Mass-Transfer Kinetics in a Shell Packing Material for

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Anal. Chem. 2007, 79, 5972-5979

Mass-Transfer Kinetics in a Shell Packing Material for Chromatography Alberto Cavazzini,†,‡ Fabrice Gritti,† Krzysztof Kaczmarski,§ Nicola Marchetti,† and Georges Guiochon*,†

Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600 and Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120, and Faculty of Chemistry, Rzeszo´ w University of Technology West Pola 2 Street, 35-959, Rzeszo´ w, Poland

A shell particle consists of a solid, nonporous core that is surrounded with a shell of a porous solid having essentially the same physicochemical properties as those of the conventional porous particles used as packing media in chromatography. The diameter of the solid core and the thickness of its shell or the external diameter of the particle characterizes the chromatographic properties of the packing material. The potential advantage of this particle structure would be the shorter average path length experienced by solute molecules during their diffusion across the particles of packing material when they are retained. Compounds having slow pore diffusion would exhibit higher efficiencies on columns packed with shell than with conventional, fully porous particles. Using columns packed with Halo, a new type of porous silica shell particles, we assess the gain achieved with this principle for peptides of moderate molecular weights and for small proteins. Although significant progress was made between the early 1980s and the early 2000s in column technology and instrument performance, the pace of changes was slow, improvements were often marginal, and few, if any, important changes took place. In the past few years, this situation has changed profoundly. Considerable pressure is applied to column and instrument manufacturers by those who need more analytical data and need them faster. This pressure is applied in various directions. Analyses of common samples should be provided faster and be less costly; more complex samples must be analyzed. For all these reasons, the ancient issue of high column performance must be revisited. Although the fundamental problems of chromatography remain the same,1 although the pressure vise has not bulged,2 technical improvements allow significant progress, pushing somewhat the borders that had long been accepted.3 Although there are practical limits to what can be achieved in a service laboratory, chromatography can be carried out under higher pressures, using shorter columns packed with finer particles.4 HPLC can be carried * Corresponding author: (e-mail) [email protected]; (fax) 865 974-2667. † University of Tennessee and Oak Ridge National Laboratory. ‡ On leave from: the Department of Chemistry, University of Ferrara via L. Borsari 46, 44100 Ferrara, Italy. § Rzeszo´w University of Technology. (1) Guiochon, G. J. Chromatogr., A 2006, 1126, 6. (2) Knox, J. H.; Saleem, M. J. Chromatogr. Sci. 1969, 7, 614. (3) Giddings, J. C. Anal. Chem. 1965, 37, 60.

5972 Analytical Chemistry, Vol. 79, No. 15, August 1, 2007

out at higher temperatures.5 Monolithic columns offer a more attractive combination of column permeability and separation power.6 Finally, significant progress have been made recently in the preparation of sophisticated particles for chromatography.7 The concepts of shell and pellicular particles and their use in chromatography is ancient.8-12 This approach was already used in the early 1970s when conventional particles had diameters in the 40-µm range and Corasil, a brand of superficially porous (3750 µm diameter) silica particles, became popular for a while8 before becoming obsolete when the generations of 10-, then 5-, and then 3-µm particles became available. Later, Zipax (30-µm diameter) had the same fate. Poroshell is an attractive packing material made of 5-µm particles with a solid core covered with a 0.25-µm-thick porous silica shell.12 Recently, a new packing material, Halo, was made available. It has particles with an average diameter of 2.7 µm, made of a solid core covered with a 0.50-µmthick shell. Horva´th preferred pellicular particles, which have a similar structure and differ from shell particles in that they have a much thinner layer of porous adsorbent coated around a solid, nonporous core.10 Shell and pellicular particles have the potential advantage of offering to the molecules of the sample components a thinner layer of porous material to diffuse through after they enter a particle and before they leave it. Using the general rate model of chromatography, Kaczmarski has recently shown that the efficiency of large molecules increases with decreasing thickness of the shell of a shell particle, at constant particle diameter.13 Shell particles should permit faster HPLC separations of biochemicals than fully porous particles made of the same silica gel. In the same time, the inlet pressure should remain reasonable, allowing the use of long columns. If we consider a series of analytes having increasing molecular weights, we observe that these compounds also have decreasing molecular diffusivities. Most contributions to the mass-transfer resistances through particles are proportional (4) Mazzeo, J. R.; Neue, U. D.; Kele, M.; Plumb, R. S. Anal. Chem. 2005, 77, 460A. (5) Yang, X.; Ma, L.; Carr, P. W. J. Chromatogr., A 2005, 1079, 213. (6) Guiochon, G. J. Chromatogr., A 2007, 1100, 00. (7) http://www.advanced-materials-tech.com/halo.html, 2007. (8) Kirkland, J. J. Anal. Chem. 1969, 41, 218. (9) Horva´th, C. G.; Preiss, B. A.; Lipsky, S. R. Anal. Chem. 1967, 39, 1422. (10) Horva´th, C.; Engasser, J.-M. Ind. Eng. Chem. Fundam. 1973, 12, 229. (11) Kalghatgi, K.; Horva´th, C. J. Chromatogr. 1990, 508, 61. (12) Kirkland, J. J. Anal. Chem. 1992, 64, 1239. (13) Kaczmarski, K.; Guiochon, G. Anal. Chem. In press (DOI: 10.1021/ ac070200w). 10.1021/ac070571a CCC: $37.00

