Influence of Molecule Size on Its Transport Properties through a

Anal. Chem. 2010, 82, 2668–2679

Influence of Molecule Size on Its Transport Properties through a Porous Medium Ve´ronique Wernert,* Renaud Bouchet, and Renaud Denoyel Universite´s d’Aix-Marseille I, II et III-CNRS, UMR 6264, Laboratoire Chimie Provence, Equipe Matdiv, Centre Saint Je´roˆme, F-13397 Marseille Cedex 20, France The mass transfer kinetics of toluene and polystyrenes (of which the Mw varies from 162 to 1.85 × 106 g mol-1) through columns filled with silica porous spheres were studied by inverse size exclusion chromatography. The mass transfer parameters were measured by modeling the band broadening of the chromatograms. The experimental height equivalent to a theoretical plate (HETP) data were analyzed using the general rate model in order to determine the effective diffusion coefficient in porous particles as a function of molecular size. The bulk molecular diffusion coefficients were experimentally determined by dynamic light scattering (DLS) and Taylor dispersion analysis (TDA). The topological tortuosity of the porous particles was determined by electrical measurements. The effective molecular diffusion coefficient through porous particles was modeled taking into account exclusion, friction, and at last tortuosity effects. A phenomenological law is proposed to model the evolution of the tortuosity experienced by a molecule in a porous particle as a function of its size. It gives a good prediction of the evolution of effective diffusion coefficient with the molecule/pore size ratio. A detailed knowledge of the transport properties of fluids through porous media is critical for the successful design, preparation, and application of adsorbents and membranes in many industrial and environmental processes that cover a very broad range of specialties (geology, engineering, chemistry, and physics). The performances of porous systems are directly related to the global kinetic of mass transfers, which is essentially determined by the architecture of the porous networks. Specific surface area, specific pore volume, porosity, and pore size distribution are well-defined microstructural parameters, which may be determined, in a standard manner, by techniques like gas adsorption or mercury porosimetry, at least in the case of rigid materials.1,2 Nevertheless, beyond the fact that very simple model of parallel pores are generally used to calculate the pore size distribution, these basic parameters cannot alone predict transport properties for which other parameters like pore connectivity or tortuosity play a crucial role. The connectivity, which is rigorously * To whom correspondence should be addressed. E-mail address: veronique. [email protected]. Fax: (33) 4 0.91.63.71.11. (1) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by powders and porous solids; Academic Press: London, 1999. (2) Giesche, H. Part. Part. Syst. Charact. 2006, 23, 9–19.

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defined in the graph theory,3 may be defined as the average number of possible fully distinct paths between two sites of the pore network. Even if its topological definition is mathematically clear, its experimental determination proves itself very difficult. The definition of tortuosity, which will be particularly studied in this paper, is not yet universally admitted. It is intuitive to geometrically define it as the ratio between the average path length followed by a probe that percolates through the porous medium to the length of the direct straight line.4 Nevertheless, this is the square of this ratio that it is generally retained.5-7 The main reason is that defined either by electrical or permeability measurements, the tortuosity of a bundle of capillaries is related to the square of this ratio.7 In a recent paper,8 the following general relationship for the definition of the tortuosity τ, was proposed: J ( DFG )

eff

)

ε J τ DGF

(

)

0

(1)

where J/DFG is the ratio between a parameter J characterizing the transport intensity (flow of matter, velocity, current, ...) and the driving force gradient DFG, responsible for the transport. The indexes “eff” and “0” indicate measurements with and without the porous medium of porosity ε, respectively. This is not a geometric definition, but the tortuosity such defined is expected to reflect the topological properties of the pores network. It implies that it is an average parameter which measures the “integral” efficiency of the fluid flow to pass trough a porous system. Therefore, the tortuosity integrates the whole complexity of the porous network which consists of labyrinths of interconnected pores with irregular shapes and cross sections. The tortuosity is a very general concept used in many different fields.7,9,10 Its measurement by electrical, molecular diffusion, or permeability methods has been carried out in many different situations. Many studies deal with the characterization of mem(3) Barett, L. K.; Yust, C. S. Metallography 1970, 3, 1–33. (4) Kozeny, J. Uber kapillare Leitung des Wassers im Boden. Sitzungberichte der Akademie der Wissenschaftung in Wein Abteilung II a, 1927; Vol. 136, pp 271-301. (5) Carman, P. C. Flow of gases through porous media; Butterworths Scientific Publications: London, 1956. (6) Adler, P. M. Porous Media; Butterworth-Heinemann: Boston, 1992. (7) Ben Clenell, M. In Developments in Petrophysics; Lowell, M. A., Harvey, P. K., Eds.; Geological Society Special Publication 122; Geological Society: London, 1997; pp 299-344. (8) Barrande, M.; Bouchet, R.; Denoyel, R. Anal. Chem. 2007, 79, 9115–9121. (9) Dullien, F. A. L. Porous Media: Fluid Transport and Pore Structure; Academic Press: San Diego, 1992. (10) Armatas, G. S.; Petrakis, D. E.; Pomonis, P. J. J. Chromatogr. A 2005, 1074, 53–59. 10.1021/ac902858b  2010 American Chemical Society Published on Web 02/25/2010

Figure 1. (i) Steric hindrance. The center of the molecule is excluded from the vicinity of the wall on a distance equal to the molecule radius rm. (ii) Tortuosity depending on the probe size. The possible pathways followed by the molecules depend on their sizes. (iii) Friction between the molecule moving within the pore and its wall.

