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Journal of Chromatography A, 1355 (2014) 164–178

Contents lists available at ScienceDirect

Journal of Chromatography A

jo ur nal ho me pag e: www.elsev ier .com/ locate /chroma

he rationale for the optimum efficiency of columns packed with new.9 �m fully porous Titan-C18 particles—A detailed investigation ofhe intra-particle diffusivity

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 15 April 2014eceived in revised form 26 May 2014ccepted 27 May 2014vailable online 11 June 2014

eywords:olumn efficiencyitan-C18 particlesarticle size distributionore size distributionntra-particle diffusivitynternal obstruction factor

a b s t r a c t

In a previous report, it was reported that columns packed with fully porous 1.9 �m Titan-C18 particles pro-vided a minimum reduced plate height as small as 1.7 for the most retained compound (n-octanophenone)under RPLC conditions. These particles are characterized by a relatively narrow size distribution with arelative standard deviation (RSD) of only 10%. A column packed with classical 5 �m Symmetry-C18 par-ticles, used as a reference RPLC column, generated a minimum reduced plate height of 2.1 for the sameretained compound. This work demonstrates that this was due to an unusually low intra-particle diffu-sivity across these particles, which leads to a small longitudinal diffusion coefficient along the column.The demonstration is based on the combination of accurate measurements of the height equivalent toa theoretical plate (HETP), inverse size exclusion chromatography (ISEC), peak parking (PP), and minordisturbance method (MDM) experiments.

The experimental results show that the reduced eddy dispersion HETP term (A = 0.8 for a reducedvelocity of 5), the internal particle porosity (�p = 0.35), and the enrichment of acetonitrile in the porevolume (75% acetonitrile in the bulk, 85% inside the mesoporous volume) are identical on both the Titan-C18 and Symmetry-C18 columns. The difference between the internal structures of these two brands ofRPLC-C18 fully porous particles lies in the values of the internal obstruction factor �p, which is 0.42 for the

Symmetry-C18 but only 0.26 for the Titan-C18 particles. This is in part related to the diffusion hindrancedue to the small average pore size of the Titan-C18 particles, around 59 A versus 77 A for Symmetry-C18

particles. A simple model of constriction along diffusion paths having the shape of a truncated conesuggests that the width of the pore size distribution (RSD of 30% and 20% for Titan-C18 and Symmetry-C18

particles) is mostly responsible for the difference in their obstruction factors.© 2014 Elsevier B.V. All rights reserved.

. Introduction

The recently introduced 1.9 �m fully porous Titan particlesre characterized by a relatively tight PSD (RSD �10%). Theyrovide unusually low minimum reduced plate heights in RPLC fortrongly retained compounds. Values of hmin between 1.7 and 1.9ere estimated for four columns of different dimensions (2.1 and

.0 × 100 mm, 2.1 and 3.0 × 50 mm) packed with these particles [1].

hese results revived the debate on the impact of the width or RSDf the PSD on the column efficiency. It has been demonstrated that:

∗ Corresponding author. Tel.: +1 865 974 0733; fax: +865 974 2667.E-mail addresses: guiochon@utk.edu, guiochon@ion.chem.utk.edu

G. Guiochon).

ttp://dx.doi.org/10.1016/j.chroma.2014.05.076021-9673/© 2014 Elsevier B.V. All rights reserved.

1. The minimum reduced HETP increases from 1.4 for core–shellparticles [2–13], which have a very tight PSD with a RSD of only5%, to 1.7 for Titan fully porous particles with a RSD of 10%, andto about 2.0 for conventional fully porous particles with a RSDaround 20%.

2. The width of the PSD has no measurable impact on the efficiencyof chromatographic columns as long as its RSD is smaller than50%. This conclusion was often investigated in the past [14–17].More recently experimental data and numerical calculationsconfirmed it [18–22].

3. Thus, the apparent relationship between column performanceand uniformity of particle size is merely coincidental. In the

case of core–shell particles, the significant improvement of thecolumn efficiency was directly related to the diminution of thetrans-column eddy dispersion as shown by the results of an accu-rate and precise experimental protocol [23–25,12,26,13,27].

F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178 165

List of symbols

Roman lettersa coefficient defined in Eq. (12)A(�) reduced eddy diffusion termb coefficient defined in Eq. (13)B reduced longitudinal diffusion coefficient with ref-

erence to the interstitial linear velocityc coefficient defined in Eq. (14)cp sample concentration in the particle mesopores

(kg/m3)cs sample concentration in the stationary phase

(kg/m3)Cp reduced solid–liquid mass transfer resistance coef-

ficientd hydrodynamic size of the polystyrene standard (m)Deff effective diffusion coefficient of the packed bed

(m2/s)Dn dean numberDp particle diffusivity (m2/s)Dpore hindrance pore diffusion coefficient (m2/s)DS diffusion coefficient in the stationary phase volume

(m2/s)dp average particle diameter (m)dSauter mean Sauter particle diameter (m)Dm bulk molecular diffusion coefficient (m2/s)Fv flow rate (m3/s)h reduced plate heightH plate height (m)k′ retention factork1 zone retention factorKc Kozeny–Carman constantk0 specific permeability (m2)K0 equilibrium Henry’s constant for the sample

adsorption–desorption between the solid phase inthe porous volume of the particle and the liquideluent at infinitely high temperature.

Ka equilibrium Henry’s constant for the sampleadsorption–desorption between the solid phase inthe porous volume of the particle and the liquideluent

Ke equilibrium Henry’s constant for the sample parti-tion between the particles and the interstitial bulkphase

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

graphic column (Pa)Qst isosteric heat of adsorption (J/mol).rc column inner radius (m)rtube PEEK capillary tube inner radius (m)tp parking time (s)tR retention time at the peak apex (s)�tp increment of the peak parking time (s)S surface area perpendicular to the diffusion direction

(m2)Sc Schmidt numberu interstitial linear velocity (m/s)uR migration linear velocity (m/s)

3

Greek letters parameter defined in Eq. (16)

parameter in Torquato’s model of effective diffusionin packed beds defined by Eq. (6)

�e external column porosity�0

p particle porosity for an infinitely small probemolecule (d = 0)

�p particle porosity�t total column porosity�e external obstruction factor caused by randomly

packed non-porous particles to the diffusion in theexternal bulk mobile phase

�c constriction factor�c,s constriction factor along a surface path�c,v constriction factor along a volume path�p internal obstruction factor to the diffusion across

the porous particles� eluent viscosity (Pa s)� parameter defined in Eq. (17)�1,t first moment (s)�′

2,t second central moment (s2)2

t second central moment measured from the half-height peak width (s2)

v RSD of the pore size distribution� reduced interstitial linear velocity

ratio of the effective diffusivity of the sample in theporous region of the particle to its bulk diffusioncoefficient

tortuosity factor� adjustable parameter in Torquato’s model of effec-

V0 column hold-up volume (m )VISEC particle volume accessible to the polymer of size d

(m3)

2tive diffusion Eq. (5)

The most plausible physical explanation is that shear friction forcesact between particles during bed consolidation and make roughcore–shell particles slip against each other and against the wall farless easily than smooth fully porous particles do. This explains themore radially uniform structure of core–shell particles beds. Thisdoes not explain the high performance of the Titan columns. Theexperimental observations are a priori consistent with their smalllongitudinal diffusion coefficient B and relatively high solid–liquidmass transfer resistance coefficient in their van Deemter equation.Remarkably, the optimum reduced velocities observed for columnsof Titan-C18 were always found between 3 and 6 [1], depending onthe retention factor. For columns packed with conventional fullyporous RPLC-C18 particles, it is usually between 8 and 10 and waseven found close to 12 for columns packed with core–shell parti-cles, consistent with their very low eddy dispersion HETP at highvelocities [7].

The goal of this work was to understand the physical reasonsfor the high kinetic performance of Titan-C18 columns (hmin = 1.7).Is it due to a low eddy dispersion term or to a small longitudinaldiffusion term in the van Deemter plot? To clarify the comparison, aconventional column packed with 5 �m fully porous Symmetry-C18particles was used as reference. This choice was motivated by thesimilar internal porosities and retentions of many analytes on thesetwo columns. In contrast, the minimum reduced HETP of the Sym-metry column is 2.0–2.1. In order to provide sound conclusions, thelow minimum reduced plate height observed on the Titan columnwas determined by combining the results of accurate and pre-

cise measurements. These include those of the van Deemter HETPplot in mixtures of acetonitrile and water, inverse size-exclusionchromatography to determine the internal porosities, the aver-age pore sizes and the width of the pore size distributions, PP

1 omato

enoootipsT

2

2

cabpgbhi

k

wrt

�

wg

u

w

2

aosd

h

t

2

f[dp

D

66 F. Gritti, G. Guiochon / J. Chr

xperiments to measure intra-particle diffusivities and the inter-al obstruction factor, and MDM to assess the enrichment of therganic modifier inside the mesoporous volume (due to the excessf adsorption of acetonitrile from water onto the silica-C18 surface)f both C18-bonded particles. Finally, a simple model of constric-ion along open-ended pores with different entrance and exit radiis proposed to account for the possible impact of the width of theore size distribution on the internal obstruction factor. The resultshould close the debate on whether the relatively narrow PSD of theitan particles affects the column kinetic performance.

