Anal. Chem. 1999, 71, 2046-2052
Symmetry Breaking during the Formation of β-Cyclodextrin-Imidazole Inclusion Compounds: Capillary Electrophoresis Study Yves Claude Guillaume* and Eric Peyrin
Laboratoire de Chimie Analytique, Faculte´ de Me´ decine et de Pharmacie, Place Saint-Jacques, 25030 Besanc¸ on Cedex, France
A mathematical model was developed for the estimation of binding constants by capillary electrophoresis. The effective electrophoretic mobility in a solute mixture is dependent on the cyclodextrin concentration in the background electrolyte (BGE) as well as the stoichiometry and the binding constant of the guest-cyclodextrin complex. As well, a determination of the degree of complexation, nc (the percent of complexed guest) could be carried out. The model was applied to a series of imidazole derivatives. Thermodynamic data for the solute complexation mechanism were calculated. Different van’t Hoff plot shapes of the degree of complexation were observed with different BGE pH values, indicating a change in the solute complexation mechanism. Enthalpy-entropy compensation revealed that the solute complexation mechanism was independent of the imidazole derivative molecular structure, the same at pH ) 4.5, 5.0, 5.5, 6.0, and 6.5 but changed at pH ) 7.0 and 7.5. Topological defects formed during a symmetry-breaking transition could be responsible for the modification of the structure of the cyclodextrin cavity and explained the nc variations in relation to pH and temperature. Cyclodextrins (CDs) are cyclic oligosaccharides containing six to eight glucopyranose units arranged in such a way as to create a cavity. These CDs are capable of forming an inclusion complex (host-guest complex) with a wide range of organic molecules or so-called guest molecules. Calculated binding stoichiometries and binding constants are used to provide general conclusions and predict the solute-CD complexation mechanism. The equilibrium constant for this molecular association can be measured using a variety of experimental techniques including spectroscopy,1 separations,2-4 calorimetry,5 potentiometry,6 and reaction kinetics.7 The response of the experimental system is commonly measured successively using different substrate or ligand concentrations while maintaining a constant solute concentration. In capillary (1) Hanna, M. W.; Ashbaugh, A. L. J. Phys. Chem. 1964, 68, 811. (2) Guttman, A.; Paulus, A.; Cohen. A. S.; Grinberg, N.; Karger. B. L. J. Chromatogr. 1992, 603, 235. (3) Wren, S. A.; Rowe, R. C. J. Chromatogr. 1992, 603, 235. (4) Rundlett, K. L.; Armstrong. D. W. J. Chromatogr., A 1995, 721, 173. (5) Bertrand, G. L.; Faulkner, J. R.; Han, S. M.; Armstrong, D. W. J. Phys. Chem. 1989, 93, 6863. (6) Gelb, R. I.; Schwartz, L. N.; Johnson, R. F.; Laufer, D. A. J. Am. Chem. Soc. 1979, 101, 1869. (7) Eadie, G. S. J. Biol. Chem. 1942, 146, 85.