branes11 or beds of nonporous particles,12 because only one scale of porosity and pore size can be considered, simplifying the analysis. In chromatography, at least two scales of porosities have to be considered because chromatographic columns are often made of a bed of porous particles. Recent monolithic silica columns exhibit also a distinct bimodal pore distribution.13,14 The whole column presents an apparent tortuosity, but inter- and intraparticle tortuosities, corresponding respectively to the transport in inter- and intraparticle porosities, may also be defined. This parameter is useful and of general interest only if it is independent of the measurement method. For this reason, the tortuosity factor of chromatographic supports estimated by different techniques, e.g. NMR,15-17 electrical measurements,5,8 gas chromatography,10 or liquid chromatography using the stop and flow method,18,19 may be realistic if, and only if, the size of the probe used is very small as compared to the pore size. For example, this is the case of electrical measurements for which the probes are small ions like Na+ and Cl- (∼0.2 nm) or of NMR relaxation methods applied to water. In most situations, the transfer depends also on the size of the probe molecule (rm) as compared to that of the pore (rp). In that case, only an apparent tortuosity is accessible. Indeed, when the size of the probe molecule is no more negligible as compared to the pore size, three distinct retarding effects must be considered as schematically represented in Figure 1: (i) The steric hindrance20,21 which corresponds to a decrease of the accessible particle porosity εp(rm) when the molecule size increases because the center of the molecule is excluded from the vicinity of the wall on a distance equal to the molecule radius rm. Topologically, it can be seen in a first approximation as an isomorphic constriction of the pores, but this could depend on the type of pore structure and organization. (ii) A more subtle factor is a change of the tortuosity of the accessible pore network with probe size. The structure seen by a molecule changes with its size simply because the pathways it (11) Mackie, J. S.; Meares, P. Proc. R. Soc., Series A 1955, 232, 498. (12) Mauret, E.; Renaud, M. Chem. Eng. Sci. 1997, 52, 1807–1817. (13) Grimes, B. A.; Skudas, R.; Unger, K.; Lubda, D. J. Chromatogr. A 2007, 1144, 14–29. (14) Miyabe, K.; Guiochon, G. J. Phys. Chem. 2002, 106, 8898–8909. (15) Tallarek, U.; Vergeldt, F. J.; Van As, H. J. Phys. Chem. B 1999, 103, 7654– 7664. (16) Stallmach, F.; Galvosas, P. Annu. Rep. NMR Spectrosc. 2007, 61, 51–131. (17) Sen, P. N. Concepts Magn. Reson., Part A 2004, 23A, 1–21. (18) Knox, J. H.; McLaren, L. Anal. Chem. 1964, 36, 1477–1485. (19) Gritti, F.; Guiochon, G. Chem. Eng. Sci. 2006, 61, 7636–7650. (20) Renkin, E. M. J. Gen. Physiol. 1954, 38, 225–234. (21) Dechadilok, P.; Deen, W. M. Ind. Eng. Chem. Res. 2006, 45, 6953–6959.

follows depend on its size. For example, some pores become inaccessible when the molecule size increases as illustrated in Figure 1. In many situations, notably when the porous structure is made of the repetition of particles with the same shape, the tortuosity increases when the porosity decreases.8 This implies that the apparent tortuosity τp(rm) is expected to increase with the size of the probe. Nevertheless, some data seem to show the reverse evolution.22,23 This point will be particularly addressed in this paper where a quantification of this effect is proposed. (iii) The friction between the molecule moving within the pore and the wall. Classically a hydrodynamical factor kf(rm) is introduced to correct the bulk diffusion coefficient from this wall drag effect. Several equations relating kf(rm) and λ ) rm/ rp can be found in the literature and have been applied to chromatography.20,21,24-27 To quantify these different contributions, a possible approach is to study the transport properties of probes of various sizes in nonadsorbing conditions. This is the type of system which is used in inverse size exclusion chromatography which is now routinely used for the pore characterization of porous chromatographic stationary phases. This is why the present paper proposes a study of the transport properties of toluene and polystyrenes in a tetrahydrofuran mobile phase through columns filled with silica porous spheres. The mass transfer parameters in those columns were measured by modeling the band broadening of the chromatograms. The experimental height equivalent to a theoretical plate (HETP) data were corrected from the solute dispersion in extra column volumes in order to properly assess the true column efficiency. These data are analyzed using the general rate model in order to determine the effective diffusion coefficient in porous particles as a function of the molecule size. This analysis needs the knowledge of the bulk diffusion coefficients of the molecules which were experimentally determined by dynamic light scattering (DLS) and Taylor dispersion analysis (TDA). The intra- and extraparticle tortuosities are determined by a method based on electrical measurements. The accessible porosity is directly determined from chromatographic measurements whereas the probe friction with the wall is modeled by the Renkin equation which relates the diffusion coefficient of a molecule in a cylindrical pore to the molecule/pore size ratio λ. A phenomenological law is proposed to evaluate the evolution of tortuosity with accessible porosity. Finally, the model predictions of the evolution of effective diffusion coefficient with λ are compared with the data obtained with the two type of porous silica analyzed. EXPERIMENTAL SECTION Chemicals. The mobile phase was tetrahydrofuran (THF, provided by Carlo Erba Reagents, SDS, HPLC grade (99.9% purity)). To explore the porous network of the chromatographic column at different scales, nonadsorbing molecules of different (22) Gustavsson, P. E.; Axelsson, A.; Larsson, P. O. J. Chromatogr. A 1998, 795, 199–210. (23) Forrer, N.; Butte´, A.; Morbidelli, M. J. Chromatogr. A 2008, 1214, 59–70. (24) Langford, J. F.; Schure, M. R.; Yao, Y.; Maloney, S. F.; Lenhoff, A. M. J. Chromatogr. A 2006, 1126, 95–106. (25) Yao, Y.; Lenhoff, A. M. J. Chromatogr. A 2006, 1126, 107–119. (26) Coffman, J. L.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. B 1997, 101, 2218–2223. (27) Gritti, F.; Guiochon, G. Anal. Chem. 2009, 81, 2723–2736.


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Table 1. Molecular Weights (Mw), Polydispersity (PDI), Bulk Diffusion Coefficient Dm Obtained by TDA for the Smallest Polymers (T, P01, P02, and P03) and by DLS for the Others Polymers (P04-P12) of the Solutes Used in ISEC and Hydrodynamic Radii rm Calculated with Stokes-Einstein Equation 5

a

polymer code

molecular weight Mwa/g mol-1

PDIa

toluene P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12

92 162 690 1380 3250 8900 19100 33500 96000 243000 546000 827000 1850000

1.00 1.09 1.05 1.05 1.03 1.03 1.03 1.04 1.03 1.02 1.08 1.05

molecular diffusion coefficient Dm (TDA and DLS measurements)/m2 s-1 2.15.10-9 b 1.51.10-9 b 7.10 10-10 b 5.00.10-10 b 3.19.10-10 2.04.10-10 1.33.10-10 8.73.10-11 5.08.10-11 3.19.10-11 2.10.10-11 1.65.10-11 1.12.10-11

probe radius (eq 5) rm/nm 0.22 0.31 0.67 0.95 1.43 2.36 3.62 5.50 9.44 15.02 22.84 29.09 42.90

Data given by the supplier. b TDA measurements.