. Theory

.1. Definitions

The external porosity of a packed bed is �e and Dm is the diffusionoefficient of the analyte in the bulk mobile phase. The apparentnalyte diffusivity through the fully porous particle is Dp = ˝Dm, ˝eing the dimensionless ratio of the analyte diffusivity through thearticle to its bulk diffusion coefficient. The reference concentrationradient taken for the definition of the diffusion coefficient Dp is theulk concentration [28]. The effective diffusion coefficient along theeterogeneous packed bed (porous particles and external eluent)

s Deff.The zone retention factor k1 is written [29,30]:

1 = 1 − �e

�e[�p + (1 − �p)Ka] (1)

here �p is the particle porosity and Ka = (cs/cp) is the equilibriumatio of the analyte concentrations in the stationary phase and inhe pore volume.

The reduced interstitial linear velocity, �, is written [29]:

= udp

Dm(2)

here u is the average interstitial linear velocity along the columniven by:

= Fv

�e�r2c

(3)

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

.2. Reduced HETP equation

The possible presence of significant frictional heating effects isccounted for by its contribution, hheat to the reduced HETP. Theverall reduced plate height is the sum of the longitudinal diffu-ion term, the trans-particle mass transfer resistance term, the eddyiffusion term, and hheat or:

= B

�+ A(�) + Cp� + hheat (4)

The explicit physico-chemical descriptions of each of these HETPerms are given in the next sections.

.2.1. The longitudinal diffusion term (B)The longitudinal diffusion term is derived from the effective dif-

usion coefficient Deff of the analyte in a heterogeneous column bed31,32]. A physically relevant model of effective diffusion in ran-om packed beds is given by the Torquato model for fully porous

articles [33,34]. It is written:

eff = 1�e(1 + k1)

[1 + 2(1 − �e) − 2�e�2ˇ2

1 − (1 − �e) − 2�e�2ˇ2

]Dm (5)

gr. A 1355 (2014) 164–178

with

= − 1 + 2

(6)

and �2 is a parameter estimated from the experimental externalobstruction factor for = 0 and k1 = 0 (non-porous particles and noretention). From Eq. (5), the expression of �e = (Deff/Dm) is given by:

�e = 2(1 − (�2/2))3 − �e(1 + �2)

(7)

For columns packed with particles having a narrow PSD and�e=0.41, �e�0.63 [35]. Therefore, �2 is estimated at 0.495. Forcolumns packed with conventional particles having a wider PSDand �e = 0.36, we have �e = 0.59 [28] and, according to Eq. (7), �2 istaken at 0.540.

The reduced B coefficient in Eq. (4) is written [31]:

B = 2(1 + k1)Deff

Dm(8)

In order to compare the values of the reduced longitudinal dif-fusion coefficient for different columns having different externalporosities, B is represented as a function of the equilibrium con-stant, Ke = (cp/cm), between the overall sample concentration in theparticle volume (cp) and the bulk concentration (cm) in the inter-particle volume. When Ke→ 0, the B coefficient tends towards 2�e.By definition,

Ke = �e

1 − �ek1 (9)

The plot of B versus Ke allows a fair comparison between thereduced longitudinal diffusion coefficients of different columns forthe same concentration distribution between the intra- and theinter-particle volumes.

2.2.2. The sample intra-particle diffusivity (˝)The coefficient = (Dp/Dm) is directly obtained from the above

coefficient B by inverting Eqs. (6) and (5). The final expression of is a function of the negative root of a polynomial equation of

degree two:

= 2 + 11 − ˇ

(10)

where is given by

= −b −√

b2 − 4ac

2a(11)

where the coefficients a, b, and c have the following expressions

a = �e�2(2 − �eB) (12)

b = −(1 − �e)(�eB

2+ 2

)(13)

c = �eB

2− 1 (14)

A model of intra-particle diffusivity was proposed some yearsago in RPLC [28]. It expresses the combination of two contrib-utions to the overall sample diffusivity: the contribution of the porevolume Dpore = �Dm (the coefficient � < 1 accounts for the overalldiffusion hindrance in the confined pore geometry) and that of sta-tionary phase diffusion in the adsorbed state Ds is described bythe effective medium theory of Landauer since the distribution ofthe mesopores in the particle can be considered as random [36].

Accordingly,

Dp = Dpore

( +

√˛2 + 1

2Ds

DporeKa

)(15)

omato

w

˛

ttiottea

D

wdK

fitfm

2

tp

C

e

2

bcpwdvcspier(n

2

emlrpticao

F. Gritti, G. Guiochon / J. Chr

here

= 14

[3�p − 1 + Ds

DporeKa(2 − 3�p)

](16)

Due to the preferential adsorption of the organic modifier (ace-onitrile in this work) onto the surface of the C18-bonded particles,he average eluent composition in the pore volume differs from thatn the external bulk volume and � is necessarily an overestimatef the true value in RPLC-C18 particles. The average composition ofhe eluent in the mesopore volume can be directly measured fromhe excess adsorption isotherm of the organic modifier. The appar-nt diffusion coefficient Ds through the stationary phase volume is

function of the equilibrium constant Ka [37]:

S = ıK�a Dm (17)

here ı and � < 0 are two constants. The equilibrium constant Ka

epends on the isosteric heat of adsorption Qst and is equal to0 exp(− Qst/RT) where K0 is a constant.

The intra-particle diffusivity coefficient is represented as aunction of the equilibrium constant Ka. The parameters � can bendependently measured from PP experiments and the remainingwo parameetrs, ı and �, can be estimated for different brands ofully porous particles by adjusting Eqs. (15) and (16) to the experi-

ental values of Dp also derived from the results of PP experiments.

.2.3. The solid–liquid mass transfer resistance term (Cp)A general expression of the solid–liquid mass transfer resis-

ance coefficient (Cp) was derived using the following coefficientrovided by [38,29]:

p = 130

�e

1 − �e

(k1

1 + k1

)2 1˝

(18)

In this equation, was estimated from the Torquato model offfective diffusion by combining Eqs. (8), (5) and (6).

.2.4. The eddy diffusion termThe eddy diffusion term A(�) results from all the sources of

and dispersion in the interparticle column volume. It includes theontributions of the homogeneous random packing or analyte dis-ersion along a hypothetical infinite diameter column (a columnith no wall, no inlet and outlet endfittings) [39,26], e.g., overistances ranging from the interparticle distance (trans-channelelocity biases) to a few particle diameters (short-range inter-hannel velocity biases). It also includes the contributions of theo-called long-range trans-column velocity biases induced by theresence of the solid cylindrical wall surrounding the bed and the

nlet/outlet (distribution/collection) endfittings. The total reducedddy dispersion HETP term, A(�), is obtained by subtracting theeduced HETP terms Cp� and B/� from the expression of h in Eq.4) when the HETP term related to frictional heating (hheat) can beeglected.

.2.5. The frictional heating termThe origin of the reduced HETP term hHeat was reported and

xplained in details in references [40–46]. When the friction of theobile phase against the surface of the packed particles becomes

arge, frictional heat is released all across the column volume; itsadial dissipation is accompanied by the formation of radial tem-erature gradients. As a result, the sample molecules located inhe central region of the column progress faster than those present

n the peripheral region of the column and their diffusion coeffi-ient is larger. This can be a significant source of band dispersionlong columns packed with fully porous particles and eluted withrganic-rich solutions.

gr. A 1355 (2014) 164–178 167

2.3. Intrinsic and apparent plate heights

The relationship between the observed plate height Happ, theintrinsic plate height Hint = (L/Nint), the column dimensions (V0 andL), the system variance in volume unit (2

v,ex), and the retentionfactor k′ is [47]:

Happ(k′) = Hint + L2

v,ex

V20

1

(1 + k′)2(19)

This equation shows that the larger k′, the closer is the observedcolumn efficiency to its intrinsic value.

2.4. Prediction of the exclusion volume of polymers in ISEC

If the plot of the derivative of the cumulative internal pore sizevolume versus the decimal logarithm of the pore size is properlyaccounted for by a Gaussian function, the accessible particle volumeVISEC(d) of a partially excluded polymer standard of size d can bepredicted by assuming that all the mesopores are either cylindricalor spherical [48]. For cylindrical pores, its expression is:

VISEC = �0p(1 − �e)Vc

ln 10√

2�v

∫ ∞

d/2

(1 − d

2r

)2 1r

× exp

[− (log 2r − log Dp)2

22v

]dr (20)

and for spherical pores it is:

VISEC = �0p(1 − �e)Vc

ln 10√

2�v

∫ ∞

d/2

(1 − d

2r

)3 1r

× exp

[− (log 2r − log Dp)2

22v

]dr (21)

where the dummy integration variable r runs from d/2 to ∞, and�0

p is the particle porosity accessible for an hypothetical probemolecule of size d = 0.

