Numerical Method for the Estimation of Column Radial Heterogeneity

Dec 7, 2010 - Many reports in the literature show that the radial distributions of the ... A degree of radial heterogeneity seems to be a common prope...
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Anal. Chem. 2011, 83, 182–192

Numerical Method for the Estimation of Column Radial Heterogeneity and of the Actual Column Efficiency from Tailing Peak Profiles Kanji Miyabe† and Georges Guiochon*,‡ Graduate School of Science and Engineering for Research, University of Toyama, 3190, Gofuku, Toyama 930-8555, Japan, and Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, United States It is probably impossible to prepare high-performance liquid chromatography (HPLC) columns that have a completely homogeneous packing structure. Many reports in the literature show that the radial distributions of the mobile phase flow velocity and the local column efficiency are not flat, even in columns considered as good. A degree of radial heterogeneity seems to be a common property of all HPLC columns and an important source of peak tailing, which prevents the derivation of accurate information on chromatographic behavior from a straightforward analysis of elution peak profiles. This work reports on a numerical method developed to derive from recorded peak profiles the column efficiency at the column center, the degree of column radial heterogeneity, and the polynomial function that best represents the radial distributions of the flow velocity and the column efficiency. This numerical method was applied to two concrete examples of tailing peak profiles previously described. It was demonstrated that this numerical method is effective to estimate important parameters characterizing the radial heterogeneity of chromatographic columns.

to the slow response of the detector; (2) the surface of the stationary phase is heterogeneous, due to the presence of different types of adsorption sites, residual silanol groups, and metal impurities; and (3) the structure of packed beds is heterogeneous, causing important radial distributions of the mobile phase flow velocity, the column efficiency, and the solute concentration. Long ago, many authors reported that HPLC columns are radially heterogeneous to a degree.5-25 It is probably impossible to prepare columns that are radially homogeneous. So, a strategy is needed that permits the derivation of reliable information on the actual characteristics of the radial heterogeneity of columns from the asymmetrical profiles of elution peaks. We previously dealt with asymmetrical (mainly tailing) peak profiles originating from the column radial heterogeneity, which is a general problem of HPLC columns.26-29 We clarified correlations between characteristics of asymmetrical peak profiles and radial distributions of the flow velocity and the column efficiency and developed a

Information on the behavior of chromatographic columns is usually derived from an analysis of recorded elution peak profiles in which these profiles are assumed to be Gaussian. However, in practice, elution peak profiles are often asymmetrical (either tailing or fronting), even under linear equilibrium isotherm conditions. A proper analysis of chromatographic behavior under such conditions is difficult because simple models of chromatography (e.g., the plate theories) are based on the assumption that Gaussian-shaped peaks and peak distortions should prevent accurate analysis of chromatographic behavior by following this simple approach. The main origins of peak distortion are considered to be1-4 (1) the profile of the sample injection pulse is tailing due to dispersion in the hold-up volumes of tube connections and

(10)

* To whom correspondence should be addressed. E-mail: [email protected]. † University of Toyama. ‡ University of Tennessee. (1) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, MA, 1994. (2) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965. (3) Sternberg, J. C. Adv. Chromatogr. 1966, 2, 205. (4) Kirkland, J. J.; Yau, W. W.; Stoklosa, H. J.; Dilks, C. H., Jr. J. Chromatogr. Sci. 1977, 15, 303.

(22) (23)

182

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

(5) (6) (7) (8) (9)

(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

(24) (25) (26) (27) (28) (29)

Horne, D. S.; Knox, J. H.; McLaren, L. Sep. Sci. 1966, 1, 531. Knox, J. H.; Parcher, J. F. Anal. Chem. 1969, 41, 1599. Knox, J. H.; Laird, G. R.; Raven, P. A. J. Chromatogr. 1976, 122, 129. Eon, C. H. J. Chromatogr. 1978, 149, 29. Baur, J. E.; Kristensen, E. W.; Wightman, R. M. Anal. Chem. 1988, 60, 2334. Bayer, E.; Mu ¨ ller, W.; Ilg, M.; Albert, K. Angew. Chem., Int. Ed. 1989, 28, 1029. Baur, J. E.; Wightman, R. M. J. Chromatogr. 1989, 482, 65. Yun, T.; Guiochon, G. J. Chromatogr., A 1994, 672, 1. Farkas, T.; Chambers, J. Q.; Guiochon, G. J. Chromatogr., A 1994, 679, 231. Tallarek, U.; Baumeister, E.; Albert, K.; Bayer, E.; Guiochon, G. J. Chromatogr., A 1995, 696, 1. Bayer, E.; Baumeister, E.; Tallarek, U.; Albert, K.; Guiochon, G. J. Chromatogr., A 1995, 704, 37. Fernandez, E. J.; Grotegut, C. A.; Braun, G. W.; Kirshner, K. J.; Staudaher, J. R.; Dickson, M. L.; Fernandez, V. L. Phys. Fluids 1995, 7, 468. Yun, T.; Guiochon, G. J. Chromatogr., A 1996, 734, 97. Farkas, T.; Sepaniak, M. J.; Guiochon, G. J. Chromatogr., A 1996, 740, 169. Farkas, T.; Sepaniak, M. J.; Guiochon, G. AIChE J. 1997, 43, 1964. Farkas, T.; Guiochon, G. Anal. Chem. 1997, 69, 4592. Guiochon, G.; Farkas, T.; Guan-Sajonz, H.; Koh, J. H.; Sarker, M.; Stanley, B. J.; Yun, T. J. Chromatogr., A 1997, 762, 83. Guiochon, G. J. Chromatogr., A 2007, 1168, 101. Mriziq, K. S.; Abia, J. A.; Lee, Y.; Guiochon, G. J. Chromatogr., A 2008, 1193, 97. Abia, J. A.; Mriziq, K. S.; Guiochon, G. J. Chromatogr., A 2009, 1216, 3185. Abia, J. A.; Mriziq, K. S.; Guiochon, G. J. Sep. Sci. 2009, 32, 923. Miyabe, K.; Matsumoto, Y.; Niwa, Y.; Ando, N.; Guiochon, G. J. Chromatogr., A 2009, 1216, 8319. Miyabe, K.; Guiochon, G. J. Chromatogr., A 1999, 830, 29. Miyabe, K.; Guiochon, G. J. Chromatogr., A 1999, 857, 69. Miyabe, K.; Guiochon, G. J. Chromatogr., A 1999, 830, 263. 10.1021/ac102195x  2011 American Chemical Society Published on Web 12/07/2010

