Volume, excluded calculation

A porous particle contains many interior voids known as open or closed pores. A pore is characterized as open when it is connected to the exterior surface of the particle, whereas a pore is closed (or blind) when it is inaccessible from the surface. So, a fluid flowing around a particle can see an open pore, but not a closed one. There are several densities used in the literature and therefore one has to know which density is being referred to (Table 3.15). True density may be defined as the mass of a powder or particle divided by its volume excluding all pores and voids. True density is also referred to as absolute density or crystalline density in the case of pure compounds. However, this density is very difficult to be determined and can be calculated only through X-ray or neutron diffraction analysis of single-crystal samples. Particle density is defined as the mass of a particle divided by its hydrodynamic volume. The hydrodynamic volume includes the volume of all the open and closed pores. Practically, the hydrodynamic volume is identified with the volume included by the outer surface of the particle. The particle density is also called apparent or envelope density. The term skeletal density is also used. The skeletal density of a porous particle is higher than the particle one, since it is the mass of the particle divided by the volume of solid material making up the particle. In this volume, the closed pores volume is included. The interrelationship between these two types of density is as follows (ASTM, 1994 BSI, 1991) ... 

The skeletal density is representative of the solid material itself, excluding its porosity. The bulk volume of catalyst minus its pore volume and the interparticle volume between discrete particles (V)) is the true skeletal volume. One calculates this term by... 

In this paper, a predictive molecular thermodynamic approach is developed to calculate the structural and compositional characteristics of microemulsions. The theory applies not only to oil-in-water and water-in-oil droplet-type microemulsions but also to bicontinuous microemulsions. The treatment is an extension of our earlier theories for micelles, mixed micelles, and solubilization but also takes into account the self-association of alcohol in oil and the volume-excluded interactions among... 

Several catalyst densities are used in the literature. True density may be defined as the mass of a powder or particle divided by its volume excluding all pores and voids. In a strict physical sense, this density can be calculated only through X-ray or neutron diffraction analysis of single crystal samples. The term apparent density has been used to refer to the mass divided by the volume including some portion of the pores and voids, and so values are always smaller than the true density. This term should not be used unless a clear description is given of what portion of the pores is included in the volume. So-called helium densities determined by helium expansion are apparent densities and not true densities since the measurement may exclude closed pores. 

All results for chain size are now written in terms of the excluded volume. To understand how the chain size changes with temperature, we simply need the temperature dependence of the excluded volume. There are two important parts of the Mayer /-function, from which the excluded volume is calculated [Eq. (3.7)]. The first part is the hard-core repulsion, encountered when two monomers try to overlap each other (monomer separation rhard-core repulsion, the interaction energy is enormous compared to the thermal energy, so the Mayer /-function for r < 6 is — I ... 

Consider a semidilute solution with volume fraction (p of chains with N Kuhn monomers of length b and excluded volume v. Calculate the free energy cost to stretch a chain to end-to-end distance R for the following cases ... 

The excluded volume of a solute molecule is the volume that is not available (because of exclusion forces or for other reasons) to the centers of mass of other similar solute molecules. As an example, let us consider the excluded volume of a spherical particle of radius, R. The position of a sphere is fully described by coordinates of its center. It is apparent from Fig. 3.7(a) that the center of one solid sphere cannot approach the center of another solid sphere closer than two radii (2i2). Hence, the volume excluded by one sphere equals 7r(2ii), that is, eight times its actual volume. The excluded volume of asymmetric particles cannot be calculated so easily. This is because the distance between their centers of mass when they are in contact depends on their orientation. Nevertheless, it has been... 

The second virial coefficient of polyelectrolytes is treated theoretically either by applying the theory for charged spherical colloids with a correction for the chain character of the polyion or as an extension of the theory of the second virial coefficient for nonionic linear polymers. An example of an extension of the theory of the second virial coefficient for nonionic linear polymers to polyelectrolytes is the above-mentioned Yamakawa approach of using perturbation theory of excluded volume to calculate A2 [42],... 

Again in flatland, exact calculations are possible for the volumes excluded by simple re-entrant polygons [Fig. 11(c) and mirror image], which while bearing little relation to projected actual molecules, help to confirm the reality of discrimination arising from space-fiUing differences in pairwise contacts. The excluded volumes for like-like cind like-unlike shapes (c) are 206.619966 a and 206.852126 a respectively, showing discrimination of about 0.1% with like-like lower. 

At present, after the development of methods of automatic design calculation (see Chapter 7), the application of simplified methods should be rejected and rigorous methods should be used. To decrease the volume of calculations, some stages can be excluded from these methods (those of iterations and of parameter optimization). 

This theory has been used to describe nematic ordering in solutions of rodlike macromolecules, such as tobacco mosaic virus (TMV) or poly(benzyl-L-glutamate). The orientational distribution is calculated from the volume excluded to one hard cylinder by another. It assumes that the rods cannot penetrate to each other. Denoting the length of the rods by L and the diameter by D, the volume fraction of the rods is expressed by O = jCkLD. (c is the concentration of rods). 

In dilute polymer solutions each polymer molecule excludes all others from its volume. Thus mean-field theories, such as Flory-Huggins theory, are inappropriate and more exact theories of dilute solutions calculate AG from the volume excluded by one polymer molecule to another, which in turn is calculated using to account for intermolecular segment-segment volume exclusion. These theories show that (jui... 

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. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

Of course, the effect of excluded volume is opposite and greatly exceeds that shown in Fig. 1.10, which is produced by uncorrelated collective interaction. Unfortunately, neither of them results in sign-alternating behaviour of angular or translational momentum correlation functions. This does not have a simple explanation either in gas-like or solid-like models of liquids. As is clearly seen from MD calculations, even in... 

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

A series of theoretical studies of the SCV(C)P have been reported [38,40,70-74], which give valuable information on the kinetics, the molecular weights, the MWD, and the DB of the polymers obtained. Table 2 summarizes the calculated MWD and DB of hyperbranched polymers obtained by SCVP and SCVCP under various conditions. All calculations were conducted, assuming an ideal case, no cyclization (i.e., intramolecular reaction of the vinyl group with an active center), no excluded volume effects (i.e., rate constants are independent of the location of the active center or vinyl group in the macromolecule), and no side reactions (e.g., transfer or termination). 

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