A comparison of 2 micron inner diameter open tubular column liquid chromatography with pressure-driven isocratic, slip-flow, and electrochromatographic modes of operation: a theoretical study

Abstract

Modifications to the flow profile used in open tube capillary liquid chromatography (OT-CLC) include using slip-flow walls and using electroosmosis as a fluid pump as practiced in electrochromatography. These modifications are implemented experimentally by changing the capillary surface and solvent conditions which results in the change of boundary conditions at the capillary wall.

In this paper we employ a theory-based study and compare the zone broadening of simple solutes using parabolic flow from a liquid pump, slip-flow from a highly hydrophobic inner surface with water eluent, and electroosmosis for the conditions of pure water and dilute salt utilizing 2 μm inner diameter OT capillaries. In general, the two types of flow other than parabolic exhibit thin zones in the early part of the chromatogram, consistent with previous studies of slip-flow and electroosmotic flow used in electrochromatography. Electrochromatography is shown to yield higher efficiency and less zone broadening than parabolic and slip-flow conditions used in this study. Nonetheless, it is found that the zone standard deviations are shown to be similar for these flow profiles as is the number of plates for these different flow profiles under the conditions utilized here. It is revealed that these modifications do not warrant the effort to maintain the special solvent conditions when compared to gradient elution OT-CLC, which gives a nearly constant peak width throughout the chromatogram, is easiest to implement, and is the method of choice for complex analysis.

1. Introduction

Capillary liquid chromatography (CLC) continues as an active area of study [1-3]. One configuration that offers very high efficiencies is open tube CLC (OT-CLC), where a capillary is coated with a thin layer of retentive material. In some cases the surface is roughened chemically by etching [4], and in other cases a thin layer of particles is attached to the surface of the capillary [5]. A number of reviews of OT columns have been issued recently [6, 7]. OT-CLC columns continue to be investigated as the quest for faster and higher-resolution columns continues. The highest resolution is needed for complex analytical tasks, especially where biomolecules are to be analyzed, such as in the digestion of proteins to characterize peptides..

A number of modifications to the basic LC technique have been published. These include using an electric field to drive the flow which is often referred to electrochromatography and has a long history. Electrochromatography employs a fluid-driving force of electroosmosis whereby the fluid boundary undergoes electrophoresis and drives the fluid with an almost flat, piston-like velocity profile. A progress report in electrochromatography [8] and a review of advancements in electroosmotic pumps [9] were recently published.

In addition to electroosmosis, slip-flow boundaries have been reported in LC experimental work [10-12]. The slip-flow boundary, which was originally postulated in the development of viscous fluid motion theory, also known as the Navier-Stokes equations [13], allows for the fluid to be mobile at the fluid-solid interface. Most applications of the Navier-Stokes theory use a “stick” boundary at this interface where the fluid is assumed to stick to the wall at the fluid-solid interface. This assumption may not always be valid as wall slippage has been observed in cases where very hydrophobic wall materials are used with water as a solvent [14]. We describe the essential mathematical development of this slip interface for use in zone dispersion calculations below. A theoretical and simulation study of slip-flow in open tube capillaries was recently published [15].

Many years ago, a number of theoretical studies [4, 16-18] examined OT-CLC, and it was realized that very small capillary inner diameters (i.d.s) were advantageous and necessary for speed and resolution. It was known during this time that very small amounts of solute could overload these columns and that detection would be problematic, if not impossible, with no porous particles to give surface area and loadability. Due to the detection problem, packed capillaries dominated the literature and OT-CLC focused on fluorescence detection and eventually mass spectrometry.

In one study, the optimal i.d. was determined to be 1 – 3 μm [4]. There are many advantages to this type of column, such as very high efficiencies, along with drawbacks: the column has very low sample loadability, detection is problematic due to this low loadability, and the detector volume must be exceedingly small to take advantage of the high column efficiency. The efficiency of these columns is potentially very large because there is no eddy (i.e. mechanical) dispersion which is found in packed beds. The loadability can be aided using fluorescence detection by derivatizing samples with fluorescence tag molecules. These columns fell out of favor and packed capillaries appeared which provided advantages over conventional columns and were commercially available. The packed capillaries were also effective for sample-limited injections.

