Anal. Chem. 2000, 72, 1307-1314
Retention of Ions on Nonporous Charged Stationary Phases Tetsuo Okada†
Department of Chemistry, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
The interactions between ion-exchange resins and counterions consist of several mechanisms, such as ion-pair formation between active sites and counterions, specific adsorption, solvation changes, and double-layer accumulation. The double-layer accumulation of ions, which is a typical nonstoichiometric mechanism, is an important factor governing overall ion-exchange chromatographic retention when a major part of the stationary-phase surface is in contact with eluent flows. Nonporous stationary phases, where solutes are accessible to the surfaces by convection as well as by diffusion, possibly highlight this nonstoichiometric contribution through the coupling of a flow profile with an electrostatic potential function. The retention of ions on nonporous stationary phases has been interpreted by a model derived on the basis of the Poisson-Boltzmann equation including solvation change terms. Unusual retention behaviors have been confirmed only for anions, and can be explained by the model including the assumption that anions undergo solvation changes in a thin layer (∼5 nm thickness) at the vicinity of the stationary phase; the thickness should be a function of eluent flow rates. This strongly suggests that there is a difference in solvation nature between cations and anions. It can be inferred that water molecules interacting with polymer domains of the stationary phase behave like single molecules and cannot form a stable hydration shell around an anion as usually seen in bulk solution. Separation mechanisms and the origin of selectivity in ion exchange have been debated on the basis of various theoretical and experimental approaches. Developments in chromatography and related techniques have allowed us to study the ion-exchange phenomena in rather simple ways, e.g., measurements of retention times or adsorption isotherms, and to compile fundamental data in regard to a variety of solutes, eluents, and ion-exchange resins.1-9 Ion-exchange chromatographic retentions have often been analyzed with a linear relation between logarithms of † Phone and fax: +81-3-5734-2612. E-mail:
[email protected]. (1) Zappoli, S.; Bottura, C. Anal. Chem. 1994, 66, 3492. (2) Okada, T. Bunseki Kagaku 1995, 44, 579 and references therein. (3) Barron, R. E.; Fritz, J. S. J. Chromatogr. 1984, 284, 13. (4) Jensen, D.; Weiss, J.; Rey, M. A.; Pohl, C. A. J. Chromatogr. 1993, 640, 65. (5) Inoue, Y.; Kawabata, K.; Suzuki, Y. J. Anal. At. Spectrom. 1995, 10, 363. (6) Hajo´s, P.; Re´ve´sz, G. J. Chromatogr., A 1997, 771, 23. (7) Nishimura, M.; Hayashi, M.; Hayakawa, K.; Miyazaki, M. Anal. Sci. 1994, 10, 321. (8) Michigami, Y.; Fujii, K.; Ueda, K. J. Chromatogr., A 1994, 664, 117. (9) Lamb, J. D.; Smith, R. G.; Jagodzinski, J. J. Chromatogr. 1993, 640, 33.