© 2007 American Chemical Society Published on Web 06/20/2007

to the molecular diffusivity.14 This explains why a given column exhibits the same or a slightly less efficiency for large rather than for low molecular weight compounds but needs to be operated at a markedly lower mobile-phase velocity.15 This smaller mobilephase flow velocity leads to longer analysis times but requires a lower inlet pressure. These properties of shell particles have already been noted for Poroshell columns by Carr et al.16 Without studying the issue in great detail, these authors reported that the efficiency of these columns was significantly better for samples of peptide digests than for samples of low molecular weight compounds. Columns packed with Halo, which is made of shell particles that are finer yet have a thicker shell than Poroshell, provide an opportunity to investigate in some detail the efficiency of shell columns. The goal of this work was to provide data on the efficiency of shell particles for compounds covering a wide range of molecular size, from compounds that have full and easy access to all the pores in the particles to some who have only restricted access. THEORY The general rate (GR) model of chromatography simultaneously considers all the possible contributions to the masstransfer kinetics. These include the intraparticle (or internal) transport mechanisms, such as pore diffusion in fluid-filled pores, surface diffusion on the adsorbed phase, adsorption/desorption kinetics, and reaction kinetics at phase boundaries; and the interparticle (or external) mechanisms, such as axial dispersion, eddy dispersion, and external film mass-transfer resistance. Mathematically, the GR model consists of two partial differential equations for each component, which express its mass balances in the mobile phase percolating through the column bed and in the stagnant fluid contained inside the particles. The detailed mathematical description of GR can be found elsewhere.17 There is no analytical solution of the GR model in the time domain, even under linear conditions, but in this case, there is a solution in the Laplace domain. The reverse transform of this analytical solution into the time domain is far too complex to be useful. However, the moments of the band profile C(t) in the time domain

mn )

∫

∞ n

0

t C(t) dt

(1)

(peak area), the first absolute moment (mu1 ) m1/m0, which is also the peak retention time), and the second central moment (which is the peak variance) are respectively defined as

m0 )

n

mn ) (- 1)n lim pf0

d C(p) dpn

(2)

∞

0

C(t) dt

∫ tC(t) dt

µ1 )

0

m0

∫ (t - µ ) C(t) dt ∞

µ′2 )

0

(14) Guiochon, G.; Felinger, A.; Katti, A. M.; Shirazi, D. Fundamentals of Preparative and Nonlinear Chromatography; Elsevier: Amsterdam, The Netherlands, 2006; Chapter 5. (15) Martin, M.; Guiochon, G. J. Chromatogr. 1985, 326, 3. (16) Wang, X.; Barber, W. E.; Carr, P. W. J. Chromatogr., A 2006, 1107, 139. (17) Guiochon, G.; Felinger, A.; Katti, A. M.; Shirazi, D. Fundamentals of Preparative and Nonlinear Chromatography; Elsevier: Amsterdam, The Netherlands, 2006; Chapters 2 and 16.

(4)

2

1

m0

(5)

These moments can be calculated as a function of the parameters that characterize the column and its packing material (for more details on these different types of moments, see ref 18). For an empirical description of chromatographic separations, the concept of theoretical plate is often used. The height equivalent to a theoretical plate (HETP, H) and the number of theoretical plates (N) are related to the column length, L, as

L)H×N

(6)

and H and N are related to the peak moments19,20 as

H ) L(µ′2/µ12)

(7)

N ) µ12/µ′2

(8)

and

Expressions relating N (or H) to the column characteristics for different kinds of columns can be derived through eqs 2 and 7 or 8.19-21 In a recent work, Kaczmarski and Guiochon used this method to derive the plate height equation of columns packed with shell or pellicular particles.13 In the framework of this model, the first absolute moment for a rectangular pulse injection of duration tp is given by

tp L (1 + k1) + ue 2

(9)

where ue is the interstitial velocity (i.e., the ratio between the superficial linear velocity, u, and the external porosity, e) and k1 is a constant defined as

k1 ) F[i + Ka(1 - i)][1 - (ri/re)3] where C(p) denotes the Laplace transform of the chromatographic elution peak and p the Laplace variable. Thus, the zeroth moment

(3)

∞

µ1 ) can be derived directly by means of the moment theorem of the Laplace transform, since

∫

(10)

where F is the phase ratio, F ) (1 - e)/e; i is the internal porosity; Ka is the Henry constant of adsorption of the analyte (18) Guiochon, G.; Felinger, A.; Katti, A. M.; Shirazi, D. Fundamentals of Preparative and Nonlinear Chromatography; Elsevier: Amsterdam, The Netherlands, 2006; Chapter 6. (19) Kubin, M. Collect. Czech. Chem. Commun. 1965, 30, 2900. (20) Kucera, E. J. Chromatogr. 1965, 19, 237. (21) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 2002, 106, 8898.

