Anal. Chem. 1998, 70, 173-181
Prediction of Electrophoretic Mobilities. 1. Monoamines Shilin Fu and Charles A. Lucy*
Department of Chemistry, The University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
The mobility of an ion is of fundamental importance in capillary electrophoresis. The size, shape, and other physicochemical parameters of monoamines are determined using molecular modeling. These parameters are used to generate regression expressions to predict absolute (infinite dilution) mobilities. Molecular volume or mass is the strongest determinant of electrophoretic mobility. However, molecular volumes calculated via molecular modeling varied systematically depending on the software used, and so molecular mass is the favored descriptor. Neither the classical spherical (Hu1 ckel) nor ellipsoidal (Perrin) models were reasonable predictors of mobility. In accord with empirical expressions, such as the Wilke-Chang equation for diffusion, the absolute mobilities correlate with mass (or volume) to a much greater power than predicted by Stokes’s law. Incorporation of the effects of hydration using the McGowan waters of hydration increments further improved the predictions. The best equation for predicting absolute mobilities of monoamines is µ0 ) [(5.55 ( 0.73) × 10-3]/[W(0.579(0.026) + (0.171 ( 0.054)H] where W is the molecular weight and H is the mean waters of hydration calculated using the McGowan increments. The uncertainties are the standard deviations of the parameters. This equation yielded an average prediction error of 4.1% for the data set used to generate the expression (literature absolute mobilities for 34 monoamines possessing no other functional groups), 7.2% for an independent data set from the literature (absolute mobilities for seven monoamines possessing other functional groups), and 3.3% for an experimentally determined data set (13 monoamines determined using capillary electrophoresis). Capillary electrophoresis (CE) has become an important separation technique in analytical chemistry. This technique has been used to separate analytes ranging from small inorganic and organic ions to macromolecular species such as DNA and proteins. Capillary zone electrophoresis (CZE) separates ions on the basis of their mobilities (µ):
µ ) q/f
(1)
where q is the electrical charge on the molecule and f is the frictional coefficient. * Author to whom correspondence should be addressed. Facsimile: 403-2899488. Internet:
[email protected]. S0003-2700(97)00663-X CCC: $14.00 Published on Web 01/01/1998
© 1997 American Chemical Society
Most previous studies of electrophoretic mobility have dealt with the influence of secondary equilibria on the effective or observed solute mobility (µobs). For instance, the observed mobility of a weak base is1
µobs )
[H+]
µB+ [H ] + Ka
(2)
+
where Ka is the acid dissociation constant and µB+ is the mobility of the protonated form of the base. Alternatively, observed mobilities can be altered by complex formation. For instance, in cyclodextrin-mediated CZE, the observed mobility is given by2
µobs ) RB+µB+ + RBCD+µBCD+
(3)
where RB+ and RBCD+ are the concentration fractions of the molecule in the free and complexed forms, and µB+ and µBCD+ are the mobilities of the free and complexed solute. In both cases, the intrinsic solute mobility (µB+) is required to model the behavior of a solute in CZE. Knowledge of the intrinsic mobility of a solute would aid in optimization and peak identification. Currently, this mobility can only be determined experimentally. This limits our ability to a priori predict whether CZE is appropriate for a given application or to identify an unknown peak within the electropherogram. Our work seeks to develop empirical expressions for the prediction of electrophoretic mobilities and to determine the underlying factors governing these mobilities. Herein, molecular modeling is used to determine the size, shape, and other physicochemical parameters of monoamines for which literature mobilities are available. Regression expressions are then generated and tested using experimentally determined mobilities. EXPERIMENTAL SECTION Molecular Modeling. Modeling was performed using Molecular Modeling Pro (version 1.44, WindowChem Software, Fairfield, CA) on a 75-MHz Pentium microcomputer. Molecules were drawn in the “extended form”,3 and then the strain energy of the molecule was minimized. An initial minimization step was performed on the molecule in the absence of hydrogen atoms to reduce overall computation time. Overall computations took from (1) Smith, S. C.; Khaledi, M. G. Anal. Chem. 1993, 65, 193. (2) Rawjee, Y. Y.; Williams, R. L.; Vigh, Gy. J. Chromatogr. A 1993, 652, 233. (3) Edward, J. T.; Waldron-Edward, D. J. Chromatogr. 1965, 20, 563.
