Volume fraction diagram

A schematic of change in the type of microemulsion with the salinity is shown in Figure 7.8, and a volume fraction diagram of the data presented in Table 7.2 is shown in Figure 7.9. The volume fraction information can also be represented by a solubility plot, as shown in Figure 7.10 (see page 254). We will see later that the solubilization ratio is a very important parameter in interfacial tension calculation. 

FIGURE 7.9 Volume fraction diagram of a salinity scan test. (Data from Table 7.2.)... 

Experimental data suggest that the optimum salinity varies linearly with the cosolvent concentration. Therefore, p7 can be estimated from the slope of the straight line of normalized optimnm salinity (C5i,op/C i,op) versns f in the case without divalent cations, as schematically shown in Figure 7.20. To obtain the effect of cosolvent on the shift in optimum salinity, P, we need to measure the volume fraction diagram for at least two different cosolvent concentrations and must know C i op. According to the definition, ff is defined as V7/(V7 + V3). 

For the first iteration, the slope parameters (mkm) are set to 0, and the intercept parameters (C33mann) are adjusted to obtain a reasonable match of the volume fraction diagrams then the slope parameters are obtained. After obtaining the slope parameters, we repeat the matching procedure for further improvements (UTCHEM-9.0, 2000). For a two-cosolvent case, UTCHEM requires input of 12 parameters mkm and C33maxm for m = 0, 1, 2, and k = 7, 8. We... 

Assnme Eq. 7.67 holds for k = 7 and 8. For the hrst iteration, set the slope parameters (mun) to 0 and adjust the intercept parameters (Cssmaxm) to obtain a reasonable match of the volume fraction diagrams. 

Obtain the slope parameters (mjan) by matching the volume fraction diagrams based on Eq. 7.68, where C33niax,ini is affected by the two cosolvents, 7 and 8. 

Fine-tune the nine parameters obtained for the improved matching of volume fraction diagrams. 

Figure 2. Volume fraction diagram for 4% sodium dihexyl sulfosuc-cinate, 8% IT A and TCE, with and without xanthan gum polymer.

The addition of xanthan gum polymer did not affect phase behavior. This was evidenced by the close match between the volume fraction diagrams for a surfactant solution (4% sodium dihexyl sulfosuccinate, 8% IPA) with 500 mg/1 xanthan gum polymer and without polymer as shown in Figure 2. As shown in Figure 4, the viscosity of the surfactant solution is increased by the addition of polymer. All samples with polymer were observed to coalesce to microemulsions in less than 20 hours, which is still fast enough to be acceptable based upon subsequent column floods. 

Figure 1.5 presents the explosion pressure (Fc)/hydrogen volume fraction diagrams of the explosion of a combustible mixture (initial pressure Po = O.l MPa and temperature T = 293 K) occurring as a result of hydrogen combustion in oxygen (1) or in air (2). The diagrams are based on experimental results [49] measured for a 6-L vessel. 

Fig. 1.5 The pressure-hydrogen volume fraction diagrams for hydrogen-oxygen (1) and hydrogen-air explosions (2) of a combustible mixture at Po = 0-1 MPa and r = 293 K [49]...

Figure B3.3.9. Phase diagram for polydisperse hard spheres, in the volume fraction ((]))-polydispersity (s) plane. Some tie-lines are shown connecting coexistmg fluid and solid phases. Thanks are due to D A Kofke and P G Bolhuis for this figure. For frirther details see [181. 182].

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

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. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

Figure C2.6.10. Phase diagram of colloid-polymer mixtures polymer coil volume fraction vs particle...

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

The lowest value of Qeff corresponds to different structures for different along the bifurcation line. The sequence of phases is always the same for various strengths of surfactant (with 7 > 27/4) and for increasing p it is L—>G—>D—>P—>C. For 7 = 50 (strong surfactant, like C10E5) the portion of the phase diagram corresponding to the stable cubic phases is shown in Fig. 14(b). For surfactants weaker than in the case shown in Fig. 14 the cubic phases occur for a lower surfactant volume fraction for example, for 7=16 cubic phases appear for p 0.45. 

Before comparing these predictions regarding the critical point with experimental results, we may profitably examine the binodial curve of the two-component phase diagram required by theory. The following useful approximate relationship between the composition V2 of the more dilute phase and the ratio y = V2/v2 of the compositions of the two phases may be derived (see Appendix A) by substituting Eq. (XII-26) on either side of the first of the equilibrium conditions (1), using the notation V2 for the volume fraction in the more dilute phase and V2 for that in the more concentrated phase, and similarly substituting Eq. (XII-32) for fX2 and y,2 in the second of these conditions ... 

Prus and Kowalska [75] dealt with the optimization of separation quality in adsorption TLC with binary mobile phases of alcohol and hydrocarbons. They used the window diagrams to show the relationships between separation selectivity a and the mobile phase eomposition (volume fraction Xj of 2-propanol) that were caleulated on the basis of equations derived using Soezewiriski and Kowalska approaehes for three solute pairs. At the same time, they eompared the efficiency of the three different approaehes for the optimization of separation selectivity in reversed-phase TLC systems, using RP-2 stationary phase and methanol and water as the binary mobile phase. The window diagrams were performed presenting plots of a vs. volume fraetion Xj derived from the retention models of Snyder, Schoen-makers, and Kowalska [76]. 

The basis of the window diagram approach is that the relative retention of a solute on a mixed phase depends only on the volume fractions of the individual phases and the partition... 

Fig. 28. NSE spectra in polyethylene melts at 509 K for three different polymer volume fractions in Rouse scaling. Upper diagram 0 = 1, central diagram O = 0.5, lower diagram O = 0.3. The solid lines correspond to a fit with the Ronca model. (Reprinted with permission from [60]. Copyright 1993 American Chemical Society, Washington)...

Fig. 10.7. Phase diagram for a homopolymer of chain length r = 8onal0xl0xl0 simple cubic lattice of coordination number z = 6. Filled circles give the reduced temperature, T and mean volume fraction, () of the three runs performed. Arrows from the run points indicate the range of densities sampled for each simulation. The thick continuous line is the estimated phase coexistence curve. Reprinted by permission from [6], 2000IOP Publishing Ltd...

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

Fig. 56 Phase diagram of blend of PS-fi-PI with PS. T0dt. o TDMt, Toot- Vertical lines separating microdomain structures are obtained from total volume fraction PS in system. Dashed line results of mean-field calculation for ODT. The OOT line which exists at volume fractions ps 5 ub was obtained during a heating process. From [174]. Copyright 2000 American Chemical Society...

Fig. 58 Lattice constants vs. volume fractions of PS phase for a blending a PI-0-PS-0-P2VP triblock terpolymer with PS homopolymer (blends I and II) and b blending a Pl-fr-PS-fr-P2VP triblock terpolymer with PI and P2VP homopolymers (blends III and IV). Arrows variations of ps with increasing volume fractions of added homopolymers. , , lattice constants of pure triblock terpolymers o, , A lattice constants of blends. Gray band between a and b expresses experimentally obtained microphase separation phase diagram for unblendend PI-6-PS-6-P2VP. From [159], Copyright 2002 Wiley...

The topological transformations in an incompatible blend can be described by the dynamic phase diagram that is usually determined experimentally at a constant shear rate. For equal viscosities, a bicontinuous morphology is observed within a broad interval of the volume fractions. When the viscosity ratio increases, the bicontinuous region of the phase diagram shrinks. At large viscosity ratios, the droplets of a more viscous component in a continuous matrix of a less viscous component are observed practically for all allowed geometrically volume fractions. 

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