Recently, a number of studies have taken place with 2 μm i.d. capillaries explored by Liu and coworkers [19-22]. These were shown to be useful for complex peptide digestions where sample complexity requires extremely high efficiency and speed. The use of a reduced capillary i.d. of 2 μm favors the reduction of the mobile phase mass transport resistance. As expected, this leads to a great detection challenge because the injected amounts must be exceedingly small as to not overload the very low surface area of the stationary phase, which is chemically bonded to the capillary surface.

It is enticing to consider augmenting the small i.d. capillary with either slip-flow boundaries or electroosmosis as practiced in electrochromatography. There is still great interest in electrchromatography as a method for increasing zone efficiency and resolution. Besides the above-mentioned review of electrochromatography [8], other recent studies using adsorbed micelles [23] and metal-organic frameworks [24] on the capillary wall are still being developed. Hence, it is of interest to compare the different boundaries and operating procedures at reduced capillary i.d. using a theoretical approach to determine if insight can be gained for these systems.

In this paper we examine open tubular column conditions used by Liu and coworkers, mentioned above, and compare the boundary conditions of normal pressure driven flow, slip-flow, and electroosmosis. Conditions are chosen similar to those used by Liu and coworkers: a capillary diameter of 2 μm, a length of 0.5 m, and an average velocity of 1 and 3 mm/s which results in a flow rate of ≈ 0.2 and ≈ 0.6 nL per minute. The plate height H is calculated for the different boundary conditions and used to determine the resulting chromatograms at different levels of retention. In addition, the mass transport resistance in the fluid phase, the number of plates, the zone width expressed in terms of the Gaussian standard deviation σ, and the ratio of mobile phase mass transport resistance to the axial diffusion plate height are given in detail. These results lead to the conclusion that while electroosmotic flow with a minimal salt concentration results in higher plate counts, the advantage is mostly lost because pressure-driven OT-CLC with gradient elution pressure-driven flow offers a more uniform zone width and quite possibly, according to experimental data [19], a narrower zone width. The gain in efficiency by augmenting the boundary with slip-flow is minimal for isocratic OT-CLC.

2. Theory

2.1. Slip-flow

This section uses nomenclature from a recent publication on slip-flow in capillaries [15]. On the macroscopic level, it is normally assumed that fluid located at the walls has zero velocity. With this no-slip boundary condition employed, the theoretical Hagen-Poiseuille velocity profile is parabolic [25, 26] and is described by

$$v(r)=-\frac{r_c^2}{4\eta }\left[1-{\left(\frac{r}{r_c}\right)}^2\right]\frac{\partial P}{\partial z}$$ (1)

where v(r) is the fluid velocity at a point located radially at r in the direction of flow (z), r_c is the radius of the capillary, η is the dynamic viscosity of the fluid, and P is pressure. The pressure drop, ∂P/∂z, is negative making the velocity positive. In terms of average velocity, this velocity profile can be written as

$$\frac{v(r)}{\overline{v}}=2\left[1-{\left(\frac{r}{r_c}\right)}^2\right]$$ (2)

where $\overline{v}$ is the mean fluid velocity in the z direction. It is assumed that there is no secondary flow. Secondary flow is where there are non-zero velocity components in the x and y coordinates of the cylindrical tube. This occurs in low Reynolds number flow in non-straight (i.e. curved) tubes [27, 28] and in certain cases of rotating open-channel ducts when flow takes place with simultaneous rotation, such as in sedimentation field-flow fractionation [29].

The no-slip assumption has proven to be accurate for flows on the macroscopic level, but it is only an assumption that is not rooted in principle [14]. For the no-slip boundary condition to apply, attractive forces between the wall and the fluid molecules must be equal to or greater than the attractive forces between mobile phase fluid molecules [12]. This is certainly not the case when the walls of the capillary are hydrophobic and the fluid is hydrophilic. In this case a “slip” boundary with a nonzero velocity at the wall may exist. A slip boundary can be captured using Navier’s boundary condition,

$$L_s{\phantom{\mid }\frac{\partial v(r)}{\partial r}\mid }_{r=r_c}=-v(r){\mid }_{r=r_c}$$ (3)

where L_s is defined as the slip length. When replacing the no-slip boundary condition with this new condition in the Navier-Stokes equation, the velocity profile for flow in the capillary with slip becomes

$$v(r)=-\frac{r_c^2}{4\eta }\left[1-{\left(\frac{r}{r_c}\right)}^2+\frac{2L_s}{r_c}\right]\frac{\partial P}{\partial z}$$ (4)

