Measurement of Desorption Rates from Octadecylsilyl Bonded-Phase

Jan 25, 2006 - An instrument is developed to measure rates of desorption of solutes from particulate HPLC packing materials for processes that are qua...
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Anal. Chem. 2006, 78, 1593-1605

Measurement of Desorption Rates from Octadecylsilyl Bonded-Phase HPLC Particles and Its Characterization in Terms of Pore, Surface, and Film Diffusion Robert Bujalski† and Frederick F. Cantwell*

Department of Chemistry, University of Alberta, Edmonton, AB, Canada T6G 2G2

An instrument is developed to measure rates of desorption of solutes from particulate HPLC packing materials for processes that are quantitatively complete in a few tenths of a second. The instrument is a modified, pressure-driven, stopped-flow device. The major modifications include positioning a very short (0.6 mm) bed of the particles just upstream of the detector cell, eliminating the mixing chamber, and adding high-speed switching valves in order to allow sequential continuous flow of individual solutions. Instantaneous rate curves are measured for the desorption of 1,2-dimethyl-4-nitrobenzene (DMNB) from 12-µm-diameter porous particles of the bonded-phase packing Luna C-18 employing high linear velocities of the eluting solvent. The same experiment is performed for the nonsorbed compound phloroglucinol (PG) The PG rate curve is used in two ways (i.e., subtraction and deconvolution) in order to correct the observed rate curve of DMNB for experimental artifacts such as bed hold-up volume and instrument band broadening. The cumulative desorption rate curve of DMNB is obtained by integration. It is accurately described (R2 > 0.999) by a theoretical model that invokes both intraparticle diffusion (including both hindered pore diffusion and surface diffusion) and external film diffusion. The surface diffusion coefficient is (3.2 ( 0.8) × 10-6 cm2/s and the diffusion film thickness is 0.5 µm. The validity, of both the experimental technique and the theoretical model, is demonstrated by excellent agreement between a predicted and an observed chromatographic elution peak for DMNB on a 25-cm-long commercial column of Luna C-18. Bonded-phase high-performance liquid chromatography (BPHPLC) is usually done on columns packed with porous silica microparticles, the pore surfaces of which are modified by attachment of a variety of alkyl chains that may or may not bear functional groups.1,2 The bound chains with their functional groups constitute the stationary phase. Among the several processes that * Corresponding author. Tel: 780-492-5927. Fax: 780-492-8231. E-mail: [email protected]. † Present address: Analytical Research and Development, Sepracor Inc., Marlborough, MA 01752. (1) Neue, U. D. HPLC Columns: Theory, Technology, and Practice; WileyVCH: New York, 1997; Chapter 4. 10.1021/ac051609r CCC: $33.50 Published on Web 01/25/2006

© 2006 American Chemical Society

are responsible for band broadening (dispersion) of a zone of solute molecules as it migrates through a BP-HPLC column are rates of sorption and desorption in to and out of the particles.1-11 Sorption/desorption rates (S/D rates) have contributions from the rates of some or all of the following steps:12 (i) “film diffusion” through the quasi-stagnant Nernst film of mobile phase surrounding each particle; (ii) diffusion through the intraparticle pores; (iii) diffusion in to and out of the layer of bonded stationary phase; and (iv) attachment and detachment of the solute in the stationary phase via noncovalent interactions (6). Steps i-iii involve mass transfer while step iv does not. Step ii can occur either by diffusion through the stagnant mobile phase that fills the pores (i.e., “pore diffusion”) or by diffusion along the surfaces that constitute the pore walls (i.e., “surface diffusion”).13-16 Because the Nernst-film surrounding the particle is thinner at higher linear velocities of mobile phase through the column (U0), the kinetic rate constant for step i is larger at higher U0,3,12,17-19 while the rate constants for steps ii-iv are independent of U0. (2) Snyder, L. R.; Kirkland, J. J. Introduction to Modern Liquid Chromatography; Wiley: Toronto, 1982; Chapter 19. (3) Horvath, C.; Lin, H. J. J. Chromatogr. 1978, 149, 43-70. (4) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965. (5) Crombeen, J. P.; Poppe, H.; Kraak, J. C. Chromatographia 1986, 22, 319328. (6) Karger, B. L.; Snyder, L. R.; Horvath, C. An Introduction to Separation Science; John Wiley & Sons: Toronto, 1973. (7) Chen, J. C.; Weber, S.G. Anal. Chem. 1983, 55, 127-134. (8) Giddings, J. C.; Bowman, L. M.; Myers, M. N. Macromolecules 1977, 10, 443-449. (9) Gzil, P.; Vervoot, N.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 62446250. (10) Knox, J. H.; Scott, H. P. J. Chromatogr. 1983, 282, 297-313. (11) Lenhoff, A. M. J. Chromatogr. 1987, 384, 285-299. (12) Gowanlock, D.; Bailey, R.; Cantwell, F. F. J. Chromatogr., A 1996, 726, 1-23. (13) Grathwohl, P. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/Desorption and Dissolution Kinetics; Topics in Environmental Fluid Mechanics; Kluwer Academic Publishers: Norwell, MA, 1998. (14) Swinton, D. J.; Wirth, M. J. Anal. Chem. 2000, 72, 3725-3730. (15) Sekine, T.; K. Nakatani, K. Langmuir 2002, 18, 694-697. (16) Miyabe, K.; Guiochon, G. J. Sep. Sci. 2003, 26, 155-173. (17) Helfferich, F. G.; Hwang, Y. L. In Ion Exchangers; Dorfner, K., Ed.; Walter de Gruyter: Berlin, 1991; pp 1277-1310. (18) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962; Chapters 6 and 9. (19) Fernandez, M. A.; Laughinghouse, W. S.; Carta, G. J. Chromatogr., A 1996, 746, 185-198.

