In the previous lesson we have learned about the classical van Deemter plate-height equation and we have seen that introducing reduced parameters (reduced plate height, h, and reduced linear velocity, \nu) allows for a much easier and better comparison of columns. Also, effects of various parameters (characteristic diameter, d, diffusion coefficient D_{\text{m}}) on the plate height can be readily predicted using reduced parameters.
While the name “van Deemter” still rings out loud in separation science, theory has progressed. Some important insights are discussed in this lesson. These include the realization that the random movement of analyte molecules cannot simply be expressed by their diffusion coefficient in the mobile phase, and the recognition that eddy diffusion (the A term) and mass transport (the C term) are coupled. In this lesson we will share some of these deeper insights.
Learning Goals #
After this course you will
- Understand some of the deeper insights arising from plate-height theory
- Grasp the concepts of effective diffusion and coupling terms.
- Express different diffusion terms in words
and in mathematical terms.
1. Decomposing the plate height #
At the basis of the original plate-height theory is the assumption that various dispersion processes (A, eddy dispersion, B, longitudinal diffusion, C, resistance to mass transfer) are independent. Statistical independence implies that variances may be added
Equation 1.85: \sigma^2_{\text{t}}=\sigma^2_{\text{t,A}}+\sigma^2_{\text{t,B}}+\sigma^2_{\text{t,C}}
or, when considering distances
The plate height is related to the variance through
Equation 1.84: H=\frac{L}{N}=\frac{L}{t^2_{\text{R}}}\sigma^2_{\text{t}}=\frac{1}{L}\sigma^2_{\text{z}}
so that we can write
=\frac{1}{L}(\sigma^2_{\text{z},A}+\sigma^2_{\text{z},B}+\sigma^2_{\text{z},C})=H_A+H_B+H_C
Thus, individual, independent contributions to the plate height may be added in the same way as variances. Specifying the contributions in the van-Deemter equation we have
Equation 1.87:
H=H_{\text{eddy diffusion}} + H_{\text{longitudinal diffusion}} + H_{\text{mass transfer}}We will now take a closer look at this contributions.
READ SECTION 1.7.4.1
Longitudinal Diffusion
EXERCISE 1
What do you know about the role of diffusion in chromatography? Select the correct statements.
This is correct! Higher analyte diffusion does not always result in broader peaks. While it leads to a higher B-term, it may reduce the C-term. The second statement is also false, because diffusion also occurs in the intraparticular (pore) space and in the stationary phase. The third statement is correct, because t_0=\frac{N \cdot h}{\nu_0} \cdot \frac{d^2}{D_{\text{m}}}, which tells us that a higher diffusion coefficient in principle allows faster analysis. The final comment is true.
This is not correct. Try again. Two statements are correct. Consider that a higher B-term, may reduce the C-term. Also consider t_0=\frac{N \cdot h}{\nu_0} \cdot \frac{d^2}{D_{\text{m}}}.
2. Longitudinal diffusion #
2.1. Molecular diffusion #
Molecular diffusion is an entropic process of Brownian motion, illustrated in the figure below. The entropy (or “chaos”) increases from stage A, where a droplet of analyte (dark-pink circles) in a sample solvent (pink) is introduced in a (miscible) liquid (light blue), to stage B, where the analyte and the sample solvent have started to diffuse into the liquid, to C, where concentration gradients have disappeared and maximum entropy has been reached.
Figure 1. Schematic illustration of diffusion of a concentrated plug of pink molecules in a medium filled with blue liquid.
The diffusion coefficient of analyte (D_i) is defined in Fick’s first law
Equation 1.21: J_{i,x}=-D_i \frac{dc_i}{dx}
where J_{i,x} is the flux of analyte i (molar flow per unit area) in the x direction and c_i its concentration. The minus sign indicates that the flux is in the opposite direction from the concentration gradient.
The dispersion in the axial direction of the chromatographic column due to random motion of molecules follows from the Einstein equation
Equation 1.88: \sigma^2_{\text{z}}=2D_{\text{m}} \cdot t
where t_0 is the time spent by the analyte molecules in the mobile phase. If we recall that
we can write
This is a very simple equation for the B term, that suggests B=2D_{\text{m}}.
Video 1. Brownian motion visible for small particles. Notice how the smallest particles migrate larger distances than larger particles.
