Compressible binary mixture

In a compressible binary mixture, the composition inside a bubble does not necessarily coincide with the composition of the vapor phase. This additional parameter is chosen as to minimize the free energy of the bubble and depends on the bubble s size. In Fig. 15 we plot the free energy AG of the bubble as a function of the excess A.5nuc of the solvent component and the excess Apnuc of the polymer. The critical bubble corresponds to a maximum of a curve. The values of AG away from the maximum depend on the way the size of the bubble is held constant. Such a dependence is rather weak in the vicinity of the maximum, but it gives rise to unphysical density profiles (and convergence problems in the numerical procedure) for very small bubble sizes. Those small sizes are excluded from Fig. 15. 

The former corresponds effectively to a one-component compressible polymer solution, while the character of a compressible binary mixture becomes more apparent at higher pressures in the vicinity of the triple line. The composition is held constant, and the temperature is varied. From Fig. 8 (b) we conclude that the composition of the coexisting phases remains almost constant in the temperature interval 0.75 < kiTje < 0.82 for both pressures. At low pressure, the nucleation barrier decreases monotonously with temperature as expected. At higher pressure, however, the nucleation barrier exhibits a non-monotonous dependence on temperature AG exhibits both a maximum and a minimum upon increasing temperature at fixed molar fraction. The inset compares the radial density distributions of the critical bubbles and planar interfaces at ksT/e = 0.7573. In both cases the solvent density at the center of the bubble is higher than at coexistence and there is an enrichment of solvent at the interface of the bubble. However, there are no qualitative differences in the structure, in agreement with the observation of Talanquer and co-workers [196] for binary Lennard-Jones mixtures. 

Fig. 19 n-< A > isotherms for 72 mol % PVP/PVAc mixed monolayers on water at 25 °C by stepwise addition and compression. Surface pressure 77 for 72 mol % polyfvinyl palmitate)/poly(vinyl acetate) binary mixture as a function of area per monomer. The surface concentration was controlled as noted in the plot. For the stepwise-addition technique, lens formation was observed in the region where the two techniques differ for 77 lOmNm-1. For the plot, the mixture required stabilization times considerably longer than the 1-2 hours allowed between points to form equilibrium films. < A > = average area per monomer... 

Appendix C Compressible Binary Blend Mixture of Stiff Polymers... 

Binary mixtures were prepared with varying drug contents (60, 70, 80, 90, and 95%) keeping constant the drug and excipient particle size. Table 24 gives the composition of the studied batches as well as the tablet thicknesses. The mixtures were compressed on an eccentric machine (Bonals A-300) without any further excipient. Cylindrical tablets with a mean dosage of 500 mg and a diameter of 12 mm were prepared at the maximum compression force accepted by the formulations. 

Schmidt, P. C., and Leitritz, M. (1997), Compression force/time-profiles of microcrystalline cellulose, dicalcium phosphate dihydrate and their binary mixtures—A critical consideration of experiments and parameters, Eur. J. Pharm. Biopharm., 44, 303-313. 

Solid-Fluid Equilibria The phase diagrams of binary mixtures in which the heavier component (the solute) is normally a solid at the critical temperature of the light component (the solvent) include solid-liquid-vapor (SLV) curves which may or may not intersect the LV critical curve. The solubility of the solid is very sensitive to pressure and temperature in compressible regions where the solvent s density and solubility parameter are highly variable. In contrast, plots of the log of the solubility versus density at constant temperature exhibit fairly simple linear behavior. 

Cook, G.P. Summers, M.P. Effect of compression speed on tensile strength of tablets of binary mixtures containing aspirin. J. Pharm. Pharmacol. 1990, 42, 462-467. 

All calculations were carried out at T = 313.15 K. The vapor-liquid equilibrium (VLB) data for the ternary mixture and the corresponding binaries were taken from [32]. The excess volume data for the ternary mixture A,A-dimethylformamide-methanol-water and binary mixtures A, A-dimethylformamide-methanol and methanol-water were taken from [33], and the excess volume data for the binary mixture A,A-dimethylformamide-water from [34]. There are no isothermal compressibility data for the ternary mixture, but the contribution of compressibility to the binary KBls is almost negligible far from the critical point [6]. For this reason, the compressibilities in binary and ternary mixtures were taken to be equal to the ideal compressibilities, and were calculated from the isothermal compressibilities of the pure components as follows ... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

The purpose of this Appendix is to provide expressions for calculating the KBIs for binary mixtures from measurable thermodynamic quantities such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. 

In equations (3-7), kj is the isothermal compressibility of a binary mixture, V. is the partial molar volume of component i, V is the molar volume of the mixture and y. is the activity coefficient of component i. The expressions at infinite dilution of the fluctuations in the number of particles can be easily derived from eqs 3-6 and are listed in the Appendix. 

Kirkwood and Buff [15] obtained expressions for those quantities in compact matrix forms. For binary mixtures, Kirkwood and Buff provided the results listed in Appendix A. Starting from the matrix form and employing the algebraic software Mathematica [16], analytical expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials for ternary mixtures were obtained by us. They are listed in Appendix B together with the expressions at infinite dilution for the partial molar volumes and isothermal compressibility. 

In the above expressions, R is the universal gas constant, 1 is the isothermal compressibility of the pure solvent, V is the partial molar volume of component i at infinite dilution in the binary mixture of component i and solvent, and q>i is the volume fraction of water. 

The calculations have been carried out for those systems for which the solubility calculations have been performed. The dilute region of sodium chloride (c < 0.3) was selected to ensure that the condition F /x = s = constant is satisfied. The partial molar volume was estimated using literature data [67-69]. According to the latter data, depends weakly on C3 and this dependence is linear in the dilute range [68,69]. For sodium chloride and potassium chloride, decreases by at most 1 cm /mol when Cj is changed from 0 to 2mol/l. In our calculations, the above decrease was taken 1 cm /mol. On this basis the composition dependence of was evaluated in the composition range 0 < C3 < 0.3. The partial molar volumes Vi and V3 of water and sodium chloride in the binary mixture water (1) + sodium chloride (3) were obtained from data available in the literature [70,71], and the composition dependence of the isothermal compressibility of the mixed solvent (water (1) + sodium chloride (3)) was taken from reference [71]. 

For binary mixtures, assuming that the volumes of constituent powders do not change during compression, the tensile strength at zero porosity can be calculated using a linear mixing rule ... 

The techniques (e.g., weighted histogram analysis, equal weight rule, etc.) utilized for one-component systems or incompressible binary mixtures can be readily carried over to compressible systems. Since the system is described by two order parameters one monitors the joint probability distribution of... 

For an analysis of the chromatographic separation of a binary mixture we will again assume the same, constant linear flow rate v = LI tm = H/t throughout all plates, as one can expect with a non-compressible mobile phase. 

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