Phase diagram calculations

Kaufman and Bernstein (1970) brought out the first book on phase diagram calculations there was not another comprehensive treatment till Saunders and Miodownik s book came out in 1998. This last covers the ways of obtaining the thermodynamic input data, ways of dealing with complications such as atomic... 

Fig. 123.—(a) Phase diagram calculated for three-component systems consisting of nonsolvent [1], solvent [2], and polymer [3] taking Xi==X2=l and Xz equal to 10 (dashed curve), 100 (solid curve), and °° (dotted curve) xi2 = xi3 = 1.5 and X23 =0. All critical points (O) are shown and tie lines are included for the xs = 100 curve. (Curves calculated by Tompa. ) (b) The binodial curve for a 3 = 100 and three solvent ratio lines. The precipitation threshold is indicated by the point of tangency X for the threshold solvent mixture. 

As seen from the figure, the stability function does not become negative at any pressure when the hydrogen sulfide mole fraction lies anywhere between 0 and 1. The phase diagram calculations at 311.5 K are shown in Figure 14.11. As seen, the correct phase behavior is now predicted by the EoS. 

Fig. 41 Theoretical phase diagram calculated by Matsen et al. [20] for diblock copolymers in intermediate segregation regime. PS-arm-P2VP miktoarm polymers PS- -P2VP linear polymer. From [119]. Copyright 2000 American Chemical Society... 

The mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

Fig. 5 Comparison of phase diagrams calculated for the melt of a proteinlike heteropolymer (b) with the phase diagram of a Markovian copolymer according to criterion II (a) and criterion I (c). Proteinlike heteropolymer consisting of / = 103 units is obtained for polymeranalogous reaction in a homopolymer globule at the value of the Thiele modulus h equal to 35... 

A significant review of several aspects of the phase diagram computation (phase diagram calculations in teaching, research and industry) has been published by Chang (2006). The relationship between the characteristic features of a phase diagram and the relative thermodynamic stabilities of the phases involved has been there underlined and exemplified. Representative examples of binary, ternary and high-order alloy systems have been presented. Moreover a number of applications have... 

Pandat software package and applications (Chen et al. 1993). Pandat is a software package for multi-component phase diagram calculations. Given a set of thermodynamic parameters for all phases in a system and a set of user constraints, Pandat automatically calculates the stable phase diagram without... 

Fig. 7.18 Phase diagram calculated for staging in a layered compound where the layers expand non-linearly with x (from Dahn et al. (1982)). The shaded regions are mixtures of phases. The solid line near a temperature of 1.3 represents a series of compounds made at constant temperature. The insert shows the Dumas-Herold picture of islands in a stage-2 structure.

Topological Features of Phase Diagrams Calculated Using... 

Integration of Ordering into Phase Diagram Calculations 210... 

Chapter S examines various models used to describe solution and compmmd phases, including those based on random substitution, the sub-lattice model, stoichiometric and non-stoichiometric compounds and models applicable to ionic liquids and aqueous solutions. Tbermodynamic models are a central issue to CALPHAD, but it should be emphasised that their success depends on the input of suitable coefficients which are usually derived empirically. An important question is, therefore, how far it is possible to eliminate the empirical element of phase diagram calculations by substituting a treatment based on first principles, using only wave-mecbanics and atomic properties. This becomes especially important when there is an absence of experimental data, which is frequently the case for the metastable phases that have also to be considered within the framework of CALPHAD methods. 

A phase diagram is often considered as something which can only be measured directly. For example, if the solubility limit of a phase needs to be known, some physical method such as microscopy would be used to observe the formation of the second phase. However, it can also be argued that if the thermodynamic properties of a system could be properly measured this would also define the solubility limit of the phase. The previous sections have discussed in detail unary, single-phase systems and the quantities which are inherent in that sjrstem, such as enthalpy, activity, entropy, etc. This section will deal with what happens when there are various equilibria between different phases and includes a preliminary description of phase-diagram calculations. 

J.7.I Topological features of phase diagrams calculated using regular solution theory... 

