Relation between the particle size distribution and the kinetic performance of packed columns

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Journal of Chromatography A, 1161 (2007) 224–233

Relation between the particle size distribution and the kineticperformance of packed columns

Application to a commercial sub-2 �m particle material

Jeroen Billen a, Davy Guillarme b, Serge Rudaz b, Jean-Luc Veuthey b, Harald Ritchie c,Brian Grady c, Gert Desmet a,∗

a Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgiumb Laboratory of Analytical Pharmaceutical Chemistry, School of Pharmaceutical Sciences, University of Geneva, University of Lausanne,

Boulevard d’Yvoy 20, 1211 Geneva 4, Switzerlandc Thermo Fisher Scientific, 112 Chadwick Road, Runcorn, UK

Received 16 January 2007; received in revised form 24 May 2007; accepted 29 May 2007Available online 2 June 2007

bstract

To study the influence of the particle size distribution (PSD), we measured the chromatographic performance of a series of sub-2 �m particleigh performance liquid chromatography (HPLC) columns packed with four different particle mixtures having a purposely imposed different sizeistribution. Using the reduced kinetic plot representation by plotting the separation impedance (E0) versus the plate number ratio (Nopt/N), theifferent columns could be classified according to their chromatographic performance without the need to specify a mean particle diameter or a

olecular diffusion coefficient, as is needed in the classical reduced plate height and flow resistance analysis. The present analysis shows that

t is not so much the width or span of the particle size distribution, but rather the presence of fines that greatly determines the chromatographicerformance of particulate columns.

2007 Elsevier B.V. All rights reserved.

bdetcsettcd

eywords: Particle size distribution; UPLC; Kinetic plot

. Introduction

The particle size distribution (PSD) of the packed bedolumns used in high performance liquid chromatographyHPLC) has always been recognised as an important factor influ-ncing their performance [1–4]. Intuitively it seems desirableo use narrow sized cuts of the HPLC packing materials [1].ommercial literature promotes this statement, but this is rarely

upported by studies appearing in the scientific literature.The subject was already discussed in an early study by Halasz

nd Naefe [2] who found that the influence of the PSD on the

late heights and the permeability is in fact very small, at leasts long as the PSD is not wider than 40% around the mean.ewaele and Verzele [3] prepared chromatographic packed beds

∗ Corresponding author. Tel.: +32 2 629 32 51; fax: +32 2 629 32 48.E-mail address: [email protected] (G. Desmet).

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021-9673/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.chroma.2007.05.088

y blending two batches of silica-based packing material withistinct particle sizes. Testing the resulting packings with differ-nt PSD, they concluded that a larger PSD has no influence onhe plate height values around the minimum of the plate heighturve, while at higher mobile phase velocities there is only amall negative effect. However, the larger PSD had a negativeffect on the column backpressure and hence on the separa-ion impedance for all velocities. These results are, however,o be considered with some caution, because the authors onlyompared columns having a clearly different average particleiameter and the packing quality at that time was worse formaller particles than for larger particles [4]. Moreover, parti-le batches with a narrow size distribution are reputed to packess homogeneously than those having a broad distribution [5,6].

ore recently, several solutions have been proposed to limit theackpressure generated by such small particles. Some providersommercialized inhomogeneous particle size distribution, witharger particles packed in the columns [7]. Apart from the stud-

Page 2

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tpathad an average pore size of 175 A and were coated with a C18layer. The primary size distributions of the two particle batchesare shown in Fig. 1 (dashed curves). Mixture D was obtainedby subsequently blending the two primary batches in a mass

J. Billen et al. / J. Chrom

es of the effect of PSDs in HPLC, a number of studies havelso been conducted in size exclusion chromatography [8–10].he general consensus from these investigations was that there

s certainly no need to strive to a monodisperse packing, becausehe column efficiency only becomes negatively affected in casesf very broad and asymmetric PSDs. For “normal” PSDs, no sig-ificant effect could be noted. All this is also clearly summarisedn [11].

Due to the recent introduction of columns packed with sub-�m particles by a number of leading column manufacturers,

he importance of the PSD of packing has, however, becomeery timely again [12]. For the width and asymmetry of theSD is known to be significantly larger in this small (i.e., sub-�m) particle range than it is for packings with the larger, i.e.,–10 �m particles. The reason for this resides in the method ofreparation and classification. Sieving of particles smaller than0 �m is inefficient and therefore air classifiers are used forhe separation of differently sized particles [11]. As the targetarticle size decreases, so does the weight of the particles and theeight difference between the target particles and the unwantedarticles also decreases. This makes the particle separation lessfficient for smaller particles.