3. Experimental

3.1. Chemicals

The mobile phase used was a binary mixture of acetonitrileand water (75/25, v/v). Tetrahydrofuran (THF) was used as themobile phase for ISEC measurements. All these solvents wereHPLC grade from Fisher Scientific (Fair Lawn, NJ, USA). Acetoni-trile was filtered before use on a surfactant-free cellulose acetatefilter membrane, 0.2 �m pore size (Suwannee, GA, USA). Theeleven polystyrene standards (MW = 590, 1100, 3680, 6400, 13,200,31,600, 90,000, 171,000, 560,900, 900,000, and 187,0000) werepurchased from Phenomenex (Torrance, CA, USA). The standardRPLC checkout sample (1 mL ampoule) was purchased from Agilenttechnologies. It contains 100.3 �g/mL (±0.5%) of acetophenone,propiophenone, butyrophenone, valerophenone, hexanophenone,heptanophenone, octanophenone, benzophenone, and acetanilide.The solvent of this sample is a mixture of acetonitrile and water

(65/35, v/v, a composition close to that of the mobile phase).Thiourea, acetophenone, valerophenone, and octanophenone (all>99% purity) were separately purchased from Sigma–Aldrich(Suwannee, GA, USA).

1 omatogr. A 1355 (2014) 164–178

3

bBa2oitp

1

2

3

4

5

Tt1ter[olwtocp

3

p(2dathcstda

wfMSm

0 200 400 600 800 10000.3

0.4

0.5

0.6

Acc

essi

ble

colu

mn

poro

sity

Polystyrene hydrody mamic dia mete r DH [Å]0 200 400 600 800 1000

0.3

0.4

0.5

0.64.6 mm x 100 mm3.3 μμm non -porous co resεεe=0.415

3.0 mm x 100 mm1.9 μm Titan-C18εe=0.385, εt=0.599 , εp=0.34 8

0 20 0 40 0 60 0 80 0 100 00.3

0.4

0.5

0.6

Acc

essi

ble

colu

mn

poro

sity

Polystyrene hydrody mamic dia mete r DH [Å]0 20 0 40 0 60 0 80 0 100 0

0.3

0.4

0.5

0.6 4.6 mm x 100 mm3.3 μμm non -po rou s core sεεe=0.415

4.6 mm x 150 mm5.0 μm Symm etry-C18εe=0.344 , εt=0.567 , εp=0.340

68 F. Gritti, G. Guiochon / J. Chr

.2. Instrument

The 1290 Infinity HPLC system (Agilent Technologies, Wald-roen, Germany) was used in this work. It includes a 1290 Infinityinary Pump with solvent selection valves and a programmableuto-sampler. The injection volume is drawn into one end of the0 �L injection loop. The instrument includes a two-compartmentven and a multi-diode array UV–vis detection system. The systems controlled by the Chemstation software. In this work, the samplerajectory in the equipment involves the successive passage of itseak through the series of:

A 20 �L injection loop attached to the injection needle. The designof this First In, Last Out (FILO) injection system is such that theentire volume of sample drawn into the loop is injected into thecolumn. 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.

A 130 �m × 50 mm long Viper capillary tube offered by the man-ufacturer (Thermo Fisher Scientific, Germering, Germany) placedupstream the column. This tube has a volume of 0.7 �L.

A 130 �m × 250 mm long Viper capillary tube offered by the samemanufacturer placed downstream the column. This tube has avolume of 3.3 �L.

A small volume detector cell, V() = 0.6 �L, VCell = 1.4 �L, 10 mmpath.

No solvent heat exchanger was used in the experiments.he elution volume of the sample from the injection port tohe detection cell in the absence of a column is estimated at.0 + 1.2 + 0.7 + 3.3 + 1.4/2 = 6.9 �L. To improve the performance ofhis Agilent system, Viper capillary tubes were used and the short-st one was placed upstream the column. They allow a bettereproducibility of the retention times and peak variances recorded49] by avoiding tube-column connection failures over the durationf the experiments that are often encountered with standard metal-ic and/or plastic endfittings at very high pressures (�1 kbar) and

ith different brands of chromatographic columns equipped withheir own inlet and outlet endfittings. Placing the short tube insteadf the long tube upstream the column allows to minimize the extra-olumn band broadening due to the impact of high pressures on there-column sample dispersion [50].

.3. Columns

A 3.0 × 100 mm column packed with 1.9 �m Titan-C18 fullyorous particles were generously given by Supelco AnalyticalBellefonte, PA, USA). The mean Sauter diameter was measured at.04 �m. For the sake of comparison, the 50% cumulative volumeiameter of the Titan-C18 particles was measured at 2.23 �m. It isn overestimate of the true mean Sauter diameter by +9%, due tohe presence of non-ionic surfactant agent necessary to disperse theydrophobic particles in an electrolyte solution during the Coulterounter PSD measurement [1]. According to the manufacturer, theurface coverage of octadecyl (C18) chains, the average pore size,he surface area and the carbon content of the particles after C18erivatization and C1 endcapping are 2.0 �mol/m2, 73 A, 410 m2/g,nd 13.5%, respectively.

The reference RPLC-C18 chromatographic column used in thisork was a 4.6 × 150 mm column packed with 5 �m Symmetry-C18

ully porous particles that was offered by its manufacturer (Waters,ilford, MA, USA). The 50% cumulative volume diameter of the

ymmetry-C18 particles was measured at 5.20 �m. Therefore, theean Sauter diameter of the Symmetry-C18 particles was estimated

Fig. 1. ISEC plots for the Titan-C18 (top graph) and Symmetry-C18 (bottom graph)columns used in this work. The black symbols represent ISEC data obtained for areference column packed with 3.3 �m non-porous silica cores.

at 4.76 �m. The surface coverage of octadecyl (C18) chains, averagepore radius, surface area and carbon content after C18 derivatizationand C1 endcapping are reported at 3.24 �mol/m2, 96 A, 340 m2/g,and 19.68%, respectively.

The PEEK capillary tube used to measure the diffusioncoefficients of thiourea, acetophenone, valerophenone, andoctanophenone (1/16 in o.d. × 0.0254 in i.d. × x50 ft long) was pur-chased from IDEX Health & Science (Oak Harbor, WA, USA).

3.4. ISEC experiments and column porosities

ISEC measurements were made on the Titan-C18 and Symmetry-C18 columns. Neat THF was used as the eluent. Eleven polystyrenestandards with molecular weights between 500 and 2 million Dal-tons were dissolved in pure THF and used as probe molecules. Theycover a wide range of molecular sizes, between 10 and 950 A. Theflow rate was set at 0.2 mL/min (3.0 mm i.d. Titan-C18 column) and1.0 mL/min (4.6 mm i.d. Symmetry-C18 column). The injected sam-ple volumes were 0.50 �L (Titan column) and 1.0 �L (symmetrycolumn). The detection wavelength was set at 254 nm, the band-width at 4 nm, and the sampling rate at 20 Hz.

The external column porosities (�e) were derived by extrapolat-ing the exclusion branches of the ISEC plots of the elution volumesof the polymers versus their hydrodynamic molecular radius to a

zero radius (see Fig. 1 for illustration). The accuracy of the ISECexperimental protocol is ca. 1%. The size of the largest mesoporeswas roughly estimated to be around 100 (Symmetry-C18) and 175(Titan-C18) ± 10 A based on the initial deviation of the ISEC plot

F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178 169

Table 1Physico-chemical property of the two columns packed with 1.9 �m Titan-C18 and 5.0 �m Symmetry-C18 particles.

Name Batchnumber

Columnnumber

Column’s dimension I.D.[mm] × length [mm]

Mean Sautersizea [�m]

Totalb porosity(εt)

Totalc porosity(εt)

Totald porosity(εt)

Externalporositye (εe)

Particleporosityf (εp)

Titan 8033 3086 3.0 × 100 2.04 0.599 0.539 0.623 0.385 0.348Symmetry 279 0279333111 4.6 × 150 4.76 0.567 0.527 0.589 0.344 0.340

a Measured from calibrated coulter counter experiments on the raw silica Titan material without dispersant agent in the electrolyte solution.b Measured from the elution of toluene in pure THF.c Measured from the elution of thiourea in CH3CN/H2O, 75/25, v/v.

fmp

�

t

3

vttcTipaeasaTip(

f

D

wtr

narccaTa1

3

iia

d Measured from the minor disturbance method (CH3CN and H2O solvents).e Measured from ISEC experiments.f Determined from Eq. (22).

rom the exclusion branches. The total column porosity (�t) waseasured from the elution time of toluene in pure THF. The internal

orosity (�p) was derived from:

p = �t − �e

1 − �e(22)

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

.5. Measurement of the diffusion coefficients of the analytes

The bulk molecular diffusivities, Dm, of thiourea, acetophenone,alerophenone, and octanophenone were measured at the constantemperature of 23 ± 0.5 ◦C. The mobile phase was a mixture of ace-onitrile and water prepared by the pump mixer (75/25, v/v). Theapillary method was used to measure these diffusivities [51–56].he five conditions required for the validity of this method are listedn reference [57]. They include the radial equilibration of the sam-le concentration along the capillary, a negligible contribution ofxial molecular diffusion to the total spatial variance, a negligiblextra-column peak variance compared to the total observed vari-nce, a product of the Dean number squared by the Schmidt numbermaller than 100 (no secondary flow circulation in the coiled tube),nd the validation of the method by using a standard sample.he diffusion coefficients measured were all validated by compar-ng the coefficient obtained for a standard compound (thiourea inure water at 298.15 K) for which its value is accurately knownDm = 1.33 × 10−5 cm2/s [58,59]).

