Simple method for the quantitative examination of extra column band broadening in microchromatographic systems

Abstract

In recent years capillary chromatography has gained popularity for trace analyses. Most often UV or electrochemical detection is employed because the small peak volumes make post-column derivatization challenging. We have developed a simple method based on flow injection for determining contributions to peak broadening from post-column reactors. The only requirement for application of our methodology is that diffusion be in the Taylor regime so that radial concentration gradients are relaxed enabling mixing purely by diffusion.

Introduction

In the past several years the use of capillary chromatography has attracted a great deal of attention. Small HPLC columns with diameters ranging from 25 to 500 μm are advantageous in trace analysis because of the increased mass sensitivity that can be achieved with a smaller column volume. Typically, these analyses are carried out with UV or electrochemical detection [1], [2], [3], [4], [5], [6], [7] with derivatization (if necessary) achieved pre-column. Post-column derivatization is challenging considering the small peak volumes associated with these analyses. Nonetheless, post-column derivatization is useful when the derivatization agent is not compatible with commercial columns or when the separation of the underivatized analytes is well understood. As post-column reactor volumes must be small, the determination of their contribution to band broadening is challenging. To determine small peak variances, a simple and reliable method is required.

We have developed such a method. In the method, we have abandoned the idea of using very small injection volumes in order to visualize band broadening from other sources. For this work, a large sample loop has been used to produce a steady-state signal that can be differentiated to yield two peaks. According to linear response theory [8], the differentiation of the response from a step function yields the response to a delta function input. The spreading of the delta function in a simple reactor can then be determined by standard methods. A capillary or channel operating in the Taylor regime [9] is the simplest reactor without special flow path geometries [10]. In such a reactor two fluid steams joined in a single channel will mix by diffusion only. Thus, we have applied the method to determine band broadening in a simple capillary.

Though a significant literature exists detailing the mathematical equations for peak shape and band broadening in traditional flow injection techniques, these experiments are not performed in the Taylor regime [11], [12]. As a result that theory is not applicable to the problem.

By using the time equivalent to a theoretical plate certain experimental and computational simplifications result. From plate height theory [13], [14] we know that the plate height in units of time, H_t, is related to the length-dimensioned plate height, H_L, as shown in Eq. (1) where σ² is the similarly dimensioned standard deviation of the zone squared (second central moment or variance), the distance that the band has traveled is L, and v is the average solute velocity.

$$\text{H}{}_{\mathrm{L}}\text{=σ}{}^2{}_{\mathrm{l}}\text{/L=(σ}{}^2{}_{\mathrm{t}}\text{/t)·v}$$

We also know that the plate height is related to the solute dispersion coefficient, D, as shown in Eq. (2).

$$\text{H}{}_{\mathrm{L}}\text{=2D/v}$$

From Taylor’s theory of dispersion in an open tube of radius a for a solute with a molecular diffusion coefficient D_mol, the dispersion coefficient is shown as Eq. (3).

$$\text{D=v}{}^2\text{a}{}^2\text{/48D}{}_{\mathrm{<mtext>mol</mtext>}}$$

Therefore, Eq. (4) expresses the plate height in units of time, which will be referred to as the “plate time” in this paper.

$$\text{H}{}_{\mathrm{t}}\text{=2D/v}{}^2\text{=a}{}^2\text{/24D}{}_{\mathrm{<mtext>mol</mtext>}}$$

The use of time units has several advantages. In applications using flow splitting, it can be difficult to maintain strictly constant fluid velocity while changing tubing lengths and diameters. The plate time is not dependent on the individual values of tubing length or fluid velocity, so the determination of an experimental plate time is straightforward, as both parameters, i.e., σ_t² and t (second central and first moments) are represented in the data and require no physical measurements of length or volume. Also, the estimation of a theoretical plate time from Eq. (4) is uncomplicated.