An obvious prerequisite to investigate the influence of theSD in depth is an accurate determination of the particle sizeistribution. Doing so, a first encountered problem is that theeasured particle size can depend quite heavily on the employedeasurement method. Different measurement methods are used

ext to each other in practice. Some of the more widely used arelectrical sensing zone method (Coulter principle), sedimenta-ion, laser light diffraction (analysis of light scattering pattern)nd analysis of scanning electron microscopic (SEM) images.ach of these has its own advantages and drawbacks [11]. How-ver, this is an issue that falls outside the scope of the presenttudy. A second problem is that there still exist no univocalr theoretically based rules that can be used to characterize thebtained PSDs. Even for the determination of something as sim-le as the average particle diameter (dp), at least four to fiveefinitions or approaches coexist in literature [11]. If the PSD isarrow and symmetrical, the mean particle size dp can be deter-ined with more confidence than for a broad and asymmetrical

istribution. In the latter case, the different mean particle sizeefinitions can lead to quite strongly differing values. This uncer-ainty on the determination and the definition of the mean particleize severely compromises the traditional column performancenalysis. In such a traditional analysis, the bed permeability isade dimensionless using the particle diameter (yielding theow resistance, φ = d2

p/Kv0) and a plot of the reduced plateeight (h = H/dp) versus the reduced velocity (ν = udp/Dm) issed to assess the band broadening. Since the values of ν, hnd φ depend heavily on the employed value for dp, it is obvi-us to expect that different conclusions on the flow permeabilityr the goodness of packing will be drawn depending upon themployed definition of dp [11].

In the present paper this is illustrated by investigating thehromatographic performance of four custom made particleixtures made from the same base material (1.9 �m Hypersilold from Thermo Fisher Scientific), but with a different, delib-

F(pw

A 1161 (2007) 224–233 225

rately induced PSD. All the mixtures have a number averagedean particle size smaller than 2 �m.In the first part of the paper, a traditional, particle size

ased column performance analysis is performed. In the secondart, the column performance is assessed using the kinetic plotethod. This is a performance assessment method that circum-

ents the need to define and measure a mean particle size [13].nother interesting method to circumvent the need to define oreasure a mean particle size has already been proposed many

ears ago by Halasz et al. [14,15]. This method uses an effectiveean particle size based on the column permeability to reduce

he plate height data. As such, the reduced h values are propor-ional to the square of Knox separation impedance. The reducedlate height curves that are obtained with this approach providen excellent measure for the packing quality. They are, however,nly partially related to the kinetic performance of the supportype, which requires a curve relating analysis time to plate num-er. As shown in [13], curves relating E0 to ν0 do not providehis information. Although E0 is directly related to the analysisime via tR = t0.(1 + k′), the link with the plate number is missingn this type of plots because the reduced velocity is not relatedirectly to the achievable value of N.

. Experimental

.1. Production of particle mixtures and columns

Four particle mixtures with a different size distribution, fur-her on referred to as particle mixture A, B, C and D, wereroduced by blending two particle batches in different percent-ges. Each primary particle batch was prepared by operatinghe air classifiers under selective conditions. The silica particles

ig. 1. Example of a particle mixture with a custom-made size distributionfull line), resulting from the blending of two particles batches with well-knownarticle size distributions (primary particle batches indicated dashed lines). Dataere smoothed with a moving average with a period of 5.

Page 3

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26 J. Billen et al. / J. Chrom

roportion of 40/60% (m:m), resulting in the curve shown byhe full line. The resulting distribution shows a higher relativeroportion of small as well as large particles. The mean value,owever, is still within the specified sub-2 �m range. The threether particle mixtures were prepared in the same manner, eachy blending different percentages of the two primary batches.

The particle mixtures were packed in columns using the samen-house developed standard procedure for every column and

ixture. A known mass of powder was evenly dispersed withltrasonication in a proprietary solvent mixture of specific vol-me. The packing pressure profile was the same for each column.o specific optimisation was carried out. The employed pack-

ng method was fully similar to that used for preparation ofommercial Hypersil Gold columns. This is a highly optimisedethod. Because the mixtures studied here lie in the same size

ange as the commercial column, the packing method is assumedo be evenly well suitable. To investigate the reproducibility ofhe experiments, three columns were tested for each of the fourarticle mixtures.

.2. Particle size measurements

The particle size distribution measurements were performedn a Beckman Multisizer 3, which utilises the electrical sensingone method to determine the number and size of the parti-les dispersed in an electrically conducting fluid. Dispersionf the particles occurred by shaking a couple of milligramsf silica powder in 5 ml of the electrolyte Isoton II, a salineolution provided by Beckman-Coulter (Fullerton, CA, USA).he instrument has two electrodes in separate compartments,onnected via a small aperture (20–100 �m diameter). In theurrent setup, the smallest particle that can be measured has aiameter of at least 600 nm. A constant current is set betweenhe electrodes. During measurement a small flow of electrolyte,ontaining the sample particles, is generated between the twoompartments. When a particle passes through the aperture, itisplaces a volume of electrolyte equivalent to the volume ofhe particle. Since the particle is non conducting the resistanceetween the two electrodes increases in proportion to the vol-me of electrolyte lost—hence proportional to the volume of thearticle. In order to maintain a constant current the instrumentas to increase the voltage between the electrodes according tohm’s law. This increase in voltage is measured as a pulse. Theumber and size of the particles are proportional to these mea-ured voltage pulses. Using porous particles, it can be expectedhat the crude particle size values obtained with the instrumentre slightly underestimated (typically some 5%). The employedethod has, however, been calibrated for this deviation by using

tandard lots and cross-checking with SEM measurements.