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(i.e., the slope of the isotherm at infinite dilution); and ri and re are the internal and the external radii of the porous shell of the pellicular particles, respectively (ri is also the radius of the solid core of the particle and re the particle radius). The length of the path effectively available for the diffusion of the analytes in to and out of the particle is (re - ri). Similarly, the second central moment of the peak is related to the average characteristics of the particles through the equation

[

(

k12re 1 L DL R 2 µ′2 ) 2 (1 + k ) + + 1 ue u 2 3F ke Deff e

)]

+

tp2 12

(11)

where DL is the axial dispersion coefficient, ke is the external mass transport coefficient, and Deff is the effective pore diffusion coefficient. R in eq 11 is a function of the only parameters ri and re

R)

[

]

re re4 + 2re3ri + 3re2ri2 - reri3 - 5ri4 5 (r 2 + r r + r 2)2 e

e i

i

(12)

Combination of eqs 5, 7, and 11, under the assumption of a δ-Dirac injection, gives the following expression for H

H)

[

2DL 2k12 uere 1 R + + 2 3F k ue D (1 + k1) e eff

]

(13)

Equation 13 can be written more concisely as

H ) H L + He + Hi

(14)

where HL represents the contribution of axial dispersion to H, with

HL ) 2DL/ue

(15)

while He and Hi are the contributions of the external and the internal mass-transfer resistances, respectively, with

He )

Hi )

2k12

uere 1 (1 + k1 ) 3F ke 2 2

2k12

uere R (1 + k1 ) 3F Deff 2 2

(16)

(17)

The overall mass-transfer resistance is

1 1 R ) + k ke Deff

(18)

where the second right-hand term is the contribution of the internal mass-transfer resistance. For totally porous particles, ri ) 0 and R ) re/5 ) dp/10 (dp ) 2re being the diameter of the particle) so that eq 18 (and eq 13) become identical to those obtained by Kubin and by Kucera.19,20 5974

Analytical Chemistry, Vol. 79, No. 15, August 1, 2007

Theoretical calculations using numerical values of the model parameters that were obtained in the past few years, on similar C18 bonded particles of B-silica that were fully porous, have shown that, for low molecular weight compounds, the dominant contribution to H is the axial dispersion one, given by eq 15, and that Hi is the smallest. In contrast, for peptides and proteins, Hi represents the dominant term to H while HL and He are almost negligible. Additionally, eqs 15-17 show that Hi decreases rapidly with increasing radius of the solid core of the porous shell particles, i.e., with decreasing shell thickness (see eq 12), whereas He remains almost constant (see eq 10) and HL, which is not a function of ri, remains constant. Consequently, it should be expected that the use of fine shell particles would offer great advantages for the analysis of large molecular weight compounds for which the resistance to internal mass transfers is the dominant contribution to H and particularly for the analysis of peptides and proteins. We investigated the range of validity of eq 13 applied to experimental HETP data acquired with several compounds analyzed on the new material Halo. In a separate report, we will compare the performance of fully porous particles to those of shell particles of comparable average size.22 The application of these columns to the separation of complex mixtures in gradient elution will be investigated separately.23 EXPERIMENTAL SECTION This work was undertaken using Halo columns (Advanced Materials Technology, Wilmington, DE). Columns. Two columns were used, a 150 × 4.6 mm and a 50 × 4.6 mm column, both packed with shell particles that have an average diameter of 2.7 µm, with a standard deviation of 0.125 µm, and a 0.50-µm-thick shell of porous B-silica with an average mesopore size of 9 nm. The silica is chemically bonded to a high density of octadecyl chains. Chemicals. Bradykinin (Bra; Mw ) 1059 Da) and [Lys0]bradykinin, also called kallidin (Kal; Mw ) 1205 Da), were from American Peptides Co. Inc. (Sunnyvale, CA); insulin (Ins, Mw ) 5700 Da) was a kind gift from Ely Lilly (Indianapolis, IN); β-lactoglobulin, (β-lact; Mw ) 14400 Da) was from Sigma (St. Louis, MO). Acetonitrile (ACN), H2O, tetrahydrofuran (THF), and dichloromethane, all HPLC grade, were from Fisher Scientific (Pittsburgh, PA). Trifluoacetic acid (TFA) was purchased from Sigma. Polymers for inverse size exclusion chromatography (ISEC) were purchased from Phenomenex (Torrance, CA) and Supelco (Bellefonte, PA). Their masses were 590, 760, 1100, 1300, 1780, 3680, 6400, 13 200, 19 300, 31 600, 90 000, 171 000, 400 000, 900 000, 1 860 000 Da. Instruments. The measurements reported here were performed on an Agilent 1100 HPLC and on a HP1090 instruments for liquid chromatography (Agilent Technologies, Palo Alto, CA). Both instruments were equipped with a DAD detector, a highpressure multisolvent delivery system, a column compartment thermostat, and a computer data station. The wavelengths used for the measurements were as follows: 210 nm for peptide and protein detection and 254 for the ISEC data acquired with (22) Gritti, F.; Marchetti, N.; Cavazzini, A.; Guiochon, G. J. Chromatogr., A. . In press. (23) Marchetti, N.; Cavazzini, A.; Gritti, F.; Guiochon, G. J. Chromatogr., A. In press.