Table 2. Properties of Silica Lichrospher Si60 (Merck) and Silica Lichrospher Si300 (Merck) Obtained by Gas Adsorption and Conductivity Measurements samples

density g cm-3

specific surface area m2 g-1

particle diameter dp′ µm

particle porosity εp(BJH)

mean pore radius rp(BJH) nm

particle tortuositya τp

Lichrospher Si60 Lichrospher Si300

2.2 2.2

721 83

12 10

0.66 0.68

2.5 22

2.1 1.9

a

Determined experimentally by electrical measurements (eq 6).

sizes made of polystyrenes (from Polymer Standards Service, Mainz, Germany) with molecular weights Mw ranging from 162 to 1 850 000 g mol-1 were used as solute probes. Also toluene (from Aldrich, 99.7% purity) was used as the smallest probe. Their characteristics, including polydispersity, are summarized in Table 1. Stationary Phases and Columns. The columns used in this work were Lichrospher Si60 and Lichrospher Si300 graciously supplied by the manufacturer (Merck, Germany), who also provide us with the corresponding stationary phases. Their dimensions are 125 mm × 4 mm. The main properties of the porous silica powders are summarized in Table 2. Pore size and pore volume were calculated by the BJH model28 applied to the adsorptiondesorption isotherms of nitrogen at 77 K (ASAP 2010, Micromeritics). The BJH method is based on the Kelvin equation which defines the equilibrium pressure for capillary condensation. The mean pore radii are 22 and 2.5 nm for Si300 and Si60, respectively, in agreement with preceding measurements on these samples.29,30 The particle porosities are derived from the pore volume assuming a density of 2.2 for silica. Densities of materials were measured by using standard picnometry with outgassed water. Inverse Size Exclusion Chromatography (ISEC) Measurements. The experiments were performed using an Agilent 1200 Series HPLC, equipped with a quaternary gradient pump, an (28) Barrett, E. P.; Joyner, L. G.; Hallenda, P. H. J. Am. Chem. Soc. 1951, 73, 373–380. (29) Iapichella, J.; Meneses, J. M.; Beurroies, I.; Denoyel, R.; Bayram-Hahn, Z.; Unger, K. K.; Galarneau, A. Micropor. Mesopor. Mater. 2007, 102, 111– 121. (30) Bayram-Hahn, Z.; Grimes, B.; Lind, A. M.; Skudas, R.; Unger, K. K.; Galarneau, A.; Iapichella, J.; Fajula, F. J. Sep. Sci. 2007, 30, 3089–3103.

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automatic sample injector with a 100 µL loop, an autosampler, and a temperate two column switch. The overall extra-column volume is 49 ± 4 µL as measured from the injection seat of the autosampler to the detector cell. All experiments were carried out at 25 °C, fixed by a column thermostat. Both the column and module were first equilibrated with the mobile phase (THF). Solutions of the polymers and toluene are prepared at a concentration of 1 g L-1. The solutes samples were then injected (1 µL), and the concentration at the outlet was recorded using the diode array detector at 262 nm. Experiments were performed at various volumetric flow rates, f, ranging from 0.04 to 2 mL min-1 which corresponds to interstitial velocities, u, ranging from 0.01 to 0.7 cm s-1. Dynamic Light Scattering (DLS) Measurements. DLS measurements were performed with a Nano ZS zetasizer (Malvern Instrument, UK) to measure the diffusion coefficients of the largest polymers (Mw varying between 3250 (P04) and 1 850 000 g mol-1 (P12)) in pure THF at 25 °C. The experimental values (Table 1) are very close to those obtained previously by DLS at 24 °C.31 For the solute concentration used in this study (1 g L-1), the impact of concentration on the molecular diffusion coefficient is small. Taylor Dispersion Analysis (TDA) Measurements. The diffusion coefficients of the smallest molecules (T, P01, P02, until P05) were determined by TDA measurements.32-36 The concen(31) (32) (33) (34) (35)

Mendema, W.; Zeldenrust, H. Polymer 1977, 18, 835–839. Taylor, G. Proc. R. Soc. London A 1953, A219, 186–203. Taylor, G. Proc. R. Soc. London A 1954, A225, 473–477. Aris, R. Proc. R. Soc. London A 1956, A235, 67–77. Sharma, U.; Gleason, N. J.; Carbeck, J. D. Anal. Chem. 2005, 77, 806– 813.

Generally, the diffusion coefficients of the smallest molecules are calculated by extrapolating the equation obtained by DLS to smaller molecules. This extrapolation overestimates the molecular diffusion coefficient. As seen in Figure 2, the differences between TDA experiments and the equation proposed by Mandela et al.31 reach 20% for the smallest molecules (T, P01, P02, P03). Hydrodynamic Radius of the Polymers. The hydrodynamic radius of the polymers is calculated by using the Stokes-Einstein equation: rm )

Figure 2. Molecular diffusion coefficients obtained by TDA and DLS measurements at 25 °C.

tration of the polymer was 1 g L-1. TDA experiments were performed on the Agilent 1200 Series HPLC replacing the column by a stainless steel tube (0.876 mm i.d., length 1.20 m). The flow rate was set at 100 µL min-1 for toluene and 10 µL min-1 for P01 to P05. The polymer diffusion coefficient is obtained using the classical relationship in TDA:

Dm )

Rc2(tr - ti) 2

2

24(σr - σi )

(2)

were RC is the capillary internal diameter, tR is the mean retention time corresponding to the peak of a given probe, and σR is the variance, σi and ti are the variance and retention time measured without capillary (see data analysis paragraph). This equation is verified as long as the two following conditions are fulfilled:32,33,35,36 first, the dimensionless residence time t′ ) Dmtr/ Rc2 should be higher than 1.4; second, the Peclet number (Pe ) usRc/Dm with us as the mean linear velocity) should be higher than 70, which is verified here. The results are in very good agreement in the zone of recovery (P04 and P05) with those obtained by the DLS methods (Figure 2). Diffusion Coefficients. The diffusion coefficients at a given temperature can be estimated from a general equation relating diffusion coefficient to molar mass: Dm ) a(Mw)-b

(3)

where Mw is the molecular mass of the polymer in grams per mole and Dm has units of squared meters per second. The parameters a and b depend on both the polymer and the solvent. The diffusion coefficients of all the molecules (toluene and polystyrenes), obtained by TDA and DLS experiments (Figure 2), may be fitted by a unique equation (R2 ) 0.9992) valid for Mw varying between 92 and 1 850 000 g mol-1 in pure THF at 25 °C: Dm ) (2.4 ( 0.1) × 10-8(Mw)-0.54(0.008(m2 s-1)

(4)

(36) Cottet, H.; Biron, J. P.; Martin, M. Anal. Chem. 2007, 79, 9066–9073.

