Separations: Colloidal Particle Column Packings

Article: B.Wei, D.S. Malkin, and M.J. Wirth, “Plate heights below 50 nm for protein electrochromatography using silica colloidal crystals,” Anal. Chem. 2010, 82, 10216-10221.

This article describes the use of colloidal particles just 330 nm in diameter as a packing material for electrochromatography. Students do not need to have been previously introduced to electrochromatography specifically, but they should have a background in reverse phase HPLC and capillary electrophoresis. The article provides an opportunity for students to apply their conceptual understanding of band broadening and the van Deemter equation to a novel experimental system. To answer several of the questions, students will need to have a detailed understanding of band broadening in separations, such as that provided by the ASDLIB Active Learning In-Class Activity on Separation Science written by Prof. Tom Wenzel. Also of interest: the small size of the particles means that when crystalline packing is achieved Bragg diffraction results in an artificial opal and the column appears blue. This can be seen in images on the website of the corresponding author, Mary Wirth.

Out-of-Class Questions

Q1. This abstract is a good example of how to present a large amount of quantitative data concisely in an abstract. Using the abstract and the body of the article as needed, report the following values for these experiments, and compare them to typical values for HPLC and CE from your text, class notes, or lab data. Don’t forget units where needed!

Parameter	This Work	Typical HPLC	Typical CE
dp	330 nm	1-5 µm	no packing
L	2 cm	15-25 cm	20-50 cm (to detector)
i.d.	75 µm	4.6 mm	50-100 µm
H	< 50 nm	~10 µm	0.5-5 µm
N	10^6	10^4	10^5-10^6

The length of the capillaries in this work is found in the Experimental section on p. 10217. Typical data for HPLC and CE should be provided previously during class lectures or laboratories, though some of the values are also provided in commonly used textbooks.

Q2. Using a form of the resolution equation, the authors remind the reader that there are two contributors to peak resolution.

1. Efficiency (peak sharpness/peak width) and selectivity are the two overall contributors to peak resolution. Many experimental variables can be changed to affect these two terms.
2. \[R_s=\dfrac{\sqrt{L/H}}{4} \dfrac{\Delta t}{<t>}\]

   The \(\sqrt{L/H}\) term is equal to \(\sqrt{N}\) and corresponds to the efficiency of the separation. The \(\Delta t\,/<t>\) term corresponds to selectivity.

Q3. Why do the authors choose to target plate height as a means to improve resolution? What other parameters could they have targeted?

The authors choose to target efficiency by improving the plate height because the experimental factors that increase plate height are determined primarily by materials science. Advances in materials science, such as formation of colloidal crystals from nanoparticles, can improve plate height. Improvements to efficiency only increase resolution by the square root, so the authors might have chosen to target selectivity instead. This requires differentially affecting the partition coefficient of each analyte between the mobile and stationary phases. This can be achieved by varying the temperature or by varying the chemical composition of either the mobile phase or the stationary phase.

Q4. Based on the final paragraph of the introduction, what was the objective of this work?

The authors purpose was to explore theoretical possibilities for this new packing, not to develop a practical method. The goal was “to characterize the contributions of [a silica colloidal crystal packing] to plate height for proteins.” (p. 10217)

Q5. Why do the authors use an electric field, rather than pressure, to drive these separations?

The pressure required to drive fluid flow through a column which such a packing would be immense. [This is the extent of the answer provided by my students; however, for those interested, a more detailed discussion is provided below.]

For a given flow rate, u, the pressure difference across the column required, ΔP, is given by

\[∆P= \dfrac{ϕηLu}{d_p^2}\]

where Φ is a flow resistance factor, η is the viscosity of the mobile phase, L is the length of the column, and d_p is the diameter of the packing particles (see, e.g., J.E. McNair, et al. Anal Chem. 1999, 71, 700). Because ΔP increases as the inverse square of the packing particle diameter, much higher pressures are required as diameter decreases. In contrast, electroosmosis is a surface-driven phenomenon that does not depend on particle diameter.