numerical method allowing the determination of the actual performance of a column from recorded tailing peak profiles, irrespective of the degree of heterogeneity of the packing structure in the column. This numerical method was applied to the analysis of peak profiles recorded in two concrete cases. The local efficiency at the column center estimated from the tailing peak profiles was in fair agreement with the actual performance of that column calculated from symmetrical peak profiles eluted in the column center, which should not be much affected by the “walleffect”. This agreement proved the validity of the numerical method. It showed that the numerical method could be effective for deriving the true column efficiency at the column center from experimental profiles. However, in this earlier study,26 we used fourth-order polynomial functions to represent the radial distributions of the flow velocity and the local column efficiency because these functions provided a proper empirical representation of the column radial heterogeneity measured.7-9,13,18-20 However, it is probable that the column radial heterogeneities could be better represented by other polynomial functions. The goal of this work was (1) to clarify the influence of the order of the polynomial functions on calculated peak profiles and (2) to develop a numerical method providing suitable information on the actual characteristics of the radial heterogeneity of columns, including the order of the best polynomial function, from recorded peak profiles. Ultimately, we show that the numerical method provides estimates of the order of the polynomial function best representing the radial distributions of the flow velocity and the local column efficiency. It also provides important information on the true column performance, i.e., the efficiency at the column center, and on the degree of radial column heterogeneity, i.e., the ratio of the HETP or the flow velocity near the wall to those at the column center. The results of this work have important consequences besides their application in chromatography because many flow operations use beds packed with different materials for the purpose of the separation of mixtures, the concentration of certain compounds, or the preparation of derivatives in analytical or industrial chemistry in the fields of filtration, adsorption, ion-exchange, solid phase extraction, catalytic reactions. An adequate control of the structures of the beds used in these processes is necessary to improve the effectiveness of these operations and to develop procedures that ameliorate their packing conditions. The elution signals of selected compounds are often analyzed to clarify some characteristics of these flow operations. Gaussian profiles are always assumed for these signals but deviations of the profiles of these signals from Gaussian shapes are frequently observed in practice. In these cases, the signal profiles contains important information on the bed characteristics that only a detailed analysis like the one provided here can accurately extract and supply. When chromatographic measurements are carried out for purposes of physical chemistry investigations in the study of the thermodynamics of retention mechanisms or, more importantly, in the analysis of different contributions to the mass transfer kinetics, it is necessary to know whether the peak distortions observed are due to some characteristics of the packing material studied or to a heterogeneous bed. This information is critical for the proper interpretation of the experimental data. This is why we studied the influence of the radial heterogeneity of packed

bed on chromatographic behavior. However, the results and the conclusions of this study are not limited to the analysis of chromatographic data. They can also be applied to all the other flow operations used in analytical chemistry. THEORY Tailing peak profiles were numerically calculated using the equilibrium-dispersive model of chromatography, as previously explained in detail26 and briefly indicated below. The calculations are based on the following assumptions: (1) The chromatographic process proceeds under linear isotherm conditions. (2) The radial distributions of the mobile phase flow velocity (u), the local column efficiency (h), and the radial dispersion coefficient (Dr) are represented by polynomial functions, which are axially symmetrical and have the same order, between two and six. (3) The column is represented by a grid of 500 coaxial annular columns of constant thickness, equal to 1/500 of the overall column radius. (4) The peaks eluted from each annular columns have a Gaussian profile because each coaxial columns is assumed to be homogeneous. The overall peak profile eluted from the radially heterogeneous column is calculated by summing up the 500 Gian profiles leaving the annular columns, weighed in proportion to their cross-sectional area. It was demonstrated that the radial distributions of the mobile phase flow velocity (u) and the reduced plate height (h) in slurrypacked columns are not flat.9,11,13,18-20 The value of u is higher in the column center (uc) than near the wall (uw). The radial distribution of h has an opposite profile, hc is lower than hw. The subscripts c and w denote the column center and the column wall, respectively. The radial distributions of u and h are represented as follows.

( Rr )

u(r) ) au

( Rr )

h(r) ) ah

n

n

+ bu

(1)

+ bh

(2)

(see definitions of variables and parameters in the Glossary) The value of n is between 2 and 6. The first terms in the right-hand side (RHS) of eqs 1 and 2 represent the influence of the “walleffect” or rather of the radial heterogeneity of the bed across the column on the distribution of u and h, respectively. According to the Aris theory of diffusion,30 the contribution due to the radial distribution of the flow velocity to band broadening is accounted by a fourth (Aris) term in the ordinary plate height equations.1,2,31-37

H)A+

dc2 B + Cu + Cm us u Dr

(3)

(30) Aris, R. Proc. R. Soc. 1959, A252, 538. (31) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271. (32) Grushka, E.; Snyder, L. R.; Knox, J. H. J. Chromatogr. Sci. 1975, 13, 25. (33) Horva´th, C.; Lin, H.-J. J. Chromatogr. 1976, 126, 401. (34) Knox, J. H. J. Chromatogr. Sci. 1977, 15, 352. (35) Horva´th, C.; Lin, H.-J. J. Chromatogr. 1978, 149, 43. (36) Antia, F.; Horva´th, C. J. Chromatogr. 1988, 435, 1. (37) Knox, J. H. Adv. Chromatogr. 1998, 38, 1.