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Accurate measurement of particle S/D rates is required for an understanding of the role that they play in solute band broadening in an HPLC column. Such an understanding would be especially desirable in studies employing new or modified types of packing particles or novel experimental conditions. Examples include the following: (a) unusual pore-size distributions such as in “perfusive” particles;20 (b) large solute molecules such as proteins;21-23 (c) thicker or more highly cross-linked layers of bonded stationary phases such as in “polymeric” alkyl bonded phases;24 (d) slow attachment or detachment reactions such as in “molecularly imprinted polymer” phases, in ligand-containing phases, in “affinity” phases,25-28 and in phases containing silanol sites;29,30 (e) unusually small particle diameters, e.g., e2 µm;31-33 (f) non-silica-based matrixes such as porous zirconia, both with34,35 and without36,37 a polybutadiene stationary phase; (g) particles embedded in porous membranes;38 (h) large retention parameters resulting from the use of weakly eluting mobile phases;15,39,40 and (i) high or low temperatures.34,41 Methods that have been used to study S/D kinetics can be loosely grouped into three categories: column chromatography, spectroscopy, and uptake/release measurements.16 In the column chromatographic approach, elution chromatograms or frontal chromatograms are obtained for the solute, observed plate heights (Hobs) are calculated from the solute zone profiles at various U0, and van Deemter plots of Hobs versus U0 are fitted with theoretical equations involving a summation of terms.3,16,42-44 Alternatively, S/D rates that are especially slow can be evaluated from the extent (20) Tallarek, U.; Paces, M.; Rapp, E. Electrophoresis 2003, 24, 4241-4253. (21) Gibbs, S. J.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. 1992, 96, 74587462. (22) Coffman, J. L.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. B 1997, 101, 2218-2223. (23) Hejtmanek, P.; Schneider, P. Chem. Eng. Sci. 1994, 49, 2575-2584. (24) McGuffin, V. L.; Lee, C. J. Chromatogr., A 2003, 987, 3-15. (25) Muller, A. J.; Carr, P. W. J. Chromaogr. 1986, 357, 11-32. (26) Deng, Q.; Watson, C. J.; Kennedy, R. T. J. Chromatogr., A 2003, 1005, 123-130. (27) Sajonz, P.; Kele, M.; Zhong, G. M.; Sellergren, B. Guiochon, G. J. Chromatogr., A 1998, 810, 1-17. (28) Ren, D. Y.; Penner, N. A.; Slentz, B. E. Inerowicz, H. D.; Rybalko, M.; Regnier, F. E. J. Chromatogr., A 2004, 1031, 87-92. (29) Ludes, M. D.; Anthony, S. R.; Wirth, M. J. Anal. Chem. 2003, 75, 30733078. (30) Wirth, M. J.; Swinton, D. J.; Ludes, M. D. J. Phys. Chem. B 2003, 107, 6258-6268. (31) Colon, L. A.; Cintron, J. M.; Anspach, J. A.; Fermier, A. M.; Swinney, K. A. Analyst 2004, 129, 503-504. (32) Patel, K. D.; Jerkovich, A. D.; Link, J. C.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5777-5786. (33) Mellors, J. S.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5441-5450. (34) McNeff, C.; Zigan, L.; Johnson, K.; Carr, P. W.; Wang, A. S.; Weber-Main, A. M. LC-GC 2000, 18, 514-529. (35) Dunlap, C. J.; McNeff, C. V.; Stoll, D.; Carr, P. W. Anal. Chem. 2001, 73, 598A-607A. (36) Lorenzanoporras, C. F.; Carr, P. W.; McCormick, A. V. J. Colloid Interface Sci. 1994, 164, 1-8. (37) Lorenzanoporras, C. F.; Annen, M. J.; Flickinger, M. C.; Carr, P. W.; McCormick, A. V. J. Colloid Interface Sci. 1995, 170, 299-307. (38) Green, C. E.; Abraham, M. H. J. Chromatogr., A 2000, 885, 41-49. (39) Bujalski, R.; Cantwell, F. F. Langmuir 2001, 7710-7711. (40) Sekine, T.; Nakatani, K. Chem. Lett. 2004, 33, 600-601. (41) Yan, B. W.; Zhao, J. H.; Brown, J. S.; Blackwell, J.; Carr, P. W. Anal. Chem. 2000, 72, 1253-1262. (42) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994. (43) Conder, J. R.; Young, C. L. Physicochemical Measurement by Gas Chromatography; John Wiley & Sons. Chichester, 1979. (44) Miyabe, K.; Guiochon, G. J. Chromatogr., A 1999, 849, 445-465.