Q6. The authors use silanes to polymerize the packing material and to form what is likely to be a very thin layer of short carbon chains on the surface (i.e., d_f is extremely low). If the stationary phase is so thin, how do the authors know that chromatography, rather than just electrophoresis, is occurring?

On p. 10218, the authors note that “[t]he retention order (Lyz, RnaseA, CytC) is different from what we observe in a CE separation (Lyz, CytC, RnaseA) by the same as what we observe in a reversed phase separation.” Because these two separation methods resolve analytes based on different chemical characteristics, they result in different retention orders.

Q7. Estimate the value of H for the lysozyme peak in Figure 2. Show your work to receive credit. Does your estimate match the authors’?

To calculate the plate height, H, from experimental data, it is easiest to first calculate the plate number, N, using

\[N=16 \left(\dfrac{t_r}{w}\right)^2=5.54 \left(\dfrac{t_r}{w_{1/2}} \right)^2\]

where t_r is the retention time, w is the width, and w_1/2 is the full-width at half-maximum, which is often easier to estimate accurate than the width of the peak at its base. For lysozyme in the Figure 2 inset, t_r appears to be approximately 107 s and w_1/2 appear to be ~1 s:

\[N=5.54 \left(\dfrac{107\: s}{1\: s}\right)^2=\textrm{63,000}\]

H is related to N by N = L/H. The caption of Figure 2 states that the separation distance for the data shown is 0.91, so

\[H= \dfrac{L}{N}=\dfrac{0.91\: cm}{63000}=1.4 \times 10^{-5}\: cm=140\: nm\]

The authors state that the plate heights for the peaks in Figure 2 are all less than 100 nm. Considering that we do not have the raw data, this calculation is in fairly good agreement.

(Note: students may estimate the values of t_r and w or w_1/2 slightly differently, resulting in varying degrees of agreement between their value and the authors’.)

Q8. Why is the plate height lower for these separations than for previous separations of dyes using the same packing material?

The authors state that the lower plate heights occur “presumably because the injected widths and diffusion coefficients are lower” (p. 10218). Students can be reminded that although we most often consider how to minimize the contribution of the column to peak width, the injection has a finite width that may also appreciably increase plate height. Additionally, proteins have much lower diffusion coefficients than dyes due to their larger size. As a result, the longitudinal diffusion term, B, in the van Deemter equation is smaller, leading to lower plate heights.

Q9. Why does heterogeneous packing increase the plate height? What term(s) in the van Deemter equation is/are affected by packing inhomogeneity like that seen in Figure 3?

Heterogeneous packing can lead to channels in which the packing particles are not well-packed, resulting in larger interstitial spaces in some parts of the column. This means that analyte molecules that pass through the void take a much less tortuous path than molecules that go through a well-packed section of column, a process called channeling. This primarily contributes to band broadening by increasing the A term of the van Deemter equation. Uneven packing also means that some molecules will travel further through the column without encountering the stationary phase than others. This results in increased band broadening due to the mass transport term, C (specifically mass transport in the mobile phase, C_m).

Q10. In Figure 4C, why does peak variance increase linearly with time? In other words, what process causes this peak broadening?

Diffusion causes the peak variance to increase linearly with time. As seen in Figure 4C, the relationship between peak variance and time is given by σ² = 2Dt. The longer the plug spends on the column, the more diffusion contributes to band broadening through the B and C terms. [Note to instructors: In well-packed columns in this paper, the C term is negligible, but in typical HPLC columns this is not the case.]

In-Class Questions

Q1. When characterizing plate height for lysozyme in Figures 4 and 5, the authors determine the width of the peak in space rather than in time. They are able to do this because they are using a camera as a detector, but why do they need to do this? How does the width of the peak in space relate to the width of the peak in time? How does the detector contribute to plate height in these experiments?

