Thermodynamic expansion factor,

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

The thermodynamic linear expansion factor has been related to Flory or thermodynamic interaction parameter, %, and the entropy of dilution parameter, Xs, through the Flory-Fox [10] equations. 

The size of a polymer molecule in solution is influenced by both the excluded volume effect and thermodynamic interactions between polymer segments and the solvent, so that in general =t= . The Flory (/S) expansion factor a is introduced to express this effect, by writing ... 

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. 

Fig. 4.18 Critical gel concentration versus l/<5, where <5, is the thermodynamic expansion factor, for PF.O-PBO diblocks (Booth et al. 1997).

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

After values of K or of (S2)0/M have been obtained as described above, and after it has been demonstrated that these values do not greatly depend on the solvent, the expansion factor a may then readily be evaluated in any given solvent from the ratio ([ ]/[ or from S2>/0)V. Thermodynamic analyses of the polymer-solvent system may then be accomplished with the aid of an appropriate equation for... 

O is the so-called Flory s constant, a is the expansion factor of the polymer molecule, which depends from the thermodynamic quality of the solvent (a = 1 in ideal solvent), ( o> is the mean-square radius of gyration, is Avogadro s number, and is the volume of the equivalent hydrodynamic sphere. 

Freire and coworkers [285,286] studied the case of miktoarm star copolymers of the type AxBf x, where f is the total functionality of the star copolymer. The conformational characteristics of these kinds of molecules were investigated as a function of molecular weight and number of the different branches, as well as the thermodynamic cross interactions between the arms and the solvent medium. Calculations based on the renormalization group and Monte Carlo methods allowed the estimation of the dimensions of each arm and of the whole molecule and the mean square distance between the two centers of mass of the different homopolymers. From these estimations different expansion factors relative to the homopolymer precursors could be calculated (Fig. 2). Different degrees of agreement were obtained by the two methods depending on the property under consideration. 

The experimental methods and the quantification of data from a TS-CST-SSR is fully delineated by the above treatment and its trivial but interesting extension to cases where the initial concentration of adsorbent in the sweeping stream is not zero. The same equations can be used to quantify commonly available TPD results as long as the reactor configuration and run conditions conform to the assumptions used in the derivations presented here. Since most TPD experiments are carried out in plug flow configurations one can take Vv=0, but the volume expansion factors remain necessary if one intends to calculate the correct quantities desorbed from the sample and/or to quantify the kinetics and thermodynamics of adsorption/desorption. 

Flory (1953) has presented a celebrated theory of the excluded volume effect that relates the expansion factor to the thermodynamic properties of the polymer-solvent system. Basically what Flory calculated was the free energy of mixing of polymer segments with solvent that is associated with the expansion of the coil dimensions in a good solvent. Such expansion is opposed, however, by the loss of configurational entropy of the chain. The latter corresponds, of course, to an elastic contractive force. Expansion proceeds until the two opposing effects are in equilibrium. Flory s result was... 

The molecular significance of the coefficient k has not yet been elucidated. Theoretical studies indicate that for coils k is composed of a hydrodynamic factor kh and a thermodynamic factor 3A iM j [17])/(a), where /(a) is a function of the expansion factor a ... 

Moreover, if we assume that a is proportional to the thermodynamic correlation length g, which is expanded as a single polyion chain in dilute solutions by an expansion factor a, we have... 

Excluded Volume Effects.—In a thermodynamically good solvent, the dimensions of a polymer molecule are greater than in the unperturbed state because of excluded volume effects. This effect is usually characterized by an expansion factor a, with... 

Polymer solution thermodynamics (Flory, 1953) also can be used to show that the expansion factor a is given by... 

As expected, in equation (37) is not the same as ao in equation (38), but the most significant feature of these relations is that the expansion factors are functions of the single excluded volume variable z. From the second form on the right hand side of equation (33), we see that z can be expressed in terms of two groups of statistical quantities nb and n p. As mentioned above, the first group is simply related to an observable property and the second is obtainable from thermodynamic measurements (see below. Section 13.4). Thus z is a function of two physically meaningful quantities, and the theory of the expansion factor outlined here is a two-parameter theory in the sense proposed by Stockmayer. ... 

As in the case of the expansion factors (Xq and the few available terms in the direct perturbation expansion of the molecular weight dependent function F(z) in equation (80) do not give a useful representation for z appreciably different from zero. Consequently a number of approximate closed form expressions for F(z), developed in much the same spirit as those for the expansion factor, have been proposed to analyze thermodynamic behavior in good solvents. A thorough discussion may be found in ref. 75. Usually, the closed form expressions are expressed in terms of a dimensionless penetration function F(z) where z = z/aj and... 

It must be kept in mind that the Goodrich treatment separates out contributions to intermolecular interactions that arise from film expansion. The differences in film expansion are a reflection of conformation and are accounted for in the pure meso- and ( )-films. However, since ( )- and meso-film components do interact, the intramolecular contribution to film compression may be altered. This effect would arise from conformational perturbations as molecules interact, thereby precluding complete separation of inter- and intra-molecular contributions to the thermodynamics of compression. However, these complicating factors can be mitigated by comparing several molecules with varying structures, as has been carried out in this instance. 

The sharply different rates at which equilibrium was reached clearly implied that kinetic factors as well as thermodynamic ones could be involved in the expansion or compression of these films. It was therefore important to test them for stability using the criteria described in Sect. IV. 

Table 5.69 lists thermal expansion and compressibility factors for some Si02 polymorphs, according to the databases of Saxena et al. (1993) and Holland and Powell (1990). Table 5.70 lists thermodynamic data for the various Si02 polymorphs according to various sources. 

From the thermodynamic standpoint, the basic components of stars can be considered as photons, ions and electrons. The material particle gas (fermions) and the photon gas (bosons) react differently under compression and expansion. Put n photons and n material particles into a box. Let R be the size of the box (i.e. a characteristic dimension or scale factor). The relation between temperature and size is TR = constant for the photons and TR = constant for the particles. This difference of behaviour is very important in the Big Bang theory, for these equations show quite unmistakably that matter cools more quickly than radiation under the effects of expansion. Hence, a universe whose energy density is dominated by radiation cannot remain this way for long, in fact, no longer than 1 million years. 

Thermodynamic perturbation theory is used to expand the Boltzmann distribution in the dipolar interaction, keeping it exact in the magnetic anisotropy (see Section II.B.l). A convenient way of performing the expansion in powers of is to introduce the Mayer functions fj defined by 1 +fj = exp( cOy), which permits us to write the exponential in the Boltzmann factor as... 

It seems necessary at this time to emphasize that the above generalizations may pertain only as concerns O-H-N-O explosives and that diametrically opposed conclusions may apply to aluminized mixtures. Evidence is available that kinetic factors (which extend far beyond the time of the C-J condition), rather than thermodynamic factors, govern the extent of utilization of aluminum m the detonation. If, as seems reasonable, the rates of the aluminum reactions are highly temperature dependent, and if aluminum reacts at different rates with HaO, CO, and COa, detonation properties of aluminized explosives should depend very strongly on exact equilibrium compositions of these species in the C-J condition and in the early stages of the gas expansion. For such reasons, Eqs. (1) and (7) may be inapplicable for use with aluminized mixtures. 

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