Retention surface

The two examples of sample preparation for the analysis of trace material in liquid matrixes are typical of those met in the analytical laboratory. They are dealt with in two quite different ways one uses the now well established cartridge extraction technique which is the most common the other uses a unique type of stationary phase which separates simultaneously on two different principles. Firstly, due to its design it can exclude large molecules from the interacting surface secondly, small molecules that can penetrate to the retentive surface can be separated by dispersive interactions. The two examples given will be the determination of trimethoprim in blood serum and the determination of herbicides in pond water. 

Interpretive methods Involve modeling the retention surface (as opposed to the response surface) on the basis of experimental retention time data [478-480,485,525,541]. The model for the retention surface may be graphical or algebraic and based on mathematical or statistical theories. The retention surface is generally much simpler than the response surface and can be describe by an accurate model on the basis of a small number of experiments, typically 7 to 10. Solute recognition in all chromatograms is essential, however, and the accuracy of any predictions is dependent on the quality of the model. 

Surface residues of parathion on peaches were 4- to 15-fold higher than for comparable schedules on apples or pears, possibly because of the higher initial deposits retained on the more retentive surfaces of these fruits. Surface residues of DDT on peaches were also higher than those which would be expected to result from comparable schedules on apples and pears. Typical residue values for peaches are shown in Table V. 

A subset of simultaneous methods which overcomes the difficulty of mapping complex response surfaces by an exhaustive series of experiments are the interpretive methods, in which retention surfaces are modeled using a minimum number of experimental data points. Retention surfaces thus obtained for the individual solutes are then used to calculate (via computer) the total response surface according to some predetermined criterion. The total response surface is then searched for the optimum. 

Although it is beyond the scope of this chapter, more sophisticated interpretive methods can be employed to compensate for the anomalous retention behavior. That is, the complex retention surfaces are broken down into smaller parts in an iterative fashion the smaller retention surfaces are more accurately modeled by simple functions. This iterative interpretive approach has recently been applied in micellar LC (MLC) for the optimization of organic modifier, surfactant concentration, and pH (57,58). 

Generation of Retention Surfaces. Two types of retention surfaces were generated (i) an isothermal (two-dimensional) surface illustrating the dependence of retention on density and (ii) a two variable (three-dimensional) surface illustrating the dependence of retention on temperature and density. 

For the density-only retention surface, retention data were collected at 4 different densities (0.1,0.2,0.3 and 0.4 g/mL) at a temperature of 80°C. Data were fit to equation 2 via nonlinear regression, and the resulting retention equations for each solute could be used to calculate the response surface. Although the goodness of fit was difficult to estimate since there was only one degree of freedom (and it is easy for R2 values to exceed 0.999 under these conditions), the good agreement of the predicted and measured retention values at the optimum (vide infra) provided additional support for the accuracy of equation 2. 

For the second retention surface, data were collected according to a three-level, two-factor (density and temperature) experimental design. Each factor was assigned three different values (0.2,0.3 and 0.4 g/mL 75,100 and 125°C), and experiments were conducted at the nine combinations. Data were fit to the model by multiple regression, and these retention surfaces were used to calculate the response surface. 

Density Optimization. The retention surfaces as a function of mobile phase density at a temperature of 80 °C are shown in Figure 4. The quadratic dependence of In k on density is apparent over a wide range in density. More importantly, however, there are numerous peak reversals which demonstrate the need for a systematic optimization approach. 

Figure 4. Isothermal retention surfaces of an eight component sample (Table HI). The diversity of their chemical structure resulted in numerous peak reversals.

Figure 5. Comparison of the threshold separation and time-corrected, normalized, resolution-product response surfaces for the eight component sample (Table ID). Response surfaces calculated via equations 9 and 10 using the isothermal retention surfaces of Figure 4.

Shown in Figure 7 is the three dimensional retention surface for quinoline, a representative sample component Retention surfaces for the 7 other solutes, not shown for the sake of clarity, were generally as smooth and continuous, and resembled each other in a fashion analogous to the two dimensional In k /density surfaces of Figure 4. Similar peak reversals were also observed, and a systematic optimization scheme is again clearly warranted. Note that the individual effects of temperature and density, as predicted by equations 2 and 3, can be inferred from the appropriate cross sections of Figure 7. 

Figure 7. Three-dimensional retention surface of quinoline, as a function of reciprocal temperature and density (equation 8). Data used to generate these surfaces were collected as described in the text...

There is a distinct advantage in the use of models to describe retention surfaces rather than response surfaces. The important characteristic of the former is that they are much more simple than the response surface. Response surfaces form a combination of many (as many as there are components in the sample) different retention surfaces. In this way, fewer experiments (chromatograms) may be required to form an overall impression of the response surface. 

We will refer to methods that try to characterize the response surface indirectly through the retention surfaces of the individual solutes as interpretive methods. They will be discussed extensively in section 5.5. 