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of asymmetric tailing of the solute zone.24 Column chromatographic methods yield S/D rates via the difference between the overall observed band broadening and the band broadening due to all of the non-S/D processes. Because they are difference measurements, they give accurate S/D rates when S/D processes dominate band broadening, but they give limited accuracy and potentially ambiguous results when non-S/D processes dominate.3,8,11,24,45 Spectroscopic methods for studying S/D kinetics include nuclear magnetic resonance21,22,36,37,46-49 and luminescence14,20,30,50-56 techniques. A limitation of these techniques is that they usually interrogate the rates of only one of the individual kinetic steps within the particle such as diffusion of nonretained solutes through pores,21,22,36,37,47 surface diffusion,14,30,54-56 or the attachment/ detachment step.50-53 Uptake/release methods for studying S/D kinetics involve monitoring the approach to equilibrium as a function of time after the concentration of solute surrounding the particle is changed abruptly. These methods measure directly the cumulative effects of steps i-iv above. They have been briefly reviewed in a recent paper.57 The shallow-bed method (SB method) of performing uptake/release rate measurements employs a very short bed (i.e., 0.4 Hz); and inverse Fourier transform to produce the deconvolved data in the time domain. RESULTS AND DISCUSSION Properties of Sorbent, DMNB, and PG. For Luna C-18, pore diameter is 10 nm (personal communication with manufacturer), estimated tortuosities are τM ) 2.0 ( 0.521,22,36,37,47,89 and τ surf ) 4 ( 1,89 and the hindrance parameter for pore diffusion is Hind ) 0.8 ( 0.113,90,91 for both DMNB and PG. Elution chromatograms for the three compounds, DMNB, PG, and the excluded molecule Blue Dextran (2 × 106 Da, 18-nm diameter),92 were each obtained in triplicate on a 250-mm × 4.6-mm HPLC column. From the mobile-phase flow rate (F ) 2.0 mL/s), the geometrical volume of the empty column (4.15 mL), and the retention times of the three compounds, the following properties were calculated for DMNB and for the bed of Luna C-18 in the HPLC column:93 retention factor of DMNB, k′DMNB ) 15.0 ( 0.1; interparticle porosity (from BD), inter ) 0.369 ( 0.003; total porosity (from PG), total ) 0.668 ( 0.002; intraparticle porosity, intra ) (total inter) ) 0.318 ( 0.004; and particle porosity, pp ) (intra/(1 - inter) ) 0.505 ( 0.009. The ratio of moles of DMNB sorbed in the ODS stationary phase to the moles of DMNB dissolved in the stagnant mobile phase filling the pores (i.e., pore-DMNB), is given by

total RDMNB ) k′DMNB‚ ) 32.4 ( 0.6 intra

(15)

and the ratio of concentrations [DMNB]part/[DMNB]0 for use in eq 11 is given by

[DMNB]part/[DMNB]0 ) (1 + RDMNB)PP ) 16.9 ( 0.5 (16) in which [DMNB]part has units of moles per liter of particle. The (86) Horlick, G.; Hieftje, G. In Contemporary Topics in Analytical and Clinical Chemistry; Hercules, D. M., Hieftje, G., Snyder, L. R., Evenson, M. A., Eds.; Plenum Publishing: New York, 1978; Vol. 3, Chapter 4. (87) Economou, A.; Fielden, P. R.; Packham, A. J. Analyst 1996, 121, 97-104. (88) Yamane, T.; Katayama, S.; Todoki, M. Thermochim. Acta 1991, 183, 329338. (89) Zalc, J. M.; Reyes, S. C.; Igselia, E. Chem. Eng. Sci. 2003, 58, 4605-4617. (90) Beck, R. E.; Schultz, J. S. Biochim. Biophys. Acta 1972, 255, 273-303. (91) Deen, W. M AIChE J. 1987, 33, 1409-1425.

