Potential phase diagram

Type 1 are potential/potential phase diagrams. The potentials considered in chemical thermodynamics are temperature (thermal potential), pressure (mechanical potential) and the chemical potentials of the N components pj, i2, ig,. ..,... 

In addition to the molecular weight of the free polymer, there axe other variables, such as the nature of the solvent, particle size, temperature, and thickness of adsorbed layer which have a major influence on the amount of polymer required to cause destabilization in mixtures of sterically stabilized dispersions and free polymer in solution. Using the second-order perturbation theory and a simple model for the pair potential, phase diagrams relat mg the compositions of the disordered (dilute) and ordered (concentrated) phases to the concentration of the free polymer in solution have been presented which can be used for dilute as well as concentrated dispersions. Qualitative arguments show that, if the adsorbed and free polymer are chemically different, it is advisable to have a solvent which is good for the adsorbed polymer but is poor for the free polymer, for increased stability of such dispersions. Larger particles, higher temperatures, thinner steric layers and better solvents for the free polymer are shown to lead to decreased stability, i.e. require smaller amounts of free polymer for the onset of phase separation. These trends are in accordance with the experimental observations. 

Numerous thermodynamic calculations of phase equilibria in the Si-C-N system have been published but only few experimental investigations are documented. Calculated isothermal sections [117, 234-237], isopleths [117, 234], different types of potential phase diagrams [234,235,238-244] and phase fraction diagrams [234, 239] were presented. Additional information is provided by [245]. No or very low solid solubilities between SiC and Si3N4 could be detected by X-ray diffraction up to 2500 K [246, 247]. Also the nitrogen solubility in SiC is low. For more experimental information see [248, 249]. 

The influence of nitrogen gas pressure on Si-C-N phase reactions is shown in Fig. 21 by the calculated potential phase diagram for Si-C-N materials C Si < 1 [244]. 

The ternary system was calculated by extrapolation from the binary subsystems (Kasper, 1996) [33]. The calculations cover phase equilibria at one bar and do not assume any solubilities as no experimental evidence for stable sohd solutions between B4+5C and BN or a-BN and graphite exist. The section between graphite and boron nitride including the invariant reactions Uj, Dj and U2 (Fig. 23) is shown in Fig. 22. A calculated potential phase diagram (logpN2-T) can be found in [244], The complete Scheil reaction scheme (P = 1 bar) is shown in Fig. 23. 

R.J. Cushman, P. M. McManus, S. C. Yang, Spectroelectrochemical study of polyaniline - the construction of a pH-potential phase-diagram, Journal of Electroanalytical Chemistry 1987, 219, 335. 

In the first two chapters, we learned about thermodynamics (free energy, osmotic pressure, chemical potential, phase diagram) of polymer solutions at equilibrium and static properties (radius of gyration, static structure factor, density correlation function) of dissolved polymer chains. This chapter is about dynamics of polymer solutions. Polymer solutions are not a dead world. Solvent molecules and polymer chains are constantly and vigorously moving to change their positions and shapes. Thermal energy canses these motions in a microscopic world. 

Fig. 30. Al-Fe-O. Chemical potential phase diagram at 1300°C. The spinel phase is treated as ideal solution. The case when Fc304 and FeAl204 are treated as different phases is shown by a dashed line...

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

Experimental results describing limited mutual solubility are usually presented as phase diagrams in which the compositions of the phases in equilibrium with each other at a given temperature are mapped for various temperatures. As noted above, the chemical potentials are the same in the equilibrium phases, so Eqs. (8.53) and (8.54) offer a method for calculating such... 

When the phase diagram for an alloy has the shape shown in Fig. 10.3 (a solid solubility that decreases markedly as the temperature falls), then the potential for age (or precipitation) hardening exists. The classic example is the Duralumins, or 2000 series aluminium alloys, which contain about 4% copper. 

In this section we study a system with purely repulsive interactions which demonstrates the importance of entropy effects on the stability of phases when the effect of the corrugation potential due to the structured surface is completely neglected. The phase diagrams are determined by finite size scaling methods, in particular the methods of Sec. IV A. 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

FIG. 8 Phase diagrams for monolayer films on surfaces with boundary potential equal to 0.25 (a), 0.50 (b), and 0.75 (c). (Reprinted with permission from Langmuir 9 2562-2568, October 1993. 1993, American Chemical Society.)... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

Fig. 1.43 Schematic potential/pH diagram for a metal M in equilibrium with water in the absence of complexing species. Line a represents equations 1.117 and 1.122. Line b represents equations 1.118 and 1.123. Line c represents equations 1.119 and 1.124. The stable phases are marked in bold. The metastable phase is in parentheses. The broken line is an extrapolation of equation 1.123 and indicates possible metastable passivity...

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