Accordingly, the diffusion coefficient is directly given by theollowing relationship [53]:

m = 124

�1,t

�′2,t

r2tube (23)

here �1,t and �′2,t are respectively the first and the second central

ime moments of the recorded peak profiles and rtube is the inneradius of the PEEK capillary tube.

The diffusion coefficients of thiourea, acetophenone, valerophe-one, and octanophenone were then accurately measureds 1.63 × 10−5, 1.77 × 10−5, 1.45 × 10−5, and 1.24 × 10−5 cm2/s,espectively. These experimental data were fitted to the empiri-al relationship Dm = (A/V˛

A ), with VA the molar volume of analytealculated from the average of the values given by the Shroedernd Le Bas group methods [60], A = 25.91 × 10−5 and = 0.542.he diffusion coefficients of propiophenone, butyrophenone, hex-nophenone, and heptanophenone were estimated at 1.64 × 10−5,.53 × 10−5, 1.37 × 10−5, and 1.30 × 10−5 cm2/s, respectively.

.6. PP experiments

The PP method was pioneered by Knox, first in gas [61], thenn liquid chromatography [62,63,28,64,34,65,66,30]. This methods applied in this work to measure the reduced coefficients B, ˝,nd Cp in the van Deemter equation.

Samples of 1.5 �L and 3.0 �L of the RPLC checkout sample wereinjected in the 3.0 mm and 4.6 mm i.d. columns at constant flowrates of 0.41 and 0.96 mL/min, respectively. For thiourea, the col-umn was eluted during the time necessary for thiourea to reachabout the middle of the column length at the constant tempera-ture of 23 ± 0.5 ◦C. For the RPLC checkout sample, the column waseluted during the time necessary for the least retained compound(acetanilide) to reach exactly three quarters of the column length.The flow was then abruptly stopped and the sample left free to dif-fuse along the column during the parking time (values of 1, 15, 30,45, and 60 min were used). Elution was resumed at the same flowrate and all samples were detected at a wavelength of 243 nm. Thedetection bandwidth was set at 4 nm and the sampling rate at 20 Hzfor all parking times. The slopes of the plots of the peak variances,2

t , versus the PP times, tp, provide a direct estimate of their trueeffective diffusion coefficients along the column [30]:

Deff = 12

�2t

�tpu2

R (24)

where uR = (u/1 + k1) is the migration linear velocity of the analyte.The B and Cp coefficients are then given by Eqs. (8) and 18. They

are listed in Tables 2 and 3 for the Symmetry and Titan columns,respectively.

3.7. HETP measurements

The retention time (tR at the peak apex) and the columnefficiency (or the time peak variance 2

t from the half-heightpeak width) of the eluted peaks were automatically processedby the integration tool of the Chemstation Software. Theintegration parameters were manually set as follows: slope sensi-tivity = 4 mAU/min, Peak width = 0.05 min, height reject = 10 mAU.The sequence of flow rates applied to the 3.0 mm I.D. Titan columnwas 0.20, 0.41, 0.61, 0.82, 1.02, 1.22, 1.43, 1.63, 1.84, 2.04, 2.24, 2.45and 2.65 mL/min. The sequence of flow rates applied to the 4.6 mmI.D. Symmetry column was 0.24, 0.48, 0.72, 0.96, 1.44, 1.22, 2.40,2.88, 3.36, 3.84, 4.32, and 4.80 mL/min. The maximum inlet pres-sure used was 990 bar (Titan) and 350 bar (Symmetry). Under suchflow rate and pressure drop conditions, the Titan column is not fullyisothermal, so the impact of the small radial temperature gradientson the column HETP are not negligible, especially for fully porousparticles [45,46]. In contrast, the impact of the frictional heatingon the efficiency of the Symmetry column is virtually negligible atmost flow rates (except for the three largest ones). Depending onthe flow rate, the sampling frequency was adjusted between 10 and160 Hz in order to record at least 150 data points for the whole bandprofile of acetanilide, the least retained compound. The injected

sample volume was set at 0.50 (Titan) and 1.0 (Symmetry) �L andthe chromatograms were recorded at a wavelength of 243 nm. Aconstant UV bandwidth of 4 nm was selected. The temperature wasset by the laboratory air-conditioning system at 23 ± 0.5 ◦C.

170 F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178

Table 2Diffusion coefficients (Dm), zone retention factor (k1), effective diffusion coefficients (Deff), reduced longitudinal diffusion coefficients (B), ratio of the particle diffusivity tothe bulk diffusion (˝), and solid–liquid mass transfer coefficient (Cp) measured for the seven compounds acetophenone, propiophenone, butyrophenone, valerophenone,hexanophenone, heptanophenone, and octanophenone at T = 23.0 ± 0.5 ◦C. The mobile phase composition is acetonitrile/water, 75/25, v/v. The column is the 4.6x150 mm onepacked with 4.81 �m Symmerty-C18 particles.

Compound Dm [cm2/s] k1 Deff [cm2/s] B Cp

Acetophenone 1.77 × 10−5 1.45 1.11 × 10−5 3.06 0.35 0.0176Propiophenone 1.64 × 10−5 1.97 9.41 × 10−6 3.41 0.42 0.0182Butyrophenone 1.53 × 10−5 2.60 7.80 × 10−6 3.67 0.48 0.0190Valerophenone 1.44 × 10−5 3.52 6.52 × 10−6 4.07 0.57 0.0186

−5 −6

h

w(

3

1l1ctattetfiw1fiptpadwcsfwM[

TDthp

Hexanophenone 1.37 × 10 4.91

Heptanophenone 1.30 × 10−5 7.01Octanophenone 1.24 × 10−5 10.16

The reduced plate height h is given by:

= L

dSauter

2t

t2R

(25)

here L is the column length and dSauter is the mean Sauter diameter2.04 �m for Titan and 4.76 �m for Symmetry particles) of the PSD.

.8. Minor disturbance experiments

The minor disturbance measurements were all acquired using a290 Infinity System (Agilent Technologies, Waldbroon, Germany)

iquid chromatograph connected to a HP1047 RI-detector with a40 �m × 220 mm metal capillary. An Agilent 35900E Interface wasonnected to the HPLC instrument and to the RI detector in ordero record the RI signal on the ChemStation Software. The excess ofcetonitrile onto the surface and inside the mesoporous volume ofhe 1.9 �m Titan-C18 and 5.0 �m Symmetry-C18 fully porous par-icles was determined from the linear perturbation of a series ofquilibrium concentrations by injecting a small volume of a solu-ion at different compositions. The chromatographic column wasrst equilibrated with a solution of known composition. Solutionsith volume fractions of acetonitrile in water having from 0 to 0.5,

, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99, 99.5, and 100% modi-er were successively pumped through the column. These mobilehases were systematically prepared by mixing the solvents withhe vHPLC pumps. After equilibration with the solution, systemerturbation peaks were generated by injection of 1 �L of eithercetonitrile or pure water into the column, recorded with the RIetector and their elution times were measured. The flow ratesere set at 0.5 mL/min (Titan column) and 1.0 mL/min (Symmetry

olumn) and the ambient temperature at 295 K ± 0.5 K for all mea-urements. The extra-column volume was minimized at 0.135 mL

rom the auto-sampler needle seat to the RI cell. All retention dataere corrected for this volume contribution. The results of theDM were also used to measure the void volume VM of the column

67,68]. The corresponding total porosities are given in Table 1.

able 3iffusion coefficients (Dm), zone retention factor (k1), effective diffusion coefficients (Def

he bulk diffusion (˝), and solid–liquid mass transfer coefficient (Cp) measured for the sexanophenone, heptanophenone, and octanophenone at T = 23.0 ± 0.5 ◦C. The mobile phaacked with 1.9 �m Titan-C18 particles.

Compound Dm [cm2/s] k1

Acetophenone 1.77 × 10−5 1.34

Propiophenone 1.64 × 10−5 1.85

Butyrophenone 1.53 × 10−5 2.47

Valerophenone 1.44 × 10−5 3.37

Hexanophenone 1.37 × 10−5 3.93

Heptanophenone 1.30 × 10−5 6.81

Octanophenone 1.24 × 10−5 9.96

5.34 × 10 4.61 0.70 0.01734.35 × 10−6 5.36 0.88 0.01523.45 × 10−6 6.22 1.11 0.0131

4. Results and discussion

The first part reports on the raw values of reduced HETPs for aseries of homologous compounds on the Titan-C18 and Symmetry-C18 columns. The second part investigates the physical reasons forthe observed differences in the efficiencies of these two columnsby measuring accurately the intra-particle diffusivity across theTitan-C18 and the Symmetry-C18 fully porous particles. Finally, thedifference between the values of the intra-particle diffusivities areinterpreted in terms of their relationships to the internal obstruc-tion factor, the average pore size and the RSD of the pore sizedistribution.

