Introduction

The logarithm of octanol–water partition coefficient (logP ow) is widely used as a general value for lipophilicity, which is an essential property of molecules, because it is highly related to solubility, membrane permeability, protein binding and so on [7]. A number of direct and indirect methods have been applied for logP ow measurement [2]. The direct methods include conventional shake-flask method, stir flask, two-phase titration, etc., and the shake-flask method has historically been considered the standard assay. These direct methods have several limitations: they are time consuming, have low reproducibility, require relatively large amounts of pure sample and are difficult to automatize. The indirect methods include reversed-phase high performance liquid chromatography (RP-HPLC) [3, 4], reversed-phase thin-layer chromatography (RP-TLC) [31] and micelle electrokinetic chromatography (MEKC) [14, 17], which use instrumental techniques and are thus faster and easier to automate than direct methods. Nevertheless, these indirect methods still have some shortcomings. For example, the range of obtainable lipophilicity is limited and dependent upon the partition chromatographic system [12, 13]. Furthermore, in RP-HPLC and RP-TLC, the hydrogen bond effects must be considered in logP ow determinations for compounds with hydrogen bond acceptor and donor, while in MEKC the electrostatic interactions between the polar solutes and the surface of the micelle must be considered.

In 1991, a branch of capillary electrophoresis, microemulsion electrokinetic chromatography (MEEKC), was developed [28], which almost possesses the required properties for estimating logP ow, because of the particular compositions of buffer. The technique uses microemulsion to separate charged or uncharged solutes based on both their electrophoretic mobility and lipophilicity, and the latter directly determines the partitioning of analyte between the microemulsion pseudostationary phase and the aqueous phase. Since its invention, MEEKC has been widely used for measurement of logP ow [1, 5, 6, 8–11, 15, 16, 18–26, 29]. However, thus far, the accuracy of logP ow measurement by MEEKC has not been evaluated and there are several questions that need to be addressed. For example, what is the range of logP ow measurement for a given microemulsion system? Is the measured value accurate in any range? How would the precision of logP ow (ΔlogP ow) be affected by the experimental uncertainty in measured migration time (Δt m)? In this paper, a theoretical model for the correlation between ΔlogP ow and Δt m is presented and experimentally verified with some microemulsion systems. Using this model, it is demonstrated that acceptable precision and accuracy can be achieved for the measurement of logP ow by MEEKC within a useful interval of a given microemulsion system.

Theoretic Calculations

The separation mechanism in MEEKC is based on the differential partitioning of analyte (uncharged ones) between the microemulsion pseudostationary phase and the aqueous phase, and the partitioning of analyte is related to its lipophilicity. Therefore, the migration time of an analyte is correlated to its lipophilicity. First, the capacity factor, k, for an uncharged analyte in MEEKC can be calculated as follows [8]:

$$ k = \frac{{t_{\text{m}} - t_{\text{o}} }}{{t_{\text{o}} \left( {1 - t_{\text{m}} /t_{\text{me}} } \right)}} $$
(1)

where t o, t me and t m are the migration times of the electroosmotic flow (EOF), the microemulsion phase and the analyte, respectively.

Then the relationship between log k and logP ow is known as the functional form [8]:

$$ \log P_{\text{ow}} = a\log k + b $$
(2)

where a and b are constants that represent the slope and intercept of a linear correlation calibration. Once a and b are established using a set of compounds with known logP ow values, Eq. 2 can be applied to determine logP ow of other compounds.

Combining Eqs. 1 and 2, the following equation can be obtained:

$$ \log P_{\text{ow}} = a\log \frac{{t_{\text{m}} - t_{\text{o}} }}{{t_{\text{o}} \left( {1 - t_{\text{m}} /t_{\text{me}} } \right)}} + b $$
(3)

In order to find the correlation between the uncertainty, or precision of logP ow (ΔlogP ow) and the uncertainty of migration time measurement (Δt m), logP ow in Eq. 3 is differentiated with respect to t m, leading to the following relationship:

$$ {\text{d}}\log P_{\text{ow}} = 0.434a\frac{{t_{\text{me}} - t_{\text{o}} }}{{\left( {t_{\text{me}} - t_{\text{m}} } \right)\left( {t_{\text{m}} - t_{\text{o}} } \right)}}{\text{d}}t_{\text{m}} $$
(4)

or

$$ \Updelta \log P_{\text{ow}} = 0.434a\frac{{t_{\text{me}} - t_{\text{o}} }}{{\left( {t_{\text{me}} - t_{\text{m}} } \right)\left( {t_{\text{m}} - t_{\text{o}} } \right)}}\Updelta t_{\text{m}} $$
(5)

