Osmotic compressibility critical

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). 

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. 

Let us fix the volume fraction cp at its critical value cpc the osmotic compressibility diverges when the reduced temperature t goes to zero. The variation law is given by the relation13... 

In view of the anomalous critical behavior of the correlation length and the osmotic compressibility, it appeared of interest to characterize the behavior of other properties. Bell-ocq and Gazeau investigated how the interfacial tension between the coexisting phases on the one hand and the difference of density of these phases on the other hand vanished at various points of the critical line P (Fig. 25) [152]. The aim of the experiments was to determine the associated critical exponents and and check whether the scaling laws that relate v,p, and f.i were valid all along the critical line. Data obtained for two critical points defined by Xc = 1.55 and Xc = 1.207 indicate that the values of the critical exponents and )U show an X dependence similar to that found for v and y. Furthermore, within the experimental accuracy, the obtained values of v, y, (i, and are in reasonable agreement with the theoretical predictions v = y 2/ = 3v (Table 2). 

Fhe same critical index, which defines osmotic compressibility in Table 1.2 ( e "t where = — Ix2 dx2ldn)r), enters into Plquation 69. 

The osmotic compressibility is divergent near the critical temperature in the form... 

The presence of a primary maximum in the potential of Figure 1 should imply an inflexion in the osmotic pressure versus volume fraction plot indicative of coagulation. The effect is indeed found in the computer simulations and to differing extents in the osmotic compression experiments. Meijer et al. have used the cell model to predict critical coagulation pressures to compare with their experimental results on the centrifugation of monodisperse polystyrene latices (diameter 0.5 jum). In the cell theory, the osmotic pressure is given by... 

The remaining bulk thermodynamic exponent a determines the rate of divergence of the constant-volume heat capacity Cv at the liquid-vapour critical point of a one-component fluid [or that of the constant-pressure heat capacity Cp, or of the literal mechanical (rather than osmotic) compressibility k, or of the coefficient of thermal expansion, of a liquid mixture near its consolute point]. At fixed p = p , the heat capacity Cv as a function of temperature has the shape shown schematically in Fig. 9.3(a) with... 

An emulsion that is, for instance, stable over many years at low droplet volume fraction may become unstable and coalesce when compressed above a critical osmotic pressure 11. As an example, when an oil-in-water emulsion stabilized with sodium dodecyl sulfate (SDS) is introduced in a dialysis bag and is stressed by the osmotic pressure imposed by an external polymer solution, coarsening occurs through the growth of a few randomly distributed large droplets [8]. A microscopic image of such a growth is shown in Fig. 5.1. 

The isothermal compressibility is made dimensionless by defining Kj = Kt/Pc, where Pc is the critical pressure. The osmotic susceptibility is defined as (dXi/d/r,)p T, where X,- is the mole fraction of component i and its chemical potential. 

Isotliennal compressibility Spreading pressure, adsorbed plrase Osmotic pressure Nrunber of plrases, plrase rale Joule/Tlromson coefficient Dipole moment Chemical potential, species i Stoiclriometric number, species i Molar or specific deirsity = 1 / V Critical density Reduced density... 

Meso-scale heterogeneities can be probed by the intensity of electromagnetic or neutron scattering at a selected wave number q, the instrumental scale. A good example of the scale-dependent meso-thermodynamic property is the isothermal compressibility of fluids or osmotic susceptibility of binary liquids near the critical point of phase separation. " In the limit of zero wave number and/or when the correlation length is small (c g 1) the intensity becomes the thermodynamic susceptibility, which diverges at the critical point as... 

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