10.1021/ac990823a CCC: $19.00 Published on Web 02/17/2000
© 2000 American Chemical Society
retention factors (k) and logarithms of eluent concentrations (X):
log k ) (-y/x)(log X) + C
(1)
where x and y are the charges of the eluent and a solute ion and C is a constant including a selectivity coefficient, the ion-exchange capacity of a resin, and a phase ratio. This linear relation, which is derived by assuming two discreet phases, i.e., a solution phase and a resin phase, and a constant ion-exchange selectivity coefficient between an eluent and a solute ion, has proven applicable to many cases. It has also been found that the slope of eq 1 (hereinafter, the slope of eq 1, -y/x, is represented by S) well reflects the ratio of the solute charge to the eluent charge, especially when both an eluent and a solute ion are monovalent.1,2,6 However, despite the simplicity and the successes of eq 1 in describing ion-exchange chromatographic retention, the assumptions of constant selectivity coefficients and two discreet phases obviously provide only approximate physical views of this important separation mode. Cantwell10-12 and Ståhlberg13 took electrostatic effects into account to explain ion-exchange chromatographic retention of ions; both research groups derived equations expressing the chromatographic retention of ions using the Gouy-Chapmann (GC) theory, which is well-known in electrochemistry and colloid chemistry.14,15 One of the most important features in Cantwell’s model is that the solvation changes are taken into account to describe ion-exchange selectivity.10-12 On the other hand, Ståhlberg13 pointed out that ions should successively distribute from the stationary-phase surface to the bulk mobile phase and that specific adsorption is solely responsible for separation selectivity; by assuming these, he avoided defining the phase boundary. Okada16 recently showed that the Stern-Gouy-Chapman theory is a better choice and the direct interaction between ion-exchange sites and counterions should be included to explain ion-exchange selectivity. The dependence of separation factors on the eluent strength was successfully explained by his model including a nonstoichiometric contribution as well as stoichiometric contribu(10) Afrashtehfer, S.; Cantwell, F. F. Anal. Chem. 1982, 54, 2422. (11) Hux, R. A.; Cantwell, F. F. Anal. Chem. 1984, 56, 1258. (12) Cantwell, F. F. In Ion-Exchange and Solvent Extraction; Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York, 1985; Vol. 9, Chapter 6. (13) Ståhlberg, J. Anal. Chem. 1994, 66, 440. (14) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985 (Japanese version, McGraw Hill: Tokyo, 1991). (15) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980. (16) Okada, T. Anal. Chem. 1998, 70, 1692.
Analytical Chemistry, Vol. 72, No. 6, March 15, 2000 1307
tions, such as the direct interaction between ions and ion-exchange sites and specific adsorption. Usual chromatographic stationary phases have porous structures because large surface areas are generally required to allow large sample loading. Only a small percentage of the total surface area of a porous stationary phase is in contact with a mobile phase flow, while a major part of the surface, to which solute and eluent components are basically accessible only by molecular diffusion, exists in a static solution. Thus, convection plays a minor role in transferring substances in separation columns. In ion-exchange chromatography, the distribution of ions basically follows the Boltzmann equation, which predicts higher counterion concentrations in the vicinity of charged surfaces than in the bulk solution. In a usual flow channel, the velocity of a flow is highest at the center of the channel and lowest at its fringe.17 Thus, ions accumulated in the electrical double layer move more slowly than those in the bulk mobile phase; this effect should be represented by an appropriate function involving both a flow profile and the distribution of ions. Porous stationary phases obviously make the observation of this effect difficult (or probably impossible) for the above reason. In this paper, the retention of ions on a nonporous charged stationary phase is interpreted by a model including a flow profile and the distribution function derived from the Poisson-Boltzmann equation. EXPERIMENTAL SECTION The chromatographic system was composed of a Tosoh HPLC pump, model CCPD, a Tosoh UV-visible detector, model UV8000, and a Rheodyne injection valve equipped with a 10 µL sample loop. A TSKgel ODS-NP separation column was employed (particle size 2.5 µm, packed in a 4.6 mm i.d. × 30 mm stainless steel column). The column was immersed in a thermostatic water bath to keep the temperature at 25 °C. Hexadecyltrimethylammonium chloride (HTAC) and hexadecyltrimethylammonium bromide (HTAB) were recrystallized from acetone-methanol. Sodium dodecyl sulfate (SDS) was recrystallized from methanol. Other reagents were of analytical grade. Solutions were prepared with distilled, deionized water. The adsorption amounts of the surfactants were spectrophotometrically determined; after the complete desorption of the surfactants from the separation column, HTA+ was extracted into 1,2-dichloroethane as an ion pair with orange II and DS- was extracted into toluene as an ion pair with ethyl violet. Linear flow velocities were determined by measuring the retention times of unretained solutes. Acetone, creatinine (for anion exchange), and iodide (for cation exchange) were used as inert solutes. No retention difference was detected between neutral and charged solutes in any case. Retention times were measured three to six times; relative standard deviations were less than 1%, even for retention times of 10 s. THEORY Flow Channel Geometry and Flow Profile. The retention model was derived basically in the same manner as in previous work.16,18,19 However, some modifications were necessary to take (17) Hatta, K.; Torii, H.; Taguchi, T. Ruutai Rikigaku no Kiso (Fundamentals in Hydrodynamics); Nisshin Shuppan: Tokyo, 1991. (18) Okada, T. J. Phys. Chem. B 1997, 101, 7814.