.3. Chromatographic analysis

.3.1. Chemicals and columns

Uracil and methyl-, ethyl-, propyl- and butyl-paraben were

rovided from Sigma–Aldrich (Munich, Germany). Acetoni-rile was of HPLC grade from Panreac Quimica (Barcelona,pain). HPLC grade water was prepared in house using a Milli-

pdr

A 1161 (2007) 224–233

Waters Purification System from Millipore (Bedford, MA,SA). Columns used in this study were provided by Thermoischer Scientific (Runcorn, UK).

.3.2. ApparatusChromatographic data were acquired with an UPLC Acquity

ystem (Waters, Milford, MA, USA), which can withstand pres-ures up to 1000 bar. This instrument includes a binary solventanager with a maximum delivery flow rate of 2 ml/min, an

uto-sampler with a 2 �l loop, an UV–vis detector with a 500 nlow cell, and a column oven set at 30 ◦C. Data acquisition, dataandling, and instrument control were performed by Empoweroftware (Waters). Samples consisting of 50 �g/ml uracil andour parabens were dissolved in a water/acetonitrile (60/40%,/v) mixture and the injection volume was reduced to 1 �l.bsorbances were measured at 254 nm, using a constant time of5 ms and a sampling rate of 40 Hz.

.3.3. MethodologyAll experiments were conducted in the isocratic mode with a

ater/acetonitrile (60/40%, v/v) mobile phase, in order to obtainppropriate retention factors for a fundamental study. For the0/40 (%, v/v) mobile phase, the viscosity η was estimated to be= 0.85 × 10−3 kg/ms [16]. In the separation of four parabensutyl-paraben was selected as model compound since it presentshe best peak shape (lowest asymmetry) and because the reten-ion factor obtained in these conditions (k′ = 6) was sufficientlyigh to perform column evaluation without being influencedy extra-column band broadening effects. All reported plateeights, u0 velocities, times and column permeabilities werebtained after correction for the system band broadening, t0ime and pressure drop, measured by removing the column fromhe system and connecting the tubing with a zero dead volumeonnection piece.

. Results and discussion

.1. Particle size measurements

The results of the particle size measurements by the Beck-an Multisizer 3 are shown in the histograms in Fig. 2. TheSD can be displayed in many different ways, two of which areepresented in Fig. 2: the number-weighted distribution (Fig. 2a)nd the volume-weighted distribution (Fig. 2b). In the former,he relative number (percentage) of particles, having a diame-er situated in a narrow particle-size class, is plotted versus theverage particle diameter of the window. In the latter, the per-entage of particles within each class is calculated on the volumehese particles take compared to the total volume of all measuredarticles. This shifts the volume averaged curves towards largerarticle sizes, because the large particles gain more weight inhis graph due to their larger volume, which is related to thearticle diameter raised to the power of 3.

A column packing is primarily characterized by the averagearticle diameter, dp. Unfortunately, the population and volumeistribution curves each have their own average dp value, hereespectively denoted by the number-averaged particle size (dp,#),

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J. Billen et al. / J. Chromatogr.

Fig. 2. Particle size distribution curves of the four tested particle mixtures (mix-ture A = black with full line, B = blue with dashed line, C = green with dotted lineand D = red with dot-dash line). Particle size distribution plotted based on (a)the population averaged distribution and (b) the volume averaged distribution.Tpt

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tflpptato(4% if averaged over all data points). The other mixtures (A–C)displayed a similar inter-column variability, with RSD valuesranging from 0.6 to 9% (n = 3) on the H values.

he curves were smoothed by a moving average with a period of 5. (For inter-retation of the references to colour in this figure legend, the reader is referredo the web version of the article.)

nd the volume-averaged particle size (dp,V). The dp,# valuesbtained from the distribution curves are shown in Fig. 2a and

iven in the left part of Table 1 (row 1). As can be noted, theifferences between the dp,# values of the different particle mix-ures are relatively small. All dp,# values, however, strongly differ

able 1haracterization parameters for the four investigated particle mixtures

Number averaged Volume averaged

ixture A B C D A B C D

p (�m) 1.82 1.88 1.90 1.85 2.33 2.24 2.13 2.39

10 (�m) 1.37 1.21 1.46 1.30 1.58 1.66 1.61 1.54

50 (�m) 1.77 1.89 1.84 1.77 2.00 2.15 2.05 2.18

90 (�m) 2.28 2.41 2.35 2.46 2.78 2.76 2.60 3.01

90/d10 (/) 1.66 1.99 1.61 1.89 1.76 1.66 1.61 1.95

90 − d10 (�m) 0.91 1.20 0.89 1.16 1.20 1.10 0.99 1.47pan (/) 0.52 0.64 0.48 0.66 0.60 0.51 0.48 0.67

Fecflto

A 1161 (2007) 224–233 227

rom the dp,V values obtained from Fig. 2b and given in the rightand side part of Table 1. In literature, there is no general consen-us on which of these two values has to be used to characterizepacking [11]. Average particle sizes cited in literature (both

cientific and commercial) hence always have to be consideredith the greatest care.In the present study, it can be expected that the average par-

icle size will not suffice to characterize the employed particleixtures. Packings with an asymmetric PSD can have a sim-

lar average particle diameter, but still a different distributionround this average. Therefore the percentiles, the percentileatio d90%/d10%, the width d90% − d10%, and the span of the PSDsave been calculated as well (Table 1). The span is defined heres the dimensionless width of the PSD, defined as:

pan = d90% − d10%

d50%(1)

