Restricted Diffusion Model for Surface Diffusion in Reversed-Phase

Anal. Chem. 2000, 72, 1475-1489

Restricted Diffusion Model for Surface Diffusion in Reversed-Phase Liquid Chromatography Kanji Miyabe and Georges Guiochon*

Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

The analysis of experimental results in reversed-phase liquid chromatography (RPLC) allows further discussion of the restricted diffusion model of surface diffusion formulated on the basis of the absolute rate theory. Chromatographic data were acquired on different RPLC systems with two series of homologous compounds, several stationary phases having different alkyl ligand densities and ligands of various lengths, and methanol/ water mobile phases of different compositions. The enthalpy-entropy compensation observed and the linear free energy relationships found for surface diffusion suggest that the surface diffusion mechanism remains probably the same in all RPLC conditions studied. Whereas the isosteric heat of adsorption approaches zero with decreasing retention, the activation energy of surface diffusion tends toward a finite limit and the surface diffusion coefficient tends toward a value near the corresponding molecular diffusivity. These results support the validity of the restricted diffusion model. The influence of different factors on the validity of this model (i.e., the activation energy and the frequency factor of surface diffusion, and the surface tortuosity) was also considered. Mass transfer kinetics in chromatographic columns contribute much to the performance achieved in both analytical and preparative separations.1 Retention data can be explained with a few simple thermodynamic models. By contrast, the mass transfer mechanisms of solutes in the stationary phase are complex, difficult to investigate, and were paid relatively little attention by chromatographers although intraparticle mass transfer was and is actively studied in gas-solid and liquid-solid systems. Most recent theoretical acquisitions in these areas remain ignored in our field. It is now well known that the contribution of surface diffusion to intraparticle diffusion is often important.2 By contrast, this phenomenon and its contribution to mass transfer kinetics and column efficiency are still not widely recognized in chromatography, although the significance of surface diffusion as one of the mass transfer mechanisms involved in this field was already described more than 30 years ago.3 For instance, there are no plate height equations including a term accounting for the (1) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 1994. (2) Kapoor, A.; Yang, R. T.; Wong, C. Catal. Rev.sSci. Eng. 1989, 31, 129. (3) Giddings, J. C. Dynamics of Chromatography, Part I, Principles and Theory; Marcel Dekker: New York, 1965. 10.1021/ac9909913 CCC: $19.00 Published on Web 02/23/2000

© 2000 American Chemical Society

contribution of surface diffusion to column efficiency, even though some kind of lumped diffusivity experienced by sample molecules in the stationary phase film is taken into account in almost all plate height equations.1,3 Recently, several groups reported experimental data for surface and lateral diffusion in RPLC.4-8 Octadecyl or methyl bonded phases and water or various methanol/water solutions were used as the stationary and the mobile phase, respectively.4-7 Only experimental values of the surface or lateral diffusion coefficient (Ds) were reported. No systematic interpretation was presented for the intrinsic characteristics of surface diffusion in RPLC, e.g., the mass transfer mechanisms and their physicochemical properties. In a recent paper,8 we reviewed surface diffusion data in RPLC measured with various solutes, stationary phases, and mobile phases, and clarified some characteristic features of surface diffusion in RPLC. It was concluded that surface diffusion played an important role in intraparticle diffusion. To date, most fundamental studies on surface diffusion have been made from the viewpoints of the dependence of Ds on temperature and the amount of solute adsorbed (q) in order to clarify the main characteristics of surface diffusion.2 The temperature dependence of Ds is usually accounted for by applying Arrhenius equation.

Ds ) Ds0 exp

( ) -Es RT

(1)

where Ds0 and Es are the frequency factor and the activation energy of surface diffusion, respectively, R is the gas constant, and T is the absolute temperature. Es is often correlated with the isosteric heat of adsorption of the solute on the stationary phase (Qst), by introducing an empirical parameter (R) which is usually smaller than unity for surface diffusion.

Es ) R(-Qst)

(2)

By combining eqs 1 and 2, the following relationship is obtained. (4) Bogar, R. G.; Thomas, J. C.; Callis, J. B. Anal. Chem. 1984, 56, 1080. (5) Hansen, R. L.; Harris, J. M. Anal. Chem. 1995, 67, 492. (6) Wong, A. L.; Harris, J. M. J. Phys. Chem. 1991, 95, 5895. (7) Zulli, S. L.; Kovaleski, J. M.; Zhu, X. R.; Harris, J. M.; Wirth, M. J. Anal. Chem. 1994, 66, 1708. (8) Miyabe, K.; Guiochon, G. Adv. Chromatogr. 2000, 40, 1.

Analytical Chemistry, Vol. 72, No. 7, April 1, 2000 1475

Ds ) Ds0 exp

[

]

-R(-Qst) RT

(3)

