Speed-resolution properties of columns packed with new 4.6μm Kinetex-C18 core–shell particles

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Journal of Chromatography A, 1280 (2013) 35– 50

Contents lists available at SciVerse ScienceDirect

Journal of Chromatography A

jou rn al h om epage: www.elsev ier .com/ locat e/chroma

peed-resolution properties of columns packed with new 4.6 �m Kinetex-C18

ore–shell particles

abrice Gritti, Georges Guiochon ∗

epartment of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA

r t i c l e i n f o

rticle history:eceived 12 November 2012eceived in revised form0 December 2012ccepted 3 January 2013vailable online 11 January 2013

eywords:

a b s t r a c t

The achievable separation speed and resolution of columns packed with the new 4.6 �m Kinetex par-ticles were characterized by their permeability and their plate heights, measured meticulously. Theirspecific permeabilities range between 1.81 and 1.95 × 10−10 cm2 and their external porosities between0.394 and 0.405. The efficiencies of the eluted peaks measured by numerical integration of the wholeband concentration provided minimum reduced plate heights for uracil (non-retained) and naphthalene(retained) between 1.3 and 1.5 (H = 6.0–6.9 �m) for two 4.6 mm × 150 mm replicate columns of Kinetexand between 1.6 and 2.0 (H = 7.4–9.2 �m) for two narrow-bore 2.1 mm × 150 mm replicate columns. The

ore–shell particlesolumn-to-column repeatabilityddy diffusionongitudinal diffusionolid–liquid mass transfer resistanceinetex-C

most efficient 4.6 mm × 150 mm Kinetex column shows a deviation of one reduced plate height h unitwith respect to the infinite diameter column (no wall, no inlet, no outlet endfitting) at high speed. Eventu-ally, the separation speed and the resolution of columns packed with 4.6 �m core–shell Kinetex particlesare better or equivalent to those of columns packed with 2.5 �m fully porous particles for hold-up timeslarger than only 10 s. These core–shell materials are virtually equivalent to the second generation of silica

the a
18 monolithic columns with
. Introduction

Since their introduction in the column market in the late 2000s,nterest for chromatographic columns packed with sub-3 �more–shell particles [1] has constantly increased [2]. Most indus-ry and academic analysts used and studied these new columnsn order to improve current separation methods that were devel-ped with columns packed of classical fully porous particles [3–5]nd to unravel their underlying mass transfer mechanism [2,6–18].ractitioners benefit from their exceptionally low minimum plateeight [4], often around Hmin = 1.4 × 2.7 = 3.8 �m (therefore, equiv-lent to that of sub-2 �m fully porous particles) and their relativelyigh specific permeability, which ranges from an average of0 = 4.6 × 10−11 cm2 for 2.7 �m Poroshell [19] to 5.8 × 10−11 cm2

or 2.6 �m Kinetex [20], and to 6.4 × 10−11 cm2 for 2.7 �m Halo21]. For the sake of comparison, the specific permeability ofolumns packed with 1.7 �m, 2.5 �m, and 3.5 �m fully porousEH particles are about 2.5, 4.0, and 7.8 × 10−11 cm2, respectively.

n summary, whereas classical 1.7, 2.5, and 3.5 �m fully porous
articles generate minimum column impedance E = ((�tH2
min)/k0)

round 3000–4000, sub-3 �m core–shell particles provide at least twice smaller column impedance at circa 1500. Only, the second

∗ Corresponding author. Tel.: +1 865 974 0733; fax: +1 865 974 2667.E-mail addresses: [email protected], [email protected]

G. Guiochon).

021-9673/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.chroma.2013.01.022

dvantage of operating well at pressure drops larger than 200 bar.© 2013 Elsevier B.V. All rights reserved.

generation of monolithic columns (Chromolith) can generate sim-ilar column impedance, with total porosity �t = 0.83, Hmin = 6.5 �m,and k0 = 2 ×10−10 cm2 but they still cannot be operated above a200 bar pressure drop [22]. In contrast, columns packed with sub-3 �m core–shell are claimed to be safe up to 600 bar back pressure.

Columns packed with core–shell particles have also some dis-advantages. While they can be operated on standard 400 bar HPLCsystems, the widths of the peaks eluted from them are oftencomparable to or even smaller than those eluted from the solechromatograph, in the absence of a chromatographic column [9].A typical system equipped with a 2.5 �L detection cell and 254 �mI.D. tubings has an extracolumn volume about 45 �L contributingto the peak variance by up to 50 �L2 [9]. A 4.6 mm × 100 mm col-umn packed with 2.7 �m core–shell particles elutes a peak with aretention factor of 1 and a variance of only 125 �L2, barely two and ahalf times larger than the system variance. The easy solution wouldbe to reduce the detection cell volume and the diameter of theconnecting tubes to 1 �L and 127 �m, respectively, which woulddecrease fivefold the system variance, to only 10 �L2 [9,23,24]. Inpractice, most application methods are currently based on usingcolumns packed with classical 5 �m fully porous particles, allow-ing the use of moderate back pressures compatible with standard400 bar HPLC systems. Decreasing the particle size to only 2.7 �m
usually requires a rather expensive system modification or thepurchase of a more costly very high pressure HPLC system. Further-more, the use of higher operating pressures can affect the resolutionof closely related compounds, if pressure affects the relative
dx.doi.org/10.1016/j.chroma.2013.01.022

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dx.doi.org/10.1016/j.chroma.2013.01.022

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6 F. Gritti, G. Guiochon / J. Ch

dsorption behavior of the analytes. To address this problem, Phen-menex (Torrance, CA) recently launched a new type of columnacked with 4.6 �m core–shell particles. These particles share theery same structural characteristics as the former 2.6 �m Kine-ex particles. Instead of a core of 1.9 �m non-porous solid silica,hey are made of a 3.3 �m non-porous silica core. The surround-ng porous shell is built layer by layer to an overall thickness ofbout 0.65 �m. Thus, the ratio of the core diameter to the particleiameter remains unchanged at about � = 0.73.

In this work, the mass transfer kinetics of 2.1 and 4.6 × 150 mmolumns packed with the new 4.6 �m Kinetex-C18 core–shell par-icles was investigated in detail and is compared to that of alassical 4.6 mm × 150 mm column packed with 5.0 �m Luna-C18ully porous particles. The eluent used was a mixture of ace-onitrile and water (65/35, v/v) at room temperature. Diffusionoefficients of uracil and naphthalene were carefully measuredsing the peak parking method and a calibrated column packedith non-porous silica particles. The exact height equivalent to a

heoretical plate (HETP) was measured with the moment methodreviously reported [25,26]. The longitudinal diffusion HETP B termas independently measured with the peak parking (PP) method

27]. The solid–liquid mass transfer resistance coefficient, Cp, waseasured by combining the PP measurements and selecting theost physically meaningful model of effective diffusion in columns

acked with core–shell particles [28,29]. The experimental overallddy diffusion A term was obtained from the subtraction method30]. The deviation of the column performance from that of the idealhaving no wall, no inlet, and no outlet endfittings) infinite diameterolumn was measured with a second subtraction method based onhe knowledge of the trans-channel and short-range inter-channelddy diffusion HETP term estimated in the bulk random structuref columns packed with solid, nonporous particles [31,32]. Finally,he speed/resolution performance of these new columns are com-ared to those of columns packed with classical 5.0, 3.5, and 2.5 �mully porous particles by constructing their respective kinetic Poppelots [33].

. Theory

.1. Definitions

Shell particles are made of a solid, non-porous, spherical coreurrounded by a concentric porous shell. The core-to-particle diam-ter ratio, �, ranges between 0.60 and 0.75 for commerciallyvailable shell particles. For instance, the 2.6 �m Kinetex particlesave a 1.9 �m non-porous core, so � = 0.73. The external porosityf the packed bed is �e, Dm is the diffusion coefficient of the analyten the bulk mobile phase. The apparent analyte diffusivity throughhe porous shells is Dp = �Dm, with � the dimensionless ratio ofhe sample diffusivity through the porous region of the particleso its bulk diffusion coefficient. The effective diffusion coefficientlong the heterogeneous packed bed (cores, porous shell, externalluent) is Deff.

We define k1 as the zone retention factor extended to the super-cially porous particle. It is written [28]:

1 = 1 − �e

�e[�p + (1 − �p)Ka][1 − �3] (1)

he zone retention factor, k1, refers to the elution time of theompound with respect to that spent by the same analyte in
he interstitial column volume (�e�r2
c ). For the sake of compar-son, the conventional retention factor, k′, refers to the samelution time but with respect to the total accessible pore volume�t�r2

c ).

ogr. A 1280 (2013) 35– 50

The reduced interstitial linear velocity, �, is written [34]:

� = udp

Dm(2)

where u in the average interstitial linear velocity along the columngiven by:

u = Fv

�e�r2c

(3)

where Fv is the flow rate and rc is the inner column radius.

2.2. Reduced HETP equation

In the absence of significant frictional heating effects (which isthe general case with core–shell particles, even at high heat frictionpower [7,8]), the overall reduced plate height is the sum of the lon-gitudinal diffusion term, the trans-particle mass transfer resistanceterm, and the eddy diffusion term or:

h = B

�+ A(�) + Cp� (4)

Next, we provide explicit physico-chemical descriptions of each ofthese three HETP terms.