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The Aris term contains the contribution of the first term in the RHS of eq 2 and decreases with increasing Dr. The faster the radial dispersion of a solute, the more homogeneous the column in the radial direction.30,38,39 The radial plate height (Hr) is correlated with Dr as follows. Hr )

2Dr u

(4)

The flow rate dependence of the radial dispersion is represented as follows in the nondimensional form. hr )

2γ B +D) +D ν ν

Dr D )γ+ ν Dm 2

[( )( ) ] n

+ 1 Dc

(7)

The combination of eqs 6 and 7 leads to the following equation, which represents the radial distribution of Dr. Dr(r) )

[[( )( ) ] ah r bh R

n

( 2ν + γ)]D

+ 1 Dc

m

(8)

The Aris term in eq 3 corresponds to the first term in the RHS of eq 2 because the influence of the radial dispersion on band broadening is not considered in the first, second, and third terms in eq 3. Gritti, F.; Guiochon, G. J. Chromatogr., A 2008, 1206, 113. Gritti, F.; Martin, M.; Guiochon, G. Anal. Chem. 2009, 81, 3365. Knox, J. H.; McLaren, L. Anal. Chem. 1964, 36, 1477. Tallarek, U.; Bayer, E.; Guiochon, G. J. Am. Chem. Soc. 1998, 120, 1494. Saffman, P. G. J. Fluid Mech. 1959, 6, 321. Saffman, P. G. J. Fluid Mech. 1960, 7, 194. Littlewood, A. B. Anal. Chem. 1966, 38, 2. Baumeister, E.; Klose, U.; Albert, K.; Bayer, E.; Guiochon, G. J. Chromatogr., A 1995, 694, 321. (46) Tallarek, U.; Albert, K.; Bayer, E.; Guiochon, G. AIChE J. 1996, 42, 3041. (38) (39) (40) (41) (42) (43) (44) (45)

184

ah r ξ R

uav )

∫

R

0

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

n

+ bh

(9)

2πr

u(r) dr πR2

(10)

The values of au and bu in eq 1 were chosen so that uav is equal to unity. The Gaussian profile of the elution peak leaving from each annular column was calculated by the following equation, a solution of the equilibrium-dispersive model of chromatography with the δ pulse boundary condition.1

(6)

In this study, the influence of the values of γ on the peak profile was studied by changing γ in the range between 0.4 and 0.8 because similar values of γ have been reported.7,8,39-41 Similarly, the value of D was changed between 0.05 and 0.4 because various values of D ranging from 0 to 0.375 have been determined in different experimental systems.5,7,8,41-46 It seems that the radial distributions of h and D must be similar because the radial heterogeneities of both the axial and the radial dispersion are attributed to the same origin, i.e., the irregularity of the packing structure across the column. As in a previous study,26 it was assumed that D is larger near the wall (Dw) than at the column center (Dc) and that the value of D at the radial position r (D(r)) is represented as follows with the parameters ah and bh in eq 2. ah r bh R

( )( )

where ξ represents the influence of the radial dispersion on the radial distribution of the local column efficiency across the column. Equation 9 was used for calculating the radial distribution of the actual local column efficiency. To calculate tailing peaks eluted from a radially heterogeneous column, the average flow velocity (uav) of the mobile phase is first calculated as follows, so that the first moment of all the calculated peaks is equal to unity.

(5)

where the coefficient D is related to the “stream splitting” mechanism, which accounts for the contribution of eddy dispersion to transverse dispersion. The following equation is derived from eqs 4 and 5.

D(r) )

h(r) )

Cd )

1 tdσd√2πtd

[

exp -

(td - 1)2 2tdσd2

]

(11)

The tailing peak from the radially heterogeneous column was calculated by summing up the Gaussian profiles eluting from all the annular columns. These calculations were carried out using a small BASIC program to obtain the overall elution peak profile and its related characteristics, the first absolute moment (µ1),and the second central moment (µ2′). RESULTS AND DISCUSSION Numerical Calculation of Tailing Peak Profiles. The first few figures show the results of a numerical study illustrating how the parameters listed earlier, uw/uc, hw/hc, n, γ, and D, influence the profiles of tailing peaks. Figure 1a shows profiles calculated under three different conditions, (1) uw/uc ) 1.0 and hw/hc ) 1.0, (2) uw/uc ) 0.985 and hw/hc ) 1.5, and (3) uw/uc ) 0.97 and hw/hc ) 3.0. In all three cases, the values of Nc, ν, n, γ, and D are assumed to be 2.0 × 104, 10, 4, 0.6, and 0.1, respectively. It was shown earlier that changes in ν do not significantly affect the peak profiles.26 The degree of column radial heterogeneity increases in the order (1) < (2) < (3). In the first case, the peak obtained is Gaussian, as expected. The asymmetry and width of the calculated peaks increase with increasing degree of radial heterogeneity of the column. In case (3), the peak profile tails obviously. These results indicate that the radial heterogeneity of the flow velocity and the local efficiency are important possible causes of the tailing phenomena. Figure 1b,c shows the influence of the order of the polynomial functions in eqs 1 and 2 that represent the radial distributions of the flow velocity and the local column efficiency. The values of Nc, ν, γ, and D are 2.0 × 104, 10, 0.6, and 0.1, respectively. Figure 1b illustrates the peak profiles calculated for uw/uc ) 0.985 and hw/hc ) 1.5. The profiles calculated for different values of n between 2 and 6 overlap and are barely distinguishable, although the peak height slightly increases with increasing n. This shows that the value of n has little influence on the profiles

Figure 2. Influence of (a) D and (b) γ on the peak profile at uw/uc ) 0.97, hw/hc ) 3.0, and n ) 4.

Figure 1. Peak profiles calculated under different conditions of the column radial heterogeneity of the mobile phase flow velocity (u), the local column efficiency (h), and the order of polynomial functions (n), i.e., (a) uw/uc ) 0.97-1.0, hw/hc ) 1.0-3.0, and n ) 4, (b) uw/uc ) 0.985, hw/hc ) 1.5, and n ) 2-6, and (c) uw/uc ) 0.97, hw/hc ) 3.0, and n ) 2-6.

of nearly symmetrical peaks. On the other hand, Figure 1c illustrates the peak profiles calculated at uw/uc ) 0.97 and hw/hc ) 3.0. All the peaks calculated have asymmetrical profiles. Although the peak height increases with increasing n as in Figure 1b, the influence of n is stronger. It seems that the more asymmetrical the peak profile, the larger the influence of the value

of n on its profile. Obviously, the peak profiles are independent of n whenever uw/uc ) 1 and hw/hc ) 1. When the packed bed is homogeneous across the column, the first term in the righthand side of eqs 1 and 2 must be eliminated. Figure 2a illustrates the influence of the value of D on the peak profiles. The values of Nc, ν, γ, n, uw/uc, and hw/hc are 2.0 × 104, 10, 0.6, 4, 0.97, and 3.0, respectively. Although the value of D was increased from 0.05 to 0.4, the profiles calculated are almost the same. Figure 2b illustrates the influence of the value of γ on the peak profiles. The values of Nc, ν, D, n, uw/uc, and hw/hc are 2.0 × 104, 10, 0.1, 4, 0.97, and 3.0, respectively. Although the peaks in Figure 2b exhibit an important degree of tailing, they overlap almost completely. Figure 2a,b indicates that the parameters D and γ have little influence on the degree of tailing of peak profiles. Influence of the Column Radial Heterogeneity on Some Characteristics of Tailing Peaks. Peak profiles were calculated for various combinations of uw/uc and hw/hc in order to illustrate how the numerical approach can account for different types of tailing peaks. In this section, the values of Nc, ν, D, γ, and n are chosen as 2.0 × 104, 10, 0.1, 0.6, and 4, respectively. The results show how some characteristics of tailing peaks are correlated with the radial distributions of u and h. Peak Retention. Figure 3a illustrates the influence of the column radial heterogeneity on the apical retention time (tR), which is Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