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Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
Figure 1. Schematic representation of a flow profile and the concentration of counterions at a cross section of a cylindrical flow channel surrounded by a charged wall.
into account the coupling of the electrostatic potential with a flow profile. Flow channels in the column densely packed with nonporous spheres can be regarded as a bundle of capillaries. Cylindrical flow channels are assumed to simplify the geometry and to facilitate the derivation of analytical solutions from the Stokes-Navie and Poisson-Boltzmann equations. Pressure measurements allow us to estimate the radius (h) of the cylindrical interparticle flow channels according to the Hagen-Poiseuille law:17
v(r) ) -
( )
h2 r2 P 1- 2 4η h
(2)
where r is the cylindrical coordinate, v(r) is the linear flow velocity at r, η is the viscosity of the medium, and P is the pressure gradient along the flow direction. The flow profile is schematically illustrated in Figure 1. We can measure average linear flow rates (eq 3), from which the radius of the cylindrical flow channel can be calculated. In
vj ) -
h2P 8η
(3)
the present experiments, h was calculated as 0.21 µm, which is ca. one-sixth as large as the radius of resin particles. Although this length is used for the following calculation, the dimension is not essential, but its ratio to the double-layer thickness is important to simplify the equation derivation as shown below. Electrostatic Potential. The linearized Poisson-Boltzmann equation in the cylindrical coordinate can be analytically solved (19) Okada, T.; Patil, J. M. Langmuir 1998, 14, 6241.
to give the electrostatic potential at r:20-22
ψr )
σSt I0(κr) κ0 I1(κh)
(4)
where I0 and I1 are the modified Bessel functions of zeroth and first orders, and 0 are the permittivities of the medium and the vacuum, σSt is the charge density at the Stern layer, and 1/κ is the Debye length. For 1:1 salt solutions, the Debye lengths vary as follows: 30 nm for 0.1 mM, 9.6 nm for 1 mM, 3 nm for 10 mM, and 0.96 nm for 100 mM. Thus, h is much larger than 1/κ, even for rather dilute salt solutions, suggesting that the electrostatic potential be calculated by assuming isolated planes. The isolatedplane assumption was supported by the fact that the electrostatic potential calculated by eq 4 was the same as that calculated by the linearized Poisson-Boltzmann equation for isolated planes:14,15
ψr )
σSt exp[-κ(h - r)] κ0
(5)
The electrostatic potential can thus be calculated on the basis of the one-dimensional Poisson-Boltzmann equation. Since the applicability of the linearized Poisson-Boltzmann equation is limited up to ψ ) (25 mV, which corresponds to ca. 0.04 C m-2 surface charge density with 50 mM ionic strength if no surface adsorption or surface ion-pair formation of ions occurs, the linearized approximation should be avoided if possible. Thus, the large ratio of h to 1/κ allows the prediction of the more precise electrostatic potential. The analytical solution of the one-dimensional PoissonBoltzmann equation is well-known as14,15
ψr )
{
}
1 + tanh(FψSt/4RT) exp[-κ(h - r)] 2RT ln F 1 - tanh(FψSt/4RT) exp[-κ(h - r)]
(6)
It is a problem common to approaches based on the PoissonBoltzmann theory that no selectivity terms other than ionic charges are included in the equation. Ions have different ionexchange selectivity coefficients, even if they have identical charges, indicating that selectivity terms other than charges should be included in a model. Ion-pair formation on the resin surface was assumed to represent ion-exchange selectivity. In our previous paper, we assumed the surface adsorption of ionic solutes as well as the surface ion-pair formation.16 In the present case, since no adsorption was detected for all solutes tested, this contribution was neglected. The retention factor of a solute is a sum of the ion-pair term (kip) and the accumulation in the electrical double layer (kDL). The former is calculated from the limiting slope of the adsorption isotherm as
(
)
A ∂Γip-Y) kip ) V0 ∂Y
Y)0
(7)
(20) Rice, R. E.; Horne, F. H. J. Chem. Phys. 1981, 75, 5582. (21) Rice, R. E.; Horne, F. H. J. Colloid Interface Sci. 1985, 105, 172. (22) Bartha, AÄ .; Ståhlberg, J. J. Chromatogr., A 1994, 668, 255 and references therein.