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998 173
5 min for ammonium (the smallest molecule investigated) to about 2 h for octadecyltributylammonium (the largest molecule studied). The modeling program calculated the van der Waals volumes (in Å3) by excluding the overlap volumes from the total volume obtained by summing volumes for the individual atoms as determined using the van der Waals increments of Bondi.4 The molecular volumes generated via Molecular Modeling Pro were manually validated versus the Bondi increments. For nine test molecules, the agreement between the manual5 and computer volumes was -0.6% ( 4.0%. The surface area was calculated using the same elementary geometry principles of the surface area of the atomic spheres of van der Waals radius minus the overlap. The semiaxes of the molecule were not automatically provided by the modeling program. The program does indicate the molecular length, width, and depth of the molecule with respect to its orientation on the screen. Therefore, the molecules were manually rotated on the screen to determine the length of the semiaxes. If any one (2a) of the three dimensions was significantly larger than the other two (2b, 2c), then the molecule was treated as prolate; alternately, if any two (2b, 2c) axes were obviously larger than the third dimension (2a), the oblate approximation was used. For axes of similar length (i.e., b and c), the geometric mean was used as b in eqs 8 and 9. The hydration number (the number of water molecules bound to the molecule in solution) was calculated using Molecular Modeling Pro based on McGowan’s fragment addition method.6 In this method, specific hydration numbers are assigned to molecular fragments (see Table 3) and are summed to give the mean waters of hydration of the molecule. Numerical Fitting of Data. The optimum constants used in the expressions were obtained using both the Solver function of Microsoft Excel (version 5.0) and the Curve Fitter function of SlideWrite Plus (version 2.0 for Windows, Advanced Graphics Software, Inc., Carlsbad, CA). Solver iteratively alters the parameters to minimize the sum of the squares of the residuals. Curve Fitter uses the iterative Levenberg-Marquardt algorithm, which yields parameters based on the minimization of the sum of the squared deviations. Both methods yielded the same fit constants. The uncertainties quoted in eqs 17-19 are the standard deviations of the fit constants determined using Curve Fitter. Such uncertainties could not be determined for eqs 21, as Curve Fitter was limited to two variables. Apparatus. The mobilities of 13 monoamines were measured using a Crystal 300 capillary electrophoresis system with a Crystal 1000 CE conductivity detector (Thermo CE, Boston, MA). A fused silica capillary of 50 µm i.d. × 52 cm (ConCap I, Thermo) and a ConTip I conductivity sensor (Thermo) were used. Before each run, the capillary was flushed with 0.1 M LiOH for 2 min at 2 bar and then with deionized water for 1 min and finally with buffer (lithium acetate/acetic acid at pH 4.75) for 2 min. Injections were hydrodynamic (20 mbar) for 0.2 min. The separation voltage was 6 kV, which was experimentally determined to be within the linear portion of the Ohm’s plot. A CHROM-1AT data acquisition board (Keithley MetraByte, Taunton, MA) and Lab Calc software (Galactic, Salem, NH) were used for data acquisition and analysis. (4) Bondi, A. J. Phys. Chem. 1964, 68, 441. (5) Edward, J. T. J. Chem. Educ. 1970, 47, 261. (6) McGowan, J. C. Tenside Surf. 1990, 27, 229.
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Chemicals. All reagents and chemicals were from Aldrich (Milwaukee, WI), BDH Inc. (Toronto, ON, Canada) or CHEM SERVICE Inc. (West Chester, PA). These were of analytical grade or higher and were used without any further purification. All solutions were prepared with distilled deionized water (18 MΩ) (NANOpure, Barnstead, NY) and filtered (0.45 µm, MilliCup, Millipore Corp., Bedford, MA) before use. The pH of the background buffer solution was adjusted to 4.75 to ensure that all amines were fully protonated. Buffers with ionic strengths from 0.100 to 0.001 02 M were used as the electrolyte. All sample concentrations were 10-4 M in analyte and adjusted to pH 4.75 with 1 M acetic acid. Determination of Absolute Mobilities. The effective mobility of an analyte was calculated using
µobs )
(
)
L2 1 1 V tm teof
(4)
