Staverman

The pressure difference between the high and low pressure sides of the membrane is denoted as AP the osmotic pressure difference across the membrane is defined as Att the net driving force for water transport across the membrane is AP — (tAtt, where O is the Staverman reflection coefficient and a = 1 means 100% solute rejection. The standardized terminology recommended for use to describe pressure-driven membrane processes, including that for reverse osmosis, has been reviewed (24). 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

The bracket (1 — 2/f) was introduced into the theory of rubber elasticity by Graessley [23], following an idea of Duiser and Staverman [28]. Graessley discussed the statistical mechanics of random coil networks, which he had divided into an ensemble of micronetworks. 

Duiser JA, Staverman AJ (1965) In Prins JA (ed) Physics of non-crystalline solids. North-Holland, Amsterdam, p 376... 

Staverman, A.J. Properties of Phantom Networks and Real Networks. Vol. 44, pp. 73-102. 

Mook, W.G., Bommerson, J.C. and Staverman, W.H. 1974 Carbon isotope fractionation between dissolved bicarbonate, and gaseous carbon dioxide. Earth and Planetary Science Letters 22 169-176. 

Duiser, J. A. Staverman, A. J. "Physics of Non-Crystalline Solids" Prins, J. A., Ed. North-Holland Publ. Corp., Amsterdam 1965. 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

Note 2 The present definition has been modified from that which appears in ref [5] to acknowledge the contributions of Staverman and to further clarify the statistical basis of the theory. 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The front factor g as defined above5 is unity in all the earlier theories (17). Recently Duiser and Staverman (233) have obtained g = j and Imai and Gordon (259) g — 0.54 with Rouse model theories which make no a priori assumptions about the junction point locations after deformation. Edwards (260) also arrives at and Freed (261) deduces that g= 1 is an upper bound by similar approaches. The front factor usually assumed in the shifted relaxation theory of the plateau modulus is g = 1, although Chompff and Duiser (232) obtain g = j through their extension of the Duiser-Staverman result to entanglement networks. The physical reasons for the different values of g in different treatments are not clear at present. 

Duiser,J.A., Staverman,AJ. On the theory of rubber elasticity. In Prins,J. A (Ed.) Physics of non-ciystalline solids, pp. 376-387. Amsterdam North Holland Publ. 1965. 

H.-J. Cantow, Freiburg i. Br. G. Dall Asta, Milano J. D. Ferry, Madison H. Fujita, Osaka M. Gordon, Colchester W. Kern, Mainz G. Natta, Milano S. Okamura, Kyoto C. G. Overberger, Ann Arbor W. Prins, Syracuse G. V. Schulz, Mainz W. P. Slichter, Murray Hill A. J. Staverman, Leiden... 

Staverman,A.J. Thermodynamic aspects of the glass-rubber transition. Rheol. Acta 5,283 (1966). 

Fig. 13. The effect of a crosslink at P on the mechanical behaviour of a two chain network between fixed points is mathematically equivalent to a fixed point on one of the chains, thus leading to three rather than four elastically effective chains [Duiser and Staverman, Chompff and Duiser 43, 32, 33)]...

Staverman, A. J. Thermodynamics of polymers. In Encyclopedia of physics (S. FlCgOE. Ed.) Vol. 13. Berlin-Gottingen-Heidelberg Springer 1962. 

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