where Γip-Y is the surface concentration of the ion pair of a solute ion, Y is the bulk concentration of a solute ion, A is the total surface area of a stationary phase, and V0 is the total void volume in a column. Since eq 6 is directly applicable to the calculation of the electrostatic potential, the contribution from the double-layer accumulation to a retention factor, kDL, is given by
kDL )
∫ rv(r) dx 2π∫ rc(r) dr -1 2π∫ r dr 2π∫ rc(r) v(r) dr
2π
h
h
0
0
h
h
0
0
(
c(r) ) Y exp -
)
zFψr RT
(8)
(9)
where z is the charge of an ion. The water structure in the vicinity of the stationary-phase surface should be different from that in the bulk solution due to the presence of ion-exchange sites and resin matrixes. It is reported that the rotational relaxation time of water molecules in organic solvents is shorter than that of those in the bulk by 1-2 orders of magnitude,23 suggesting that water molecules behave not as clusters but as single molecules in such media. A similar situation possibly holds for water molecules bound by the polymer domain of a stationary phase; they should thus have either stronger or weaker solvation ability than those in the bulk. It is assumed that the standard chemical potential changes stepwise at a particular distance (d) from the surface as shown in Figure 2, where d is assumed to be 1 nm but varies over some range in an actual system. It should be noted that the double-layer thickness characterized by 1/κ (also indicated for selected salt concentrations in Figure 2) is not necessarily related to d. This simplification facilitates the solution to the Poisson-Boltzmann equation. The mathematical derivation for the electrostatic potential is shown in the Appendix, where the standard chemical potential differences (∆µ°) are included in the starting PoissonBoltzmann equation. Equation 8 should thus be modified to calculate the contribution of double-layer accumulation according to this concept because the distribution of ions near the surface (h - d e r e h) is affected by the ∆µ° term but that outside this thin layer (0 e r e h - d) is not; hereinafter, the former region is named the interfacial layer and the latter the central layer. Finally, we obtain
∫ rv(r) dx ) × 2π∫ r dr dr + ∫ 2π(∫ rc(r) 2π(∫ rc(r) v(r) dr + ∫ h
2π
kDL
0
h
0
h-d
h
cent
0
h-d h
h-d
0
rc(r)int dr)
- 1 (10)
rc(r)int v(r) dr) h-d
cent
(
c(r)cent ) Y exp -
(
c(r)int ) Y exp -
)
zFψcent RT
(11)
)
zFψint - ∆µ° RT
(12)
where the subscripts “cent” and “int” denote the central and interfacial layers, respectively. (23) Nakahara, M.; Wakai, C. J. Chem. Phys. 1992, 97, 4413.
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Figure 2. Solvation energy changes in the vicinity of the stationaryphase surface, which occur stepwise as shown in the upper diagram. The lower diagram the electrostatic potential changes as functions of the distance from the charged surface for ∆µ° ) 2 kJ mol-1, 0, and -2 kJ mol-1. The Debye lengths (1/κ) at 10 and 100 mM concentrations of a monovalent salt are also shown in the figure for comparison.
RESULTS AND DISCUSSION Calculations. The retention natures of the monovalent ions were studied according to the above theory because the behaviors of monovalent ions are described by the Poisson-Boltzmann theory better than those of multivalent ions, and it has been confirmed in many cases that plots based on eq 1 for monovalent ions are linear with S ) 1. To compare our calculations with experimental results, some parameters were taken from chromatographic experiments, e.g., h ) 0.21 µm, total surface area of the stationary phase ) 0.81 m2, and surface charge density ) 0.48 C m-2. In the present calculations, since the adsorption of ions does not have to be taken into account as stated above, the Stern layer thickness is not very important; it was assumed to be 0.3 nm, which is almost the same as a typical hydrated ion radius. Linear log k-log X plots with S ) 1 ( 0.12 were predicted by our previous calculations including Ståhlberg’s approximation for the electrical double-layer accumulation.16 Interestingly, eq 8 also predicts the linear log k-log X plots with S ) 1 ( 0.05 in almost all usual instances. In contrast, if the solvation changes for ions are taken into account (eq 10), rather different features emerge. Figure 3A shows the effect of ∆µe° on the S values for a solute ion of ∆µs° ) 0. The S values are almost unity when ∆µe° ) 0 to -4 kJ mol-1 but smaller than unity for ∆µe° < -4 kJ mol-1 and 1310 Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
Figure 3. Effects of solvation changes on the slopes of log k-log X plots. (A) Effect of solvation changes of eluent ions. ∆µe° ) 0, d ) 5 nm, and Kip (ion-pair formation constant) ) 0.01 m3 mol-1 for both eluent and solute ions were assumed. (B) Effect of solvation changes of solute ions. ∆µe° ) 0 was assumed, and other parameters were the same as those for part A. The log k-log X plots were calculated by assuming eleven points; X ranged from 0.11 to 102 mM.