Several models were derived from eq 3 to explain the dependence of Ds on q. Equation 3 is frequently used to analyze the influence of the temperature and the concentration on Ds in both gas- and liquid-phase adsorption systems.2 Obviously, eq 3 suggests that Ds approaches Ds0 as the interactions between adsorbate molecules and adsorbent surface decrease, i.e., when Qst tends toward zero. Values of Ds0 between 10-4 and 10-1 cm2 s-1 were reported for various liquid-solid-phase systems.2,8-14 This suggests that Ds is several orders of magnitude larger than the molecular diffusivity (Dm) when -Qst is small because Dm is usually estimated to be of the order of 10-5 cm2 s-1 in liquid-solid systems, including RPLC.1,15,16 This conclusion seems unreasonable, suggesting that eq 3 should not be used for the study of surface diffusion mechanism when |Q|st is small. The model provides no information regarding the conditions of validity of eq 3, not even the acceptable range of Qst. As shown later, it also leads to a contradictory situation when one tries to explain some of its thermodynamic properties. Surface diffusion is a mass transfer process of the adsorbate molecules while they remain in the adsorbed state. It is considered to be an activated process. A gain of Es is necessary for the adsorbate molecules to migrate because they must overcome the energy barrier that exists between two close adsorption sites. However, it is unnecessary that Es be larger than -Qst because adsorbate molecules do not need to be completely desorbed from the surface into the bulk liquid phase to migrate in the sorbed layer. The ratio Es/|Qst| should be smaller than unity. In many cases of surface diffusion in gas-solid adsorption, Es/|Qst| values smaller than unity were reasonably determined.8,17 In contrast, Es values larger than -Qst were frequently found for liquid-solid adsorption systems.8,9,12,18 The occurrence of surface diffusion is not expected under such conditions because it would be energetically more advantageous for the adsorbate molecules to be desorbed from the surface to the bulk phase rather than to migrate on the surface. On the other hand, there are a few reports of Es values smaller than -Qst in liquid-phase systems.10,14,19 Equation 3 provides no appropriate explanation for these contradictory correlations between Es and Qst in liquid-solid-phase adsorption. The mass transfer inside porous adsorbents is frequently represented by assuming that the contributions of pore and surface diffusion to intraparticle diffusion are parallel,17,20 hence (9) Awum, F.; Narayan, S.; Ruthven, D. Ind. Eng. Chem. Res. 1988, 27, 1510. (10) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 728. (11) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 1110. (12) Ma, Y. H.; Lin, Y. S.; Fleming, H. L. AIChE Symp. Ser. 1988, 84, 1. (13) Suzuki, M.; Kawazoe, K. J. Chem. Eng. Jpn. 1975, 8, 379. (14) Suzuki, M.; Fujii, T. AIChE J. 1982, 28, 380. (15) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. (16) Treybal, R. E. Mass-Transfer Operations; McGraw-Hill: New York, 1980. (17) Suzuki, M. Adsorption Engineering; Kodansha/Elsevier: Tokyo/Amsterdam, 1990. (18) Chiang, C. B.; Hidajat, K.; Uddin, M. S. Sep. Sci. Technol. 1989, 24, 581. (19) Muraki, M.; Iwashita, Y.; Hayakawa, T. J. Chem. Eng. Jpn. 1982, 15, 34. (20) Ruthven, D. M. Principles of Adsorption & Adsorption Processes; John Wiley and Sons: New York, 1984.

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Analytical Chemistry, Vol. 72, No. 7, April 1, 2000

De )Dp + FpKDs

(4)

where De is the intraparticle diffusivity, Dp is the pore diffusivity, Fp is the density of the packing material, and K is the adsorption equilibrium constant. When K tends toward zero, the contribution of the second term in the right-hand side (rhs) of eq 4 to De becomes negligibly small, irrespective of the Ds value. Because the contribution of surface diffusion is small in the range of low K values, few detailed studies have been made to date on the dependence of Ds on the magnitude of the interactions between adsorbate molecules and adsorbent surface, although temperature and concentration dependences of Ds have been extensively investigated.2 The main disadvantage of eq 3 results probably from the insufficient number of studies made on the correlation between surface diffusion and the intensity of the interactions between the adsorbate molecules and the adsorbent surface. In previous papers,8,21,22 we analyzed the correlation between Ds and the parameters characterizing retention, K and Qst, and derived the restricted diffusion model based on an approximation of the mass transfer mechanism of surface diffusion. Surface diffusion was regarded as molecular diffusion restricted by the adsorptive interactions between the adsorbate molecules and the adsorbent surface. The restricted diffusion model is based on the absolute rate theory23 and gives an equation for Ds which is different from eqs 1 and 3. Some intrinsic characteristics of Ds could be interpreted with this new model. In addition to the dependence of Ds on temperature and the retention parameters, the concentration dependence of Ds in different systems, following the Langmuir, Freundlich, and Jossens adsorption isotherms, could be consistently interpreted.24 (Jossens isotherm is represented by the following equation: C ) (q/H) exp(KJqp), where C is the solute concentration, q is the amount adsorbed, H is the Henry constant, and KJ and p are the numerical parameters.) Attempts were also made to develop an estimation procedure for Ds.8,21,25-28 The restricted diffusion model provides an appropriate interpretation for the contradictory correlations described earlier, between Es and Qst in liquid-solid adsorption systems including RPLC, i.e., for cases in which the Es/|Qst| values are larger than unity and in other cases for which they are smaller. In this model, Es is assumed to consist of two contributions originating from a hole-making step and a jumping step, respectively. The activation energy of these steps may be correlated with the evaporative energy (∆Ev) of the solvent and the isosteric heat of adsorption of the adsorbate, respectively. The proportionality coefficients were respectively estimated at ca. 0.47 and 0.4.22 Because ∆Ev for various common solvents (e.g., water, alcohols, and aromatic (21) Miyabe, K.; Guiochon, G. Anal. Chem. 1999, 71, 889. (22) Miyabe, K.; Takeuchi, S. J. Phys. Chem. B 1997, 101, 7773. (23) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGrawHill: New York, 1964. (24) Miyabe, K.; Takeuchi, S. AIChE J. 1997, 43, 2997. (25) Miyabe, K.; Takeuchi, S. Ind. Eng. Chem. Res. 1998, 37, 1154. (26) Miyabe, K.; Takeuchi, S. Can. J. Chem. Eng. 1998, 76, 887. (27) Miyabe, K.; Takeuchi, S.; Tezuka, Y. J. Chem. Eng. Jpn. 1998, 31, 347. (28) Miyabe, K.; Takeuchi, S.; Tezuka, Y. Bull. Chem. Soc. Jpn. 1998, 71, 1755. (29) Jossens, L.; Prausnitz, J. M.; Fritz, W.; Schlunder, E. U.; Myers, A. L. Chem. Eng. Sci. 1978, 33, 1097. (30) Itaya, A.; Kato, N.; Yamamoto, J.; Okamoto, K. J. Chem. Eng. Jpn. 1984, 17, 389.

