Theory, size exclusion

Poly(ethylene oxide). The synthesis and subsequent hydrolysis and condensation of alkoxysilane-terniinated macromonomers have been studied (39,40). Using Si-nmr and size-exclusion chromatography (sec) the evolution of the siUcate stmctures on the alkoxysilane-terniinated poly(ethylene oxide) (PEO) macromonomers of controlled functionahty was observed. Also, the effect of vitrification upon the network cross-link density of the developing inorganic—organic hybrid using percolation and mean-field theory was considered. 

In fundamental SEC studies retention is often described in terms of a distribution coefficient. The theoretical distribution coefficient Kg is defined as the ratio of solute concentration inside and outside of the packing pores under size exclusion conditions. The experimental distribution coefficient as defined in Equation 1, is a measurable quantity that can be used to check the theory. 

The degree of size exclusion observed for a given solute is directly related to the size and geometry of the solute. The theory advanced by Giddings et al. [182] predicts the proportionality... 

Figure 12 Theoretical predictions of solute partitioning in a gel with a fiber radius of 0.4 nm for solutes of different hydrodynamic radii, based on the size exclusion theory of Eq. (30). (From Ref. 174.)...

It took a long time before the percolation theory could be proved to be the better one in most cases. The reason for this delay resulted in part from the fact that until ten years ago the size exclusion chromatography (SEC) with on-hne light scattering was not sufficiently well developed. A direct molar mass determination is, however, imperative, since the separation in SEC is due to the hydrodynamic volume of the particles. A branched macromolecule has, however, a significantly higher molar mass than a linear one of the same hydrodynamic volume. Since 1989 a number of results have been reported which all strongly supported the percolation theory [109,111-116]. 

Since the technique was introduced (JJ in 1964, gel permeation chromatography (GPC), or size exclusion chromatography (SEC), has played an increasingly important role for the characterisation of polymers. The theory and practice of this chromatographic method have been extensively reported and a comprehensive text has recently been published on modern size exclusion chromatography (2). 

Fractionation of proteins according to size utilizing cross-linked dextran or polyacrylamide gel columns was first demonstrated by Porath and Flodin 63 in 1959. This technique has become the most widely accepted method for separation and molecular weight determination of hydrophilic and some hydrophobic macromolecules using aqueous buffers with or without organic modifier. While this technique might not be unique in its ability to resolve and separate proteins, it is one additional simple and effective tool in the chemist s armamentarium. The theories behind size-exclusion HPLC and size-exclusion chromatography at low pressure are identical and are described in several publications. 31 34 36 39 44 64 65 ... 

Separation can now be achieved by using any one and/or a combination of the various modes available. These modes include size exclusion, normal phase, reversed-phase, paired-ion reversed-phase and ion-exchange. The principles, theories and the... 

Sources of error in this approach arise from both experimental and theoretical grounds. Modern theories of SEC retention mechanism are based on the assumption that the size exclusion process uniquely determines the elution volume, and yet the possibility of reversible adsorption is difficult to dismiss and, where it occurs, errors in the interpretation may easily result. As a warning for the application of universal calibration methodology, Cas-sassa (29) indicates in a later paper that the quantity rj M is not a truly... 

Both the Yasuda et al. and Schnitzer theories indicate that the key variables defining permeability in a gel are the swelling degree of the gel and the average dimension of the solute. Thus the dependence of permeability on swelling degree for a responsive gel inert to solutes can be predicted using the theory of Yasuda et al. to obtain the ratio D/Dc (the ratio of the diffusion coefficient of a solute in the gel to its value in free solution) and the theory of Schnitzer for ideal size exclusion to obtain K. A dimensionless permeability can be defined as the... 

Fig. 14. Prediction of solute permeability for solutes of different hydrodynamic radii in swollen gels. Permeability is calculated as the product of the partition coefficient, K, using the size exclusion theory of Schnitzer and the ratio of the solute diffusion coefficient in the gel, D, to its value in solution, D , using the theory of Yasuda et al. [123, 159, 160]...

The experiments which yielded the diffusion coefficients for acetaminophen in PNIPAAm gel in Fig. 16 also yielded the corresponding partition coefficients. While the diffusion coefficients fit theory, the partition coefficients as plotted in Fig. 18 do not at all. In fact, a trend opposite to theory is observed as the partition coefficients are seen to increase as the gel swelling decreases. In fact, above the transition temperature of the gel, at 35 °C, the partition coefficient is seven times the maximum possible size exclusion coefficient, 1. This implies the dominance of hydrophobic effects over steric effects, since acetaminophen is a relatively small, nonionic but hydrophobic solute, and while the gel mesh size shrinks with increasing temperature, its level of hydrophobicity increases with temperature. 

Fig. 18. The partition coefficients of acetaminophen in 10 x 4 PNIPAAm gels as a function of the swelling degree of the gel. The trend is opposite to that estimated by the ideal size exclusion theory of Schnitzer, shown as a solid line [17]. The dashed line is to guide the eye. Reprinted from the Journal of Controlled Release (1992) 18 1, by permission of the publishers, Elsevier Science Publishers...

As the pore diameter increases in size (s decreases) relative to molecular or colloidal dimensions, less restrictions are imposed on the motions of contained species. Thus the exclusion effect gradually subsides as the pore size increases and consequently K-+1. For the separation of two molecules of different size, it is important to pick a pore diameter that will substantially exclude one species but not another. Pore size selection is thus of utmost importance in membrane science and in choosing a support for size exclusion chromatography (SEC). Aspects of pore size optimization in SEC based on the above partitioning theory have been developed [28]. 

Note that the physicochemical mechanisms that enables us to perform the chromatographic bioseparations are not always adsorption-like but can involve ion exchange, ion exclusion, or size exclusion. Even if it is generally possible to fit experimental data with a mathematical function derived from the adsorption theory, it is strongly advisable to refer to the proper physicochemical process before modeling the separation. For instance, ion exchange can be modeled with selectivity coefficients (derived from the mass action law) that can be constant or not,18,19 ion-exclusion can be modeled thanks to theories based on the Donnan exclusion, etc. 

Because of the inadequacies of QED a fundamental theory of electrode processes is still lacking. The working theories are exclusively phenomenological and formulated entirely in terms of ionic distributions in the vicinity of electrode interfaces. An early, incomplete attempt [54] to develop a quantum mechanical theory of electrolysis based on electron tunnelling, is still invoked and extensively misunderstood as the basis of charge-transfer. It is clear from too many superficial statements about the nature of electrons that the symbol e is considered sufficient to summarize their important function. The size, spin and mass of the electron never feature in the dynamics of electrochemistry. 

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