[30]. In terms of average velocity, the velocity profile with slip is now expressed as:

$$\frac{v(r)}{\overline{v}}=\frac{2}{1+2\gamma }\left[1-{\left(\frac{r}{r_c}\right)}^2+\gamma \right]$$ (5)

where γ = 2L_s/r_c [31] and γ is referred to as the nondimensional slip length.

Slip lengths for various surface/liquid boundaries reported in the literature are typically on the order of tens of nanometers [32]. For this reason, the phenomenon of slip becomes significant on the nanoscale using devices such as carbon nanotubes with radii on the order of nanometers through tens of nanometers [30, 33, 34] and with sub-micron silica particles [10, 12]. It also follows that slip would be negligible for macroscopic flows resulting in the no-slip boundary condition being a valid assumption.

The contact angle between surface and fluid directly impacts slip length as contact angle is a measurement of attraction between different materials. Thus, larger slip lengths will require very weak interactions between surface and fluid such as a highly hydrophobic material and water.

At the wall, the fluid velocity is

$$v(r){\mid }_{r=r_c}=2\gamma {\overline{v}}_{no-slip}$$ (6)

We consider two values of γ for this study based on studies summarized in Lauga and Stone [35] where slip lengths ranged from the nanometer scale to the micron scale and system size ranged from the nanometer to the millimeter scale. Despite this wide range, values of L_s/r_c differed by a factor of 40 from 0.005 to 0.2, corresponding to 0.01 ≤ γ ≤ 0.4 [35].

Fig. 1A summarizes the above concepts by illustrating how increasing the slip length L_s (and increasing γ) leads to an increase in v(r)∣_r=rc. This is discussed in more detail in section 3 below.

Figure 1:

A. Velocity profiles for varying slip lengths referenced to the average velocity. The γ = 0 curve is for pressure-driven flow with no-slip boundaries. B. Velocity profiles for eletroosmotic flow. The κr_c = 1 curve represents conditions for water and the κr_c = 520 curve represents conditions for 0.025 M KCl.

2.2. Electroosmotic flow

Electroosmotic flow takes place when the boundaries of a container possess mobile charged species, typically protons, that can be driven by electrophoresis. These charged species move the fluid at the tube boundary and result in a piston-like flow. This type of flow is one of the reasons that capillary electrophoresis (CE) can achieve very narrow zones because the solute(s) travel at relatively the same rate, unbroadened by lateral (i.e. radial) diffusion processes [36]. This assumes in CE that the solutes do not adsorb to the wall, which destroys the high resolution that CE is capable of [37].

The theory of electroosmotic flow in capillaries has been described previously [38, 39]. We restrict our treatment to the Rice and Whitehead treatment as it is simple and is quite easily calculated. As we will show and is well-known, at relatively low salt concentration, the flow velocity profile is well approximated as flat. The starting point here is the electroosmotic velocity profile, v_eo(r), given in [38]:

$$v_{eo}(r)=-\Omega E_z\left(1-\frac{I_0(\kappa r)}{I_0(\kappa r_c)}\right)$$ (7)

where I₀ is the zero-order modified Bessel function of the first kind [40], κ is the inverse double layer thickness, also known as the Debye length [41], Ω is a constant which contains the dielectric constant, permitivity, and other constants, and E_z is the electric field strength, also in the z direction. The minus sign is used to give a positive velocity as the electric field strength is related to the potential difference E_z = −dV/dz where dV/dz is the gradient of the electric potential in the z direction.