185

Figure 3. Influence of the column radial heterogeneity on (a) the peak retention time (tR), (b) the apparent column efficiency (N), (c) the asymmetry factor of elution peaks ((wR/wF)0.1), and (d) the vertical distortion of elution peaks (w0.5/w0.1).

normalized by µ1. As explained in eq 10, the value of µ1 is equal to unity for all the peaks calculated in this study. For symmetrical peaks, the apical retention time and the first moments are equal. Although the ratio tR/µ1 decreases with increasing column radial heterogeneity, the magnitude of this change is initially quite small. As illustrated in Figure 1a, the peak profile calculated for uw/uc ) 0.97 and hw/hc ) 3.0 is obviously asymmetrical. However, even in this case, tR differs from unity by only ∼0.5%. This means that tR cannot be used to characterize the column radial heterogeneity. On the other hand, the ratio tR/µ1 at uw/uc ) 1.0 is only 0.9998, not unity, in Figure 3a. This deviation was attributed to a numerical calculation error.26 Apparent Column Efficiency. Figure 3b illustrates the influence of the column radial heterogeneity on the apparent column efficiency (N) calculated from the moments, µ1 and µ2′, of the peak eluting from the column.

N)

µ12 µ2

(12)

The value of N decreases rapidly with an increasing degree of column radial heterogeneity and is reduced to about a quarter of its true value (at 2.0 × 104) for uw/uc ) 0.97 and hw/hc ) 3.0. It is, therefore, difficult to assume that N is equal to Nc when recorded peaks exhibit asymmetrical profiles. Nevertheless, 186

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

valuable information on the column radial heterogeneity can be derived from N, due to the large amplitude of its variation with uw/uc. Peak Asymmetry. Peak asymmetry is another parameter that characterizes peak tailing. There are many definitions of the peak asymmetry.47-49 We use a conventional one, the ratio of the width of the rear part to that of the front part of the peak profile at 10% height, (wR/wF)0.1, as the asymmetry factor.4,50-55 Figure 3c illustrates how the asymmetry factor increases with increasing column radial heterogeneity. For example, the value of (wR/wF)0.1 for the obviously tailing peak in Figure 1a at uw/uc ) 0.97 and hw/hc ) 3.0 is about 1.9. Thus, the peak asymmetry is another effective mean to characterize column radial heterogeneity. On the other hand, the value of (wR/wF)0.1 is equal to 1.02 at uw/ uc ) 1.0 in Figure 3c. The deviation of this value from unity is also attributed to a numerical calculation error.26 (47) Grushka, E.; Myers, N. M.; Schettler, P. O.; Giddings, J. C. Anal. Chem. 1969, 41, 889. (48) Cuso´, E.; Guardino, X.; Riera, J. M. J. Chromatogr. 1974, 95, 147. (49) Pa´pai, Zs.; Pap, T. L. J. Chromatogr., A 2002, 953, 31. (50) Barber, W. E.; Carr, P. W. Anal. Chem. 1981, 53, 1939. (51) Foley, J. P.; Dorsey, J. G. Anal. Chem. 1983, 55, 730. (52) Bidlingmeyer, B. A.; Warren, F. V., Jr. Anal. Chem. 1984, 56, 1583A. (53) Foley, J. P.; Dorsey, J. G. J. Chromatogr. Sci. 1984, 22, 40. (54) Anderson, D. J.; Walters, R. R. J. Chromatogr. Sci. 1984, 22, 353. (55) Jeansonne, M. S.; Foley, J. P. J. Chromatogr. Sci. 1991, 29, 258.

Other parameters could be used to assess the degree of peak asymmetry. For example, the third moment of a peak accounts for its deviation from cylindrical symmetry, another property of a cylindrical bed. However, it is not practically useful because it is quite difficult to accurately measure the third moment of experimental peaks. The signal noise severely prevents accurate measurements of third moments. The higher the moment order, the more difficult the accurate determination of its value because (1) signal noise prevents the accurate determination of the times when integration of the profile must begin and end, and (2) the contribution of signal noise to the moment in the regions far from the peak center increases with increasing moment order. Thus, the logical validity of the numerical approach proposed here is independent of the parameter selected to represent the peak asymmetry. If a parameter other than (wR/wF)0.1 were used, a similar algorithm could be established. Vertical Distortion. The asymmetry factor represents the degree of horizontal distortion of tailing peaks. These profiles deviate also vertically from Gaussian ones, for which the peak width at any fractional height is proportional to the standard deviation of the distribution. For example, the ratio of the peak widths at 50% (w0.5) and 10% (w0.1) of the peak height is 0.549. The subscripts 0.1 and 0.5 denote 10% and 50% peak heights, respectively. In this study, the ratio w0.5/w0.1 was used as a parameter to represent the vertical distortion of tailing peaks. Figure 3d illustrates the influence of the column radial heterogeneity on the vertical distortion. As stated above, the vertical distortion is equal to 0.549 at uw/uc ) 1.0 and hw/hc ) 1.0. However, the value of w0.5/w0.1 is widely spread in the range from ca. 0.51 to 0.57, depending on the combination of uw/uc and hw/hc. This means that the numerical approach used in this study may represent various types of tailing peak profiles, which are vertically skewed from a Gaussian profile. Thus, the vertical distortion can also be used as an informative characteristic of tailing peak profiles. Characteristic Network Map of Tailing Peak Profiles. Intrinsic characteristics of tailing peak profiles can be represented by different parameters. Three of them effectively characterize tailing peaks, not tR, but N, (wR/wF)0.1, and w0.5/w0.1. This is fortunate because chromatographers are familiar with them since they are commonly used in calculations of the column efficiency and the peak asymmetry factor.4,50-55 Figure 4a-e shows network maps useful to characterize tailing peak profiles and illustrates correlations between w0.5/w0.1 and (wR/wF)0.1 for different combinations of uw/uc and hw/hc. The order, n, of the polynomial functions representing the radial distributions of u and h was varied between 2 and 6. The arrow indicates the point corresponding to a Gaussian profile. The values of w0.5/w0.1 and (wR/wF)0.1 for Gaussian profiles are 0.549 and 1.0, respectively. Other data points scatter away from the Gaussian point in a wide range. The network maps in Figure 4a-e show quite different profiles, depending on the value of n, when the values of uw/uc and hw/hc are changed in the same range, i.e., uw/uc ) 0.97-1.0 and hw/hc ) 1.0-5.0. As illustrated in Figure 4a, when parabolic distributions of u and h are assumed, the values of (wR/wF)0.1 are relatively small and range from 1.0 to 1.4. This means that peaks are only moderately skewed in the horizontal direction. On the other