.2. Traditional column performance analysis

In Fig. 3a an overlap of three chromatograms is shown. Inhis example, the separation of four parabens was achieved at aow rate of 1000 �l/min using the three columns packed witharticles from mixture D. For these separations, retention times,eak width and column backpressure were close (i.e., RSD lowerhan 0.2%, 5% and 3%, respectively) demonstrating an accept-ble inter-column variability. Fig. 3b compares the results ofhese columns over the entire velocity range for butyl-parabennly. In this comparison, all RSD values were lower than 8%

ig. 3. (a) Overlay of three chromatograms for the separation of uracil, methyl-,thyl-, propyl- and butyl-paraben. Each chromatogram corresponds to one of theolumns packed with particle mixture D. All experiments were performed at aow rate of 1000 �l/min. (b) Plate height curves showing the reproducibility of

he experiments over the entire velocity range. Each curve corresponds to onef the three investigated columns packed with particles of mixture D.

Page 5

228 J. Billen et al. / J. Chromatogr.

Fig. 4. Plate height curves obtained for butyl-paraben in 60/40 (%, v:v) wateraofi

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cetonitrile mobile phase for the four tested particle size mixtures. The indicationf the curves is in accordance with that used in Fig. 2. Experimental data weretted to the Knox plate height equation.

Fig. 4 shows the average plate height values (averaged overhe three columns investigated for each particle size mixture) forutyl-paraben measured on the different columns for pressuresp to 700 bar. Throughout the manuscript, these averaged valuesill be used as being representative for mixtures A, B, C and. The curves of mixtures A, C and D lay close to each otherver the entire velocity range, while particle mixture B clearlyisplayed higher H values. As shown in Table 2, mixtures C and

have the lowest minimal plate height values, with an Hminf 4.60 �m and 4.65 �m, respectively. The C-term dominatedanges for mixtures A, C and D lie very close to each other andave about the same slope. The plate height curve for mixturehas a high velocity range slope that is clearly less steep than

or the other three mixtures, but this is essentially due to the facthat mixture B apparently has a higher A-term band broadeningindicated by the higher curve minimum), which always tendso weaken the slope in the high velocity range (see also furthern: discussion of Table 3).

By measuring the pressure drop over the column, the bedermeability can be derived using Darcy’s law. The data for theressure measurements had a maximum inter-column variability

f 5% per particle mixture. Table 2 gives the average bed per-eability values (Kv0) of the columns. The differences between

he mixtures are very small, with mixture A presenting a slightlyower KV0 value.

iedl

able 2hromatographic parameters for the four studied particle size mixtures, obtained from

ixture A

min (�m) 4.8

v0 (10−15 m2) 3.9

Number averaged

ixture A B C D

p (�m) 1.82 1.88 1.90 1.85

min (/) 2.6 3.0 2.4 2.5(/) 840 800 850 800

min (/) 5800 7300 5000 510

A 1161 (2007) 224–233

Traditionally, the quality of packing is assessed by calculatinghe reduced plate height and flow resistance values. As explainedn the introduction, the calculation of ν, h and φ requires a valueor the average particle diameter. In the case of a broad and asym-etric PSD, the selection of this average diameter is, however,

ot straightforward, as can be noted from the fact that the valuesbtained for dp,# and dp,V and given in Table 1 differ strongly.iven these differences, it is not surprising that the obtained val-es for h and φ differ strongly (Table 2), depending on whetherne uses the population weighted or the volume weighted case.his in turn leads to different conclusions on the packing quality.or example: based on the number-averaged particle size anal-sis (left hand side of Table 2) the reduced plate height valuef particle mixture B is significantly larger than the other par-icle mixtures. From the mutual hmin values of mixtures A, Cnd D one would conclude that mixture C yields the best packedolumns, closely followed by mixtures D and A. However, look-ng at the right hand side of Table 2 (volume-averaged particleize), particle mixture B still has the highest hmin value, but thelassification of the three other particle mixtures is different.ixture D now has the lowest minimal reduced plate height

alue, followed by mixtures A and C. Admittedly, the differ-nces between mixtures A, C and D are, however, quite small.

more significant difference between the dp,# and dp,V valuess found for the flow resistance values, which is logical becauseere the mean particle size has an effect to the second power.sing dp,# to reduce the bed permeabilities, the obtained φ val-es lie relatively close to each other, with mixture B and mixtureapparently yielding columns with the smallest flow resistance.

he reduction based on dp,V on the other hand, suggests a biggerifference in flow resistance, with now mixtures C and B hav-ng the lowest flow resistance (nearly 30% difference between

ixture C and A). Switching from the dp,#-based flow resistanceo the dp,V-based flow resistance, mixture D demotes from theixture yielding the lowest flow resistance to that yielding the

ighest flow resistance.The above discussion clearly shows that conclusions on the

acking quality and the flow resistance of a given column dependighly on the definition of the average particle diameter. This

mplies that care has to be taken if comparing data from differ-nt studies found in literature. If these studies employ differentefinitions for the average dp value, a comparison will inevitablyead to false conclusions. The same problem arises when trying

a classical column performance analysis

B C D

5.7 4.6 4.74.4 4.2 4.2

Volume averaged

A B C D

2.33 2.24 2.13 2.392.1 2.5 2.2 2.01380 1130 1070 1350

0 5800 7300 5000 5100

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J. Billen et al. / J. Chromatogr. A 1161 (2007) 224–233 229