Table 1. Physical Properties of the RPLC Columns column/stationary phase no. main alkyl ligand particle density, rp (g cm-3) porosity, ep (-)

1 C18 0.64 0.64

2 C18 0.71 0.57

3 C18 0.79 0.50

tortuosity factor, k2 (-)

3.8

4.1

4.4

carbon content (%) before end-capping after end-capping main ligand density (µmol m-2)c distance between ligands (nm) reaction ratio of silanol group (%) mass of adsorbent (g) void fraction of column, e (-)

3.6 6.6 0.59 1.9 7.3 1.8 0.35

6.4 8.6 1.1 1.4 13 1.8 0.41

12.8 13.7 2.3 1.0 29 1.9 0.42

4 C18 0.86 0.52(40)a 0.47(60)a 0.46(70,80)a 4.1(40)a 4.5(60,70)a 4.6(80)a

5 C1 0.74 0.62

6 C4 0.73 0.61

3.9

4.1

17.1 17.1 3.2 0.81 40 2.1 0.42

4.1 -b 13.4 0.69 56 1.8 0.44

6.7 -b 3.7 0.76 46 1.8 0.42

a The numbers in parentheses represent the volumetric fraction of methanol in the mobile phases. b No end-capping treatment was made. Calculated from the carbon content before end-capping, the surface area of the base silica gel (290 m2 g-1), and the density of silanol group on the surface of the base silica gel (assumed to be 8 µmol m-2).

c

hydrocarbons) is between ca. 30 and 40 kJ mol-1, the activation energy of the hole-making step is about 15-20 kJ mol-1. When Qst is -20 kJ mol-1, the contribution of the jumping step is estimated to about 8 kJ mol-1. (As explained later, the jumping step requires that the interactions between the molecule which is diffusing and all its neighbors be broken, to free it temporarily.) Accordingly, Es is about 23-28 kJ mol-1. In this case, Es would be larger than -Qst. By contrast, if Qst ) -50 kJ mol-1, Es is approximately 35-40 kJ mol-1 because the activation energy of the jumping step is now about 20 kJ mol-1, suggesting that the ratio Es/(|Q|st) would be smaller than unity. It is probable that the contradictory correlations regarding the ratio Es/Qst depend on the value of Qst. When -Qst is larger than about 35 kJ mol-1, Es seems to be smaller than -Qst. This hypothetical calculation using the proportionality coefficients reported above is supported by previous experimental data, in which -Qst is larger than about 40 kJ mol-1 and Es/(|Q|st) is smaller than unity.10,14,19 On the other hand, ratios Es/(|Q|st) larger than unity are frequently observed in RPLC because |Q|st is usually smaller than this critical value, i.e., about 35 kJ mol-1. Thus, the restricted diffusion model provides a suitable framework for the comprehensive interpretation of the characteristic features of surface diffusion. However, there remains issues with this model which must be clarified to increase its usefulness. This paper discusses the application of the restricted diffusion model to an important set of experimental results obtained with various RPLC stationary phases, mobile phase compositions, and solutes. The goal of this work is to provide a quantitative analysis of these data in terms of the restricted diffusion model, to clarify the influence of different parameters on the characteristics of surface diffusion, and to show how this diffusion explains many important and useful features accompanying separations on alkylbonded silicas in RPLC. EXPERIMENTAL SECTION Apparatus. Measurements of elution peaks under various RPLC conditions were made with a high-performance liquid chromatograph system (LC-6A, Shimadzu). A valve injector was used to introduce small amounts of sample solutions into the

column. The temperature of the column was maintained constant by placing it in a jacket swept by a stream of temperaturecontrolled water. The concentration of the sample in the eluate at the column exit was monitored by the ultraviolet detector (SPD6A) of the HPLC instrument. Further details regarding the instrument can be found elsewhere.8 Column and Reagents. The physical properties of the six RP columns (YMC) used in this study are shown in Table 1. These columns (6 × 150 mm) were packed with different RP stationary phases (1-6), four with different densities of C18 ligand and two with other alkyl ligands (C1 and C4). The carbon contents of these packing materials range from 4.1 to 17.1%. The density of the alkyl ligands chemically bonded to the packing materials was estimated from these carbon contents. The carbon content of the phases 1-3 increased 1-3% upon end-capping treatment with trimethylsilyl ligands. A substantial increase of carbon content was not observed for the stationary phase 4. No end-capping was carried out for the RP stationary phases 5 and 6. It was estimated that between 7.3 and 40% of the silanol groups reacted with the octadecyl bonding reagent on the phases 1-4, assuming a surface density of the silanol groups of about 8 µmol m-2.31 The particle diameter and the surface area of the base silica gel determined by nitrogen BET were 45 µm and 290 m2 g-1, respectively. The average distance between two bonded C18 ligands was calculated from the ligand density. Compared with the molecular size of the solutes, between 0.7 and 0.9 nm, this distance is about 1-2.5 times larger. The radii of the benzene and hexylbenzene molecules were estimated from their molar volume at their normal boiling point, at 0.34 and 0.45 nm, respectively. The mobile phases were methanol/water solutions of different compositions (volumetric fractions of methanol (φ), 40, 60, 70, and 80%). The samples were selected among two homologous series of organic compounds, n-alkylbenzenes and p-alkylphenols. Uracil was used as an inert tracer. Procedures. Pulse response experiments (i.e., elution chromatography) were made at zero surface coverage of the samples (31) Unger, K. K. Porous Silica; Elsevier: Amsterdam, 1979.