where tm is the migration time of analyte (in seconds), and teof is the migration time of electroosmotic flow (in seconds). The effective mobilities at different ionic strengths were then plotted versus the square root of corresponding ionic strength,7-9,11 and linear least-squares regression was performed to extrapolate the mobility to zero ionic strength (infinite dilution, µ0). RESULTS AND DISCUSSION Numerous solution conditions, such as ionic strength, counterion interaction, and viscosity, affect the observed mobility. Further, the effect of these factors can be complex.10,11 In this initial work, the complexities of ionic strength are circumvented by working with the absolute mobility, µ0 (infinite dilution), of the ion. Absolute mobilities are a constant characteristic of the ion. Typically, absolute mobilities (µ0) are measured experimentally either by extrapolating the mobilities observed over a range of ionic strength to infinite dilution or by measuring their limiting equivalent conductance (λ0).12 The equivalent conductance can be converted to absolute mobility by13
µ0 ) λ0/F
(5)
where F is the Faraday constant. Literature absolute mobilities for a number of monoamines were compiled from numerous sources14-18,21 and are presented (7) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry, Vol. 1; Plenum Press: New York, 1970; Section 4.6.1. (8) Reijenga, J. C.; Kenndler, E. J. Chromatogr. A 1994, 659, 403. (9) Issaq, H.; Atamna, I.; Muschik, G.; Janini, G. Chromatographia 1991, 32, 155. (10) Compton, B. J.; O’Grady, E. A. Anal. Chem. 1991, 63, 2597. (11) Friedl, W.; Reijenga, J. C.; Kenndler, E. J. Chromatogr. A 1995, 709, 163. (12) Adamson, A. W. Physical Chemistry, 2nd ed.; Academic Press: New York, 1979; pp 442-449. (13) Jandik, P.; Bonn, G. Capillary Electrophoresis of Small Molecules and Ions; VCH: New York, 1993; p 15. (14) Handbook of Chemistry and Physics,1st Student ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1991; p D-106. (15) Moor, T. S.; Winmill, T. F. J. Chem. Soc. 1912, 101, 1635 (16) Hirokawa, T.; Kiso, Y. J. Chromatogr. 1982, 252, 33. (17) Kuhn, D. W.; Kraus, C. A. J. Am. Chem. Soc. 1950, 72, 3676. (18) Beckers, J. L.; Everaerts, F. M. J. Chromatogr. 1989, 470, 277. (19) Hiemenz, P. C. Principles of colloid and surface chemistry, 2nd ed.; Marcel Dekker: New York, 1986; p 747.
Table 1. Physicochemical Parameters and Electrophoretic Mobilities of Monovalent Amines Possessing No Other Functional Groups molecules ammonium methylammonium dimethylammonium ethylammonium trimethylammonium tetramethylammonium isobutylammonium propylammonium diethylammonium piperidinium pentylammonium ethyltrimethylammonium dipropylammonium propyltrimethylammonium triethylammonium butyltrimethylammonium tetraethylammonium tripropylammonium hexyltrimethylammonium benzyltrimethylammonium octyltrimethylammonium tetra-n-propylammonium n-dodecylammonium decyltrimethylammonium dodecyltrimethylammonium tetra-n-butylammonium tetradecyltrimethylammonium hexadecyltrimethylammonium tetraisopentylammonium tetra-n-pentylammonium octadecyltrimethylammonium octadecyltriethylammonium octadecyltripropylammonium octadecyltributylammonium a
MW (W) mol volume (V), Å3 hydr no. (H) kappa 2 (K) 18.04 32.06 46.09 46.09 60.12 74.15 74.15 74.15 74.15 86.16 88.17 88.17 102.20 102.20 102.20 116.23 130.25 144.28 144.28 150.24 172.33 186.36 186.36 200.39 228.44 242.47 256.49 284.55 298.57 298.57 312.60 354.68 396.76 438.84
23.28 42.02 61.3 58.4 80.4 99.7 91.6 91.3 94.1 99.3 107.6 116.1 126.9 132.4 129.6 149.3 163.8 177.3 182.2 171.9 215.3 232.4 222.5 247.6 281.0 297.9 314.5 347.8 362.2 363.5 378.2 428.7 479.6 528.0
12 9 6 9 3 3 9 9 6 6 9 3 6 3 3 3 3 3 3 3 3 3 9 3 3 3 3 3 3 3 3 3 3 3
0.00 0.00 2.00 2.00 1.33 1.00 2.25 4.00 4.00 2.22 5.00 1.63 6.00 2.34 4.17 3.11 3.92 7.11 4.76 3.60 6.51 7.41 12.00 8.32 10.17 11.11 12.06 13.96 10.68 14.92 15.88 18.78 21.70 24.64
f/f0 1.002 1.003 1.016 1.016 1.006 1.004 1.011 1.039 1.047 1.005 1.058 1.006 1.078 1.016 1.018 1.027 1.007 1.038 1.053 1.021 1.084 1.013 1.177 1.114 1.143 1.016 1.176 1.203 1.014 1.022 1.208 1.195 1.142 1.107
shape factor (S) µ0 (lit.), cm2/(V‚s) × 104 1.0 2.0 2.0 3.0 3.0 4.0 5.0 4.0 2.0 3.0 6.0 6.0 2.0 8.0 3.0 10.0 4.0 3.0 14.0 16.0 18.0 4.0 13.0 11.0 26.0 4.0 30.0 34.0 3.3 4.0 38.0 25.3 19.0 15.2
7.62a 6.20,b 6.08c 5.46,b 5.37c 4.85a 5.03,b 4.89c 4.43,d 4.65c 3.94c 4.30,a 4.23c 3.89a 3.86c 3.83c 4.20,c 4.27a 3.31,b 3.12c 3.80a 3.52a 3.48,c 3.44a 3.39,a 3.38c 2.79,b 2.71c 3.07c 3.59c 2.75c 2.43c 2.47a 2.53c 2.34c 2.02c 2.23c 2.17c 1.86c 1.81c 2.08,e 2.06c 1.86c 1.78c 1.72c
Data obtained from ref 21. b Data obtained from ref 15. c Data obtained from ref 14. d Data obtained from ref 18. e Data obtained from ref 17.