polystyrenes. The extracolumn volumes were (averaged values, see below) 0.026 (for the HP 1090) and 0.035 mL (for the Agilent 1100). The column temperature was kept at 60 ( 2 °C. Measurements. The mobile phases were obtained by mixing, through the solvent delivery system, 0.1 v/v % solutions of TFA in pure H2O and ACN. The following data were obtained (compound: retention factor, mobile-phase composition as H2O/ ACN v/v %): (Kal: 1.7 ( 0.1, 80/20; 3.2 ( 0.1, 82/18), (Bra: 2.6 ( 0.1, 80/20; 4.6 ( 0.1, 82/18), (Ins: 3.0 ( 0.1, 68/32), and (β-lact: 4.5 ( 0.1, 59.5/40.5). The retention factors (calculated through the peak first moments) were measured at a flow rate of 1 mL/min, ignoring the influence of pressure on the adsorption equilibria, although it is not entirely negligible. Zhou et al., found that for Bra and Kal the capacity factors increase by nearly 20% when the average column pressure increases by 200 bar.24 A similar study conducted on three different insulin variants showed a 2-fold increase of the retention factor when the average column pressure increased from 55 to 250 bar.25,26 The sample sizes injected were 1 µL of 10 mg/L solutions of each compound studied. Operation of the column under linear conditions in this sample size range was verified by comparing the retention times of increasingly more dilute analyte solutions (10-200 mg/L) injected under the same experimental conditions. The retention times were constant. The column holdup volume, V0, was measured by pycnometry. The values found were 0.450 ( 0.002 and 1.345 ( 0.002 for the 5and the 15-cm columns, respectively. The solvents used for pycnometry were THF and dichloromethane. The total porosity of the columns were found to be 0.541 (5 cm) and 0.540 (15 cm). The internal pore size distribution was investigated using ISEC at room temperature.27-29 Samples of polystyrene of known average molecular weight and narrow molecular weight distributions were injected into the columns. Their retention volumes are plotted versus the cubic root of their molecular weight (see Figure 1, data obtained with the 5-cm column). This method allowed estimates of the internal and external porosities of the column. The external porosities were 0.433 for the 5-cm and 0.432 for the 15-cm column. Calculations. The column efficiency was measured in a wide range of flow velocities, using the method of moments. The first absolute moment and the second central moment were calculated numerically by evaluating the first and second moments using eqs 4 and 5, respectively. These values were compared with those obtained for these moments by fitting the experimental profiles to an exponential modified Gaussian function and deriving the moments from the corresponding analytical equation.30 A satisfactory agreement (with differences smaller than 5%) was obtained between the two approaches. The first absolute moments and the second central moments were corrected for the extracolumn (24) Zhou, D.; Liu, X.; Kaczmarski, K.; Felinger, A.; Guiochon, G. Biotechnol. Prog. 2003, 19, 945. (25) Liu, X.; Zhou, D.; Szabelski, P.; Guiochon, G. J. Chromatogr., A 2003, 988, 205. (26) Szabelski, P.; Liu, X.; Guiochon, G. J. Chromatogr., A 2003, 1015, 43. (27) Hala´sz, I.; Martin, K. Angew. Chem., Int. Ed. Engl. 1978, 17, 901. (28) Guan, H.; Guiochon, G. J. Chromatogr., A 1996, 731, 27. (29) Al-Bokari, M.; Cherrak, D.; Guiochon, G. J. Chromatogr., A 2002, 975, 275. (30) Felinger, A. Data Analysis and Signal Processing in Chromatography; Elsevier: Amsterdam, 1998.