2.2. Relation to molecular weight #
Smaller molecules have higher diffusion coefficients because their lower molecular mass and smaller hydrodynamic radius allow them to move more rapidly through a medium. This is made visual for particles in Video 1.
Table 1. Diffusion coefficients in water at 25 oC for different compounds of varying molecular weights.
| Solute | Molecular weight | D_{\text{m}} (10^{-9} \text{m}^2 \text{s}^{-1} ) |
|---|---|---|
|
Oxygen |
32 |
2.0 |
|
Urea |
60 |
1.4 |
|
Glucose |
180 |
0.7 |
|
Lysozyme |
14 000 |
0.12 |
|
Albumin |
66 000 |
0.06 |
|
Myoglobin |
820 000 |
0.02 |
|
Tobacco mosaic virus |
39 000 000 |
0.004 |
This emphasizes that the B-term is more significant for molecules of higher molecular weight. Indeed, we will see in Lesson 16 that for separations of very large polymers, the B-term is so low that one could simply turn off the flow, come back the next day and turn on the flow, to find the same separation still intact! The diffusion coefficients of such macromolecules are simply too low! This is also illustrated by Table 1.
EXERCISE 2
Rank the following molecules based on their diffusion coefficients from high (top) to low (bottom)
This is correct! This is precisely why SFC is fundamentally so attractive related to LC. It profits from higher diffusion coefficients relative to LC, and a wider applicability than GC.
This is not correct. Try again. Consider the estimated sizes/weights of the molecules, the conditions (temperatures) and physical phases of each of these techniques.
2.3. Determining diffusion coefficients #
A method to obtain D_{\text{m}} values that is close to our hearts as separation scientists is the Taylor-Aris method. It requires the kind of experiment that chromatographers often perform, i.e. measuring the time-based standard deviation (\sigma_{\text{t}}) of a peak. To obtain a true measure of D_{\text{m}}, a capillary tube should be used instead of a chromatographic column. The capillary should be uncoated or coated (or deactivated) with an inert material, so as to avoid any interactions of the analyte with the surface. Under ideal conditions (sufficiently high velocities, sufficiently long and wide and not tightly coiled capillaries, negligible extra-capillary band broadening), the following simple equation applies [1]
Equation 1.90: D_{\text{m}}=\frac{L \cdot d^2_{\text{c}}}{96 u \sigma^2_{\text{t}}}
where t_{\text{e}} is the elution time of the analyte.
EXERCISE 3
Taylor-Aris experiments can be conveniently performed on equipment designed for capillary electrophoresis, because the length of the capillary (e.g. 0.5 m) and its diameter are appropriate for such instruments. Moreover, the pressure needed to transport the mobile phase through the capillary is extremely low (e.g. 10 kPa). The following peak is obtained for Peptide X, from a Taylor-Aris experiment on a 50-µm i.d. capillary.
Estimate the diffusion coefficient of Peptide X. Pick the answer that lies closest to yours. The values shown below are in 10-10 m2 s-1.
This is correct! The elution time is about 310 seconds, and the standard deviation of the peak about 6 seconds. This yields: 2.24 \cdot 10-10 m2 s-1.
This is not correct. We roughly measure 310s as elution time and 6 seconds as the standard deviation from the peak. This should be sufficient to fill in Equation 1.90.
However, it is questionable whether the molecular diffusion coefficient of the analyte in the mobile phase is truly characteristic of the diffusion coefficient in the column.
Van Deemter already included a “labyrinth factor” \gamma so that B=2\gamma D_{\text{m}} to account for obstructed diffusion. In contemporary literature [1] an effective diffusion coefficient (D_{\text{eff}}) is used, that is a weighted average of the diffusion coefficients in various zones, as indicated in the figure below. As a first approximation we may write
Here we assume that the diffusion coefficient outside the particles is equal to the bulk diffusion coefficient (D_{\text{m}}). D_{\text{pores}} is the diffusion coefficient of the analyte in the pores and D_{\text{s}} is its diffusion coefficient in or on the stationary phase. The exclusion time (t_{\text{excl}}) is accessible by measuring the elution time of a fully excluded analyte.
Figure 2. Different contributions to the longitudinal diffusion in a column packed with porous particles. The blue areas are a schematic (two-dimensional) representation of a porous particle.