4.1. Experimental Van Deemter plots

The reduced plate heights, happ, were measured for sevenhomologous compounds as a function of the reduced interstitiallinear velocity, on both the column packed with 5.0 �m Symmetry-C18 particles and the one packed with 1.9 �m Titan-C18 particles.Fig. 2 shows the plots of these data uncorrected for the contribu-tion of the extra-column volume to the observed peak variancesand plate heights. The 1290 Infinity vHPLC instrument used inthis work was optimized in order to minimize the contribution ofthe extra-column band broadening to the overall measured bandbroadening. Its volume variance 2

v,ex is roughly around 10 �L2

near the optimum velocity. Yet, its effect on band dispersion isclearly measurable for the smallest column volume (V0 = 425 �L,Titan) packed with the finest particles, which provides the high-est maximum intrinsic plate count Nint � (100000/2 × 2.0) = 25,000.The ratio of the measured to the intrinsic plate heights is a functionof the retention factor k′. From Eq. (19), this ratio is written

happ

hint= 1 + Nint

2v,ex

V20

1

(1 + k′)2(26)

For the Titan-C18 column, at optimum velocity, the ratio

Nint(2

v,ex/V20 ) � 1.4. The retention factors of the seven homolo-

gous compounds increases from 0.50 to 0.83, 1.23, 1.81, 2.69, 4.02and to 6.04. According to Eq. (26), close to the optimum veloc-ity for ��5, the apparent plate height is larger than the intrinsic

f), reduced longitudinal diffusion coefficients (B), ratio of the particle diffusivity toeven compounds acetophenone, propiophenone, butyrophenone, valerophenone,se composition is acetonitrile/water, 75/25, v/v. The column is the 3.0x100 mm one

Deff [cm2/s] B Cp

8.13 × 10−6 2.15 0.18 0.03746.87 × 10−6 2.39 0.24 0.03725.87 × 10−6 2.66 0.30 0.03524.83 × 10−6 2.91 0.36 0.03443.29 × 10−6 3.28 0.46 0.03123.18 × 10−6 3.82 0.60 0.02672.48 × 10−6 4.39 0.76 0.0228

F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178 171

0 10 20 30 40 500

2

4

6

happ

4.6 mm x 150 mmSymmetry-C18

Reduced velocity νν

Acetophenone, k1=1.5 Propiophenone, k1=2.0 Butyrophenone, k1=2.6 Valerophenone, k1=3.5 Hexanophenone, k1=4.9 Heptanophenone, k1=7.0 Octanophenone, k1=10.2

0 10 20 30 40 500

2

4

6

3.0 mm x 100 mmTitan-C18

happ

Reduced veloci ty νν

Acetophenone, k1=1.3 Prop iophenone, k1=1.9 Bu tyrophenone, k1=2.5 Valerophenone, k1=3.4 Hexanophenone, k1=4.7 Heptanophenone, k1=6.8 Octanoph enon e, k1=10.0

Fig. 2. Plots of the apparent reduced plate heights of seven homologous compounds(n-alkanophenones) as a function of the reduced interstitial velocity for the 15 cml1t

owsc00stittStc

rotantt

vcTc

0 10 20 30 40 500

1

2

3

4

4.6 mm x 150 mmSymmetry-C18

Ace tophenone, k1=1.5 Propiophenone, k1=2.0 Butyrophenone, k1=2.6 Valerophenone, k1=3.5 Hexanophenone, k1=4.9 Heptanophenone, k1=7.0 Octanophenone, k1=10.2

B/νν + Cpνν

Reduced ve loc ity νν

0 10 20 30 40 500

1

2

3

4

3.0 mm x 100 mmTitan-C18

B/νν + Cpνν

Reduced ve loc ity νν

Acetophenone, k1=1.3 Prop ioph enon e, k1=1.9 Bu tyrophenone, k1=2.5 Valerophenon e, k1=3.4 Hexanophenone, k1=4.7 Heptanophenone, k1=6.8 Oc tanophenone, k1=10.0

Fig. 3. Plots of the sum of the experimental reduced longitudinal diffusion (B/�) and

ong column packed with 5.0 �m Symmetry-C18 particles (top graph) and for the0 cm long column packed with 1.9 �m Titan-C18 particles (bottom graph). Notehe impact of the retention factor on the observed Titan column performance.

ne by 62, 42, 28, 18, 10, 5.6, and 2.8%, respectively. This explainshy the seven reduced h versus � plots do no overlay but are

hifted toward lower reduced plate heights with increasing k′. Inontrast, for the Symmetry-C18 column, V0=1414 �L2, Nint � (150,00/2 × 4.8) = 15,625, Nint(2

v,ex/V20 ) � 0.08 and k′ increases from

.48 to 0.80, 1.18, 1.74, 2.58, 3.86 to 5.77. Accordingly, the mea-ured plate heights are only 3.6, 2.5, 1.7, 1.1, 0.6, 0.3 and 0.2% largerhan the intrinsic plate heights. The extra-column band broadenings then virtually negligible for the large volume column packed withhe largest particles. This explains why the reduced plate heights ofhe seven homologous are very close to each other at high velocities.ome differences can be observed at the lowest reduced velocities;hey are due to the significant variation of the longitudinal diffusionoefficient with the retention factor.

The most remarkable result is that the minimum intrinsiceduced plate height observed for the the most retained compound,ctanophenone, is lower on the Titan column (hmin = 1.7 for �opt = 5)han on the Symmetry column (hmin = 2.1 for �opt = 9). This result is

priori surprising because it is more challenging to uniformly packarrow (3.0 mm i.d.) columns with fine particles (2.0 �m in diame-er) than wider (4.6 mm) columns with large (5 �m) particles. Yet,his observation is indisputable.

This result seem to be consistent with the relatively small

alue of the optimum reduced velocity observed for the Titan-C18olumn, 5 versus 9 for the conventional Symmetry-C18 column.his could suggest that the reduced longitudinal diffusion coeffi-ient B of the Titan column is smaller than that of the Symmetry

solid–liquid mass transfer resistance (Cp�) HETP terms for the 15 cm long columnspacked with 5.0 �m Symmetry-C18 particles (top graph) and for the 10 cm longcolumns packed with 1.9 �m Titan-C18 particles (bottom graph).

column and that the reduced solid–liquid mass transfer resistancecoefficient Cp is larger for the Titan column. This suggestion was val-idated with the PP measurements. The principle and quantitativedetails of the PP experiments are given in the experimental section.The results are listed in the fifth (B) and seventh (Cp) columns ofTable 2 (Symmetry column) and Table 3 (Titan column). Fig. 3 com-pares the sum of the experimental reduced HETP terms (B/�) + Cp�of these two columns. For the lowest reduced interstitial veloc-ity (� � 1) on the far left side of these two graphs, the B term isclearly larger for the Symmetry-C18 than for the Titan-C18 column.Inversely, the Cp coefficient, which is inversely proportional to theintra-particle diffusivity (Dp = ˝Dm), is larger for the Titan column(see the far right side of the two graphs). These two observationsdemonstrate indirectly but clearly that the intra-particle diffusivityis smaller across the Titan-C18 particles than across the Symmetry-C18 particles. This results is also supported by the results of the PPexperiments given in Tables 2 and 3 (sixth column). It is all the moresurprising that the internal porosities �p of these two brands of fullyporous C18-bonded fully porous particles are virtually the same (�0.35, Table 1). Further investigations are necessary to justify thisunusually low value of the coefficient for the 1.9 �m Titan-C18

fully porous particles.