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at different column temperatures (288-308 K) and mobile phase flow rates (1.0-2.0 mL min-1). Once recorded, the chromatographic peaks were analyzed by the method of moments. The first and second moments of the elution peaks provide information concerning the phase equilibrium and the mass transfer rates in the columns, respectively. Relevant details on the moment analysis method are found in the references cited.1,8,17,20 The first moment (µ1) provides the adsorption equilibrium constant (K). The Ds value is estimated from the second moment (µ2′) by subtracting the contributions of axial dispersion, fluid-to-particle mass transfer, and pore diffusion to peak spreading. The fluid-toparticle mass transfer coefficient was estimated with the WilsonGeankoplis equation.32 The Wilke-Chang equation was used to estimate the molecular diffusivity, Dm, of the samples.1,15,16 The contributions of axial dispersion and intraparticle diffusion were separated by taking advantage of the difference in their flow rate dependence. According to eq 4, Ds was calculated by subtracting the contribution of pore diffusion from intraparticle diffusion. According to the parallel pore model,17,20 Dp was estimated from Dm, the intraparticle porosity of the adsorbent, and the tortuosity factor of the pores. The tortuosity factor was determined from the pulse response experiments made with the inert tracer, uracil.8 The influence of the extra-column volume on µ1 and µ2′ was measured by the chromatographic method, by running similar experiments with the same tracer, with and without any column. The true µ1 and µ2′ values of the elution peaks were derived by correcting the data for the influence of the extra-column volumes and of peak distortion. Theories and models are available for interpreting some sources of peak asymmetry (tailing and fronting).1 In this study, only the radial heterogeneity of the columns was regarded as the origin of the skewed peaks.33-35 The presence of heterogeneous mass transfer kinetics was not taken into account because experimental results demonstrated the energetic uniformity of the RP surfaces (the isosteric heat of adsorption was independent of the surface coverage).8 The phase equilibrium could be accounted for by the simple Langmuir isotherm. Nearly constant values of the thermodynamic properties, i.e., Qst and Es, were observed, irrespective of the surface coverage, q.8 Although the moments µ1 and µ2′ were corrected for the contributions of the sample pulses introduced into the columns by assuming rectangular profiles for these injected pulses, the correction was negligibly small. (In the worst case, 100 µL of a n-hexylbenzene solution was injected. The values measured for µ1 and µ2′ were 100 and 30 min2, respectively, at F ) 1 mL/min. The first and second moments contributions of such a pulse, assuming that it has a rectangular shape, are 5 × 10-2 min and 8.3 × 10-4 min2, respectively.) After all of these corrections, Ds is determined with a probable error of approximately a few percent.8 RESULTS AND DISCUSSION We first discuss some general properties of surface diffusion and experimental correlations established by various authors between the parameters of this diffusion. Then, we analyze the characteristic features of the steps involved in the surface diffusion (32) Wilson, E. J.; Geankopolis, C. J. Ind. Eng. Chem. Fundam. 1966, 5, 9. (33) Miyabe, K.; Guiochon, G. J. Chromatogr. A 1999, 830, 263. (34) Miyabe, K.; Guiochon, G. J. Chromatogr. A 1999, 830, 29. (35) Miyabe, K.; Guiochon, G. J. Chromatogr. A 1999, 857, 69.

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Figure 1. Correlation of the frequency factor and the activation energy of surface diffusion.

mechanism , the hole-making process, and the jumping process. We show how our present lack of understanding of some parameters of these processes affects the quantitative accuracy of the restricted diffusion model. Finally, we analyze the surface tortuosity and the frequency factor of surface diffusion and correlations involving their parameters. General Properties of Surface Diffusion as a Mass Transfer Mechanism. In a previous paper,8 we demonstrated an enthalpy-entropy compensation between Ds0 and Es and a linear free energy relation between Ds and K. These results suggest a similar mechanism of surface diffusion in RPLC systems under different chromatographic conditions. The dashed and dotdashed straight lines in Figure 1 represent the correlations for nonpolar and polar analytes in RPLC systems, using various alkylbonded silica gels (C1, C4, C8, and C18) and methanol/water mixtures of different compositions as stationary and mobile phase, respectively. Similar correlations were also observed between Ds0 and Es with acetonitrile/water and ethanol/water solutions as mobile phases and a C18-silica gel as the stationary phase.36,37 The dotted line in Figure 1 illustrates the correlation in a gassolid adsorption system, using a C18-silica gel and helium.8 The symbols in this figure show experimental data reported in the same study for several stationary phases with different density of C18 ligand and length of alkyl ligands. The Ds0 and Es values were determined by analyzing the temperature dependence of Ds according to eq 1. Although the data in Figure 1 are somewhat scattered, they group around the three straight lines. Experimental data were determined with compounds of the two homologous series and methanol/water solutions of various compositions grouped also around the three linear correlations (not shown). (36) Miyabe, K.; Takeuchi, S. Ind. Eng. Chem. Res. 1997, 36, 4335. (37) Miyabe, K.; Takeuchi, S.; Tezuka, Y. Adsorption 1999, 5, 15.

Figure 2. Correlation of the surface diffusion coefficient and the adsorption equilibrium constant (logarithmic coordinates).

When enthalpy-entropy compensation is established, a linear free energy relationship is expected to hold. Figure 2 illustrates the correlation between Ds and K values measured at 298 K, with stationary phases bonded to different alkyl chains or C18 chains with different bonding densities. The same straight line (solid line) represents the correlation between ln Ds and ln K for alkylbenzene derivatives, irrespective of the nature and density of the bonded alkyl ligands. The same correlation applies also to data measured with methanol/water solutions of various compositions (not shown). Although there may not be any theoretical proof of the existence of such a linear free energy relationship, a similarity between the mechanism of chemical reactions or between physical phenomena is empirically assumed when a linear relationship between two free energies is observed. For instance, the retention increases and the surface diffusivity decreases with decreasing methanol concentration in the mobile phase. Similar changes are observed when the length or density of the bonded alkyl ligands increases. Even though the origins of these changes are different, the stronger their retention, the more restricted the surface diffusivity of the sample components. The results in Figure 2 indicate that the variations of the surface diffusivity and the retention are linearly correlated. The nature of the surface diffusion mechanism seems independent of the importance of the retention. It seems that, at least in RPLC, the surface diffusion mechanism is the same, irrespective of the chemical nature of the samples, the type and concentration of the organic modifier in the mobile phases, and the length and density of the alkyl chains bonded to the surface of the support (silica gel). The values of the surface diffusivity and the retention depend