2046 Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
electrophoresis (CE), the estimation of binding constants involves measuring the change in electrophoretic mobility of a solute through buffer solutions containing dissolved complexation agents or ligands. There are several requirements that must be met to estimate a binding constant. First, the solute must undergo a change in electrophoretic mobility upon complexation. Either the solute or the ligand must be charged under experimental conditions in order to satisfy this criterion. Second, the equilibrium time scale must be faster than the CE separation time scale. Third, there must be sufficient concentrations of both free ligand and ligand-solute complex present.8-11 It is usually assumed in general studies that the stoichiometry of the complex is 1:1.12-15 Several inconsistencies in the assignment of binding stoichiometries appear in the literature. For example, p-nitroaniline is reported to form either a 1:1 16 or a 1:2 17 inclusion complex with R-CD. Prostaglandin B1 is reported to form either a 1:1 18 or a 1:2 16 inclusion complex with R-CD. Many imidazole derivatives are widely used or recommended for pharmaceutical use as antimycotics.19-22 Nevertheless, these hydrophobic compounds have a weak penetration into human hydrophilic nails. Their inclusion in the apolar CD cavity could improve this penetration, considering the hydrophilic character of the exterior of the CD, which is made up of a great number of hydroxyl groups. In this paper, a novel EC mathematical model was developed for assigning the complex stoichiometries, the binding constants for a series of imidazole derivatives and the (8) Neerink, D.; Van Audenhauge, A.; Lamberts, L.; Huyskens, P. Nature 1968, 218, 461. (9) Connors, K. A.; Lipari, J. M. J. Pharm. Sci. 1976, 65, 379. (10) Miyaji, T.; Kwrono, K.; Vekama, K.; Ikeda, K. Chem. Pharm. Bull. 1976, 24, 1115. (11) Rogan, M. M.; Altria, K. D.; Goodall, D. M. Chirality 1994, 6, 25. (12) Terabe, S.; Otsuka, K.; Ichikawa, K.; Tsicjura, A.; Ando, T. Anal. Chem. 1984, 56, 111. (13) Terabe, S.; Otsuka, K.; Ando, T. Anal. Chem. 1985, 57, 834. (14) Wren, S. A.; Rowe, R. C. J. Chromatogr. 1992, 609, 363. (15) Wren, S. A.; Rowe, R. C. J. Chromatogr. 1993, 635, 113. (16) Armstrong, D. W.; None, F.; Spino, L.; Golden, T. J. Am. Chem. Soc. 1986, 108, 1418. (17) Wong, A. B.; Lin, S. F.; Connors, K. A. J. Pharm. Sci. 1983, 72, 388. (18) Kawaguchi, Y.; Tanaka, M.; Nakae, M.; Funazo, K.; Skono, T. Anal. Chem. 1983, 55, 1852. (19) Reynolds, J. E. F., Ed. The Extra Pharmacopoeia/Martindale, 30th ed.; Pharmaceutical Press: London, 1993; pp 315-368, 523, and 917. (20) Pedersen, M.; Edelsten, M.; Nielsen, V. F.; Scarpllini, A.; Skytte, S.; Slot, C. Int. J. Pharm. 1993, 90, 247. (21) Van Doorne, H.; Bosch, E. H.; Lerk, C. F. Pharm. Weekl. Sci. Ed. 1988, 10, 80. (22) Pedersen, M. Drug Dev. Ind. Pharm. 1993, 19, 439. 10.1021/ac980848u CCC: $18.00
© 1999 American Chemical Society Published on Web 04/20/1999
percentage of the guest that contributes to the formation of the intermediate complex (degree of complexation nc). The shapes of van’t Hoff plots of nc were used to assess changes in the solute complexation process in relation to temperature and background electrolyte (BGE) pH. To understand the dependence of these changes in relation to pH and temperature, a topological model, as an analogue of vortex creation in liquid helium following a transition into the superfluid state, was proposed. METHODS Mathematical Model. Consider a background electrolyte in which the solute molecule M (any guest imidazole derivative) at a total concentration [Mt] and the CD at a total concentration [CDt] form a multiple x-y complex (MxCDy). This complex could be further defined as a MCDn complex where n ) y/x. The successive complexation equilibria of M with CD are
