Anal. Chem. 2001, 73, 3051-3058
Chromatographic Probing of Electrostatic Potential Tetsuo Okada* and Yutaka Sugaya
Department of Chemistry, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
The electrostatic potential of a charged surface and its vicinity has received increasing attention because it plays important roles in biological systems, behaviors of colloids, electrochemical processes, separations, sensors, etc. Various theories and experimental approaches have been employed to probe the electrostatic potential and to explain surface and interfacial phenomena.1-11 The Gouy-Chapman theory has, for example, provided successful
interpretations of various features and behaviors of ionic solutes existing in the vicinity of a charged surface. In colloid chemistry, the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory is extensively employed to explain the surface forces and coagulation of particles.1 These theories, which are derived from the PoissonBoltzmann equation, predict the surface potential as well as the spatial change in electrostatic potential. A number of effective experimental approaches have been advanced to evaluate the surface potential, for example, on the basis of electrokinetic phenomena, such as electrophoresis, streaming potential, and electroosmosis.12-19 These electrokinetic phenomena are characterized by the ζ-potential, which is the electrostatic potential at a shear plane, but are often recognized as equal to the surface potential without relevant local information. A disadvantage of the electrokinetic approach is thus that it is not clear whether the shear plane is identical to the real surface. Another general approach to the electrostatic potential measurements is based on equilibrium shifts of an appropriate probe molecule as a result of electrostatic effects.3,20 Acid-base dyes or fluorescent molecules have been employed for this purpose. To properly employ this approach, one should pay special attention to where the probe molecules exist and how their equilibria are affected by the electrostatic potential. Thus, existing experimental approaches involve some ambiguity in probing the electrostatic potential. Electrostatic theories have successfully explained various separation phenomena and particularly made significant contributions to membrane science, capillary electrophoresis, electrochromatography etc.14-16,19,21,22 In contrast, such concepts have not been well taken into consideration in the advancement of usual chromatographic theories, even when electrostatic interactions comprise a major sector of the overall separation mechanism. Ståhlberg,23,24 Cantwell,25,26 and Ho¨ll and Horst27,28 have pointed
* Phone and Fax: +81-3-5734-2612. E-mail:
[email protected]. (1) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: 1985; Japanese version, McGraw Hill: Tokyo, 1991. (2) Fujihira, M.; Yanagisawa, M.; Kondo, T. Bull. Chem. Soc. Jpn. 1993, 66, 3600. (3) Cassidy, M. A.; Warr, G. G. J. Phys. Chem. 1996, 100, 3237. (4) Graham, I. S.; Cohen, J. A.; Zuckermann, M. J. J. Colloid Interface Sci. 1990, 135, 335. (5) Johnson, S. B.; Drummond, C. J.; Scales, P. J.; Nishimura, S. Langmuir 1995, 11, 2367. (6) Senden, T. J.; Drummond, C. J.; Ke´kicheff, P. Langmuir 1994, 10, 358. (7) Campbell, S. D.; Hillier, A. C. Langmuir 1999, 15, 891. (8) Hu, K.; Fan, F.-R. F.; Bard, A. J.; Hillier, A. C. J. Phys. Chem. B 1997, 101, 8298. (9) Cevc, G. Biochim. Biophys. Acta 1990, 1031-3, 311. (10) Lue, L.; Zoeller, N.; Blankschtein, D. Langmuir 1999, 15, 3726. (11) Moncelli, M. R.; Becucci, L.; Buoninsegni, F. T.; Guidelli, R. Biophys. J. 1998, 74, 2388.
