Excluded volume, definition

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 identifier HC means that this is the solvation free energy for the case of a hardcore solute with the excluded-volume region established by the inner-shell definition. 

Thus, there are two limitations of the pycnometric technique mentioned possible adsorption of guest molecules and a molecular sieving effect. It is noteworthy that some PSs, e.g., with a core-shell structure, can include some void volume that can be inaccessible to the guest molecules. In this case, the measured excluded volume will be the sum of the true volume of the solid phase and the volume of inaccessible pores. One should not absolutely equalize the true density and the density measured by a pycnometric technique (the pycnometric density) because of the three factors mentioned earlier. Conventionally, presenting the results of measurements one should define the conditions of a pycnometric experiment (at least the type of guest and temperature). For example, the definition p shows that the density was measured at 298 K using helium as a probe gas. Unfortunately, use of He as a pycnometric fluid is not a panacea since adsorption of He cannot be absolutely excluded by some PSs (e.g., carbons) even at 293 K (see van der Plas in Ref. [2]). Nevertheless, in most practically important cases the values of the true and pycnometric densities are very close [2,7],... 

The lower cycle represents the chemical changes occurring during polymerization and relates them to the free volume of the system. In general, free volume of a polymer system is the total volume minus the volume occupied by the atoms and molecules. The occupied volume might be a calculated van der Waals excluded volume [139] or the fluctuation volume swept by the center of gravity of the molecules as a result of thermal motion [140,141]. Despite the obscurity in an exact definition for the occupied volume, many of the molecular motions in polymer systems, such as diffusion and volume relaxation, can be related to the free volume in the polymer, and therefore many free volume based models are used in predicting polymerization behavior [117,126,138]. 

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

We recall that we are interested in universal features, i.e. properties that are independent of the microstructure. Our model simple as it is - still has a definite microstructure n = no Gaussian segments of length = (h interacting via an excluded volume of strength q. Taking universality... 

By definition the excluded volume limit is reached if under renormalization the coupling constant approaches the fixed point so closely that we can replace it by 8. ... 

Thus these ratios are observable quantities, taking a well defined numerical value in the excluded volume limit. This is the definition of a universal ratio ... 

As already stated in the Introduction, a problem that sometimes arises in pharmacophore approaches is the need to take into account possible adverse steric interactions between inactive compounds in a dataset and the target protein counterpart In these situations, the definition of ligand-forbidden zones by means of the addition of excluded volume spheres to a pharmacophore is nowadays considered a reasonable and effective improvement. 

As described above VWS and SAS are easily defined as sets of spheres centred on atoms. This definition, however, does not apply to SES in this case in fact, the pair of surfaces delimiting the boundary between the excluded volume and the solvent cannot be defined using spheres. There are several algorithms which translate the abstract definition of the SES into a complex solid composed of simple geometrical objects from which the surface can be easily tessellated. 

The last method which will be considered here is the GEPOL, which was first elaborated in Pisa by Tomasi and Pascual-Ahuir [15]. GEPOL will be presented in two steps in this section we will treat the excluded volume filling, whereas the definition of surface elements will be given in the next section. 

The definition of excluded volume in GEPOL which is exact only if we consider an infinite generation of supplementary spheres, replaces the complex geometrical structure of torus and curvilinear prisms used in BLSURF, Connolly and DEFPOL by simply extending the set of atomic spheres. This aspect is very important from the computational point of view, because it allows an easy development and implementation of well-defined tessellations. 

The origins of the present three-dimensional molecular-level branching concepts can be traced back to the initial introduction of infinite network theory by Flory [62-65] and Stockmayer [66, 67], In 1943, Flory introduced the term network cell, which he defined as the most fundamental unit in a molecular network structure [68]. To paraphrase the original definition, it is the recurring branch juncture in a network system as well as the excluded volume associated with this branch juncture. Graessley [69, 70] took the notion one step further by describing... 

Let us begin with the estimation of the polymer volume fraction inside the coil formed by one long semiflexible macromolecule. It is well known that this estimation depends essentially on the strength of the excluded volume effect, i.e. on the value of the parameter z = vN /a, where v is the excluded volume of a monomer and a is the spatial distance between two neighbouring monomers. To be definite let us adopt for a moment the model shown in Fig. 1 b. Then, if we choose one segment as an elementary monomer, v 6. (see Eq. (2.4)) and a i.e. z p Consequently, the excluded volume effect is pronounced at N p and negligible at N [Pg.77]

Steric descriptors and/or -> size descriptors representing the volume of a molecule. The volume of a molecule can be derived from experimental observation such as the volume of the unit cell in crystals or the molar volume of a solution or from theoretical calculations. In fact, analytical and numerical approaches have been proposed for the calculation of molecular volume where the measure depends directly on the definition of - molecular surface-, -> van der Waals volume and -> solvent-excluded volume are two volume descriptors based on van der Waals surface and solvent-accessible surface, respectively. 

Here we calculate the size of ideal randomly branched polymers, ignoring excluded volume interactions and allowing each molgcule to achieve the state of maximum entropy (recall the discussion of ideal chains in Chapter 2). Since branched molecules have many ends, the mean-square end-to-end distance used to characterize the size of linear chains is not appropriate for them. The simplest quantity describing the size of branched molecules is their mean-square radius of gyration j g [see Eq. (2.44) fonthe definition]. 

The second virial coefficient f3i is actually the excluded volume of the at molecule to another rod with orientation a3. According to the definition, the exclusion volume is the volume occupied by one molecule in which the mass centers of other molecules are not allowed to touch. For cylindrical rods with length L and diameter I). the exclusion volume is schematically shown in Figure 2.2. In this figure the two particular cases are depicted, i.e., two cylinders are parallel or perpendicular to each other, fti is dependent on the shape of the rigid rods. The expression for cylinders can be approximately expressed by... 

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