Table 3Van Deemter-curve fitting parameters for the four studied particle mixtures obtained using Eq. (2)

Number averaged Volume averaged

Mixture A B C D A B C D

A-constant 1.12 (±0.18) 1.68 (±0.31) 0.90 (±0.20) 1.04 (±0.08) 0.82 (±0.15) 1.44 (±0.26) 0.84 (±0.18) 0.81 (±0.05)B 0.40)C 0.01

T

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Fig. 5 has been obtained by applying Eqs. (4) and (5) to theexperimental plate height data shown in Fig. 4. Plotting the Nvalues in reverse order has the additional advantage that it not

-constant 6.50 (±0.12) 6.54 (±0.34) 6.56 (±0.02) 6.50 (±-constant 0.089 (±0.010) 0.071 (±0.020) 0.086 (±0.005) 0.087 (±he absolute standard deviation of these values is indicated between brackets.

o discuss the packing quality from the A- and C-term con-tants of any of the existing plate height expressions, because thebtained fitting parameters also depend heavily on the appliedeference diameter. As an example, we fitted our experimentaleduced plate height data with the original Van Deemter equation17]:

= A + B

ν+ Cν (2)

The obtained fitting parameters and the accompanying stan-ard deviations are given in Table 3. The data illustrate that the-term constant, traditionally employed as the measure for theoodness of the column packing [18,19], yields different con-lusions depending upon whether dp,# or dp,V is used as theeduction parameter. Using dp,#, there is a small yet signifi-ant difference between mixtures C, D and A (listed in orderf decreasing order of “goodness” of packing). Using dp,V ashe reduction parameter, the three mixtures seem to yield equallyell-packed columns. Regardless of whether dp,# or dp,V is beingsed as the reduction parameter, particle mixture B stands outs yielding the columns with the poorest packing homogene-ty (highest A-term constant). Given that the plate height datare collected over a relatively short velocity range (not reallyeep into the C-term dominated range), mixture B also standsut as the mixture yielding the lowest C-term band broaden-ng, but this is merely due to the fact that the velocity pointsn the right hand side of the plate height curve minimum (seeig. 4) are still strongly influenced by the high A-term bandroadening, and therefore appear less steep, hence yielding amaller apparent C-term constant. If the columns would haveeen tested at much higher velocities (which is impossible withhe current generation of instruments because of the high pres-ure drop), the fitted C-term constant for mixture C undoubtedlyould have lied more closely to the C-term values obtained for

he three other mixtures.This type of conflicting conclusions is not only obtained for

he Van Deemter plate height equation, but it is common to alleduced plate height models (data not shown), because they allely on the introduction of a reference diameter. Using eitherp,# or dp,V in a reduced plate height and flow resistance anal-sis leads to conflicting results. Since there is no real theoryictating which one of the two (or any of the many other exist-ng mean particle size definitions) should be used, analysing the

hromatographic performance of a column solely based on its hnd φ values is insufficient.

Knox separation impedance number, E0,min, on the other handan be calculated without first having to measure and define a

Fs

6.62 (±0.12) 6.48 (±0.46) 6.72 (±0.06) 6.50 (±0.4)0) 0.060 (±0.006) 0.049 (±0.016) 0.066 (±0.003) 0.052 (±0.008)

haracteristic size:

0,min = H2min

Kv0(3)

As a consequence, the E0,min number gives a much morerustworthy indication of the chromatographic performance andacking quality of the different columns [20]. From the E0 valueshown in Table 2, it can be concluded that particle mixtures Cnd D yield the lowest separation impedances and hence havehe best packing structure.

.3. Kinetic plot analysis

The E0,min number analysis is however incomplete, becausehe minimal separation impedance is only achieved for one givenolumn length, i.e., for one plate number. Other column lengthsyielding a different plate number) inevitably constitute differenteparation impedance. This has been more completely explainedn [13], showing that a simple recombination of the experimentalu0,H)-data couples together with the bed permeability sufficeso obtain a series of data couples linking a minimal separationmpedance to each N value:

0 = H2

Kv0(4)

= �pmax

η

(Kv

u H

)(5)

ig. 5. Kinetic plot of the four studied particle mixtures. Curves indicated iname way as in Fig. 2.