M + CD h MCD
rium with one another, migrated in the electric field as one uniform substance.23-25 Thus, the effective mobility µe was given by n
µe )
∑R
M
µiM
i
(8)
i)0
where µiM is the mobility of the MCDi forms. Combining eqs 7 and 8 gives n
µe ) (
∑
n
βiM[CD]iµiM/
i)0
∑β
M
i
[CD]i)
(9)
i)0
The total bound cyclodextrin concentration noted [CDb] can be defined as
[CDb] ) [MCD] + 2[MCD2] + ... + i[MCDi] + ... +
MCDi-1 + CD h MCDi
n[MCDn] (10)
MCDn-1 + CD h MCDn
Thus, combining eqs 4 and 10 gives n
For these complex formation equilibria, the following constants were defined:
KiM ) [MCDi]/[MCDi-1][CD]
[CDb] )
∑iβ
M i
[CD]i[M]
(11)
i)0
(1) The free (unbound) cyclodextrin concentration [CD] is
[CD] ) [CDt] - [CDb]
The total solute concentration [Mt] was
(12)
[Mt] ) [M] + [MCD] + ... + [MCDi] + ... + [MCDn] (2) The average number nc of CD molecules bound per molecule is The molar fractions of the individual form MCDi noted RiM were given by
RiM ) [MCDi]/[Mt]
(3)
The global constants of the MCDi complex formation corresponding to the equilibrium M + iCD h MCDi were M
βi ) [MCDi]/[M][CD]
i
nc ) [CDb]/[Mt]
(13)
[CD] ) [CDt] - nc[Mt]
(14)
Therefore
(βiM)
(4)
) K1MK2M ... KiM ... KnM
Substitution of eq 14 into eq 9 provides a polynomial equation which describes the behavior of the guest complexing n cyclodextrin molecules. If β2 ) β3 ) ... ) βn ) 0, the final formula describing the formation of 1:1 complexes is related to the following equation:
n
βiM )
∏K
M
(5)
i
i)0
µe ) (µ0M + β1Mµ1M([CDt] - nc[Mt]))/ (1 + β1M([CDt] - nc[Mt])) (15)
with
β0M ) 1
(6)
Thus, in combining eqs 2-4, the following was obtained:
The proposed nonlinear model involves no approximation of the concentration of the two species CD and M and correlates their initial, total concentration with the pH of the BGE and the effective electrophoretic mobility of the free µ0M and complexed µiM solute. Refinements of the treatment outlined here include the use of
n
RiM ) (βiM [CD]i)/(
∑β
M i
[CD]i)
(7)
i)1
The different MCDi forms, which were in rapid dynamic equilib-
(23) Grossmann, P. D.; Colburn, J. C. Capillary Electrophoresis: Theory and Practice; Academic Press: San Diego, 1992. (24) Li, S. F. Y. Capillary Electrophoresis: Principles, Practice and Applications; Elsevier: Amsterdam, 1992. (25) Foret, F.; Krivankova. L.; Bocek, P. Capillary Electrophoresis; VCH Verlagsgesellschaft: Weinheim, 1993; pp 15-19.
Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
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corrections to mobility to account for the variation in solution viscosity with increasing selector concentration. The µe value was replaced by the corrected mobility µ′:
µ′ ) µeη/η0
(16)
where η and η0 are the viscosities of the BGE with and without a chiral selector.3,26-34 Therefore, if 1:1 stoichiometry has already been established, eqs 15 and 16 can be used to describe the effective electrophoretic mobility of a ionizable solute. The unknown global binding constants β1M in the simple case of the 1:1 complexes or β1M and β2M in 1:2 complexes can be calculated according to this model by using weighted nonlinear regression.35 Thermodynamic Considerations. Considering a 1:1 complex, if ∆GM°, ∆HM°, and ∆SM° are the Gibbs free energy, enthalpy, and entropy, respectively, for the inclusion complex formation between the solute molecule M and the cyclodextrin molecule, the van’t Hoff plot equations are
ln β1M ) - ∆GM°/RT
(17)
∆GM° ) ∆HM° - T∆SM°
(18)
ln β1M ) -∆HM°/(RT) + ∆SM°/R
(19)
with
then,
