While the linear elution theory is more generally applicable, the focus here will be on reversed-phase chromatography. After the definitions of the variants of peak capacity in gradients, I will examine the maximization of the peak capacity for specific analytical problems. Finally, I will briefly discuss the phenomenon of peak compression, which enhances the resolving power of gradient chromatography.

In this section, we will occupy ourselves with the fundamentals of the gradient equation. A more in-depth analysis can be found in textbooks, especially in reference [1]. The treatment here is a subtle simplification of Snyder’ treatment, but the elements are very similar.
 
In a gradient, the retention factor k of a compound is constantly changing with the mobile phase composition (see Sidebar 1 for more detailed equations). The dependence of the retention factor on the solvent composition c can often be described by a simple log-linear relationship: 

(1) 

k₀ is the retention factor in a reference composition, in our case here in the solvent composition at the beginning of the gradient, and S is the slope of this relationship. ∆c is the difference in the solvent composition between the reference state with the retention factor k₀ and the composition that delivers the retention factor k. The slope S depends on the molecular weight of the analyte. Its value increases roughly with the square root of the molecular weight of the analyte. 

While such a simple linear relationship is not accurate for a very wide range of solvent compositions [2], it is definitely a suitable description for the range of solvent compositions in which the peak is actually moving through the column (as compared to the time when it “sits” at the column inlet with little migration). If we then change the solvent composition linearly with time, the natural logarithm of the retention factor will then also change linearly with time. This permits a simple integration and derivation of the retention of an analyte as a function of the way we carry out the gradient [1, 3].

Click here to see in the sidebar how we come to the following formula. The derivation shown in the sidebar results in the following equation for the retention factor under gradient conditions k_g: 

 (2) 

We have used the following abbreviations in this equation: 

 (3a) and  (3b) 
 
G is the generalized gradient slope, comprising the slope S of equation 1, the difference in the solvent composition from the beginning to the end of the gradient ∆c, and the inverse ratio of the gradient run time t_g to the column dead time t₀. To complete the information, we also need to know the peak width under gradient conditions, which is primarily determined by the retention factor at the point of elution k_e. This retention factor is calculated by simply introducing the gradient retention factor into the equation S1.3:

   (4a)

  (4b) 

Equations 2 and equation 4b are a complete description of the behavior of a peak under gradient conditions using the linear solvent strength model [1]. The gradient parameter G from equation 3b contains all the elements of the gradient separation: the slope S of the relationship between the natural logarithm of the retention factor and the solvent composition, which is a property of the analyte, the column and the mobile phase chosen, and the experimental gradient slope  that is open to the manipulation of the investigator.

If the analytes elute in the developed gradient (and for example not still in the isocratic section caused by a gradient delay; see the section on Gradient Transfer) then a constant gradient slope will result in a constant elution pattern. For example, we can change the difference in the solvent composition from the beginning to the end of the gradient, if we change the gradient run time by the same factor without affecting the retention pattern and resolution of the analytes. This is for example useful, if our initial gradient is too long and no peaks are eluting any more beyond a particular time. Then we should simply reduce the gradient run time and the difference in the solvent composition by the same factor, and the elution pattern is preserved (Figure 1). Also, if we change from one column to another, as long as we keep the ratio of column dead time to gradient run time constant, we will get the same retention pattern (for a fixed change in solvent composition, of course), and only the peak widths will change in parallel with the changes in the plate count (Figure 1). Chromatogram scaled from a shorter column (top) to a longer column (bottom). The elution pattern is preserved to take advantage of the improvement in plate count for the longer column. (Click here to open pdf)
This is discussed in more detail in the section on gradient scaling. 

Conversely, changing the gradient run time alone, while keeping a constant difference in solvent composition from the beginning to the end of the gradient is the first tool recommended for changing the selectivity of a separation [1]. Now the value of the gradient slope changes for all compounds, and the resolution will change. If the value of S varies randomly throughout the chromatogram, many individual changes in the selectivity can be observed. If the value of S changes in a regular pattern, such as for a group of oligomers, where S increases with the molecular weight of the analyte, then easily predictable changes in the elution pattern occur with a change in the gradient slope. A good rule of thumb is that the slope S changes with the square root of the molecular weight of the analytes. 

The same changes in retention and selectivity can be achieved by changing the flow rate, while keeping the gradient run time constant. While this is not common practice, it is a simple approach to decrease the run time required during method development. A limitation to this approach can be the pressure limit of the instrument, or to a much lesser degree, the deterioration in plate count. It needs to be pointed out clearly that this situation is completely different from isocratic chromatography, where retention times and peak widths change with the flow rate, while the selectivity of the separation remains constant. 