It is more difficult to distinguish between the time analyte molecules spend in the pores (t_{\text{pores}}) and the time they spend on or in the stationary phase (t_{\text{s}}). If we assume that all molecules spend equal times outside the particles (t_{\text{excl}}) and in the pores (t_{\text{pores}}) we may write
where t_{\text{R}} is the retention time of the peak and t_0 can be obtained from the elution time of an unretained component. However, the assumption of a constant t_{\text{pores}} is questionable, due to possible exclusion of larger analyte molecules and different kinetics inside the pores, and the above two equations have been disputed [2].
2.4. Peak parking #
To measure D_{\text{eff}} the longitudinal diffusion the peak-parking method may be used. When the flow is stopped eddy dispersion and resistance to mass transfer no longer contribute to band broadening. The only process that continues is longitudinal diffusion. As a result, the B-term can be characterized by measuring peak widths with and without stopping the flow for a period of time (t_{\text{park}}). Two factors determine the observed peak width (\sigma_{\text{t}}), i.e. the dispersion in length units (\sigma_{\text{z}}) when the peak reaches the end of the column emerging from the column and the speed (u_i) at which the peak of analyte emerges, i.e.
Equation 1.92: u_i=\frac{u_0}{1+k_i}
which yields
Equation 1.93: \sigma^2_{\text{t}}=\frac{\sigma^2_{\text{z}}(1+k)^2}{u^2_0}=\frac{2 D_{\text{eff}}(1+k)^2}{u^2_0}(t_{\text{R}}+t_{\text{park}})
The effective diffusion coefficient can be obtained from the slope (S_{\text{pp}}) of a plot of \sigma^2_{\text{t}} versus the total time spent in the column (t_{\text{R}}+t_{\text{park}})
Equation 1.94: D_{\text{eff}}=\frac{S_{\text{pp}} \cdot u^2_0}{2(1+k)^2}
and we obtain
Equation 1.95: H_{\text{B}}=\frac{\sigma^2_{\text{z}}}{L}=\frac{2 D_{\text{eff}}}{u_0} (1+k)
In terms of reduced parameters, we have
Equation 1.96: h=\frac{2}{\nu_0} \cdot \frac{D_{\text{eff}}}{D_{\text{m}}} (1+k)=\frac{2 \gamma_{\text{eff}}}{\nu_0} (1+k)
where \gamma_{\text{eff}}=D_{\text{eff}}/D_{\text{m}} is an effective obstruction factor. For unhindered diffusion, as in an inert open capillary tube, k=0, \gamma_{\text{eff}} =1 and h=2/ \nu_0. For hindered diffusion \gamma_{\text{eff}}. To obtain a value for \gamma_{\text{eff}}, peak-parking experiments (yielding D_{\text{eff}}) can be combined with Taylor-Aris experiments (yielding D_{\text{m}}).
READ SECTION 1.7.4.2
Mass Transfer Resistance
3. Resistance to mass transfer #
Next we will consider the resistance-to-mass-transfer (C) terms. These are illustrated in Figure 3. The C_{\text{s}} term arises from incomplete equilibration, as illustrated by the icon on the top left of the figure. The C_{\text{m}} term arises from variations in the mobile-phase velocity, as indicated by the icon on the bottom left.
Figure 3. Illustration of the C-term contributions to the plate height.
Note that in open-tubular (OT) chromatography it is customary to use f(k) and g(k) instead of f_{\text{m}}(k) and f_{\text{s}}(k), respectively, and the column diameter d_{\text{c}} takes the place of the particle diameter d_{\text{p}}. Also, u_{\text{int}} can be replaced by the average velocity u. The resulting Golay equation
Equation 1.125: H_{\text{C, OT}}=\frac{2D_{\text{m}}}{u}+f(k)\frac{d^2_{\text{c}} \cdot u}{D_{\text{m}}}+g(k) \frac{d^2_{\text{f}} \cdot u}{D_{\text{f}}}
will be discussed in Lesson 10.
In packed-column (PC) chromatography we consider convective motion to be confined to the interstitial space. Hence, we use u_{\text{int}} and we may substitute d_{\text{p}} for d_{\text{f}} and introduce an effective intra-particle diffusion coefficient, D_{\text{part}}.
