Concentration shock

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

As SIAB is water-insoluble, it must be first dissolved in organic solvent prior to addition to an aqueous reaction medium. The most commonly used solvents for this purpose include DMSO and DMF. Typically, a concentrated shock solution is prepared in one of these solvents and an aliquot added to the protein conjugation solution. Long-term storage of the reagent in these solvents is not recommended, however, due to slow uptake of water and breakdown of the NHS ester end. 

Equation 6.49 is strictly valid only for the disperse part of the peak (Chapter 2.2.3). Depending on the shape of the isotherm, this is the rear part ( Langmuir ) or the front part ( anti-Langmuir ) of the peak (Fig 2.6). The sharp fronts ( Langmuir ) or tails ( anti-Langmuir ) of the peaks are called concentration discontinuities or shocks. To describe the movement of these shocks, the differential in Eq. 6.49 has to be replaced by discrete differences A, the secant of the isotherm, which describe the amplitudes of the concentration shocks in the mobile and stationary phases ... 

The method of elution of an isotopic pulse on a plateau was developed by Helfferich and Peterson [136] (see Figure 3.38). If labeled and unlabeled molecules are injected simultaneously on a concentration plateau, the unlabeled molecules travel at the velocity associated with the plateau concentration (Eq. 3.94), while the labeled molecules travel at the velocity associated with the concentration shock ... 

Furthermore, the theoretical analysis of the single-component problem in the ideal model provides some of the fimdamental concepts in nonlinear chromatography, such as the notions of the velocity associated with a concentration, of concentration shocks, and of diffuse bormdaries [1,2]. It also provides an understanding of the relationship between the thermod5mamics of phase equilibria, the shape of the isotherm (i.e., convex upward, linear, convex downward, or S-shaped) and the band profiles. Finally, it provides an explanation of the relative importance of the influences of the thermodynamics and the kinetics on the band profile. These concepts will provide a most useful framework for imderstanding the phenomena that occur in preparative chromatography. 

The study of the properties of this equation shows that its solution has a continuous or diffuse boundary, but also that the equation propagates discontinuities (or shocks) and that a stable concentration shock must take place on one side of the profile [22]. 

The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

The migration velocity of the stable concentration shock is given by the following equation... 

Thus, while the band widens and spreads, the shock height is constantly eroded, and the band area remains constant. This is in agreement with the second law of thermodynamics. Equation 7.7 cannot be used directly to calculate the trajectory of a concentration shock in elution chromatography, however, since the height... 

This equation cannot be used directly to calculate the retention time of a concentration shock of known height, as will be discussed later but is useful to find the retention time of a tracer pulse on a plateau of concentration C. 

In the following section (Section 7.3, we will analyze the trajectory of a stable concentration discontinuity injected into the column, i.e., as one of the boxmdaries of a rectangular injection profile. This derivation could be done directly in the case of a Langmuir isotherm [1,2,31] (see Section 7.3.3). It can also be derived as a particular case of a more general solution. First, however, in a last subsection, we determine the time it takes for a concentration shock to form on a simple continuous profile. 

For the asymptotic solution [20], the concentration is different from 0 between the following two times, corresponding to C = 0 at one end of the peak and the elution of the concentration shock at the other... 

In summary, the wide rectangular profile is characterized by two concentration shocks, at times and fR,2/ for the first and the second component, respectively, by a residual of the injection plateau, and by a concentration plateau at C having a length At2- These characteristics define the three zones of the elution chromatogram of a binary mixture (Figure 8.1) the pure first component zone, the mixed zone, and the pure second component zone. Analytical solutions are provided to calculate the individual band profiles for a binary mixture. Table 8.1. We now study the elution profile of a narrow injection pulse, when Eq. 8.36 is no longer verified. 

The difference between the elution of a narrow band and the elution of a wide band, discussed in the previous section, is that in the former case the injection plateau has eroded during elution and, therefore, tR,2 (Eq- 8.19) is larger than (Eq. 8.23a), making Eq. 8.36 no longer valid at z = L. After the injection plateau has disappeared, the heights of the two concentration shocks decrease continuously. Accordingly, the velocities of these shocks decrease and the solution of the problem becomes more complex. We will show that the first shock continues to move faster than the second shock and that the two bands eventually separate. 

Since the equations giving the rear diffuse profiles of the two components in the mixed zone and of the second component in the third zone where it is pure are the same for a narrow or for a wide injection band, it seems logical to begin here the description of the chromatogram. Equations 8.25 to 8.29 and 8.33 apply to both cases. By contrast, the retention times of the two concentration shocks, tRg and tRg, and the elution profile of the pure first component between the two shocks are different and must be calculated separately. 

When the injection plateau has eroded, the maximum concentration of the second component begins to decrease below Cj, and the second component shock slows down. The pure first component plateau concentration at is no longer stable, because the rear concentration shock of the first component also has to slow down. A rear diffuse profile of the first component appears between the two shocks (see... 

Figure 8.15 Elution of a wide injection band of a ternary mixture, (a) Experimental chromatogram and profile predicted by the model, (b) Trajectories of the concentration shocks and simple wave regions, (c) Hodograph transform in the Cj,C2 plane, (d) Hodograph transform in the C2, C3 plane, (e) Hodograph transform in the Ci,C3 plane. Reproduced with permission from R. Zenhdusem, Ph.D. Thesis, Eidgenosische Technische Hochschule, Zurich, Switzerland, 1993.

Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the... 

The column is equilibrated with the carrier mobile phase (I), and the injection of a sample (F) of a binary mixture (components A and A2) is performed diu-ing a finite period of time (tq — utj,J L), after which the displacer (P3) stream is pumped into the column. There are two concentration shocks one at the origin (0, 0), the stable front shock of the first component, the other at the end of the injection (0, To), the front shock of the displacer. Thus, two wave solutions appear in the x, t) plane. According to Eq. 9.25, for an initially empty bed, the characteristic parameters are I ai,a2, cif). State F represents the injection of A and A2 into the column according to Eq. 9.26, the characteristic parameters are F(co, ui2,as), where... 

Using the propagation velocity associated with a concentration and the node equations, it is easy to calculate the concentration plateau of each component in the columns 111 and IV and the location of the concentration shocks at both ends... 

We demonstrate in the next subsection that, although we are under linear conditions, the boundaries of the concentration plateaus opposed to the feed port are true, stable concentration shocks. 

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