(12) Dunstan, D. E.; White, L. R. J. Colloid Interface Sci. 1990, 134, 147. (13) Hely, T. W.; Drummond, C. J.; Grieser, F.; Murray, B. S. Langmuir 1990, 6, 506. (14) Cai, J.; Rassi, Z. E. J. Chromatogr. 1992, 608, 31. (15) Lee, C. S.; McManigill, D.; Wu, C.-T.; Patel, B. Anal. Chem. 1991, 63, 1519. (16) Hayes, M. A.; Kheterpal, I.; Ewing, A. G. Anal. Chem. 1993, 65, 27. (17) Yoon, R.-H.; Yordan, J. L. J. Colloid Interface Sci. 1986, 113, 430. (18) Morini, M. A.; Schulz, P. C. Colloid Polym. Sci. 1997, 275, 802. (19) Iso, K.; Okada, T. Langmuir 2000, 16, 9199. (20) Fernandez, M. S.; Fromherz, P. J. Phys. Chem. 1977, 81, 1755. (21) Rathore, A. S.; Horva´th, C. J. Chromatogr. A 1997, 781, 185. (22) Rathore, A. S.; Horva´th, C. Anal. Chem. 1998, 10, 3069. (23) Ståhlberg, J. Anal. Chem. 1994, 66, 440. (24) Ståhlberg, J. J. Chromatogr. A 1994, 668, 255. (25) Afrashtehfer, S.; Cantwell, F. F. Anal. Chem. 1982, 54, 2422. (26) Hux, R. A.; Cantwell, F. F. Anal. Chem. 1984, 56, 1258. (27) Ho ¨ll, W. H.; Horst, J.; Franzreb, M. In New Developments in Ion-Exchange; Abe, M., Kataoka, T., Suzuki, T., Eds.; Kodansha: Tokyo, 1991; p 277.
Electrostatic potential in the vicinity of the surface of a cation-exchange resin has been evaluated by modeling chromatographic retention. Binary mixtures of K+ and its crown ether complex in methanol are used as mobile phases, and two types of solutes, that is, cationic and crown ether probes, have been examined. The cationic probes show the sigmoidal retention changes with increasing concentration of a crown ether incorporated into the mobile phase, whereas crown ether probes give retention maximums. The model derived from the Poisson-Boltzmann theory well explains these specific changes in probe retention and gives the electrostatic potential at the closest approach of each probe molecule. The closest approaches for probe molecules correlate well with their molecular sizes. In addition, changes in retention of cationic probes also correlate well with the electrostatic potential changes at the closest approaches of probe molecules, indicating that simple sensing of the electrostatic potential is feasible using probe retention. The reduction of crown ether complexation occurs in the vicinity of the cation-exchange resin surface and causes the specific retention behaviors of crown ether probes in the mobile-phase systems composed of K+ and its complex with a modifier crown ether.
10.1021/ac010020z CCC: $20.00 Published on Web 05/17/2001
© 2001 American Chemical Society
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001 3051
out important roles of the electrostatic potential in the determination of chromatographic retention of ionic compounds and exploited chromatographic models based on the Gouy-Chapman or Gouy-Chapman-Stern theory. We have also developed electrostatic chromatographic models and elucidated several features involved in ion-exchange or its related separation modes.29-33 Thus, chromatography is also a useful probe for electrostatic phenomena taking place at charged surfaces and interfaces. One may criticize the use of chromatography for such studies, because chromatographic processes comprise several complex mechanisms, and a complete understanding is difficult. We believe that high-resolution chromatography would be a great advantage and the chromatographic approach can, therefore, continue to contribute to the understanding of electrostatic phenomena if the separation systems are appropriately designed and relevant theories can be advanced. We found that electrostatic phenomena can be highlighted in cation-exchange systems by using methanolic binary mobile phases. A chromatographic model developed in this paper can interpret probe retention, which specifically changes with a function of the binary mixture compositions. Analyses of probe retention on the basis of the developed model allow sensing of the electrostatic potential and its effects on equilibria. EXPERIMENTAL SECTION The chromatographic system was basically the same as that used in our previous work.29-33 The stationary phase was silica gel-based cation-exchange resin TSKgel IC-Cation-SW (packed in a 4.6-mm-i.d. × 50-mm PTFE column, particle size 5 µm). This resin was used as the K+ form. The ion-exchange capacity of the column was determined by determining the K+ concentration in an effluent using atomic absorption spectrometry after the complete replacement of K+ in the column by H+. The ionexchange capacity was 0.082 mmol in the entire column. The charge density of the resin surface was 0.066 C m-2 (the total surface area of the stationary phase was 120 m2). The separation column was immersed in water thermostated at 25 °C. Methanol was distilled after being refluxed with magnesium. Benzo- and dibenzocrown ethers [benzo-15-crown-5 (B15C5), benzo-18-crown-5 (B18C6), dibenzo-24-crown-8 (B24C8), and dibenzo-30-crown-10 (DB30C10)] were synthesized according to the literature,34 and 18-crown-6 (18C6) and 15-crown-5 (15C5) were purchased from Tokyo Kasei and used as received. Other reagents were of analytical grade. RESULTS The methanolic binary systems containing K+ and its 18C6 or 15C5 complex were used as mobile phases throughout this study. The ion-exchange mechanism is very complex and has several origins; electrostatic, hydrophobic, specific adsorption should usually be involved. In most ion-exchange research, aqueous solutions have been used in which various interactions simultaneously (28) Ho ¨ll, W. H.; Franzreb, M.; Horst, J. In Ion-Exchange and Solvent Extraction; Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York, 1993; Vol. 11, Chapter 3. (29) Okada, T. Anal. Chem. 1998, 70, 1692. (30) Okada, T. Anal. Chem. 2000, 72, 1307. (31) Okada, T. J. Phys. Chem. B 1997, 101, 7814. (32) Okada, T.; Patil, J. M. Langmuir 1998, 14, 6241. (33) Okada, T. Phys. Chem. Chem. Phys. 2000, 2, 3669. (34) Pedersen, C. J. J. Am. Chem. Soc. 1967, 89, 7017.