Figure 1. ISEC plot. b, experimental data. (---): linear regression calculated for the volumes of the totally excluded PS samples. V0 (picnometry): holdup volume through picnometry. Ve: external volume. Vi: internal (pore) volume. VPS (6.4 kDa): elution volume of a PS of Mw ) 6.4 kDa. Ve (6.4 kDa) and Vi (6.4 kDa) are the external and the internal volumes for this polystyrene. Column: 5 cm. See text for details.

contributions measured at the different flow rates. The small sample amounts injected to avoid overloading limited the accuracy of the measurements made at low flow rates due to the low signalto-noise ratio achieved. The data set obtained for each compound represents the set of HETP data of the compound studied. The Marquardt-Levenberg algorithm was used for the nonlinear fitting of the experimental data to eq 13. RESULTS AND DISCUSSION In the conventional modeling of chromatography, the sum of the contributions of all the phenomena responsible for band broadening is assumed to be the sum of three contributions. Two different empirical HETP equations are used in the literature:

H ) A + B/u + Cu

(19)

H ) B/u + Au1/3 + Cu

(20)

The first equation is the Van Deemter equation,31 the second the Knox equation.32 The B/u term accounts for axial diffusion in the external porosity of the column. Axial diffusion in the stationary phase is usually neglected. The A term accounts for eddy dispersion, due to the wide variations of the average velocity of the mobile phase between the anastomosed channels available for the percolation of the mobile phase along the column.3,33,34 The C term accounts for the lumped sum of all contributions to the mass-transfer resistances. This makes the conventional approach unable to explain the differences in band spreading observed for a given compound in columns packed with particles of identical sizes made of fully porous, shell, or pellicular particles using the same adsorbent. Table 1 lists the best coefficient (31) Van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271. (32) Knox, J. H. Anal. Chem. 1966, 38, 253. (33) Giddings, J. C. Anal. Chem. 1962, 34, 885. (34) Weber, S. G.; Carr, P. W. In High Performance Liquid Chromatography; Brown, P. R., Hartwick, R. A., Eds.; Wiley: New York, 1989; Chapter 1.

Analytical Chemistry, Vol. 79, No. 15, August 1, 2007

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Table 1. A, B, and C Coefficients Obtained by Fitting the Experimental Data (Figures 2, 3, 5, and 6) to Eq 19a Kal A (cm) B (cm2/s) C (s) a

10-5

1.8 × 7.8 × 10-6 5.8 × 10-3

Bra

Ins

10-5

10-5

4.8 × 7.4 × 10-6 5.1 × 10-3

5.6 × 2.1 × 10-6 9.5 × 10-3

β-lact 2.4 × 10-3 3.6 × 10-8 7.1 × 10-2

Experimental conditions as in the corresponding figures.

obtained by fitting eq 19 to the experimental data shown in Figures 2, 3, 5, and 6, which are the Van Deemter plots obtained for Kal, Bra, Ins, and β-lact, respectively, on the 5-cm column (see further on). Although eqs 19 and 20 are usually presented in a physicochemical cloak, these equations are essentially empirical correlations. The numerical values of the coefficients A, B, and C derived from a set of HETP data have only a general, qualitative physical meaning. A more fundamental approach is necessary. In fundamental theories of chromatography, the two contributions to band broadening, axial diffusion and eddy dispersion, are usually lumped into a single axial dispersion coefficient but the contributions of the different mass-transfer resistances are expressed separately. The axial dispersion coefficient can be estimated by using different empirical or semiempirical equations.13 The main correlations available are those of Gunn35 and of Wen and Fan.36 In this work, we used the Gunn correlation, written as

(

DL ) dpue Q + Z +

)

e γ Re Sc

Z)

1.09 0.33 0.33 Re Sc e

ke ) (Dm/dp)

Re Sc (1 - p)2 2 4R1 (1 - e)

[ (

(22)

) ]

- 4R12(1 - e) 3 p(1 p) exp -1 p(1 - p)Re Sc 16R14(1 - e)2 (23) (Re Sc)2

where R1 is the first root of the zero-order Bessel function (2.405) and p ) 0.17 + 0.33 exp(-24/Re). In eq 21, the variance of the distribution of the ratio between the local fluid linear velocity and the average velocity over the column cross section was assumed to be equal to zero.13,18,35 The retention of molecules in HPLC proceeds by a series of successive steps: (1) the passage of the molecule from the mobile phase percolating through the particles to the mobile phase (35) Gunn, D. Chem. Eng. Sci. 1987, 42, 363. (36) Wen, C. Y.; Fan, L. S. Chem. Eng. Sci. 1973, 28, 1768. (37) Tallarek, U.; Dusschoten, D. V.; As, H. V.; Bayer, E.; Guiochon, G. J. Phys. Chem. B 1998, 102, 3486.

