Displacement isotherms

The graphical interpretation of Eq. (16-197) is shown in Fig. 16-37 for the conditions of Example 12. An operating hne is drawn from the origin to the point of the pure displacer isotherm at = cf. For displacement to occur, the operating hne must cross the pure component isotherms of the feed solutes. The product concentrations in the iso-tachic train are found where the operating hne crosses the isotherms. When this condition is met, the feed concentrations do not affect the final product concentrations. 

An adsorption-desorption transition is illustrated schematically in Figure 1, where we plot a displacement isotherm, i.e. the adsorbed amount of a polymer as a function of the composition of a mixture of solvent and displacer. At the left in Figure 1, where the concentration of displacer is low, the polymer surface excess is positive. As we increase the proportion of displacer in the mixture, we observe a decrease in the adsorbed amount. At a certain composition the adsorbed amount of polymer becomes zero. The concentration at which the polymer surface excess just vanishes will be denoted as the critical displacer concentration cr. Beyond 4>cr, the surface excess of the polymer is negative (and very small if the polymer concentration is low). 

Recent polymer adsorption theories, such as those of Roe (3) and of Scheutjens and Fleer (h) allow the calculation of displacement isotherms, so that we could study the dependency of these isotherms on various parameters by numerical methods. However, all the essential features of displacement can also be demonstrated by means of a simple analytical expression for the critical point, which can be derived in a straightforward way. 

The critical adsorption energy. A critical adsorption energy is predicted by many theories (2-6). Its value is dependent on conformational properties of the polymer and usually estimated as a few tenths ofkT (7). Yet, a method to determine x c experimentally has never been suggested. Displacement studies provide such a method. Inspection of Equation 5 bears out that Xsc is obtained from a displacement isotherm, provided that Xgd and the solvency terms vanish. This condition is met by taking as the displacer a molecule which is (nearly) identical to the repeating unit of the polymer, i.e. the polymer is displaced by its own monomer. Such... 

Figure 1. General shape of displacement isotherm and location of the critical point (schematically).

Figure 2. Theoretical displacement isotherms for various values, calculated from the Roe theory Athermal systems (Ax = 0, Xpd = 0). The dotted line is the extrapolation of the isotherm into the physically inaccessible region...

Figures 3 and 4 give displacement isotherms of PVP adsorbed on pyrogenic silica. The polymer surface excess is plotted versus the volume fraction of displacer < >d. Results in Figure 3 are for water as the solvent and in Figure 4 for dioxane as the solvent. Various displacers were used, as indicated in the figures. One of the displacers was N-ethyl pyrrolidone (NEP), which can be considered as the monomer of PVP. From the isotherm in dioxane, it was found that In = 0.14. The aqueous system shows strong...

Determination of the segmental adsorption energy. The determina-tion of x ° is also possible. Since x d can be found from Equation 5 if Xsc an the solvency terms are known, we can add xf° and find x ° by Equation 1. The determination of xf° calls for a separate experiment, e.g., an adsorption isotherm of the displacer from solvent, in the absence of polymer. Following such a scheme we used the values of cr obtained from the displacement isotherms of Figure 3 and 4 to determine segmental adsorption energy parameters Xg° for PVP on silica. The required additional information on xdo was obtained from the initial slopes of dis-... 

Figure 3. Experimental displacement isotherms of poly (vinyl pyrrolidone) from silica in water/displacer mixtures. NMP N-methyl pyrrolidine PYR Pyridine NEM N-ethyl morpholine DMSO dimethylsulfoxide...

Even complicated, charged macromolecules like proteins can be succesfully displaced. As an example we give in Figure 6 the displacement isotherm for human plasma albumin from silica by morpholine (21). Of course, in this case where charge effects and a variety of segment/surface interactions play a role, our simple Equation 5 does not apply. Nevertheless, for practical work it is important to realize that most macromolecules, often thought to be irreversibly adsorbed, can be removed completely from the adsorbent surface by the concerted action of a large number of small molecules. 