Page 7

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otcurve minima, the four particle mixtures can now be clearlyclassified in increasing order of packing quality, yielding: mix-ture B < A < D < C, with particle size mixtures C and D lyingvery close to each other, and with mixture A and especially

30 J. Billen et al. / J. Chrom

nly contains the desired kinetic information (E0 versus N) butlso still clearly reflects the characteristics of the original plateeight plot, i.e., the data points on the left hand side of the curveinimum correspond to the B-term dominated mobile phase

elocities and the data points on the right hand side of the curveinimum correspond to the C-term dominated velocities.Considering now for example the bottom curve of Fig. 5, it

an readily be seen that the minimal Knox impedance numberiven by Eq. (3) and cited in Table 2 is indeed only relevant forhe curve minimum, i.e., at N = Nopt. For the presently consid-red particle mixtures, this corresponds to separation requiringetween N = 20 000 and 25 000 plates. Using a column with theame packing quality but now with a length optimized to yieldhe fastest possible N = 10 000 plates separation, the correspond-ng separation impedance is no longer equal to the E0,min valueited in Table 2. Considering particle mixture C for example,his value shifts from E0,min = 5000 at the curve minimum to0 = 7100 at N = 10 000. Fig. 5 therefore clearly shows that, iftating a separation impedance value, one should also alwaysutomatically also specify the plate number this impedance cor-esponds to. Or inversely, if trying to select the packing typeielding a given number of plates in the smallest possible time,ne should consider the E0 number corresponding to this spec-fied plate number instead of the minimal E0 number.

Showing at a glance the complete overview of all possibleeparation impedances, a plot of E0 versus N allows makingglobal comparison of the packing quality. As shown in [13],

he plot of E0 versus N is in fact only one of the many pos-ible kinetic performance plots, all allowing to compare thehromatographic performance of different packings on a size-nd geometry-independent basis. The E0 number is for exampleirectly related to the required analysis time, and hence also tohe t0/N2 quantity that was used in previous kinetic plot methodapers [13] via:

t0

N2 = η

�pmax

(H2

Kv0

)= η

�pmaxE0 (6)

If all studied columns are operated for the same η and �p, ass the case in the present study, the t0/N2 values vary in a linearlyroportional way with the separation impedance. Eq. (6) hencehows that the vertical differences between the E0 values in Fig. 5re equal to the differences in analysis times needed to achievegiven plate number with the considered particle mixtures.

Comparing now the four different investigated packings,ig. 5 shows that the curves for particle mixtures C and D lieignificantly below those for particle mixtures A and B over thentire plate number range. For separations requiring for exam-le N = 40 000 plates, particle mixture B will need 40% morenalysis time than particle mixture C (cf. double-headed arrowt N = 40 000). Below N = 20 000 plates, the difference betweenhe E0 curves becomes smaller (cf. double-headed arrow at

= 20 000). Or, put in other words, the differences between the

urrently investigated particle mixtures have a smaller effect inhe C-term dominated range of fluid velocities than near the u0,optelocity. The latter can be understood from the link between und N (cf. Eq. (5)), showing that the range of small plate numbers

Fc

A 1161 (2007) 224–233

right hand side of Fig. 5) correspond to the C-term dominatedange of velocities.

A drawback of the E0 versus N plot as shown in Fig. 5 ishat, despite the fact that the two quantities are dimensionless,he N value still depends on the size of the particles [21]. Thearger the particles, the more the minimum of the curve shiftso the left, or, minding that the x-axis is reversed, the more the

inimum of the curve shifts to higher N values.This size-dependency can, however, be completely elimi-

ated by switching to the reduced kinetic plot method describedn [22], wherein E0 is no longer plotted versus N but versus

opt/N. The advantage of this approach is that the horizontalosition of the curves is independent of the actual size of thearticles. This is demonstrated in Appendix A, together withhe fact that the Nopt/N ratio can be directly calculated from thexperimental (u0,H)-couples (once Hmin and the corresponding0,opt are determined):

Nopt

N= u0H

u0optHmin(7)

In Appendix A, it is also shown that a plot of E0 versusopt/N depends only on the dimensionless variables h, ν and φ.his implies that a reduced kinetic plot yields the same typef information as a reduced Knox plot: packings with the sameacking quality and the same retention factor but a different sizeill yield coinciding curves. The lower this curve, the betteracked the column. The reduced kinetic plot, however, has twomportant advantages over the reduced Knox plot. The first ones that it also incorporates the flow resistance information. Theecond one is that it can be established without having to definer measure a reference diameter, or without having to know theobile phase diffusivity. This obviously excludes a number of

mportant error sources.Using Eqs. (4) and (A.4) to transform the plate height data

f Fig. 5 into a reduced kinetic plot (Fig. 6), hence excludinghe influence of the mean particle size on the position of the

ig. 6. Reduced kinetic plot of the four studied particle mixtures. Curves indi-ated in same way as in Fig. 2.

Page 8

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J. Billen et al. / J. Chrom

ixture B performing clearly much less well. The order of theurves is clearly maintained over the entire possible applicationange (i.e., the range of possible column lengths) of the particleixtures.Now that this packing quality order has been clearly estab-

ished, the question remains why the different particle mixturesie in this order. In other words why does one particle mix-ure perform better than the other? An obvious parameter toonsider if trying to answer this question is the width of theSD. Considering the span data (cf. Eq. (1)) given in Table 1,

wo classifications can be made depending on the mean parti-le size definition that is applied. Using the number-averagedean particle size, the particle mixtures can be classified from

mallest to broadest span as mixture C ∼= A < B ∼= D, whileor a volume-averaged mean particle diameter, the order is∼= B < A < D. The same order is obtained if considering the

ercentile ratio d90%/d10%. None of these two classificationss, however, consistent with the packing quality order comingut of Figs. 5 and 6. For example, mixture D is consistentlymong the PSDs having the largest width and mixture C isonsistently among the PSDs having the smallest width, andevertheless they yield kinetic performances that are nearlyndistinguishable. It is also striking to note that whereas mix-ure B displays the highest peak in the volume-based particleverage plot (Fig. 1b), and would hence be selected as the oneith the narrowest distribution, yields by far the worst kineticerformance.