It is often believed that particles with a narrow PSD can bepacked more uniformly than those having a wide PSD. Thus, toascertain that the superior performance of the Titan column is not

172 F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178

0.0 0.2 0.40

2

4

6

8

10

3.0 mm x 100 mmTitan- C18

0.20 mL/min 0.41 mL/min 0.61 mL/min 0.82 mL/min 1.02 mL/min 1.22 mL/min 1.43 mL/min 1.63 mL/min 1.84 mL/min 2.04 mL/min 2.24 mL/min 2.45 mL/min

(1+k')-2

Hap

p [μμ

m]

020100

1

2

3

4

5 3.0 x 100 mm Titan-C18, k1~ 2

hint-B/ν - Cpν

4.6 mm x 150 mmSymme try-C18

Aceto phen one , k1=1.5 Propiophenone, k1=2.0 Butyrophenone, k1=2.6 Vale rophenone, k1=3.5 Hexa nophen one , k1=4.9 Hepta nophen one , k1=7.0 Octanophe none, k1=10.2

happ-B/νν - Cpνν

Reduced velo city νν

Fig. 4. (Top graph) Plots of the observed plate height Happ versus (1 + k′)−2 for the twelve applied flow rates on the Titan column and three homologous n-alkanophenonesamples (acetophenonone, propiophenone, and butyrophenone). The linear extrapolation of these plots to an infinitely large retention factor k′ provides the intrinsic HETPa termsp ve tow

dmsHo(eiptTsrtbfphvpmlb

t

t a given flow rate. (Bottom graph) Plots of the reduced eddy dispersion HETP

ropiophenone on the Titan column. Note the parabolic deviation of the purple cur

ue to a smaller reduced eddy dispersion term, A(�), it was esti-ated experimentally by subtraction, as done in the experimental

ection. The top graph in Fig. 4 shows plots of the apparent HETPapp (non-corrected for extra-column contributions) as a functionf the reciprocal of (1 + k′)2 for the least three retained compoundsacetone, propiophenone, and butyrophnone). These plots are lin-ar at all applied flow rates, from 0.20 to 2.45 mL/min because thentrinsic plate heights and the diffusion coefficients of these com-ounds are relatively close. This step was necessary to eliminatehe contribution of the instrument to the observed band dispersion.he minimum reduced HETP was then measured at a low 1.7. Afterubtraction of the B/� and Cp� reduced HETP terms from the totaleduced plate height, hint, the bottom graph in Fig. 4 shows thathe eddy dispersion of the Titan column (full purple square sym-ols) is matching well with those measured by the same methodor the Symmetry column. Note that the deviation of the eddy dis-ersion term above � = 5 for the Titan column is due to frictionaleating [1]. For the sake of comparison, in the range of reducedelocity between 0 and 20, frictional heating affects very little theerformance of the Symmetry column due to its much higher per-eability. For � = 20, the frictional heat power per unit column

ength is only about 7 W/m for the 15 cm long Symmetry columnut it is 40 W/m for the 10 cm long Titan column.

In the next section, the effective intra-particle diffusivity acrosshe Titan-C18 and Symmetry-C18 fully porous particles is analyzed

versus � measured for each n-alkanophenone on the symmetry column and forards high values due to the effect of frictional heating on the column efficiency.

in details by combining the results of the PP, ISEC, and MDM mea-surements.

4.2. Investigation and measurement of the intra-particlediffusivity

The intra-particle diffusivity is a property directly related to theinternal structure of the particle (porosity, pore size, tortuosity,constriction, pore connectivity). ISEC experiments are first carriedout in order to estimate the internal porosity (�p), the average poresize (Dp), and the width of the pore size distribution (v).

4.2.1. ISEC dataFig. 5 shows plots of the measured fraction of the particle volume

accessible to a series of polystyrene standards as a function of theirsize ranging, from 10 to 1000 A (logarithmic scale). The two curvesare similar, although they intersect for a hydrodynamic diame-ter of about 80 A. The PS standards smaller than 80 A are slightlymore excluded on Titan-C18 than on Symmetry-C18 columns andvice versa for PS sizes larger than 80 A. However, the difference inaccessible volume fractions of the two brands of particles is relative

small. For molecules smaller than 10 A (toluene, 6.5 A), the accessi-ble porosity is even slightly larger in the Titan particles at �p=0.35versus 0.34 for the Symmetry particles. The ISEC data show that thesmall values of n-alkanophenones intra-particle diffusivity across

F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178 173

0001001010.0

0.1

0.2

Crossing point

Polystyrene hydrod yma mic diameter DH [Å]

Symmetry-C18 Titan-C18

Part

icle

por

osot

y ac

cess

ible

Fig. 5. Experimental plot of the fraction of the particle volume accessible to a seriesofp

ti

adc

Fdp

00101

0.0

0.1

0.2

Part

icle

por

osot

y ac

cess

ible

n=3εp=0.34

Dp=96 Åσv=16%

Polystyrene hyd rody mamic diamete r DH [Å]

ISEC data, 5 μμm Symmetry-C18Best fit

0.2

soty

acc

essi

ble

n=3εp=0.35

ISEC data, 1.9 μm Titan-C18Best fit

f polystyrene standards as a function of their hydrodynamic diameter DH rangingrom 10 to 1000 A (in a logarithmic scale) for the RPLC-C18 symmetry and Titanarticles.

he Titan particles is unrelated to an unusually small value of theirnternal porosity.

The fits of Eq. (20) (in which pores are assumed to be cylinders)

nd Eq. (21) (in which pores are assumed to be spheres) to the ISECata was successful for both the Symmetry-C18 and the Titan-C18olumn data. In order to limit the number of adjustable parameters

00101

0.0

0.1

0.2

Part

icle

por

osot

y ac

cess

ible

n=2εp=0.34

Dp=69 Åσv=20%

Polystyrene hydrodymamic diameter DH [Å]

ISEC data, 5 μμm Symm etry-C18Best fi t

00101

0.0

0.1

0.2

Part

icle

por

osot

y ac

cess

ible

n=2εp=0.35

Dp=55 Åσv=30%

Polystyrene hydrod ymam ic diameter DH [Å]

ISEC data , 1.9 μm Titan-C18Best fit

ig. 6. Fit of the ISEC data to the model Eq. (20). The pores are assumed to be cylin-ers. (Top graph) 5.0 �m Symmetry-C18 particles. (Bottom graph) 1.9 �m Titan-C18

articles.

00101

0.0

0.1Pa

rtic

le p

oro Dp=78 Åσv=25%

Polystyrene hydrod ymam ic diameter D [Å]

H

Fig. 7. Same figure as in Fig. 6, except the pores are assumed to be spheres (Eq. (21)).

in these two models (�0p , v, n, and Dp), the results of the best fits

are shown in Fig. 6 (for n = 2, cylindrical pore shape) and Fig. 7 (forn = 3, spherical pore shape) in which �0

p was set at the experimentalvalues given in Table 1. Only the average pore diameter Dp and thestandard deviation v of the pore size distribution were optimizedduring the fitting procedure.

The results of these fits suggest that the average pore size ofthe Titan-C18 particles is about 15–20 A smaller than that of theSymmetry-C18 particles. They also suggest that the pore size dis-tribution of the Titan-C18 particles is wider, with v of the orderof 25–30% rather than 15–20%. Note that the average pore sizediameter estimated by assuming that the pores are spheres provideexactly the median pore diameter of the unbonded Symmetry parti-cles as reported by the manufacturer (96 A). Since the average porediameter decreases after C18 derivatization and surface endcap-ping by about 19 A for a surface coverage of 3.2 �mol/m2 [69,70],the expected value of Dp = 96 − 19 = 77 A is intermediate betweenthose estimated by assuming n = 2 (69 A, cylindrical pores) and n = 3(96 A, spheres). For C18 surface coverages around 2.0 �mol/m2, thedecrease of the pore diameter after derivatization and endcappingof the silica Titan particles is likely around 14 A [69,70]. Followingthis simple rule of thumb, the expected value of Dp for Titan-C18particles is close to 59 A. Inversely, by fixing Dp = 77 and 59 A in thefitting procedure, the best n value of the pores was found equal to2.3 and 2.2, and v = 19% and 29% for Symmetry-C18 and Titan-C18

particles, respectively. All these results are summarized in Table 4.

These results are highly important but need to be confirmedby BET low temperature nitrogen adsorption experiments. TheTitan-C18 and Symmetry-C18 particles have the same internal

174 F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178

Table 4Best average pore size diameter Dp, relative standard deviation v , pore dimensionality n of the 1.9 �m Titan-C18 and 5.0 �m Symmetry-C18 particles estimated from ISECdata and model equations.

Name Batch number Column number Particle porosity ε0t Dp

a (A) Fixed parameter Dp (A) v (%) n

Titan-C18 8033 3086 0.35 79 n = 2 55 30 –n = 3 78 25 –Dp = 59 A – 29 2.2

Symmetry-C18 279 0279333111 0.34 96 n = 2 69 20 –n = 3 96 16 –

the r

vildnrsw

4

taepttmtbfrtisitodf

Fp(git

a Median pore diameter measured by the manufacturer from BET experiments on

olume (�0.35) but a clearly different average pore size. This maympact the hindrance diffusion factor F(�m) (the ratio of the ana-yte size to the average pore diameter) of the small analytes whichiffuse through the mesoporous network. Additionally, the inter-al obstruction factor, �p, of these two particles, which is mainlyelated to their porosity, must be measured. The goal of the nextection is to estimate these parameters from PP experiments madeith a non-retained compound.