kT 6πηbDm

(5)

where T is the absolute temperature, k is the Boltzmann constant, ηb is the viscosity of the mobile phase (for THF, ηb ) 0.46 cP at 298 K), and rm is the radius of the tracer. The size of the polymers calculated from the Stokes-Einstein equation using the experimental results of Dm is given in Table 1. The hydrodynamic radius of all the molecules (rm in nanometers) obtained by TDA or DLS experiments may be fitted by a unique equation (R2 ) 0.9996) valid for Mw varying between 92 and 1 850 000 g mol-1 in pure THF at 25 °C: rm ) (0.024 ( 0.002)(Mw)0.518(0.005(nm)

(5b)

Conductivity Measurements. The particle tortuosities are determined by conductivity measurements as described by Barrande et al.8 Briefly, the effective conductivity of a homogeneous suspension of porous particles is measured as a function of the total porosity. The effective particle conductivity, κeff p , is obtained by extrapolation at infinite dilution of the Maxwell equation37 which is only valid for a dilute suspension. Then, we applied the general definition (eq 1) for conductivity measurements to deduce the particle tortuosity (seen by an infinitely small probe): εp κpeff ) κ° τp

(6)

where κ° is the bulk conductivity of the solution, and εp is the particle porosity. Values are given in Table 2. RESULTS AND DISCUSSION Data Analysis. Peak Analysis. The classical method to characterize mass transport limitations inside a packed bed of porous particle relies on the calculation of the so-called height equivalent to a theoretical plate (HETP). This procedure is based on the simplified description of a column into a number of identical equilibrium plates, introduced by Martin and Synge in 1941.38 The HETP is representative for the separation capacity of a chromatographic column, and it provides a normalized measure of the elution peak variance. The HETP can be calculated from the characteristic parameters of the elution peaks as follows: HETP ) L

σR2 tR2

(7)

(37) Maxwell, J. C. A treatise on electricity and magnetism; Clarendon Press, London, 1873. (38) Martin, A. J. P.; Synge, R. L. M. J. Biochem. 1941, 35, 1359–1368.


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where L is the length of the column, tR is the mean retention time corresponding to the peak of a given probe, and σR is the variance of the peak. The mean retention time and the variance of a chromatogram peak may be obtained by two methods: (a) a fit of the peaks with Gaussian functions or (b) by using the theory of moments.39,40 The first and the second central moment of the elution peak are the retention time and the peak standard deviation (variance), respectively. The theory of moments is more rigorous than the modeling by a Gaussian function because no assumption is made on the shape of the peaks. With the method of the moment, it is necessary to define the cutoff points19 for the integration method in order to limit the influence of the other peaks. As an example, the peak profile obtained at a flow rate of 0.5 mL min-1 for the polymer P05 and P12 with the silica Si300 is shown in Figure 3. At the end of the profile in Figure 3a, there is a second small peak which corresponds to the THF or impurities (for example residual monomer). Due to the low signal-to-noise ratio experienced with small sample sizes19 because of the presence of impurities, a large error is experienced with the determination of the first and second moments. Our results show that the values of the moments are very sensitive to the choice of the cutoff points. So, in order to compare the results obtained for toluene and the twelve polymers, we decided to fit our peaks with a Gaussian function. All the chromatograms are then corrected from the solvent peak (Figure 3b) by fitting it with a Gaussian function. It appears that the experimental peaks always exhibit a slight tailing at long time certainly due to the presence of some death volumes as well as turbulences on the analysis loop. However, the Gaussian fit represents an accurate estimate of the chromatographic peaks even for the largest polymer (Figures 3c). Dispersion in the Extra-Column Volume. The actual injection profile of a sample into a chromatographic column is far from the ideal Riemann (i.e., a rectangular pulse) injection profile. The profile of the injected sample is broadened due to axial diffusion and radial friction inside the connecting volumes and the turbulences at any section change. For example, the toluene peak width at half-maximum is about 7.2 s without a column. If the injection process was ideal, the duration of the injection would be only 0.6 s. Due to dispersion effects in the “dead volumes” of the analysis loop, the actual peak is 12 times larger than the ideal injection. The theoretical prediction of the exact contributions of these volumes is difficult.32–34,41,42 The effect of the analysis loop used in chromatographic systems on the dispersion cannot be described by Taylor analysis of dispersion because of turbulences. The conditions necessary to use eq 2 are not fulfilled in this case. As a confirmation, our results showed clearly that it is necessary to evaluate the dispersion experimentally. Therefore, to correct our data from the instrumental dispersion, the peak profiles for each solute at each velocity were also recorded without column; i.e., a zero death volume connector is used to connect the inlet to the outlet capillaries of the column. The experimental peak broadening being a convolution of physical effects in the porous bed of the column and of instrumental effects (i.e., extracolumn and column (39) (40) (41) (42)

Kucera, E. J. Chormatogr. A 1965, 19, 237–248. Kubin, M. Collect. Czech., Chem. Commun. 1965, 30, 2900. Atwood, J. G.; Golay, M. J. E. J. Chromatogr. A 1981, 218, 97–122. Grzna´rova´, G.; Polakovicˇ, M.; Acˇai, P.; Go ¨rner, T. J. Chromatogr. A 2004, 1040, 33–43.

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Figure 3. (a) Experimental band profile recorded for the polymer P05 and silica Si300 at a volumetric flow rate of 0.50 mL/min. The Gauss profile used to correct for the solvent peak is given. (b) Corrected P05 peak and its superimposed fit with a Gauss function. (c) Experimental band profile for the polymer P12 and silica Si300 at 0.50 mL/min and Gauss function.

contributions are independent so their variance is additive), the Gaussian peak profiles analysis allows writing the truth HETP as: HETP ) L

(σR2 - σi2) (tR - ti)2

(7bis)

Figure 4. Comparison, in the case of Lichrospher Si300, between the HETP curves corrected (eq 7bis) or not (eq 7) from the contribution of the extracolumn volume for (a) a small molecule like toluene, molecules with intermediate size like (b) polymer P04 and (c) polymer P07, and (d) a molecule having a size larger than the pore size like the polymer P09.