hand, most values of w0.5/w0.1 are larger than the standard value for a Gaussian profile, 0.549. This means that, compared to Gaussian profiles, the peak width is relatively larger at high fractional height (50% peak height) than at lower ones (10% height). In contrast, Figure 4e illustrates the network map at n ) 6. Compared with the network map at n ) 2 in Figure 4a, the network map in Figure 4e shows quite different profiles. The values of w0.5/w0.1 are smaller than 0.549. The peak width at 50% peak height is relatively smaller than that at 10% height. Additionally, peak tailing is significantly enhanced. The maximum value of (wR/wF)0.1 is around 2.0. These results are consistent with the peak profiles in Figure 1c. Figure 4a-e includes two series of solid symbols, which represent the plots at uw/uc ) 0.985 and hw/hc ) 1.5 and at uw/ uc ) 0.97 and hw/hc ) 3.0. The solid circles, triangles, squares, pentagons, and hexagons correspond to the values of n ) 2, 3, 4, 5, and 6, respectively. Compared with the plots at uw/uc ) 0.985 and hw/hc ) 1.5, those at uw/uc ) 0.97 and hw/hc ) 3.0 are more widely spread, showing that the extent of the peak distortion depends much on the value of n. In addition, the more significant the peak distortion, the more important the influence of n on the peak profile. The exponentially modified Gaussian (EMG) function has been suggested as a model for tailing peaks.47,50-62 In this model, a Gaussian profile is exponentially modified through a suitable convolution integral, which contains two additional parameters, the time constant of the exponential and a dummy variable of integration. The values of the moments of tailing peaks are functions of the two parameters of the original Gaussian peak, its mass center, and its variance and of the time constant.57 Other equation models were also proposed on the basis of a numerical analyses of various characteristics of tailing peaks. These equations correlate some graphically measurable parameters of the tailing peak at 10, 30, or 50% of its maximum height with the parameters of the EMG function.51 The following equations were proposed for (wR/wF)0.1 larger than 1.09 in order to represent the retention time (tG) and the standard deviation (σG) of the parent Gaussian peak from which the skewed EMG peak is derived.51 The value of σG is correlated with the peak width and its asymmetry factor at 10% and 50% of the maximum peak height as follows.

σG )

w0.1

( )

wR 3.27 wF

σG )

(13) + 1.2

0.1

w0.5 wR 2.5 wF

( )

(14)

0.5

The following equations were also proposed to calculate tG. (56) (57) (58) (59) (60) (61) (62)

Maynard, V.; Grushka, E. Anal. Chem. 1972, 44, 1427. Grushka, E. Anal. Chem. 1972, 44, 1733. Yau, W. W. Anal. Chem. 1977, 49, 395. Pauls, R. E.; Rogers, L. B. Anal. Chem. 1977, 49, 625. Pauls, R. E.; Rogers, L. B. Sep. Sci. Technol. 1977, 12, 395. Jeansonne, M. S.; Foley, J. P. J. Chromatogr. 1992, 594, 1. Li, J. J. Chromatogr. Sci. 1995, 33, 568.

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187

[

( )

tG ) tR - σG -0.193

wR wF

2

( )

+ 1.162

0.1

wR wF

0.1

[ ( ) ( )

wR tG ) tR - σG -1.46 wF

2

0.5

wR +5 wF

]

]

- 3.14 0.5

γ, n, uw/uc, and hw/hc. For a Gaussian peak, hp is given as follows.

- 0.545

(15) hp ) (16)

The dashed lines added in Figure 4a-e were calculated using eqs 13-16, which represents the correlation between w0.5/w0.1 and (wR/wF)0.1 in the range of (wR/wF)0.1 from 1.09 to 2.0. As shown in Figure 4a-e, there are various types of tailing peaks, even though the source of peak asymmetry is limited to the radial heterogeneity of the column. However, the dashed line seems to pass through a limited region. The EMG functions approximately describe the characteristics of tailing peaks and may be useful under limited conditions. However, the EMG model does not properly characterize all the types of tailing peak profiles which originate from the column radial heterogeneity. The idea of calculating the peak moments from an equation which would fit the profile recorded would be sound if there was a correct profile model. Unfortunately there is no such model, and recorded profiles cannot be accurately fitted to any known function, let alone the invalid EMG model.63 Admittedly, this method gives precise results but it sells accuracy for precision and this is not acceptable. Figure 4a-e indicates that the characteristics of the various types of tailing profiles can be characterized by two parameters, w0.5/w0.1 and (wR/wF)0.1, which are easy to measure from recorded chromatograms. They suggest that each experimental profile corresponds to a set of values of uw/uc and hw/hc for given values of Nc and n, and that the tailing profile can be calculated numerically. This means that useful information on the degree of radial heterogeneity of the flow velocity and the local column efficiency can be derived by the numerical method proposed in this study from the three important characteristics of the tailing profiles, i.e., N, w0.5/w0.1, and (wR/wF)0.1, that are easily measured from experimental peak profiles. Deviation of the Peak Width from That of Gaussian Profiles. Another parameter is needed to determine the value of n. As illustrated in Figure 4a-e, roughly speaking, when n increases, the peak profile becomes sharper and the peak height higher. However, the peak also exhibits a stronger horizontal distortion for the 10% peak height. On the contrary, when n is relatively small, the difference in peak widths at 10 and 50% peak heights becomes smaller. The peak becomes broader and its height lower. In addition, the asymmetry factor decreases. These correlations between the peak profiles and n are also illustrated in Figure 1c. In this study, the parameter w0.1/w0.1ph was used to represent the intrinsic characteristic of tailing profiles illustrated in Figures 1c and 4a-e. The value of w0.1 is the peak width at 10% peak height, which is calculated for any given set of Nc, ν, D, γ, n, uw/uc, and hw/ ph hc. The value of w0.1 is also a peak width at 10% peak height, but it is estimated by assuming a Gaussian profile having the same area and maximum peak height (hp) as the peak considered and it is calculated for the same values of Nc, ν, D, (63) Gritti, F.; Guiochon, G. J. Chromatogr., A, submitted.