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stagnant inside a particle and its penetration inside this particle; (2) the diffusion of the molecule inside the particle, through the mesopore network or (3) its diffusion along the adsorbent surface; and (4) the adsorption followed by the desorption of the molecule. The kinetics of equilibration between the mobile phases in the internal and external porosities of the column is essentially controlled by the external film mass-transfer resistance and intraparticle diffusion (pore and surface diffusion). The external mass-transfer coefficient was evaluated through the WilsonGeankoplis correlation

(21)

where γ is the bed tortuosity factor (this factor has a numerical value close to 1.437); dp, the particle diameter, was assumed to be equal to 2.7 µm; Re ) ueFdp/η is the Reynolds number (with F the density and η the viscosity of the mobile phase); and Sc ) η/FDm is the Schmidt number (with Dm the molecular diffusion coefficient in the bulk phase). Q and Z in eq 21 are defined as

Q)

Figure 2. Van Deemter plot for Bra. b, experimental points. (---): fitting of the experimental data points to eq 13. Column: 5 cm. MP: H2O/ACN/TFA 82/18/0.1 v/v %. Right axis, h: reduced HETP.

Analytical Chemistry, Vol. 79, No. 15, August 1, 2007

(24)

This correlation is valid provided that the Reynolds number is in the range 0.0015 < Re < 55. This assumption is valid in chromatography since Re is typically less than 0.1 and larger than 0.001. Modeling of internal mass-transfer resistances is complex. For instance, adsorption/desorption can take place several times before the molecule eventually diffuses out of the particle. Experimental measurements of the kinetics of the individual steps is difficult, and in most cases, the results obtained include the contributions of more than one of these steps. Consequently, the derivation of a reliable estimate of Deff is most difficult. In many cases, kinetic parameters are estimated using parameter identification, that is through the nonlinear fitting of the experimental data to a given model. In this work, for the sake of simplicity, we fitted the HETP data measured for a series of compounds to the HETP equation for pellicular/shell materials (see eq 13) taken as the fitting model, Deff being the fitting variable. To do this, the knowledge of the internal and external porosities of the column and an estimation of molecular diffusion coefficients for the compounds under investigation is needed. The correlation derived by Young et al.38 has been used to estimate molecular diffusivity: (38) Young, M. E.; Carroad, P. A.; Bell, R. L. Biotechnol. Bioeng. 1980, 22, 947.

T Dm ) 8.31 × 10-8 ηMw1/3

(25)

where Dm is given in cm2/s, η in cP, Mw in grams, and T is the absolute temperature (K). Dm values for Bra, Kal, Ins, and β-lact are listed in Table 2. The internal and external porosities of a column are easily determined using ISEC. In Figure 1, the elution volumes of a series of 15 samples of polystyrenes (PS) having average molecular masses ranging from 590 to 1.8 × 106 Da are plotted versus the cubic root of their molecular mass. The mobile-phase volume proportional to the column total porosity obtained by pycnometry measurements (see Experimental Section) is indicated on the plot (V0 ) 0.450 mL). The ISEC estimate of the external volume, Ve, is derived from the extrapolation to Mw ) 0 of the linear regression calculated for the volumes of the totally excluded PS samples, as indicated in the figure. For the column studied in this work, Ve was found to be 0.360 cm3. From this value, the internal volume, Vi, is calculated as the difference between V0 and Ve. The vertical dashed line in Figure 1 is drawn at Mw ) 500, which can be taken as an estimate of the upper limit of the molecular mass of compounds that have full access to the pores inside the particle. The compounds that have molecular volumes between those corresponding to the totally excluded and the totally included compounds, have only partial access to the internal pore volume. For these compounds, the values of Ve and Vi have to be calculated in each case. This can be done as shown in Figure 1 for a PS of Mw ) 6.4 kDa (VPS ) 0.371 cm3) as example. The external volume for this component is the value of the linear regression at Mw ) 6400 (0.350 mL). The portion of the pore volume that is accessible to this molecule, Vi (6.4 kDa), is the difference between VPS (6.4 kDa) and Ve (6.4 kDa) that is 0.021 mL (see Figure 1). The molar volume of a polymer of given molecular mass depends on its conformation, itself a function of its chemical structure and of the polymer-solvent interactions taking place, interactions that depend on several physicochemical parameters such as the temperature. For polystyrene in THF, the following semiempirical equation was derived between the hydrodynamic radius, rh, and Mw, at 298 K39

rh ) 0.177Mw0.564

(26)

with rh in angstroms. This relationship assumes that the polymer has a random coil structure. Thus, for instance, for a polystyrene of Mw ) 6.4 kDa in THF at 298 K, rh can be estimated to be 24.8 Å. Since the average diameter of the pores of the packing material is 90 Å,7 this means that the ratio between hydrodynamic and pore radii is 0.55.15 The random coil model is a reasonable description for polystyrene structures in THF that, however, cannot be applied to biomolecules, such as proteins, as it will be discussed later on. Figures 2 and 3 show the van Deemter plots obtained for Bra and Kal with a H2O/ACN/TFA (82:18:0.1 v/v %) mobile phase at 60 °C on the 5-cm column. H is expressed both in length unit (cm, left axis) and as the dimensionless reduced plate height, (39) Kurata, M.; Tsunashima, Y. In Polymer Handbook, 3rd ed.; Brandroup, J., Immergut, E. H., Eds; Wiley: New York, 1991.