2.2.1. The longitudinal diffusion termThe longitudinal diffusion term is derived from the effective

diffusion coefficient of the analyte in a heterogeneous packingmaterial [35–40]. It includes the contributions of the diffusionprocesses taking place in the external and in the internal eluentvolumes and on the surface of the stationary phase,

For columns packed with core–shell particles, an empiricalmodel of effective diffusion is provided by the time-averaged modeladjusted for the obstruction caused by the presence of the non-porous core [29]. It is written [39]:

Deff = 1�e(1 + k1)

[�e�e + (1 − �e)

1 − �3

1 + (�3/2)�

]Dm (5)

A physical relevant model of effective diffusion in randompacked beds is given by the composite Garnett–Torquato model[35,38]. The Garnett model accounts for the effective diffusion inthe core–shell particles while the Torquato model describes theeffective diffusion of packed spheres immersed in a homogeneousmedium. Accordingly, it is written [39,40]:

Deff = 1�e(1 + k1)

[1 + 2(1 − �e) − 2�e�2ˇ2

1 − (1 − �e) − 2�e�2ˇ2

]Dm (6)

with

ˇ = ((1 − �3)/(1 + (�3/2)))� − 1((1 − �3)/(1 + (�3/2)))� + 2

(7)

and �2 is an adjustable parameter that is estimated from theexperimental external obstruction factor for � = 0, � = 0, and k1 = 0

(non-porous particles). From Eq. (6), the expression of �e = DeffDm

isthen given by:

�e = 2(1 − (�2/2))3 − �e(1 + �2)

(8)

According to total pore blocking measurements (in which themesopores were filled with liquid n-nonane), the obstruction fac-tor of a 4.6 mm × 150 mm column packed with 5 �m fully porousGemini-C18 particles is 0.570 for an external porosity of 0.364[41]. Accordingly, the value of �2 given by Eq. (8) and used for
columns packed with fully porous particles will be 0.627. Similarly,the obstruction factor of a 4.6 mm × 100 mm column packed with3.3 �m non-porous silica cores (these cores have the same particlesize distribution as that of the 5 �m Kinetex particles) were found

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qual to 0.633 for an external porosity of 0.415 [42]. Therefore, thealue of �2 given by Eq. (8) and used for columns packed with the

�m superficially porous particles will be 0.493.The reduced B coefficient in Eq. (4) is then written [39]:

= 2(1 + k1)Deff

Dm(9)

In order to validate the best model of effective diffusion, theffective diffusivity of a non-retained compound was estimatedrom independent experimental data. This estimate was obtainedrom NMR measurements of the effective diffusion coefficient ofyclohexane (small molecule) in various porous silica with dif-erent average pore sizes [43]. For an average pore size of 85 Acorresponding to that of 100 A particles derivatized with octadecylhains), the product �pF(m) � 0.48 and, so,

= 0.48�p (10)

here �p is the volume fraction of the particle (Luna) or of the shellarticle (Kinetex) accessible to the non-retained small moleculeuracil).

.2.2. The solid–liquid mass transfer resistance termThe general expression of the solid–liquid mass transfer resis-

ance coefficient (Cp) has been derived for core–shell particles. Thisoefficient is given by [44–47]:

p = 130

�e

1 − �e

(k1

1 + k1

)2[

1 + 2� + 3�2 − �3 − 5�4

(1 + � + �2)2

]1�

(11)

n this equation, � was estimated from the model of effective diffu-ion that gives the best agreement between the calculated (Eq. (5)r (6)) and the empirically estimated (Eq. (10)) values of �. Notehat the finite rate at which the molecules diffuse through the thintagnant layer of eluent surrounding the particles before they enternto the particle volume or exit to the moving inter-particle eluents infinitely fast. The largest thickness of the stagnant layer can bessessed from the surface roughness, ı, of the core–shell particle.or current smooth core–shell particles, it is about 0.03 �m, a dis-ance negligible with respect to the thickness of the porous shell (�.5 �m) or that of the particles (5 �m) [48].

.2.3. The eddy diffusion termThe eddy diffusion term results from all sources of sample

ispersion in the interparticle column volume. It includes theontribution of the homogeneous random packing or the analyteispersion along the hypothetical infinite diameter column (noall, no inlet and outlet endfitting) [31,32], e.g., distances ran-

ing from the interparticle distance (trans-channel velocity biases,TS) to a few particle diameters (short-range inter-channel veloc-ty biases, hIC). It also includes the contribution of the so-calledong-range trans-column velocity biases (hTC) induced by the pres-nce of the solid cylindrical wall surrounding the bed and thenlet/outlet (distribution/collection) endfittings. The presence of allhese boundaries induces physical constraints during the forma-ion of the chromatographic bed (slurry process) preventing thetreamlines of the eluent during chromatographic analysis fromeing parallel to the column axis. This leads to trans-column veloc-

ty biases which take place from the bulk center to the region inhe vicinity of the column wall or column ends. In practice, fortandard HPLC analytical columns, because the transverse disper-ion coefficient cannot reach its asymptotic value, which shoulde expected for infinitely long column [15,49]. So far, there is no
vailable theoretical model for standard 2.1–4.6 mm I.D. chromato-raphic columns which could predict accurately this long-rangeddy diffusion term during the transient dispersion regime beforehe asymptotic regime is reached [32]. It can only be measured by
ogr. A 1280 (2013) 35– 50 37

a second subtraction method and is reported as such. Eventually,the overall eddy dispersion HETP term, A(�), is simply written:

A(�) = hTS + hIC + hTC (12)

= 1(1/21) + (1/ω1�)

+ 1(1/22) + (B/(2�eω2�))

+ hTC (13)

where 1, ω1, 2, and ω2 are the eddy dispersion parameters thatpredict most accurately the sample dispersion in the homogeneous,random, bulk region of the bed packed with impermeable solidspheres under asymptotic conditions [31] according to the cou-pling theory of eddy diffusion of Giddings [50]. These parameterswere obtained as a function of the bed porosity (external porosity),for monosized and non-porous particles (B = 2�e) and for differentpacking protocols [31]. Because the effect of the particle size dis-tribution on the dispersion of the analyte in the bulk region of apacking is small [51] and the bed porosity is always in the porosityrange 0.35 < �e < 0.41, accurate estimates of these four parametersis provided by the so-called S × 2 packings in which each particlewas randomly placed in n equal cubic cells that define the simula-tion box volume. For more details about the computer-generatedbeds, the reader is referred to reference [31].

3. Experimental

3.1. Chemicals

The mobile phase was a mixture of acetonitrile and water(65/35, v/v). Small volumes of tetrahydrofuran (THF) anddichloromethane were used in the pycnometry experiments. Allthese solvents were HPLC grade from Fisher Scientific (Fair Lawn,NJ, USA). Acetonitrile was filtered before use on a surfactant-freecellulose acetate filter membrane, 0.2 �m pore size (Suwannee,GA, USA). The eleven polystyrene standards (MW = 590, 1100,3680, 6400, 13,200, 31,600, 90,000, 171,000, 560,900, 900,000, and1,870,000) were purchased from Phenomenex (Torrance, CA, USA).Uracil, toluene, and naphthalene were purchased from Fisher Sci-entific, with a minimum purity of 99%.

3.2. Apparatus

All the measurements were performed on a 1290 Infinity HPLCsystem (Agilent Technologies, Waldbroen, Germany) liquid chro-matograph. The system includes a 1290 Infinity Binary Pump withSolvent Selection Valves and a programmable auto-sampler. Theinjection volume is drawn into one end of the 20 �L injection loop.The instrument includes a two-compartment oven and a multi-diode array UV-VIS detection system. The system is controlled bythe Chemstation software. The sample trajectory in the equipmentinvolves the successive passage of its band through the series of:

• A 20 �L injection loop attached to the injection needle. The designof the First Injected Last Outlet (FILO) injection system is such thatthe entire volume of sample drawn into the loop is injected intothe column. This ensures an excellent injection repeatability.

• A small volume needle seat capillary (115 �m I.D., 100 mm long),�1.0 �L, located between the injection needle and the injectionvalve. The total volume of the grooves and connection ports inthe valve is around 1.2 �L.

• Two 130 �m × 250 mm long Viper capillary tubes offered by the
manufacturer (Dionex, Germering, Germany), placed one before,the second after the column. Each tube has a volume of around3.3 �L.
• A small volume detector cell, V(�) = 0.6 �L, 10 mm path.

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The total dwell volume of this instrument was measured fromtep gradient experiments made by the manufacturer at 170 �L. Itncludes the volume contributions of the connecting tubes (greenubings, I.D. = 254 �m, � 75 �L), of the jet weaver V35 mixer (35 �L),f the loop (40 �L), and of the syringe (20 �L). No solvent heatxchanger was used in the experiment. The extra-column volumes close to 10 �L and generates an extra-column peak variance thataries between 1 and 12.5 �L2 for small molecules when the flowate is increased from 0.01 to 2.25 mL/min (injection volume <

�L).

.3. Columns

A 4.6 mm × 100 mm column packed with 3.3 �m bare fused coreilica cores was kindly given by Phenomenex (Torrance, CA, USA).he external obstruction factor and porosity of this column aree = 0.633 and �e = 0.415, respectively [42]. �e was measured by theeak parking method at a flow rate of 0.4 mL/min, using thiourea ashe reference probe. �e was measured from five consecutive injec-ions of thiourea corrected from the extra-column contributionsing a zero dead volume (ZDV) union connector. Thiourea has a dif-usion coefficient known independently at 298.15 K, in pure waterDm = 1.326 × 10−5 cm2/s [52,53]). This column was also used to

easure the diffusion coefficient of uracil (Dm = 1.11 × 10−5 cm2/s)nd naphthalene (Dm = 1.59 × 10−5 cm2/s) at 298.15 K in a mixturef acetonitrile and water (65/35, v/v).

A set of four columns generously offered by the manufac-urer (Phenomenex, Torrance, CA, USA) was used. They were allacked with the same batch of 4.6 �m Kinetex-C18 shell parti-les. Two of them are 4.6 mm × 150 mm columns, the two othersre narrow-bore 2.1 mm × 150 mm columns. For the sake of com-arison, a standard 4.6 mm × 150 mm column packed with 5.0 �muna-C18(2) provided by the same manufacturer was also tested.able 1 lists the physico-chemical properties of these five columns,ccording to the manufacturer data and to data acquired in ouraboratory.