Eq. (7) can be normalized by the average electroosmotic velocity, ${\overline{v}}_{eo}$, [42] such that:

$$\frac{v_{eo}(r)}{{\overline{v}}_{eo}}=\left(\frac{1-\frac{I_0(\kappa r)}{I_0(\kappa r_c)}}{1-\frac{I_1(\kappa r_c)}{\kappa r_c{I}_0(\kappa r_c)}}\right)$$ (8)

where I₁ is the first-order modified Bessel function of the first kind [40]. The flow profiles for a range of κr_c values are shown in Fig. 1B. Two κr_c values are important: κr_c = 1, which is found for pure water and κr_c = 520 which is found for 0.025 M univalent salt such as KCl [41]. This figure is discussed in more detail in section 3.

2.3. Plate height theory

The theoretical plate height for open tube systems with parabolic flow, slip-flow boundaries, or electroosmotic flow is determined using the expression [25, 43]

$$H=\frac{2D_m}{\overline{v}}+C_m\frac{r_c^2\overline{v}}{D_m}+C_s\overline{v}$$ (9)

where C_m and C_s are the mobile and stationary phase mass transport coefficients, respectively. D_m is the solute diffusion coefficient in the mobile phase. For all of the cases in this paper we use a diffusion coefficient D_m = 1.00 × 10⁻⁹ m²/s, typical of a small molecule [44]. In the case of open tube systems with no particle separation media, there is no stationary phase mass transport resistance and thus for the case we will study here it is assumed that C_s = 0. The mathematical development of C_s has been discussed in many papers [16, 45, 46] and C_s = 0 is a limiting condition for a stationary phase bonded to the capillary wall. Referring to the C_s term, Knox and Gilbert [16], have suggested that “Thus, to a first approximation the last term in … can be assumed to be negligible in LC.” For finite-thickness particle layers coating a capillary surface known as a porous layer open-tubular (PLOT) column, a finite C_s is apparent and quantitatively lowers the performance for small to medium-sized molecule solutes [46]. This paper also emphasizes the very good agreement between theory and experiment with few assumptions. PLOT column efficiency is further discussed in the Discussion section.

For no-slip boundary conditions used in this ideal capillary column, it is established that [25]:

$$H=\frac{2D_m}{\overline{v}}+\frac{1}{24}(6R^2-16R+11)\frac{r_c^2\overline{v}}{D_m}=H_{\text{axial diffusion}}+H_{\text{mobile phase}}$$ (10)

where R is the retention ratio [47] and is equal to t₀/t_R where t₀ is the void time and t_R is the retention time. The retention factor k′ and R are related as k′ = (1/R) – 1. As stated by Giddings “… nearly every mathematical expression containing R is more complex when written in terms of k′ ” [47]. The retention factor k′ indicates the number of eluted column volumes minus 1; hence the void peak has k′ = 0. For the limiting case of no wall retention, R = 1, and Eq. (10) reduces to the well-known relationship:

$$H=\frac{2D_m}{\overline{v}}+\frac{r_c^2\overline{v}}{24D_m}$$ (11)

The mass transport term in the mobile phase, C_m, is determined via the method laid out by Martin and Guiochon [43] using the generalized dispersion theory of Aris [48] whereby the radial flow profile v(r) is specified and C_m determined from this theory such that:

$$C_m=I_1^{\star }-2R{I}_2^{\star }+\frac{R^2}{4}$$ (12)

where $I_1^{\star }$ and $I_2^{\star }$ are defined as,

$$I_1^{\star }={\int }_0^1\frac{A^2(\rho )}{\rho }\mathrm{d}\rho$$ (13)

$$I_2^{\star }={\int }_0^1\rho A(\rho )\mathrm{d}\rho$$ (14)

$$\rho =r∕r_c$$ (15)

and

$$A(\rho )={\int }_0^{\rho }2{\rho }^{\prime }\frac{v(\rho )}{\overline{v}}\mathrm{d}{\rho }^{\prime }$$ (16)

