Theory of Gradient Elution Liquid Chromatography with Linear Solvent Strength: Part 2. Peak Width Formation

Abstract

The process of formation of the width (σ _b) of a solute band migrating along a column and its effect on the width (σ) of a corresponding peak in a chromatogram are quantified along with the extra-column contributions (Δσ _b and Δσ) to these parameters due to insufficiently narrow injection plugs. Previously unknown expressions for σ _b and Δσ _b as functions of the band migration distance and time were found. The negative gradients in the solvent strength cause the fronts of the solute bands to travel slower than their tails. This compresses the bands (reduces their widths). Previously unknown expressions describing the band compression process as functions of the band migration distance and time are found. The band compression tends to narrow the peaks. However, as shown here, the gradients that compress the bands also reduce their elution speeds. This tends to broaden the peaks (typically ignored phenomenon) and, as shown here, can cause a slight net peak broadening under normal conditions (in spite of general expectations that the gradients should narrow the peaks). On the other hand, as shown here, the gradients can significantly suppress the harmful effect of the extra-column peak broadening.

Appendix

Elution Mobility at Uniform Solvent Strength Increase

Let ζ and τ be dimensionless distance and time, Eq. (1), traveled by a solute band. A band migration in gradient LC can be described by differential equation [1]

$$ \frac{{{\text{d}}\tau}}{{{\text{d}}\zeta}} = 1 + k_{\text{init}} e^{{r_{\phi} (\zeta - \tau \,)}} $$

(32)

where k _init is the initial retention factor of the injected solute.

Suppose that the solvent strength is increasing along the column uniformly (without gradient) and, therefore, at any time, the strength is the same at any distance from the column inlet as it is at the inlet. As a result [1], the derivative in Eq. (32) at any ζ is the same as it is at ζ = 0, then Eq. (32) becomes

$$ \frac{{{\text{d}}\tau_{\text{U}}}}{{{\text{d}}\zeta}} = 1 + k_{\text{init}} e^{{- r_{\phi} \tau_{\text{U}}}} $$

(33)

where τ _U is τ at ζ = 0. Solving Eq. (33) for τ _U, one has:

$$ \tau_{\text{U}} {\kern 1pt} = \frac{{\ln ((1 + k_{\text{init}})e^{{r_{\phi} \zeta}} - k_{\text{init}})}}{{r_{\phi}}} $$

(34)

The solute mobility (μ) relates to the solute retention factor (k) as μ = 1/(1 + k) [1, 17, 19, 20]. In gradient LC [1],

$$ k = k_{\text{init}} \exp \left({r_{\phi} \cdot \left({\frac{z}{L} - \frac{t}{{t_{\text{M}}}}} \right)} \right) $$

(35)

In the (imaginary) case of a uniform (gradientless) solvent strength increase, k at any z is the same as it is at z = 0. Assuming in Eq. (35) that z = 0, assigning Eq. (35) to the solute band (t = t _b), and using dimensionless time, Eq. (1) yields for the retention factor (k _b,U) of a solute band at uniform solvent strength increase:

$$ k_{\text{b,U}} = k_{\text{init}} e^{{- r_{\phi} \tau_{\text{U}}}} $$

(36)

Due to Eq. (34), this yields for the eluting solutes (ζ = 1):

$$ k_{\text{R,U}} = \frac{{k_{\text{init}}}}{{(1 + k_{\text{init}})e^{{r_{\phi}}} - k_{\text{init}}}} $$

(37)

The solute elution mobility (μ _R,U) becomes:

$$ \mu_{\text{R,U}} = \frac{1}{{1 + k_{\text{R,U}}}} = 1 - \frac{{k_{\text{init}} e^{{- r_{\phi}}}}}{{1 + k_{\text{init}}}} = 1 - \omega_{\text{init}} e^{{- r_{\phi}}} $$

(38)

where [1, 17, 19, 20] $\omega_{\text{init}} = k_{\text{init}}/(1 + k_{\text{init}})$ is the solute fraction in the stationary phase at the column inlet.

