Osmotic compressibility factor

It is well known that the osmotic pressure of a solution of one polymer can be scaled with a single dimensionless variable "S" which is proportional to polymer concentration at least for the case of mixtures with good solvents in the dilute to semidilute regime (12, 17). This implies that the osmotic compressibility factor (ti/cRT) can be expressed as some function of "S" only as shown in Equation 11. 

At this point we note that Equation 13 is the McMillan-Mayer (16) expansion for the osmotic compressibility factor which is fundamentally different from the analogous expansion that was obtained from the formalism of Hill (Equation 10). We also identify B as a McMillan-Mayer osmotic virial coefficient. 

One should notice here, that according to the compressibility equation and Eq. (35), the osmotic compressibility is given by the long wavelength limit (k - 0) of the static structure factor, i.e. 

These approximations can then be used in the osmotic equation of state to obtain the compressibility factor. Monte Carlo simulations using the above-discussed Monte Carlo techniques have been performed to assess the approximations inherent in the generalized Flory theory of hard-core chain systems. This theory does quite well in predicting the equations of state of hard-core chains at fluid densities. The question then arises, why does it do so well since the theory typically only incorporates information from a dimer fluid as a reference state ... 

For small particles, g r) is substantially different from unity only at interparticle distances r Then, the structure factor is equal to the inverse osmotic compressibility of the suspension ... 

The solution analogue of the compressibility factor of a gas is the reduced osmotic pressure (I7/C2). This quantity is shown sch aticaOy in Fig. 3.3 for polymer molecules under different solvency conditions. In a poor solvent for the polymer, negative deviations from ideality are apparent. This can be envisag as arising because the polymer molecules are in dynamic association under such solvency conditions. Since osmotic pressure is a coUigative... 

This is known as osmotic virial expansion and is analogous to the virial expansion of the compressibility factor. At very low solute concentrations, the linear and higher-order terms in Ci on the right-hand side are negligible and eq. reverts to The coefficients B(T), C(T), and others are the osmotic virial... 

The osmotic coefficient which is analogous to the compressibility factor... 

Fig. 8.8. Osmotic compressibility against volume fraction for various polymers and solvents. Osmotic compressibility means change of osmotic pressure with volume fraction. Horizontal scale is volume fraction 0, relative to 4>. Vertical scale is osmotic compressibility relative to its value in the dilute limit. The scales are logarithmic. Points are combined data on three samples of poly a-methyl styrene in toluene differing by a factor of five in molecular weight [8.14] and one sample of polyisoprene (natural rubber) in cyclohexane [8.1]. The superposition of his data suggests that all long polymers in good solvents have the same osmotic-pressure behavior. The line indicates the expected behavior in the semidilute regime...

The stracture factor S(q) is the Fourier transform of the pair correlation function, which depends on interaction between macromolecules in solution. In the limit of 0 (forward scattering), it is related to the osmotic compressibility of the solution ... 

In the disordered state, hard spheres of different sizes pack together slightly more efficiently than identical hard spheres. At constant number density, therefore, the osmotic pressure of the polydisperse hard-sphere fluid is lower than that of the monodisperse system with the same volume (packing) fraction. Figure 8 shows the effect on the compressibility factor as a function of... 

Osmotic Compressibility In Section 2.4, we learned that the molecular weight and concentration-dependent factor in the excess scattering intensity of the polymer solution is c/(511/5c). The denominator is the osmotic compressibility. See Eqs. 2.104-2.107. At low concentrations, 5ll/5c = NjJcqT/M, and therefore... 

From self-diffusion data, we have found that the structure is composed of closed droplets. NMR data and SANS data, analysed at higher q values, have demonstrated that the droplets have a concentration-independent size. The SANS data have also given us one size of the droplets, i.e. the hydrocarbon radius (and the shape), while collective and self-diffusion data have provided us with a value for the hydrodynamic radius. Knowing that we have identified a dilution line of spherical droplets of constant size, we can now turn to investigate the interactions. The latter affect properties such as osmotic pressure, diffusion and viscosity. The osmotic pressure, jr, can be measured, for example, in a membrane osmometer. A more common experiment for colloidal systems, however, is to measure the osmotic compressibility (97t/90) For a binary system, the osmotic compressibility is proportional to the structure factor at = 0, as follows ... 

In a static fight scattering experiment the effective structure factor, S(0) at zero scattering vector was obtained from the extrapolated forward scattering. In Fig. 4 is shown the variation of the excess Rayleigh ratio dR(0) extrapolated zero scattering vector with the volume fraction of droplets, (f> [3]. In the monodisperse case, S(0) is finked to the osmotic compressibility for which, in the case of hard spheres, an accurate expression exists due to Carnahan and Starling [17]. The experimental data were... 

Here Ap(q) denotes a Fourier transformation %(g) is a factor of osmotic compressibility of the system ... 

The theory of simple electrolytes provides a scheme for calculating S(q) for a given interaction potential U(r) where the interaction strength varies via a screening parameter k. From the structure factor S(q) important properties may be derived, e.g., the charge-charge structure factor which is related to the spatial distribution of the ionic charges in the solution, the osmotic compressibility and the shear viscosity. The diffusion coefficient which is connected to S(q) will be treated in Sect. 3. 

A third system that is claimed to behave as a model hard sphere fluid is a dispersion of colloidal silica spheres sterically stabilized by stearyl chains g ted onto the surface and dispersed in cyclohexane ". Experimental studies of both the equilibrium thermodynamic and structural properties (osmotic compressibility and structure factor) as well as the dynamic properties (sedimentation, diffusion and viscosity) established that this system can indeed be described in very good approximation as a hard sphere colloidal dispersion (for a review of these experiments and their interpretation in terms of a hard sphere model see Ref. 4). De Kruif et al. 5 observed that in these lyophilic silica dispersions at volume fractions above 0.5 a transition to an ordered structure occurs. The transition from an initially glass like sediment to the iridescent (ordered) state appears only after weeks or months. 

As explained in Section 4.2, the osmotic pressure can be expressed as a product of two independent terms, the derivative of the free energy with respect to the amount of interface a = ( f and a geometrical factor ( f representing the effect of compression (see Eq. (4.1)). Since in solid-stabilized emulsions the interfaces... 

In this process, water crosses the outer semi-permeable membrane of the pump. The characteristics of the semipermeable membrane including permeability, pore size, and thickness are key factors determining the rate at which water molecules enter the osmotic sleeve. The water that is drawn across the semipermeable membrane causes the osmotic chamber to expand. This force compresses the flexible drug reservoir, discharging the drag solution through the flow moderator. 

The strong osmotic pressure is also the decisive factor for the understanding of the interaaion of the SPEB in solution. A simple model was developed that is based on the compression of the surface layer upon mutual interaction of two SPEBs. Since the counterions cannot escape, the... 

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