∆µe° > 0. Large ∆µe° values tend to result in small S values. The effects of ∆µs° is shown in Figure 3B, where ∆µe° is kept to zero; the S values increase and approach unity when ∆µs° changes to positive values. These results indicate that the S values are smaller than unity when a solute ion rather than an eluent ion is stabilized in the interfacial layer. The S values, calculated from 11 points differing in eluent concentration, were plotted in Figure 3 only when the correlation coefficients were larger than 0.95. The linearity became worse when ∆µs° was assumed to be lower than -10 kJ mol-1. Figure 4 shows the effect of the thickness of the interfacial layer on the S values, where ∆µs° ) -5 kJ mol-1 and ∆µe° ) 0 are assumed. The S values decrease with an increase in the interfacial layer when d is smaller than 5 nm but slightly increase when d exceeds 5 nm. Since 5 nm is almost 10 times as large as the diameter of a solvated ion, this thickness must be reasonable if the solvation changes are caused by water molecules interacting
Figure 4. Effect of the thickness of the interfacial layer (d) on the slopes of the log k-log X plots. ∆µs° ) 5 kJ mol-1, ∆µe° ) 0, and Kip ) 0.01 m3 mol-1 for both eluent and solute ions were assumed.
Figure 5. Relationships between calculated retention factors (k) and the surface densities of charged groups (Γ) and between k/Γ and Γ (inset). Parameters for calculations were the same as those for Figure 4. Eluent concentration was 21 mM.
with the stationary-phase matrixes. Hence, the assumption regarding the interfacial layers predicts S values smaller than unity. Chromatographic Retention of Selected Ions. The retention of ions was studied with an ion-pair (ion-interaction) chromatographic mode. HTA+ was added to the mobile phases for anion separation, while SDS was added for cation separation. The addition of surfactants to the mobile phases was necessary to avoid desorption of the surfactants from the stationary-phase surface. However, this procedure brings about additional complexity in the interpretation of results: (1) the adsorption of the surfactants is enhanced by adding a salt to the mobile phases, and thus it is difficult to keep the surface density of active groups constant; (2) if micellization occurs, the partition of solutes into the micelle should be taken into account. Figure 5 shows the relation between retention factors calculated with the developed model and surface densities of charged groups (Γ). Retention factors linearly increase with increasing Γ over a wide range of surface densities of charged groups; almost the same trend was confirmed for different retention ranges, that is, for different eluent concentrations and
different eluent and/or solute natures. The ratios of retention factors to Γ are also plotted against Γ in Figure 5; the ratios are almost constant for Γ > 3 µmol m-2. Γ for HTA+ ranged from 6.33 to 7.29 µmol m-2, and that for DS- ranged from 5.83 to 13.3 µmol m-2. In both cases, since k/Γ ratios can be regarded as constant from Figure 5, effects of adsorption enhanced by adding salts can be corrected simply by using the surface densities of charged groups. The micellar partition effects can be minimized by keeping the added surfactant concentration below its critical micellar concentration (cmc). Although