188

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

Ap σ√2π

(17)

In this study, the numerical calculations were carried out so ph is calculated from that Ap is equal to unity. The value of w0.1 hp as follows because w0.1 is equal to 4.29σ for a Gaussian profile.

ph w0.1 )

4.29 hp√2π

(18)

Figure 5a illustrates the influence of the column radial heterogeph for Nc ) 2.0 × 104, ν ) 10, D ) 0.1, neity on the ratio w0.1/w0.1 ph is unity γ ) 0.6, and n ) 4. As predicted, the value of w0.1/w0.1 at uw/uc ) 1.0 and hw/hc ) 1.0. However, it varies significantly depending on the exact combination of the values of uw/uc and hw/hc, although the variation does not exceed a few percent. ph and n for Figure 5b illustrates the correlation between w0.1/w0.1 five different combinations of uw/uc and hw/hc. The value of ph increases with increasing n. However, when the peak w0.1/w0.1 ph is almost equal to asymmetry is low (open circle), w0.1/w0.1 unity, irrespective of n. On the other hand, when the column radial heterogeneity of the flow rate and the local efficiency ph varies more signifiare important (open diamond), w0.1/w0.1 ph cantly. For example, w0.1/w0.1 varies by more than ∼15% for uw/uc ) 0.97 and hw/hc ) 3.0. The results in Figure 5b suggest that the order of the polynomial function representing the column radial distributions of u and h can be estimated by analyzing the ph . value of w0.1/w0.1 Numerical Approach to Estimate the Best Radial Distributions of the Flow Velocity and the Local Efficiency in a Radially Heterogeneous Column. Figures 4a-e and 5b indicate that the different types of peak tailing profiles can be characterized by a combination of three parameters, w0.5/w0.1, (wR/wF)0.1, and ph , which can easily be derived from the recorded peak w0.1/w0.1 profiles. It should be possible to obtain information on the degree of radial heterogeneity of the flow velocity (uw/uc), the local column efficiency (hw/hc), and the order of polynomial function representing their radial distribution (n) by analyzing these three parameters of the experimental tailing profiles. Our results suggest the following approach for the numerical estimation of the values of uw/uc, hw/hc, and n from recorded tailing peak profiles. A similar approach was already proposed.26 It is modified in order to estimate not only uw/uc and hw/hc but also n. (1) First, an arbitrary value of n is assumed. An initial value of 4 seems reasonable because it is close to the average of experimental results and a fourth-order polynomial functions is most consistent with the column radial distributions of the flow velocity and the local column efficiency experimentally measured.7-9,13,18-20 However, the best value of n depends on the experimental conditions, which is why repetitive (trial and error) calculations as described below are required in order to determine the best value of n.

Figure 4. Network maps representing the correlation between w0.5/w0.1 and (wR/wF)0.1 at Nc ) 2.0 × 104. The radial heterogeneity of u and h is represented by (a) second-, (b) third-, (c) fourth-, (d) fifth-, and (e) sixth-order polynomial functions. The plots indicate the results calculated for various combinations of the values of uw/uc and hw/hc. The dotted lines represent the interpolation between the neighboring data points. Solid symbols indicate the sets of values of w0.5/w0.1 and (wR/wF)0.1 at uw/uc ) 0.985 and hw/hc ) 1.5 and uw/uc ) 0.97 and hw/hc ) 3.0. The symbols of solid circle, triangle, square, pentagon, and hexagon show the values calculated by assuming that the radial heterogeneity of u and h is represented by second-, third-, fourth-, fifth-, and sixth-order polynomial functions, respectively. The dashed line indicates the correlation between w0.5/w0.1 and (wR/wF)0.1 calculated by the exponentially modified Gaussian (EMG) model.

(2) Then, the original value of the apparent whole column efficiency (N) is derived from a recorded elution peak profile. A hypothetical value of Nc is assumed from N. As shown in Figure

3b, the value of Nc is usually several times larger than that of N. However, the true ratio of Nc to N depends on the chromatographic conditions. So, it is required to repeat the Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

189

Figure 5. Correlation between the peak width discrepancy ph (w0.1/w0.1 ) and the column radial heterogeneity, i.e., (a) uw/uc ) 0.97-1.0, hw/hc ) 1.0-5.0, and n ) 4; (b) uw/uc ) 0.97-0.995, hw/hc ) 1.2-3.0, and n ) 2-6.

calculations described below for different values of Nc to determine its true value. (3) A network map between w0.5/w0.1 and (wR/wF)0.1 like those in Figure 4a-e is prepared from peak profiles numerically calculated by changing the values of uw/uc and hw/hc for the hypothetical value of Nc. (4) Similar to Figure 3b, a graph representing the influence of the column radial heterogeneity (uw/uc and hw/hc) on N is prepared from the profiles calculated numerically for the hypothetical value of Nc. The value of N is derived from either µ1 and µ2′ or tR and w0.5 of the calculated peak profiles. (5) The values of w0.5/w0.1 and (wR/wF)0.1 are measured from recorded elution peaks. The values of uw/uc and hw/hc corresponding to the set of measured values of w0.5/w0.1 and (wR/ wF)0.1 are read from the network map prepared in step (3). This means that the tailing peak profile, the characteristics of which are represented by w0.5/w0.1 and (wR/wF)0.1, is reproduced by the numerical calculation at uw/uc, hw/hc, and the hypothetical value of Nc. (6) In order to prove the validity of the estimated values of uw/uc, hw/hc, and Nc, the value of N calculated from the simulated tailing peak at the values of uw/uc, hw/hc, and Nc is compared with the original experimental value of N described 190