3052 Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
Figure 1. Changes in retention of cationic probes with the mobile phases containing K+ and 18C6. Solid curves were calculated using a model developed in this work, whereas broken curves show the results of a simplified model (see the text for more details).
occur. In contrast, the use of nonaqueous solutions allows the reduction of hydrophobic interaction and specific adsorption, but when their polarities are too low, it facilitates association between ion-exchange sites and counterions. MeOH has relatively high polarity, polarizability, and donor and acceptor abilities. For these reasons, MeOH was selected as the solvent in which the overall ion-exchange mechanism is more dominantly governed by electrostatic interaction than it is in water and most nonaqueous solvents. Cationic Probes. The affinity of crown ether complexes to the cation-exchange resin is much lower than that of K+. If the probe cations form no complexes with crown ethers, their retention should increase with increasing crown ether concentration in the mobile phase containing constant concentration of K+. Results obtained for three probe cations are shown in Figures 1 and 2, where ethylviolet (EV+), N-dodecylpyridinium ion (C12py+), and phenyltrimethylammonium ion (PTMA+) were selected as the probes. The structures and approximate molecular sizes of probe cations are illustrated in Figure 3. These molecules have rather
Figure 4. Changes in the retention of crown ether probes with the mobile phases containing K+ and 18C6. Solid curves were calculated using a model developed in this work (see the text for more details). Figure 2. Changes in the retention of cationic probes with the mobile phases containing K+ and 15C5. Solid curves were calculated using a model developed in this work (see the text for more details).
Figure 3. Structures and approximate molecular sizes of cationic probes and mobile phase components.
Figure 5. Changes in retention of crown ether probes with the mobile phases containing K+ and 15C5. Solid curves were calculated using a model developed in this work (see the text for more details).
bulky cationic groups that prevent the formation of complexes with crown ethers. The elution order reflects the approximate molecular sizes of the probes; the larger the molecular sizes, the weaker the affinity to the cation-exchange resin. As reported previously,33 retention changes induced by 18C6 addition are noteworthy (Figure 1). Probe retention is almost constant for a lower concentration of 18C6, steeply increases when the concentration of added 18C6 approaches to that of K+, and then reaches a plateau; its overall change is, thus, sigmoidal. In contrast, retention changes are rather gradual for the addition of 15C5 to the mobile phase (Figure 2), reflecting a difference in the complex formation ability between 18C6 and 15C5. Changes in retention
seem similar for all of the probe compounds tested except for their retention intensity. Crown Ether Probes. Figures 4 and 5 show the retention changes of probe crown ethers with increasing concentration of 18C6 and 15C5, respectively, in the mobile phase. Retention of probe crown ethers first increases and then decreases with increasing concentrations of the crown ethers that are added to the mobile phases. This trend is clearly seen for 18C6 because of its high complexation ability; all of the probe crown ethers show maxima at CCr ) CK (CCr and CK are the bulk concentrations of a crown ether and K+). In contrast, retention of B18C6 and DB30C10 simply increases with increasing concentration of Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
3053
approach and ion-association constant can be independent parameters, the Fuoss equation was employed to relate these to each other and to avoid too many adjustable parameters. Retention of a cationic solute should be given by a sum of the contributions from the surface ion-association and double-layer condensation. Although, in our previous paper,33 the latter effect was ignored for simplicity, it has been proven that this simplification causes the deviation of the theoretical retention from the experimental one. The contribution from the surface ion-association to probe retention can be written as
(Vr - V0)ia )
(
A
(
ΓtotalKP exp -
1 + KK[K+] exp -
)
)
FψP RT
(
)
FψK FψCrK + KCr-K[Cr-K+] exp RT RT (2)
Figure 6. Schematic representation of the present model.