.2.2. PP results for a non-retained compoundThe measurement of the product of the obstruction factor and

he hindrance diffusion factor of the analyte molecules diffusingcross porous particles must be made with a non-retained solute toliminate the contribution of surface diffusion to the overall intra-article diffusivity [34]. Thiourea was selected. Fig. 8 shows theotal column porosity measured with thiourea in a mixture of ace-onitrile and water (75/25, v/v). It slightly underestimates the value

easured with toluene in pure THF, an observation consistent withhiourea being less soluble in pure acetonitrile than in the externalulk phase (75% v/v acetonitrile) because an adsorbed multi-layerorms on the surface of C18-bonded silica [67,68,71,72]. The sameesult was recently obtained with the non-retained uracil, knowno be insoluble in pure acetonitrile at room temperature [73]. Thiss why hold-up volumes should always be measured with a pureolvent so that the mobile phase composition is the same in thenternal (mesopores) and the external (macropores) volumes. Note

hat the total column porosity determined by MDM with mixturesf acetonitrile and water slightly overestimates that given by ISECata (+4%). The same relative difference was previously reportedor values measured by pycnometry, which provides the true value

0.0

0.2

0.4

0.6

Titan-C18 Symmetry-C18

Tota

l col

umn

poro

sity

Thiourea injection (CH3CN/H2O, 75 /25, v/v )ISE C, Toluene i nject ion ( THF)Mino r Disturbance Method (CH3CN/H2O)

ig. 8. Comparison between the results of the measurements of the total columnorosity �t from (1) the injection of thiourea in a mixture of acetonitrile and water75/25, v/v), as shown in red color; (2) that of toluene in pure toluene, shown inreen; and (3) the MDM data made with binary acetonitrile/water mixtures. (Fornterpretation of the references to color in this figure legend, the reader is referredo the web version of the article.)

Dp = 77 A – 19 2.3

aw silica material.

of the hold-up column volume [74], (the true value of the hold-upvolume referring to the column volume accessible to small eluentmolecules such as methanol and dichloromethane used in the pyc-nometry experiment) and for those measured from toluene elutedin pure THF [73]. Thus, due to the negative difference between themesopore and macropore concentrations of thiourea, the apparentparticle porosities accessible to thiourea are �p,thiourea=0.279 and0.250 for the Symmetry-C18 and Titan-C18 particles, respectively.

The determination of the coefficient = �p,thiourea�pF(�m) fornon-retained compounds (Ka=0) was explained in details in the the-ory section. was measured at 0.106 (Symmetry-C18) and 0.056(Titan-C18). The products �p × F(�m) for thiourea are 0.380 and0.224, respectively. The hindrance diffusion factor F(�m) can beestimated from the [75] correlation:

F(�m) = (1 − �m)2(1 − 2.1044�m + 2.089�3m − 0.948�5

m) (27)

where �m is the ratio of the molecular size of thiourea (5.5 A) to theaverage pore size (77 and 59 A). So, �m = 0.071 (F(�m) = 0.735) and0.093 (F(�m) = 0.663) for the Symmetry-C18 and Titan-C18 porousparticles.

Accordingly, the PP measurements applied to the non-retainedcompound thiourea provide a good estimate of the obstruction fac-tor through the two fully porous particles studied in this work:�p is equal to 0.517 and 0.338 for the Symmetry-C18 and Titan-C18 fully porous particles, respectively. The PP data confirm thenthat the sample diffusivity through the mesopores of the Titan-C18 particles is more hindered than that through the mesoporesof the Symmetry-C18 particles due to 1) the enhanced obstructionfor diffusion along the internal channels of the Titan particles (�p

decreases by −35%) and 2) the increasing diffusion hindrance duethe confinement of the analyte into narrower mesopores (F(�m)decreases relatively by −10%). This explains why the B term of theTitan column is smaller and its Cp coefficient is larger.

4.2.3. MDM: actual internal eluent compositionIn the previous section, the mobile phase composition in the

mesopores of the two fully porous particle brands were assumedto be the composition of the bulk eluent (acetonitrile/water, 75/25,v/v). However, the C18 surface coverages of the Titan and Sym-metry particles are 2.0 and 3.2 �mol/m2, respectively. Therefore,the acetonitrile concentration in the mesoporous volume exceedsthe bulk composition. Thus, the diffusion coefficient of free analytemolecules in the internal porous volume of the two particle brandsmay differ. Fig. 9 shows plots of the excess concentration of acetoni-trile ce

A in the stationary phase (VC − VM) as a function of the volumefraction of acetonitrile xACN. Remarkably, these plots are similar forSymmetry-C18 and Titan-C18 particles despite their difference insurface coverage. The larger concentration excess (+17%) observedfor the Titan-C18 column is most likely explained by the specific sur-face area of Titan particles (410 m2/g) being larger (+20%) than that

of Symmetry-C18 particles (340 m2/g). For the bulk mobile phasecomposition used (xACN = 0.75), the excess amount of adsorbed ace-tonitrile is 1.06 mmol in the Symmetry column and 0.33 mmolin the Titan column. The respective total mesoporous volumes of

F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178 175

0.0 0.2 0.4 0.6 0.8 1.0

0.000

0.001

0.002

0.003

xACN

n Ae /(Vc-V

M)

[mol

/cm

3 ]

Titan-C18Symmetr y-C18

F e

vp

tftltScaAttndafopoT

ddomdw

˝

a

˝

vatSC

0 2 4 6 8 100.0

0.4

0.8

1.2

Best Landaue r mode l, δδ=0.42 , ββ=-0.3 4 Best Landauer model, δ=0.21, β=-0.19

Part

icle

diff

usiv

ity ΩΩ

=Dp/D

m

Equilibrium constant Ka

Data Symmetry-C18Data Tita n-C18

Fig. 10. Plots of the intra-particle diffusivity coefficient as a function of the equi-librium constants Ka for the symmetry (full black squares) and the Titan (full red

ticles. This difference of intra-particle diffusivity directly explainsthe difference observed for the reduced longitudinal coefficient Bshown in Fig. 11 and listed in Tables 2 and 3.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Long

itudi

nal c

oeffi

cien

t B

Equili bri um consta nt Ke=εεek1/(1-εεe)

Symmetry-C18 Tit an-C18

ig. 9. Plots of the excess concentration of acetonitrile cA

in the stationary phaseolume (VC − VM) as a function of the volume fraction of acetonitrile xACN in the bulkhase.

hese columns are 0.556 and 0.151 cm3. Thus, the average volumeraction of acetonitrile in the mesoporous volumes of the Symme-ry and Titan particles are 85.0% and 86.4%, respectively. Due to itsarger excess in the stationary phase volume (Fig. 9), the acetoni-rile concentration is slightly larger in the Titan-C18 than that in theymmetry-C18 particles. The difference in intra-particle diffusivityannot be due to differences in the surface coverages nor in thecetonitrile enrichment (+10%, v/v) inside the mesoporous volume.n adsorbed multilayer of pure acetonitrile exists at the surface of

he C18-bonded layer. Calculations show that it extends far abovehe surface since it occupies 40% and 46% of the particle inter-al volume for Symmetry and Titan, respectively. Given the poreimensionalities estimated from the ISEC data at D=2.3 and 2.2, thecetonitrile multilayer extends up to 7.7 and 7.2 A, above the sur-ace, respectively. Assuming an acetonitrile monolayer thicknessf 2.5 A, this means that, irrespective of the nature of the RPLC-C18articles, three monolayers of acetonitrile are stacked up on the topf the C18-bonded layer. There is no significant difference betweenitan-C18 and Symmetry-C18 fully porous particles.

The most important result of this section is that the averageiffusion coefficients of analytes in the mesopores differs from theiffusion coefficient in the external bulk eluent. The relative changef the viscosity from acetonitrile–water eluent (ACN 75%) to theesoporous eluent (ACN 85%) should be taken into account in this

iscussion. Accordingly, the coefficient for thiourea should beritten for the Symmetry-C18 particles as:

= �(75.0%)�(85.0%)

× �p,thioureaF(�m)�p

= 1.246 × 0.279 × 0.735 × �p (28)

nd for Titan-C18 particles, it should be:

= �(75.0%)�(86.4%)

× �p,thioureaF(�m)�p

= 1.288 × 0.250 × 0.663 × �p (29)

The excess amount of acetonitrile adsorbed in the mesoporousolume of both particles affects the internal diffusion coefficient of

nalytes. Due to this excess, the internal obstruction factors haveo be corrected. They were decreased from 0.517 to 0.415 (for theymmetry-C18 column) and from 0.338 to 0.262 (for the Titan-18 column). In conclusion, the obstruction to internal diffusion

circles) RPLC-C18 columns. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of the article.)

through Titan-C18 particles is 1.6× larger than through Symmetry-C18 particles.

4.2.4. PP results using a homologous series of retainedn-alkanophenones

Figs. 10 and 11 show the plots of the coefficients and B asa function of the equilibrium constants Ka and Ke, respectively, ofthe Symmetry (full black squares) and the Titan (full red circles)RPLC-C18 columns. When the analyte mass increases from those ofacetophenone to octanophenone, the intra-particle diffusivity andthe longitudinal coefficients increase both, due to the increase ofsurface diffusion along the silica-C18 surface [37]. The most remark-able observation in Fig. 10 is that the intra-particle diffusivity isclose to 40% larger across the Symmetry-C18 than across the Titan-C18 particles for retained analytes. This confirms a previous resultfound for the non-retained compound thiourea showing enhancedobstruction for diffusion along the mesopores of the Titan-C18 par-

Fig. 11. Plots of the reduced longitudinal coefficient B as a function of the equilib-rium constant Ke for the symmetry (full black squares) and the Titan (full red circles)RPLC-C18 columns. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of the article.)