Figure 3. Van Deemter plot for Kal.b, experimental points. (---): fitting of the experimental data points to eq 13, using Dm calculated with the correlation of Young et al.38 Column: 5 cm. MP: H2O/ACN/ TFA 82/18/0.1 v/v %. Right axis, h: reduced HETP. Table 2. Molecular Diffusion Coefficient (Dm, cm2/s) Calculated through Young’s Correlation (Eq 25)a Kal (80/20)

Bra (80/20)

Ins (68/32)

10-6

10-6

10-6

4.73 ×

4.93 ×

2.98 ×

β-lact (59.5/40.5) 1.75 × 10-6

aMP composition: (X/Y) X, H O v/v %; Y, ACN v/v %; TFA 0.1 v/v 2 % in all cases. T ) 60 °C.

Table 3. Effective Pore Diffusion Coefficient (Deff, cm2/s) Calculated on the Two Columns (5 and 15 cm)a Kal (80/20) 5 cm 15 cm

10-7

1.7 × 2.6 × 10-7

Bra (80/20) 10-7

2.6 × 2.4 × 10-7

Ins (68/32) 10-7

2.1 × 2.5 × 10-7

β-lact (59.5/49.5) 1.0 × 10-8 7.4 × 10-9

a MP composition: (X/Y) X: H O v/v %, Y: ACN v/v %, TFA 0.1 2 v/v % in all cases. T ) 60 °C.

h ) H/dp. The symbols show the experimental data. The dashed line represents the fitting of the experimental data to eq 13. The scattering of the data at low velocities is due to the experimental errors, which become significant at low flow velocities, and to the use of a HETP equation (eq 13) that neglects eddy diffusion. The unusually low value of h observed for the two peptides (1.4 for Kal and 1.6 for Bra, both at ∼0.02 cm/s) is remarkable. The best values obtained for the effective pore diffusion coefficient Deff are 3.0 × 10-7 and 5.7 × 10-7 cm2/s for Kal and Bra, respectively (see Table 3). The relatively high temperature at which these measurements were carried out explains the important decrease of the mobile-phase viscosity observed, ∼0.55 cP at 60 °C for the H2O/ACN 80/20 solution instead of ∼1 cP at 25 °C. This decrease of the velocity permitted the use of relatively large flow rates (up to 4.0 mL/min). Obviously, the column efficiency decreases at the highest flow rates achieved, but the column still performs efficiently up to flow rates between 1 and 1.5 mL/min (corresponding to flow velocities of 0.1-0.15 cm/s and a value of h close to 2) and h is still only 3.6 or so at 2.0 mL/min. The correlation derived by Wilke and Chang40 is probably the most popular equation giving the molecular diffusivities of low (40) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264.

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Figure 4. Same as Figure 3 but fitting (---) of the experimental data points (b) for Kal to eq 13 using instead the Wilke and Chang40 correlation to estimate Dm (same experimental conditions as in Figure 3). Right axis, h: reduced HETP.

Figure 5. Van Deemter plot for Ins. b, experimental data points. (---): fitting of experimental points to eq 13. Column: 5 cm. MP: H2O/ ACN/TFA 68/32/0.1 v/v %. Right axis, h: reduced HETP.

molecular weight compounds in conventional solvents. It does not give reasonable estimates of this parameter for proteins, but it could still be used for small peptides. For Kal, under the experimental conditions of this work, the Wilke and Chang equation gives Dm ) 1.7 × 10-6 cm2/s, while eq 25 gave 4.7 × 10-6 cm2/s (see Table 2). However, when the experimental data are fitted to eq 13 using the value of Dm provided by the Wilke and Chang correlation, the value obtained for the pore diffusion coefficient is not satisfactory (see Figure 4), as the curve obtained fits the data poorly and leads to an estimate of Deff ) 0.0491 cm2/ s, which is several orders of magnitude larger than the one derived from the fit of the same data to the same eq 13 when using the diffusion coefficient provided by the correlation of Young et al.38 Figure 5 shows similar results obtained for Ins. The value of Dm derived from the correlation of Young et al. for a small protein like Ins is ∼40% smaller than that obtained for the peptides Bra and Kal. The mobile-phase velocity corresponding to the minimum of the van Deemter curve shifts to a smaller value (∼0.01-0.015 cm/s or 0.1-0.2 mL/min), and the slope of the asymptotic branch of H at high velocity is steeper than for the peptides. Again, the minimum value of h is unusually small and it 5978 Analytical Chemistry, Vol. 79, No. 15, August 1, 2007