For slip-flow, beginning with Eq. (5) in terms of ρ [31],

$$\frac{v(\rho )}{\overline{v}}=\frac{2}{1+2\gamma }(1-{\rho }^2+\gamma )$$ (17)

which yields

$$A(\rho )={\int }_0^{\rho }2{\rho }^{\prime }\frac{2}{1+2\gamma }(1-{\rho }^{\prime 2}+\gamma )\mathrm{d}{\rho }^{\prime }=\frac{{\rho }^2}{1+2\gamma }[2(1+\gamma )-{\rho }^2]$$ (18)

and performing the integrations in Eqs. (13) and (14) results in

$$I_1^{\star }=\frac{24{\gamma }^2+32\gamma +11}{24(1+2\gamma {)}^2}$$ (19)

$$I_2^{\star }=\frac{2+3\gamma }{6(1+2\gamma )}$$ (20)

This leads to an expression for C_m with slip flow boundaries by substituting $I_1^{\star }$ and $I_2^{\star }$ into Eq. (12):

$$C_m=\frac{24{\gamma }^2+32\gamma +11}{24(1+2\gamma {)}^2}-2R\left[\frac{2+3\gamma }{6(1+2\gamma )}\right]+\frac{R^2}{4}$$ (21)

Finding a common denominator and grouping terms with like orders of γ,

$$C_m=\frac{24(R^2-2R+1){\gamma }^2+8(3R^2-7R+4)\gamma +(6R^2-16R+11)}{24(1+2\gamma {)}^2}$$ (22)

Further details of analytically deriving C_m for slip-flow conditions is given in [15].

For electroosmotic flow, it is convenient to substitute Eq. (8) into Eq. (16) and then calculate the integrals numerically for the determination of C_m using Simpson’s rule [40]. C_m for a flat flow profile was derived previously [36]:

$$C_m=\frac{(1-R{)}^2}{4}$$ (23)

Once H is determined from Eq. (9), given C_m, the number of plates N_j of the j^th peak was determined by N_j = L/H_j where L is the column length. The Gaussian standard deviation of the j^th peak, σ_j, a measure of peak width, is related by $N_j=t_{R_j}^2∕{\sigma }_j^2$, where t_Rj is the retention time of the j^th peak, so that

$${\sigma }_j=t_{R_j}\sqrt{\frac{H_j}{L}}=\frac{t_{R_j}}{\sqrt{N_j}}$$ (24)

2.4. Synthetic chromatograms

The construction of synthetic chromatograms is well-documented in a recent publication [49]. By summing Gaussian single-component peaks (SCPs) the signal of the j^th SCP, g_j, is composed of discrete data points having index i at equally spaced times t_i leading to

$$g_j(h_j,t_i,t_{R_j},{\sigma }_j)=h_j\text{exp}\left[-\frac{(t_i-t_{R_j}{)}^2}{2{\sigma }_j^2}\right]$$ (25)

where h_j and σ_j are the j^th SCP amplitude and SCP standard deviation. The maximum signal of g_j is the amplitude h_j and occurs at retention time t_Rj. The total chromatogram, H(t_i), is the sum of M SCP’s:

$$H(t_i)=\sum _{j=1}^M{g}_j(h_j,t_i,t_{R_j},{\sigma }_j)$$ (26)

The peak amplitudes h_j are all the same in this study and arbitrarily set to unity value.

2.5. Software

The software used in this study was programmed in MATLAB (Mathworks, Natick, Massachusetts).

3. Results

The various flow profiles are shown in Fig. 1. The slip-flow velocity profiles given in Fig. 1A show that the flow velocity is finite at the wall with velocity specified in Eq. (6) noting that this equation uses the no-slip (i.e. stick) average velocity. As the nondimensional slip length γ is increased, the velocity at the wall increases. Intuitively, this should reduce the variation in the flow profile and result in narrower zones. The electroosmotic flow profiles are shown in Fig. 1B. As shown for the κr_c = 520 curve, which represents 0.025 M of a univalent salt like KCl, the flow profile is nearly flat and should offer little variation in solute velocity due to radial diffusion until the solute is delayed by wall retention. Without salt, as is the case for the κr_c = 1 curve, the velocity profile is quite similar to pressure-driven flow.

The plate height H as a function of average velocity $\overline{v}$ for three different values of retention is shown in Fig. 2. As retention increases, the higher κr_c curves show significantly lower plate heights compared to the other flow profiles as the fluid velocity is increased. This is expected as the variation in radial velocity is minimized for this case. The differences between the parabolic flow and electroosmotic flow plate height results at low salt concentration are less at zero retention (i.e. k′ = 0).