Figure 5.5 Example of a response surface for the optimization of the mobile phase in RPLC. Horizontal axis ternary mobile phase composition. Drawn line response surface using the resolution product as the criterion. Dashed lines retention surfaces for individual solutes (In k). For further details see section 5.5.2. Figure taken from ref. [504], Reprinted with permission. 

For chromatographic separations it is more sensible to compare k values, because retention surfaces are easier to characterize than response surfaces. Hence, Jf in eqn.(5.2) is the capacity factor of the solute at the jth data point. Each solute will have its own values for and hence a different mean effect can be defined for each sample component and each parameter. The results for the four solutes and four parameters studied are given in table 5.2b. 

The chromatographic data is interpreted in terms of the retention surfaces of the individual components. 

Interpretive methods owe their existence to the relative simplicity of the retention surfaces in comparison to the response surface. Indeed, attempts to describe the latter by a mathematical model [539,540,541,542] have never been successful. The general idea behind interpretive methods is is that whereas many experiments are necessary to describe... 

In this section we will describe several optimization procedures which are simultaneous in the sense that all experiments are performed according to a pre-planned experimental design. However, unlike the methods described in section 5.2, the experimental data are now interpreted in terms of the individual retention surfaces for all solutes. The window diagram is the best known example of this kind of procedure. 

Figure 5.17 Application of the window diagram method for optimizing the pH in RPLC. Solutes S = scopoletin, U = umbelliferone, TF = trans-ferulic acid, TC = trans-p-coumaric acid, CF = cis-feruiic acid and CC = cis-p-coumaric acid, (a) retention surfaces, (b) window diagram. Figure taken from ref. (552J. Reprinted with permission.

However, a simple linear relationship does not usually exist. A clear example is the optimization of the pH in RPLC. The window diagram approach was applied to this problem by Deming et al. [550,551,552]. They measured the retention of each solute at a series of pH values (9 in ref. [550], 4 in refs. [551,552]) and fitted the experiments to eqn.(3.70). This is a three-parameter equation and hence a minimum of three experiments is required for it to be applied as a description of the retention surface. If more data points are available, the equation can be fitted to the data by regression analysis. 

The retention surfaces (lines) for all solutes need to be known. 

The response surface needs to be calculated from the retention surfaces. 

Colin et al. [SSS] have described a different method to construct a diagram that allows the prediction of optimum conditions. Their approach is based on the calculation of so-called critical bands. If the retention surface of a solute j is known, then a forbidden zone may be defined below the capacity factor kj. If the preceding solute i has a capacity factor kp which falls in this critical band, then the resolution between i and j is insufficient. Eqn.(1.20) relates the resolution to the capacity factors of the individual solutes ... 

Figure 5.18 (a) Figure showing the retention surfaces for some aromatic solutes in RPLC. Critical bands have beat constructed according to eqn.(5.16) below each solute. The dashed line indicates the optimum ternary mobile phase composition, (b) Chromatogram obtained at the predicted optimum composition. Figures taken from ref. [555], Reprinted with permission. 

Window diagrams and related methods may in principle be applied to optimization problems in more than one dimension. The main difference compared with one-parameter problems is that graphical procedures become much more difficult and that the role of the computer becomes more and more important. Deming et al. [558,559] have applied the window diagram method to the simultaneous optimization of two parameters in RPLC. The volume fraction of methanol and the concentration of ion-pairing reagent (1-octane sulfonic acid) were considered for the optimization of a mixture of 2,6-disubstituted anilines [558]. A five-parameter model equation was used to describe the retention surface for each solute. Data were recorded according to a three-level, two-factor experimental... 

Experiments were performed at the nine possible combinations of the values of the two parameters (see eqn.5.1) The model was fitted to the data by regression analysis. From the retention surfaces, the response surface was calculated. 

Weyland et al. [560,561] used this method to optimize ternary mobile phase compositions for the separation of sulfonamides by RPLC. They fitted the retention surfaces to a quadratic model similar to eqn.(3.39), and also used a combination of a threshold resolution and minimum analysis time (min tm fl / vmin> 1.25 eqn.4.24) [560]. This criterion may yield a good optimum if the optimization is performed on the final analytical column (see table 4.11). 

Once the retention surfaces are known, any criterion may in principle be used to calculate the response surface and to locate the optimum composition. One of the criteria used by Glajch et al. is the threshold minimum resolution criterion (section 4.3.3). This is done by means of a graphical procedure, referred to as overlapping resolution mapping or ORM. This procedure involves the location of areas in the triangle where the resolution Rs exceeds a certain threshold value. This is repeated for all pairs of solutes and the results are combined to form a single figure. 

An example of the procedure and the resulting overlapping resolution map is shown in figures 5.24 and 5.25. Because of the simple retention surfaces, each pair of peaks yields... 

Because the retention surfaces are known, the calculation step of the procedure can he repeated using different conditions, such as another criterion or another column. 

---