The construction of the kinetic plots of the 2.5 and 3.5 �mully porous BEH-C18 particles was made using data provided by.6 mm × 100 mm and a 4.6 mm × 150 mm columns generouslyffered by the manufacturer (Waters, Milford, MA). The physico-hemical properties of these two columns can be found in [54,55].he kinetic plot of the silica monolith column of the second gen-ration was built from the HETP and permeability data previouslyeasured and reported for a 4.6 mm × 100 mm research sample

iven to us by Merck Millipore (Darmstadt, Germany) [22].

.4. ISEC experiments and column porosities

ISEC measurements were made on all six columns. Neat THF wassed as the eluent. Eleven polystyrene standards with moleculareights between 500 and 2 millions Dalton were used as probeolecules. They cover a wide range of molecular sizes, between

0 and 950 A. The average mesopore size before C18 derivatiza-ion of the fully porous (Luna) and of the superficially porousKinetex) particles was 100 A according to the manufacturer. After6/C18-derivatization for Luna-C18(2) [56] and C18-derivatizationor Kinetex, followed by trimethylsilane (TMS) endcapping, theverage size decreased to about 85 A [57–59]. The flow rates wereet at 0.48 and 0.10 mL/min for the 4.6 and 2.1 mm I.D. columns,espectively, and the sample sizes at 2 and 0.5 �L, respectively.he detection wavelength was set at 254 nm with a bandwidth of

nm. The external porosities (�e) were derived by extrapolation to
zero molecular weight of the exclusion branches of the ISEC plotsf the elution volumes of the polymers versus their hydrodynamicolecular radius. The accuracy of the ISEC experimental protocol
ies within 1%.

ogr. A 1280 (2013) 35– 50

The total porosity (�t) was measured according to three dif-ferent methods. First, the elution time of non-retained uracil(MW = 112 g/mol) was measured, using a mixture of acetonitrileand water (65/35, v/v) as the mobile phase. Second, the elu-tion time of toluene in pure tetrahydrofuran was also measured.Third, pycnometry was performed at 295.2 K using pure THFand dichloromethane, with respective densities of �THF = 0.886and �CH2Cl2 = 1.322 g/cm3 at this temperature. The hold-up vol-ume, V0, is then measured from the total weights of the column,mTHF and mCH2Cl2 , immediately they were filled with THF anddichloromethane, respectively:

V0 = mCH2Cl2 − mTHF

�CH2Cl2 − �THF(14)

The internal porosity (�p) was derived from:

�p = �t − �e

(1 − �e)(1 − �3)(15)

All the results for the total, the external, and the internal porosi-ties are listed in Table 1.

3.5. Peak parking (PP) experiments

The PP method was pioneered by Knox, first in gas [60], thenin liquid [27] chromatography. This method was recently used tomeasure the internal obstruction factor of columns packed withporous silica-C18 particles [61] and the bulk diffusion coefficientsof various solutes in different liquid mobile phases [62,63].

The bulk molecular diffusivities, Dm, of uracil and naphthalenewere measured with the PP method in a mixture of acetonitrile andwater (65/35, v/v) at several temperatures according to:

Dm(T)Dm(298.15)

= T

298.15�(298.15)

�(T)(16)

The eluent viscosity is a function of the temperature provided inreference [64]. The relative error made on the true value of the bulkdiffusion coefficient is smaller than 4% [11,20,30]. It is mostly dueto the errors made in the measurements of the external obstructionfactor of the column packed with non-porous particles and of theeluent viscosities.

The measurements of the reduced longitudinal coefficients,B = 2(1 + k1)(Deff/Dm), of uracil and naphthalene were carried out forthe four Kinetex and the Luna columns. All details on these mea-surements are given in [29]. The B coefficient is obtained from theexperiments according to [20]:

B = 1Dm

�2PP

tp

L

�1,PP − �1,PP,ex

Fv,PP

�e�r2c

(17)

where �2PP/ tp is the best slope of the plot of the peak variance

(�2PP) versus the parking time (tp). This variance is measured by the

numerical integration method, Fv,PP is the applied flow rate deliv-ered by the 1290 Infinity system, and �1,PP − �1,PP,ex is the firstmoment corrected from the extra-column contribution also mea-sured by the numerical integration method. rc is the column innerradius and L is its length.

The ratio of the shell to the bulk diffusivities, �, is obtained bycombining experimental data (the experimental B value given byEq. (17)) and a model of effective diffusion along the column (Deffis theoretically given by either Eq. (5) or Eq. (6)). The relationshipbetween B and Deff is given by Eq. (9). Therefore, � is given by the
unique solution of the following equations in � or ˇ:
2�e

[�e�e + (1 − �e)

1 − �3

1 + (�3/2)�

]= B (18)

F. Gritti, G. Guiochon / J. Chromatogr. A 1280 (2013) 35– 50 39

Table 1Physico-chemical property of the five columns packed with 4.6 �m Kinetex-C18 shell particles (batch number 5701-10) and 5.0 �m Luna-C18(2) particles.

Silica particle Column’s serialnumber

Column’s dimensionI.D. (mm) × length(mm)

Totala

porosity(εt)

Externalporosity (εe)

Particle/shellporosity (εp)

Permeability(cm2) (k0)

Kozeny–Carmanconstant (Kc)

4.6 �m Kinetex 632901-13 4.6 × 150 0.548 0.405 0.382 1.95 × 10−10 203632901-20 4.6 × 150 0.549 0.406 0.381 1.95 × 10−10 20641573 2.1 × 150 0.532 0.394 0.360 1.81 × 10−10 19541575 2.1 × 150 0.540 0.401 0.368 1.86 × 10−10 205

5.0 �m Luna 233512 4.6 × 150 0.628 0.363 0.416 1.84 × 10−10 161

a Measured from the elution of toluene in THF.

Table 2Temperatures (T), diffusion coefficients (Dm), zone retention factor (k1), effective diffusion coefficients (Deff), reduced longitudinal diffusion coefficients (B), ratio of the shelldiffusivity to the bulk diffusion (�), and solid-liquid mass transfer coefficient (Cp) measured for the non retained compound uracil.


Column’s dimension I.D.(mm) × length (mm)

T (K) Dm (cm2/s) k1 Deff (cm2/s) B � Cp

4.6 �m Kinetex-C18 632901-13 4.6 × 150 296.0 1.11 × 10−5 0.20 7.20 × 10−6 1.56 0.14 0.0021632901-20 4.6 × 150 296.5 1.13 × 10−5 0.20 7.20 × 10−6 1.54 0.13 0.002241573 2.1 × 150 297.1 1.14 × 10−5 0.21 7.06 × 10−6 1.49 0.11 0.0026

296.8

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f

fs

�

cluddrsa

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r00a1fltc

a

TS

41575 2.1 × 150

5.0 �m Luna-C18(2) 233512 4.6 × 150

or the time-averaged model of effective diffusion, or

2�e

[1 + 2(1 − �e) − 2�e�2ˇ2

1 − (1 − �e) − 2�e�2ˇ2

]= B (19)

or the Garnett–Torquato model of effective diffusion where theolution � is given by reversing Eq. (7), e.g.,

= 1 + (�3/2)1 − �3

1 + 2ˇ

1 − ˇ(20)

Finally, the reduced solid–liquid mass transfer resistance coeffi-ient Cp is provided by Eq. (11). Tables 2 (uracil) and 3 (naphthalene)ist the complete set of values of the temperatures T, the bulk molec-lar diffusivities Dm, the zone retention factors k1, the effectiveiffusion coefficients Deff, the best ratio � of the shell to the bulkiffusion, the reduced longitudinal diffusion B coefficients, and theeduced solid–liquid mass transfer resistance Cp coefficients mea-ured during the complete series of experiments for all five Kinetexnd Luna columns.

.6. HETP measurements

For both uracil and naphthalene, two different sequences of flowates were applied to the 2.1 mm I.D. (0.010, 0.021, 0.031, 0.042,.063, 0.083, 0.104, 0.135, 0.167, 0.208, 0.261, 0.313, 0.365, 0.417,.469, 0.573, 0.677, 0.782, 0.886, 0.990, 1.094, and 1.198 mL/min)nd the 4.6 mm I.D. (0.05, 0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.65, 0.80,.0, 1.25, 1.50, 1.75, 2.00, and 2.25 mL/min) columns. The maximumow rate was chosen so that the frictional power dissipated through
he column did not exceed 2 W/m. Under such flow conditions, theolumns can be considered as isothermal.
Depending on the flow rate, the sampling frequency wasdjusted between 1.25 and 160 Hz in order to record at least 150

able 3ame as in Table 2, except the compound, naphthalene.



T (K)

4.6 �m Kinetex-C18 632901-13 4.6 × 150 296.0632901-20 4.6 × 150 296.541573 2.1 × 150 297.141575 2.1 × 150 296.8

5.0 �m Luna-C18(2) 233512 4.6 × 150 296.9

1.14 × 10−5 0.20 6.97 × 10−6 1.48 0.11 0.0027

1.13 × 10−5 0.40 7.11 × 10−6 1.75 0.11 0.0146

data points for the whole band profile of uracil, the thinnest andleast retained compound. Samples of 0.5 and 2 �L of each solution(concentration < 0.5 g/L) were injected into the four columns andthe chromatograms recorded at a wavelength of 257 nm. A con-stant UV bandwidth of 4 nm was selected. The temperature wasset by the laboratory air-conditioning system at 296.3 ± 1 K. Foreach HETP measurement, the average temperature was recordedduring the programmed sequence of flow rates. They are listed inTables 2 (uracil) and 3 (naphthalene).

The HETP data were all measured using two differentapproaches. The first method is based on the measurement of thehalf-height time peak width (ω1/2) and the elution time taken atthe peak apex (tR). The reduced plate height h1 is then calculatedaccording to:

h1 = L

5.545dp

ω21/2

t2R

(21)

The precision of this method is excellent but its accuracy is poorand, unless the peak shape is rigorously Gaussian, it can lead tovery large relative errors (up to 50%) with respect to the true columnreduced HETP for peaks showing obvious peak tailing due to cracksand heterogeneities of the column packing [20].