Figure 2:

H as a function of $\overline{v}$ for k′ = 0, 2, and 4 for the 6 different flow profiles. Conditions are: r_c = 1 μm, D_m = 1.00 × 10⁻⁹ m²/s.

The plate height minima present at k′ of 2 and 4 appear at different average velocities. In all cases the electroosmotic flow results at low salt concentration have a minimum plate height at larger velocity than the other flow profiles. This can be explained by solving for the minimum in Eq. (9) with respect to $\overline{v}$ which gives:

$${\overline{v}}_{\text{min}}(k^{\prime })=\frac{D_m}{r_c}\sqrt{\frac{2}{C_m(k^{\prime })}}$$ (27)

noting that C_m and ${\overline{v}}_{\text{min}}$ are functions of k′. At constant k′, D_m, and r_c, the lower the mass transport resistance C_m, the larger the velocity will be at the plate height minimum. The slip-flow result shows little difference from the pressure-driven parabolic flow profile.

Examining the resistance to mobile phase transport through the coefficient C_m, shown in Fig. 3, electroosmotic flow with small salt concentration, as embodied in the κr_c = 520 and the flat-profile curves, appears to offers a substantial speed advantage here. This is due to the low C_m which, as viewed by Eq. (27), provides a higher ${\overline{v}}_{\text{min}}$ allowing faster operation throughout the k′ range of interest. Electroosmotic flow at low salt concentration appears to give approximately half the mobile phase resistance to mass transport as does pressure-driven flow. Slip-flow boundaries are shown to reduce this resistance compared to normal stick boundary flow, but these are far less effective than electroosmotic flow at low salt concentration.

Figure 3:

The mass transport coefficient, C_m, in the mobile phase for each flow profile as a function of k′. C_m is only a function of the flow profile and k′. The electroosmotic flow flat profile and the κr_c = 520 result are superimposed.

From the results shown in Fig. 2, we choose to study velocities of 1 and 3 mm/s and this is consistent with the experimental results of Liu and coworkers [19-22]. As mentioned previously, these average velocities correspond to a flow rate of ≈ 0.2 and ≈ 0.6 nL per minute, which is lower than the definition of “nano-LC” given in 1996 [50] and may not be experimentally accessible by most commercially-available pumps used for capillary LC. Nonetheless, flow-splitting can reliably achieve these flow rates.

Figs. 4 and 5 show synthetic chromatograms with the different flow profiles at these two velocities. All flow profile results have small peak widths at the front of the chromatograms and these peak widths increase rather dramatically as retention increases. These chromatograms show that for electrochromatography with κr_c = 520 and the flat flow profile cases, which are nearly identical, zone widths are smaller than the pressure-driven flow case.

Figure 4:

The chromatograms showing zone widths of each flow profile as a function of k′. $\overline{v}=1.00$ mm/s, L = 0.5 m, with other conditions listed in the caption to Fig. 2.

Figure 5:

The chromatograms showing zone widths of each flow profile as a function of k′. $\overline{v}=3.00$ mm/s, L = 0.5 m, with other conditions listed in the caption to Fig. 2.

The higher velocity result, depicted in Fig. 5 shows narrower zone widths for most zones. This is not surprising when looking at the results in Fig. 2 where for retained zones the lower velocity, 1 mm/s, is to the left of the plate height minimum and the higher velocity, 3 mm/s, is to the right of the plate height minimum. The zone width, embodied in the Gaussian standard deviation, σ, appears as a square root function of the plate height, as show in Eq. (24), so the differences are not that profound between velocity results. The higher velocity allows 66% reduction in analysis time with similar zone widths for pressure-driven flow with no boundary modifications. For the electroosmotic flow case with small salt concentration, the higher velocity results in narrower zones than the same flow profile at lower velocity.

The consequences of velocity choice are illustrated clearly when plotting the ratio H_{mobile phase} to H_{axial diffusion} as shown in Fig. 6. For $\overline{v}=1$ mm/s the scale of this ratio shows that the axial diffusion contribution to the plate height is larger than the resistance to mass transport in the mobile phase, i.e. H_{axial diffusion} > H_{mobile phase}. However, with $\overline{v}=3$ mm/s, this ratio is larger and indicates that the mobile phase mass transport resistance is now comparable with the axial diffusion term. At this velocity and k′ > 1, H_{axial diffusion} ≈ H_{mobile phase}.