on the combination of analytes, stationary phases, and mobile phases selected but their influence on the equilibrium and kinetic parameters are linearly correlated, irrespective of the RPLC system studied. If the migration mechanism of surface diffusion remains the same, it may be possible quantitatively to account for its characteristics. Correlation of Some Properties in RPLC with K. The retention of sample molecules on the surface of a stationary phase in RPLC is mainly based on hydrophobic interactions. Changes in the hydrophobicity of the surface of an RP stationary phase are accompanied with changes in both the retention and the mass transfer kinetics of the sample molecules. In the experimental study, we changed the hydrophobic adsorption of the RP packing materials by changing the density or the length of the alkyl ligands, assuming that the mechanism of surface diffusion would remain the same, irrespective of the changes made to the surface of the adsorbent. Figure 2 suggests that new information concerning surface diffusion in RPLC could be derived by considering the correlation between Ds and the retention constant. Figure 3 shows a plot of the ratio Ds/Dm versus K (instead of ln K in Figure 2). Although the extrapolation is approximate, it is likely that, for all the compounds, when K decreases, Ds approaches a limit near the corresponding Dm value. The dotted line in Figure 2 corresponds to the solid line in Figure 3, illustrating another possible correlation of the experimental data. The symbols in Figure 2 could fit properly to the dotted curved line. However, both the solid straight line and the dotted curved line in Figure 2 show nearly the same trend for the correlation between ln Ds and ln K, at least under the Analytical Chemistry, Vol. 72, No. 7, April 1, 2000

1479

Figure 3. Ratio of the surface diffusion coefficient to the molecular diffusivity versus the adsorption equilibrium constant. Inset: enlargement of the plots in the range of small K values.

experimental conditions used in this study. Because the K value is usually between about 0.5 and 50 cm3 g-1, it is likely that the correlation between ln Ds and ln K is apparently linear under conventional RPLC conditions, as illustrated in Figure 2, although the discrepancy between the solid and the dotted lines seems to increase in the low range of K. In addition, experimental data were measured with methanol/water solutions of different compositions because the mobile phase gave approximately the same correlation (not shown). According to the van’t Hoff equation, Qst was derived from the temperature dependence of K.

K ) K0 exp

( ) -Qst RT

(5)

where K0 is K at T ) 4 or Qst ) 0. Figure 4a illustrates the plot of -Qst against K at 298 K. The -Qst value should tend toward zero with K. It seems unreasonable that temperature could affect the elution time of an unretained substance or that there would be a heat of adsorption without actual adsorption. Although the plots in Figure 4a are scattered, the curved correlation between -Qst and K seems to approach the origin. Similar trends of the dependence of Qst on K were observed under various RPLC conditions, using the different compounds and methanol/water solutions studied (not shown). Es is plotted against K at 298 K in Figure 4b. It is unlikely that Es tends toward zero for K ) 0. The same conclusion was obtained for similar correlations between Es and K for different compounds and mobile phase compositions (not shown). The comparison of the results in Figure 4a and b suggests that an energy contribution to Es, probably independent 1480


of Qst, must be taken into account in order to interpret the difference in characteristics of Es and Qst. These results seem to substantiate the validity of the concept of the restricted diffusion model,8,21,22,24 which was proposed as an approximation of the migration mechanism of surface diffusion, to interpret its intrinsic characteristics. In the absolute rate theory,23 the mechanism of molecular diffusion is assumed to consist of two processes, a hole-making step and an interactionbreaking or jumping step. For a molecule to gain the activation energy of molecular diffusion (Em), a hole must be made in the solvent (Ehm) and the interaction between the molecule and the surrounding solvent must be broken (Ebm).

Em ) Ehm + Ebm

(6)

It was shown that the contribution of Ehm to Em is larger than that of Ebm, that it is about a third of the evaporation energy of the solvent (∆Ev), and that Ebm is about 10-20% of Em. As for molecular diffusion, the restricted diffusion model for surface diffusion assumes that a hole is first formed by removal of the proper number of solvent molecules located in the potential field of adsorption. Then, the sample molecule transfers from a neighbor adsorption site into this hole, overcoming two kinds of interactions, that between sample and solvent molecules (solvation) and that between sample molecule and surface of the stationary phase (adsorption). As described later, the former has no significant influence on Es, whereas the latter is correlated with Qst. In the restricted diffusion model, surface diffusion is regarded as molecular diffusion restricted because of the additional attractive interactions resulting from the adsorption of the sample

Figure 4. Dependence of (a) the isosteric heat of adsorption and (b) the activation energy of surface diffusion on the adsorption equilibrium constant.

molecule onto the surface of the stationary phase. Like in eq 6, Es is assumed to consist of two contributions corresponding to the hole-making (Ehs) and the jumping (Ebs) steps.

Es ) Ehs + Ebs

(7)

The Ehs value is correlated with ∆Ev of the solvent. From considerations based on the absolute rate theory, Ehs was assumed to be almost equal to Em in previous papers.8,22 On the other hand, Ebs is probably predominantly correlated with Qst. On the basis of these considerations, the following equations were derived previously.8,22

Es ) Ehs + β(-Qst)

(8a)

Es ≈ Em + β(-Qst)

(8b)

The Es value is the sum of Ehs and the contribution of the adsorption interactions, β(-Qst). The β value was estimated to be about 0.4 for surface diffusion in RPLC.8,22 In the following subsection, we discuss the correlation of Es with Qst measured under different experimental conditions, to verify the validity of the restricted diffusion model. As shown in eqs 7 and 8a, the intercept and slope of the linear relation between Es and Qst should correspond to Ehs and Ebs, respectively. Correlation between Qst and Es. Figure 5a illustrates the correlation between Qst and Es for alkylbenzene derivatives, determined using data from the six RP packing materials. For each material, Es increases almost linearly with increasing -Qst.

