Compressibility specific free volume

As discussed, the intuitive notion that there should be a connection between the statistics of the free volumes of a fluid and its measurable macroscopic properties has a long history in studies of the liquid state. In fact, it turns out that this connection is precise in the case of the thermodynamics of the single-component hard-sphere fluid. Specifically, Hoover, Ashurst, and Grover77 and Speedy82 have provided independent derivations that predict the relationship between the hard-sphere compressibility factor Z = P/pksT and the geometric properties of its free volumes, as follows ... 

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

In many production routes, and also during processing, polymer systems have to undergo pressure. Changes in the volume of a system by compression or expansion, however, cannot be dealt with in rigid-lattice-type models. Thus, non-combinatorial free volume ( equation of state ) contributions to AG have been advanced [23 - 29]. Detailed interaction functions have been suggested (but all of them are based on adjustable parameters, for blends, e.g., Mean-field lattice gas [30], SAFT [31], specific interactions [32]), and have been succesfully applied, for example, by Kennis et al. [33]. 

At first glance, one might consider the effect of compressed CO2 on the phase behavior of multi-component polymer systems to be a simple combination of the known effects of liquid solvents and hydrostatic pressure. Solvent effects are primarily enthapic in nature and typically manifest in upper critical solution behavior. Common solvents mitigate unfavorable interactions between dissimilar segments and enhance miscibility. In blends, the addition of highly selective solvents, e.g. a non-solvent for one component, can lead to precipitation of the unfavored species at high dilution. In block copolymers, the effect of selective solvents is less clear, but studies to date reveal a collection of the solvent at the domain interface, selective dilation of one phase, and stabilization of the disordered phase via depression of the UODT. The systems we have studied each exhibit a lower critical transition. For these specific systems, previous work indicates the hydrostatic pressure suppresses free volume differences between the components and expands the region of miscibility. 

As the pressure exerted by the tip is localized and limited in time, a shift in Tg due to the applied load is not expected. The sanple is typically scanned over several pm with a scan rate of 0.5 Hz. Assuming values of 3 to 5 nm and 30 to 80 nm for the contact radius and area, the volume affected would be equal to a cylinder that is 5 times the length of the contact radius with a volume, approximately, 450 to 2000 nm (25). In our case, the radius of gyration Rg, is about 5 nm and the volume occupied by one molecule is F = A 3%Rg - 500 nm (27). This leaves only 1 to 4 molecules in the compression zone. Such a limited volume could not have such a dramatic effect over the properties of the polymer. In fact, the contact radius is so small compared to the scan length, the time of permanence over a specific area is only a few milliseconds per cycle. For the rest of the scan cycle, the molecules are unconstrained and can relax. If creep or thermal drift is taken into consideration, it is quite likely that the tip never passes over the same molecule twice. Hence, the pressure exerted by the tip cannot be considered hydrostatic. In addition, the lateral force is measured after the temperature has stabilized (20, 22, 28-30). Therefore, imder these isothermal conditions, the polymer can be considered incon ressible, i.e. it undergoes mechanical deformations but its density and free volume remain constant. 

The main excess properties are the free energy gE, enthalpy hB, entropy sE, and volume v (per molecule) data on other excess properties (specific heat, thermal expansion or compressibility) are rather scarce. In most cases gE, hE, sE, and vE have been determined at low pressures (<1 atm) so that for practical calculations p may be equated to zero their theoretical expressions deduced from Eqs. (33) and (34) are then as follows ... 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

The distance between chemically bound atoms in many molecules is shorter than the sum of the radii of the same atoms when free, and the specific volume of the compound may be actually smaller than the total covolume of its gaseous products. If, as seems plausible, the drastic compression within the detonation front ruptures chemical bonds, many atoms suddenly expand, exerting forces like those by which solids resist compression. Such forces could result in a spike pressure much higher than the peak pressure of the non-reactive shock front, exert a brisant effect on the surroundings, and expedite the progress of the detonation wave. This view accords with observations of cases in which... 

Here ceff represents the effective speed of sound, c0 is the actual speed of sound in free space, y is the specific heat ratio, and kt is the isothermal compressibility of the fluid. A fraction of shell volume occupied by tubes, solidity o can be easily calculated for a given tube pattern. For example, o = 0.9069(d lp,)2 for an equilateral triangular tube layout, and o = 0.7853(d /p,)2 for a square layout. Coefficients a, are the dimensionless sound frequency parameters associated with the fundamental diametrical acoustic mode of a cylindrical volume. For the fundamental mode al = 1.841, and, for the second mode, a2 = 3.054 [122],... 

Clay particles occur abundantly in the soil. They are mostly colloidal aluminosilicates. To learn how their extensive surfaces react with water, we have studied the following properties of water associated with montmorillonite and other clay minerals threshold gradient [1], thermal expansibility [2, 3], isothermal compressibility [4], frequency of O—H stretching [5, 6], molar absorptivity [7], freezing point depression [8, 9] specific volume [10], specific heat capacity [11], heat of compression [12], viscosity [13], and free energy, enthalpy and entropy [6,14,15]. Not all of these properties will discussed here. Instead, we will discuss only a few of them to illustrate the kind of results obtained. 

Nor can the theory of regular solutions based on the simplified lattice model (cf. Ch. Ill) give any indication on the excess properties related to the equation of state such as the excess volume, the excess compressibility and hence the excess entropy and the excess specific heat all of which are closely related to the equation of state. In fact, no equation of state at all is introduced in this model. The lattice model can only be used to calculate the excess free energy and the excess enthalpy which should be equal in the zerbth approximation. However the experimental data invalidate this conclusion. 

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