the addition of salts lowers the cmc, very small concentrations of micelles never affect the retention of ions. The added surfactant concentrations were 1 mM HTAC, 0.5 mM HTAB, and 1 mM SDS; these are lower than their cmc’s, 1.3 mM for HTAC,24 0.5 mM for HTAB,24 and 8 mM for SDS.25 NO3- and I- were selected as UV-absorbing anionic solutes, and the phenyltrimethylammonium ion (PTA+) was chosen as the cationic solute. Although all solutes gave linear log k-log X plots under every condition tested (R > 0.997, for six or seven experimental points), slopes for the anionic solutes were much smaller than expected from eq 1 (at 1 mL min-1 flow rate, -0.816 for NO3- and -0.635 for I- with the mobile phase containing HTAC and -0.876 for NO3- and -0.875 for I- with the mobile phase containing HTAB), while the slope for PTA+ was -1.006, which almost agrees with the prediction of eq 1, similar to what is seen in usual ion-exchange chromatography using porous resins. Thus, the unusual behaviors were confirmed only for anions. Bartha and Ståhlberg22 and Li and Cantwell26 interpreted ion-pair chromatographic retention behaviors of simple ions by models including the adsorption constant of an ion-pair reagent as well as electrostatic potential contributions. When the ionic strength was kept constant and the surface charge density was low, the following simple relation was derived:22
log k ) C -
xy log X 1 + x2
(13)
This equation indicates that log k-log X plots have slopes of 1/2 when x ) y ) (1. It should be noted that this equation is not applicable to the present case, where the ionic strength effect on the surface charge was corrected by the procedure stated above. The present results should thus be interpreted by a model for ion-exchange rather than one for ion-pair chromatography. Other interesting retention features were found in the retention dependence on flow rates. Figure 6 shows the effects of flow rates on the retention volumes relative to that obtained at a flow rate of 0.1 mL min-1. Retention volumes are not constant but decrease with increasing flow rates. Peak shape changes and sample concentration dependence were not detected, suggesting that this flow rate effect is due neither to overloading nor to sorption kinetics. Decreases in retention with increasing flow rate are also more marked for the anions than for the cation; obviously, the flow rate effects decrease in the order I- > NO3- . PTA+. Although the flow rate effects were studied for the same solutes with usual porous ion-exchange resins, no obvious flow rate (24) Rosen, J. J. Surfactants and Interfacial Phenomena; Wiley: New York, 1978. (25) Corrin, M. L.; Harkins, W. D. J. Am. Chem. Soc. 1947, 69, 683. (26) Liu, H.; Cantwell, F. F. Anal. Chem. 1991, 63, 2032.
Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
1311
Figure 6. Effects of mobile phase flow rates on the relative retention volumes of selected solutes. Retention volumes relative to those obtained at 0.1 mL min-1 are plotted for facile comparison. Mobile phases: 0.5 mM HTAC; 20 mM NaCl for anion separation and 1 mM SDS and 15 mM Na2SO4 for cation separation.
Figure 7. Effects of flow rates on the slopes of log k-log X plots for anions. Mobile phases: 0.5 mM HTAC and NaCl (10-80 mM). Slopes were corrected with the amounts of adsorbed HTAC.