15C5, and DB24C8 retention shows a small maximum. Differences in retention between probes reflect not only their molecular sizes but also their complexation ability. Additionally, it is obvious that the much stronger complexation ability of 18C6, as compared with that of 15C5, results in the drastic changes in probe retention. According to the literature,35 log K ) 3.3 for K+-15C5 complexation in MeOH, which is lower than log K ) 6 for K+-18C6 complexation by more than 2 orders of magnitude. THEORY An electrostatic model was developed to explain the above results and to understand electrostatic phenomena involved in the separation mechanism. A schematic representation for a model is given in Figure 6. The electrostatic potential in the system is established by the mobile phase components, that is, K+ and its crown ether complex, and is not disturbed by introducing small amounts of solutes into the system. K+ must have a smaller molecular size and shorter interaction distance than its crown ether complex. Two Stern layers can, thus, be assumed in the vicinity of the cation-exchange resin surface. As shown in the Appendix section, if appropriate surface ion-association constants are assumed for these two cationic components present in the mobile phase, we can calculate the spatial distribution of the electrostatic potential from the surface to the bulk solution according to the Stern-Gouy-Chapman theory. The ion-association constants (K) can be related to the interaction distance; in this work, the Fuoss equation was used for their estimation.36
K)
(
e2 4πNa3 exp 3 4π0akT
)
where Vr and V0 are the retention volume of a solute and the void volume of the column; A is the total stationary phase area; KK, KCrK, and KP are the ion-association constants of a metal ion, its crown ether complex, and a probe; ψK, ψCrK, and ψP are the electrostatic potential at the closest approaches for these three cations; and Γtotal is the total surface concentration of sulfonate groups. The contribution from the double-layer accumulation of a probe is given by
(Vr - V0)dl ) A
∫ {exp(- RT ) - 1} dx Fψx
∞
aP
(3)
where ap is the closest approach for a probe, and ψx is the electrostatic potential at x (the distance from the surface). The retention volume is equal to the sum of eqs 2 and 3. For crown ether probes, their complexation should be taken into account. Equation 2 can be modified for a crown ether probe as
(Vr - V0)ia )
(
ΓtotalKP exp -
)
FψP RT
‚ A FψK FψCrK 1 + KK[K+] exp + KCr-K[Cr-K+] exp RT RT 1 (4) 1 + Kcomp [K+] P
(
)
(
)
where Kcomp is the complexation constant of a probe. The P double-layer accumulation occurs only for crown ether complexes, not for neutral crown ethers; thus, the electrical double-layer accumulation for crown ether complexes is given by
(1) (Vr - V0)dl ) A
∫{ ∞
ap
(
exp -
) }
[K+] Fψx Kcomp P - 1 dx RT 1 + Kcomp [K+] P (5)
where a is the interaction distance, and e, k, T, N, 0, and are the elementary charge, the Boltzmann constant, absolute temperature, the Avogadoro’s number, the permittivity of a vacuum, and that of the medium, respectively. Although the closest
The retention volume of the crown ether probe is equal to the sum of eqs 4 and 5.
(35) Izatt, R. M.; Bradshaw, S. J.; Nielsen, S. A.; Lamb, J. D.; Christensen, J. J. Chem. Rev. 1985, 85, 271 and references therein. (36) Okazaki, S.; Sakamoto, I. Ion to Yobai (Ions and Solvens); Taniguchi Insatsu: Matue, Japan, 1990, and references therein.