Equation (5) shows that the extent of effects of Δt m on ΔlogP ow is given by the coefficient \( 0.434a\frac{{t_{\text{me}} - t_{\text{o}} }}{{\left( {t_{\text{me}} - t_{\text{m}} } \right)(t_{\text{m}} - t_{\text{o}} )}}. \) For convenience, a function f(t m) that describes the importance of the analytes migration time upon the correlation is defined as:

$$ f\left( {t_{\text{m}} } \right) = \frac{{t_{\text{me}} - t_{\text{o}} }}{{\left( {t_{\text{me}} - t_{\text{m}} } \right)\left( {t_{\text{m}} - t_{\text{o}} } \right)}} \, \left( {t_{\text{o}} < t_{\text{m}} < t_{\text{me}} } \right). $$
(6)

Thus, ΔlogP ow and Δt m are simply correlated as:

$$ \Updelta \log P_{{{\text{ow}}\,}} = \,0.434 \cdot a \cdot f\left( {t_{\text{m}} } \right) \cdot \Updelta t_{\text{m}} $$
(7)

Equation (7) shows that as the value of f(t m) increases, ΔlogP ow is more strongly affected by Δt m.

Figure 1 shows the curve of f(t m) versus t m for the microemulsion system (ME) used in our experiment, where t o and t me are 6.50 min and 32.00 min, respectively. It is obvious that within a relatively wide range of migration times (t m), the value of f(t m) is small and almost constant, but as t m approaches either t o or t me, the value of f(t m) increases rapidly and leads to unacceptable uncertainty in the prediction of logP ow using Eq. 2.

Fig. 1
figure 1

The curve of \( f\left( {t_{\text{m}} } \right) \) versus t m for an MEEKC system with t o = 6.50 min and t me = 32.00 min

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To find the useful interval of analyte migration times, f(t m) in Eq. 6 is differentiated with respect to t m and the requirement is implemented that df(t m)/dt m must be numerically less than 1. Thus,

$$ f^{\prime}\left( {t_{\text{m}} } \right) = \frac{{{\text{d}}f\left( {t_{\text{m}} } \right)}}{{{\text{d}}t_{\text{m}} }} = \frac{1}{{\left( {t_{\text{me}} - t_{\text{m}} } \right)^{2} }} - \frac{1}{{\left( {t_{\text{m}} - t_{\text{o}} } \right)^{2} }} $$
(8)

and the lower and upper limits of t m (t m1, t m2) can be calculated by inserting either \( f^{\prime}\left( {t_{\text{m}} } \right) = - 1, \) or \( f^{\prime}\left( {t_{\text{m}} } \right) = + 1 \) into Eq. 8 and numerically solving the equation. For the microemulsion system with t o = 6.50 min and t me = 32.00 min, t m1 and t m2 are 7.66 and 30.84 min, respectively.

When \( f^{\prime}\left( {t_{\text{m}} } \right) = 0, \) f(t m) is minimal. In this case, \( t_{\text{m}} = \frac{{t_{\text{me}} + t_{\text{o}} }}{2}, \) and \( f\left( {t_{\text{m}} } \right)_{\min } = \frac{4}{{t_{\text{me}} + t_{\text{o}} }}. \) For our microemulasion system, f(t m)min is calculated as 0.104 at t m = 19.25 min, which is the center of the useful interval.

Experimental

Reagents

Fluorene (98%), pyrene (99%), benzanthracene (99%) and coronene (97%) were purchased from Aldrich Chemical Company, Inc. (USA). Biphenyl (AR) was made in the Department of Chemistry, University of Copenhagen (Denmark). 1-Heptane (AR), 1-butanol (99%), sodium dodecyl sulfate (SDS) and dimethyl sulfoxide (DMSO, AR) were purchased from Ferak (Germany), Sigma (USA) and E. MERCK (Germany), respectively. The water for solution preparation was deionized and purified through a Millipore (Water class 2) system.