Despite the success of phase-diagram calculations, there is still a considerable reluctance by sections of the scientific community to accept that TC lattice stabilities represent a real physical entity as distinct from an operational convenience. This inevitably creates doubts concerning the ultimate validity of the calculations. It is therefore important to verify that TC lattice stabilities, largely derived by extrapolation, can be verified by ab initio calculations and placed on a sound physical basis. 

Thermochemical methods generate lattice stabilities based on high-temperature equilibria that yield self-consistent multi-component phase-diagram calculations. However, as they are largely obtained by extrapolation, this means that in some cases they should only be treated as effective lattice stabilities. Particular difficulties may occur in relation to the liquid — glass transition and instances of mechanical instability. 

Rapidly solidified in-situ metal matrix composites. A design project for alloys based on the Fe-Cr-Mo-Ni-B system, and produced by rapid solidification, was undertaken by Pan (1992). During processing a mixture of borides is formed inside a ductile Fe-based matrix which makes the alloys extremely hard with high moduli. These alloys provide a good example of how phase-diagram calculations were able to provide predictions which firstly helped to identify unexpected boride formation (Saunders et al. 1992) and were ultimately used in the optimisation of the modulus of a shaft material for gas turbines (Pan 1992). 

Fig. 10. Theoretical ternary phase diagram calculated from the scaled particle theory for worm-like hard spherocylinders with (Ni, N2) = (0.930,0.070), d = 1.52nm, q = 200nm, and ML = 2150 nm-1 [17]...

Fig. 2.43 Phase diagrams calculated (Hamley and Podneks 1997) within the Landau-Brazovskii approximation, applied to block copolymers by Fredrickson and Helfand (1987) (a) for a degree of polymerization N = 5 x 103, a direct transition between gyroid and dis phases occurs near f = 0.43 (b) for N = 106, there is no direct transition between these phases (Hamley and Podneks 1997).

Fig. 6.36 Phase diagram calculated using SCFT for a blend of a symmetric diblock with a homopolymer with fl = 1 (see Fig. 6.32 for a blend with a diblock with / = 0.45) as a function of the copolymer volume fraction

Fig. 6.40 A phase diagram calculated using SCFT for a mixture containing equal amounts of two homopolymers and a symmetric diblock, all with equal chain length (Janert and Schick 1997a). A-rich and B-rich swollen lamellar bilayer phases are denoted LA and LH respectively whilst the corresponding disordered phases are denoted A and B. The con-solute line of asymmetric bilayer phases LA and Lu, shown dotted, is schematic.The dashed line is the unbinding line. The arrows indicate the locations of the unbinding transition X jN and multicritical Lifshitz point, cMiV " 6.0.

Fig. 6.49 Phase diagram calculated using SCFT for blends of strongly segregated diblocks (both symmetric, N, = N2 = 400, yN = 100). The blend composition

Liquid-Solution Models. The simple-solution model has been used most extensively to describe the dependence of the excess integral molar Gibbs energy, Gxs, on temperature and composition in binary (142-144, 149-155), quasi binary (156-160), ternary (156, 160-174), and quaternary (175-181) compound-semiconductor phase diagram calculations. For a simple multicomponent system, the excess integral molar Gibbs energy of solution is expressed by... 

Fig. 8.1. Original diagrams of the first COSMO-RS phase-diagram calculations by Iven Clausen [96] for four alcohol-water mixtures (methanol at 60 C, ethanol at 55 °C, 1-propanol at 60 °C and 1-butanol...

Detailed theoretical calculations establish that phase diagrams separating homogeneous from phase-separated regions will depend on intermolecular electrostatic interactions, the strength of the applied poling field, and on the dielectric constant of the host polymer. In Fig. 16, we show phase diagrams calculated as a function of N, 8, and E. 

Figure 16 shows the gas-liquid phase diagrams calculated with DHH+ DHHDS [65, 81], compared to the ones calculated from MSA and from BB+ODS. Also include the data of Loth et al. [102] and those obtained from the Gibbs ensemble method by Panagiotopoulos [103]. Note that the reported values obtained from DHH+DHHDS are only estimates, because as said above, this approximation, even if accurate, exhibits little thermodynamic inconsistencies. As... 

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