where σi and ti are the variance and retention time measured without column. Some comparison examples are shown in Figure 4. At low flow rate, there is practically no difference between the measured and the corrected HETP values for all molecules. The time spent in the column is long and the extent of band spreading due to axial diffusion in the column exceeds the one that takes place in the extracolumn volume. On the contrary, when the flow rates increases, the importance of the correction depends on the size of the molecule. For very small molecules (λ , 1 with λ ) rm/rp) or for molecules completely excluded (λ > 1) from the pores of the particles, the dispersion in the extra-column volume is in the same range as in the column. The amplitude of the correction can be seen in Figure 4a and d. These results show that it is necessary to take into account the dispersion in the extracolumn volumes, and it must be determined with the same molecule. Similar results were shown by Kaczmarski et al.43 with a small nonretained molecule. For molecules having a size similar to that of pores (Figure 4c), the dispersion, which is controlled by intraparticle diffusion and (43) Kaczmarski, K.; Gritti, F.; Guiochon, G. Chem. Eng. Sci. 2006, 61, 5895– 5906.

fluid-phase mass transfer, is much higher than the extent of axial dispersion that takes place in the extracolumn volume of the analysis loop. In this specific case, the effect of the extracolumn dispersion becomes negligible. Finally, all the HETP data were corrected from the extracolumn band broadening contribution. Accessible Porosities and Pore Sizes. The knowledge of the column porosity accessible to a given solute is essential for the prediction of the elution time and to understand the origin of the mass transport limitations. Tracers of different molecular weight, and hence of different radius rm, are injected under nonadsorbing conditions and the total accessible volume, Vt(rm), is measured. The total porosity accessible to a probe of rm radius is defined as:

εt(rm) )

Vt(rm) VC

(8)

where VC is the volume of the column. The average retention volume Vt(rm) is calculated from the mean time retention obtained by the Gaussian fit of the eluting peak. The term εt(rm) ranges between two limits: (i) the total column porosity, εt, that Analytical Chemistry, Vol. 82, No. 7, April 1, 2010

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Figure 5. Measured and calculated particle porosity of polystyrenes as a function of the probe size rm for (a) Si60 and (b) Si300. The best fits are obtained with rp(Si60) ) 1.95 nm and rp(Si300) ) 17.5 nm (eq 11).

corresponds to the volume accessible to a solute small enough to enter every intraparticle pore (λ , 1, the smallest used molecule is toluene) and extraparticle pores and (ii) the extraparticle porosity, εe, which corresponds to the volume accessible by the largest tracer supposed to be entirely excluded from all the particle pores (here, the largest polymer used is P12, λ . 1). Finally, the accessible particle porosity for the probe of size rm, εp(rm), is calculated by the following equation: (εt(rm) - εe) εp(rm) ) (1 - εe)

(9)

The total porosity of the column εt was determined by the double weight method: the column is filled by two solvents with different densities and then weighted. Knowing the density of silica and the column volume, the total porosity is deduced. The determination of pore size is a critical issue since it is then used in most models of mass transport. In the present study, two methods can be used for that. The first one is gas adsorption: the corresponding average pore size values are given in Table 2. The second method is ISEC. If we assume that the pore size is monodisperse and the center of the molecule is excluded from the vicinity of the wall on a distance equal to the molecule radius (rm), the probed volume would correspond, in the case of a cylindrical pore, to εp(rm) ) εp(1 - λ)2

(10)

where εp should be the value obtained by ISEC for the smallest probe. Nevertheless, because the size of the smallest probe used here, toluene, is not negligible as compared to the pore size, notably in the case of Si60, εp was derived from the total porosity determined by the double weight method and from the external porosity obtained with the excluded molecules following the equation: εp ) 2674

(εt - εe) (1 - εe)


(9bis)

The obtained values (εp(Si60) ) 0.65, εp(Si300) ) 0.67) are in very good agreement with the gas adsorption measurements. If a distribution of cylindrical pores is considered, the relationship between the accessible porosity and the pore size distribution is given by44

εp(rm) ) εp

∫

∞

rm

(

f(r) 1 -

rm r

)

2

dr

(11)

By using an Excel software procedure (the solver function of Excel is used after establishing a matrix corresponding to the integral equation), this equation is inverted in order to get f(r): f(r) dr values for each solute probe can be evaluated by fitting the experimental values of εp(rm) with the model eq 11. The pore size distribution obtained assumes only the pore shape but not the distribution law. In Figure 5, the experimental values obtained experimentally and those derived by eq 10 and 11 are plotted for both columns. The average pore radius corresponding to the best fitting are 1.95 and 17.5 nm for Si60 and Si300, respectively. The agreement between experiments and both eqs 10 and 11 is good in the case of Si300. It does not mean that the pores are monodisperse in size, but it simply comes from the fact that the difference of size between two successive ISEC probes is larger than the actual pore size distribution. The agreement with eq 10 in the case of Si60 is not so good. The particle porosities accessible to toluene obtained by ISEC are smaller than that obtained by N2 adsorption (compare Table 3 with Table 2). It confirms that the size of the toluene is not negligible compared to the pore size, and the difference between the two types of measurement corresponds to the excluded volume, but the equation εp(rm) ∼ εp(BJH)(1 - rm/rp(BJH))2 is not verified in the case of Si60 (εp(toluene) ) 0.41 whereas εp(BJH)(1 - λ)2 ) 0.50). It means that the exclusion is higher than that expected for toluene. The more probable explanation is that not only toluene is not adsorbed, but it does not displace THF molecules from the surface. This creates a supplementary excluded volume which decreases the accessible porosity as (44) Knox, J. H.; Scott, H. P. J. Chromatogr. A 1984, 316, 311–332.

Table 3. Porosities and Pore Radius of Silica Lichrospher Si60 (Merck) and Silica Lichrospher Si300 (Merck) Obtained by ISEC Measurements Samples

