Anal. Chem. 2003, 75, 6975-6986
Moment Analysis of Mass-Transfer Kinetics in C18-Silica Monolithic Columns Kanji Miyabe,† Alberto Cavazzini,‡ Fabrice Gritti,§ Marianna Kele,| and Georges Guiochon*,§
Department of Applied Chemistry, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466-8555, Japan, University of Ferrara, Via L. Borsari 46, I-44100 Ferrara, Italy, Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600, Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, and Waters Corporation, Milford, Massachusetts 01757
The moment analysis of elution peak profiles based on new moment equations provides information on the masstransfer characteristics of C18-silica monolithic columns. The flow rate dependence of the HETP data was analyzed using the generalized van Deemter equation, after correction of these data by subtraction of the external masstransfer contribution to band broadening. Kinetic parameters and diffusion coefficients related to the mass-transfer processes in monolithic columns were derived by taking advantage of the different flow velocity dependence of their contributions to band broadening. At high flow rates, axial dispersion and diffusive migration across the monolithic C18-silica skeleton contribute much to band broadening, suggesting that it remains important to reduce the influence of eddy diffusion and the mass-transfer resistance in the stationary phase to achieve fast separations and a high efficiency. Surface diffusion plays a predominant role for molecular migration in the monolithic stationary phase. Although the value of the surface diffusion coefficient (Ds) depends on an estimate of the external masstransfer coefficient, Ds values of the order of 10-7 cm2 s-1 were calculated for the first time for the C18-silica monolithic skeleton. The value of Ds decreases with increasing retention of sample compounds. Analysis of a kind of time constant calculated from Ds suggests that the “chromatographic corresponding particle size” is ∼4 µm for the C18-silica monolithic stationary phase used in this study. The accuracy of the Ds values determined was discussed. Monolithic (i.e., continuous and porous) media are attracting much attention as superior tools to perform chromatographic separations. HPLC columns made of a C18-silica monolith are now commercially available for RPLC. It is well known that the chromatographic behavior of these columns differs from that of the conventional columns packed with spherical particles of packing materials.1-6 One of their advantages is their high external * Corresponding author. E-mail:
[email protected]. † Nagoya Institute of Technology. ‡ University of Ferrara. § The University of Tennessee and Oak Ridge National Laboratory. | Waters Corp. (1) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. Anal. Chem. 1996, 68, 3498-3501. 10.1021/ac0302206 CCC: $25.00 Published on Web 11/15/2003
© 2003 American Chemical Society
porosity resulting from the network of through-macropores. Another one is the stationary-phase network consisting of a small, thin silica skeleton. These two networks twist around each other and provide the intricate structure of the monolithic media. These two structural characteristics allow both a low hydraulic resistance of the column to the mobile-phase flow and an enhancement of the mass transfer of sample molecules through the column. Consequently, monolithic HPLC columns exhibit both a high permeability and a high separation efficiency at high-flow velocities. Several advantageous characteristics of monolithic columns seem to originate from the fast molecular migration through their stationary-phase network. The small and thin structure of their silica skeleton contributes to the fast mass transfer of sample molecules in these columns. It is essential for a good understanding of the intrinsic characteristics of their chromatographic behavior to study the fundamentals of the mass-transfer kinetics in monolithic columns. In many previous papers,1-3,5,6 kinetic studies on monolithic columns were carried out by analyzing the dependence of their efficiency on the mobile-phase flow velocity. The profiles of the van Deemter plots of monolithic and conventional packed columns were compared, and the van Deemter equation was used to account for the flow rate dependence of their efficiency.2,3,6 However, there was little quantitative, detailed discussion of the values of the coefficients of the van Deemter equation and few experimental data were reported. Gritti et al.7 studied the mass-transfer kinetics of butylbenzoate on a C18-silica monolithic column, using a methanol/water (65/ 35, v/v) solution as the mobile phase. They made perturbation experiments at different mobile-phase flow velocities and plateau concentrations of the sample compound. The flow rate dependence of the HETP was analyzed by applying the classical random-walk model of Giddings.8 They derived physically meaningful informa(2) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr., A 1997, 762, 135-146. (3) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr., A 1998, 797, 121-131. (4) Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Soga, N.; Tanaka, N. J. Chromatogr., A 1998, 797, 133-137. (5) Minakuchi, H.; Ishizuka, N.; Nakanishi, K.; Soga, N.; Tanaka, N. J. Chromatogr., A 1998, 828, 83-90. (6) Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Soga, N.; Nagayama, H.; Hosoya, K.; Tanaka, N. Anal. Chem. 2000, 72, 1275-1280. (7) Gritti, F.; Piatkowski, W.; Guiochon, G. J. Chromatogr., A 2003, 983, 5171.
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tion on the structure of the monolith from an analysis of axial dispersion, estimating the average characteristic distance in the monolithic bed as the sum of the average through-macropore diameter and the average skeleton element size. Minakuchi et al.2,3 used the same definition of the combined domain size as the characteristic parameter of silica monolithic rods. Gritti et al.7 also analyzed the skeleton/through-macropore mass-transfer kinetics. They demonstrated that the combination of the external mass-transfer kinetics and the pore diffusivity accounts well for the actual mass-transfer kinetics in their monolithic column at relatively low plateau concentrations and that pore diffusivity is the major contribution to band broadening. To the best of our knowledge, this work7 was the first detailed study of the masstransfer kinetics in monolithic columns. A few other papers deal with retention equilibrium in monolithic columns.9-11 More fundamental kinetic studies of the chromatographic behavior of monolithic columns including an analysis of experimental data on their mass-transfer kinetics in connection with thermodynamic equilibrium are now necessary. The analysis of the moments of elution peak profiles based on the solution of the general rate model is a useful strategy of study of retention equilibrium and mass-transfer kinetics in chromatographic columns.12-20 In previous papers,19,20 we used this method, studied the chromatographic behavior in RPLC, and derived information on the mass-transfer processes in RPLC columns, i.e., axial dispersion, external mass transfer, intraparticle diffusion, and the actual adsorption/desorption kinetics. Surface diffusion phenomena in the intraparticle pore space could be studied in detail. On the basis of these results, a surface-restricted molecular diffusion model was proposed to explain the mechanism and some of the intrinsic characteristics of surface diffusion.19-29 It is expected that the same method will provide new information on the chromatographic behavior of monolithic columns. In a previous paper,30 we derived new equations that correlate the peak moments and some parameters characterizing the (8) Giddings, J. C. Dynamics of Chromatography, Part I, Principles and Theory; Marcel Dekker: New York, 1965. (9) Cavazzini, A.; Bardin, G.; Kaczmarski, K.; Szabelski, P.; Al-Bokari, M.; Guiochon, G. J. Chromatogr., A 2002, 957, 111-126. (10) Gritti, F.; Piatkowski, W.; Guiochon, G. J. Chromatogr., A 2002, 978, 81107. (11) Cavazzini, A.; Felinger, A.; Guiochon, G. J. Chromatogr., A 2003, 1012, 139. (12) Kucera, E. J. Chromatogr. 1965, 19, 237-248. (13) Kubin, M. Collect. Czech. Chem. Commun. 1965, 30, 2900-2907. (14) Grushka, E.; Myers, M. N.; Schettler, P. D.; Giddings, J. C. Anal. Chem. 1969, 41, 889-892. (15) Grushka, E. J. Phys. Chem. 1972, 76, 2586-2593. (16) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 1994. (17) Suzuki, M. Adsorption Engineering; Kodansha/Elsevier: Tokyo/Amsterdam, 1990. (18) Ruthven, D. M. Principles of Adsorption & Adsorption Processes; John Wiley and Sons: New York, 1984. (19) Miyabe, K.; Guiochon, G. Adv. Chromatogr. 2000, 40, 1-113. (20) Miyabe, K.; Guiochon, G. J. Sep. Sci. 2003, 26, 155-173. (21) Miyabe, K.; Guiochon, G. Anal. Chem. 1999, 71, 889-896. (22) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 1999, 103, 11086-11097. (23) Miyabe, K.; Guiochon, G. Anal. Chem. 2000, 72, 1475-1489. (24) Miyabe, K.; Guiochon, G. J. Chromatogr., A 2000, 903, 1-12. (25) Miyabe, K.; Guiochon, G. Anal. Chem. 2001, 73, 3096-3106. (26) Miyabe, K.; Guiochon, G. J. Chromatogr., A 2001, 919, 231-244. (27) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 2001, 105, 9202-9209. (28) Miyabe, K.; Guiochon, G. Anal. Sci. 2001, 17 (Suppl), i209 - i212. (29) Miyabe, K.; Guiochon, G. J. Chromatogr., A 2002, 961, 23-33. (30) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 2002, 106, 8898-8909.
