Theta pressure

Hsiao et al. reported a cloud point pressure at approximately 150-170 bar and 40 C for 4 wt% of a PFOA sample (Af = 1,000,000 g/mol) [30], An LCST-type behavior was also noted as a result of the positive dependency of pressure upon increasing temperatures. Through neutron scattering experiments, this high solubility of PFOA could be related to a positive second virial coefficient (A2) in conditions as mild as 207 bar and 45°C in the case of sample with an equal to 36,500 g/mol [31]. However, the theta pressure was not determined. In another set of experiments, the positive A2 was found to be a decreasing function of molecular weight, as expected from the case of polymer solutions in incompressible solvents [32]. A2 was also estimated to be proportional to at the power of -0.4. 

In addition to studying the solubilization of C02-insoluble polymers by means of emulsifying agents [25], as described in Section 7.1.3, SANS has also been used to study C02-soluble systems such as fluoropolymers and PDMS [24]. In particular, Melnichenko et al. [239] have studied the dimensions of PDMS molecules in CO2 in order to test the prediction of Kiran and Sen [240] that they will adopt ideal configurations, unperturbed by excluded-volume effects, at a critical theta pressure (Pe) as they do in polymer solutions at the theta temperature (T ). Results of experiments on PDMS in CO2 (Figs. 7.39 and 7.40) confirm that the system exhibits both of these phenomena at P 540 bar and T 55 °C. 

Fig. 7.40. Observation of the theta pressure in supercritical CO2. As is observed for organic solvents (see Fig. 7.38), the chains do not collapse below Te. (Reproduced with permission from [239]. Copyright 1999 American Chemical Society.)...

The SANS experiments on PDMS in COj were designed to compare the chain dimensions with those in organic solvents and to test the prediction [7] that they will adopt ideal configurations, unperturbed by excluded volume effects, at a critical theta pressure XPo) as they do in polymer solutions at the theta temperature. The effect of temperature on the thermodynamic state of PDMS - COj solution is illustrated in Fig. 4. The concentration fluctuations diverge as the temperature approaches the phase boundaries, which are themselves a function of pressure. When the correlation length decreases to the theoretical value (0) Rg(0)/3 s 41 A, the condition is reached and this demonstrates that, unlike PS-AC, the solvent quality may be changed from the "poor to theta solvent by varying the temperature, as in PS-... 

Fig.6 shows that PDMS may tdso enter the good solvent domain at a theta temperature Te = 65 5 C, as observed in PS-CH solutions [21]- However, unlike organic solvents, diis transition can also be made to occur at a critical tiieta pressure (P - 52 4 MPa) in COj, as illustrated in the inset in Fig.6. To our knowledge, this is die first time that the existence of a theta pressure has been demonstrated experimentally and a more detailed description of tilts phenomenon is given in reference [22].

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

It is now recognized that a continuum of architecture and properties, which begins with the classical branched polymers, resides between these two classes. Typical branched structures such as starch or high pressures polyethylene are characterized by more than two terminal groups per molecule, possessing substantially smaller hydrodynamic volumes and different intrinsic viscosities compared to linear polymers, yet they often exhibit unexpected segmental expansion near the theta state . 

X-ray diffraction powder patterns were recorded on a CGR Theta 60 instrument, using monochromated CuKa radiation. The adsorption capacities for several adsorbates were measured at room temperature by gravimetry, using a Cahn RH microbalance as proposed by Vaughan and Lussier (3 ). The samples were first treated in air for 5 hours at 480°C. The experiment was performed by passing, over the sample, a stream of nitrogen saturated by the vapor pressure of the sorbate at room temperature, the relative pressure P/Po was then equal to 1. 

Most liquid-crystalline polymer solutions have a large second virial coefficient ( > 10 4 cm 3mol/g2) [41], which means that it is rather difficult to find poor or theta solvents for these polymers and that liquid-crystalline polymers in solution interact repulsively. This fact is essential in formulating their static solution properties (osmotic pressure, phase separation, etc.). 

Theta conditions are identified experimentally as the situation in which the second virial coefficient of the osmotic pressure is zero. 

Molecular weight from osmotic pressure measurements Degree of polymerization and molecular weight distribution Excluded volume from osmotic pressure measurements Theta temperature from second virial coefficient data Evaluation of charges on macroions from osmotic pressures... 

In order to get the real mass flow from the map, multiply by delta and divide by the square root of theta. The pressure and temperature condition immediately before the turbomachineiy is used, not the ambient measurement. For instance, at the turbine expander, the inlet temperature and pressure will be quite elevated and will change dramatically with a change in some process condition such as combustor firing rate therefore, the look-up value on the map will change. 

The free energy change which takes place when polymer chains interpenetrate is influenced by factors such as temperature, pressure and solvent composition. The point at which this free energy change is equal to zero is known as the 0 (theta)-point and such a solvent is called a 0-solvent. More formally, a 0-point is defined as one where the second virial coefficient of the polymer chains is equal to zero. It can be determined by light scattering and by osmometry. 

Leiva et al. [65] have reported for poly(itaconates) monolayers the surface behavior at the air - water interface at different surface concentrations. They have found that for these type of polymers, the air - water interface at 298 K, is a bad solvent, very close to the theta solvent. At the semidilute region concentration, the surface pressure variation was expressed in terms of the scaling laws as a power function of the surface concentration. According to equation (3.3), the log it vs log T plot shows a linear variation with slope 2 v/(2 u-1). 

Theta-pinch concepts belong to the so called high-0 devices since the ratio of the kinetic pressure of the plasma to the pressure of the external confining magnetic field is relatively high ... 

VAPOR-LIQUIO EQUILIBRIA for ORGANIC PHASE PRESSURE 44.815 TEMP 154,450 COMP X Y K ALPHA GAMMA THETA VR/TWETA... 

Figure 10.18 is a schematic representation of depletion stabilization in which the polymer is prevented from the zone of close approach between two particles. As a result of this low polymer concentration between the particles due to size exclusion, there is a lower osmotic pressure, which results in (1) an attractive force for greater than theta solvents and (2) a repulsive force for less than theta solvents. Theta solvents will be discussed in the section on the thermodynamics of polymer solutions, but first a discussion of pol3naaer properties. 

FOR AQUEOUS PHASE PRESSURE 44,814 TEMP 154.450 K ALPHA GAMMA THETA VP/THETA... 

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