Modeling of Stationary Phases

Hyver (Ed.) High Resolution Gas Chromatography, 3rd ed., Hewlett-Packard, Palo Alto, CA, 1989, Chap. 2. 

Dandeneau and E. H. Zerenner, Proc. 3rd International Symposium on Glass Capillary Gas Chromatography, Hindelang, Germany, 1979, pp. 81-97. 

Jennings, Comparisons of Fused Silica and Other Glasses Columns in Gas Chromatography, Alfred Heuthig, Publishers, Heidelberg, Germany, 1986, p. 12. 

Macomber, P. Nico, and G. Nelson, LC/GC, The Apphcation Notebook, June, p. 

---

Kinetic Modeling of Stationary Phase of Growth The logarithmic or exponential law, given by the equation... 

Chapter 9 consists of the following in Sect. 9.2 the physical model of two-phase flow with evaporating meniscus is described. The calculation of the parameters distribution along the micro-channel is presented in Sect. 9.3. The stationary flow regimes are considered in Sect. 9.4. The data from the experimental facility and results related to two-phase flow in a heated capillary are described in Sect. 9.5. 

Solvent selectivity is a measure of the relative capacity of a solvent to enter into specific solute-solvent interactions, characterized as dispersion, induction, orientation and coaplexation interactions, unfortunately, fundamental aiq>roaches have not advanced to the point where an exact model can be put forward to describe the principal intermolecular forces between complex molecules. Chromatograidters, therefore, have come to rely on empirical models to estimate the solvent selectivity of stationary phases. The Rohrschneider/McReynolds system of phase constants [6,15,318,327,328,380,397,401-403], solubility... 

Figure 4.16 Model of reversed-phase ion-pair liquid chromatography in which counterions are held on the stationary-phase surface.

Kollie, T.O., Poole, C.F., Abraham, M.H., and Whiting, G.S., Comparison of two free energy of solvation models for characterizing selectivity of stationary phases used in gas-liquid chromatography. Anal. Chim. Acta, 259,1-13, 1992. 

Cluster analysis (numerical toxonomic aggregation) is applied to arrange phases according to their chromatographic behaviour. A set of retention data for 16 monofimctional benzenes, 110 difunctional benzenes and 15 trifunctional benzenes was subjected to analysis. Three groups of stationary phases can be distinguished polar, non-polar, and polyfluorinated. A linear relationship between the retention data of two stationary phases of the same class can be worked out. This linear relationship fits the model... 

A mathematical model was developed by Smith et al. (1998) to extract data on the elastic Young s modulus and some criteria of failure of stationary phase Baker s yeast from compression testing data. A mean Young s modulus of 150+ 15 MPa with a corresponding mean von Mises strain at failure of 0.75 + 0.08 and a mean von Mises stress at failure of 70+ 4 MPa (Smith et al., 2000b) were obtained. 

Thus, the absorption factor or the conversion increases. The relationship between increased conversion and mass transfer surface can be clarified with a simple model of gas-phase-limited absorption. Integration of Eq. (13) over the height coordinate provides the stationary solution of the component which can be absorbed and also... 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

FIGURE 3.6. (a) Reaction scheme that corresponds to the decrease and increase in capillary frequency, (b) Model of the phase transfer catalytic reaction in the stationary state. Reaction proceeds between the adsorbed chemical species at the interface after adsorption from the water phase. 

This equation has been derived for the model of the analyte distribution between mobile and stationary phases and is the same as expression (2-30) in Section 2.6. To be able to use this equation, we need to dehne (or independently determine) the volumes of these phases. The question of the determination or definition of the volume of stationary phase is the subject of significant controversy in scientihc literature, especially as it is related to the reversed-phase HPLC process [19]. 

I. D. Wilson, Investigation of a range of stationary phases for the separation of model dmgs by HPLC nsing superheated water as the mobile phase, Chro-matographia 52 (SuppI) (2000), S28-S34. 

Retention of Rohrschneider-McReynolds standards of selected chiral alcohols and ketones was measured to determine the thermodynamic selectivity parameters of stationary phases containing (- -)-61 (M = Pr, Eu, Dy, Er, Yb, n = 3, R = Mef) dissolved in poly(dimethylsiloxane) . Separation of selected racemic alcohols and ketones was achieved and the determined values of thermodynamic enantioselectivity were correlated with the molecular structure of the solutes studied. The decrease of the ionic radius of lanthanides induces greater increase of complexation efficiency for the alcohols than for the ketone coordination complexes. The selectivity of the studied stationary phases follows a common trend which is rationalized in terms of opposing electronic and steric effects of the Lewis acid-base interactions between the selected alcohols, ketones and lanthanide chelates. The retention of over fifty solutes on five stationary phases containing 61 (M = Pr, Eu, Dy, Er, Yb, n = 3, R = Mef) dissolved in polydimethylsiloxane were later measured ". The initial motivation for this work was to explore the utility of a solvation parameter model proposed and developed by Abraham and coworkers for complexing stationary phases containing metal coordination centers. Linear solvation... 