Instruments

A Beckman P/ACE System 5000 (Beckman Instruments Inc., Fullerton, CA) equipped with P/ACE UV absorbance detector at 200 nm was employed for all MEEKC measurements. Data were collected at a rate of 5 Hz using System Gold (Beckman Instrments Inc. Fullerton, CA), and a presented chromatogram was redrawn using Microsoft Excel and Origin 8.0. An uncoated fused silica capillary (SGE, SGE Analytical Science Pty Ltd., Australia) of 75 μm i.d. and 56.5 cm length (50.3 cm to the detector), which was thermostated by a liquid coolant, was employed in the experiments. A PHM220 Lab pH Meter (MeterLab™, Radiometer, Villeurbanne, CEDEX-Lyon, France) was used in the measurement of pH for running media and sample solutions.

Preparation of Microemulsions and Sample Solutions

The microemulsion containing 45 mM SDS, 80 mM 1-heptane, 850 mM 1-butanol and 5 mM phosphate at pH 7.0 was prepared according to our previous work [30].

A tracer solution was prepared in the microemulsion system with 1/1,000 (v/v) of DMSO as the EOF tracer. Weighted amounts of the respective samples were dissolved in the microemulsion solution to give concentrations between 0.025 and 0.05 mg mL−1.

Experiment of MEEKC

Before running of the MEEKC, the capillary was treated with 1 mol L−1 NaOH for 15 min, followed by a purge of 0.1 mol L−1 NaOH for 10 min, and then rinsed with water for 10 min and the microemulsion system for 10 min sequentially. Purges with 0.1 mol L−1 NaOH and water were performed periodically to remove contaminants from the capillary wall. Between the electrophoretic runs, the capillary was rinsed with microemulsion system for 5 min. All logP ow determinations were based upon five successive electropherograms.

Results and Discussion

Determination of logP ow Values Using MEEKC

In a previous paper [30], six compounds (sulfanilamide, aniline, p-toluidine, phenol, β-naphthol and naphthalene) with known logP ow values from −1.05 to 3.33 were used to establish a calibration equation between logP ow and measured migration times (t m) as

$$ \log P_{\text{ow}} = 2.42\left( { \pm 0.25} \right)\log k + 1.53\left( { \pm 0.14} \right);\,r = 0.9796 $$
(9)

This equation was used again in this paper to predict logP ow values for a range of compounds with values of logP ow known in the literature from −1.74 to 7.38. Although the original calibration equation was established using a MEEKC instrument different from that used here, the operating parameters for the two instruments were set up as identical as possible. Figure 2 shows the MEEKC electropherogram of five polyaromatic hydrocarbons (PAHs) with logP ow values higher than those used previously to establish the calibration equation. The new results, together with those from our previous work [30], are displayed in Table 1. It is seen that the error between the logP ow values measured by MEEKC and those of the literature is about or below 0.3 logarithm units in most cases, except for the compounds with logP ow values greater than 5.00, indicating that the measurements were fairly accurate.

Fig. 2
figure 2

MEEKC electropherograms for five PAHs. Conditions Microemulsion: 45 mM SDS, 80 mM 1-heptane, 850 mM 1-butanol and 5 mM phosphate, pH 7.0; separation voltage: 16.67 kV; injection: 34.5 mbar × 3 s; wavelength: 200 nm; peaks: 1 DMSO, 2 biphenyl, 3 fluorene, 4 pyrene, 5 benzanthracene, 6 coronene

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Table 1 Comparison of logP ow obtained by MEEKC versus literature values for 11 uncharged compounds
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Figure 3 shows the logP ow values determined by MEEKC and those from the literature correlated by the linear least-square fitting, as well as a perfect fit of Y = X. The slope and intercept by the linear least-square fitting are 0.952 (±0.026), −0.170 (±0.107), respectively, and the linear correlation coefficient r is 0.9963. This is a surprisingly good correlation between the MEEKC-determined values and those of the literature considering the fact that two different MEEKC instruments were used and the calibration equation obtained by one MEEKC instrument was transferred to the other. The observed deviation from the ideal correlation between measured logP ow values and literature values seems to become larger at the extremes in agreement with the theoretical expectations found in “Theoretic Calculations”. The measurement error includes both systematic error and experimental uncertainty. The systematic error is represented by the deviation of the linear least-square fitting from the perfect fit between the MEEKC-determined values and those of the literature. The experimental uncertainty of logP ow can be evaluated by the residual between the measured values and the linear least-square fitting when the systematic errors are removed by the fit, which is called ΔlogP ow (Experiment) here. This will be further discussed in detail in the next section.