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retention equilibrium and the mass-transfer kinetics in monolithic columns, by solving the basic equations of the general rate model of chromatography in the Laplace domain. In this work, it was assumed that the unit structure of the continuous porous packing material in monolithic columns consists of a cylindrical fiber of stationary phase (including meso- or micropores), surrounded by a through-macropore space filled with the mobile phase. The goal of this study is to derive information on the mass-transfer kinetics in C18-silica monolithic columns by applying these new moment equations to a set of experimental data. THEORY: MOMENT ANALYSIS Derivation of New Moment Equations for Monolithic Columns. The analysis of the moments of elution bands is based on the general kinetic model of chromatography. It is one of the most effective strategies to elucidate the chromatographic behavior from the viewpoints of the retention equilibrium and the masstransfer kinetics. Information regarding the retention equilibrium and the mass-transfer kinetics is derived from the first absolute moment (µ1) and the second central moment (µ2′) of the elution bands, respectively. Relevant details on the moment analysis method can be found in the literature.12-20 So far, however, the equations relating the moments of elution bands and the experimental parameters have been derived only for conventional columns, packed with spherical particles of the packing material.16-20,31-33 These moment equations cannot be used to analyze the profiles of the bands eluting from monolithic columns because the structural characteristics of continuously porous rods are different from those of conventional beds of particulate packing materials. We need a set of moment equations suitable for a detailed analysis of the chromatographic behavior of monolithic columns. In a previous paper,30 we derived the moment equations for monolithic columns, filled with a continuously porous rod. Only basic information is presented below. Monolithic silica rods have a complicated spongy-like structure with thin silica membranes, filaments, and globs surrounding large through-macropores, as shown in the scanning electron micrographs found in the literature.1-6,34-37 It is difficult to design an accurate model of the structure of such a continuous porous medium because of its extremely complicated and tortuous morphology. We assumed, as a first approximation, that the unit structure of the continuous porous packing material in monolithic columns consists of a cylindrical skeleton of the stationary phase (including the meso- or micropores), surrounded by a throughmacropore space filled with the mobile phase.30 Then, the basic equations of the general rate model of chromatography can be written as follows for a monolithic column.30
DL
∂2C ∂C As ∂C - N0 ) -u 2 ∂z ∂t ∂z e
N0 ) kf(C - Ci,Rss) ) De(∂Ci/∂r)Rss
(
De
∂2Ci ∂r
2
+
)
∂Ci 1 ∂Ci - N i ) i r ∂r ∂t
Ni ) (1 - i)(∂q/∂t) ) (1 - i)ka(Ci - q/Ka)
(1) (2) (3) (4)
where C is the concentration of the sample compound in the
through-macropore space, z the longitudinal distance along the column, t the time, DL the axial dispersion coefficient, u the average interstitial velocity of the mobile phase, As the ratio of the surface area of the stationary-phase skeleton to the column volume, e the column void fraction (external porosity), N0 the mass flux of the sample compound from the mobile phase to the external surface of the stationary-phase skeleton, Ci the concentration of the sample compound within the mesopores inside the stationary phase, Rss the radius of the cylindrical stationary-phase skeleton, r the radial distance from the center of the stationaryphase cylinder, kf the external mass-transfer coefficient, De the diffusion coefficient of the sample compound in the pore space inside the stationary-phase skeleton, Ni the mass flux of the sample compound from the stagnant mobile phase in the mesopore space to the surface of the stationary phase, i the internal porosity of the stationary phase skeleton, q the sample concentration adsorbed on the stationary phase, ka the adsorption rate constant, and Ka the adsorption equilibrium constant. This model cannot be solved in the time domain, but there is an analytical solution, C h , in the Laplace domain for the system of eqs 1-4, with the appropriate initial and boundary conditions of elution chromatography.30
C h ) (C0/p)[1 - exp(-pτ)] exp[-M(p)z]
(5)
analysis of the elution peak profiles. On the other hand, the band moments can be easily derived from the solution in the Laplace domain. The first two moments, µ1 and µ2′, give important information on the elution position (retention time and equilibrium thermodynamics) and width (variance and mass-transfer kinetics) of the chromatographic peak, respectively. It is rarely useful to analyze the zeroth moment, i.e., the peak area, because no chemical reactions take place in conventional HPLC. In principle, higher-order moments, e.g., the third and fourth moments, can also be interpreted within the framework of the general rate model. However, these moments cannot be analyzed in practice because it is extremely difficult to measure them with accuracy.20 The analysis of the moments µ1 and µ2′ of elution peaks is an effective approach to obtain detailed information on the equilibrium thermodynamics and the kinetic characteristics of chromatographic separations. This analysis requires the equations relating µ1 and µ2′ to the experimental parameters. The following new moment equations are derived from the analytical solution in the Laplace domain, eqs 5-12.30
µ1 )
∫Ce(t)t dt z ) [ ∫Ce(t) dt u 0
e + (1
M(p) )
[x
u 2DL
1+
4DL G(p) - 1 u2
]
z δ u0 0 (13)
∫Ce(t)(t - µ ) ∫Ce(t) dt
2
µ2′ )
where
- e)[i + (1 - i)Ka]] )
1
dt
)
2z (δ + δf + δd) u0 ax
(14)
(6)
δ0 ) e + (1 - e)[i + (1 - i)Ka]
(15)
(7)
δax ) (DL/u02)δ02
(16)
F(p) ) (E(p)/Bi)I1(E(p)) + I0(E(p))
(8)
δf ) (1 - e)(Rss/2kf)[i + (1 - i)Ka]2
(17)
Bi ) kfRss/De
(9)
δd ) (1 - e)(Rss2/8De)[i + (1 - i)Ka]2
(18)
E(p) ) RssxB(p)
(10)
B(p) ) A(p)/De
(11)
pkaKa A(p) ) (1 - i) + ip pKa + ka
(12)
G(p) ) p +
[
]
I0(E(p)) Askf 1e F(p)
where C h denotes the Laplace transform of C, C0 is the sample concentration of the injection pulse, p is the Laplace transform variable, τ is the width of a rectangular sample pulse introduced into the column, and I0(x) and I1(x) are the modified Bessel function of the zeroth and first order, respectively. In principle, the inverse transform of C h from the Laplace (eqs 5-12) to the time domain should provide the analytical solution of C. Unfortunately, this approach is not practical. The inversion is too complicated, and its solution would not permit a practical (31) Suzuki, M.: Smith, J. M. Chem. Eng. Sci. 1971, 26, 221-235. (32) Suzuki, M.: Smith, J. M. J. Chem. Eng. Jpn. 1973, 6, 540-543. (33) Suzuki, M.; Smith, J. M. Adv. Chromatogr. 1975, 13, 213-263. (34) Cabrera, K.; Lubda, D.; Eggenweiler, H.-M.; Minakuchi, H.; Nakanishi, K. J. High Resolut. Chromatogr. 2000, 23, 93-99. (35) Nakanishi, K.; Soga, N. J. Am. Ceram. Soc. 1991, 74, 2518-2530. (36) Nakanishi, K.; Soga, N. J. Non-Cryst. Sol. 1992, 139, 1-13. (37) Nakanishi, K.; Soga, N. J. Non-Cryst. Sol. 1992, 139, 14-24.