amount of DMNB in the pores is equal to less than 3% (i.e., (100 (1 + RDMNB)-1) of the sorbed DMNB. Linearity of the sorption isotherm of DMNB in the concentration range of interest was demonstrated by running both a frontal chromatogram1,6,42,81 in which a 6.1 × 10-5 mol/L solution of DMNB in 50% methanol was the mobile phase and an elution chromatogram in which 5 µL of the same 6.1 × 10-5 mol/L DMNB solution was injected into a 50% methanol mobile phase. The measured k′DMNB values were 15.2 ( 0.1 and 15.07 ( 0.04, respectively. For the unretained compound, PG, k′PG ) 0 and RPG. ) 0. Diffusion coefficients for PG and DMNB in 50% methanol mobile phase are DM,PG ) (6.0 ( 0.7) × 10-6 cm2/s and DM,DMNB ) (5.0 ( 0.6) × 10-6 cm2/s, as calculated from the Wilke-Chang correlation.94 IRF for Porous Particles. An ideal IRF marker molecule would not be sorbed by the ODS stationary phase, would be large enough to be excluded from the pores of the Luna C-18 particles, and would have a diffusion coefficient similar to that of the solute being studied (i.e., DMNB). The exclusion requirement exists because the IRF marker must compensate for interparticle DMNB, but not for pore-DMNB. The exclusion requirement and the diffusion coefficient requirement for the IRF marker tend to be mutually exclusive because a very large excluded molecule will have a much smaller diffusion coefficient than will a small solute molecule. Therefore, in this study, the small molecule PG was used as the IRF marker, and the resulting IRF was corrected for the effects of PG in the pores. Details of the iterative corrections are given in ref 81. For run 1, the PG desorption rate curve and the IRF are shown in Figure 4A and Figure 4B, respectively. Also shown are their versions after the first and second iterations. The centers-of-gravity (i.e., first moments) of the initial, the first-iteration and the second-iteration IRFs in Figure 4B are 0.4916, 0.4810, and 0.4809, respectively, and their variances (i.e., second central moments) are 0.00400, 0.00399, and 0.00398. This indicates that the width of the IRF in Figure 4B is due, nearly exclusively, to extraparticle band broadening and only very slightly to intraparticle diffusion out of the pores of Luna C18. Although the iterations make very little change either to the initial curves in Figure 4A and 4B or to the final kinetic parameters obtained after all the data processing steps, in the interest of rigor, the versions from the second iterations are used in the subsequent calculations. Instantaneous Rate Curve for DMNB. The short bed contained ∼0.5 mg of Luna C-18 mixed with ∼1.5 mg of glass beads. Three desorption runs each, for both DMNB and PG, were performed at three flow rates. The A(t) versus t data for each run were obtained by averaging four to seven sets of five cycles each (i.e., 20-35 cycles/run). For every cycle, A(t) values were acquired for at least 4.0 s at an acquisition rate of 1000/s. By way of example, data for the first 0.5 s of run 1 are shown in Figure 5. Panel A shows the normalized observed curves. The solid line is for the DMNB solute, and the dashed line is for the PG IRF marker. Panel B shows the result of subtracting the dashed curve from the solid line. Panel C repeats the IRF marker curve from panel A and also shows its negative derivative, which is the IRF. (92) Wang, Y. Ph.D. Dissertation, University of Alberta, Edmonton, Alberta, 2000; Appendix A. (93) Horvath, C.; Lin, H. J. J. Chromatogr. 1976, 126, 401-420. (94) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264-270.

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Table 1. Parameters from Nonlinear Regression Fit of Eq 5 (for Spherical Diffusion) to the Cumulative Rate Data for Desorption of DMNB from the Bed of Luna C-18a result/parameter

run 1

run 2

run 3

[DMNB]0 (× 104 mol/L) F (× 105 L/s) nDMNB,O (× 1010 mol)b β (s-1)b DIP,DMNB (× 107 cm2/s)c R2 d

0.61 27.5 5.50 ( 0.01 9.84 ( 0.04 3.77 0.9839

0.61 37.3 5.37 ( 0.01 10.06 ( 0.05 3.86 0.9843

0.61 49.0 5.41 ( 0.01 10.99 ( 0.04 4.21 0.9860

a n DMNB,O and β are fitting parameters. The regression fit lines are shown in gray in Figure 6. b Standard deviations come from the individual nonlinear regressions. c DIP,DMNB is obtained from β via eq 6 using 6.15 × 10-4 cm for particle radius. d R is correlation coefficient for regression.