Fig. 3
figure 3

Plot of experimentally obtained logP ow values using MEEKC against literature logP ow values, and the perfect fitting of Y = X

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Evaluation of the Accuracy of MEEKC Using the Deduced Theory

Equation (7) shows that the uncertainty in predicting logP ow (ΔlogP ow) is given by a constant multiplied by a function of the migration time f(t m) multiplied by the uncertainty in migration time (Δt m). The importance of the function f(t m) in ΔlogP ow has been discussed thoroughly in “Theoretic Calculations”. However, in order to evaluate the full extent of uncertainty of our MEEKC system (ΔlogP ow), we need to know Δt m as a function of t m. To obtain that value, we recorded the electropherogram of all compounds five times and calculated the standard deviation of t m for each compound. With the migration time for each compound given as the mean value of five determinations, we can estimate Δt m as 1.24 times the standard deviation for a 95% confidence interval (CI), since:

$$ {\text{CI}} = {\text{mean}} \pm {\text{CL}} $$
(10)

where CL is confidence level, and for n = 5,

$$ {\text{CL}} = s \times 2.776/\sqrt 5 = s \times 1.24 $$
(11)

Therefore, the ΔlogP ow values for each compound can be obtained using Eq. 7, which is called ΔlogP ow (Model) here. All the data, including the experimentally measured Δt m values, together with the calculated values of f(t m) and estimated ΔlogP ow (Model) for all compounds are shown in Table 2.

Table 2 Experimentally measured t m values and estimation of ΔlogP ow by our model for 11 uncharged compounds based upon Eq. 7
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To have a fair comparison for the model and experiment, the experimental uncertainty should be compared to CL,

$$ \Updelta \log P_{\text{ow}} \left( {\text{model}} \right) = {\text{CL}} $$
(12)
$$ \Updelta \log P_{\text{ow}} \left( {\text{experiment}} \right) = {\text{Uncertainty}} $$
(13)

Table 2 gives the experimentally measured Δt m values, together with a calculated value of f(t m) and estimated ΔlogP ow using Eq. 7 for all compounds. The curves of ΔlogP ow including ΔlogP ow (Model) and ΔlogP ow (Experiment) as a function of logP ow are shown in Fig. 4. These two curves show the same trend as the curve for f(t m) (Fig. 1), except that the rise at the extremes is slower, suggesting that the MEEKC system does not show an extremely abrupt transfer from a high precision to a low precision method, when the migration times approach either t o or t me. Instead, a somewhat smoother deterioration in the accuracy is observed. Nevertheless, there is still a wide range of logP ow values that can be measured with fairly high precision. It is also noted in Fig. 4 that while following the same trend, most of the experimentally obtained ΔlogP ow (Experiment) are somewhat greater than the pseudo-theoretical ΔlogP ow (Model) versus logP ow. This thereby verifies the model for MEEKC accuracy. Based on the above observation, the high accuracy range of the system is estimated to be between a logP ow value of 0.50 and 5.50, which is narrower than the range of 0–6.50 as estimated by the f(t m) curve alone. Outside the range of 0.50–5.50, the uncertainty increases rapidly.

Fig. 4
figure 4

The error of logP ow (ΔlogP ow) versus logP ow. The solid line ( ) is polynormal of 95% confidence interval for random uncertainty(CI) versus logP ow, which represents the ΔlogP ow obtained from our model versus logP ow, i.e., ΔlogP ow (Model); see Table 2. The dashed line ( ) is polynormal of residual versus logP ow, which represents the errors between logP ow values obtained experimentally by MEEKC and those from the linear least-square fitting, i.e., ΔlogP ow (Experiment); see Table 1

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Conclusions

A theoretical model for the dependence of the precision of logP ow (ΔlogP ow) determinations upon deviations in migration time (Δt m) in MEEKC was deduced. This model shows that in a given microemulsion system, there is a relatively wide range of logP ow values that can be measured precisely and accurately. However, as the migration time of analyte approaches either the migration time of the EOF (t o) or of the microemulsion phase (t me), the precision of logP ow measurement by MEEKC deteriorates relatively fast. For a system with t o equal to 6.50 min and t me equal to 32.00 min, respectively, the useful range of logP ow for accurate measurement is from 0.50 to 5.50, which has been established by both theoretic calculation and experimental verification.