Osmotic compression

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

Accordingly to (19) the osmotic compressibility dlt / dc into diluted solutions does not depend on the concentration of macromolecules (dft / dc = RT) on the contrary, in semi-diluted solutions it becomes (as it follows from (25)) as linear function of relative concentration ... 

It follows that the osmotic compressibility C7i / dc = von / dtp will be equal to... 

In all hydrodynamic methods we have the effect of both the hydrodynamic and thermodynamic interactions and these do not contribute additively but are coupled. This explains why the theoretical treatment of [77] and of the concentration dependence of has been so difficult. So far a satisfactory result could be achieved only for flexible linear chains [3, 73]. Fortunately, the thermodynamic interaction alone can be measured by static scattering techniques (or osmotic pressure measurement) when the scattering intensity is extrapolated to zero scattering angle (forward scattering). Statistical thermodynamics demonstrate that this forward scattering is given by the osmotic compressibility dc/dn as [74,75]... 

As demonstrated in a previous section the osmotic compressibility can be obtained from the forward scattering of light... 

In order to have a suitable connection to the well understood dilute solutions it was suggested by Debye to use the reciprocal of the osmotic compressibility, which for convenience will be called the osmotic modulus... 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Calculated from d = (4vspMj,/itNA)i 2 with the Avogadro constant NA b Calculated from the fitting to osmotic pressure and osmotic compressibility data (cf. Sect. 3) e Calculated from intrinsic viscosity or sedimentation coefficient data d Obtained from crystallographic data... 

For a binary polymer solution, the reciprocal of the osmotic compressibility 0n/0c at constant T and the solvent chemical potential p0 can be determined by sedimentation equilibrium through the relation [58,59] ... 

Fig. 3. Comparison between the scaled particle theory and experiment for reciprocal osmotic compressibility of PHIC-DCM [60]. Vertical segments indicate C. ...

Figure 4 compares osmotic compressibility data for isotropic schizophyllan-water solutions [63] with the scaled particle theory. The ratios of the z-average to the weight-average molecular weights of these schizophyllan samples are ca. 1.2. The solid curves, calculated with d taken to be 1.52 nm and other molecular parameters (Lc, v, and c ) estimated from Mw and the wormlike chain parameters in Table 1, are seen to come close to the data points for all samples. 

Therefore the scattering intensity defined in equation (5.20) is directly proportional to the osmotic compressibility ... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

For colloidal liquids, Eqs. (19-21) refer to the excess energy [second term of the right-hand side of Eq. (19)], the osmotic pressure and osmotic compressibility, respectively. They show one of the important features of the radial distribution function g(r), namely, that this quantity bridges the (structural) properties of the system at the mesoscopic scale with its macroscopic (thermodynamic) properties. 

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. 

The OZ equation depends on the scattering vector q and involves the osmotic compressibility % and the correlation length of the fluctuations, which temperature dependencies are given by scaling laws of the form analogous to Eq. (20),... 

It was proved by scattering studies (1-9) that water in oil microemulsions at low water content are dispersions of identical spherical droplets in a continuous oil phase. Dilution procedure and light scattering measurements (both the intensity and the correlation function of the scattered light) allow to measure the osmotic compressibility and the diffusion coefficient of the droplets ... 

As the attractive potential V between droplets increases, g decreases. The normalized osmotic compressibility curves v/kT.3tt/3tf> versus cf pass closer and closer to the axis and become tangent to... 

Figure 2. Normalized osmotic compressibility versus droplets volume fraction for various microemulsions.

The dilution procedure (2) associated to the measurements of the intensity of the scattered light allows to study the osmotic compressibility of the samples as a function of the volume fraction of the dispersed phase. In that case the scattered intensity I is given by (9-10) ... 

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