Equation 1.97: H_{\text{C, PC}}=f_{\text{s}}(k)\frac{u_{\text{int}} \cdot d^2_{\text{p}} }{D_{\text{part}}}+f_{\text{m}}(k) \frac{u_{\text{int}} \cdot d^2_{\text{p}} }{D_{\text{m}}}
or
3.1. Mobile phase contribution #
In the 70 years since van Deemter et al. wrote their seminal paper, numerous groups of researchers have investigated the band broadening in packed columns and a number of different plate-height equations have been proposed. We discuss just one such detailed approach here.
Desmet et al. propose the following equation for the mobile-phase contribution to the C term (in dimensionless terms) [1]
Equation 1.98: h_{C_{\text{m}}}=\frac{1}{3} (\frac{k"}{1+k"})^2 (\frac{\epsilon_{\text{int}}}{1-\epsilon_{\text{int}}}) \frac{\nu_{\text{int}}}{\text{Sh}}
Where the zone retention factor k" is a replacement for the conventional retention factor k when using u_{\text{int}} instead of u, which is defined as k"=(t_{\text{R}}-t_{\text{excl}})/t_{\text{excl}}, \epsilon_{\text{int}} is the interstitial porosity and \text{Sh} is a dimensionless form of the mass-transfer coefficient in the mobile zone (\kappa_{\text{m}}, in m\cdots-1) defined as
The faster the mass transfer in the mobile phase, the higher are \kappa_{\text{m}} and \text{Sh} and the lower is the mobile-phase contribution to the plate height. \kappa_{\text{m}} cannot easily be measured, but it can be conveniently approximated by
Equation 1.102: \text{Sh}=\frac{13}{1+2.1 \cdot \nu_{\text{int}}}+8.6 \cdot \nu^{0.21}_{\text{int}}
3.2. Stationary phase contribution #
For the stationary-phase contribution to the mass transfer band broadening we have
Equation 1.104: h_{C_{\text{s}}\text{,PC}}=f_{\text{s}}(k) \frac{D_{\text{m}}}{D_{\text{part}}} \cdot \nu_{\text{int}}=\frac{1}{30} \cdot \frac{k"}{(1+k")^2} \cdot \frac{D_{\text{m}}}{D_{\text{part}}}\cdot \nu_{\text{int}}
Under some assumptions D_{\text{part}} can be estimated if D_{\text{eff}} is measured using the peak-parking method [1].
READ SECTION 1.7.4.3
Eddy Diffusion
4. Eddy diffusion #
We mentioned a few times the ambiguity surrounding the eddy diffusion. We will first explore why the van Deemter A[/later] is plausible.
4.1. Radom-walk model #
The classicial treatment follows the random walk model, where we assume that an analyte travels through the chromatographic column in a number of steps n of length l, reaching the outlet when
Equation 1.106: L=n \cdot l
It is reasonable to assume that l is of the order of d_{\text{p}} leading to
and that variations in l (i.e. \sigma_l are proportional to d_{\text{p}}, leading to
Equation 1.107: \sigma_l=q \cdot d_{\text{p}}
or
Equation 1.108: \sigma^2_l=q^2 \cdot d^2_{\text{p}}
where q is a constant. The total variance in column length units involves \sigma_{\text{z}} involves n independent steps, so that
=\frac{L}{d_{\text{p}}}\cdot q^2 \cdot d^2_{\text{p}}=L \cdot q^2 \cdot d_{\text{p}}
and finally
Equation 1.111: H_A=A=\frac{\sigma^2_{\text{z},n}}{L}=q^2 \cdot d_{\text{p}}=\lambda \cdot d_{\text{p}}
where we have replaced q^2 by the “turtuosity factor” \lambda from the treatment of van Deemter.
4.2. Criticism #
You may wonder why we have kept the Eddy dispersion (A) term for last when treating the different van Deemter terms in this lesson. There are several reasons for this:
- Reports have disputed the plausibility of the original van Deemter A-term.
- It is actually the most difficult process to describe.
- It differs most from the original van Deemter equation.
- It is not – as was first thought – independent of the linear velocity.
- Divergent approaches have been published, ranging from simple and pragmatic to highly complex.
Rather than a mental picture in which analyte molecules follow different parts along or around particles, the band broadening that is not accounted for by longitudinal diffusion or mass transfer is collected in an eddy dispersion term as follows
The main source of eddy dispersion in a packed column are thought to be due to inhomogeneities, which are encountered on at least five different scales, i.e.