T is the temperature and R the gas constant. Equation 19 shows that ln β1M versus 1/T is a van’t Hoff plot. For a linear plot, the slope and the intercept are -∆HM°/R and ∆SM°/R, respectively. EXPERIMENTAL SECTION Apparatus. CE separations were carried out using an automated CE apparatus (Beckman, Pace 550, Paris, France). The capillaries used were 57 cm (50 cm to the detector) × 75 µm i.d. The following conditions were applied: voltage 30 kV; capillary thermostated at 25 °C unless otherwise specified; UV detection at 230 nm; 2-s pressure injection of imidazole derivative solution dissolved in the BGE. Solvents and Samples. Water was obtained from an Elgastat option I water purification system (Odil, Talant, France) fitted with a reverse osmosis cartridge. The BGE buffer consisted of a (26) Penn, S. G.; Goodall, D. M.; Loran, J. S. J. Chromatogr. 1993, 636, 149. (27) Penn, S. G.; Bergstrom, E. T.; Goodall, D. M.; Loran, J. S. Anal. Chem. 1994, 66, 2866. (28) Baumy, P.; Morin, P.; Dreux, M.; Viaud, M. C.; Boye, C.; Guillaumet, G. J. Chromatogr., A 1995, 707, 311. (29) Sadek, M. J. Electroanal. Chem. Interfacial Electrochem. 1983, 144, 11. (30) Ipono, T.; Takahashi, K.; Tamamushi, R. Bull. Chem. Soc. Jpn. 1981, 54, 2183. (31) Valko, I. E.; Billiet, H. A. M.; Frank, J.; Luyben, K. C. A. M. Chromatographia 1993, 35, 419. (32) Shibukawa, A.; Lloyd, D. K.; Wainer, I. W. Chromatographia 1993, 35, 419. (33) Vespalek, R.; Fanali, S.; Bocek, P. Electrophoresis 1994, 15, 1523. (34) Wren, S. A.; Rowe, R. C.; Payne, R. S. Electrophoresis 1994, 15, 774. (35) Bevington, P. R. Data reduction and error analysis for the physical sciences; McGraw-Hill: New York, 1969.
2048 Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
Figure 1. Imidazole derivative structures.
phosphate buffer with CD concentrations varying from 0 to 14 mmol/L (15 values were included in this range). β-CD was obtained from the Roquette Laboratories (Lestrem, France). The phosphate buffer was composed of 0.05 M diammonium hydrogen phosphate and 0.05 M ammonium dihydrogen phosphate (ionic strength 0.2 M). The BGE pH were adjusted to values equal to 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, and 7.5 with ammoniac or phosphoric acid. The BGE, at all pH values, were stocked for 1, 2, and 4 h at ambient room temperature to study the accuracy of their pH values. No fluctuations were observed; the maximum relative difference of the pH value of the different mobile phases was always 0.5%. Bifonazole (1), clotrimazole (2), econazole (3), sulconazole (4), miconazole (5), and oxiconazole (6), obtained from Sigma (Saint-Quentin, France), were dissolved in pure acetone to obtain a concentration varying from 0.1 to 1 mM. The chemical structures of these compounds are given in Figure 1. Mobilities were calculated from migration times of analytes and a neutral marker mesityl oxyde (Aldrich, Paris, France).
Table 1. β1M and nc Values for the Six Imidazole Derivatives at pH ) 4.5 and T ) 25 °C
bifonazole (1) clotrimazole (2) econazole (3) sulconazole (4) miconazole (5) oxiconazole (6)
β1M
nc
348 32 203 295 54 61
0.95 0.56 0.82 0.85 0.72 0.79
Temperature Studies. Compound migration times were determined at the temperatures 20, 25, 30, 35, 40, 45, 50, and 55 °C. The electrophoretic system was equilibrated at each temperature for at least 1 h prior to each experiment. To study this equilibration, the compound migration time of the bifonazole was measured every hour for 7 h and again after 22, 23, and 24 h. The maximum relative difference in the migration times of this compound between these different measurements was always 0.6%, making the electrophoretic system sufficiently equilibrated for use after 1 h. All the solutes were injected in triplicate at each temperature and pH value. RESULTS AND DISCUSSION Validation of the Electrophoretic Model. To obtain the constants β1M and nc at 25 °C and pH ) 4.5, the effective electrophoretic