Panel D shows the normalized result of deconvolving the IRF from panel C with the difference curve from panel B. Cumulative Rate Curves for DMNB. The deconvolved instantaneous rate curves for each of the three runs were digitally integrated for 4.0 s as per eq 2, taking the absorbances between 2.5 and 3.5 s as the baseline. Integration was performed with Microsoft Excel software. The 1000 data points covering the first 1 s are shown in Figure 6 for runs 1-3, respectively. Prism Software Version 4.0 (GraphPad, San Diego, CA) was used to fit eq 5 (spherical intraparticle diffusion) to the data by nonlinear regression. Since only the first 15 exponential terms in eq 5 were used, only data points above 0.009 s were included, as discussed in ref 57. The fit lines are shown in gray in Figure 6A-C. Values for the two fitting parameters nDMNB,0 and β are presented in Table 1 as is the value of DIP,DMNB, which is calculated from β via eq 6. Also included in Table 1 are the known value of [DMNB]0 and the three known values of F. DIP,DMNB and nDMNB,0 in eq 5 are expected to be independent of F. Although there is reasonably good agreement among the means and standard deviations obtained at the three flow rates (DIP,DMNB ) (3.9 ( 0.2) × 10-7 cm2/s, and nDMNB,0 ) (5.43 ( 0.06) × 10-10 mol), the squared correlation coefficients for the fits at each flow rate are low (R2 ≈ 0.98) and systematic deviations between the experimental data and the fit lines are visible, especially at times less than ∼0.3 s. These deviations arise from incomplete attainment of shallow-bed conditions, as is shown below. The fit lines in Figure 7 are the result of nonlinear regression of eq 8 to the data points from runs 1-3, performed with the Solver macro in Microsoft Excel software, utilizing the Newton gradient search method with forward derivatives and tangent-based estimates.95 Values of the three fitting parameters, nDMNB,0, DIP,DMNB, and L, are presented in Table 2. (Solver does not calculate standard deviations for fitting parameters obtained in a given run.) The agreement among the three values of DIP,DMNB is close ((8.8 ( 0.3) × 10-7 cm2/s), and based on the t-test at the 99.999% confidence level (Excel software), their mean value is significantly different from the mean value obtained from the fit of eq 5 to the data. Also, the squared correlation coefficients are high (R2 ≈ 0.999 for each), suggesting that the data are described well by eq 8. (95) Billo, E. J. Excel for Chemists: A Comprehensive Guide; Wiley-VCH: Toronto, 1997; Chapter 17.

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Figure 7. Cumulative desorption rate curves for DMNB. (A) From run 1. (B) From run 2. (C) From run 3. The apparently solid lines are actually 500 closely spaced experimental data points. The gray line is for the best fit of eq 8 to the data points. Fitting parameters are reported in Table 2. Table 2. Parameters from Nonlinear Regression Fit of Eq 8 (for Spherical Diffusion with Incomplete SB Conditions) to the Cumulative Rate Data for Desorption of DMNB from the Bed of Luna C-18a result/parameter 104

[DMNB]0 (× mol/L) F (× 105 L/s) 10 nDMNB,O (× 10 mol) DIP,DMNB (× 107 cm2/s) L δ (× 104 cm)b R2 c

run 1

run 2

run 3

0.61 27.5 5.42 8.6 4.1 0.50 0.9998

0.61 37.3 5.28 9.0 4.1 0.50 0.9996

0.61 49.0 5.34 9.1 4.5 0.46 0.9995

a n Dmnb,O, DIp,dmnb, and L are fitting parameters. The regression fit lines are shown in gray in Figure 7. b The Nernst film thickness δ was obtained from L and DIP,DMNB via eq 11. c R is correlation coefficient for regression.

The low value of the fitting parameter L suggests that there is incomplete achievement of SB conditions.81 The reason for this can be elucidated by plotting L versus F, as shown in Figure 8. If failure of the zero concentration criterion were the cause of the low L then, from eq 10, the plot would be linear with zero intercept as in eq 17.

L ) (slope)F

(17)

Table 3. Theoretical Expressions for Plate Height Contributions from the Terms in Eq 20a term

expression

HLDb 2γDM,i/Uo HSc 2ki′Uo/(1 + ki′)kD Hcoupled (1/λd + D /ωd 2U )-1 p M,i p o HSMe dp2Uo (1 - f + ki′)2 ‚ ′ 2 (D (H /τ )) + ((k ′/fτ 30(1 - f)(1 + ki ) M,i ind M i surf)Dsurf,i) Hfilmf

Figure 8. Evidence for film diffusion being the primary contributor to the deviation of the DMNB experimental data in runs 1-3 from purely spherical diffusion as described by eq 5. The data points are values of L obtained by fitting the rate data with eq 8, as reported in Figure 7 and Table 2. The solid line for eq 18 and the dashed line for eq 17 are assumed to pass through the origin.

