Column Porosity

Figure 2.7: (A) Histograms of volume-based size distributions for the 2.6 µm Kinetex core–shell particles derived from a CLSM reconstruction of 5213 completely imaged particles, and comparison with a particle size distribution determined from scanning electron microscopy images. (B) Analysis of packing gaps with a minimum of 2 µm diameter in the underlying reconstruction (gray). The green volumes represent void spaces inaccessible by particles contained in the particle size dis-tribution. Yellow volumes are defective positions accessible by 5% of the particles, whereas the red volumes can be accessed by at least 95% of the particles.

(to intentionally create defects) results in preferential flow paths and introduces new time and length scales for solute dispersion. These inhomogeneities were found to decrease strongly the chromatographic separation efficiency. In comparison to the number and qual-ity of defects evaluated in that principal study [22], the particles analyzed in our recon-struction packed extremely dense and well, with only minor defects. In total, only four pores were found in the reconstruction that were large enough to be filled by a particle tak-en from the determined PSD. This is equivaltak-ent to a particle removal of less than 0.1%

from a perfect packing. The smallest amount of particles removed in the study of defective packings by Schure and Maier [22] was 0.4%. However, the location of the gaps in our reconstructed packing cannot be considered as random. Three of the gaps were directly located at the column wall (Fig. 2.7B), a region of particular interest when analyzing the column cross-sectional heterogeneity of packing microstructure in confined packings and the origins of hydrodynamic dispersion, as we discuss further below.

In HPLC practice wehave accepted to “jam-pack” columns using a slurry-packing pro-cess that experience has told us to be most appropriate in terms of the traditionally meas-ured (post-column) separation efficiency [6–8]. The packing process is complex and in-volves several, often strongly interrelated, parameters, among them the physicochemical properties of the stationary-phase particles (including PSD, mechanical strength, surface roughness, chemical surface modifications), interparticle forces (electrostatic, van der Waals), slurry preparation (concentration, slurry liquid, ionic strength), the application of pressure and ultrasound, as well as the coupled stress–strain-flow behavior [68]. Due to the difficulty in probing the packing microstructure systematically as a function of all relevant process parameters, column packing and consolidation are largely treated phenomenologically and considered an art rather than a science.

Confined cylindrical packings of spherical particles consist of an ordered wall region, with high porosity fluctuations over a distance of 4–5 dp from the wall, and a random, densely packed core region [20,25,40,69,70]. These porosity oscillations result from the inability of the hard particles to form a close packing against the hard surface of the cylin-drical column as particles can touch, but not penetrate the wall. The first particle layer of the bed in contact with the wall is not only highly ordered, but differs from subsequent layers, because the interstitial space between the wall and the first layer cannot be partially occupied by other particles. Subsequent particle layers towards the column center do not retain this level of order and the degree of randomness increases with the distance from the wall. This wall effect is a purely geometrical effect existing in immediate vicinity of the

column wall and is distinct from a second and more extended wall effect caused by friction between the particles of the bed and the column wall [71–73]. The latter effect is tradition-ally discussed in HPLC in connection with relatively large column-to-particle diameter ratios. Here, the packing density near the wall is higher than in the core region. The effect is related to the relatively high compressibility of pulverulent materials and the complex distribution of axial and radial stress during the compression of the bed. The extent of this wall effect strongly depends on packing procedure, column-to-particle diameter ratio, and operational conditions.

The geometrical wall effect was envisioned early in the chromatographic literature [74]

and later carefully studied by Jorgenson and co-workers with packed capillaries [75–77].

For example, Kennedy and Jorgenson [75] and subsequently Hsieh and Jorgenson [76]

have demonstrated that the performance of fused-silica capillaries packed with 5 µm parti-cles improves significantly with decreasing capillary inner diameter between 12 and 50 µm. At these low column-to-particle diameter ratios the core region ultimately disap-pears and the packing structure is dominated by the wall region, i.e., the packing structure becomes effectively more ordered and homogeneous over the whole cross-section. The effects of the geometrical wall effect at increasing column-to-particle diameter ratio on hydrodynamic dispersion and chromatographic band broadening (compared with bulk packings which mimic infinitely wide, randomly packed beds “without walls”) are striking and have been quantified by extensive numerical simulation studies [20,25].

