Osmotic modulus

Munns, R., Greenway, H., Stelter, T.L. Kuo, J. (1983). Turgor pressure, volumetric elastic modulus, osmotic volume and ultrastructure of Chlorella emersoni grown at high and low external NaCl. Journal of Experimental Botany, 34,144-55. 

Alexander S, Chaikin P M, Grant P, Morales G J, Pincus P and Hone D 1984 Charge renormalisation, osmotic pressure, and bulk modulus of colloidal crystals theory J. Chem. Phys. 80 5776-81... 

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

Tbe discussion of the semi-chlute properties remains confined mainly to the osmotic modulus which in good solvents describes the repulsive interaction among the macromolecules as a function of concentration. After scaling the concentration by the overlap concentration c = A2M.Yf) and normalizing the osmotic modulus by the molar mass, universal masteS" curves are obtained. These master curves differ characteristically for the various macromolecular architectures. The branched materials form curves which lie, as expected, in the range between hard spheres and flexible linear chains. 

Keywords. Solution properties. Regularly branched structures. Randomly and hyperbranched polymers. Shrinking factors. Fractal dimensions. Osmotic modulus of semi-di-lute solutions. Molar mass distributions, SEC/MALLS/VISC chromatography... 

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... 

The expression in brackets represents the interparticle interactions. It gives the contribution by which the true molar mass is modified to yield the measurable apparent molar mass M pp(c), which is a function of the concentration. A dimensionless quantity can be obtained by multiplying Eq. (78b) with which will be called the reduced osmotic modulus... 

Fig.32a,b. Plot of the reduced osmotic modulus MJM (c) as a function of reduced concentration X = = cIc A2 I. The solid lines represent theoretical curves for the... 

At present, there are only a few experimental results known on the osmotic modulus of randomly branched macromolecules or randomly cross-Hnked chains in the semi-dilute regime. One possible explanation for this lack of data may be based on the prejudice that the universaHty predicted by de Geimes [4] for Hnear chains will hold in the same maimer also for branched materials. In particular it is expected that the individual characteristics of the macromolecules are lost due to the strong overlap of the segments from different macromolecules. The following data, mainly from the author s own research group, revealed however, that the characteristics of the special architectures are not lost. 

Fig. 35. Plot of the reduced osmotic modulus from anhydride cured linear and cross-linked phenylglycidylethers [165,173-175]...

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

This makes the catastrophic break-down of the osmotic modulus somewhat softer. 

A second reason for the turn-over in the osmotic modulus may arise from a decrease in A2 until it becomes zero or even negative. This would be the classical situation for a phase separation. The reason why in a good solvent such a phase separation should occur has not yet been elucidated and remains to be answered by a fundamental theory. In one case the reason seems to be clear. This is that of starches where the branched amylopectin coexists with a certain fraction of the linear amylose. Amylose is well known to form no stable solution in water. In its amorphous stage it can be brought into solution, but it then quickly undergoes a liquid-solid transition. Thus in starches the amylose content makes the amylopectin solution unstable and finally causes gelation that actually is a kinetically inhibited phase transition [166]. Because of the not yet fully clarified situation this turn-over will be not discussed any further. 

In some case, however, only a flattening of the osmotic modulus curve is observed. Such a case is found with star-branched macromolecules. This observation has rather comprehensively been investigated by Roovers et al. with stars of 64 and 128 arms [172]. The authors give the following explanation. At the point of coil overlap and at somewhat higher concentrations the stars feel the interaction as a quasi colloidal particle. Hence, a steeper increase of the osmotic mod-... 

Rgure 4.4. The plateau storage modulus (small solid symbols) and the minimum of the loss modulus G (small open symbols) as a function of the effective oil volume fraction. a = 0.25 um (circles), a = 0.37 j,m (triangles), a = 0.53 um (squares), and a = 0.74 qm (diamonds). The large circles are the measured values for the osmotic pressure. All data are normalized by yint/a (Adapted from [10].)... 

Rgure 4.5. The computed shear modulus G (stars) and osmotic pressure (line), compared with the experimental values for (squares) and n (full circles). All data are normalized by Kint/a- (Adapted from [21].)... 

The osmotic modulus becomes zero at the spinodal point. Therefore, the spinodal temperature, Ts, is obtained by substituting K = 0 into Eq. (220)... 

The scaling arguments simplify the expression of the osmotic pressure (or the osmotic modulus) by extracting the important terms in Eq. (2.12) and by taking account of the excluded volume effect of the network chains as follows... 

Equation (2.22) enables one to predict the asymptotic behavior of the osmotic pressure (and/or the osmotic modulus). However, since Eq. (2.22) does not predict the phase transition, it is an expression appropriate only for a good solvent system. 

The osmotic modulus, K, the frictional coefficient, f, and the diffusion coefficient, D, are related to density-density correlation function of the network, g(r), by [62]... 

Depending on the enrichment term (E0) of the membrane, the modulus can be larger or smaller than 1.0. For reverse osmosis E0 is less than 1.0, and the concentration polarization modulus is normally between 1.1 and 1.5 that is, the concentration of salt at the membrane surface is 1.1 to 1.5 times larger than it would be in the absence of concentration polarization. The salt leakage through the membrane and the osmotic pressure that must be overcome to produce a flow of water are increased proportionately. Fortunately, modem reverse osmosis membranes are extremely selective and permeable, and can still produce useful desalted water under these conditions. In other membrane processes, such as pervaporation or ultrafiltration, the concentration polarization modulus may be as large as 5 to 10 or as small as 0.2 to 0.1, and may seriously affect the performance of the membrane. 

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