If, on the other hand, failure of the zero film thickness criterion were the cause of the low L, then the plot would be described by the equation

L ) (const)F1/3

(18)

This equation is obtained by combining eqs 11-13. The use of eq 13 is justified because Re, from eq 14, is ∼1.5-2.0 in these experiments. The fit of the solid regression line for eq 18 to the data points in Figure 8 yields a value for (const) of 5.87 with a relative 95% confidence range of (8%, while the fit of eq 17 yields a value for (slope) of 10.7 with a relative 95% confidence range of (35%. The much smaller relative confidence range for eq 18 indicates a better fit, which shows that it is the presence of a Nernst diffusion film, which is largely responsible for the deviation of the cumulative rate curves of DMNB from the shape predicted by eq 5. Following this line of reasoning, the film thicknesses δ shown in Table 2 were calculated via eq 11, employing the respective fit values for L and DIP,DMNB. They are close to the value of 0.3 × 10-4 cm, which is predicted for this range of Fs by the (approximate) eq 12. Fitting the desorption rate curve for DMNB with eq 8 not only yields the Nernst film thickness that prevails under the conditions of the shallow-bed experiment, but more importantly, it yields an accurate value of the intraparticle diffusion coefficient DIP,DMNB. At the same temperature and mobile-phase composition, this value of DIP,DMNB applies to diffusion of DMNB within the rigid Luna C18 particles regardless of whether they are relatively loosely packed in the presence of glass beads, as in the shallow-bed experiment, or tightly packed under high pressure, as in an HPLC column. The thickness of the Nernst film surrounding a particle depends on the linear velocity (and flow rate) of the mobile phase, as shown by eqs 12 and 13. The presence of glass beads probably does not influence the Nernst film thickness,77 but whether it does is irrelevant when it comes to predicting the value of Hfilm in the HPLC column in Table 3, below. This is because the Nernst film thickness that prevails in the HPLC column is totally independent of any values in Table 2 that are obtained from fitting eq 8 to the (96) Miller, J. M. Separation Methods in Chemical Analysis; Wiley-Interscience: New York, 1975; Chapter 8.

ξ(ko + ki′ + koki′)2 dp5/3Uo2/3 (1 + ko) (1 + ki′) 2

2

value (µm) ≈ 0.3 ≈0 ≈ 30 10 ( 1 (148) 20 ( 2

DM,i2/3

a

Values predicted from the expressions are for DMNB, identified by subscript i, chromatographed on a 0.46-cm-i.d. × 25.0-cm-long column of Luna C-18 using 50% MeOH mobile phase at a linear velocity of 0.292 cm/s (i.e., 2.00 mL/min). Value in parentheses for HSM is for the assumption of no surface diffusion (Dsurf,DMNB ) 0 cm2/s). b Obstruction factor γ ) 0.5-0.8.3 c H ≈ 0 because rate constant k S D is assumed to be very large. d λ ≈ 2.5 1 and ω ≈ 196 are packing factors. e Individual values of HEddy ≈ 30.8 µm and HM ≈ 880 µm; see eq 21. f ≡ intra/total ) 0.46; dp is particle diameter (12.3 × 10-4cm). f ξ ≈ 153 is a packing factor; ko ≡ intra/inter ) 0.86.

shallow-bed desorption rate curve. Therefore, no errors are introduced in the predicted plate height term Hfilm as a result of either the presence of glass beads or the looser packing density of Luna C18 particles in the shallow bed. Pore and Surface Diffusion of DMNB. Intraparticle diffusion in Luna C-18 occurs by means of a combination of pore diffusion and surface diffusion, as expressed, respectively, by the first and second terms on the right-hand side of eq 7. Inserting the values of DM,DMNB, Hind, τM, and RDMNB from above yields a value of (6 ( 2) × 10-8 cm2/s for the pore diffusion term. Subtracting this from the value of DIP,DMNB (≡ DIP,i in eq 7) ) (8.8 ( 0.3) × 10-7 cm2/s, which was obtained in the desorption rate studies described in the preceding section, yields a value of (8.2 ( 0.4) × 10-7 cm2/s for the surface diffusion term in eq 7. Thus, for DMNB, the rate of surface diffusion is more than 1 order of magnitude faster than the rate of pore diffusion through the stagnant mobile phase. Inserting the values of τsurf and RDMNB into the surface diffusion term of eq 7 yields the value of the surface diffusion coefficient, Dsurf,DMNB ) (3.2 ( 0.8) × 10-6 cm2/s which is well within the range of 0.2 × 10-6-7 × 10-6 cm2/s that has been measured for small molecules by luminescence methods on ODS-derivatized flat plates of fused silica in various solvents14,30,55,56 and by chromatographic methods on ODS particles in a methanol/water mobile phase.16 Contribution to Band Broadening in HPLC. The phenomenon of band broadening in a particle-packed chromatographic column has been treated extensively in the literature.1-11 In this section, the theoretically predicted plate height (Htheory) is compared to the observed plate height (Hobs) for a chromatographic peak of DMNB measured in triplicate on a 0.46-cm-i.d. Luna C18 HPLC column of length Z ) 25.0 cm, using 50% MeOH mobile phase at a linear velocity U0 of 0.292 cm/s (i.e., F ) 2.00 mL/ min). For the (triplicate) observed peak, the retention time (i.e., first moment) is tR ) 1377 ( 3 s, and the standard deviation (i.e., square root of the second central moment) is σ ) 20.3 ( 0.2 s. The observed plate height is calculated via eq 19 to be Hobs ) 54 ( 2 µm. Analytical Chemistry, Vol. 78, No. 5, March 1, 2006