The superficial velocity (u0) is used in eqs 13, 14, and 16 instead of the interstitial velocity (u). Equations 13-18 show how the combination of several fundamental parameters related to the equilibrium thermodynamics and the mass-transfer kinetics influence the shape of the elution peak profiles under linear isotherm conditions. As indicated in eq 13, µ1 of the elution peak depends only on the equilibrium constant (Ka) while µ2′, which characterizes the bandwidth, is the sum of contributions due to the different mass-transfer processes in the column, i.e., axial dispersion, external mass transfer, and diffusive migration inside the stationary phase skeleton. Equations 14 and 16-18 indicate that band broadening is related to three main kinetic parameters, DL, kf, and De. Several important corrections must be made for an accurate analysis of the bandwidth. The contribution of the sample injection volume to µ2′ is usually calculated by assuming that the sample pulse has a rectangular profile, but it can be neglected when the sample size is sufficiently small. The contribution of the adsorption/desorption kinetics can most often be neglected in RPLC because the rate of adsorption is sufficiently high.19,20,38 Equations (38) Miyabe, K.; Guiochon, G. Anal. Chem. 2000, 72, 5162-5171.
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13-18 are rearranged from the original form of the moment equations given previously,30 by assuming that the influence of these last two contributions is negligible. Moment Analysis of Elution Peak Profiles. From eqs 1318, the HETP (Htotal) is given by
2DL 2u0 2u0 Htotal ) + 2 δf + 2 δd ) Hax + Hf + Hd (19) u0 δ δ 0
0
As shown in eq 19, the contribution of the axial dispersion (δax) can be separated from those of both the external mass transfer (δf) and the diffusive migration of the sample molecules inside the stationary-phase skeleton (δd), by taking advantage of the difference in their flow rate dependence. The following correlation is derived from eq 19.
Htotal - Hf ) Htotal -
2u0δf δ02
2DL 2u0δd ) Hax + Hd ) + u0 δ02 (20)
The axial dispersion coefficient (DL) of the conventional HPLC columns packed with spherical particles is usually accounted for by assuming that axial dispersion consists of two main mechanisms, molecular diffusion and eddy diffusion.16
DL ) γ1Dm + γ2dpu
(21)
where γ1 and γ2 are two geometrical coefficients and dp is the particle diameter. When eq 21 is applied to the analysis of the chromatographic behavior of monolithic columns, an appropriate value must be used as dp. Gritti et al.7 and Minakuchi et al.2,3 used the combined domain size, i.e., the sum of the average throughmacropore diameter and the average skeleton element size, as the average characteristic distance in the monolithic bed. For the C18-silica monolithic columns used in this study (Chromolith Performance, Merck), Gritti et al.7 estimated the eddy diffusion length to be 3.7 µm. Kele and Guiochon39 similarly assigned them a “chromatographic apparent particle size” of 4 µm. These values correspond to the average through-macropore diameter of 2 µm and the average skeleton size ranging between 1.3 and 1.6 µm of the Chromolith Performance columns.7,39 When the radius of the element of silica skeleton (Rss) and the superficial velocity of the mobile phase (u0) are used instead of dp and u in eq 21, DL becomes
DL ) γ1Dm + γ3Rssu0
(22)
where γ3 is another geometrical coefficient. Substituting eq 22 into eq 20,
Htotal - Hf )
2γ1Dm 2u0δd B + 2γ3Rss + ) + A + Cu0 u0 u0 δ02 (23)
Equation 23 indicates that, after subtracting the contribution of (39) Kele, M.; Guiochon, G. J. Chromatogr., A 2002, 960, 19-49.
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the external mass transfer (δf), the HETP is given by a conventional equation with three coefficients, i.e., A, B, and C. The external mass-transfer coefficient (kf) was estimated on the basis of the penetration theory.40
kf ) x4Dmu/2πRss
(24)
where Dm is the molecular diffusivity. The equation of WilkeChang was used to estimate Dm of the sample molecules.16,18,40-42 The value of kf was also derived from two literature correlations between the Sherwood (Sh ) kfdp/Dm), Reynolds (Re ) u0dpF/ η), and Schmidt (Sc ) η/FDm) numbers.43,44
Sh )
( )
kf(2Rss) 1 - e ) 1.85 Dm e
1/3
Re1/3Sc1/3 (Kataoka et al.43) (25)
Sh ) (1.09/e)Re1/3Sc1/3
(Wilson and Geankoplis44) (26)
As results from eq 23, the accuracy of the data analysis made in this study depends on that of the estimate of kf. Thus, external mass-transfer phenomena should be studied carefully in this work because they have been so far the topic of only few publications in chromatography, none yet dealing with monolithic columns. Equation 23 shows that the effective diffusivity (De) in the pores located inside the stationary-phase skeleton must be derived from the value of the coefficient C, suggesting that information on the mass-transfer kinetics in the C18-silica skeleton should also be derived from this coefficient. The value of De is usually accounted for by assuming that there is parallel combination of the contributions of pore and surface diffusion.17,18
De ) Dp + (1 - i)KaDs
(27)
The surface diffusion coefficient (Ds) was calculated by subtracting the contribution of pore diffusion (Dp) from De. The value of Dp was estimated using the following equation.