As suggested by Figure 4, Es is not equal to zero at Qst ) 0. From the intercept of the linear correlations in the figure, Ehs is estimated at approximately 15-19 kJ mol-1 or about 70-90% of Em (21 kJ mol-1) and 35-44% of ∆Ev (43 kJ mol-1), except for the C1-silica gel. The Em and ∆Ev values for a 70/30 methanol/ water solution were derived from the temperature dependence of Dm estimated by the Wilke-Chang equation and from the activation energy of its viscosity (Evis).1,15,16 These results agree approximately with those derived by applying the absolute rate theory to molecular diffusion.23 The Ehs value is close to the Evis value (ca. 16 kJ mol-1 at φ ) 70%). The mechanism of molecular diffusion is analogous to that of viscosity which originates from solvent molecules migrating through other solvent molecules. In molecular diffusion, the sample and solvent molecules are different. Because the sample molecules are usually larger than the solvent molecules, it may be necessary to make a larger hole for molecular diffusion than for viscosity. An Em value slightly larger than Evis could be explained by the difference in hole size needed by the two types of migration. However, as seen in Figure 5a, it is likely that the intercept depends on the changes made to the surface of the RP stationary phases. Because Ehs is the activation energy of the hole-making process in a potential field of adsorption, different intercepts are explained by different structural situations of the solvent in the vicinity of the hydrophobic surface of the adsorbent. The influence of the hydrophobicity of the packing materials on the intercept in Figure 5a will be discussed later. A value of about 0.3-0.6 was calculated for the ratio Ebs/(|Q|st) on the various adsorbents. The contribution of the interactions between sample molecules and surface of the packing material Analytical Chemistry, Vol. 72, No. 7, April 1, 2000

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Figure 5. Correlation of the activation energy of surface diffusion with the isosteric heat of adsorption in RPLC systems using (a) six different stationary phases (chemically bonded porous silica); (b) compounds from two homologous series; and (c) methanol/water solutions of four different compositions.

to Es is represented by the term Ebs. The ratios Ebs/(|Q|st) in Figure 5a are close to the β value (0.4) previously reported.8,22 However, the agreement is insufficient to reach a definitive conclusion. To obtain more information on the β value, the correlation between Es and Qst is analyzed in more detail for C181482


silica gel (carbon content, 17.1%), the most popular packing material in RPLC. Figure 5b illustrates the correlation between Qst and Es for the compounds of the two homologous series. Values of approximately 0.55 and 0.67 were calculated for the ratio Ebs/(|Q|st),

for alkylbenzene and p-alkylphenol derivatives, respectively. These Ebs/(|Q|st) values are larger than the β value (0.4) reported in previous papers.8,22 This difference arises probably from the first term in RHS of eqs 8a and 8b. In previous studies,8,22 Es was analyzed with eq 8b in which Em (21 kJ mol-1) was taken as equal to Ehs. In another previous paper,38 we discussed the β value by analyzing correlations relating the phase equilibrium and the surface diffusion in RPLC, i.e., correlations between Ds and K, between Ds and the boiling point of the compounds, and between Ds/Dm and Qst. It was concluded that β is smaller than unity and close to 0.5. Finally, the intercepts in Figure 5b are respectively 17 and 14 kJ mol-1 for the two homologous series. There is no information on the correlation of the β and Ehs values with the chemical properties of the compounds. The influence of the chemical and physical properties of the sample on the β and Ehs values will be discussed later. Figure 5c shows the correlation between Qst and Es for benzene, toluene, and ethylbenzene determined with methanol/ water solutions of different compositions. Alkylbenzenes with short alkyl chains were used as compounds to measure experimental data at low methanol concentrations. Although Figure 5c shows some scatter, the correlations between Es and Qst exhibit the same profile for all φ values. The slope is about 0.4-0.6, in agreement with the results in Figure 5. The intercept is about 17-18 kJ mol-1, also close to the Evis value (ca. 16-17 kJ mol-1 at φ ) 40-80%). The results in Figure 5 support the validity of the concept of the restricted diffusion model and the representation of Es given in eq 8a. However, as shown in Figure 5, the β and Ehs values vary somewhat with the experimental conditions. Accuracy in the determination of Ehs and β (or Ebs) affects the conclusion derived by the restricted diffusion model. In the following, we discuss some factors influencing the Ehs and β values. Factors Influencing the Hole-Making Process. As indicated in eq 8a, the intercept and slope of the linear correlations between Es and Qst in Figure 5 are related with Ehs and Ebs, respectively. We consider now the two factors influencing Ehs: the hydrophobicity of the alkyl ligands on the surface of the packing materials, and the molecular size of the compounds studied. Influence of the Hydrophobicity of the Alkyl Ligands on Ehs. When an aqueous solution of an organic solvent is used as the mobile phase in RPLC, this organic solvent is preferentially adsorbed onto the hydrophobic surface of the stationary phase and its concentration is higher close to the surface than in the bulk mobile phase. The difference between the concentration of the organic modifier near the hydrophobic surface and in the bulk mobile phase, or surface excess, was calculated by the partition-displacement model.39,40 The concentration of acetonitrile is about 10-30% higher near the surface of a C18-silica gel than in the bulk mobile phase for φ between 40 and 80%.40 Similarly, we calculated that the surface excess of methanol on the C18-silica gel (carbon content, 17.1%) is about 10-20% in the same range of φ (not shown). When φ is higher than about 40-50%, Evis for methanol/ water solutions decreases with increasing φ. Evis was calculated as 17.4 (40%), 16.8 (60%), 16.3 (70%), 16.0 (80%), and 15.6 kJ mol-1 (38) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 1999, 103, 11086. (39) Jaroniec, M. J. Chromatogr. A 1993, 656, 37. (40) Jaroniec, M. J. Chromatogr. A 1996, 722, 19.