values, (4) the S values approach unity with increasing flow rates, (5) the higher flow rates cause the lower retention, and (6) the flow rate effects are more marked for anions than for cations, and for poorly solvated anions than for better solvated ones. These unusual retention behaviors of ions observed with a nonporous charged stationary phase can be explained according to the model developed above. Small S values are predicted by calculations using the developed model as shown in Figures 3 and 4. The most important aspect of these calculations is the assumption that there is a very thin layer in the vicinity of the stationary-phase surface, where ions undergo solvation changes. The calculation shows that smaller S values are predicted when ∆µs° < ∆µe°; i.e., the solute solvation relative to the eluent solvation is enhanced in the interfacial region. In bulk water, smaller ions are solvated better than larger ions when they have identical charges. However, dispersion forces contribute more significantly to the overall solvation of large ions in organic solvents, which usually have larger molecular sizes and polarizabilities than water. Thus, the Gibbs free energy of transfer from water to an organic solvent for a smaller ion is more positive than that for a larger ion.27 In the present case, although ions must still be solvated by water in the interfacial layer, the structure of water herein is different from that in the central layer. If water molecules behave as single molecules in the interfacial region rather than as those bound to clusters as in the bulk, the hydration energy in the interfacial layer is similar to that in the gas phase. The energies for 1:1 complex formation between a water molecule and an anion in the gas phase are, for example, -34 kJ mol-1 for Cl-, -29 kJ mol-1 for Br-, and -24 kJ mol-1 for I-; these values are more than 10 times smaller than the hydration energies for these anions, -376 kJ mol-1 for Cl-, -345 kJ mol-1 for Br-, and -283 kJ mol-1 for I-.27 In contrast, the 1:1 interaction energies for cations are rather large: -113 kJ mol-1 for Li+, -73 kJ mol-1 for Na+, and -48 kJ mol-1 for K+. In addition, differences in the hydration enthalpies for metal cations are almost explained by a sum of the enthalpic changes of consecutive interactions between a water molecule and a metal ion:
∆∆H°hydr(A,B) )
∑[∆H°
n-1,n(B)
- ∆H°n-1,n(A)]
(14)
n)1
dependence was detected (retention volumes were constant within 2-3% variation upon changing the flow rate). Thus, the flow rate dependence of retention volumes should be related to the small slopes of log k-log X plots for anions. The slopes of log k-log X plots were studied again for varying flow rates. The results for HTAC mobile phases are summarized in Figure 7. It appears that S values approach unity as the flow rate increases. If small S values have a kinetic origin, the higher flow rates should cause more marked deviations from behaviors predicted eq 1. Thus, this unusual retention behavior must be characteristic of a nonporous stationary phase and have a mechanistic origin; detailed discussions are presented below. Interpretation of Chromatographic Behaviors by the Developed Model. Experimental results obtained with our current nonporous stationary phase can be summarized as follows: (1) the S values for anions are smaller than unity, unlike those for cations or those predicted by eq 1, (2) the larger and less hydrated the solute anions, the smaller the S values, (3) the smaller and better hydrated the eluent anions, the smaller the S 1312
Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
where ∆H°n-1,n(A) denotes the enthalpic change for the equilibrium
A‚(n - 1)H2O + H2O ) A‚nH2O
(15)
The difference in hydration enthalpy between Li+ and Na+ or between Li+ and K+ is, for example, almost equal to the above cumulative sum of enthalpic changes up to n ) 6. However, the corresponding differences between Cl- and Br- (∆H°(Br-) ∆H(Cl-) ) 31 kJ mol-1) and between Cl- and I- (∆H°(I-) ∆H°(Cl-) ) 76 kJ mol-1) are much larger than the sum represented by eq 14; ∆∆H°hydr(Cl-,Br-) ) 5 kJ mol-1 and ∆∆H°hydr(Cl-,I-) ) 42 kJ mol-1 for n ) 1-4.27 This must imply a more significant contribution from the second and outer hydration shells to the overall solvation energy for an anion than for a cation; (27) Marcus, Y. Ion Solvation; Wiley: Chichester, U.K., 1985.