Figure 6:

The ratio H_{mobile phase} to H_{axial diffusion} as a function of k′. Top: $\overline{v}=1.00$ mm/s and bottom: $\overline{v}=3.00$ mm/s with other conditions listed in the caption to Fig. 2. The electroosmotic flow flat profile and the κr_c = 520 result are superimposed.

The zone width expressed as the Gaussian standard deviation, σ, is shown in Fig. 7 for the two velocities chosen for study. This figure displays small difference in the zone widths for the various flow profile modifications. The almost linear behavior of these curves suggests that these flow profile modifications offer smaller zone widths, but as is typical with isocratic systems, the best performance is at lower retention values. These calculations are consistent with experimental results utilizing 2 μm capillaries [19] where the full-width at half maxima (fwhm) for the 10 largest peaks of an amino acid separation is ≈ 2.3 s. Assuming Gaussian peaks, this would result in σ ≈ 1 s noting that $\sigma =\text{fwhm}∕\sqrt{(8\text{ln}2)}$. These experimental results were obtained using gradient elution. From the viewpoint of zone width, as shown in Fig. 7, there is very little advantage in implementing slip-flow or electroosmotic flow to lower zone width with the conditions used here. As was suggested in Figs. 4 and 5, the zone width differences between the different flow profile modifications are not that large.

Figure 7:

The standard deviation of each peak for the various flow profiles, as a function of peak retention, in units of k′. Top: $\overline{v}=1.00$ mm/s and bottom: $\overline{v}=3.00$ mm/s, L = 0.5 m, with other conditions listed in the caption to Fig. 2. The electroosmotic flow flat profile and the κr_c = 520 result are superimposed.

These relatively small changes between flow profile results are reflected in Fig. 8, where the number of plates is plotted as a function of k′ at the two chosen average velocities. Clearly, the most plates are obtained with the low salt electroosmotic flow for both average velocities. But the differences here are minuscule with ≈ 1 – 2% between electroosmotic flow with minimal salt and pressure-driven flow at zero retention for the $\overline{v}=1$ mm/s case. This difference increases with retention at the smaller average velocity where there is ≈ 10% difference in plates at k′ ≈ 4. The additional difficulties in using electroosmotic flow are hard to justify for a gain this small. Some aspects of the compatibility of electroosmosis with gradient elution are discussed in the Discussion section.

Figure 8:

The number of plates for each flow profile as a function of k′. Top: $\overline{v}=1.00$ mm/s and bottom: $\overline{v}=3.00$ mm/s, L = 0.5 m, with other conditions listed in the caption to Fig. 2.The electroosmotic flow flat profile and the κr_c = 520 result are superimposed.

Slip-flow boundaries, while showing more plates than parabolic pressure-driven flow, are hard to justify in terms of the difficulty of implementation and small reward. For slip-flow, this is especially a difficult situation because gradient elution is fundamentally incompatible with the physics of maintaining slip-boundaries. Although slip-boundaries have been reported in the context of gradient elution LC with very small particles [11], the addition of an organic modifier would be expected to destroy the slip layer as organic solvent is introduced. As has been suggested [51] the presence of dissolved gas, nucleated gas, or microbubbles [52] may explain these results.

The differences in plate count increase between pressure-driven flow and the electroosmotic flow case with low salt concentration at the higher velocity shown in Fig. 8. For this case the differences are ≈ 20% at k′ = 0 and ≈ 33% at k′ ≈ 4. Note that the predicted plate counts are impressive for the faster average velocity with nearly a million plates per meter for the electroosmotic flow case at small salt concentration and k′ ≤ 4. Even pressure-driven parabolic flow for this isocratic case will yield over 600,000 plates per meter for k′ ≤ 4. As will be discussed below, gradient elution conditions may allow thinner and relatively constant-width zones and provide more plates here suggesting that if one can utilize this small diameter capillary column, highly efficient separations could be achieved with standard pressure-driven parabolic flow with retentive boundaries.