The second approach used the numerical integration method.The results were systematically corrected for the dispersion dueto extra-column band broadening in absence of chromatographiccolumns (replacement with a ZDV union connector). All the detailsfor the measurement of the HETP data and their accuracy are given
in [25,26,65]. The first and second central moments of the elutedband profiles were measured from the following discreet sums. Themoments are calculated based on the decomposition of the peakarea into a series of elementary trapezes, which is more accurate
Dm (cm2/s) k1 Deff (cm2/s) B � Cp

1.51 × 10−5 2.91 7.56 × 10−6 3.92 1.26 0.0046 1.53 × 10−5 2.90 7.60 × 10−6 3.88 1.24 0.0047 1.55 × 10−5 3.00 7.33 × 10−6 3.78 1.13 0.0049 1.54 × 10−5 2.97 7.41 × 10−6 3.82 1.18 0.0048

1.54 × 10−5 6.39 6.60 × 10−6 6.32 1.24 0.0115

4 romatogr. A 1280 (2013) 35– 50

tiw

�

�

tttc

t

a

t

wasN6

slflipsaacmt

h

wac

rpr

3e

irrTtm�otta(

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

LunaKine tex#7 5Kine tex#7 3Kine tex#2 0

To

tal p

oro

sit

y ε

t

Kine tex#1 3

Pycnometry (THF/CH2Cl

2)

Toluene marker (THF )Uracil mark er (CH

3CN/H

2O, 65 /35, v/v)

0.0

0.1

0.2

0.3

0.4

0.5

LunaKinetex#7 5Kine tex#7 3Kine tex#2 0Kine tex#1 3

Pycnometr y (THF /CH2Cl

2)

Toluene marker (THF )Uracil mark er (CH

3CN/H

2O, 65 /35 , v/v)

Inte

rnal p

oro

sit

y ε

p

Fig. 1. Total (top) and internal (bottom) porosities of the columns packed with4.6 �m Kinetex core–shell particles and the 5.0 �m fully porous particles. The inter-


han using the rectangle approach. However, if the number of pointss large enough (>90), both approaches give the very same results

ithin 0.001%. These moments are written:

1 =∑i=N−1

i=1 (Ci + Ci+1)((ti + ti+1)/2)∑i=N−1i=1 Ci + Ci+1

(22)

′2 =

∑i=N−1i=1 (Ci + Ci+1)(((ti + ti+1)/2) − �1)2

∑i=N−1i=1 Ci + Ci+1

(23)

The first (t1, left cut) and the last (tN, right cut) elution times inhese sums were unambiguously determined from the two elutionimes recorded when the sample concentration first increased (tleft)hen decreased (tright) to 1.5% of the peak height after linear baselineorrection:

1 = tR − ˛(tR − tleft) (24)

nd

N = tR + ˛(tR − tright) (25)

here ˛ was fixed at 2.5 so that the sample concentration below therbitrary 0.3% of the maximum recorded sample concentration wasystematically discarded for all compounds and for all flow rates.ote that the signal-to-noise levels in these experiments exceeds500 (4.6 mm I.D. Luna-C18 column, naphthalene, 2.25 mL/min).

This method has the tremendous advantage of properly andystematically adjusting the integration window ([t1, tN]) regard-ess of the degrees of the peak fronting and peak tailing that mayuctuate in various ways depending on the nature of the sample

njected and on the flow rate applied. It is highly sensitive to anyacking heterogeneity, which would not be perceptible by the clas-ical half-height peak width method. The second method takes intoccounts the very first and very last eluted sample molecules, elutedt concentrations larger than only 0.3% of the maximum elutedoncentration. In contrast, the first method ignores the early for-ation of the adsorption front and the late vanishing boundary of

he desorption rear under strictly linear conditions.Accordingly, the reduced plate height h2 is given by:

2 = L

dp

�′2 − �′

2,ex

(�1 − �1,ex)2(26)

here L is the column length, dp the mean particle size, and �1,exnd �′

2,ex are the first and second central moments of the extra-olumn band profiles measured by the same integration method.

We emphasize that this method provides an excellent accu-acy, although its precision is necessarily smaller than that of therevious method. This accuracy is around 10% and 4% for the non-etained and the retained compounds, respectively.

.7. Accuracy and precision of the measurements of thexperimental HETP terms

The accuracies of the B, C, and A terms obtained from the exper-mental protocol used in this work were estimated in a previouseport [20]. The largest relative error made on the true value of theeduced longitudinal diffusion coefficient, B, is smaller than ±9%.he relative accuracy of the experimental reduced C coefficient isypically smaller than ±20%; it is due essentially to the model error

ade on � and to the uncertainty on the core-to-particle size ratio. Finally, the accuracy of the reduced A term decreases continu-usly with increasing reduced velocity because the importance of
he longitudinal diffusion term decreases as B/�. Typically for � > 5,he relative error is smaller than 10%. The precisions of the B, C,nd A terms provided by this protocol are ±2.5%, ±3%, and ±3.5%� > 3), respectively, given the level of precision of the measurement
nal porosity of the core–shell particles exclude the volume of the solid non-porous3.3 �m cores. The pairs of Kinetex columns #13/#20 and #73/#75 are replicatecolumns packed with the same batch of core–shell particles.

of the first (±0.1%) and second central (±1%) moments and of thelaboratory temperature (±0.2%).

4. Results and discussion

4.1. Porosities of the columns

Three different methods were used to estimate the true col-umn hold-up volume and the total porosity. Two of them used themarkers toluene (eluted from pure THF) and uracil (eluted from thesame eluent composition as that used in this study, an acetonitrile-water mixture, 65/35, v/v). The third one is based on pycnometricmeasurements using THF and dichloromethane as the two pureliquids [66]. The results are summarized in Fig. 1. It is remark-able that the column porosities provided by the elution volumeof uracil corrected from the extra-column volume differ signifi-cantly from those provided by the retention volume of tolueneand by the pycnometric measurements. These results suggest that
uracil does not come in contact with the C18-bonded surface areaand then is not retained. In fact, it seems to be excluded from asignificant fraction of the internal volume. Most likely, uracil isexcluded from the region close to the hydrophobic layer, where a


0 200 400 600 800 1000

0.4

0.5A

ccessib

le c

olu

mn

po

rosit

y

Polystyrene hydrodymamic diameter DH [Å]

0 200 400 600 800 1000

0.4

0.5

4.6 mm x 100 mm3.3 m non-porous cores

e=0.415

4.6 mm x 150 mm4.6 m Kinetex-C

18 #13

e=0.405

0 200 400 600 800 1000

0.4

0.5

Accessib

le c

olu

mn

po

rosit

y


0 200 400 600 800 1000

0.4

0.5


e=0.415


18 #73

e=0.394

0 200 400 600 800 10000.3

0.4

0.5

0.6

Accessib

le c

olu

mn

po

rosit

y


0 200 400 600 800 10000.3

0.4

0.5

0.6


e=0.415

4.6 mm x 150 mm5.0 m Luna-C

18(2)

e=0.363

0 200 400 600 800 1000

0.4

0.5

Accessib

le c

olu

mn

po

rosit

y


0 200 400 600 800 1000

0.4

0.5


e=0.415


18 #20

e=0.406

0 200 400 600 800 1000

0.4

0.5

Accessib

le c

olu

mn

po

rosit

y


0 200 400 600 800 1000

0.4

0.5


e=0.415


18 #75

e=0.401

F compw

moptmw(t0#t

ura

ig. 2. ISEC results reported for the five columns used in this work. For the sake of

ith the 3.3 �m non-porous silica cores.

ultilayer of acetonitrile forms due to the preferential adsorptionf acetonitrile with respect to water [59,67–73]. This exclusion isrobably explained by the extremely small solubility of uracil inhe acetonitrile-rich multi-layer of the eluent stagnant inside the

esopores. The experimental data shown in Fig. 1 are consistentith this interpretation. So, the true values of the total porosity

�t) of the columns are better provided by either pycnometry orhe elution volume of toluene in THF. They were equal to 0.548,.549, 0.532, and 0.540 for the Kinetex columns #13, #20, #73, and75, respectively. The total porosity of the Luna column was found

o be larger at 0.628, as expected.
In order to measure the external porosity (�e) of the five columns
sed, ISEC experiments were performed for each column. All theesults are shown in Fig. 2. The external porosities were measureds 0.405, 0.406, 0.394, and 0.401 for the Kinetex columns #13,

arison, the black symbols represents the ISEC data obtained for the column packed

#20, #73, and #75, respectively. It is significantly smaller for theLuna column at 0.363, in excellent agreement with all previousmeasurements made on columns packed with fully porous [74,75]and core–shell particles [2,6,10,19–21]. The external porosities ofthe columns packed with fully porous particles are in the range0.35–0.37 whereas those of columns packed with core–shell parti-cles lies typically within 0.39–0.41.

The internal porosity (�p) of the porous stationary phase wasderived from Eq. (15) (with � = 0.717 for the Kinetex particles, � = 0for the Luna particles). The results are shown in the bottom graphof Fig. 1. They are equal to 0.382, 0.381, 0.362, 0.368, and 0.416
for the Kinetex columns #13, #20, #73, #75, and the Luna column,respectively. Again, it is noteworthy that uracil has access to only57% of the total internal volume of these particles when it is elutedwith a mixture of acetonitrile and water (65/35, v/v).