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

in step (2). Figure 3b indicates that, due to the column radial heterogeneity of uw/uc and hw/hc, N is much smaller than Nc. Also, N is a composite parameter, calculated from both the parameters relating to the retention equilibrium (tR or µ1) and the mass transfer kinetics (w0.5 or µ2′). This is why a graph similar to Figure 3b prepared in step (4) is used to prove the validity of the estimated values of uw/uc, hw/hc, and Nc at this step. If the values of uw/uc, hw/hc, and Nc estimated above are valid, the two values of N should agree with each other. If they are not sufficiently close, the hypothetical value of Nc assumed in step (2) is incorrect. Then, the hypothetical value of Nc must be adjusted and the whole process from steps (3) to (6) repeated until the N value in step (6) agrees with the experimental value of N in step (2). (7) The set of values of uw/uc, hw/hc, and Nc obtained was derived for n ) 4. However, it may be possible that a different order of polynomial functions accounts for the column radial heterogeneity. Also, in some cases, an appropriate value of Nc cannot be determined in step (6) because the network map prepared in step (3) cannot cover the characteristic point representing the set of experimental values of w0.5/w0.1 and (wR/ wF)0.1. Then, the n value must be changed. As illustrated in Figure 4a-e, the direction of the change of n (increase or decrease) can be predicted by considering the position in the network map of the characteristic point corresponding to the experimental values of w0.5/w0.1 and (wR/wF)0.1. The whole process from steps (1) to (6) is repeated for a different value of n. (8) Finally, several sets of appropriate values of uw/uc, hw/hc, and Nc corresponding to different values of n may be derived because the experimental profile, the characteristics of which are represented by w0.5/w0.1 and (wR/wF)0.1, is reproduced by numerical calculations made for several sets of values of uw/ uc, hw/hc, Nc, and n. To determine the most appropriate set of ph uw/uc, hw/hc, Nc, and n, the ratio w0.1/w0.1 is calculated for the experimental peak and the profiles calculated for these different ph sets. As illustrated in Figure 5b, the ratio w0.1/w0.1 depends on the combination of uw/uc, hw/hc, and n. The most appropriate set of values of uw/uc, hw/hc, Nc, and n is the one for which ph the ratio w0.1/w0.1 of the calculated peak agrees best with that of the experimental peak. Concrete Examples of the Estimation of the Radial Heterogeneity and True Efficiency of the Column. Similar to our previous paper,26 tailing peak profiles reported in two published papers9,19 were analyzed by applying the numerical approach proposed in this study in order to prove the validity and effectiveness of the method for estimating the radial heterogeneity and true performance of columns. Although there are several papers that report the results of experimental measurements of column radial heterogeneity,7-9,11,13,18-20 we applied the numerical method to the two concrete examples9,19 because we need to analyze the profiles of three elution peaks, i.e., the elution peak for the whole column recorded with a bulk detector and the peaks at the column center and near the column wall detected with a local detector. We could not apply the numerical method to several other concrete examples. The three experimental peak profiles necessary for the data analysis are not reported.11,18 The elution peak profiles could not be accurately analyzed

because they overlapped and/or the baseline could not clearly be defined.20 Several parameters, such as tR, w0.5, w0.1, wF0.1, wR0.1, and hp, were graphically read from the tailing peak profiles in refs 9 and 19 because they are required for the numerical data analysis. Some items of additional information about the column radial heterogeneity, which cannot be obtained in the previous study,26 were derived from the tailing peak profiles.9,19 For example, it was tried to determine the order of polynomial functions, which represent the distributions of the mobile phase flow velocity (u) and the local column efficiency (h) in the radial direction of the columns. The results of the numerical analysis are explained in detail in the Supporting Information. In the following, the analytical results are briefly explained. First Example. Three chromatographic peaks reported in ref 19 were analyzed. As listed in Table S1 in the Supporting Information, the original values of tR, w0.5, w0.1, w0.5/w0.1, (wR/ wF)0.1, and N are, respectively, 298, 17.1, 31.7, 0.530, 1.42, and 1.7 × 103 for the elution peak for the whole column detected by the bulk detector. From these data, as listed in Table S2 in the Supporting Information, the two sets of values of uw/uc, hw/ hc, Nc, and n are estimated as follows: n ) 4, Nc ) 3.0 × 103, uw/uc ) 0.955, and hw/hc ) 2.5 and n ) 5, Nc ) 2.8 × 103, uw/uc ) 0.948, and hw/hc ) 1.4. The value of N was calculated as 1.7 × 103 for the two sets of parameters, which is in agreement with the original value of N. ph In addition, the value of w0.1/w0.1 of the experimental tailing peak is calculated as 1.03. On the other hand, as listed in Table S2 in the Supporting Information, w0.1/wph 0.1 is calculated as 1.02 at n ) 4 and 5. The most appropriate value of n cannot be ph definitively determined. However, the value of w0.1/w0.1 calculated for both the sets of parameters is reasonably close to the ph original value of w0.1/w0.1 . The order of polynomial functions representing the radial distributions of u and h was at least estimated as fourth or fifth. The validity of n was also checked on the basis of the results in Figure 4a of ref 19, which illustrates the radial distribution of u experimentally measured. On the whole, the radial distributions of u calculated by assuming n ) 4 and 5 are similar to those experimentally measured, suggesting that the value of n is around 4 or 5. Although the estimated value of Nc (2.8 × 103 or 3.0 × 103) is not completely in agreement with the experimental value of Nc (3.4 × 103) calculated from the peak profile detected at the column center, they are reasonably close to each other. The numerical approach proposed in this study can provide the information about Nc and n from only one tailing peak profile experimentally measured. At least, it was demonstrated that we can more accurately analyze experimental data of elution chromatography by using the numerical method because the estimated value of Nc (2.8 × 103 or 3.0 × 103) is closer to the experimental value of Nc (3.4 × 103) than to the original value of N (1.7 × 103). Second Example. The peak profiles reported in ref 9 were also analyzed in the same manner. As listed in Table S3 in the Supporting Information, the original values of tR, w0.5, w0.1, w0.5/ w0.1, (wR/wF)0.1, and N are, respectively, 133.6, 3.51, 6.6, 0.555, 1.34, and 8.0 × 103 for the elution peak for the whole column detected by the bulk detector. From these data, as listed in