Dp ) iKpDm/k2
(28)
The tortuosity factor (k) was estimated from the moment µ2′ of the elution peak of a nonretained tracer, i.e., thiourea.39 The following equations have been proposed to estimate the hindrance parameter (Kp).45,46
log Kp ) -2.0λm Kp )
(Satterfield et al.45)
(29)
1 + (9/8)λm ln λm - 1.593λm (1 - λm)2 (Brenner and Gaydos46) (30)
where λm is the ratio of the diameter of the sample molecule to (40) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 2002.
Table 1. Physical Properties of the Monolithic Columns monolithic column
total porosity external porosity, e internal porosity, i average pore size of macropores (µm) average pore size of mesopores (nm) radius of cylindrical skeleton, Rss (µm)
19
21
22
0.83 0.69 0.44 2.0 13 0.68
0.84 0.69 0.50 2.0 13 0.67
0.86 0.71 0.52 2.0 13 0.63
the average pore diameter. The molecular diameter of the sample compound was calculated from its molar volume at its normal boiling point, on the assumption that the shape of the sample molecule is spherical. In this study, eq 29 was used to calculate the value of Kp because both equations provide almost the same values of Kp. For instance, in the case of the pore diffusion of thiourea, values of Kp of 0.79 and 0.83 were supplied by eqs 29 and 30, respectively, in mesopores of 12.5-nm diameter. EXPERIMENTAL SECTION Columns and Reagents. The concentration perturbation method was used in this study to acquire thermodynamic and kinetic data in the same time. The elution peak profiles of small perturbations were measured at different mobile-phase flow velocities, under three sets of experimental conditions (stationary and mobile phases and sample compounds, sets 1-3, below). The experimental data previously published by Kele and Guiochon39 were also analyzed in this study (sets 4-6). The C18-silica monolithic columns used are Chromolith Performance (Merck, Darmstadt, Germany) RP-18e columns (19, 21, and 22). The mobile phases were methanol/water mixtures of different compositions. The sample compounds were butylbenzene, butylphenol, butylbenzoate, and triphenylene. Thiourea was used as the inert tracer. The experimental conditions were as follows: set 1, Chromolith Performance RP-18e column 19, methanol/water (80/ 20, v/v), butylbenzene, 22.5-24.5 °C; set 2, Chromolith Performance RP-18e column 21, methanol/water (60/40, v/v), butylphenol, 22.5-24.5 °C; set 3, Chromolith Performance RP-18e column 22, methanol/water (65/35, v/v), butylbenzoate, 23 °C; set 4, Chromolith Performance RP-18e column 19, methanol/water (80/20, v/v), butylbenzene, 25 °C’ set 5, Chromolith Performance RP-18e column 19, methanol/water (80/20, v/v), triphenylene, 25 °C; set 6, Chromolith Performance RP-18e column 19, methanol/ water (80/20, v/v), thiourea, 25 °C. Table 1 lists the physical properties of the C18-silica monolithic columns. Some parameters listed in Table 1 were calculated from the porosity data determined by inverse size-exclusion chromatography.47 For instance, the hypothetical radius of the C18-silica (41) Treybal, R. E. Mass-Transfer Operations; McGraw-Hill: New York, 1980. (42) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. (43) Kataoka, T.; Yoshida, H.; Ueyama, K. J. Chem. Eng. Jpn. 1972, 5, 132136. (44) Wilson, E. J.; Geankoplis, C. J. Ind. Eng. Chem. Fundam. 1966, 5, 9-14. (45) Satterfield, C. N.; Colton, C. K.; Wayne, H. P., Jr. AIChE J. 1973, 19, 628635. (46) Brenner, H.; Gaydos, L. J. J. Colloid Interface Sci. 1977, 58, 312-356. (47) Al-Bokari, M.; Cherrak, D.; Guiochon, G. J. Chromatogr., A 2002, 975, 275284.
skeleton (Rss) was calculated from the external porosity (e) and the average radius of the through-macropores. The value of Rss was estimated as 0.6-0.7 µm, on the assumption that the unit element of the stationary-phase skeleton is a porous cylinder containing the mesopores and located in the center of a coaxial, cylindrical through-macropores. This value of Rss is consistent with the data on the average skeleton size ranging from 1.3 to 1.6 µm, which was provided from the manufacturer (Merck).7 The value of the tortuosity factor (k2) was estimated to be between 5.5 and 6.7 by analyzing the elution peak profiles of thiourea,39 which is not retained. These values of k2 are reasonable. Apparatus. Measurements of the elution peak profiles were carried out using a Hewlett-Packard (now Agilent Technologies, Palo Alto, CA) HP-1090 liquid chromatograph. This instrument was equipped with a multisolvent delivery system, an automatic sample injector with a 25-µL loop, a diode-array UV detector, and a computer data acquisition station. Procedure. Perturbation pulses were injected under steadystate conditions, at mobile-phase flow rates set in the range from 0.1 to ∼3.0 mL min-1 and at constant concentration of the sample in the mobile phase. Because the amount of sample compound injected was quite small, the elution peaks were recorded under locally linear conditions, i.e., corresponded to linear perturbations. It was separately confirmed that the retention of the elution peaks properly corresponded to the slope of the tangent of the equilibrium isotherm previously measured by the frontal analysis method and that the perturbation experiments were actually carried out under locally linear isotherm conditions. All these experiments were carried out at ambient temperature. RESULTS AND DISCUSSION Plot of (Htotal - Hf) versus u0. Panels a-c of Figure 1 show the dependence of (Htotal - Hf) on the mobile-phase flow velocity in the first three sets of experiments (sets 1-3). According to eq 23, (Htotal - Hf) was plotted versus u0. Experimental data previously published (sets 4-6) were similarly analyzed, as illustrated in Figure 2a-c. The experimental results (symbols) exhibit the conventional profiles of the flow rate dependence of the HETP, that is, are convex downward. Each one of the six parts shows three lines that were obtained by fitting the experimental data to each one of the three equations (eqs 24-26) used to estimate the value of kf because no more appropriate correlation has yet been proposed for the external mass transfer of the sample molecules between the mobile phase and the surface of monolithic stationary phases. The difference between these three curves depends on the estimation results of eqs 24-26. Two of these curves, those based on the penetration theory40 and on the equation of Kataoka et al.43, are similar over the whole range of u0 in Figures 1 and 2. On the other hand, the lines derived from the WilsonGeankoplis equation (eq 26)44 deviate from the other two curves, except in Figure 2c. When eq 26 is used to estimate kf, an appropriate value of the average size of the element of the monolithic skeleton must be selected for dp. Gritti et al.7 and Minakuchi et al.2,3 used the combined domain size, which is the sum of the average diameter of the through-macropores and the average thickness of the silica skeleton elements. Gritti et al.7 estimated the eddy diffusion length as 3.7 µm for the monolithic column 22. Kele and Guiochon39 similarly assigned the “chroAnalytical Chemistry, Vol. 75, No. 24, December 15, 2003
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Figure 1. Dependence of the value of (Htotal - Hf) on the superficial velocity, u0, for three experimental sets. (a) Set 1. Penetration theory (b, s): A ) 4.1 × 10-4 ( 4 × 10-5, B ) 2 × 10-5 ( 6.3 × 10-7, C′ ) 1.95 × 10-3 ( 1.7 × 10-4, R2 ) 0.99 147. Kataoka equation (2, - -): A ) 4.3 × 10-4 ( 4 × 10-5, B ) 2 × 10-5 ( 6.3 × 10-7, C′ ) 1.67 × 10-3 ( 1.7 × 10-4, R2 ) 0.99 184. Wilson-Geankoplis equation (9, ‚‚‚): A ) 3.8 × 10-4 ( 4 × 10-5, B ) 2 × 10-5 ( 6.7 × 10-7, C′ ) 1.1 × 10-3 ( 1.8 × 10-4, R2 ) 0.99 209. (b) Set 2. Penetration theory (b, s): A ) 5.4 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.4 × 10-7, C′ ) 2.52 × 10-3 ( 9 × 10-5, R2 ) 0.98 753. Kataoka equation (2, - -): A ) 5.6 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.4 × 10-7, C′ ) 2.08 × 10-3 ( 9 × 10-5, R2 ) 0.98 676. Wilson-Geankoplis equation (9, ‚‚‚): A ) 4.9 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.2 × 10-7, C′ ) 1.26 × 10-3 ( 9 × 10-5, R2 ) 0.99 046. (c) Set 3. Penetration theory (b, s): A ) 1.5 × 10-4 ( 2 × 10-5, B ) 2 × 10-5 ( 3.9 × 10-7, C′ ) 2.13 × 10-3 ( 1 × 10-4, R2 ) 0.99 244. Kataoka equation (2, - -): A ) 1.7 × 10-4 ( 2 × 10-5, B ) 2 × 10-5 ( 3.9 × 10-7, C′ ) 1.56 × 10-3 ( 1 × 10-4, R2 ) 0.9932. Wilson-Geankoplis equation (9, ‚‚‚): A ) 7 × 10-5 ( 2 × 10-5, B ) 2 × 10-5 ( 4.2 × 10-7, C′ ) 4.2 × 10-4 ( 1 × 10-4, R2 ) 0.99 452.