4 romatogr. A 1280 (2013) 35– 50

scfinuLofttcps

S

T01ctplsttc

4

t

k

wrciecmdssasii1#ta

K

wttam2rL

0.0 0.1 0.20

10

20

30

40

50

Co

lum

n p

ressu

re d

rop

[b

ar]

uinterstitial

[cm/s ]

Kinetex#13Kinetex#20Kinetex#73Kinetex#75

Luna


Fig. 2 confirms that the absolute value of the slope of the exclu-ion branch is systematically smaller for the Kinetex and the Lunaolumns than for the column packed with solid silica cores. As arst approximation, this slope is equal to one third of the exter-al surface area, Sex generated by the particles per column volumenit [76]. Since the average particle sizes of both the Kinetex anduna particles are larger than that of the non-porous silica cores, thebservation shown in Fig. 2 makes sense. Better, the external sur-ace area available in each column was estimated from the slope ofhe exclusion branches at 3.29, 3.36, 0.65, 0.70, 2.83, and 2.67 m2 forhe Kinetex columns #13, #20, #73, #75, the Luna column, and theolumn packed with the silica cores, respectively. They were com-ared to the expected surface area assuming uniform (dp = Const),pherical particles from:

ex = 6(1 − �e)�r2c L

dp(27)

his equation provides theoretical surface area of 1.93, 1.93, 0.41,.41, 1.91, and 1.77 m2 which are 1.70, 1.74, 1.58, 1.72, 1.49, and.51 times smaller than the respective experimental values. Thisan be explained in part by the finite particle size distribution ofhese real materials and by the existence of smaller than averagearticles inside the column. Remarkably, the ratio is systematically

arger for the core–shell particles (�1.7) than for the Luna and theilica core particles (�1.5). Possibly, the slurry packing process ofhe core–shell particles might generate more fines (shell breakage)han the process used to pack the fully porous Luna and solid silicaores.

.2. Permeabilities of the columns

The permeabilities of the columns were measured according tohe following equation:

0 = �L1

�r2c (( (Ptot − Pex))/ Fv)

(28)

here � is the eluent (acetonitrile-water, 65/35, v/v) viscosity atoom temperature, L is the column length (15 cm), rc is the innerolumn radius (1.05 and 2.3 mm), Ptot is the total pressure dropncluding the system contributions, Pex is the pressure drop gen-rated by the HPLC system in absence of the chromatographicolumn replaced by a zero dead volume (ZDV) connector. Theaximum flow rate, Fv, was set so that the column inlet pressure

oes not exceed 50 bar and the column temperature remains con-tant. The compression of the eluent is smaller than 0.3% underuch conditions and the linear velocity of the mobile phase can bessumed to remain constant along the whole column length. Fig. 3hows the plots of the corrected pressure drop as a function of thenterstitial linear velocity for all columns. The specific permeabil-ties were then measured at 1.95 × 10−10 cm2, 1.95 × 10−10 cm2,.81 × 10−10 cm2, and 1.86 × 10−10 cm2 for the Kinetex columns13, #20, #73, and #75, respectively. k0 = 1.84 × 10−10 cm2 for

he Luna column. The Kozeny–Carman constant Kc was measuredccording to:

c = �3e d2

S

k0(1 − �e)2(29)

here dS is the mean Sauter diameter. Because we did not knowhe exact particle size distribution of the packing materials used inhis work, we simply assumed dS = 4.6 and 5.0 �m for the Kinetexnd the Luna particles, as provided by the manufacturer (Pheno-
enex). Thus, the Kozeny–Carman constant are estimated at 203,
06, 195, and 205 for the Kinetex columns #13, #20, #73, and #75,espectively. It is found to be smaller at 161 for the fully porousuna particles. This result agrees well with previous findings that

Fig. 3. Plots of the column pressure drop as a function of the interstitial linearvelocity for the five columns used in this work.

showed that Kc is significantly larger for core–shell than for classicalfully porous particles [19–21,74–76].

Overall, it is noteworthy that, despite the average particle sizeof the Luna particles being larger than that of the Kinetex parti-cles (5.0 �m for Luna versus 4.6 �m for Kinetex) and the values ofits Kozeny–Carman constant being smaller (�160 for Luna versus200 for Kinetex), the specific permeability of the Luna column issimilar to or even smaller than those of the Kinetex columns. If thecolumns had the same external porosity, the specific permeabilityof the Luna column would be circa +50% larger than those of theKinetex columns. This observation is obviously explained by thesignificant difference between the values of the external porositiesof the columns (�0.36 for Luna versus 0.40 for Kinetex), which com-pensates the effect of the differences between the particle sizes andtheir Kozeny–Carman constant.

4.3. 1290 Infinity system contributions

In the early developments of 5 �m fully porous particles, theretention and band broadening contributions of instruments tothe observed retentions and peak widths were neglected due tothe column dimension (4.6 mm × 250 mm, hold-up volume 2.7 mL).Assuming a total porosity of �t = 0.65 and an optimum plate heightof 10 �m, the contributions to the elution volume and the varianceof a non-retained compound due to the sole column were 2.70 cm3

and 292 �L2, respectively. The contributions of a standard HP1100instrument were typically of the order of 0.05 cm3 and 50 �L2, thusaffecting the retention and the column efficiency of retained com-pounds by less than +0.9 and −4.1%, respectively, if the retentionfactor were to be larger than 1. In conclusion, they were negligible.

For shorter (15 cm long) and thinner (2.1 mm I.D.) columnspacked with 4.6 �m core–shell particles, which could potentiallydeliver minimum plate height as small as 1.4 × 4.6 �m � 6.5 �m[4], and have a total porosity of only 0.55, the extra-column con-tributions become critical and harmful for the column performance.For instance, assume L = 15 cm, rc = 0.105 cm, �t = 0.55, H = 6.5 �m,and k′ = 1, the system contributions affect the elution volume andthe column efficiency by as much as +8.8% and −72%. This is clearlynot acceptable.

Therefore, to perform this study, we used an Agilent 1290 Infin-ity system, as described in Section 3.2. The contributions of thisinstrument are 0.01 cm3 and 10 �L2 only reducing the increase ofthe elution volume and the decrease of the column efficiency to


0.00 0.08 0.16 0.240.0

0.2

0.4

0.6

0.8

1.0

uS [cm/s]

2,e

x/

2,t

ot

4.6 x 150 mm Kinetex 5 m #13

Uracil (k=-0.11)Naphthalene (k=1.89)

0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

2.1 x 150 mm Kinetex 5 m #73


uS [cm/s]

2,e

x/

2,t

ot

0.00 0.08 0.16 0.240.0

0.2

0.4

0.6

0.8

1.0

2,e

x/

2,t

ot

uS [cm/s]


4.6 x 150 mm Luna 5 m

0.00 0.08 0.16 0.240.0

0.2

0.4

0.6

0.8

1.0

uS [cm/s]

2,e

x/

2,t

ot

4.6 x 150 mm Kinetex 5 m #20


0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

uS [cm/s]

2.1 x 150 mm Kinetex 5 m #75

2,e

x/

2,t

ot


F of urf r velor

ousog

taflsv

ig. 4. Relative contributions of the 1290 Infinity system to the total peak variancesor the five different columns used in this work as a function of the superficial lineaeader is referred to the web version of the article.)

nly +1.7% and −42%. So, the impact of the HPLC system on the col-mn efficiency is significant and its band broadening contributionhould be systematically subtracted from the total peak variancebserved in order to obtain the sole contribution of the chromato-raphic column.

Fig. 4 shows the relative contributions of the 1290 Infinity sys-em to the total peak variances of uracil (full black square symbols)nd naphthalene (full red circle symbols) recorded for the five dif-
erent columns used in this work as a function of the superficialinear velocity applied (the superficial linear velocity was cho-en to present these graphs because the interstitial velocity mayary from one to another column and the extra-column variance
acil (full black square symbols) and naphthalene (full red circle symbols) measuredcity applied. (For interpretation of the references to color in this figure legend, the

contributions are not affected by the column properties). For4.6 mm I.D. columns packed with 4.6 �m Kinetex particles, the 1290Infinity system accounts at most for 16% and 2% of the peak varianceof uracil (k′ � −0.1) and naphthalene (k′�1.9), respectively. Theserelative variance contributions increases to about 63% and 15% forthe 2.1 mm I.D. narrow-bore columns packed with the same mate-rial. For the 4.6 mm I.D. Luna column, they are minimized to only10% (k′ � −0.2) and 0.5% (k′ � 3.3). In the next section we report on
and compare the intrinsic performance of these five columns, basedon the results of our measurements of the individual HETP contrib-utions of the B/�, A(�), and C� after correction for the contributionsof the HPLC system.

4 romatogr. A 1280 (2013) 35– 50

4

4p

eotatmfrc

ctctpt

maeap#g

fwGptC

4

hna0rg45aKh(wthfatIfd(pptdt

0 4 8 120

1

2

3

4

5

Red

uced

pla

te h

eig

ht h

URACIL

Reduced interstitial velocity

4.6 mm I.D. Kinetex#134.6 mm I.D. Kinetex#202.1 mm I.D. Kinetex#732.1 mm I.D. Kinetex#754.6 mm I.D. Luna

0 4 8 12 160

1

2

3

4

5

Red

uced

pla

te h

eig

ht h



NAPHTHALENE

Fig. 5. Plots of the reduced plate heights versus the reduced velocities for the fivecolumns studied in this work and two different compounds, uracil (excluded andnon-retained, top graph) and naphthalene (adsorbed onto the stationary phase,


.4. Intrinsic reduced plate height

.4.1. Physically meaningful model of effective diffusion in theacked columns

The model of effective diffusion was selected among several oth-rs, according to its ability to predict the intra-particle diffusivityf the non-retained and partially excluded uracil [29]. Accordingo independent measurements, � = 0.48�p where �p stands for thectual particle or shell porosity accessible to this analyte. Fromhe combination of pycnometric, ISEC, and uracil elution measure-

ents it was found that �p = 0.22, 0.22, 0.21, 0.21, 0.21, and 0.23or the Kinetex columns #13, #20, #73, #75, and the Luna column,espectively. In conclusion, the model should predict a � value aslose as possible as 0.48 × 0.22 = 0.11 for both columns.

For column packed with fully porous particles such as the Lunaolumn, the Torquato model is a priori the best model because itakes into account the random distribution of the particles in theolumn. In contrast, the time-averaged model is a mere approxima-ion of the sophisticated effective diffusion mechanism and is nothysically reliable. The Torquato model predicts � = 0.11 whereashe time-averaged model gives � = 0.17.