in Table 1. These data are restricted to monoamines possessing no other functional groups. The data in Table 1 were used for development of the empirical expressions for electrophoretic mobility. Literature absolute mobilities for monoamines possessing other functional groups (Table 4) were subsequently used for evaluation of these expressions (see below). For a number of the solutes in Table 1, literature absolute mobility values were obtained from two different sources. On the basis of these duplicates, it is estimated that the uncertainty in the literature values for the monoamines is 2.4%. If literature values for carboxylates and sulfonates (not given) are included, the uncertainty of the literature values is 3.8%. Table 1 also presents the physicochemical parameters calculated for these molecules as described in the Experimental Section. Both fundamental and empirical relations have been developed to relate the electrophoretic mobility to physical parameters. Compton and O’Grady provide an excellent discussion of many of these expressions with an emphasis on the effect of ionic strength on charge suppression of proteins.10 Our discussion will focus on the factors affecting the frictional coefficient (f in eq 1) for small organic molecules. A summary of the major electro(20) Grossman, P. D.; Colburn, J. C.; Lauer, H. H. Anal. Biochem. 1989, 179, 28. (21) Edward, J. T. In Advances in Chromatography; Giddings, J. C., Keller, R. A., Eds.; Marcel Dekker: New York, 1966; Vol. 2, pp 63-98.
phoretic mobility equations developed previously and derived herein is given in Table 2. The area of application of each expression, along with its underlying assumptions, is also included in this table. Hu1 ckel Equation. The most commonly quoted expression for electrophoretic mobility is the Hu¨ckel equation,19,20
µ ) Ze/6πηrs
(6)
where Z is the charge on the molecule, e is the charge of an electron, η is the viscosity of the solution, and rs is the Stokes’s radius. The denominator is the frictional coefficient as predicted by Stokes’s law for moving a hypothetical spherical molecule of radius rs at uniform velocity through a continuum of viscosity η. The Stokes’s radius can only be determined experimentally on the basis of diffusion, sedimentation, or electrophoretic mobility. Figure 1 shows the absolute mobilities, calculated using the Hu¨ckel equation (eq 6), plotted versus the literature mobilities listed in Table 1. The Stokes’s radius used in eq 6 was obtained by treating the molecule as a sphere (rs ) [V/(4/3π)]1/3, where V is the van der Waals volume of the molecule, as calculated using molecular modeling). The line in Figure 1 indicates the ideal correlation (i.e., if the calculated mobilities precisely matched the literature mobilities). Clearly, the mobilities determined using the Hu¨ckel equation (eq 6) deviate significantly from the literature Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
175
Table 2. Summary of Electrophoretic Mobility Relationships equation
application
assumption
average relative error (%) molecule19,20
eq 6
general expression for zero ionic strength
derived from Stoke’s law using a spherical
eq 7
small molecules
molecule approximated as ellipsoid23,24
eq 11
both hydrodynamic and dielectric frictions considered27-29
eq 13
ions with a symmetrical charge distribution moving in a continuum media peptides in paper electrophoresis and CZE32
empirically based31
eq 14
small ions in paper electrophoresis
empirically based37
eq 19a
monoamines at zero ionic strength
derived herein
µ)
13.9 (34 amines) 10.0 (34 amines) 12.9 (34 amines) 15.7 (34 amines) 8.6 (47 ions) 4.1 (34 amines)
Ze 6πηrs(f/f0)
(7)
The ellipsoid is defined by three semiaxes, a, b, and c. The semiaxes are the axes of rotation, at 90° angles to one another, going through the center of mass of the molecule. Molecules may be modeled as a prolate or an oblate ellipsoid. Prolate refers to a cigar-shaped ellipsoid. More rigorously, the semiaxes of a prolate ellipsoid have a > b ) c, where b/a < 1. For a prolate molecule, the frictional correction is given by25
f ) f0 Figure 1. Correlation of mobilities predicted using the Hu¨ ckel equation (eq 6) with the absolute mobilities for the 34 monoamines in Table 1. The line is the expected behavior for perfect correlation between the literature and predicted values.