A plot of the interparticle (or external) porosity against the distance from the capillary inner surface also revealed these fluctuations characteristic of the geometrical wall effect in our reconstructed packed-bed segment (Fig. 2.8A, “external porosity”). Notably, these near-wall oscillations, over a distance of about 4 dp from the wall, remain mostly above the mean external porosity (0.362). Within these regions, the porosity fluctuations seem to be damped by the presence of gaps in the packing (cf. Fig. 2.7B). When we add the shell vol-ume of the particles to the interstitial void space, a porosity profile is obtained for a hypo-thetical packing with only the particles’ solid cores (Fig. 2.8A, “particle cores only”). This increases the mean porosity (void space between the solid cores) to a fictitious value of 0.783. Thereby, however, the relative influence of gaps in the packing is reduced and more symmetrical oscillations (around that mean porosity of 0.783), characterizing a geometrical wall effect, become visible (Fig. 2.8A).

In general, locally increased oscillations in the porosity profile suggest deviations from

Figure 2.8: (A) Radial porosity profiles of the reconstructed core–shell packing plotted for the interparticle or external porosity (black line) and the hypothetical loose packing of the particles’ cores only, i.e., without their shell (red line). (B) Lowpass of the external porosity illustrating the second wall effect, beyond the geometrical wall effect in direct vicinity of the column wall, most prob-ably related to friction during column packing. The mean porosity is represented by the dashed line (at 0.362).

direct vicinity of the column wall. Similarly, parts of the porosity profile of our reconstruc-tion that are shaped like beat frequencies suggest the presence of increased order (Fig. 2.8A), in this case, e.g., due to some agglomerated particles which were not dispersed during the slurry preparation and therefore could transfer this higher order into the final bed structure. Though these porosity fluctuations appear to be insignificant, their presence denotes variations in packing disorder on the scale of several particles, which inevitably increases time and length scales of dispersion in the bulk packing and generally will ad-versely affect separation efficiency.

In addition to the variation of packing density and disorder in the direct vicinity of the column wall and in the identified bulk regions (Fig. 2.8A), another origin of systematic radial heterogeneity in the external porosity profile of the reconstruction becomes evident by taking a look at its lowpass: Fig. 2.8B provides physical evidence for the earlier-mentioned second wall effect caused by the high radial stress applied by the bed to the wall as a consequence of friction between particles during the slurry-packing process. Because of these frictional forces, of the resulting radial stress that forces the particles against the wall and of the friction between the bed and the wall [71], a higher packing density is es-tablished in the wall region. Consequently, the bed permeability is higher in the central region than near the wall and a heterogeneous radial flow velocity distribution will devel-op.

For the reconstructed capillary packing (Fig. 2.8B), the external porosity profile exhib-its a minimal porosity at a distance of 4–5 dp from the wall and gradually increases, almost symmetrically, from both sides up to a distance of 10.5 dp from the wall, until the dashed line representing the mean external porosity (0.362) is crossed for a second time and bulk packing properties seem to be approached. We are not aware of any systematic study of this wall effect in columns with a column-to-particle diameter ratio that is characteristic for capillary type stationary phases. Thus, a comparison can only be made with results pub-lished for analytical and larger bore columns where already the packing process varies markedly. Therefore, this comparison should be made with care and only aim at resolving similar physical origins of wall effects in packed beds, not their magnitudes and spatial dimensions. For example, using optical on-column visualization Shalliker et al. [72] ana-lyzed wall effects by tracking the migration of sample bands through a 17 mm i.d. glass column packed with 21 µm particles. Their results demonstrated that two wall effects take place in chromatographic columns. The first is purely geometrical, the second is related to

friction, as explained. Our results well confirm their conclusions by a direct physical re-construction of the chromatographic bed for a 100 µm i.d. capillary column packed with 2.6 µm particles, resulting in a column-to-particle diameter ratio of ~40: a relatively ho-mogenous core region of roughly 19 dp is surrounded by heterogeneous wall regions of 21 dp including both wall effects (Fig. 2.8B).

The porosity profile in Fig. 2.8B implies that in a narrow (~2 dp wide) region next to the column wall the mobile phase velocity is higher than its average in the whole packing, as a consequence of the geometrical wall effect, and that in a directly neighbored, but wid-er region (~7 dp) towards the center of the column the mobile phase velocity is lower, most likely due to the packing process-specific second wall effect related to friction. Further analysis requires confirmation of these results, but already promises great potential in the comparative study of packing conditions, particle properties, and column-to-particle di-ameter ratios, particularly in view of characterizing and minimizing the packing process-specific heterogeneities in the final bed structure.

2.3.2.3 Chord Length Distributions. Giddings [1] has divided local velocity