For the core–shell particles, neither the modified Torquatoodel (so-called Garnett–Torquato) nor the time averaged model

re rigorously physically sound because they do not account prop-rly for the obstruction caused by the solid cores [39,40]. For thatll, they represent very fair approximations [29]. The latter modelredicts values of 0.19, 0.18, 0.14, and 0.15 for the Kinetex columns13, #20, #73, and #75, respectively. The Garnett–Torquato modelenerates more reliable values at 0.14, 0.13, 0.11, and 0.11.

In conclusion, because the time-averaged model of effective dif-usion overestimates the true value of � for uracil, the � valuesill be estimated according to either the Torquato (for Luna) or thearnett–Torquato (for Kinetex) model of effective diffusion and theeak parking data (see data in Tables 2 and 3 for uracil and naph-halene, respectively). This also provides the value of the coefficientp from Eq. (11).

.4.2. Reduced plate height contributionsFig. 5 shows plots of the experimental total reduced plate height

versus the reduced interstitial linear velocity � for uracil andaphthalene. The two graphs of this figure zooms in the regionround the minimum reduced plate height for 0 < � < 12 (uracil),

< � < 16 (naphthalene), and 0 < h < 5. As expected, the minimumeduced plate height h2 measured with the accurate numerical inte-ration method is smaller for the columns packed with core–shell.6 �m Kinetex particles than for those packed with fully porous

�m Luna particles. For uracil, the minimum reduced plate heightsre 1.43 and 1.31 (4.6 mm I.D. Kinetex), 2.05 and 1.62 (2.1 mm I.D.inetex), and 1.97 (Luna). For naphthalene, the minimum of the

values are 1.42 and 1.51 (4.6 mm I.D. Kinetex), 1.87 and 1.582.1 mm I.D. Kinetex), and 1.99 (Luna). These data are consistentith results obtained for another brand of 5 �m core–shell par-

icles (core diameter of 3.8 �m or � = 0.76) with a minimum plateeight of 6 �m [77]. The minimum reduced plate heights measured

rom the inaccurate half-height peak width method, h1, are 1.04nd 1.06 (4.6 mm I.D. Kinetex), 2.03 and 2.23 (2.1 mm I.D. Kine-ex), and 1.42 (Luna) for uracil. They are 1.22 and 1.34 (4.6 mm.D. Kinetex), 1.87 and 1.65 (2.1 mm I.D. Kinetex), and 1.80 (Luna)or naphthalene. The differences between these two sets of hminata is due to the non-Gaussian shape of the eluted band profilesfronting and/or tailing at the base of the peaks, down to 0.3% of theeak height) which the half-height method ignores. The half-height
eak width method assesses the column/system performance fromhe sole observation of the peak width at half height. Remarkably,espite that the h1 data are not corrected for extra-column con-ributions, they can be found smaller that the h2 data, which are
bottom graph).

corrected for the system contributions. Obviously, this makes nophysical sense and illustrates that the integration method shouldalways be preferred when measuring the true intrinsic column (cor-rection for extra-column effects) or column/system (no correctionfor extra-column effects) performance.

Figs. 6 and 7 show the plots of the relative contributions mea-sured for the longitudinal diffusion (red color), the eddy diffusion(blue color), and the solid–liquid mass transfer resistance (greencolor) to the overall reduced plate height. The measurementswere made in the ranges of reduced plate heights and velocitiesshown. The total reduced HETPs is given by the height of the barsin the graphs of these two figures. As previously demonstrated[2,11,17,18], the performances of columns packed with core–shellparticles are better than those of columns packed with fully porousparticles. This is due to (1) the diminution of the reduced B coef-ficient especially for retained analytes (−14% for uracil and −40%for naphthalene), (2) the diminution of the reduced eddy diffusionHETP term A for weakly retained analytes (−42% for uracil, −10%for naphthalene), and (3) the large decrease of the Cp coefficient
for all analytes (−86% for uracil, −60% for naphthalene). Due to thelarge diffusion coefficients of small molecules, the contribution of


0 10 20 300

1

2

3

4

5

4.6 x 150 mm Kinetex 4.6 m #13R

ed

uced

pla

te h

eig

ht h


Eddy diffusionSolid-liquid mass transfer resistanceLongitudinal diffusion

0 20 40 600

2

4

6

8

Red

uced

pla

te h

eig

ht h


2.1 x 150 mm Kinetex 4.6 m #73

0 10 20 300

1

2

3

4

5

Red

uced

pla

te h

eig

ht h


4.6 x 150 mm Luna 5.0 m

0 10 20 300

1

2

3

4

5

Red

uced

pla

te h

eig

ht h


4.6 x 150 mm Kinetex 4.6 m #20

0 20 40 600

2

4

6

8

2.1 x 150 mm Kinetex 4.6 m #75


Red

uced

pla

te h

eig

ht h

Fig. 6. Reduced plate height h of uracil (excluded, non-retained) as a function of the reduced interstitial linear velocity � showing the contributions of longitudinal diffusion(red), eddy diffusion (blue), and solid–liquid mass transfer resistance (green) to the overall reduced plate height. (For interpretation of the references to color in this figurel

Cv

ptrtdnb

egend, the reader is referred to the web version of the article.)

p� to h remains small compared to those of B/� at small reducedelocities and of A(�) at high reduced velocities.

The most remarkable fact about the two 4.6 mm I.D. columnsacked with 4.6 �m Kinetex-C18 core–shell particles is the rela-ive smoothness of the increase of the A(�) term with increasingeduced velocity for the non-retained uracil (Fig. 6) compared with
hat for the other three columns. For the retained naphthalene, theifferences observed between the five different columns used is sig-ificantly less due to the enhanced intra-particle diffusivity causedy surface diffusion inside the porous structure of the particles.
Table 3 shows that the effective diffusion coefficient of naphtha-lene (Deff) is comparable to that of non-retained uracil despite azone retention factor, k1, which is about fifteen times larger. Thiseffect results from the larger intra-particle diffusion coefficient �of naphthalene than that of uracil (1.2 versus 0.1) and by its largerbulk diffusion coefficient Dm (1.5 cm2/s versus 1.1 × 10−5 cm2/s).
Therefore, short-range inter-channel and trans-column velocitybiases are relaxed faster for naphthalene than for uracil becausethe residence time of the former compound inside the column istypically three (Kinetex) and four (Luna) times larger than that

46 F. Gritti, G. Guiochon / J. Chromatogr. A 1280 (2013) 35– 50

0 5 10 15 200

4

8

12

Eddy diffusionSolid-liquid mass transfer resistanceLongitudinal diffusion

4.6 x 150 mm Kinetex 4.6 m #13


Red

uced

pla

te h

eig

ht h

0 10 20 30 400

4

8

12


Red

uced

pla

te h

eig

ht h 2.1 x 150 mm Kinetex 4.6 m #73

0 5 10 15 200

4

8

12

Red

uced

pla

te h

eig

ht h


4.6 x 150 mm Luna 5.0 m

0 5 10 15 200

4

8

12


Red

uced

pla

te h

eig

ht h 4.6 x 150 mm Kinetex 4.6 m #20

0 10 20 30 400

4

8

12 2.1 x 150 mm Kinetex 4.6 m #75


Red

uced

pla

te h

eig

ht h

Fig. 7. Same as in Fig. 6, except for the retained analyte naphthalene. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of the article.)

Table 4Estimated parameters 1, ω1, 2, ω2, and �e for the determination of the trans-channel and short-range inter-channel eddy diffusion h terms of the five columns packed with4.6 �m Kinetex-C18 shell and 5.0 �m Luna-C18(2) particles.



εe 1 ω1 2 ω2 �e Buracil Bnaphthalene

4.6 �m Kinetex 632901-13 4.6 × 150 0.405 0.46 0.0043 0.23 0.12 0.63 1.56 3.92632901-20 4.6 × 150 0.406 0.46 0.0043 0.23 0.12 0.63 1.54 3.8841573 2.1 × 150 0.394 0.45 0.0042 0.23 0.12 0.63 1.49 3.7841575 2.1 × 150 0.401 0.46 0.0043 0.23 0.12 0.63 1.48 3.82

5.0 �m Luna 233512 4.6 × 150 0.363 0.42 0.0037 0.21 0.11 0.57 1.75 6.32

romat

otbF

eovblsfToptcwc(fwcom[

4o

tedtlaodd

nc

h

Fu


f the latter compound. Any source of structural heterogeneity ofhe packing structure tend to be concealed and all five columnsehave similarly. This is supported by the five h plots shown inig. 7.

For uracil in contrast, the influences of the column inner diam-ter (2.1 mm versus 4.6 mm for Kinetex particles) and of the naturef the particles (4.6 mm I.D. columns packed with fully porous Lunaersus core–shell Kinetex particles) on the column performanceecomes obvious. Decreasing the column inner diameter usually

eads to a degradation of the column performance because the bedtructure is heterogeneous near the wall region, where it differsrom the random structure of the bulk packing in the central region.he effect is critical in this range of column diameters [78]. Evidencef this behavior was already reported for both core–shell and fullyorous particles [13,32,54]. For instance, the difference betweenhe h versus � plots reported here for the two replicate 2.1 mm I.D.olumns and for the two replicate 4.6 mm I.D. columns, all packedith 4.6 �m Kinetex particles, is noteworthy. For the 2.1 mm I.D.

olumns, the increase of h with increasing � from 2.5 to 25 is fasterfrom 2 to 6) than for the 4.6 mm I.D. columns, for which h increasesrom 1.5 to only 2. Similarly, for the 4.6 mm I.D. column packedith 5.0 �m particles, h increases from about 2 to only 3.5. In con-

lusion, more than the structure of the particle itself (fully porousr core–shell particles), the inner diameter of the column controlsost of the column performance of weakly retained compounds

32,79].

