n coefficient of molecule
r factor to consider is the diffusion of the molecule through the mobile phase. ase the larger the molecule, in general, the greater will be the resistance to ent and so the diffusion coefficient will be lower.
actors
persion within the chromatographic system does not just come from the attributable to the volume of the tubing, but can also arise from the injection , the void volume of the column, the volume associated with an ill-fitting tor as well as the detector volume and can be summarised as;
75 µm + Vcol + Vtubing + Vconnectors + Vdetector cell
umes that each of these components contributes to the dispersion process he chromatographic system can also be expressed in terms of a system e with the individual variances having the same form as the above equation.
σ2 inj + σ2 col + σ2 tubing + σ2 connectors + σ2 detector cell
ariance associated with the column is the major component of analyte n time then changing the particle size to reduce this will have a beneficial however if wider bore tubing is being used, or a large injection volume then ance or dispersion associated with these components will dwarf the ution from the column and thus changes in the particle size will not be ble. The impact of using inappropriate connector s and tubing can be seen in 1.
170 µm Flow Rate [ml/min]
Figure 1. Effect of column tubing diameter and length on the dispersion in a chromatographic system. Reproduced with kind permission from Chromatography Today May/June 2016.
plots er, this is not the only reason why smaller particles may not be beneficial and better understanding for this interpretation it is necessary to look at the ns of the initial modelling work associated with investigating dispersion within mn. The most notable work in this area was initially done by Jan Jozef van er, a Dutch physicist working at the Royal Dutch Shell company as a her. His modelling work split the dispersion process into three key nents, referred to as eddy dispersion (A term), longitudinal dispersion (B term) persion associated with the resistance to mass transfer (C term) [3]. The which bears his name has the form of;
Kinetic Plots
However, this is not the only reason why smaller particles may not be benefi cial and to get a better understanding for this interpretation it is necessary to look at the limitations of the initial modelling work associated with investigating dispersion within a column. The most notable work in this area was initially done by Jan Jozef van Deemter, a Dutch physicist working at the Royal Dutch Shell company as a researcher. His modelling work split the dispersion process into three key components, referred to as eddy dispersion (A term), longitudinal dispersion (B term) and dispersion associated with the resistance to mass transfer (C term) [3]. The model which bears his name has the form of;
= + +(% +&).
Where;
s the height equivalent to a theoretical plate, and is a measure of the peak maller numbers are better.
linear velocity of the mobile phase eddy diffusion constant, and is inversely related to particle size longitudinal diffusion
and Cm stationary phase, and is related to the particle size.
Cm are the diffusional constants of the analyte in the mobile phase and in the ary phase, and is related to the particle size.
This work has been reviewed and updated on numerous occasions; however, the underlying structure of the equation has not altered. The above model does not take into consideration pressure and it does not consider time, both of the parameters are important from a practical perspective. In order to better understand these practical limitations, it is necessary to look at a different modeming approach, namely kinetic plots [4-10].
To perform the modelling three equations will be used.
Figure 3. Constrained kinetic plots derived by modelling for different particle sizes and different maximum operating pressures.
Conclusions Where;
N is the effi ciency of the column defi ned by the equation. L is the length of the column u0 t0
is the linear velocity of the mobile phase is time of a peak, in this case an unretained peak.
dp is the particle size ∆Pmax is the maximum operating pressure Φ is the column resistance factor η is the dynamic viscosity of the mobile phase
versus log N. It uses the same data but also looks at the impact of column length and pressure in accordance with the three equations given previously. In these plots we will focus on the maximum log N that can be achieved.
In the van Deemter plots HETP is plotted against the linear velocity u, with Figure 2 showing a plot of three different particle sizes, 1.8, 3.0 and 5.0 μm, with a clear benefi t in smaller HETP values as the particles are reduced. On the right-hand side for Figure 2, shows a plot of log t0
The solid line represents the point of infl exion, or the point where the most effi cient chromatography, sharpest peak relative to retention time, is obtained. As with the van Deemter plots the smaller particles clearly provide sharper peaks. The dashed line, however, represents where the column would be operating above the maximum operating pressure of the chromatographic system and so would not be obtainable in a practical sense. It can be clearly seen that for the smaller particles this has quite a signifi cant effect on the performance capability when practical considerations are taken into account.
For many separation scientists today there is a tendency to focus on moving to smaller particles to drive separation performance, and the marketing hype associated with the use of small particles has been very intense. Indeed, the use of smaller particles can drive chromatographic performance under certain conditions, namely highly effi cient separations performed in a short time performed on highly effi cient chromatographic systems, the data provided in this article demonstrates this. It is not however, the panacea to solving separation scientists’ challenges, and having a thorough understanding of the theory can help ensure that the best operating conditions are chosen for a separation given the practical limitations of the instrumentation. As with most things we are far too often lured in with the headline fi gures and have our opinions swayed without really having a full understanding of all of the issues. Unfortunately, the world is not a simplistic place and having an ability to understand the nuances of an issue will hopefully ensure that we have a better understanding of what is happening but also ensure that we may make the correct choices.
References
1. Fick, A. Ann. der. Physik (in German) 94 (1855) 59 2. Aris, R., Proc. Roy. Soc. A., 235 (1956) 67–77 3. van Deemter J.J., Zuiderweg F.J. and Klinkenberg A., Chem. Eng. Sci., 5 (6): (1956) 271–289 4. Giddings, J.C., Anal. Chem. 37 (1965), 60-63 5. Knox, J.H. and Saleem, M.,J. Chromatogr. Sci. 7 (1969), 614-622 6. Knox, J.H., Annual Review of Physical Chemistry 24 (1973), 29-49 7. Guiochon, G., High-Performance Liquid Chromatography: Advances and Perspectives, Horvath,C., Ed.; Academic Press: New York, 1980; Vol. 2, pp 1-56 8. Hans, P., J. Chrom. A 778.1 (1997), 3-21 9. Desmet, G., Clicq, D. and Gzil, P., Anal. Chem. 77.13 (2005), 4058-4070 10. Carr, P.W., Wang, X., Anal. Chem. 2009, 81, 5342–5353
Figure 4. Experimental constrained kinetic plot.
HETP is the height equivalent to a theoretical plate, and is a measure of the peak width, smaller numbers are better. u is the linear velocity of the mobile phase A is the eddy diffusion constant, and is inversely related to particle size B is the longitudinal diffusion Cs
are the diffusional constants of the analyte in the mobile phase and in the
Figure 2. Van Deemter plots and unconstrained kinetic plots showing where the theoretical optimal performance for a column exists
Taking the analysis further it is possible to plot the maximum practical performance of the column as shown in Figure 3. This plot is referred to as a constrained kinetic plot and takes into consideration real world parameters such as pressure. It can now be seen that the lines generated by the three different particle sizes cross, which then means that under different conditions different particle sizes should be considered to give the optimal performance. Thus the ability of a sub 2 μm to provide more chromatographic effi ciency at increased analysis times is dramatically reduced, and that for longer analysis smaller particles will provide more chromatographic effi ciency.
However, it could be said that the above work is all theoretical and so may not translate into a practical environment, however Figure 4 is based on experimental data and shows exactly the same trend as was seen with the theoretical data, demonstrating that bigger can truly be better for some forms of separations.
110 µm 5
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variance (5s) [µl2]
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