matographic apparent particle size” as 4 µm for column 19. These values correspond to the sum of an average through-macropore diameter of 2 µm and an average skeleton size ranging between 1.3 and 1.6 µm for the Chromolith Performance columns used in this study. In this study, kf was calculated using eq 26 and assuming that the corresponding value of dp was equal to 4 µm. As indicated in eq 23, the experimental data points of coordinates (Htotal - Hf) and u0 in Figures 1 and 2 were fitted to the classical van Deemter equation, including the three coefficients A, B, and C.8,16,48 On the other hand, eq 23 indicates how these three coefficients are correlated with Rss, Dm, and De, respectively. The values of the three coefficients of each curve were determined as indicated in the figure captions, using the least-squares regression method. The solid, dashed, and dotted lines show the (48) Knox, J. H. J. Chromatogr., A 1999, 831, 3-15.
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best van Deemter profiles derived from the experimental data represented by the solid circles, triangles, and squares, respectively. Mass Transfer in the C18-Silica Monolithic Column. Contribution of Each Mass-Transfer Process to the Column Efficiency. Equation 19 shows that the HETP (Htotal) is the sum of the contributions of the three mass-transfer processes taking place in the column. However, the overall column efficiency, Htotal, is the only parameter that can be measured experimentally. No information on each individual contribution is directly accessible. The values of the coefficients A, B, and C′ are determined from the data in Figures 1 and 2, by curve-fitting. As indicated in eq 23, the coefficients A and B correspond to the contribution of eddy and axial dispersion, respectively. The coefficient C′ is correlated with the mass-transfer resistance in the C18-silica monolithic stationary phase.
Figure 2. Dependence of the value of (Htotal - Hf) on u0 for three experimental sets. (a) Set 4. Penetration theory (b, s): A ) 6.3 × 10-4 ( 2 × 10-5, B ) 2 × 10-5 ( 4.4 × 10-7, C′ ) 1.96 × 10-3 ( 8 × 10-5, R2 ) 0.97 847. Kataoka equation (2, - -): A ) 6.5 × 10-4 ( 2 × 10-5, B ) 2 × 10-5 ( 4.4 × 10-7, C′ ) 1.61 × 10-3 ( 8 × 10-5, R2 ) 0.97 882. Wilson-Geankoplis equation (9, ‚‚‚): A ) 5.7 × 10-4 ( 2 × 10-5, B ) 2 × 10-5 ( 4.0 × 10-7, C′ ) 8.8 × 10-4 ( 7 × 10-5, R2 ) 0.98 564. (b) Set 5. Penetration theory (b, s): A ) 8.4 × 10-4 ( 3 × 10-5, B ) 1 × 10-5 ( 5.7 × 10-7, C′ ) 2.75 × 10-3 ( 1 × 10-4, R2 ) 0.95 148. Kataoka equation (2, - -): A ) 8.6 × 10-4 ( 3 × 10-5, B ) 1 × 10-5 ( 5.7 × 10-7, C′ ) 2.28 × 10-3 ( 1 × 10-4, R2 ) 0.9382. Wilson-Geankoplis equation (9, ‚‚‚): A ) 7.6 × 10-4 ( 3 × 10-5, B ) 1 × 10-5 ( 5.4 × 10-7, C′ ) 1.3 × 10-3 ( 9 × 10-5, R2 ) 0.93 403. (c) Set 6. Penetration theory (b, s): A ) 6 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.7 × 10-7, C′ ) 5.7 × 10-4 ( 6 × 10-5, R2 ) 0.95 362. Kataoka equation (2, - -): A ) 6 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.7 × 10-7, C′ ) 5.4 × 10-4 ( 6 × 10-5, R2 ) 0.954. Wilson-Geankoplis equation (9, ‚‚‚): A ) 5.9 × 10-4 ( 2 × 10-5, B ) 1 × 10-5 ( 3.7 × 10-7, C′ ) 4.7 × 10-4 ( 6 × 10-5, R2 ) 0.95 686.
The contributions of the three kinetic processes to Htotal are compared in Figure 3, which shows the plots of Htotal and of the three to Htotal as functions of u0, taking as an example the case of the solid line in Figure 1a. In the low flow rate range (u0 < ∼0.25 cm s-1), the contribution of axial dispersion (Hax) to Htotal is always larger than those of the external mass transfer (Hf) and the diffusive mass transfer inside the stationary-phase skeleton (Hd). Both contributions, Hf and Hd, increase with increasing flow rate. However, the slope of Hd is about twice (or even more) as large as that of Hf. In the high flow rate range (u0 > ∼0.25 cm s-1), the contribution of Hd becomes larger than that of Hax. As illustrated in Figure 3, the relative contributions of the three kinetic processes to Htotal change with the flow rate. This figure corresponds only to the experimental set 1. Figure 4 summarizes the comparison of the contributions to Htotal of the three masstransfer processes under different experimental conditions, including those of set 1. The values in parentheses indicate the
adsorption equilibrium constant (Ka). The values of δax, δf, and δd were calculated at three different flow rates. The values at u0 ) 1.0 cm s-1 are the results of hypothetical calculations because no experimental data were actually measured at this flow rate condition. The total length of the bars give the sum of δax, δf, and δd. As indicated in eq 16, δax depends on u0. The value of DL also shows the flow rate dependence as indicated in eq 22. Although eq 17 does not include u0, the value of δf also depends on u0 because kf is a function of u0 as indicated in eqs 24-26. Although the values of both δax and δf vary with u0, it is noteworthy that δax decreases faster than δf with increasing u0, for all the experimental sets 1-5. Contribution of the A and B Terms to Hax. Figure 5 shows the variations of the A and B terms as a function of u0. As indicated in eq 23, the contribution of the A term is independent of u0. The B term is inversely proportional to u0. As is well known, this term accounts for the contribution of molecular diffusion in the mobile Analytical Chemistry, Vol. 75, No. 24, December 15, 2003
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Figure 3. Contribution of the three mass-transfer processes in the C18-silica monolithic column (contributions of axial dispersion, Hax, external mass transfer, Hf, and intraparticule diffusive transfer, Hd) to the correlation between HETP and u0 for the experimental set 1. The value of the external mass-transfer coefficient, kf was calculated using the penetration theory.