total porosity of fixed bed (toluene) εt

extraparticle porosity (P12) εe

particle porosity (toluene) εp

pore radius (ISEC) rp/nm

Lichrospher Si60 Lichrospher Si300

0.63 0.77

0.38 0.38

0.41 0.64

1.95 17.5

compared to nitrogen adsorption. The diameter of the THF molecule is around 0.3 nm. The value of the pore radius obtained by ISEC (rp(ISEC) ) 1.95 nm) is comparable to the value calculated by correcting the BJH pore radius (2.5 nm) with the solvent diameter. Therefore, the relationship εp(rm) ∼ εp(BJH)(1 - rm/rp(ISEC))2 gives a better correlation with experimental results (Figure 5). However, the agreement is not very good because, contrary to the Si300, the pore size distribution is larger than the difference of size of two successive probes. In this approach, we did not use the concept of constriction. Its effect is indeed to exclude the molecules from a part of the porosity, which is already taken into account by the accessible porosity experimentally measured using eq 9. Modeling the Mass Transfer. In a physical point of view, the HETP curves should be expressed in terms of the physical parameters governing the mass transport inside the particles.18,45-47 The different contributions involved in the elution peak spreading are: (i) The so-called “eddy diffusion” process, which is due, contrary to what its name suggests, to a convection process. The flow paths through the column are not of equal length and size; therefore depending to the path taken by the probes, they will arrive at the end of the column at different times leading to a band broadening. This contribution depends on the homogeneousness of the column packing as well as on the particle size and is essentially independent of the mobile phase velocity. Hence, it is expected to be a constant contribution to the overall HETP. (ii) The longitudinal diffusion, which is due to the molecular self-diffusion of molecules in the porous bed, leads to an important contribution to the band dispersion, mainly at low velocity. This contribution is generally considered as inversely proportional to the velocity. The proportionality coefficient is the so-called B term. At zero velocity, it is clearly related to the tortuosity of the porous bed, including intra- and extraparticle porosity. The best way to measure it by chromatography is the peak parking method.18,19 At other velocities, there is a complex coupling between convection and diffusion which is not really taken into account in the standard models as discussed in the following. (iii) The mass transfer limitation which is supposed to correspond to a nonequilibrium partition of the solute concentration between the mobile phase and the stationary phases. As the mobile phase velocity increases, the rate of equilibration becomes slower which means that the spatial deviation and time delay between the solute in the mobile phase (which is always ahead) and the solute in the stationary phase increases, thus broadening (45) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271–289. (46) Piatkowski, W.; Gritti, F.; Kaczmarski, K.; Guiochon, G. J. Chromatogr. A 2003, 989, 207–219. (47) Parris, N. A. Instrumental liquid chromatography, 2nd ed., Journal of Chromatography Library; Elsevier: New York, 1984; Vol. 27.

the eluting peak. Three kinetic terms are generally considered: a term of transfer at the outer surface of particles, a term of adsorption, and, finally, the contribution on which we particularly focused our attention, the intraparticle diffusion. This latter corresponds to the fact that the solute within the pore structure of the stationary phase will move mainly by molecular diffusion, because the convection inside particle pores is negligible as compared to the outside particle convection. The time spent to go through a particle by molecular diffusion becomes quickly important as compared to the time spent for convective elution when the velocity increases. The total contribution of mass transfer is proportional to the velocity. In this study, the contribution of polymer dispersity is not considered. As shown in Gritti and Guiochon,48 this contribution is constant with velocity, thus it will not modify the calculation of the mass transfer coefficient. The mathematical modeling of the band broadening is generally done by the general rate model (GRM),36,37 which takes into account most of the contributions to the mass transfer resistance. The HETP curves for a molecular probe of radius rm can be described in nonadsorption conditions by the following expression:18,39,40

HETP )

[

]

2 2Dax dp′ 2 1 (1 - εe)εp[rm] εe dp′ + + u u 3 km εt[rm]2 10Dpeff[rm]

(12)

where Dax is the axial dispersion coefficient of the column, u is the interstitial (or linear) velocity, dp′ is the particle diameter, Dpeff[rm] is the effective intraparticle diffusion coefficient of the probes of radius rm, and km is the mobile-phase mass transfer coefficient. Before applying this equation, the conditions of its establishment should be discussed.39,40 It is obtained, in conditions of nonadsorption, from three basic equations that are solved by a Laplace transformation. The first one is the differential equation of mass balance in the interparticle porosity, the second one is the same but inside a porous particle, and the third equation corresponds to the transfer rate through the interface between inter- and intraparticle porosities. The extension of this interface is deduced from particle size and external porosity. This separation in three equations is reflected in the final eq 12 which contains three independent terms. Theoretically, the axial dispersion coefficient in eq 12 should correspond only to effects (self-diffusion and eddy diffusion) in the external porosity. The axial diffusion coefficient is commonly calculated by assuming that the longitudinal diffusion and the eddy diffusion contributions are additive,45 since these phenomena are simultaneous: Dax)γDm + δdp′ u

(13)

(48) Gritti, F.; Guiochon, G. Anal. Chem. 2007, 79, 3188–3198.


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where γ is a coefficient (often defined as obstructive factor) which should correspond to the diffusion transport efficiency45 through the external porous network, and the δ coefficient corresponds to the eddy diffusion contribution. In literature, these coefficients are fixed to values between 0.5 and 1. Using eq 13 in eq 12 leads finally to the classical expression for HETP: HETP ) A +

B + Cu u

(14)

with A ) 2δdp′

(15)

B ) 2γDm

(16)

2

C)2

[

2

dp′ (1 - εe)εp[rm] εe dp′ + 2 6km εt[rm] 60Dpeff[rm]

]

εDm τ

(17)

(18)

This equation follows the general definition given by eq 1. Nevertheless, because in the GRM model, the interstitial velocity is introduced (by dividing the linear velocity by the bed porosity), the contribution of diffusion to the Dax value corresponds to Deff defined by eq 18 divided by the external porosity. Comparing this to eq 13 leads to γ ) 1/τ, which preserves the condition τ > 1. This is in agreement with Gritti and Guiochon48 where the obstruction factor is equal to the reverse of tortuosity in the absence of friction, which is reasonable when considering the diffusion in the external porosity. This means that B may be written: B ) 2γDm )

2Dm τe

(19)

where τe is the tortuosity of the interparticle porosity. When the velocity is very small, making the C term negligible, the spreading of the peaks is due only to diffusion but both in external and internal porosities. In that case, B is related to the tortuosity of the whole accessible porosity by B ) 2γDm )

2εt[rm]Dm εeτt[rm]

(20)

which is equivalent to eq 30 of ref 49 but without detailing either the inter and intraparticle contributions to tortuosity or friction factors in the narrowest pores. As underlined in the same paper, the GRM model may not be correct in the whole velocity range (49) Gritti, F.; Guiochon, G. Anal. Chem. 2006, 78, 5329–5347.