[1 - (b/a)2]1/2 (b/a)
2/3
{
}
1 + [1 - (b/a)2]1/2 ln (b/a)
(8)
An oblate molecule is discus-shaped (b ) c > a; b/a >1). Under these conditions, the frictional correction is given by25
values. The average relative error using this expression is 13.9%. Edward5,21 has suggested that the numerical constant in the denominator of eq 6 should be other than 6. Indeed, the 6 in the demoninator of eq 6 arises solely from the assumption of “stick” between the moving sphere and the surrounding continuum solvent. It has alternatively been proposed that it would be more appropriate to assume that the solvent “slips” past the surface of the solute sphere, in which case the constant should be 4.22 Allowing this constant to be an adjustable parameter within our regression resulted in a value of 5.00, in agreement with Edward, but did not significantly alter the correlation. Ellipsoidal Approximation. The use of an ellipsoidal rather than a spherical approximation would provide more flexibility in modeling the shape of the molecule. In the 1930s, Perrin derived expressions for calculating the frictional resistance to moving an ellipsoid through a continuum medium.23,24 Generally, the frictional resistance of the ellipsoid is expressed relative to the resistance of moving a sphere of the same volume (i.e., f/f0, where f is the frictional coefficient of the ellipsoid and f0 is the frictional coefficient for a sphere of the same volume). Thus, Perrin’s equation has the form
Graphical representations21 and tabulations26 of the relationship between the axial ratio b/a and the frictional correction term have been given in the literature. Figure 2 shows the relationship between mobilities predicted using Perrin’s ellipsoidal model and the literature absolute mobilities. Perrin’s ellipsoidal model provides a better approximation of the mobilities than the Hu¨ckel model. The average relative error is 10.0%. Interestingly, in the low-mobility region (∼2 × 10-4 cm2/(V‚s)) the mobilities predicted by Perrin’s model are more scattered than those observed using the Hu¨ckel model. Inspection of these points indicates that the deviations are related to the shape of the molecules. The upper points are symmetrical quaternary amines, such as tetra-n-propylammonium and tetra-nbutylammonium. The lower points are amines with long main chains and relatively short side chains, such as octadecyltrimethylammonium and tetradecyltrimethylammonium. Thus, not only does Perrin’s model not provide precise approximations of
(22) Cussler, E. L. Diffusion: mass transfer in fluid systems; Cambridge University Press: New York, 1984; Section 5.2. (23) Perrin, F. J. Phys. Radium 1934, 5, 497. (24) Perrin, F. J. Phys. Radium 1936, 7, 1.
(25) Eisenberg, D.; Crothers, D. Physical Chemistry with Applications to the Life Sciences; Benjamin/Cummings: Menlo Park, CA, 1979; pp 718-724. (26) Edsall, J. T.; Mehl J. W. In Proteins, Amino Acids and Peptides as Ions and Dipolar Ions; Cohn, E. J., Edsall, J. T., Eds.; Reinhold: New York, 1943; p 405.
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[(b/a)2 - 1]1/2 f ) f0 (b/a)2/3 tan-1[(b/a)2 - 1]1/2
(9)
and high-frequency dielectric constants of the medium, respectively. Substituting literature values30 for these parameters (in centimeter-gram-second units) for water at 25 °C yields
µ0 ) 0.168ri +
Figure 2. Correlation of mobilities predicted with Perrin’s ellipsoidal model (eq 7) with the absolute mobilities for the 34 amines in Table 1. The line is the expected behavior for perfect correlation between the literature and predicted values.
the mobility, but it also induces a shape-dependent deviation not previously noted with the Hu¨ckel equation. Dielectric Friction. It has previously been noted that the agreement between the Hu¨ckel equation and the behavior of small (