mobilities of the six imidazole derivatives were determined for a wide range of total concentrations of CD and M. All the experiments were repeated three times. The variation coefficients of the µe values were less than 2% in most cases, indicating a high reproducibility and good stability for the electrophoretic system. Using the weighted nonlinear regression, the β1M and nc values were determined (Table 1). The correlation between values predicted by the model (eq 15) and experimental µe values is presented in Figure 2. The slope (0.998; ideal is 1.00) and r2 (0.987) indicate that there is an excellent correlation between the predicted and the experimental effective mobilities. Furthermore, by examining the behavior of eq 15, it becomes clear that by changing the [Mt] value and assuming that all the guest molecules are complexed (avoiding the term nc), the β1M value changes significantly. On the contrary, when the term nc was used, a balance was kept between the [Mt] value and the calculated β1M values; for instance, when [Mt] was doubled, the value of nc was halved with the β1M values remaining almost unchanged. nc can also be used as an indication of the extend to which the pH and temperature value changes the complexation mechanism. To investigate the dependence of the pH and temperature on the µe values, the previous experiments carried out at 25 °C and pH ) 4.5 were repeated at other temperatures and pH, and the model parameters corresponding to each temperature and pH were calculated using the same methodology. Complex Formation Constant β1M Values. The β1M values calculated from the model equation are given in Table 1. The values again agree with values reported in the literature,36,37 showing the high reliability of the CE method. Two groups of compounds were distinguished according to the β1M values. (36) Sadlej-Sosnowska. N. Eur. J. Pharm. Sci. 1995, 3, 1. (37) Sadlej-Sosnowska. N. J. Chromatogr., A 1996, 728, 89.
Figure 2. Correlation between the predicted (eq 15) and the experimental electrophoretic mobilities for the six imidazole derivatives. The slope is 0.998 with a correlation coefficient of 0.987, as determined by linear regression.
Bifonazole, econazole, and sulconazole, which constitute the first group, had the highest formation constant values in comparison with those of the second group composed of miconazole, oxiconazole, and clotrimazole. This result demonstrates that the β-CD size is more appropriate for bifonazole, econazole, and sulconazole than for miconazole, oxiconazole, and clotrimazole. Van’t Hoff Plots. The curves were all linear for the six imidazole derivatives and for all pH values. The correlation coefficients for the fits were over 0.986. The typical standard deviations of the slope and the intercept obtained were respectively 0.008 and 0.05. Figure 3 shows the van’t Hoff plot for bifonazole at pH ) 4.5. Table 2 contains a complete list of ∆HM° and ∆SM° values for all solutes for all the pH values. ln(nc) vs 1/T Plots. The degree of complexation was fitted to the equation
nc ) nco exp(-∆H ˜ /kT)
(20)
where nco is a preexponential factor, T is the temperature, k is the Boltzman constant, and ∆H ˜ is a term called activation enthalpy (∆H ˜ ) ∆U + P∆V, where ∆U is activation energy, ∆V is activation volume, and P is pressure). In all cases, nc values decreased with increasing temperature. ln(nc) versus 1/T plots were linear at pH ) 7 and 7.5 (Figure 4) for all solutes but present a break for the other pH values at T ) Tc ≈ 35 °C (Figure 5). Enthalpy-Entropy Compensation. Investigation of the enthalpy and entropy compensation temperature is a thermodynamic approach to the analysis of physicochemical data.38 Enthalpy(38) Sander, L. C.; Field, L. R. Anal. Chem. 1980, 52, 2009.
Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
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Table 2. Thermodynamic Parameters with Standard Deviations (in Parentheses): (A) ∆H°M and (B) ∆S°M for the Six Imidazole Derivatives at All pH Values pH ) 4.5 1 2 3 4 5 6 1 2 3 4 5 6
-17.2 (0.5) -9.7 (0.1) -12.3 (0.2) -16.1 (0.3) -10.3 (0.2) -10.8 (0.1) -1.80 (0.05) -0.30 (0.02) -1.50 (0.01) -1.30 (0.02) -1.21 (0.03) -0.60 (0.02)
pH ) 5.0
pH ) 5.5
-16.8 (0.6) -9.5 (0.1) -11.9 (0.1) -15.6 (0.1) -10.0 (0.2) -10.10 (0.08)
-16.1 (0.3) -9.0 (0.1) -11.0 (0.2) -15.4 (0.2) -9.87 (0.1) -9.76 (0.1)
-1.60 (0.05) -0.25 (0.01) -1.45 (0.06) -1.25 (0.03) -1.00 (0.03) -0.55 (0.01)
pH ) 6.0 (Α) ∆H°M (kJ/mol) -15.9 (0.3) -8.7 (0.2) -10.8 (0.4) -15.0 (0.4) -9.5 (0.2) -9.6 (0.1)
(B) ∆S°M (J mol-1 K-1) -1.50 (0.01) -1.45 (0.01) -0.20 (0.01) -0.15 (0.01) -1.40 (0.03) -1.35 (0.04) -1.20 (0.02) -1.00 (0.03) -0.95 (0.02) -0.85 (0.02) -0.50 (0.02) -0.45 (0.01)
pH ) 6.5
pH ) 7.0
pH ) 7.5
-15.0 (0.1) -8.4 (0.2) -10.5 (0.2) -14.9 (0.3) -9.4 (0.1) -9.0 (0.1)
-14.9 (0.1) -8.0 (0.1) -10.0 (0.2) -14.0 (0.3) -9.0 (0.2) -8.5 (0.2)
-14.1 (0.2) -7.9 (0.1) -9.9 (0.1) -13.6 (0.2) -8.5 (0.2) -8.7 (0.2)
-1.20 (0.05) -0.11 (0.02) -1.25 (0.01) -0.90 (0.02) -0.82 (0.03) -0.40 (0.03)
-1.10 (0.01) -0.10 (0.01) -1.15 (0.01) -0.85 (0.02) -0.65 (0.02) -0.35 (0.02)
-1.00 (0.03) -0.09 (0.01) -0.80 (0.02) -0.70 (0.02) -0.55 (0.03) -0.20 (0.01)
Figure 5. ln (nc) versus 1/T for bifonazole at pH ) 4.5. Figure 3. Van’t Hoff plot for bifonazole at pH ) 4.5.
Figure 4. ln (nc) versus 1/T plot for bifonazole at pH ) 7.0.
entropy compensation was used to test the variation in the complexation mechanism of a solute with its molecular structure. A plot of ln β1M (for T ) 303 K) against -∆H°M calculated for each of the six solutes, when the pH had a value of 7.0 and 7.5, was drawn. The correlation coefficients r for the fits were at least 2050 Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
equal to 0.900. This can be considered adequate to verify enthalpy-entropy compensation. Nevertheless, when clotrimazole was suppressed, the linear fit was better; r was at least equal to 0.986. Therefore, the retention mechanism can be thought to be independent of the solute molecular structure. For a set of compounds where there is enthalpy-entropy compensation, the slope of ln k′ versus -∆H° will be the same for the same type of reaction.39-41 For pH ) 4.5, 5.0, 5.5, 6.0, and 6.5, the relative difference in the slope values obtained was less than 5% and for pH ) 7 and 7.5 less than 6%. This comparison indicated that the imidazole complexation process was the same for these two groups of pH values. Topological Model. Another model was developed to explain the existence of a break on the ln(nc) versus 1/T curves observed at pH ) 4.5, 5.0, 5.5, 6.0, and 6.5. This model was based on the structure of β-cyclodextrin (Figures 6 and 7). It has been demonstrated by differential scanning calorimetry and thermogravimetric analysis that when the pH were different from physiological pH (between 7.0 and 7.5) the β-CD cavity structure was in equilibrium between an ordered and disordered state.42 (39) Boots, H. M. J.; de Bokx, P. K. J. Phys. Chem. 1989, 93, 8243. (40) Tchapla, A.; Heron, S.; Colin, H.; Guiochon, G. Anal. Chem. 1988, 60, 1443. (41) Guillaume, Y. C.; Guinchard, C. J. Phys. Chem. 1997, 101, 8390.
Figure 8. Theoretical potential V for a CD field with two temperatures. For T < Tc (domain 1), the only minimum was Φ ) 0. For T > Tc (domain 2), a circle (3) with several minimums appear. Figure 6. β-Cyclodextrin structure.
Figure 7. β-Cyclodextrin functional structure representation.