Figure 4. Comparison of the mass-transfer resistance contributions of the three kinetic processes in the C18-silica monolithic column (contributions of axial dispersion, δax, external mass transfer, δf, and intraparticule diffusive transfer, δd) to the second central moment, µ2′, at different mobile-phase flow velocities for the experimental sets 1-5. The value of the external mass-transfer coefficient, kf, was calculated using the penetration theory.
phase and has a large influence on band broadening at low flow rates. Since it is constant, the A term becomes predominant in the high flow rates. As shown in Figure 5, the contribution of Hax to Htotal reaches ∼37% at u0 ) 0.3 cm s-1. Then, the value of Hax is almost entirely due to the contribution of the A term. The results in Figure 5 suggest that an improvement of the column efficiency at high flow velocities could consist in reducing the contribution of eddy diffusion to band broadening. Figure 5 also illustrates the calculation of DL, which is of the order of 10-5 cm2 s-1. As indicated in eqs 22 and 23, the value of DL is derived from the values of the coefficients A (4.1 × 10-4) and B (2 × 10-5), both obtained by curve fitting of the experimental data in Figure 1a. Mass Transfer in the C18-Silica Monolithic Stationary Phase. Figure 4 shows that the value of δd does not exhibit any flow rate dependence, as indicated in eq 18. The mass-transfer 6982 Analytical Chemistry, Vol. 75, No. 24, December 15, 2003
Figure 5. Contributions of the A and B term of the van Deemter equation to the axial dispersion, Hax, for experimental set 1. The straight line shows the flow rate dependence of the axial dispersion coefficient, DL, which correlates with Hax. The value of the external mass-transfer coefficient, kf was calculated using the penetration theory.
kinetics in the monolithic skeleton should be independent of u0 because it is the result of the diffusive flux of the sample molecules in the stagnant mobile phase contained in the mesopores. In Figure 4, the value of δd is constant, regardless of u0. However, the relative importance of δd increases with increasing u0, as illustrated by the total length of the bars decreasing with increasing u0. The relative importance of δd reaches approximately 60-70% at u0 ) 1.0 cm s-1, irrespective of Ka. The main purpose of the development of the monolithic columns is the achievement of high-efficiency, high-speed separations. This advantage arises from the fast mass-transfer kinetics of the sample molecules in the monolithic columns, suggesting that the intrinsic mass-transfer characteristics of the monolith pore space should be studied in more detail. Contribution of Pore Diffusion and Surface Diffusion to the Mass Transfer in the C18-Silica Skeleton. As indicated in eq 27, it is widely assumed that the mass transfer in porous adsorbents consists of two parallel mechanisms, pore and surface diffusion. Figure 6 compares these two contributions to the overall mass-transfer kinetics in the monolithic stationary phase. The numbers in parentheses on the ordinate axis are the values of Ka. The total length of each bar is the value of De. The length of the hatched part corresponds to that of Dp. In Figure 6, each set of three bars corresponds to each set of experiments 1-5. As shown in Figure 6, the values of De and Dp depend on the value of kf, estimated from eqs 24-26. The value of De exceeds that of Dp by a factor of ∼3.3-7.4 in the five experimental sets of this study, suggesting that a large fraction of the mass flux inside the monolith, between 77 and 88%, is due to surface diffusion. It seems, however, that the ratio De/Dp in set 3, derived using the Wilson-Geankoplis equation (eq 26), is exceptionally too large. We may conclude that surface diffusion plays the predominant role in the mass transfer inside the monolithic stationary phase. To the best of our knowledge, however, there has yet been no investigations of surface diffusion in monolithic columns. Surface diffusion is a migration process of the sample molecules that takes place while they are sorbed on the stationary-
Figure 6. Comparison of the contributions of pore diffusion (Dp) and surface diffusion (Ds) to the mass transfer of the sample molecules in the C18-silica monolithic stationary phase for experimental sets 1-5. The three different literature correlations were used for estimating the external mass-transfer coefficient, kf.
phase surface.17-20 Molecules can migrate in the vicinity of the surface, in the adsorbed state. Surface diffusion is an activated process. Its thermodynamic properties, for example, its activation energy and frequency factor, can be derived by analyzing the temperature dependence of Ds. The flux (Js) of the sample molecules due to surface diffusion is given by
Js ) -DsFp(dq′/dx)
(31)
where Fp is the density of the particles of stationary phase. The ratio of dq′ to dx is the gradient of the adsorbate concentration, which is the driving force of surface diffusion. According to eq 31, Ds is a proportionality coefficient, the surface diffusion coefficient. Surface diffusion contributes considerably to intraparticle diffusion; hence, it plays an important role in the masstransfer kinetics in chromatographic columns, at least in RPLC.17-20 Surface diffusion informs also on the adsorption characteristics of the mobile-stationary-phase interface because it is a molecular migration process that takes place in the adsorbed state. Its detailed study allows a better understanding of the retention mechanism of chromatography. Although the significance of surface diffusion as one of the important mass-transfer mechanisms had already been described more than 30 years ago,8 surface diffusion and its contributions to mass-transfer kinetics and to column efficiency are still insufficiently widely recognized in liquid chromatography. Surface Diffusion Coefficient. Figure 7 shows the correlation between Ds and Ka. The open, solid, and open with central vertical line symbols represent the Ds values calculated using eqs 24-26. The circle, pentagonal, square, diamond, and hexagonal symbols represent the results of the experimental sets 1-5, respectively. The vertical error bar corresponds to the variation in the value of the coefficient C′, which is indicated in the figure captions in Figures 1 and 2. Close values of Ds were calculated using eqs 24 and 25. The Ds values based on eq 26 are 2-5-fold larger than the other ones. It is likely that the value of Ds in the monolithic stationary phase studied is of the order of 10-7 cm2 s-1. The solid
Figure 7. Surface diffusion coefficient, Ds, versus the equilibrium constant, Ka, for experimental sets 1-5. The straight line represents the correlation between Ds and Ka for spherical RP silica gel particles bonded with alkyl ligands of different lengths.