increases slowly with increasing mobile-phase velocity, although markedly faster than for Bra and Kal. At a flow rate of 1 mL/min, h is 3.7 against ∼1.9 for the two peptides. Surprisingly, however, the value of Deff calculated for Ins does not significantly differ from those obtained for Bra and Kal (see Table 3). A possible explanation for this can be given by referring again to Figure 1. The solid vertical line labeled Ins in Figure 1 was drawn at an abscissa equal to the cubic root of the Mw of Ins (17.3). The ISEC plot in this figure can be used to derive an approximate estimate of an equivalent to VPS for Ins, as if it were in a random coil structure. Most proteins, however, are globular macromolecules, which have a markedly smaller radius and a much better defined structure than the average random coil of a polystyrene, resulting in a larger diffusion coefficient. For instance, if we assume the specific volume of Ins to be equal to 0.73 cm3/g (a typical specific volume for proteins), this would give it a radius of 13.9 Å while the hydrodynamic radius of a polystyrene of the same molar mass, 5.7 kDa, would be 23.2 Å. As a consequence, the pore volume of the column for Ins is larger than the equivalent to the VPS value indicated in Figure 1 (it would correspond, thus, to a vertical line shifted to the right with respect to that in Figure 1). The molecules of Ins can access a wider portion of the intraparticle mesopore network of the particle than a molecule of polystyrene having the same mass. The ratio between the hydrodynamic radius of Ins and the average radius of the mesopores of the particles is ∼0.3. This explains why Ins is retained, still having access to a large fraction of the internal pore volume. Because it has a larger molecular mass than the peptides Bra and Kal, Ins may have access only to a smaller fraction of the internal porosity than these peptides. A somewhat reduced ability of Ins at penetrating the pores of the packing material could explain the lower value of its pore diffusion coefficient, Deff, obtained through the fitting of the experimental HETP data to eq 13. The value obtained is close to the value of Deff determined for Bra and Kal, even though, as it was noticed, Dm is ∼40% smaller for Ins than for the two peptides (see Tables 2 and 3). The behavior of larger protein molecules, such as β-lact, also presents some interesting features. As was done for Ins, the equivalent VPS for this molecule was plotted in Figure 1 (see vertical line labeled β-lact). Among the samples of polystyrenes injected, the one having the molecular mass closest to that of this protein is the one with Mw ) 19 300. It is in a random coil configuration with an estimated radius of 46 Å38,39,41 that is nearly identical to the nominal average radius of the pores of the packing material. This molecule should be largely excluded. On the other hand, the globular protein has a radius of only 17 Å. Experimentally and notwithstanding the significantly large value of the ratio between the molecular and the pore radius (0.4), no clear sign of exclusion effect was observed for this molecule. However, the data (see Figure 6) shows that, for this large molecule, the minimum of the Van Deemter curve could not be reached because it corresponds to a very slow velocity, too low for accurate measurements. This is clearly evidenced from the trend of the low-velocity data in Figure 6, where hred is 3.35 at u ) 0.005 mL/min. (41) Yau, W. W.; Kirkland, J. J.; Bly, D. D. Modern size-exclusion liquid chromatography; Wiley: New York, 1979.

be useful for the separation of protein mixtures, most notably the availability of a similar material with larger mesopores. The optimum velocity for maximum efficiency decreases steadily with increasing molecular weight, so the operation of advanced columns at high efficiency can be done only at relatively low mobile-phase velocities.15 Important work remains to be done to clarify the optimum combination of the properties and characteristics of packing materials depending on the goal pursued, fast separations, high efficiency, and low detection limits. Shell particles will contribute in improving the terms of the compromises.

Figure 6. Van Deemter plot for β -lact. b, experimental data points. (---): fitting of experimental points to eq 13. Column: 5 cm. MP: H2O/ ACN/TFA 59.5/40.5/0.1 v/v %. Right axis, h: reduced HETP.

CONCLUSIONS The new generation of shell particles offers an extremely attractive combination of properties. Their relatively high porosity, moderate permeability, and low values of the A term of their Van Deemter equation allow the operation of these columns at high mobile-phase velocities. This could be most useful for the separation of low molecular mass compounds. However, while it is possible to use the current columns for high-efficiency separations of complex peptide mixtures, further improvements would

ACKNOWLEDGMENT This work was supported in part by grant CHE-06-08659 of the National Science Foundation, by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory, by the Italian University and Scientific Research Ministry (2005037725_002), and by the University of Ferrara (Progetto Giovani Ricercatori and ex-60%). The authors acknowledge fruitful, informative discussions with and the generous gifts of columns by J. J. Kirkland (Advanced Materials Technology, Wilmington, DE).

Received for May 16, 2007.

review

March

22,

2007.

Accepted

AC070571A

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