4. Discussion

For the condition of very small capillary i.d. and low flow velocity, it is suggested through simple theoretical relationships that pressure-driven isocratic liquid chromatography differs little from slip-flow and electroosmotic flow performance. Wall modifications (as in the case of slip-flow) and driving force modifications (in the case of electroomosis) appear to be very difficult if not impossible to implement with gradient elution conditions. Electrochromatography would be most difficult to control with gradient elution because the zone velocity would vary as a function of the wall charge density which would change with the composition of the solvent. Method development here would be very complicated as the parameters of the gradient composition, the variation in flow at different parts of the gradient, and, if the sample is charged, the electrophoretic mobilty of the solute zones would be changing dynamically. These added complexities make the method development and analysis exceedingly challenging.

In addition, gradient elution LC operation is well-known to keep zone width relatively constant across the chromatogram. The results demonstrated here show zones getting wider with time. Gradient elution LC has a well-known compressive effect on zone width [53-56] that produces narrow zones. This effect is due to the solvent composition being stronger in the rear of the zone than the zone front, and this causes a thermodynamic compression effect reducing the overall zone width. This is not a focusing effect, such as that found in isopycnic and isoelectric focusing separations [57], but can substantially narrow zones making gradient elution techniques indespensible in providing high resolution of complex samples.

There are other aspects to this comparison with respect to column length. Electrochromatography must use higher voltages to maintain the same field strength as the column length increases. This may result in using exotic, higher voltage power supplies although the current is typically low for these applications. In contrast, open tube columns with pressure-driven parabolic flow have relatively small pressure drops and increasing column length to increase resolution is straightforward with pressure. Modern pumps are able to deliver higher pressure at low flow rates and while increasing analysis time, longer columns can easily be made and utilized as needed. Using 2 μm capillaries, the pressure drop was noted to be less than 50 bar for a capillary less than a meter [19]. Capillaries multiple times longer than this are easily driven by reasonable pressures.

The use of electroosmotic flow [43] and slip-flow [15] in capillaries have noted a similarity in that low retention zones have potential advantages in reducing the mobile phase resistance to mass transport. For example [43]: “…it appears that the electroosmotic flow will prove advantageous compared to the parabolic forced flow principally at k′ values lower than 2.” For ultra-narrow capillaries with 400 nm capillary i.d.s utilizing slip-flow boundaries [15]: “… the main reduction in plate height by utilizing slip-flow takes place in the region where retention is in the range 0 ≤ k′ ≤ 1.” These results suggest that isocratic techniques may be advantageous in extremely fast LC analysis but only within a limited retention range.

One of the tradeoffs to using a PLOT column, as mentioned previously, is that C_s is no longer zero. As the layer thickness increases, the plate count decreases from the OT result, as expected from theory [46]. Thicker layers increase detectability, as one can load more solute onto the larger surface area per unit length. This has been known for years [58, 59]. Nonetheless, OT capillaries with a bonded stationary phase provide a high level of performance once instrumental problems like detectability are overcome.

Finally, not all aspects of OT-CLC with very small i.d. are ideal. Again, the loadability of sample and the injectability and detectability are problematic and may require extremely specialized equipment and/or techniques. The small i.d. can also lead to clogging from sample residue and particulate contamination. This is a problem for solvents and samples as any impurities can render a capillary with small i.d. useless if clogged by a particle. This gets extremely problematic for biological samples where sample preparation can become critical. Nonetheless, with better injectors, detectors, and advanced column hardware, this style of column can be the answer for samples of biological origin and high complexity. Although currently exotic, the use of 2 μm capillaries can be developed into a viable method which is part of the separation scientist’s toolkit for small to medium size molecule samples like peptides, especially when mass spectrometry is utilized as a sensitive detector.

Highlights.

• Theoretical investigations show 2 um open tube capillaries produce high performance.

• The performance is compared with normal, slip-flow boundaries and electrochromatography.

• These techniques are suggested to not be worth the effort, compared to gradient elution operation.

• The zone standard deviations are shown to be similar for these different flow profiles.

• The number of plates for the different flow profiles are also shown to be similar.