It can hence be clearly concluded that there is no direct rela-ion between the span of the PSD and the packing quality. Thiss in good agreement to the earlier studies referred to in thentroduction, claiming that the PSD (if not exaggeratingly broadnd/or asymmetric) only has a minor influence on the columnfficiency.

Considering all other possible PSD characteristics (not allepresented here), the only plausible explanation for the columnerformance classification coming out of Fig. 6 is to be found inhe presence of fines in the particle mixtures. A visual inspectionf the number averaged PSD plot (Fig. 2a) shows that the besterforming particle mixture, i.e., mixture C, indeed contains themallest amount of particle fines, closely followed by mixture Das is the case for their kinetic performance). Mixture A alreadyontains significantly more fines, and mixture B contains theargest amount of fines. This order is in perfect agreement withhe order observed in Fig. 6. For a packed bed, the diameters ofhe through pores lie, depending on the packing quality of theolumn, between 0.2 and 0.4 of the size of the particle diameters23]. A particle can hence be reckoned as a fine when it fitsn the through pores of the packing. Apart from the packinguality, the through pore size is also determined by the PSD.efining a particle as a fine when it has a particle diameter

hat is less than 0.4 times the average particle diameter size,he number averaged PSD data of Fig. 2a shows that particleixture C only has 0.25% of fines, closely followed by mixture

(0.95% fines), mixture A (3% fines) and finally mixture B

5% of fines), being the same order of classification as foundn the reduced kinetic plot. Using an alternative definition forhe presence of fines, for example by defining a fine as 0.3 or

fi

sd

A 1161 (2007) 224–233 231

.2 times the average particle diameter, still the same order isbtained.

The explanation for the important effect the fines have onhe chromatographic performance is that, despite their rela-ively low amount, they are able to position themselves in thehrough pores of the packing. This can be expected to leado a partial blocking of the pores, which in turn results in aigher pressure drop over the column. Also the size distri-ution of the through pores will increase (not all pores willet blocked to the same extent), leading to a broader velocityistribution, which can be expected to lead to a larger Eddyispersion.

The presence of fines is not visible from the volumetric distri-ution (Fig. 2b). Due to the increased relative importance of thearger particle sizes, the fines that are present in a given particle

ixture are not visible anymore. It is hence important to alwaysonsult the particle number PSD plot to verify for the presencef fines.

. Conclusions

The traditionally employed mean particle size based columnharacteristics such as the reduced plate height, h, and the flowesistance, φ, are very sensitive to the errors and uncertaintieshat exist on the determination of the mean particle size. Asuch, different definitions of the mean particle diameter canead to different conclusions on the goodness of packing. Inhe present study, the problem is enlarged compared to morelassical particle mixtures because we purposely increased theidth of the employed PSDs. The kinetic plot representation

and especially its reduced variant) of E0 versus N circum-ents this problem and allows the unambiguous identificationf the best performing particle mixture as a function of theequired plate number. The identification can be said to be unam-iguous because the kinetic plots of E0 versus N or Nopt/Nllow to evaluate the packing quality in direct terms of sepa-ation speed without having to specify a mean particle size oro measure the molecular diffusivity of the employed compo-ents.

In the present study, the kinetic plot approach leads to theonclusion that, provided that all other packing conditions aredentical, the particle mixtures containing the smallest amountf fines yield the best overall chromatographic performance. Theroblem of particle fines can be expected to be especially impor-ant in columns packed with sub-2 �m particles, because these

ixtures tend to display a broader PSD, due to the productionnd particle selection processes used. The presence of the finess most important when the columns are pushed to their limit,.e., used in columns that are sufficiently long to yield a plateumber corresponding to their Nopt value. For high throughputpplications requiring only 10 000 plates or less, i.e., for appli-ations where the columns should ideally be used in the C-termominated range of their Van Deemter curve, the presence of

nes seems to matter less.

Given the importance of the presence of fines, PSDshould preferably be represented using a particle number basedistribution plot rather than by a volume or weight based distri-

Page 9

2 atogr.

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N

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32 J. Billen et al. / J. Chrom

ution. Although both histograms essentially contain the samenformation, the latter tends to camouflage the presence ofnes.

omenclature

, B, C constants in Eq. (2) (–)p,# mean particle diameter, based on the number averaged

PSD (m)p,V mean particle diameter, based on the volume averaged

PSD (m)x xth percentile, x% of the particles has a diameter

smaller then this valuem molecular diffusion coefficient in mobile zone (m2/s)0 t0-based separation impedance, Eq. (4) (–)0,min minimal E0 value (–)

reduced plate height, Eq. (2) (–)min minimal h value in a reduced plate height curve (–)

height equivalent of a theoretical plate (m)min minimal H value in a plate height curve (–)

′ phase retention factor (–)v0 u0 based column permeability (m2)

plate number (–)opt N value corresponding to the minimum of the kinetic

plot curve (/)P pressure drop (Pa)