The physical parameter called the “cyclodextrin field”, FCD, which is linked to the “high-energy water” inside the CD cavity, was introduced.43 The phase transition is simply understood as occurring through competition between a state where FCD can be supposed to go in all directions (disordered state of the CD) and a second state that favors it being oriented (ordered state of the cyclodextrin). Similarly with “Higgs potentials”, in particle physics theories,44 at a sufficiently low temperature the potential V from which FCD is derived had only one minimum Φ ) 0 which respected the symmetry. When the temperature increased above a critical temperature Tc, V changed its form and a great variety of minimums appeared where Φ was not equal to zero. This set where Φ was not equal to zero was called the “CD variety”.44 As an example, the potential for a “CD field” with two components is shown in Figure 8. During the symmetry-breaking phase transition, the “CD variety” had closed loops, which in this variety did not contract into one point, and a topological default appeared. This one was similar to an energetic rope (Figure 9) which brought about a deformation of the cyclodextrin cavity (disordered state). This formation of energetic rope loops is analogus to vortex (42) Morin, N.; Guillaume, Y. C.; Peyrin, E.; Rouland. J. C. Anal. Chem. 1998, 70, 2819. (43) Steiner, T.; et al. Angew. Chem., Int. Ed. Engl. 1995, 34, 1452. (44) Hill, C. T.; Schramm, D. N.; Fry, J. N. Commun. Nucl. Particle Phys. 1989, 19, 25.
Figure 9. FCD (“CD field”) pointing in all directions at the time of the phase transition. An energetic rope is created at this time.
creation in liquid helium following a rapid transition into the superfluid state45,46 (the superfluid an analogue of energetic ropes). Application of This Topological Model to the Solute Complexation Mechanism. (a) For pH ) 4.5, 5.0, 5.5, 6.0, and 6.5. At a temperature T < Tc, the β-CD cavity, following the topological model, was in a more ordered state than that for T > Tc. The creation of energetic ropes implied that the water molecules inside the CD cavity were more constrained for T < Tc than for T > Tc. Thus, the water molecules at T < Tc were released more quickly from the hydrophobic cavity and replaced more easily by the apolar imidazole derivative than at T > Tc. This result was objectified by the fact that nco(T nco(T>Tc) and ˜ (T Tc ≈ 35 °C nco ) 1.59 × 10-5 ∆H ˜ (eV) ) -0.29
molecules from the cavity and the inclusion of the solute in the cavity decreased and resulted in higher ∆HM° and ∆SM° values. In summary, in this paper, the solute inclusion process was studied for six imidazole derivatives by HPCE over a range of BGE, pH, and temperature. A nonlinear model was developed for the calculation of the binding constants in multiple complex formation of isomers with CDs. The calculation was based on the effective electrophoretic mobility of a guest being affected by the presence of CDs. As this model requires only the initial concentrations of the two species (M and CD), without any limitations and approximations, experimental and theoretical shortcomings are avoided. Enthalpy-entropy compensation revealed that the com-
2052 Analytical Chemistry, Vol. 71, No. 10, May 15, 1999
plexation process of imidazole derivatives was independent of the molecular structure. It was identical at pH ) 4.5, 5.0, 5.5, 6.0, and 6.5 and changed at pH ) 7.0 and 7.5. The solute complexation isotherms (ln(nc) versus 1/T) calculated at different pH values confirm the DSC and TGA measurements;42 i.e., this behavior is a result of a phase transition in the CD cavity structure. A topological model was described to elucidate this phase transition. It was based on the existence of topological defects inside the cavity. Variations of column temperature and mobile-phase pH tended to cause a change in the topological default density and, thus, a transition between an ordered and a disordered state of the CD cavity. This confirms the need to control temperature for the formation of inclusion complexes, especially at pH ) 4.5, 5.0, 5.5, 6.0, and 6.5 where the CD phase transition takes place. ACKNOWLEDGMENT We thank M. Thomassin for her technical assistance.
Received for review July 30, 1998. Accepted February 6, 1999. AC980848U