(90% methanol) from the temperature dependence of the viscosity of the solutions.1 However, the variation of Evis with φ is not large. The nearly constant Ehs values in Figure 5c might be related with the small difference in the Evis values. The surface excess of methanol probably depends on the hydrophobicity of the surface. The effect of the alkyl chain length on the surface excess of acetonitrile was already studied.41 The surface excess of acetonitrile on an RP2 stationary phase (bonded ethyl chain) was reported to be larger than that on an RP18 stationary phase (bonded octadecyl chain). The larger surface excess of acetonitrile on the RP2 phase was explained in terms of a higher bonded ligand density on the RP2 than on the RP18 phase. The C1-silica gel used in this study probably shows a similarly larger surface excess of methanol than do the RP packing materials with longer alkyl chains because the density of its ligands is also higher than those of the other stationary phases (Table 1). Thus, the lower value of the intercept of the straight line for the C1-silica gel in Figure 5a might be due to the larger surface excess of methanol. In addition, the contribution of an adsorption potential (Eap) to Ehs must be taken into account when solvent molecules are also adsorbed on the surface of an adsorbent. The Eap value is the free energy for the transfer of the adsorbable solvent from the potential field of adsorption to a bulk phase, and is estimated by the following equation:

Eap ) RT ln

() Cs C

(9)

where C and Cs are the real and saturation concentrations of a solute in a solution, respectively. However, Eap for methanol in this study is probably small in comparison with ∆Ev of methanol, i.e., ca., 35 kJ mol-1, because the concentration of methanol in the mobile phase solvents (methanol/water mixtures) is sufficiently high. When polar solvent molecules interact with hydrophobic alkyl ligands on the surface of RP packing materials, the ligands are probably solvated in a manner similar to that of hydrophobic solvation. Then, mutual repulsion takes place between them and this structure-promoting effect may compress the polar solvent molecules near the hydrophobic ligands. The interactions between polar solvent molecules may be amplified and their mobility restricted. Mobile phase solvents near longer alkyl chains are probably more structured than those near a shorter chain. If we connect (not shown) the symbols in Figure 5a for each compound (e.g., all the squares for ethylbenzene), the curved line obtained is the correlations between Es and Qst for this compound. Similar curves are observed for ethyl-, butyl-, and hexylbenzene, although this correlation is unclear for benzene. Even though the hydrophobicity of the RP packing materials varies, Qst does not change markedly. Es varies by about 10 kJ mol-1, always more than -Qst. The results in Figure 5a imply that Ehs is more sensitive than Ebs to changes in the hydrophobicity of the RP stationary phases. The larger value of Ehs for the phases with longer alkyl chains or higher densities of C18 ligand illustrated in Figure 5a might result from the higher structure-promoting effect of a more hydrophobic surface. (41) Slaats, E. H.; Markowski, W.; Fekete, J.; Poppe, H. J. Chromatogr. 1981, 207, 299.


1483

However, this probably cannot explain the very small intercept observed for the C1-silica gel in Figure 5a. In addition to the larger surface excess of methanol justified earlier, other effects, e.g., a structure-breaking effect due to methyl groups, should be taken into account. For instance, a Stokes radius of 0.204 nm is reported for tetramethylammonium, a value smaller than its crystal radius, 0.347 nm. This suggest that tetramethylammonium ion could be regarded as a structure-breaker. In this section, we assumed that Ehs depends on both the surface excess of the organic modifier (here, methanol) and structure-promoting or -breaking effects in the vicinity of the surface of the stationary phase. The hydrophobicity of RPLC phases probably causes these phenomena. To reach clearer conclusions, more fundamental studies on the composition and structure of aqueous solutions of organic modifiers close to hydrophobic surfaces are necessary. Influence of the Molecular Size of Sample Compounds on Ehs. As shown in Figure 5b, β is between 0.55 and 0.67, which is slightly larger than reported previously (0.4),8,22 although these values are close. The straight lines in Figure 5b were drawn with the assumption that Ehs was the same for each compound in the alkylbenzenes or p-alkylphenols homologous series. However, Ehs probably depends on the molecular size of the compounds. For example, Ehs is probably larger for hexylbenzene than for benzene. Incidentally, the molar volumes of benzene and hexylbenzene at their normal boiling point are 96 and 229 cm3 mol-1, respectively. If we take into account the influence of the molecular size of the compounds on Ehs, β values smaller than 0.55 and 0.67 might be obtained. In addition, a cause of the difference between β values calculated from Figure 5b and those previously reported may be the difference in the Ehs values in the rhs of eq 8a. Although Ehs was assumed to be equal to Em (21 kJ mol-1) in previous papers,8,22 the intercept in Figure 5b was taken to be Ehs (17 and 14 kJ mol-1 for alkylbenzenes and p-alkylphenols, respectively). As mentioned earlier, the difference of Ehs for each compound should also be considered. From these calculation results, it seems that β is smaller than 0.55 or 0.67. Thus, an accurate value of Ehs for each solute should be taken into account in order to analyze the correlation between Es and Qst. Factors Influencing the Jumping Process. In this section, we discuss the influence of two factors on Ebs: the reduction in hydrophobic surface area due to adsorption, and the molecular size of the compound. Influence on Ebs of the Reduction in Hydrophobic Surface Area Due to Adsorption. In the solvophobic theory,8,42 it is assumed (1) that the contact area between polar solvent molecules and the hydrophobic surfaces of both the sample molecules and the alkyl groups on the surface of an RP stationary phase decreases upon adsorption of the sample molecules on the alkyl ligands; and (2) that the decrease of total hydrophobic surface area (∆A) exposed to the polar solvent is proportional to the surface area of the sample molecule (As). The ratio ∆A/2As and the value of (1 ∆A/2As) represent the relative fractions of hydrophobic surface of the sample molecule in contact with the alkyl ligands and the polar mobile phase solvent, respectively.8 The former is related with the interactions between sample molecules and surface of the stationary phase, the latter with the solvation energy of the sample molecules and the solvent molecules. In this study, the (42) Horva´th, C.; Melander, W.; Molnar, I. J. Chromatogr. 1976, 125, 129.