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Hobs ) (σ/tR)2Z

(19)

One of the three experimentally measured peaks is shown as the gray line in Figure 9. Based on the frequently invoked rate theory of Giddings, Horvath, and others,3,4,6 the plate height can be predicted by an equation of the form

Htheory ) HLD + Hcouple + HS + HSM + Hfilm

(20)

in which the coupled term arising from the nonuniform flow profile in the mobile phase is

Hcouple ≡ (1/Heddy + 1/HM)-1

(21)

The subscripts LD, M, S, and SM are abbreviations for longitudinal diffusion, mobile phase, stationary phase, and stagnant mobile phase, respectively. Shown in the second column in Table 3 are the theoretical expressions, containing system parameters, for the plate height terms in eqs 20 and 21. Known values of those system parameters that have not already been presented are given in the footnotes to the table. Shown in the third column of Table 3 are the calculated values for each of the plate height terms. The Nernst film thickness that prevails in the HPLC column is implicit in the second term of the theoretical expression for Hfilm in Table 3. Its value is δ ≈ 0.9 × 10-4 cm.93 Summing the values of the five terms in eq 20 gives a theoretically predicted value of Htheory ) 60 µm, which is close to the experimentally measured value of Hobs ) 54 ( 2 µm. By substituting into eq 19 the theoretical value Htheory ) 60 µm in place of Hobs and rearranging, the square root of the second moment of the theoretically predicted chromatographic peak is σtheory ) 21.3 s. The solid line in Figure 9 is the peak that is predicted by inserting tR and σtheory into the equation for a Gaussian distribution.54 It is in good agreement with the experimentally measured peak. From the numbers in Table 3, it can be seen that the phenomena associated with Hcouple, HSM, and Hfilm are the only ones that make significant contributions to chromatographic band broadening of DMNB on Luna C18 under these conditions, which involve relatively large values of k′DMNB and U0. The contributions of HLD and HS are, as expected, very small. When U0 is changed, as in a van Deemter plot, the value of HSM will change in a way that can now be predicted accurately because the necessary diffusion coefficient, Dsurf,i, is known from the S/B desorption rate experiments. Often in the literature, the role of surface diffusion is ignored in the HSM term by assuming that Dsurf,i is equal to zero, and equally often, the contribution of film diffusion is ignored by omitting the Hfilm term from eq 20. Omitting these two terms when dealing with the present set of data results in Htheory ) 178 µm and σ ) 33.5 s, which is farther from the experimentally measured value of 20.3 ( 0.2 s. Thus, ignoring surface diffusion under the present experimental conditions leads to a larger discrepancy between the predicted and measured chromatographic peak width, as can be seen by the poorer agreement of the dashed-line Gaussian peak in Figure 9 with the gray line experimentally measured peak. Future applications of the shallow-bed technique might include systematic studies of the role of surface diffusion 1604 Analytical Chemistry, Vol. 78, No. 5, March 1, 2006

Figure 9. Experimentally observed and theoretically predicted elution peaks for DMNB chromatographed on a 0.46-cm-i.d. × 25.0cm-long column of Luna C-18 using 50% MeOH mobile phase at a linear velocity U0 ) 0.292 cm/s. Gray line is experimental chromatogram. Solid, black line is predicted by eq 20, which includes the effects of surface and film diffusion. Dashed line is predicted by a more conventional equation in which Dsurf,i and Hfilm are both considered to be zero. Values of plate-height terms are given in Table 3.