line in Figure 7 illustrates the typical correlation of Ds with Ka derived from previous experimental data, measured on spherical RP packing materials.19,20 Figure 7 suggests that the Ds values in the C18-silica skeleton are several times smaller than those in spherical particles. At this stage, however, it is not possible to conclude that there is a significant difference between these Ds values because the number of experimental data is limited and the estimates of kf are uncertain. A suitable correlation must be developed to estimate kf in monolithic columns, and a larger number of reliable experimental results must be acquired. The solid line in Figure 7 indicates that Ds decreases with increasing Ka. The other plots in Figure 7 exhibit similar trends. These results suggest that the moment analysis method based on the new moment equations (eqs 13-18) provides correct results on surface diffusion in C18-silica monoliths although the absolute value of Ds must be checked in more detail. As indicated in eq 23, the coefficient C′ of the van Deemter equation corresponds to the ratio of 2δd to δ02. Equation 18 indicates that the contribution of δd to band broadening is proportional to the ratio Rss2/(8De) in the case of the monolithic stationary phase. The value of De is usually derived by assuming parallel contributions of pore and surface diffusion as shown in eq 27.17,18 As indicated in Figure 6, however, surface diffusion has the largest contribution to the mass transfer in the monolithic stationary phase. As a first approximation, the ratio Rss2/(8De) corresponds to the ratio Rss2/[8(1-i)KaDs]. This value is a similar to a time constant. The plots in Figure 8 show the dependence on Ka of the ratio Rss2/[8(1-i)KaDs], calculated from the Ds values in Figure 7. The contribution of δd to band broadening in conventional columns packed with spherical particles is represented as follows.17,18
δd ) (1 - e)(Rp2/15De)[i + (1 - i)Ka]2
(32)
Similarly, the ratio Rp2/(15De) is close to the ratio Rp2/[15(1 i)KaDs] because of the large contribution of surface diffusion to the intraparticle mass transfer.19,20 Figure 8 illustrates five linear Analytical Chemistry, Vol. 75, No. 24, December 15, 2003
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Figure 8. Ratio Rss2/[8(1 - i)KaDs] versus the equilibrium constant, Ka, for experimental sets 1-5. The straight lines represent the correlation between Rp2/[15(1 - i)KaDs] and Ka for spherical RP particles of different diameters from 1 to 5 µm.
correlations between Ka and Rp2/[15(1 - i)KaDs], derived from the straight line in Figure 7, for values of dp from 1 to 5 µm and assuming a value of i equal to 0.4, a typical value for conventional porous silica gel particles.17,18 The symbols are located in the region between the lines for dp ) 3 and 5 µm. This is consistent with the earlier result reported by Kele and Guiochon,39 who measured the HETP of the monolithic column 19 as a function of the mobile-phase flow velocity under various experimental conditions. They found that the minimum of the van Deemter plots was between about 7.5 and 13 µm. These values are comparable to those of conventional columns packed with spherical particles of dp ) 3-5 µm. From this result, they estimated a chromatographic apparent particle size of 4 µm for this column. Consistent with this result of Kele and Guiochon,39 in Figure 8, the plots scatter around the linear correlation for dp ) 4 µm. The lines in Figure 8 are based on the linear correlation between ln Ds and ln Ka shown in Figure 7 and concerning surface diffusion in various RP packing materials made of spherical silica gel particles bonded to alkyl ligands of different lengths (C1, C4, C8, C18).19,20 Finally, the plots in Figure 8 show the results obtained for the C18-silica monolithic stationary phases considered in this study. Although these plots show some scatter, the results in Figure 8 suggest that the time constants of the mass transfer in the C18-silica skeleton of the monolithic columns 19, 21, and 22 are similar to that in spherical silica gel particles having ∼4-µm diameter. This conclusion is also consistent with the observations of Gritti et al.,7 Minakuchi et al.,2,3 and Kele and Guiochon,39 probably suggesting the reliability of the Ds values derived in this study for the C18-silica monolithic stationary phase. Accuracy of the Ds Values. As shown in Figure 7, it was possible to determine values of Ds in this study. This required that several corrections be made, using related kinetic parameters, which were estimated using literature correlations. The influence of these corrections on the accuracy of Ds is now considered. First, the contribution of the external mass transfer to µ2′ was subtracted beforehand, during the determination of DL and De. As previously indicated several times, an uncertainty in the estimate of kf affects the results of the moment analysis of the 6984 Analytical Chemistry, Vol. 75, No. 24, December 15, 2003
mass-transfer kinetics in the monolithic columns. In this study, kf was estimated in three different manners, i.e., using the penetration theory (eq 24),40 the equation proposed by Kataoka et al. (eq 25),43 and the Wilson-Geankoplis equation (eq 26),44 because there are no available correlations to estimate kf for the external mass transfer of the sample molecules between the mobile phase percolating through the monolith and the surface of the porous elements of the monolith. At this point, however, we do not have enough information to discuss quantitatively the accuracy of the estimates made of kf in the monoliths by these three equations. However, we can consider at least the following three points. First, in Figures 1 and 2, the experimental data and the curves derived using eqs 24 and 25 have similar profiles while those based on eq 26 deviate from these lines. The relative error made on the values of the coefficient C′ calculated using eqs 24 and 25 ranges between about 5 and 27%, as indicated in the captions of Figures 1 and 2. Kataoka et al.43 reported that most experimental data previously published fit the correlation of eq 25 with an error of (20% if the modified Reynolds number (Re′) is smaller than 100. The Wilson-Geankoplis equation gives a suitable correlation to estimate kf in fixed beds packed with spherical particles. However, it seems difficult to use it to obtain an accurate estimate of the average size, dp, of the structural elements of the monolithic silica. In this study, eq 26 was used to estimate kf by assuming that the corresponding value of dp is equal to 4 µm.39 Second, as explained later, the error made in estimating kf and due to the propagation of the error made in estimating Dm could be several percent. Finally, as shown in Figures 3 and 4, the contribution of the kinetics of external mass transfer to band broadening is relatively small, i.e., approximately 10-30%. When this contribution was calculated in Figures 3 and 4, the value of kf was estimated using eq 24, based on the penetration theory. It is likely that the results of this study are not significantly influenced by variations of the estimated value of kf because of the small contribution of δf to µ2′. As a result, we consider that the value of the coefficient C′ was estimated with a relative error between 5 and 27%, as explained above. In future work, however, it will be necessary to develop an estimation method providing more accurate values of kf in monolithic columns. Although we can use several literature correlations to estimate kf, the structural characteristics of monolithic columns are quite different from those of conventional beds of particulate packing materials having a spherical, cylindrical, or ellipsoidal shape. The accuracy achieved in the measurement of the mass-transfer kinetics coefficients using the moment analysis method depends on the error made in estimating kf. The achievement of an accurate estimate of kf is thus necessary for a good understanding of the mass-transfer kinetics in monolithic columns. This subject is currently investigated. The contribution of axial dispersion to µ2′ was separated from those of the mass-transfer resistances in the porous packing material by taking advantage of the different flow rate dependence of these two contributions. As indicated previously, the value of De was derived from that of the coefficient C′ of the classical van Deemter equation. The value of Ds was derived from De by subtracting the contribution of Dp to De. The value of Dp was calculated from Dm, i, Kp, and k, according to eq 28, which is based on the parallel pore model and is one of the most popular
correlations for estimating Dp. The accuracy of the estimate of Ds depends on the errors made in estimating Dm and Kp. The two other physical parameters, i and k, are directly measured and are not calculated by any estimation procedure. The value of Dm was derived from the Wilke-Chang equation, one of the most popular correlations for estimating Dm.16,18,40-42 It is considered that the average error made on Dm by this correlation is usually between 10 and 20%.42 Equations 29 and 30 have been proposed for estimating Kp.45,46 Close values of 0.79 and 0.83 are calculated for Kp using eqs 29 and 30, respectively, in the case of the pore diffusion of thiourea molecules in mesopores of 12.5-nm diameter. Equation 29 proposed by Satterfield et al.45 was used for estimating the other value of Kp in this study. As shown in Figure 6, the contribution of surface diffusion to the overall mass transfer inside the C18-silica monolithic stationary phase is much larger than that of pore diffusion. Most of the sample molecules migrate inside the C18-silica skeleton by surface diffusion. Because of the major contribution of surface diffusion, the influence of small variations in Dp (hence in Dm and Kp) on the estimation of Ds is extremely small. The errors made in the estimations of Dm, Kp, and Dp have only a small influence on the results obtained here concerning surface diffusion. Finally, eqs 24-26 indicate that the value of kf is proportional to Dm1/2 or Dm2/3. Because Dm is estimated with an error of 1020%, as explained above,42 the error made in estimating kf by eqs 24-26 and due to the propagation of the error made on Dm should be of only a few percent. All the corrections described above are responsible for errors made in the determination of Ds. At this stage, however, we cannot assess the accuracy of the values obtained for Ds. We can only conclude that the values of Ds that have been determined in this first detailed study of the masstransfer resistances of a C18-silica monolithic stationary phase are most probably reasonable estimates.