Figure 6. Displacement isotherm of Human Plasma Albumin from silica in aqueous morpholine solutions at pH = 8.5.

The original treatment of this type of displacement isotherm was developed by Flory and Huggins separately, back in 1942. They developed the isotherm for adsorption of large molecules (polymers). However, applicability to small molecules (e.g., ions) displacing only few water molecules has been proved. 

Sigurskjold, B.W. (2000) Exact analysis of competition ligand binding by displacement isothermal titration calorimetry, Anal.Biochem. 277, 260-266 ... 

The watershed point of component i is the displacer concentration (C = w ) corresponding to the intersection of the displacer isotherm and the initial tangent to the isotherm of that component. This concept was first identified by Glueckauf [6,7]. This critical concentration is given by Eq. 9.7. No displacement of component i is possible if the displacer concentration is below its watershed point (Figure 9.5a). Under conditions where C = w, the tail of the first component will end at the front of the second component band. As the displacer concentration falls below the watershed point, the rear boimdary of the first component separates completely from the isotachic train. In this case, which is not exceptional, one or a few early eluting components appear as independent elution bands before the isotachic train. This may be impossible to avoid—for example, if the solubility of the displacer in the carrier is insufficient. 

Operating line Straight line between the origin and the point of the displacer isotherm at the displacer concentration used. Its intersection with the component... 

Figure 13.21 Indicator displacement assay for tartrate sensing (upper panel). Displacement isotherms at 597 nm for the addition of (R,R)- and (S,S)-tartaric acid 74 to the complex of 72 and bromophenol blue (73) (c72 and 73=1 x 10 4mol/dm3)...

In displacement chromatography, a pulse of mixture is injected, and this is followed by a step of a single component called a displacer, which is adsorbed more strongly than any of the mixture components. After a certain period of time, during which the profiles of each pulse become reorganized, an isotachic pulse train is formed. In the isotachic train, each component forms its own concentration boxcar, with a height that depends on the component and displacer isotherms as well as on the displacer concentration, and a width proportional to the amount of the corresponding component in the sample [14], [15]. 

FIG. 4 Schematic representation of the enthalpy of displacement isotherm in binary mixtures in the case of U-shaped adsorption excess isotherms. 

FIG. 5 (a) Adsorption excess and (b) enthalpy of displacement isotherms in benzene... 

Enthalpy of displacement isotherms were determined by the flow technique. The heat effects recorded on dodecylammonium and dodecyldiammonium ver-miculites are found to be endothermic in both cases, i.e., the measured heat exchange process results in heat extraction. Since the adsorption isotherms unambiguously indicate positive adsorption of n-butanol, the question arises as to why an exothermic exchange enthalpy is not recorded. In our opinion, the reason for this is the endothermic enthalpy of dilution [59-61,69], which overcompensates for the interlamellar adsorption of butanol. When, knowing the adsorption excesses, the enthalpy isotherm A21// = f Xy,) characteristic of the solid/Uquid interfacial adsorption layer can be calculated, it is indeed the exothermic adsorption enthalpy isotherm specific for the surfacial interaction that is obtained (Fig. 30). This measurement suggests that the interlamellar adsorption of n-butanol is thermodynamically preferred and is accompanied by the liberation of a very large amount of heat (A21// = 16.0-16.5 J/g). 

If a side-wall of the cube is displaced isothermally and reversibly so as to increase the area by 8A, then the (tangential) work done on the system is... 

Figure 4.4. Simulation of the displacement train under ideal conditions from a set of individual substance isotherms and the displacer isotherm. The operating line is found by connecting the origin of the plot with the point on the displacer isotherm, defined by the displacer concentration in solution. The intersection points of the operating line with the substance isotherms determined the substance concentration in the displacement train. Substances whose isotherms do not intersect with the operating line elute ahead of the displacement train. Top Multiisotherm-plot with operating line. Bottom Corresponding chromatogram (displacement train).

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