.4.3. Trans-column eddy diffusion: deviation from performancef the infinite diameter column

Figs. 6 and 7 shows that, at high flow rates, the performance ofhe three different sets of chromatographic columns studied differssentially because of the variation of their respective reduced eddyiffusion terms, A(�), with the reduced velocity. A(�) is the sum ofhe trans-channel (hTS), the short-range inter-channel (hIC), and theong-range trans-column (hTC) eddy diffusion terms. The last termccounts for all sources of band dispersion caused by the presencef the column wall, the inlet and the outlet endfittings. A noticeableeviation from the dispersion expected for homogeneous infiniteiameter columns can be measured as follows.

The trans-channel reduced eddy diffusion term of uracil andaphthalene, due to the velocity biases taking place between adja-
ent particles is given by [50]:
TS = 1(1/21) + (1/ω1�)

(30)

0 20 40 600

1

2

3

4

5

Tra

ns-c

olu

mn

ed

dy d

iffu

sio

nre

du

ced

pla

te h

eig

ht h T

C


URACIL


ig. 8. Plots of the trans-column eddy diffusion reduced plate height versus the reducedracil (excluded and non-retained) and naphthalene (adsorbed onto the stationary phase

ogr. A 1280 (2013) 35– 50 47

In this equation, the coefficients 1 and ω1 are independent of theanalyte retention factor because lateral transfer diffusion betweenslow and fast streamlines involves only diffusion in the bulk elu-ent. The values of 1 and ω1 can be found from Ref. [31] forexternal porosities between 0.366 and 0.46, using the data givenby the S × 2 computer generated packing. In contrast, the short-range inter-channel reduced eddy diffusion term depends on theanalyte retention factor in RPLC because the apparent lateral dif-fusion transfer involves diffusivity of the analyte through thewhole bed (Deff = (B/(2(1 + k1)))Dm). The short-range inter-channelreduced eddy diffusion term of uracil and naphthalene should thenbe physically corrected to [32]:

hIC = 1(1/22) + (B/(2�eω2�))

(31)

where B is the reduced longitudinal diffusion coefficient mea-sured for the analytes studied in this work (see Tables 2 and 3).The coefficients 2 and ω2 were obtained for non-porous par-ticles (B = 2�e) using the computer-generated S × 2 packings for0.366 < �e0.46 [31]. They are listed in Table 4. The �e values weremeasured from the combination of the total pore blocking and peakparking methods [30,41]. They are equal to 0.63 for the columnspacked with core–shell particles and 0.57 for fully porous Lunaparticles.

By simple difference, the trans-column reduced eddy dispersionHETP term is measured from:

hTC = A(�) − hTS − hIC (32)

The results are shown in Fig. 8 for uracil and naphthalene.The difference between the five columns are remarkable for thenon-retained analyte, uracil. This confirms that non-retained com-pounds are the least forgiving compounds regarding the columnperformance because their residence time is the shortest. So, theinfluence of long-range trans-column velocity biases tend to relaxless during their migration [80,81], usually providing the largestpossible values observed for hTC. Even for the smallest reducedvelocity, hTC converges to about 0.5, the incompressible eddy diffu-sion reduced HETP term of actual columns. The column that havethe behavior closest to the infinite diameter column is the 4.6 mmI.D. column #20 packed with 4.6 �m Kinetex core–shell particles.The one that exhibits the largest deviation from this behavior is the2.1 mm I.D. column #73 packed with the same core–shell particles.
The 4.6 mm I.D. column packed with 5 �m Luna particles showsan intermediate behavior. When the retained compound naphtha-lene is injected, the trans-column eddy diffusion term is found tobe smaller than that of uracil for the reasons explained. The dif-
0 20 40 600

1

2

3

4

5

NAPHTHALENE


Tra

ns-c

olu

mn

ed

dy d

iffu

sio

nre

du

ced

pla

te h

eig

ht h T

C


velocity for the five columns studied in this work and two different compounds,).

48 F. Gritti, G. Guiochon / J. Chromat

103

10 4

10 5

10-3

10-2

100 s10 s1 s

t0/N

N

5.0 m Luna3.5 m BEH2.5 m BEH

Monolith 2nd

generatio n4.6 m Kinetex

103

10 4

10 5

10-3

10-2

L=12.1 cm

L=14.6 cm

L=6.1 cm

L=8.5 cm

L=13.2 cm

Naphthalene

=0.67 cP

P=400 bar

0.0 0.1 0.2 0.3 0.4 0.50.000

0.001

0.002

0.003 Best van Deemter plots

5.0 m Luna3.5 m BEH2.5 m BEH

Monolith 2nd

generatio n4.6 m Kinetex

H [cm]

uS [cm]

Fig. 9. Theoretical Poppe plots at 400 bar pressure drop built from the experimentalH5e

fetImr

ptfpo

4

Le(cd01tv

ETP (retained naphthalene, see bottom graph) and specific permeability of the Luna �m, Kinetex 4.6 �m, BEH 3.5 �m, and BEH 2.5 �m chromatographic columns. Theluent is a mixture of acetonitrile and water (65/35, v/v) at room temperature.

erences between the columns behavior are reduced and hTC neverxceeds 2.5 for the 2.1 mm I.D. columns. Note that the decrease ofhe reduced HETPs at high linear velocity for uracil and the 2.1 mm.D. columns is not due to errors of the corrected second central

oments (maximum ± 10%). Possibly, it could be due to a betteradial mixing at extremely high linear velocity.

In summary, 4.6 mm I.D. columns packed with 4.6 �m Kinetexarticles are more efficient than 2.1 mm I.D. columns packed withhe same packing materials because their trans-column eddy dif-usion term is minimum. Also, this work confirms that core–shellarticles are more uniformly packed than fully porous particlesver the distances covering the inner diameter.

.5. Poppe plots

Fig. 9 shows the kinetic Poppe plots for columns packed withuna 5 �m, Kinetex 4.6 �m, BEH 3.5 �m, and BEH 2.5 �m. The elu-nt was a mixture of acetonitrile and water at room temperatureviscosity � = 0.67 cP), at a constant inlet pressure set at 400 bar. Theonstruction of these plots necessitates additional experimentalata for these columns, their total porosity (�t = 0.63, 0.55, 0.65, and
.65, respectively), their specific permeability (k0 = 1.84 × 10−10,.95 × 10−10, 0.78 × 10−10, and 0.40 × 10−10 cm2, respectively), andheir HETP H(uS) expressed as a function of the superficial linearelocity uS. The y-axis of the kinetic Poppe plot represents the
ogr. A 1280 (2013) 35– 50

chromatographic speed measured by the hold-up time per theo-retical plate. It is given by [33,82,83]:

t0

N= �tH(uS)

uS(33)

The x-axis of the kinetic Poppe plot represents the resolution powerby the number of plates of the column:

N = k0

H(uS)uS

P

�(34)

It is important to emphasize that these plots are strictly virtual,since they assume that the HETP was recorded at the very flow rateleading to a pressure drop of exactly 400 bar. For instance, this wasnot the case for both the Luna and the Kinetex columns since thepressure drop was kept smaller than 200 bar in the present work.Additionally, these plots assume that the plate height is indepen-dent on the column length, that the chromatographic system isisothermal, that the eluent density and viscosity do not vary withthe pressure, and that the extra-column volume contribution isnegligible. This latter assumption is acceptable since we used aretained analyte, naphthalene, and wide I.D. columns.

For infinitely small linear velocity (long columns), uS → 0,t0→ ∞, and H(uS)uS → ((BDm)/�e) where B is the reduced longi-tudinal diffusion coefficient. So, a vertical asymptote is expectedat Nmax = ( P/�)((�ek0)/(BDm)). In conclusion, the maximum col-umn efficiency expected increases in proportion to the square ofthe particle diameter due to the increase of k0 with increasingdp. This explains why in Fig. 9, the 5 �m Luna (fully porous) and4.6 �m Kinetex columns can generate the highest plate counts.Inversely, the 2.5 �m BEH column provides the smallest plate countand should obviously not be used at low speed.

For infinitely fast separations (short columns), uS→ ∞, N → 0,and (H(uS))/(uS) → Cp((d2

p)/(�eDm)). So, a horizontal asymptote isexpected at ((t0/N))min = Cp((�td2

p)/(�eDm)). Therefore, the short-est hold-up time per unit plate height is proportional to the squareof the particle diameter times the coefficient Cp and the total poros-ity �t. As expected, the 2.5 �m BEH column provides the lowesthorizontal asymptote (small dp), followed by the 4.6 �m Kinetex(small Cp and �t), the 3.5 �m BEH, and the 5.0 �m Luna particles.

Interestingly enough, such kinetic representation demonstratesthat under the conditions listed above (standard pressure drops,moderate eluent’s viscosity, small compounds), the columnspacked with 4.6 �m Kinetex particles and with 2.5 �m BEH par-ticles provide similar plate counts for hold-up times around 10 s.Their speed-resolution performance exceeds systematically thoseof columns packed with either 5.0 or 3.5 �m fully porous parti-cles. It is almost equivalent to the speed-resolution properties ofthe second generation of silica monolithic columns [22,84–86].These four columns can provide the same analysis speed (t0 = 10 s at400 bar) but with different plate counts. The 4.6 �m Kinetex, mono-lithic, 3.5 �m BEH, and 5.0 �m Luna columns could theoreticallyachieve such a speed by delivering 8990 plates (L = 14.6 cm), 8120(L = 12.1 cm), 7255 plates (L = 8.5 cm), and 3610 plates (L = 13.2 cm),respectively.

5. Conclusion

This in-depth investigation of the mass transfer resistancesof long (15 cm), wide (4.6 mm I.D.) or narrow-bore (2.1 mm I.D.)columns packed with the new 4.6 �m Kinetex particles for lowmolecular weight analytes (uracil and naphthalene) confirms the
important advantages of the core–shell structure over the tradi-tional fully porous one. These new columns are more efficient thanclassical columns of the same size packed with 5 �m fully porousparticles. Their minimum reduced HETPs were found to be between

romat

1p

srnte2tKfhmTtdrfufa

stcblp(omce(aeaim

N

RAB

CCD

DddDEFFFhhhhHH


.3 and 1.5 (Hmin� 6.4 �m), making them as efficient as columnsacked with 3.2 �m but twice as permeable (k0 � 1.9 × 10−10 cm2).