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km )

( )

Dm 1.09 uεedp′ Dm d′ εe p

This shows that the B term derived from the GRM model depends mainly on the efficient diffusion through the external porosity. In the absence of adsorption and provided the solute is small as compared to the pore size, the diffusion coefficient in a porous bed is related to the bulk diffusion coefficient by the following equation: Deff )

because it is not able to describe the shift of the B value from eq 19 to eq 20. Nevertheless, the use of eq 20 is probably more appropriate since it is correct, at least, for low velocities. Moreover, it may be directly determined by peak parking experiments.50,51 For intermediate velocities, the coupling convection-diffusion is not really taken into account. At high velocity, the term containing B becomes negligible (then, its exact value is not important) and the HETP should vary linearly with the velocity u. The slope C at high u values of the HETP curves comprises two terms, the mobile-phase mass transport and the intraparticle diffusion. The mobile-phase mass transport coefficient can be estimated in the case of nonadsorption conditions by the equation of Wilson and Geankoplis:52 1/3

(21)

The characteristic HETP curves obtained with the silica Si60 and silica Si300 for molecules of different size are plotted versus the linear velocity in Figure 6, in the case of intermediate probe molecular size, whereas some examples for toluene and large polymers are given in Figure 4. The relative importance of the different contributions depends on the size of the molecule compared to the pore size. For small molecules like toluene (λ < 0.1) and for large molecules (λ > 1), the transport seems to be governed only by axial dispersion at small u values and by the eddy diffusion contribution at higher linear velocities, since in the high velocity regime the HETP curve is parallel to the velocity axis. Consequently, for small probes the B term should be well represented by eq 20; from the HETP experimental data, the calculated tortuosity value of the column (τt ) 1.36 for Si 60 and 1.35 for Si300) is close to that obtained by electrical conductivity8 (τt ) 1.46) for both silicas. Further experiments at very low velocities and peak parking experiments are necessary in order to better describe the terms A and B and to determine the total tortuosity. For the two extreme sizes, the fluid-phase mass transfer coefficient and the intraparticle contribution seems to be very small. This is logical in the case of large molecules (λ > 1) since they cannot diffuse in the intraparticle porosity and in the absence of adsorption there is no mobile-phase mass transfer contribution. The only effect at high speed corresponds to convective effects (eddy diffusion) related to the permeability of the column. For the smallest molecules (λ < 0.1), a slope should exist because of the intraparticular diffusion, but the data scattering due to the instrumental band broadening correction does not allow its measurement. For intermediate sized molecules (0.1 < λ < 1), the mass transport is clearly governed by the C term and an almost linear behavior is observed in the whole analyzed velocity range. For both silica columns, the slope of the HETP curves increases when the probe size tends toward the pore size (Figure 6a and b). The C values are calculated from the linear portion of the HETP plots at high flow rates. The numerical estimation of the mobile-phase (50) Desmet, G.; Broeckhoven, K.; De Smet, J.; Deridder, S.; Baron, G. V.; Gzil, P. J. Chromatogr. A 2008, 1188, 171–188. (51) Broeckhoven, K.; Cabooter, D.; Lynen, F.; Sandra, P.; Desmet, G. J. Chromatogr. A 2008, 1188, 189–198. (52) Wilson, E. J.; Geankopolis, C. J. Ind. Eng. Chem. (Fundam.) 1966, 5, 9.

Figure 6. Plots of the experimental HETP curves obtained for molecules smaller than the mean pore radius vs the interstitial velocity for (a) silica Si60 and (b) silica Si300.

mass transfer coefficient using eq 21 shows that it could be neglected43 in the C term (eq 17) compared to the intraparticle diffusion contribution. Then by applying eq 17, the intraparticle diffusion coefficient Dpeff(rm) can be evaluated. The values are plotted in Figure 8. As expected,20,21,25 the intraparticle diffusion coefficient strongly decreases when the probe size tends to the pore size. As stated in the introduction, this decrease has several origins: steric hindrance, friction, and modification of the pore structure seen by the probe. Therefore, for a probe infinitely small as compared to the pores size (λ ∼ 0) for which friction can be neglected, the effective molecular diffusion coefficient through the particle should obey a law similar to that of eq 18:

Dpeff )

εpDm τp

(22)

kf[rm] )

This equation was used by other authors,10,18,23,26,43 but whatever the probe size, εp and τp are often fixed to constant values, τp being notably fixed to values ranging between 1.4 and 2. But, because in the general case the probes are not infinitely small as compared to pore size, to calculate Dpeff(rm), eq 22 must be modified to take into account that the accessible porosity53 εp(rm) as well as the tortuosity, τp(rm), depend on the size of the probe. There is also a contribution of friction which slows down the “bulk” diffusion of the probe in the pores. In a first approach, one may consider that this is equivalent to consider that the reference diffusion coefficient Dm is reduced by a friction factor kf(rm), leading to the following relationship: Dm[rm] ) kf[rm]Dm

(23)

Finally, one can deduce for the effective intraparticular diffusion coefficient the following general expression:

Dpeff[rm] )

εp[rm]kf[rm]Dm τp[rm]

Our principal objective in this paper is to evaluate the respective contributions of topological and friction effects on Dpeff(rm). Various models describe how the probe friction with the wall of the pore decreases the diffusion coefficient Dm. The most simple and used model was proposed by Renkin;20 it corresponds to the diffusion of a sphere in a cylindrical pore of radius rp as a function of sphere size. The theory combines two expected phenomena, the exclusion effect, and the wall drag effect. The exclusion effect is based on the solute molecule being excluded from the region near the pore wall. The wall drag effect is modeled using centerline hydrodynamics. The Renkin equation, which is generally restricted to λ < 0.4, gives the following numerical relationship for the friction factor kf[rm]:

(24)

(53) Lightfoot, E. N.; Athalye, A. M.; Coffman, J. L.; Roper, D. K.; Root, T. W. J. Chromatogr. A 1995, 707, 45–55.

Dm[rm] ) 1 - 2.104λ + 2.09λ3 - 0.956λ5 Dm (25)

The pioneering work of Renkin was recently extended using an off-axis hydrodynamic method.21 However, the results are very close to eq 25 in the whole λ range, and therefore, eq 25 was used. There is no simple method to follow the evolution of the structure of the accessible porous network with probe size. When the probe size increases, it is clear that the hindrance pore diffusion becomes important but the evolution of apparent tortuosity is not clear. In most of the publications, the tortuosity factor is taken as a constant value and the probe size effect is only related to the friction factor.25,26,43 Our approach to model the evolution of the tortuosity factor with λ is based on two main assumptions. The first one is that the porous silica particles may be seen as a random distribution of nonporous particles creating a particle with porosity εp. In that case, the particle tortuosity may be reasonably calculated by the following equation, which is theoretically found54 for spheres and experimentally verified8,55 for some other shapes: (54) Weissberg, H. L. J. App. Phys. 1963, 34, 2636–2639. (55) Comitti, J.; Renaud, M. Chem. Eng. Sci. 1989, 44, 1539–1545.


