribution of the bare susceptibility yoo(Q> ) he invisible matrix component 0 . Finally, combining Eq. 6.6 with Eq. 6.1 the response function in the interacting system is given by ... 

The geometrical structure of the chain in Fig. 2 is determined entirely by 6, (j), and B. The model only describes the linker geometry and does not account for excluded volume effects and other forms of nucleosome-nucleosome interaction it assumes that the core particles are point-like and that they are located at the joints of the linkers, which are straight rods. 

Alternatively, sodium nitro prusside gives a value, close to that of dispersed silica, and, in practice, could be used to determine the total excluded volume or the kinetic dead volume without incurring serious error. Values for the kinetic dead volume measured in this way could be... 

These are repulsive interactions of short range (0.2-0.3 nm) arising from the highly unfavourable overlap of full electron clouds. They are sometimes called steric interactions because they restrict the relative spatial arrangement of pairs of segments on the same or different macromolecules. The excluded volume expression arises from the fact that the volume occupied by one biopolymer molecule in solution is not available to other biopolymer molecules. Thus the size and shape of the biopolymer molecule/particle (as determined by the macromolecular conformation/ flexibility or aggregate architecture) is of prime importance in relation to steric5 interactions. 

To a first approximation, one can assume that the excluded volume term (A/xc) is determined only by the physical volume occupied by the biopolymer molecules/particles (see equations (5.32), (5.34) and (5.35)). The electrostatic term, A if1 = A V i T2, is determined by the ideal Dorman contribution, A corrected by factors to take into account the electrostatic interactions amongst the ions (Ti) and the chain-like character of the polyelectrolyte (T2) (Nagasawa and Takahashi, 1972). 

In the final section, we build on the thermodynamic theories of polymer solutions developed in Chapter 3, Section 3.4, to provide an illustration of how a thermodynamic picture of steric stabilization can be built when excluded-volume and elastic contributions determine the interaction between polymer layers. 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

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