To obtain a simple interpretation of the experimental findings in IIC, theoretical chromatographers first adopted a stoichiometric strategy that pioneered this separation mode. Unfortunately, the reaction schemes of stoichiometric models in both the mobile phase (ion pair model) and stationary phase (dynamic ion exchange model) lack a firm foundation in physical chemistry because they are not able to account for the stationary-phase modification that results from the addition of the HR to the eluent, and they fail to properly describe experimental results, as pointed out by Bidlingmeyer et al. " Key insights on these retention models were also provided by... 

Figure 14.16 Elution peaks of 3-phenyl-l-propanol for various loadings. Experimental data (symbols) and best fit of the kinetic model (line). Stationary phase Vy-dac 201THP ODS silica (10 particles, 300 Apores, 56 m /g) column dimensions 150 x 4.6 mm. Mobile phase Fn = 1 mL/min methanol-water (25 75) 30°C sample volume 100 WL. The figure on each curve is the sample size (jimol). Reproduced with permission from C.A. Lucy, J.L. Wade and P.W. Carr,. Chromatogr., 484 (1989) 61 (Fig. 1).

We now consider the entire column rather than a small section thereof, as was done in the plate model. Let the column contain a total volume Vs of stationary phase, and a total volume Vm that can be occupied by the mobile phase. Since the column is uniformly coated or packed , the ratio Vm / will be equal to the ratio vml vs we used earlier for a single plate. Therefore the... 

An example will be given on the use of three-way models to explore chromatographic data. The data have the structure of stationary phases x mobile phases x solutes and the purpose of the analysis is to understand the differences between different stationary phases and... 

This resulted in a three-way array X (6 x 6 x 8) of retention factors with mode one holding the stationary phase, mode two the mobile phase and mode three the solutes. The purpose of the analysis was two-fold (i) to investigate the differences between the types of stationary-phase material and (ii) to develop a calibration model for transfer of retention factors from one stationary phase to another one. Only the first part of the investigations will be discussed. 

The master retention equation of the solvation parameter model relating the above processes to experimentally quantifiable contributions from all possible intermolecular interactions was presented in section 1.4.3. The system constants in the model (see Eq. 1.7 or 1.7a) convey all information of the ability of the stationary phase to participate in solute-solvent intermolecular interactions. The r constant refers to the ability of the stationary phase to interact with solute n- or jr-electron pairs. The s constant establishes the ability of the stationary phase to take part in dipole-type interactions. The a constant is a measure of stationary phase hydrogen-bond basicity and the b constant stationary phase hydrogen-bond acidity. The / constant incorporates contributions from stationary phase cavity formation and solute-solvent dispersion interactions. The system constants for some common packed column stationary phases are summarized in Table 2.6 [68,81,103,104,113]. Further values for non-ionic stationary phases [114,115], liquid organic salts [68,116], cyclodextrins [117], and lanthanide chelates dissolved in a poly(dimethylsiloxane) [118] are summarized elsewhere. 

The system of stationary phase constants introduced by Rohrschneider [282,283] and later modified by McReynolds [284] was the first widely adopted approach for the systematic organization of stationary phases based on their selectivity for specific solute interactions. Virtually all-popular stationary phases have been characterized by this method and compilations of phase constants are readily available [28,30]. Subsequent studies have demonstrated that the method is unsuitable for ranking stationary phases by their selectivity for specific interactions [29,102,285-287]. The solvation parameter model is suggested for this purpose (section 2.3.5). A brief summary of the model is presented here because of its historical significance and the fact that it provides a useful approach for the prediction of isothermal retention indices. 

Optimization of the combination of stationary phases using the PRISMA model. 

An alternative for a more general measure of orthogonality should use specific parameters of the columns being considered, instead of analyte retention data. The solvation parameter model [55] characterizes column retention by using five parameters. These parameters define for each column a vector in a five-dimensional space. If d is the angle between two column vectors, cos 6 will be nearly 1 for very similar pair of stationary phases, while values of cos 6 close to 0 will correspond to column sets of high orthogonality [58]. 

Because of the complexity of the phenomena, it is difficult to derive a generally valid stationaiy-phase mass-transfer term for the packed bed. This is the reason why we used the simplified model above to introduce the basic concepts of stationary-phase and mobile-phase mass transfer. 

---