- the length flow channels between particles (the trans-channel range);
- the size of a single particle (the trans-particle range);
- clusters of particles that may be more-tightly or more-loosely packed (the short-range interchannel range);
- large domains across multiple particles (the long-range interchannel range);
- the diameter of the column, for example between the centre regions and regions near the wall (the trans-column range).
Understanding these different sources of inhomogeneity has helped the design of columns with minimal inhomogeneity, such as pillar-array columns (which are commercially available already) or 3D-printed columns (which are still at a research stage). However, the multiple timescales make it virtually to quantitatively describe the eddy dispersion term.
4.3. Coupling term #
The suggestion from the van-Deemter equation and from the random-walk model discussed above that the eddy-dispersion term is independent of the flow rate has long been discarded. To overcome inhomogeneities diffusion in directions perpendicular to the flow rate is desirable. Such radial or lateral dispersion is aided by the same processes that lead to band broadening in the axial direction. Thus, C-term dispersion actually reduces eddy diffusion. This implies that the A and C terms in van-Deemter type equations are not independent. Coupling terms have been proposed to describe the combined effects of the two terms.
Giddings and Desmet et al. have both come up with sets of equations to describe eddy dispersion as the sum of contributions at different length scales, but the parameters needed to fully describe a column cannot be obtained.
The Giddings equation for the A-term reads
Equation 1.112: H_{AC \text{,Gid}}=\sum_{i=1}^{m} (\frac{1}{A_i} + \frac{1}{C_{\text{m},i} \cdot u_{\text{int}}})^{-1}
where the index i indicates the m different length scales on which path-length differences or packing inhomogeneities are considered. The Desmet equation reads
H_{AC \text{,Des}}=C_{AC \text{,Des}} \cdot u_{\text{int}} \Big( 1-\frac{C_{AC \text{,Des}} \cdot u_{\text{int}}}{2 \cdot A_{AC \text{,Des}}} \big(1-e^{-\frac{2 \cdot A_{AC \text{,Des}}}{C_{AC \text{,Des}} \cdot u_{\text{int}}}}\big)\Big)
A more practical, empirical approach has been described by Knox. He suggested a simple coupling term in terms of the reduced velocity \nu_{\text{int}} with an exponent n_{\text{Knox}}.
Desmet et al. have refined this empirical approach and they included include the effect of the zone retention factor. The proposed the following equations for the parameters in the Knox equation
Equation 1.118: n_{\text{Knox}}=0.55- \frac{0.11}{(k")^{0.93}}
and
Equation 1.119: A_{\text{Knox}}=0.19+\frac{0.25}{(k")^{0.46}}
The different approaches are compared in Figure 5.
Concluding remarks #
Current plate height theory superseded the classical van Deemter approach. Most notably, eddy diffusion is no longer considered independent of the linear velocity. Instead, the classical A and C terms are coupled.
Longitudinal diffusion can be measured using the peak-parking method. Models exist to determine the C_{\text{s}} and C_{\text{m}} contributions.
The eddy dispersion is usually the determined as the residual between the observed plate height and the B and C contributions.
References #
[1] G. Desmet, K. Broeckhoven, S. Deridder, D. Cabooter, Review of recent insights in the measurement and modelling of the B-term dispersion and related mass transfer properties in liquid chromatography, Anal. Chim. Acta, 2022, 1214, 339955, DOI: 10.1016/j.aca.2022.339955
[2] G. Desmet, K. Broeckhoven, J. De Smet, S. Deridder, G.V. Baron, P. Gzil, Errors involved in the existing B-term expressions for the longitudinal diffusion in fully porous chromatographic media: Part I: Computational data in ordered pillar arrays and effective medium theory, J. Chromatogr. A, 2008, 1188(2), 171-188, DOI: 10.1016/j.chroma.2008.02.018
[3] M. Callewaert, J. O. De Beeck, K. Maeno, S. Sukas, H. Thienpont, H. Ottevaere, H. Gardeniers, G. Desmet and W. De Malsche, Analyst, 2014, 139, 618–625, DOI: 10.1039/C3AN02023A
[4] C. Salmean, S. Dimartino, 3D-Printed Stationary Phases with Ordered Morphology: State of the Art and Future Development in Liquid Chromatography, Chromatographia, 2019, 82, 443-463, DOI: 10.1007/s10337-018-3671-5