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τp ) 1 - p ln εp

(26)

The p value is specific to the pore topology (p ) 0.5 for a distribution of hard spheres). The experimental values of τp for the two silicas Si60 and Si300 as well as the one obtained with the Si100 silica8 are plotted in Figure 7 versus the particle porosity. Fitting the data with eq 26, a value of p ) 2.4 ± 0.3 is obtained. The higher this value, the higher the tortuosity. This means that the particle structure is more complex than a simple aggregation of spheres, for which p should be around 0.5. Mauret and Renaud12 have shown that a porous bed made of particles with a similar shape can be described by this equation: p is then a characteristic of this shape. The second assumption is that the evolution of tortuosity with probe size in a pore network is simply given by the same equation applied to the accessible porosity of the corresponding probe. This is indeed shown by random walk simulations of the diffusion process through a porous bed of spheres.56 The picture is that when the probe size increases, for the center of the molecule it corresponds to a pore size constriction but the topological properties of the pores should remain mainly unaffected. This assumption seems to be verified quite well by a set of silicas which were synthesized by the same process.8 In a recent paper, Gritti and Guiochon27 proposed a relationship between tortuosity and accessible porosity which is in fact the Maxwell equation valid only at very high porosity known as less efficient than eq 26.8 Therefore, the effective tortuosity of the accessible porosity seen by a probe of rm size may be reasonably given by τp[rm] ) 1 - p ln εp[rm]

(27)

with p ) 2.4 ± 0.3 for the samples used here. Finally, if one combines eqs 10, 25, and 27, the effective diffusion coefficient is given by

Dpeff[rm] )

εp(1 - λ)2(1 - 2.104λ + 2.09λ3 - 0.956λ5)Dm 1 - p ln[εp(1 - λ)2] (28)

Figure 8. Plot of the effective diffusion coefficient as a function of molecule/pore size ratio. Comparison of the experimental data and the model showing the cumulated contribution of each of the physical parameters beginning by the friction factor kf(rm) (eq 25) and the accessible porosity εp(rm) (eq 10). Then, one can see that if constant tortuosity factor (τp ) 1.4) is applied, the agreement with experimental data is insufficient, whereas if eq 27 is used the agreement is much better (continuous line). Only polymers P01 to P04 and polymers P04 to P07 are presented in the case of Si60 and Si300, respectively. The biggest polymers are not presented because they are too large to enter the pores. In the case of the smallest ones, the error on the C slope is too high due to the contribution of dispersion in experimental extracolumn devices (Figure 4a).

This equation is established using an average value of the pore size to express the contribution of each of the parameters. A more accurate approach would need to take into account the pore size distribution. It is difficult to introduce it without making a hypothesis on the arrangement of pores, serial or parallel.25 This problem will be addressed in the future. The agreement of our model with the experimental data for the columns Si60 and Si300 is better than considering a constant value of tortuosity (Figure 8). These results show that it is possible to model the transfer properties of the column starting from basic material structure parameters. Notably, the contribution of eq 27 to the calculations clearly improves the result. Quantitatively, this effect may decrease the effective diffusion coefficient of a factor 3 compared with the use of a constant tortuosity factor. CONCLUSIONS From an experimental point of view, this work has confirmed that to extract the physical parameters from the HETP curve careful attention has to be paid to the instrumental peak spreading, in particular at high velocities. This effect cannot be easily modeled and has to be measured experimentally for each probe. It was also useful to directly measure the bulk diffusion coefficients of the probes used in this study, especially in the case of the small molecules. This allowed us to propose a new law, based on both DLS and TDA analysis, reliable from the monomer to the largest polymers. From a theoretical point of view, the influence of the probe size on the different contributions (accessible porosity, tortuosity, and friction) which allow predicting the mass transport kinetics has been evaluated using a new approach and compared with the

Figure 7. Plots of the particle tortuosity as function of their porosities for the silica studied. 2678


(56) Kim, A. S.; Chen, H. J. Membr. Sci. 2006, 279, 129–139.

experimental results. A fixed value of tortuosity leads to a strong overestimation of the effective diffusion coefficient, notably when the molecule radius is not negligible compared to the pore radius. The phenomenological law which is here proposed to describe the evolution of apparent tortuosity with probe size clearly improves the prediction of the evolution of effective diffusion coefficient with the molecule/pore size ratio. It seems, therefore, possible to model the transfer properties of a column starting from basic material structure parameters. Moreover, provided the pore size distribution is known, the proposed eq 28 used to predict the effective diffusion coefficient has only one adjustable parameter (p), which means that the properties of the column (or any porous medium in which transfer is studied) can be predicted from a measurement with only one probe. The prediction of transport in nonadsorbing conditions thus needs only three types of parameters: pore accessibility, tortuosity versus probe size, and friction effects.

k km kf(rm) L Mw rm rp T u us VC Vp Vp(rm) Vt(rm)

Boltzmann constant film mass transport coefficient [m s-1] friction factor [-] column length [m] molecular weight [g mol-1] molecule radius [nm] pore radius [nm] absolute temperature [K] interstitial velocity ) us/εe [m s-1] superficial velocity () f/A) [m s-1] total column volume [mL] total particle volume [mL] particle pore volume accessed by a probe of radius rm [mL] total liquid volume accessed by a probe of radius rm [mL]

Greek Letters

ACKNOWLEDGMENT We thank Merck for the gift of the Lichrospher Si60 and Si300 columns. We thank Dr. Trang Phan (Universite´s d’Aix-Marseille I, II et III-CNRS, UMR 6264, Laboratoire Chimie Provence, Equipe CROPS, Marseille, France) for help in DLS analysis. NOMENCLATURE A column cross-sectional area [m2] dp′ particle diameter [µm] Dax axial diffusion coefficient [m2 s-1] Dm molecular diffusion coefficient [m2 s-1] Deff (r ) effective pore diffusion coefficient of a probe of radius p m rm [m2 s-1] f flow rate [m3 s-1] HETP height equivalent to a theoretical plate [µm]

εe εp(rm) εt(rm) εt γp ηb τp(rm) τt(rm) τe

bed porosity [-] accessible particle porosity for a probe of radius rm [-] total accessible porosity for a molecule of radius rm [-] total porosity of the column [-] obstruction factor [-] solvent viscosity [cP] particle tortuosity seen by a probe of radius rm [-] total tortuosity seen by a probe of radius rm [-] external tortuosity [-]

Received for review October 13, 2009. Accepted February 14, 2010. AC902858B


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Influence of Molecule Size on Its Transport Properties through a

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