Table S4 in the Supporting Information, the two sets of values of uw/uc, hw/hc, Nc, and n are estimated as follows: n ) 3, Nc ) 1.6 × 104, uw/uc ) 0.978, and hw/hc ) 1.8 and n ) 4, Nc ) 1.5 × 104, uw/uc ) 0.976, and hw/hc ) 1.1. The value of N was estimated as 7.9 × 103 for the two sets of parameters, which is in excellent agreement with the original value of N, i.e., 8.0 × 103. Although the estimated value of Nc (1.5 × 104 or 1.6 × 104) is not in agreement with the experimental value of Nc (9.6 × 103) calculated from the peak profile detected at the column center, this deviation seems to be attributed to the obvious tailing profile of the peak. As listed in Table S3 in the Supporting Information, in spite of this peak being detected at the column center, its asymmetry factor is large, i.e., (wR/wF)0.1 ) 1.4, suggesting that the experimental value of Nc is not accurate. ph In addition, the value of w0.1/w0.1 of the experimental tailing peak is calculated as 1.02. On the other hand, as listed in Table S4 in the Supporting Information, w0.1/wph 0.1 is calculated as 1.00 at n ) 3 and 4. Again, the most appropriate value of n cannot ph be definitively determined. However, the value of w0.1/w0.1 calculated for both the sets of parameters is reasonably close ph . The order of polynomial to the original value of w0.1/w0.1 functions representing the radial distributions of u and h was estimated as third or fourth in the case of the second example, although again it was not definitively determined at this time. However, the results at n ) 4 seem to be inappropriate because the estimated values of the radial heterogeneity are uw/uc ) 0.976 and hw/hc ) 1.1. This combination of values of uw/uc and hw/hc is somewhat unrealistic because they assume that the radial heterogeneity of the column bed affects only the distribution of the flow velocity while the value of hw/hc ) 1.1 means that the local column efficiency is almost constant, irrespective of the radial position in the cross-section of the column. Yet, the radial heterogeneity of both the flow velocity and the local column efficiency must originate from the same cause, a radial heterogeneity of the packing structure. Consequently, radial variations of both the flow velocity and the local efficiency must simultaneously take place because they have the same cause. This result seems to imply that the radial distributions of u and h should be represented by third-order polynomial functions. Similar to the first example, the validity of n was checked on the basis of the results in Figure 4A of ref 9, which illustrates the distributions of u and h experimentally measured. The radial distributions of u and h calculated for n ) 3 are in good agreement with those experimentally measured. This means that the radial distributions of u and h are probably represented by third-order polynomial functions. It was also demonstrated in the second example that the numerical method can provide appropriate estimates of Nc and n from a single tailing peak profile experimentally measured. The numerical method is effective for the accurate analysis of experimental data of elution chromatography because the value of Nc can appropriately be derived even from a tailing peak profile. The results presented in this section demonstrate the validity of the analysis procedure of tailing peak profiles proposed in this study. Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

191

CONCLUSIONS The results obtained in this study demonstrate that if most peaks recorded in column chromatography often exhibit a moderate degree of tailing, this phenomenon is explained by the radial heterogeneity of the beds. A detailed analysis of this phenomenon shows how it is possible to derive accurate estimates of the true column efficiency or efficiency of the peaks eluted at the column center, of the degree of radial column heterogeneity or ratios of the local HETP, of the flow velocity near the wall to those at the column center, and of the best profiles for the radial distributions of the local column efficiency and the mobile phase velocity. The method proposed was validated by comparing the data derived with this method from the analysis of recorded chromatograms and the few true values reported in the literature. The agreement between calculated and measured column properties validate our method. This method will be useful to investigate column performance, to support the development of new methods allowing better column packing and providing more radially homogeneous beds, and for accurate measurements of the physicochemical parameters necessary in the study of mass transfer kinetics, which are conveniently derived from pulse response peaks. Systematic errors due to the tailing profiles often exhibited by these peaks could be largely reduced.

Dr(0)

ACKNOWLEDGMENT This work was supported in part by a Grant-in-Aids for Scientific Research (Grant No. 21350040) from the Ministry of Education, Science and Culture of Japan.

Greeks

GLOSSARY Symbols

ah au A Ap bh bu B C Cd Cm dc D D(r) Dm Dr Dr(r)

192

coefficient in eq 2 coefficient in eq 1 (cm s-1) coefficient in eq 3 (cm) area of sample pulse injected (s mol dm-3) coefficient in eq 2 coefficient in eq 1 (cm s-1) coefficient in eq 3 (cm2 s-1) concentration (mol dm-3), coefficient in eq 3 (s) dimensionless concentration coefficient in eq 3 inner diameter of column (cm) coefficient accounting for the contribution of eddy dispersion to transverse dispersion coefficient D at the radial position r molecular diffusivity (cm2 s-1) radial dispersion coefficient (cm2 s-1) radial dispersion coefficient at the radial position r (cm2 s-1)

Analytical Chemistry, Vol. 83, No. 1, January 1, 2011

h hr hp h(r) H Hr k n N r R td tG tR u uav us u(r) w wF wR γ µ1 µ2′ ν ξ σ σd σG

radial dispersion coefficient at the column center (cm2 s-1) reduced plate height reduced radial plate height peak height (mol dm-3) reduced plate height at the radial position r height equivalent to a theoretical plate (µm) radial plate height (µm) retention factor order of polynomial function number of theoretical plates radial distance from the center of column (mm) column radius (mm) dimensionless time retention time of parent Gaussian peak (s) retention time (s) interstitial velocity of mobile phase solvent (cm s-1) average mobile phase flow velocity (cm s-1) propagation velocity of a retained sample compound () u/[1+k]) (cm s-1) linear velocity at a radial position r (cm s-1) peak width (s) front half-width of elution peak (s) rear half-width of elution peak (s)

tortuosity coefficient first absolute moment (s) second central moment (s2) reduced interstitial velocity ratio of Dr(r) to Dr(0) standard deviation (s) dimensionless standard deviation standard deviation of parent Gaussian peak (s)

Subscripts

c w 0.1 0.5

at the center of column near the wall of column at 10% of maximum peak height at 50% of maximum peak height

Superscript

ph

calculated from the peak height

SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review August 20, 2010. Accepted November 12, 2010. AC102195X