When increasing the gradient run time while keeping the change in composition and the flow rate constant, the gradient is flattened. As a consequence, all compounds elute in a weaker eluent strength. This means that the retention factor at the point of elution k_e becomes larger, the peaks become wider and the signal height decreases. This is seen in Figure 2 for a 3-fold change in the gradient run time, from 10 minutes to 30 minutes. For our analytes, all small molecules, the gradient retention increased, but not in direct proportion to the increase in the gradient duration. In addition, as pointed out by the lines in Figure 2, the changes in retention are different for different compounds, and these differences are due to differences in both the retention factor at the beginning of the gradient k₀ and the slope S. The solvent composition at the point of elution changes less for compounds with a steep slope than for those with a flat slope. For instance, the change in solvent composition is smaller for a peptide sample than for a sample of molecules with a low molecular weight. In our example, compounds 1, 2, 4, 5 and 7 have the same slope S, so their relative position in the gradient will always remain the same no matter how we run the gradient. In a steep gradient, they will be located closer together, and in a flat gradient, they will be further apart. On the other hand, for the compound pair 5 and 6, the slopes S are different. The slope is flatter for analyte 6 compared to analyte 5. Therefore, as the gradient is flattened, analyte 6 will move towards a solvent composition with more water compared to analyte 5. The difference in the slope is the cause of the change in the elution sequence. For a steep gradient, compound 5 elutes before compound 6. As the gradient is flattened, both peaks coelute, and finally, for a still flatter gradient, compound 5 elutes behind compound 6. Furthermore, the ratio of the gradient retention factors between the steep and the flat gradient is about the same for compounds 3, 4 and 6, which show a regular increase in the S-value with increased retention. Such a scenario can be encountered in many realistic situations, as the slope S often slowly increases with the retention of a group of analytes, especially as this increase is caused by an increase in the hydrophobic area (for example, methyl-, ethyl-, propyl- substitutions on a parent molecule, or even more so for oligomers). 

(Figure 2: Change in retention with an expansion of the gradient. Top: 10 minute gradient. Bottom: 30 minute gradient. (Click to open pdf))

In these examples in figure 1 and 2, we have seen the three possibilities of changes in the elution pattern with changing gradients:

1. The analytes have the same slope S and their retention changes in parallel. In this case, the resolution improves as the gradient expands (dotted lines).
2. Two analytes with a different slope S elute close to each other, and a change in the gradient steepness will affect their resolution and relative position (compounds 5 and 6).
3. A group of analytes have slopes S that increase with the retention factor at the beginning of the gradient (solid lines). These compounds will show a synchronized shift in the gradient elution pattern as the gradient steepness is changed (example: oligomers).

The elution pattern in a gradient also depends on the initial conditions. This is less pronounced, when large molecules like peptides are the analytes. Also, for a given gradient slope, the later eluting compounds are less affected than the early eluting compounds, or even do not change their relative positions at all. Nevertheless, the starting conditions to the gradient are an additional tool in method development especially for the early eluting analytes.  

The elution pattern in a gradient also depends on the initial conditions. This is less pronounced, when large molecules like peptides are the analytes. Also, for a given gradient slope, the later eluting compounds are less affected than the early eluting compounds, or even do not change their relative positions at all. Nevertheless, the starting conditions to the gradient are an additional tool in method development especially for the early eluting analytes. 
 
Figure 3 shows an example of shifts in retention for a fairly complex sample containing 20 analytes. The separation at the bottom was completely optimized in a simulation for a high-performance UPLC column (N ~ 25 000) and contains all 20 peaks. The gradient at the bottom was executed with a gradient slope of 50% over 22 minutes with a starting composition of 20% organic. The gradient at the top started with 0% organic with the same gradient slope.  One can see that the retention pattern at the end of the chromatogram did not change, while more significant changes in peak position and elution order are observed early in the chromatogram. Specifically, the small peak indicated by the solid line coeluted with its neighbor in the top chromatogram.

(Figure 3: Changes in the elution pattern due to changes in the initial condition of gradients with the same gradient slope. )

In general, or on average, resolution in gradient chromatography can be treated in the same way as in isocratic chromatography [1, 4]. In isocratic chromatography, the resolution equation looks as follows:  (5) 

with the usual definitions: N is the plate count, k is the retention factor, and α is the relative retention between the two closely eluting compounds under consideration. If α does not change with retention, one can consider the influence of these three parameters on the resolution separately.
 
A very similar equation can be derived for gradients [1, 4]: 

(6) 

α is the same relative retention as considered for isocratic chromatography above. The derivation included a simplification of equation 4b for a large

k₀:  (7) 

and the similarity increases further: 

  (8) 

As above for the resolution equation for isocratic chromatography, if α does not change with retention, the resolution increases with an increase in k_e, the retention factor at the point of elution. Since k_e always increases with a flattening of the gradient, the resolution generally increases under these conditions, as pointed out above. However, as in isocratic chromatography, this may not be true for a specific pair of analytes, due to the specific changes in the elution order with solvent composition. Only the average resolution increases with a flattening of the gradient.