this may be due to the unclear hydration geometry of an anion. Six water molecules can form a stable coordination shell for a cation, while more water molecules are necessary for the complete solvation of an anion. Thus, the transfer of an anion from the bulk water to a different phase, where enough water molecules to form a large but loose hydration shell around the anion are not available, is energetically less favorable than the analogous transfer of a cation. In the interfacial layer, most of the water molecules should interact with the charged groups of adsorbed surfactants and the polymer domain of the stationary phase, and thus anions are insufficiently solvated by water molecules. The above calculation of the hydration enthalpies for anions implies that the transfer of smaller anions from the bulk water to the interfacial layer is less preferable than that of larger anions because the larger desolvation energy loss for smaller anions cannot be compensated by solvation in the interfacial layer. In the present work, the retention of large anions was studied with smaller eluent anions (i.e., ∆µs° < ∆µe°) because of the weak retention ability of the stationary phase and the simplicity of the detection. This results in unusually small S values. The contact of a liquid flow with a charged layer generates the streaming potential (Es), which is related to the electrostatic ζ potential at the shear plane as in
Esηk ζ) 0∆P
(16)
that the ζ potential measurements are not very sensitive to the shift of the shear plane, and thus it has been assumed immobile. In conclusion, the developed model successfully explained unusually small S values obtained with nonporous ion-exchange chromatography. The studies on the nature of the S values and the flow rate effects strongly suggested that there is a very thin layer on the surface of the stationary phase, where ions undergo solvation changes. Also, it was indicated that the solvation nature of anions is different from that of cations; this should be studied by different approaches as well. Although, in the present model, stepwise changes were assumed, changes in solvation in the vicinity of surfaces should be gradual. Appropriate experimental tools capable of detecting such changes should be developed. This is a future task. APPENDIX The Poisson-Boltzmann equation including solvation effect terms for the isolated plane can be written as
∑z n ° exp(
d2ψ F )2 dx 0
)
-ziFψ - ∆µi° RT
i i
(A1)
where x is the coordinate and ni° (mol m-3) is the bulk concentration of the ionic species, i. For monovalent ions, eq A1 can be simplified by the Debye-Hu¨ckel approximation:
d2ψ κ2 RT + (Ma + Mc)ψ ) (Ma - Mc) 2 2 F dx
[
where k is the solution conductivity and η is the solution viscosity. Fo´ti et al.28 pointed out the possibility that the streaming potential is related to the ion-exchange selectivity. However, the excess charges carried by a flow are so small that the ion-exchange retention of ions is not directly influenced. The ζ potential is usually determined on the basis of this equation, i.e., using the measurements of the steaming potentials upon varying ∆P. This method is obviously based on the assumption that the shear plane does not shift upon changing the pressure difference and thus the flow rates. However, it has been reported that the ζ potential determined by this method changes even for surfaces having constant charge density. This was explained by the shift of the shear plane due to the swelling of the surface layer.29 Although the ambiguous structures and nature of the interfacial layer make the interpretation of flow effects difficult, it is possible to infer that high flow velocity prevents the development of the interfacial layer; this layer becomes thinner as the flow rate increases. The flow rate effects on the S values can thus be explained by assuming changes in the thickness of the interfacial layer. The ζ potential measurements largely depend on the immobile shear layer assumption as represented by eq 16. The calculations with the same parameters listed in the captions of Figure 4 indicate that the potential at d (the boundary between the interfacial and central layer) is not varied very much, e.g., 0.102 V for d ) 3 nm, 0.099 V for d ) 4 nm, 0.096 V for d ) 5 nm, and 0.093 V for d ) 6 nm. Thus, a 1 nm shift of the shear plane results only in a 3 mV h shift of the ζ potential, if it is defined as the potential at d. This implies
1 sinh-1 xR
(28) Fo´ti, G.; Re´ve´sz, G.; Hajo´s, P.; Pellaton, G.; Kova´ts. E. Anal. Chem. 1996, 68, 2580. (29) Mo ¨ckel, D.; Staude, E.; Dal-Cin, M.; Darcovich, K.; Guiver, M. J. Membr. Sci. 1998, 145, 211.
The boundary conditions are
Ma ) exp Mc ) exp κ)
]
(A2)
( ) ( ) ∆µa° RT ∆µc° RT
( ) 2F2n° 0RT
0.5
∆µa° and ∆µc° are the differences in standard chemical potential for an anion and a cation. Equation A2 can be transformed to
dψ ) -(Rψ2 + βψ + C)0.5 dx R) β)
(
(A3)
κ2 (M + Mc) 2 a
2Fn° (Ma - Mc) 0
)
where C is an integral constant. The solution of eq A3 finally gives
ψint +
x
β 2R
- sinh
β2 C - 2 R 4R
-1
ψSt +
x
β 2R
β2 C - 2 R 4R
)x
Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
(A4)
1313
σSt dψ |r)h ) dx 0
(A6)
σr)h-d dψ | )dx r)+(h-d) 0
(A7)
Analytical Chemistry, Vol. 72, No. 6, March 15, 2000
The integral constant C is determined from eq A5 as
C)
ψ|r)-(h-d) ) ψ|r)+(h-d)
dψ | ) ψxf∞ ) 0 dx xf∞
1314
(A5)
(A8)
( ) σ 0
2
- RψSt2 - βψSt
(A9)
Thus, we can represent ψint as a function of x. Received for review July 26, 1999. Accepted November 29, 1999. AC990823A