To investigate band broadening in their columns, the authors needed to determine the variance, σ², of their peaks. The variance is found by fitting the peak to the equation for a Gaussian; however, the authors could not fit the peaks as a function of time because they were only a few points wide. This occurred because the camera needed to use a relatively long exposure time (0.2 s, given as the acquisition time on p. 10218) to capture the image of each peak because the channels are small and the analytes are dim. As a result, by the time the 0.2 s exposure time has passed, the peak is almost passed the detector. The authors can only capture a couple images of the peak as it passes.

However, each picture of the peak is many pixels wide on the camera sensor. This means the authors can fit the Gaussian peak in space very well, as seen in Figure 5. Because the width of the peaks in space is related to their width in time by the velocity of the peaks, the authors can then convert between these two data domains.

[Students can be prompted to sketch the peak as a function of time and as a function of space to help bring them to the conclusion described above.]

One caveat is that the authors must account for the fact that the detector is contributing some variance, \(σ_{detector}^2\). This occurs because the peak is moving as the image is acquired. As a result, the peaks look a bit wider than they really are. (Image using a camera with a slow shutter speed to take an image of a sprinter. The picture of the runner looks smeared out wider than it should because the runner is moving as the image is exposed.) The authors account for this using Equation 3 on p. 10219 which relates the variance to the exposure time, τ, and the velocity of the peak, v.

Q2. In discussing Figure 6, the authors assert that the A and C terms of the van Deemter equation are negligible for their separations. In Figure 7 and the latter part of the Results & Discussion, the authors address whether the extremely low plate heights observed could be due to focusing rather than to the achievement of a diffusion-limited separation. Why would the A and C terms be negligible under the conditions used in this work? What evidence supports the authors’ assertion that the efficiency of their separations is limited only by diffusion?

The A term, Eddy diffusion, is negligible because the particle diameter is so small. Recall that A decreases with d_p. The C term is negligible because C_s is minimized by the very thin stationary phase since it is inversely related to d_f and because the C_m term is minimized by the small packing particles which result in very small interstitial spaces for analyte to diffuse across before encountering the stationary phase.

The authors present quite a bit of evidence that their peak widths are diffusion-limited. One of the easiest pieces of evidence for students to appreciate is the shape of the curve in Figure 6. This is a van Deemter curve, but rather than having the characteristic “Nike Swoosh” shape of a curve that includes a C term, it has the shape of the B/u term, which depends only on longitudinal diffusion. The authors also check to make sure that the diffusion coefficient obtained from the data in Figure 6 matches the diffusion coefficient in the absence of an electric field (Figure 7). This rules out the possibility of focusing that could occur if aqueous solution entered during the injection, increasing the retention of the front of the sample plug (since this is more polar than the mobile phase, which contained some acetonitrile). Finally, the authors calculated the expected diffusion-limited peak width based on the diffusion coefficient of lysozyme and the velocity of the peaks (Equation 5) and found good agreement with their experimental data.*

*This calculation is fairly hard to follow. I do not usually go through it in detail in class. We simply discuss how it is related to the authors’ assertion that their separation is diffusion-limited.

Q3. The authors specifically state that their goal for this work was not to achieve a practical method for protein separations. That would have been outside the scope of this paper because many practical considerations would need to be addressed before this type of packing could be made available in commercial columns. Imagine that an instrument manufacturer wants to use columns like these in a commercial HPLC instrument. What changes to the instrument and practical improvements in the column would be needed?

The most important improvement to the column would be to eliminate the channeling issue by improving the packing process. On p. 10218, the authors state that about two-thirds of the columns they packed had gaps in packing at the walls. This would be unacceptable for commercial preparation of columns.

A traditional HPLC instrument would need to be significantly modified to accept these columns. The pressure system would need to be replaced with a high voltage power supply to produce electroosmotic flow. The injection system would also need to be modified for electrically-driven injections. The detector would need to be replaced with a highly sensitive camera, and the software would need to be re-written to accept the output of this detector and convert it into a chromatogram. Ultimately it would probably make more sense to have a dedicated instrument for electrochromatography rather than to retrofit these columns into an existing HPLC design.