We confirm also that the efficiency of columns packed with theame particles depends on the column diameter, the minimumeduced plate height is significantly smaller for wide bore than forarrow-bore columns. The difference between the performance ofhese columns is explained by a larger contribution to band broad-ning of the trans-column eddy dispersion reduced HETP term in.1 mm I.D. columns than in 4.6 mm I.D. ones. In spite of this impor-ant progress, the efficiency of the columns packed with the 4.6 �minetex particles are still remote from the performance expected

or the infinite diameter column. The “best” current column (#20)as a reduced HETP higher by about 1 unit than the theoreticalaximum, obtained for a reduced interstitial linear velocity of 20.

he importance of the progress already achieved is measured byhe importance of the eddy diffusion term that accounts for theifference between the efficiency of actual columns and the theo-etical infinite diameter one. This deviation is 2.5 and 4–5 h unitsor the Luna and narrow-bore Kinetex columns, respectively, usingracil as the analyte. As the retention increases in RPLC, these dif-erences fades away with deviations of only 1.5 h units for the Lunand narrow-bore Kinetex columns.

From a practical viewpoint, this study showed that, at a con-tant 400 bar pressure drop, the speed-resolution properties ofhe best Kinetex column was systematically better than those ofolumns packed with 3.5 �m fully porous particles and compara-le to that of columns packed with 2.5 �m fully porous particles, as

ong as the hold-up time is kept larger than 10 s, which alreadyermits the achievement of analyses in less than one minutefor k < 5 as it is the case in most isocratic separation meth-ds) when using standard 400 bar HPLC systems with inexpensiveodifications (for 4.6 mm I.D. columns). In summary, the 4.6 �m

ore–shell Kinetex columns (�e = 0.40) provide the same maximumfficiency as columns packed with 3.2 �m fully porous particlesHmin = 6.4 �m) and the same permeability (k0 = 1.9 × 10−10 cm2)s columns packed with 5.1 �m fully porous particles. They arequivalent to the second generation of monolithic columns with thedvantage of allowing operations at pressure drops up to 600 barnstead of 200 bar for the monolithic column, hence the achieve-

ents of faster analyses or the use of long columns.

omenclature

oman letters(�) reduced eddy diffusion term

reduced longitudinal diffusion coefficient with referenceto the interstitial linear velocity

i sample concentration at time ti (kg/m3)p reduced solid–liquid mass transfer resistance coefficientp effective sample diffusivity across the porous volume of

the particle (m2/s)eff effective diffusion coefficient of the packed bed (m2/s)p average particle diameter (m)S mean Sauter particle diameter (m)m bulk molecular diffusion coefficient (m2/s)

column impedance(m) hindrance diffusion factorv flow rate (m3/s)v,PP flow rate applied during the peak parking method (m3/s)

reduced plate heightTS reduced trans-channel eddy diffusion plate height
IC reduced short-range eddy diffusion plate heightTC reduced trans-column plate height
plate height (m)min minimum plate height (m)

ogr. A 1280 (2013) 35– 50 49

k′ retention factork1 zone retention factorKc Kozeny–Carman constantk0 specific permeability (m2)Ka equilibrium Henry’s constant for the sample

adsorption–desorption between the solid phase inthe porous volume of the particle and the liquid eluentphase

N column efficiencyL column length (m)Ptot total pressure drop (Pa)Pex system pressure drop in absence of the chromatographic

column (Pa)rc column inner radius (m)Sex external surface area of all the particles packed in the

chromatographic column (m2)t1 first discretized elution time (s)tN last discretized elution time (s)tleft elution time (adsorption) at 1.5% of the peak height (s)tright elution time (desorption) at 1.5% of the peak height (s)ti discretized elution time (s)tp parking time (s)tR retention time at the peak apex (s) tp increment of the peak parking time (s)u interstitial linear velocity (m/s)uS superficial linear velocity (m/s)V0 Column hold-up volume (m3)

Greek letters˛ numerical coefficient applied in Eqs. (24) and (25)

parameter in Torquato’s model of effective diffusion inpacked beds defined by Eq. (7)

�2PP increment of the peak variance measure in the peak par-

king method (s2)ı external roughness of the particle (m)�e external column porosity�p particle porosity�t total column porosity�e obstruction factor caused by randomly packed non-

porous particles to the diffusion in the external bulkmobile phase

�p internal obstruction factor� eluent viscosity (Pa s)m ratio of the probe diameter to the average pore size�1 first moment (s)�′

2 second central moment (s2)�1,PP,ex first moment recorded during the peak parking method

in absence of the chromatographic column (s)�1,PP first moment recorded during the peak parking method

(s)� reduced interstitial linear velocityω1/2 half-height peak width (s)� ratio of the effective diffusivity of the sample in the porous

region of the particle to its bulk diffusion coefficientω1 diffusion eddy dispersion coefficient related to trans-

channel velocity bias1 flow eddy dispersion coefficient related to trans-channel

velocity biasω2 diffusion eddy dispersion coefficient related to short-

range inter-channel velocity bias in columns packed withnon-porous particles

2 flow eddy dispersion coefficient related to short-range
inter-channel trans-column velocity bias in columnspacked with non-porous particles
� ratio of the solid non-porous core diameter to thecore–shell particle diameter

5 romat

�

A

bLU

R

[[[[[[

[[[[[[[[[[[[[[[[

[[[

[

[[[[[[[[[[[[[[[

[[[[[[[

[[[[[[[[[[[[[[[[[[[[

[

[[[[82] F. Gritti, G. Guiochon, J. Am. Pharm. Rev. 14 (3) (2011) 26.


2 adjustable parameter in Torquato’s model of effective dif-fusion Eq. (6)

cknowledgements

This work was supported in part by the cooperative agreementetween the University of Tennessee and the Oak Ridge Nationalaboratory. We thank Tivadar Farkas (Phenomenex, Torrance, CA,SA) for the generous gift of the four Kinetex columns.

eferences

[1] J.J. Kirkland, T.J. Langlois, US Patent Application 20070189944 a1 (2007).[2] G. Guiochon, F. Gritti, J. Chromatogr. A 1218 (2011) 1915.[3] J.J. DeStefano, T.J. Langlois, J.J. Kirkland, J. Chromatogr. Sci. 46 (2007) 254.[4] F. Gritti, G. Guiochon, J. Chromatogr. A 1157 (2007) 289.[5] F. Gritti, G. Guiochon, J. Chromatogr. A 1217 (2010) 1604.[6] F. Gritti, I. Leonardis, D. Shock, P. Stevenson, A. Shalliker, G. Guiochon, J. Chro-

matogr. A 1217 (2010) 1589.[7] F. Gritti, G. Guiochon, J. Chromatogr. A 1217 (2010) 5069.[8] F. Gritti, G. Guiochon, Chem. Eng. Sci. 65 (2010) 6310.[9] F. Gritti, C. Sanchez, T. Farkas, G. Guiochon, J. Chromatogr. A 1217 (2010) 3000.10] F. Gritti, I. Leonardis, J. Abia, G. Guiochon, J. Chromatogr. A 1217 (2010) 3219.11] F. Gritti, G. Guiochon, Chem. Eng. Sci. 65 (2010) 6327.12] S. Bruns, U. Tallarek, J. Chromatogr. A 1218 (2011) 1849.13] F. Gritti, G. Guiochon, J. Chromatogr. A 1218 (2011) 1592.14] F. Gritti, J. Omamogho, G. Guiochon, J. Chromatogr. A 1218 (2011) 7078.15] A. Daneyko, D. Hlushkou, S. Khirevich, U. Tallarek, J. Chromatogr. A 1257 (2012)

98.16] F. Gritti, K. Horvath, G. Guiochon, J. Chromatogr. A 1263 (2012) 84.17] F. Gritti, G. Guiochon, LC–GC North Am. 30 (2012) 586.18] F. Gritti, Chromatogr. Today 3 (2012) 4.19] F. Gritti, G. Guiochon, J. Chromatogr. A 1252 (2012) 46.20] F. Gritti, G. Guiochon, J. Chromatogr. A 1252 (2012) 31.21] F. Gritti, G. Guiochon, J. Chromatogr. A 1252 (2012) 35.22] F. Gritti, G. Guiochon, J. Chromatogr. A 1225 (2012) 79.23] F. Gritti, G. Guiochon, J. Chromatogr. A 1217 (2010) 7677.24] F. Gritti, G. Guiochon, J. Chromatogr. A 1218 (2011) 4632.25] P. Stevenson, F. Gritti, G. Guiochon, J. Chromatogr. A 1218 (2011) 8255.26] P. Stevenson, H. Gao, F. Gritti, G. Guiochon, J. Sep. Sci., in press.27] J. Knox, H. Scott, J. Chromatogr. 282 (1983) 297.28] F. Gritti, G. Guiochon, J. Chromatogr. A 1221 (2012) 2.29] F. Gritti, G. Guiochon, Chem. Eng. Sci. 66 (2011) 6168.30] F. Gritti, G. Guiochon, J. Chromatogr. A 1217 (2010) 5137.31] S. Khirevich, A. Daneyko, A.A. Holtzel, S.-M.U. Tallarek, J. Chromatogr. A 1217

(2010) 4713.32] F. Gritti, G. Guiochon, Anal. Chem., accepted for publication.
33] H. Poppe, J. Chromatogr. A 778 (1997) 3.34] G. Guiochon, A. Felinger, A. Katti, D. Shirazi, Fundamentals of Preparative and
Nonlinear Chromatography, 2nd ed., Academic Press, Boston, MA, 2006.35] S. Torquato, Random Heterogeneous Materials. Microstructure and Macro-

scopic Properties, Springer, New York, 2002.

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Speed-resolution properties of columns packed with new 4.6μm Kinetex-C18 core–shell particles

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Transcript of Speed-resolution properties of columns packed with new 4.6μm Kinetex-C18 core–shell particles