176 F. Gritti, G. Guiochon / J. Chromatogr. A 1355 (2014) 164–178

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Symm etry-C18Titan-C18

Equilibrium cons tant Ka

DS/Dm

Fig. 12. Plots of the ratio of the diffusion coefficient in the stationary phase to thebulk diffusion coefficient, DS/Dm , as a function of the equilibrium constant Ka for thesit

(im(mpC

tiStRiatiopitctlat

4t

cf[

�

ioauc

ymmetry (full black squares) and the Titan (full red circles) RPLC-C18 columns. (Fornterpretation of the references to color in this figure legend, the reader is referredo the web version of the article.)

The effective model of intra-particle diffusivity (Eqs. (15) and16)) was fitted to experimental data in Fig. 10. The parameter �n the expression of diffusion in a confined pore geometry was

easured earlier for thiourea at � = 1.246 × 0.415 × 0.735 = 0.380Symmetry) and 1.288 × 0.262 × 0.663 = 0.224 (Titan). The agree-

ent between the model and the data is good for the Symmetry-C18articles with ı = 0.42 and � = −0.34 and is excellent for the Titan-18 particles with ı = 0.21 and � = −0.19.

The ratios of the diffusion coefficients of retained analytes inhe stationary phase volume and in the eluent, DS/Dm, are shownn Fig. 12. They are obviously smaller for the Titan-C18 than for theymmetry-C18 columns because these values are directly relatedo the smaller pore obstruction factor measured for Titan particles.etained analyte molecules do not physically enter the solid and

mpermeable silica-C18 phases. Instead, they either adsorb in thecetonitrile-rich multilayer [76,71], at the very interface betweenhe C18-bonded layer and the acetonitrile layer [77], or at the sil-ca surface when its access is not hindered [78,77,71]. Therefore,bstruction to diffusion in the stationary phase that actually takeslace along the surface of the mesopore walls is related to that

nside the mesoporous volume [63]. The tortuosity (or ratio ofhe actual to the shortest path length) and the constriction (theross-section area or contour varies along the diffusion path) ofhe surface and the volume diffusion paths are necessarily corre-ated. A simple model of constriction is proposed next in order toccount for the impact of the width of the pore size distribution onhe internal obstruction factor.

.2.5. Possible effect of the width of the pore size distribution onhe constriction factor

The overall obstruction factor �p results from the simultaneousombination of tortuous (tortuosity ) and constricted (constrictionactor �c) diffusion paths. Giddings suggested for �p the expression79]:

p = �c

2(30)

The goal of this section is to assess from qualitative and theoret-cal viewpoints the possible impact of a wide pore size distribution

n the constriction factor �c. The pores are assumed to be straightnd open-ended, which gives them a tortuosity factor, , equal tonity. The cross-section area along the pores are those of a trun-ated cone with two end radii r and R. Fluctuations of the relative

Fig. 13. Plots of �c,s and �c,v as a function of the ratio of the smallest radius r to thelargest radius R (0 < (r/R) < 1) at both ends of open-ended pores. The shape of thepores is a truncated cone.

distance between r and R is assumed to account for the RSD of thepore size distribution. If this distribution is uniform then r

R = 1 andthe constriction factor is unity. The general expression of the con-striction factor is given along the pore volume (representing porediffusivity) by [79]:

�c,v = 1〈S〉〈1/S〉 (31)

where 〈S〉 and 〈1/S〉 are the mean cross-section area and the meanreciprocal cross-section area, respectively. This surface area is per-pendicular to the pore axis.

For diffusion along the surface wall of the pores (representingsurface diffusion), the constriction factor is written:

�c,s = 1〈l〉〈1/l〉 (32)

where 〈l〉 and 〈1/l〉 are the mean contour length and the mean recip-rocal contour length, respectively. The surface generated by theclosed contour length is equal to S.

Consider the diffusion process along an open-ended truncatedcone with end radii r and R. The tortuosity factor is unity. After a fewelementary integration steps, the constriction factors are expressedas the ratio of the end radii r

R :

�c,v = 3(r/R) + 1 + (R/r)

(33)

and

�c,s = 2ln(R/r)

1 − (r/R)1 + (r/R)

(34)

Fig. 13 shows the plots of �c,s and �c,v as functions of the ratio ofthe smallest radii r to the largest radius R at both ends of the openpore (0 <(r/R)< 1). This graph confirms that the two constriction fac-tors are correlated and both increase from 0 (cone-shaped pore) to1 (cylindrical pore). Analysis of the ISEC data suggests that the RSDof the volume pore size distribution of the Titan-C18 and Symmetry-C18 particles were around 30% and 20%, respectively. The averagepore sizes were Dp = 59 A and 77 A. For the sake of simplicity, let usassume that the complex and actual pore size distribution can berepresented by those of truncated pores with their end radii equalto the average pore size plus one (R(+v)) and minus one (r(−v))

standard deviation. Therefore:

log[r(−v)] = (1 − v) log[

Dp

2

](35)

omato

a

l

(pf�dtwp

s0ai8rtctndpt

5

mwoetrTstteapwmebas

Dcmvot

CiaP5bio

[

[

[

[

[

[

[

[

[

[

[

[

[

[

F. Gritti, G. Guiochon / J. Chr

nd

og[R(+v)] = (1 + v) log[

Dp

2

](36)

One can now define two particular values for the ratior(−v))/(R(+v)) = 0.08 (Titan-C18) and 0.18 (Symmetry-C18). Theredicted constriction factor �c,v along the pore volume increasesrom 0.24 (Titan) to 0.46 (Symmetry). The experimental values ofp are 0.262 and 0.415. The relative values agree well with thoseerived from this simple model but they certainly underestimatehe true values by about 50–70% because the tortuosity factorsould be close to unity for both particles. In contrast, for an internalorosity around 0.35, is typically equal to 1.3 [63].

According to this model, the constriction factor �c,s along theurface area of the pore wall would increase from 0.68 (Titan) to.82 (Symmetry). So, surface constriction would not be as severes volume constriction along the same pore. Regarding the exper-mental surface diffusion for the most retained compounds (Ka�), the experimental ratio of DS,Symmetry/DS,Titan is close to 1.4. Theatio of the estimated constriction factors is 1.2. This confirms therend observed and the simplistic character of the present model ofonstriction along a surface. Overall, however, this model supportshe suggestion that a wide pore size distribution or the intercon-ections between very small and very large mesopores is somehowetrimental in terms of constriction effect. This could be the mainhysical explanation for an internal obstruction factor smaller forhe 1.9 �m Titan-C18 than for the 5.0 �m Symmetry-C18 particles.

. Conclusion

This work explored the reasons for the exceptionally low mini-um reduced plate heights (hmin = 1.7) of the RPLC columns packedith 1.9 �m fully porous Titan-C18 particles. By combining results

f accurate HETP, ISEC, MDM and PP experiments, it gave anxperimental demonstration that this reason is due to an excep-ionally low intra-particle diffusivity of non-retained (thiourea) andetained (series of homologous n-alkanophenones) compounds.he sample diffusivity across the Titan particles is mostly due to amall internal obstruction factor (�p = 0.26) and, to a lesser extent,o a relatively small average pore size (59 A). It is important to addhat the internal porosity (�0.35) of the Titan particles and that thexcess adsorption of acetonitrile onto the C18-bonded layer (threedsorbed monolayers of pure acetonitrile) are conventional, com-arable to those measured for the 5 �m Symmetry-C18 particles forhich �p = 0.42 and the average pore size is 77 A. Finally, a simpleodel of diffusion along truncated cones suggests that the differ-

nce in the internal obstruction factors of the particles of these tworands is due to the enhanced constriction of the diffusion pathscross the Titan particles. This is likely related to their wide poreize distribution, with a RSD of about 30%.

Finally, the longitudinal diffusion coefficient B in the vaneemter equation is found to be 30% smaller for the Titan-C18olumn than that for the Symmetry-C18 column; its solid–liquidass transfer coefficient Cp is 75% larger and its optimum reduced

elocity has unusually small values, between 3 and 5, dependingn the retention factor of the analyte, much smaller than those ofhe Symmetry-C18 column, between 6 and 10.

To summarize, the excellent kinetic performance of the Titan-18 column at low reduced velocity is not due to the narrow PSD of

ts particles, with a RSD around 10%. This confirms a wealth of pastnd recent literature data that demonstrate the lack of impact of theSD on the column efficiency provided that the RSD is lower than

0% as was the case in this work. Definitely, Titan-C18 columns cane recommended for the analysis of small molecules. Short, wide

.d. columns should be preferred to minimize the negative effectsf frictional heating on the column efficiency and to improve the

[

gr. A 1355 (2014) 164–178 177

uniformity of the packed bed by minimizing the wall effects onsample dispersion.

Acknowledgements

This work was supported in part by the cooperative agreementbetween the University of Tennessee and the Oak Ridge NationalLaboratory, with the support of the Waters Technologies Corpo-ration. We thank David Bell (Bellefonte, PA, USA) for the gift ofTitan-C18 column and for fruitful discussions. We thank MartinGilar (Milford, MA, USA) for the gift of the Symmetry-C18 column.

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