0 dead time of a column (s)0 mean velocity of permeating, but non-retained solute

(m/s)opt u0 value corresponding to the Hmin value in a plate

height curve (m/s)

reek lettersexternal or through pore porosity (–)

0 u0-based flow resistance, Eq. (3) (–)dynamic viscosity (kg m−1 s−1)reduced velocity, Eq. (2) (–)

cknowledgements

J.B. is supported through a specialization grant from the Insti-uut voor Wetenschap en Technologie (IWT) of the Flandersegion (grant No. SB/1279/00). J.-L.V. and his group wish to

hank Rosanna Russo for the technical assistance. Thanks tohermo Fisher Scientific for providing the studied columns.

ppendix A

Demonstrating that the ratio Nopt/N is a quantity that onlyepends on the reduced parameters h, ν and φ can be done bytarting from the kinetic plot expression for N (Eq. (5)) and

eplacing H, u0 and Kv0 by their dimensionless counterparts:

= d2p

(�pmax

ηDm

) (1

hνφ

)(A.1)

[

[[

A 1161 (2007) 224–233

Knowing that the minimum of the E0 curve is obtained for= hmin and ν = νopt, Eq. (A.1) can also be used to write the Noptalue in terms of the dimensionless variables h, ν and φ:

opt = d2p

(�pmax

ηDm

) (1

hminνoptφ

)(A.2)

Subsequently dividing Nopt by N, it can be readily noted thathe obtained ratio only depends on the reduced plate height vari-bles h and ν, and is hence totally independent of the value or theefinition of the support size, dp, and also of the physicochemicalarameters η, �P and Dm:

Nopt

N= hν

hminνopt(A.3)

Transforming the right hand side of Eq. (A.3) back into abso-ute plate height curve parameters, it is found that the Nopt/N ratioan also be directly calculated, i.e., without first having to makehe transition to reduced plate heights:

Nopt

N= u0H

uoptHmin(A.4)

The latter also implies that Nopt/N can be calculated withoutrst having to define or measure a characteristic size. Since E0an also be written as E0 = h2φ, it follows from this fact and fromq. (A.3) that E0 and Nopt/N are both independent of the sizef the packing and only reflect the values of h, ν and φ. Chro-atographic systems that can be expected to yield the same φ

nd the same h versus ν curve should hence also yield coincid-ng curves on a plot of E0 versus Nopt/N. The advantage of theatter approach, however, is that h, ν and φ do not need to bealculated explicitly, so that no characteristic size needs to beefined.

eferences

[1] R.E. Majors, H.G. Barth, C.H. Lochmuller, Anal. Chem. 54 (1982)323.

[2] I. Halasz, M. Naefe, Anal. Chem. 44 (1972) 76.[3] C. Dewaele, M. Verzele, J. Chromatogr. 260 (1983) 13.[4] R. Endele, I. Halasz, K. Unger, J. Chromatogr. 99 (1974) 377.[5] R. Ohmacht, I. Halasz, Chromatographia 14 (1981) 155.[6] L.-I. Kulin, P. Flodin, T. Ellingsen, J. Ungelstad, J. Chromatogr. 514 (1990)

1.[7] W. Barber, A. Broske, T. Langlois, Presented at the 29th Interna-

tional Symposium on High Performance Liquid Phase Separationsand Related Techniques, Stockholm, 26–30 June, 2005 [posterP5:04].

[8] K. Nakanishi, S. Yamamoto, R. Matsuno, T. Kamikubo, Agric. Biol. Chem.42 (1978) 1943.

[9] N.W. Han, J. Bhakta, R.G. Carbonell, AIChE J. 31 (1985) 277.10] A.M. Athalye, S.J. Gibbs, E.N. Lightfoot, J. Chromatogr. 589 (1992)

71.11] U.D. Neue, HPLC Columns: Theory, Technology and Practice, Wiley-

VCH, New York, 1997.12] R.E. Majors, LC–GC N. Am. 23 (2005) 1248.13] G. Desmet, D. Clicq, P. Gzil, Anal. Chem. 77 (2005) 4058.

14] I. Halasz, R. Endele, J. Asshauer, J. Chromatogr. 112 (1975)

37.15] R. Endele, I. Halasz, K. Unger, J. Chromatogr. 99 (1974) 377.16] D. Guillarme, S. Heinisch, J.-L. Rocca, J. Chromatogr. A 1052 (2004)

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[

[[[

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17] J.J. van Deemter, F.J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 5 (1956)271.

18] J.H. Knox, J. Chromatogr. A 831 (1999) 3.19] J.H. Knox, J. Chromatogr. A 960 (2002) 7.20] M. Golay, Gas Chromatography, Butterworths, London, 1958, 195.

[

[

[

A 1161 (2007) 224–233 233

21] G. Desmet, D. Cabooter, P. Gzil, H. Verelst, D. Mangelings, Y. VanderHeyden, D. Clicq, J. Chromatogr. A 1130 (2006) 158.

22] D. Cabooter, S. Heinisch, J.L. Rocca, D. Clicq, G. Desmet, J. Chromatogr.A 1143 (2007) 121.

23] A.-M. Siouffi, J. Chromatogr. A 1126 (2006) 86.