DISCUSSIONS Sigmoidal Changes in Cationic Probe Retention. As the first approximation, a single Stern layer was considered; that is,
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Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
ψM ) ψCrM ) ψp. Although this simple model can predict sigmoidal changes for cationic probe retention, the predicted probe retention shows a much steeper increase than the experimental retention. Broken curves in Figure 1 show cationic probe retention changes calculated by this simplified model. Disagreements between experiments and this approximation are apparent. In particular, marked deviations can be seen for PTMA+. For better fittings, we had to assume a formation constant for the K+-18C6 complex in MeOH that was much smaller than the reported values. Although as stated above, many instances indicate that log K ) 6 for this complexation, log K should be 4 mM, indicating that we can roughly discuss probe retention solely on the basis of eq 2. Thus, when C18C6 is larger than CK, a relative retention volume can be written as
Vr - V0 [Vr - V0]ref
{
) exp -
}
F(ψP - ψP,ref) RT
the total countercation, which were determined from the breakthrough curves, are also listed. There are very good correlations between these ratios and ∆ψ for individual probe cations, implying that surface potential is well-reflected in the Stern-layer potential of a probe cation. In addition, the larger the molecule, the smaller the electrostatic potential changes. This indicates that the condition of the stationary phase surface more strongly affects its vicinity; the effect is weaker as the interaction distance becomes longer. Maximum Formation in Retention of Crown Ether Probes. As shown in Figures 3 and 4, crown ether probes give the retention maxima when 15C5 or 18C6 is added to the mobile phase. This phenomenon is more obvious for the addition of 18C6 than it is for the addition of 15C5. As shown by solid curves in these figures, the theory developed above follows well the experimental changes in probe retention. The interaction distances are assumed to be 0.24, 0.30, 0.285, and 0.37 nm, and log K ) 2.9, 5.1, 3.4, and 4.5 for B15C5, B18C6, DB24C8, and DB30C10, respectively. There are some inconsistencies between the interaction distances and molecular sizes, which may come from structures, configurations, or orientations of interacting molecules. The maximum formation can be intuitively understood. Both the probe and the mobile-phase crown ethers form complexes with the K+ added to the mobile phase. When the concentration of the modifier crown ether is low, a probe crown ether can form a complex with the K+ in the mobile phase, and its retention becomes weaker. An increase in the concentration of the modifier crown ether brings about two effects: one is the reduction of the equilibrium concentration of uncomplexed K+, and the other is the complexation with the K+ in the resin. The former effect enhances probe retention, but the latter reduces it. Because the complexation in the resin is electrostatically eliminated ,as discussed above, complexation in the mobile phase preferentially occurs in comparison with that in the resin; thus, the probe retention basically increases when C18C6 < CK. Further increases in the modifier crown ether cause saturation in mobile-phase complexation (this situation is attained when C18C6 > CK), and excess crown ether forms a complex with K+ in the resin. The complexation of K+ in the resin causes the exclusion of the probe crown ethers from the resin, and thus, probe retention decreases. The electrostatically reduced complexation of probe crown ether in the vicinity of the resin can be quantitatively evaluated by
Kresin )
) (6)
[Cr-K+]s [K+]s[Cr]s
(
[K+]b exp -
( (
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Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
) Kbulk
)
FψCrK RT
) ) )
FψK [Cr]b RT
FψCrK RT FψK exp RT
exp where ref denotes the reference values. Thus, if the retention data obtained with a particular C18C6 (e.g., C18C6 ) CK) is taken as a reference, potential differences (∆ψ ) ψP - ψP,ref) can be determined. Selected results are summarized in Table 1. It should be noted that eq 6 involves no unknown parameters. In this table, the molar ratios of the 18C6 complex in the stationary phase to
(
[Cr-K+]b exp -
(7)
where subscripts s and b denote the resin surface and bulk. Thus, the complexation constant at the resin surface (Kresin) is related
surface and the closest approach for K+(C0) and between the closest approaches for K+ and a K+-crown ether complex (C1) are assumed as circuit elements equivalent to the present case. When the charge densities of the surface, the closest approaches for these two cations, and the diffuse layer are defined as δs, δK, δCr-K, and δd, these are related to the capacitor and electrostatic potential
δs ) C0(ψs - ψK)
(A1)
δs + δK ) C1(ψK - ψCr-K)
(A2)
δs + δK + δCr-K + δd ) 0
(A3)
where ψs is the surface potential. Eq A3 represents electroneutrality. The charge density of the diffuse layer, δd, is given by the Gouy-Chapmann theory
(
δd ) sinh -
Figure 10. (A) Changes in the electrostatic potential at the closest approaches of probe crown ethers and (B) reduction of their apparent complexation constants by adding 18C6 to the 4 mM K+ mobile phase.