results obtained with conventional packing materials. The “chromatographic corresponding particle size” of the C18-silica monolithic stationary phase studied was estimated at ∼4 µm by analyzing the pseudo time constant of molecular migration in mesopore space, a result also consistent with values previously reported, suggesting that the Ds values derived in this study are reliable. Finally, the values of Ds determined in this study depend on the estimate made of kf. Their accuracy cannot be assessed because of the limited number of experimental data and the uncertainty in the estimate of kf. It is necessary to develop a suitable correlation, of known accuracy, valid for monolithic columns and to acquire reliable experimental data. Work is in progress in this area.
CONCLUSION The kinetics of mass transfers in a commercial C18-silica monolithic columns were studied for the first time by the method of moment analysis. The elution profiles of perturbation pulses were recorded at a series of increasing mobile-phase flow velocities. The HETP were measured and corrected by subtracting the contributions of the instrument and of the external mass transfer. Three equations from the literature (eqs 24-26) were used to estimate kf since no suitable correlation is available yet for monolithic columns. At high flow rates, the contribution of axial and eddy dispersion to band broadening is large, although the contribution of the diffusive migration of the solute through the monolithic C18-silica skeleton is the largest at high flow rates. A significant reduction of either of these two contributions would markedly improve the resolutions observed for high-speed separations. As for conventional particulate RP packing materials, surface diffusion plays the predominant role for the mass transfer inside the porous monolithic stationary phase. Approximately 75-90% of the pore diffusion flux is carried by surface diffusion. A first estimate of the value of Ds in a C18-silica monolithic stationary phase at ambient temperature is of the order of 1 × 10-7 cm2 s-1, i.e., several times less than typical values measured for spherical RP stationary phases. It was also observed that Ds decreases with increasing retention of the sample compounds, a result consistent with experimental
DL
axial dispersion coefficient (cm2 s-1)
Dm
molecular diffusivity (cm2 s-1)
Dp
pore diffusivity (cm2 s-1)
Ds
surface diffusion coefficient (cm2 s-1)
H
height equivalent to a theoretical plate, HETP (µm)
Js
mass flux due to surface diffusion (g cm2 s-1)
k
tortuosity factor
ka
adsorption rate constant (s-1)
kf
external mass-transfer coefficient (cm s-1)
GLOSSARY A
coefficient of the van Deemter equation (cm)
As
ratio of the surface area of the stationary phase to the column volume (cm-1)
B
coefficient of the van Deemter equation (cm2 s-1)
Bi
Biot number defined in eq 9
C
concentration of the sample molecules in the throughmacropore space (g cm-3)
C′
coefficient of the van Deemter equation (s)
Ce(t)
concentration of sample molecules in the mobile-phase solvent leaving from the column as a function of t (g cm-3)
Ci
concentration of sample molecules within the mesopores inside the stationary-phase skeleton (g cm-3)
C0
sample concentration of the injected pulse (g cm-3)
dp
particle diameter (µm)
De
diffusion coefficient of the sample molecules in the stationary phase (cm2 s-1)
Ka
adsorption equilibrium constant
Kp
hindrance parameter
mn
nth moment of a chromatographic peak (g cm-3 sn+1)
m0
zeroth moment of a chromatographic peak (g cm-3 s)
Ni
mass flux of sample molecules from the mobile-phase stagnant in the mesopore space to the surface of the stationary phase (g cm-3 s-1)
N0
mass flux of sample molecules from the mobile phase to the external surface of the stationary-phase skeleton (g cm-2 s-1)
p
Laplace transform variable (s-1)
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sample concentration adsorbed on the stationary phase (g cm-3)
µ1
first absolute moment (s)
µ2′
second central moment (s2)
r
radial distance from the center of the stationary-phase cylinder (µm)
F
density (g cm-3)
F
particle or stationary-phase density (g cm-3)
Re
Reynolds number () dpeuF/η or 2RsseuF/η)
τ
injection time of a sample pulse (s)
Rp
radius of the spherical stationary-phase particles (µm)
Rss
radius of the cylindrical stationary-phase skeleton (µm)
Sc
Schmidt number () η/(FDm))
ax
axial dispersion
Sh
Sherwood number () dpkf/Dm or 2Rsskf/Dm)
d
diffusive migration in the stationary phase
t
time (s)
f
external mass transfer
Rss
at the surface of monolithic skeleton
total
total value
q
s-1)
u
interstitial velocity (cm
u0
superficial velocity (cm s-1)
x
distance along the surface (cm)
z
longitudinal distance along the column (cm) ACKNOWLEDGMENT
Greek Letters δ
contribution of mass-transfer resistance to µ2′ (s)
δ0
defined in eq 15
e
void fraction (external porosity) of the column
i
internal porosity of the stationary phase materials
γ1, γ2, γ3
geometrical coefficients in eqs 21 and 22
η
viscosity (Pa s)
λm
ratio of the diameter of the sample molecule to the average pore diameter (sn)
µn
nth absolute moment
µn′
nth central moment (sn)
6986
Subscripts
Analytical Chemistry, Vol. 75, No. 24, December 15, 2003
This work was supported in parts by Grant-in-Aids for Scientific Research (12640581 and 14340234) from the Japanese Ministry of Education, Science and Culture, by Grant CHE-00-70548 of the National Science Foundation, and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
Received for review May 28, 2003. Accepted September 17, 2003. AC0302206