in ODS- and other reversed-phase bonded-phase particles, including different percentages of a particular organic modifier in the mobile phase, different organic modifiers and different alkyl-chain lengths of the stationary phase.16 CONCLUSIONS In this work, a shallow-bed instrument and associated method have been developed for measuring desorption rate curves from ODS particles, which are complete in less than 1 s. It has been possible to take into account, via eq 8, the contribution of Nernst film diffusion and, therefore, to obtain the rate constant for intraparticle diffusion. It has been possible, furthermore, to obtain the relative contributions of pore and surface diffusion to intraparticle diffusion. For any future studies, it is desirable to know how the instrument can be adapted for use with smaller ODS particles such as those of 5- and 3-µm diameter that are commonly used in HPLC. For solutes having the same k′, desorption rates will be faster from smaller diameter particles (see radius in eq 8). Therefore, attainment of the zero concentration condition and minimization of the Nernst film thickness will require higher linear velocities of the mobile phase. Since the pressure required to achieve a given linear velocity is inversely proportional to the square of particle diameter and directly proportional the bed length, working with smaller particles will necessitate the use of some combination of higher pressure and shorter, though still physically stable, beds. Experience with the 12-µm Luna particles showed that beds much shorter than 0.6 mm yielded noisy desorption rate curves, presumably because of local fluctuations in bed thickness due to instability of the bed (i.e., movement of the particles) under the stress of the pressure wave of mobile phase at the start of the desorption step. It was also found that in the absence of glass beads a 0.6-mm-long bed of 12-µm Luna particles exhibited a much higher flow resistance, which continued to increase with time, presumably because of an initially higher, and increasing, packing density. Thus, for work with smaller particles, the bed holder should include shorter microfabricated grids having smaller-width perforations; the bed should include nonporous glass beads of suitable size; the metal screens should have suitably smaller openings; and the pressure limit of pumps, valves, and shallowbed assembly may have to be increased. (Current pressure limit

is 250 psi.) In addition, it will be necessary to reduce the magnitude of instrument band broadening because the time-based width of the desorption rate curves will be narrower. Recent theoretical simulations and experiments performed with a bed containing only nonporous glass beads81 have shown that the main source of instrument band broadening in the present version of the shallow-bed instrument is flow through that 2.5mm-long section of the cylindrical quartz tube located between the woven stainless steel outlet screen (Figure 2E) and the outflow end of the window aperture (Figure 2A). It is the nonuniform flow profile in the low Reynolds number laminar regime, rather than longitudinal molecular diffusion of solute, that is responsible. Thus, for work with smaller particles, a window aperture smaller than 2 mm should be used. Furthermore, in light of this finding, it would be expected that a large-molecule solute could be used as the IRF marker. This has been confirmed for Blue Dextran by reversefrontal elution experiments on a bed filled with glass beads.81 The resulting IRF is virtually identical to that which is obtained using the small molecule phloroglucinol. These findings suggest that a large, pore-excluded molecule such as Blue Dextran might be usable as an IRF marker in future shallow-bed experiments involving porous ODS particles. GLOSSARY

Sh

Sherwood number

tR

chromatographic retention time, s

U0

linear velocity of mobile phase, cm/s

Z

length of chromatographic column, cm

Greek Letters β

intraparticle desorption rate constant

δ

Nernst film thickness, cm



porosity in bed

pp

particle porosity

η

viscosity of mobile phase, P

F

density of mobile phase, g/cm3

σ

square root of second central moment of chromatographic peak, s

τM

tortuosity for M-phase in particle pores

τsurf

tortuosity for surfaces of particle pores

Subscripts film

pertaining to Nernst diffusion film

i

pertaining to solute i

inter

pertaining to interparticle property

[i]

concentration of i in mobile phase, mol/L

intra

pertaining to intraparticle property

[i]0

concentration of i in preequilibration mobile-phase solution, mol/L

IP

pertaining to intraparticle process

LD

[i]part

initial concentration of i in particles, in units of moles per liter of particle

pertaining to longitudinal diffusion in the chromatographic column

M

pertaining to mobile phase

[DMNB]0

concentration of DMNB in preequilibration mobile phase solution, mol/L

obs

pertaining to an experimentally observed parameter

[DMNB]part

concentration of DMNB in units of moles per liter of particle

S

pertaining to the stationary phase (e.g., ODS)

SM

pertaining to stagnant (in pores) mobile phase

A(t)

absorbance as fct(t)

surf

pertaining to pore surfaces

A0,i

absorbance of equilibrating solution of i

theory

pertaining to a theoretically derived parameter

A0,IRF marker

absorbance of equilibrating solution of IRF marker

D

diffusion coefficient, cm2/s

F

volumetric flow rate, L/s

H

plate height, cm

Hind

pore hindrance parameter

k′

chromatographic retention factor

L

parameter in eq 8, values given by eq 10 or 11

ni(t)

moles of i desorbed as fct(t)

ni,0

moles of i in particles at time zero

r

radius of particle, cm

rbed

radius of bed, cm

Re

Reynolds number

Ri

ratio of moles of i in ODS stationary phase to moles in pore liquid

ACKNOWLEDGMENT This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the University of Alberta. R.B. thanks NSERC also for a postgraduate scholarship. Eric Carpenter is gratefully acknowledged for assistance with Labview. Dieter Starke of the U of A Chemistry Machine Shop is acknowledged for fabricating several parts of the shallow-bed assembly. Dr. Monica Palcic and Dr. Brian Dunford are thanked for donating the stopped-flow equipment. Received for review December 19, 2005.

September

8,

2005.

Accepted

AC051609R

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