Band Width Equation in Gradient LC

The width (σ _b) of a band located at z _b can be found from differential equation [16]:

$$ \frac{{{\text{d}}\sigma_{\text{b}}^{2} (z_{\text{b}})}}{{{\text{d}}z_{\text{b}}}} = {\fancyscript{H}} + \frac{{2\sigma_{\text{b}}^{2} (z_{\text{b}})}}{v(z,t)} \cdot \left. {\frac{\partial v(z,t)}{\partial z}} \right|_{\begin{subarray}{l} z = z_{\text{b}}, \\ t = t_{\text{b}} \end{subarray}} $$

(39)

where v is the solute velocity relating to the solvent velocity (u) as

$$ v = \frac{u}{1 + k} $$

(40)

There are some constraints [16] to variations of quantities ${\fancyscript{H}}$ and v in Eq. (39). However, there is no need to discuss them here because the constraints assumed in this report are stricter. Let u be a uniform quantity, i.e., ∂u/∂z = 0. Substitution of Eq. (40) in Eq. (39) and differentiation yields for uniform u:

$$ \frac{{d\sigma_{b}^{2} (z_{b})}}{{dz_{b}}} = {\fancyscript{H}} - \frac{{2 a k_{b} (z_{b})\,\sigma_{b}^{2} (z_{b})}}{{1 + k_{b} (z_{b})}} $$

(41)

where k _b is the solute retention factor at the band’s center of mass, and

$$ a = \frac{1}{{k_{\text{b}} (z_{\text{b}})}}\left. {\cdot \frac{\partial k}{\partial z}} \right|_{\begin{subarray}{l} z = z_{\text{b}}, \\ t = t_{\text{b}} \end{subarray}} $$

(42)

Equation (41) is known from Poppe et al. [4]. It has been derived for gradient LC with a uniform u and fixed ${\fancyscript{H}}$ (the LSS model of solute retention was not required). It should be noted that as u is not necessarily uniform in all separation techniques, Eq. (41) might be invalid outside gradient LC. A case in point is GC where, due to the gas decompression along a column, the gas velocity (u) might be a strong function of z changing from finite value at the column inlet to possibly infinity (in GC–MS) at the outlet. On the other hand, there are no specific restrictions on u in Eq. (39), and Eq. (39) is suitable for GC [16, 17, 27] and other separation techniques including LC as it has been shown here.

Substitution of Eq. (42) in Eq. (41) yields:

$$ \frac{{{\text{d}}\sigma_{\text{b}}^{2} (z_{\text{b}})}}{{{\text{d}}z_{\text{b}}}} = {\fancyscript{H}} - \frac{{2\sigma_{\text{b}}^{2} (z_{\text{b}})}}{{1 + k_{\text{b}} (z_{\text{b}})}}\left. {\frac{\partial k}{\partial z}} \right|_{\begin{subarray}{l} z = z_{\text{b}}, \\ t = t_{\text{b}} \end{subarray}},\,\,\left({\frac{\partial u}{\partial z} = 0} \right) $$

(43)

The derivation of this equation required a uniform solvent velocity (u) and did not require that retention factor (k) should be governed by LSS model. However, only the LSS gradient LC is considered in this report. In this case [1], u is a fixed quantity and k can be expressed as in Eq. (35). Substitution of Eq. (35) in Eq. (43) and differentiation yields:

$$ \frac{{d\sigma_{b}^{2} (z_{b})}}{{dz_{b}}} = {\fancyscript{H}} - \frac{{2r_{\phi} \omega_{b} (z_{b})\,\sigma_{b}^{2} (z_{b})}}{L},\quad ({\text{LSS}}) $$

(44)

where [1] ω _b(z _b) = k _b(z _b)/(1 + k _b(z _b)) is the solute immobility (its fraction in the stationary phases [1]) at location z _b when the band is there. The quantity ω _b in gradient LC can be found as [1]

$$ \omega_{\text{b}} (z_{\text{b}}) = \frac{{L\,\omega_{\text{init}}}}{{L + r_{\phi} {\kern 1pt} \omega_{\text{init}} {\kern 1pt} z_{\text{b}}}} $$

(45)

where ω _init is the initial solute immobility, i.e., ω _init = ω _b(0). Substitution of Eq. (45) in Eq. (44) yields Eq. (4) of the main text.