to the corresponding intrinsic complexation constant in bulk (Kbulk). Figure 10 shows changes in the electrostatic potential at the closest approach for probe crown ethers (ψCr-K) with varying 18C6 concentrations in a 4 mM K+ mobile phase and those in their Kresin/Kbulk ratios. The reduction of complexation constants at the resin surface becomes obvious as the 18C6 concentration increases, because an increase in 18C6 concentration in the mobile phase enlarges a difference between ψK and ψCr-K. Although the developed model is applicable to the description of retention of crown ether probes, these probes are not suitable for evaluating the electrostatic potential for the following two reasons: retention of probe crown ethers is given by more complicated equations than that of cationic probes and the doublelayer accumulation becomes dominant for probe retention when an excess amount of the modifier crown ether is added to the mobile phase. Thus, the simplification made for cationic probes is not possible for crown ether probes. In conclusion, the developed model describes well the retention behaviors of cationic and crown ether probes on a cation-exchange resin and provides the information about the electrostatic potential on a molecular scale. It has been pointed out that the PoissonBoltzmann theory has limitations in describing the electric double layer and predicting the electrostatic potential therein, especially for systems including multivalent ions. Although the present model derived from this classical theory may have similar limitations and ambiguities, it should be noted that the electrostatic information on the molecular scale is given by measuring chromatographic retention. We believe that separation approaches continue to provide insights into the molecular interactions taking place at an interface. APPENDIX The electrostatic potential at the closest approach was calculated by the following procedure; two capacitors between the
)x
FψCr-K 2RT
8RT0[K+]b
(A4)
and δK and δCr-K are calculated with ion-association constants for these cations given by eq 2. C0 and C1 are given by
0 aK
(A5)
0 aCr-K - aK
(A6)
C0 ) C1 )
where aCr-K and aK are the distance between the surface and the closest approaches for K+ and the K+-crown ether complex. Thus, we can calculate ψs, ψK, and ψCr-K by solving simultaneous equations. The potential sensed by a probe molecule, ψP, is calculated by
ψP )
ψCr-K - ψK (a - aK) + ψK aCr-K - aK P
2RT ψP ) ln F
[
( (
when aK < aP < aCr-K
]
) )
FψCr-K exp{-κ(aP - aCr-K)} 4RT FψCr-K 1 - tanh exp{-κ(aP - aCr-K)} 4RT when aP > aCr-K (A7) 1 + tanh
where κ is the Debye shielding parameter. The corrected retention volume of the probe is given by a sum of the contributions from surface association and double-layer accumulation. The former is given by the limiting slope of the adsorption isotherm of the probe
(Vr - V0)ia ) A
( ) ∂ΓP ∂CP
(A8)
CP)0
where ΓP and CP denote the equilibrium concentrations of the probe at the surface and in solution, respectively. Substituting ionAnalytical Chemistry, Vol. 73, No. 13, July 1, 2001
3057
association constants and surface potential into this equation gives eqs 2 and 4 for a cationic and crown ether probe, respectively. The total excess concentration of a charged solute accumulated in the electrical double layer per unit area (ΓP,dl) is given by
ΓP,dl ) CP
∫ [exp(- RT ) - 1] dx ∞
Fψx
Because this equation is valid for a monovalent cation, eq A10
(Vr - V0)dl ) A
CP
∫ [exp(- RT ) - 1]dx ∞
Fψx
aP
CP
(A10)
(A9)
should be multiplied by a ratio of the cationic species concentration to the total one for a crown ether probe; thus, eqs 3 and 5 can be derived.
The corrected retention volume originating from the electrical double-layer accumulation is given by the ratio of eq A9 to the corresponding one for an uncharged solute.
Received for review January 5, 2001. Accepted March 30, 2001.
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aP
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
AC010020Z