1484


ratio ∆A/As was calculated from the retention data by applying the solvophobic theory in order to analyze the influence of ∆A/ As on Ebs. Figure 6a shows the correlations between K and As for the various RP stationary phases. The As value was obtained by summing up the surface area increments of each group of the molecules of alkylbenzene derivatives.43 Figure 6a shows different linear relations, with a slope depending on the type of stationary phase. The ratio ∆A/As calculated from these slopes ranges from 0.20 to 0.35. Values of ∆A/As also reported in RPLC with C18silica gel are 0.30-0.35 in 70% methanol,8 0.18 in 70% acetonitrile,8,44 0.26 in 70% ethanol,8,37 and 0.35 in an aqueous buffer.42 A ∆A/As value of 0.20-0.30 was reported for a system consisting of activated carbon and water.45 Similarly, Figure 6b shows linear correlations between ln K and As for the two homologous series on one system. The parallelism of the correlations in Figure 6b suggests that the contribution of one methylene group to K is the same, irrespective of the chemical properties and structure of the compound series. From the slope of the lines, ∆A/As is 0.35 for both homologous series. Figure 6c shows linear correlations between ln K and As at different φ values. The slope of these lines increases with decreasing φ, suggesting that the decrease in φ is accompanied with a strengthening of the interactions between the sample molecules and the C18-silica gel. The change in structural properties of the C18 ligands due to that in φ may also affect this phenomenon. The results in Figure 6 correspond to the dependence of the retention factor on the number of methylene groups in the alkyl chain of the solutes. The slopes of the straight lines represent the so-called methylene selectivity. Ratios ∆A/As ranging from 0.20 to 0.43 were calculated from the slope of the linear correlations in Figure 6. In RPLC, part of the solvation between sample molecules and the surrounding molecules of mobile phase seems to be reduced to about 80-90% of its initial value because the adsorption of the sample molecules on the stationary phase is accompanied with a reduction of the hydrophobic surface area of the sample molecule exposed to the polar solvent. The additional interactions between the sample molecule and the alkyl ligand groups may act as a restriction force for surface diffusion, despite the reduction in solvation energy. As described earlier, the absolute rate theory concluded that Ehm had a major contribution to Em and was about one-third of ∆Ev, and that the contribution of Ebm to Em was small, about 10-20% of Em.23 Because Em for sample molecules in 70% methanol is about 21 kJ mol-1, Ehm and Ebm are estimated to be about 17-19 and 2-4 kJ mol-1, respectively. The reduction in solvation energy of the sample due to its adsorption on the stationary phase is about 0.2-0.8 kJ mol-1, and from these results, ∆A/As ) 0.2-0.4 and Ebm ) 2-4 kJ mol-1. It is likely that the influence of the reduction in solvation energy on Es is small. Influence of the Molecular Size of Sample Compounds on Ebs. As shown in Figure 6a, the logarithm of K increases linearly with the number of methylene groups in the adsorbate molecule. The methylene increment of ln K is larger on packing materials having (43) Bondi, A. J. Phys. Chem. 1964, 68, 441. (44) Miyabe, K.; Takeuchi, S. Anal. Chem. 1997, 69, 2567. (45) Belfort, G.; Altshuler, G. L.; Thallam, K. K.; Feerick, C. P., Jr.; Woodfield, K. L. AIChE J. 1984, 30, 197.

Figure 6. Correlation of the adsorption equilibrium constant with the hydrophobic surface area of the compounds studied on RPLC systems using (a) six different stationary phases; (b) compounds from two homologous series; and (c) methanol/water solutions of four different compositions.

longer alkyl chains and on C18-silica gels of higher bonding density. This effect arises from differences in the steric interactions between bonded alkyl ligands and sample molecules. Only planar interactions seem possible in the case of the C1-silica gel. The penetration of sample molecules into a layer of long alkyl-bonded

ligands on RP packing materials was suggested by Tchapla et al.46,47 on the basis of the results of their detailed studies of the retention of many homologous series. Figure 6 indicates that (46) Tchapla, A.; Colin, H.; Guiochon, G. Anal. Chem. 1984, 56, 621. (47) Tchapla, A.; Heron, S.; Colin, H.; Guiochon, G. Anal. Chem. 1988, 60, 1443.


1485

∆A/As ranges from 0.27 to 0.35 for C18-silica gels with a carbon content between 6.6 and 17.1%. These ∆A/As values are larger than those for the C1-silica gel. A similar conclusion is also derived from a comparison of the linear correlations in Figure 6a for the C18-silica gel of 6.6% carbon content with that for the C4-silica gel, which has almost the same carbon content, 6.7%. Despite this, the slope of the line is slightly larger for the C18-silica gel than for the C4-silica gel, suggesting that the solute is more strongly attracted by the C18- than by the C4- bonded silica. The ratio ∆A/As, i.e., the magnitude of the hydrophobic interactions between solute molecules and alkyl ligands, increases with increasing density and length of the alkyl ligands and so do the slopes of the lines in Figure 5a, as expected because these slopes depend on the energy of these interactions, as is well known in RPLC. The significant influence of the retention on Ebs explains its dependence on the structural characteristics of the solute molecules and the alkyl ligands. However, this does not explain the slope of the linear correlation observed for the C1-silica gel. Restriction Factors of Surface Diffusion. It is likely that surface diffusion of sample molecules takes place in a potential field of adsorption, in a mixed phase consisting of the alkyl ligands and the mobile phase solvents, in a region near the surface of the adsorbent which is influenced by adsorption interactions. In this case, obstruction of the migration of the molecules may be amplified by an increase of the concentration and/or the length of the ligands. In addition, surface diffusion probably takes place on an irregular surface. The influence of this tortuosity on Ds should also be taken into account. However, it is difficult to obtain reliable information concerning the influence of the obstruction and tortuosity of diffusion paths on Ds. It is well known that Dp is approximately 1 order of magnitude smaller than Dm in many cases, because of the influence of the tortuosity of the micropores. If we assume that Ds is similarly reduced by a factor of 10 due to the restriction factors, Es may be in error by about 5.7 (RT ln 10) kJ mol-1 at 298 K because the restricted diffusion model includes no parameter describing the effect of obstruction and tortuosity on Ds. This error may be related with the difference between Es for different stationary phases shown in Figure 5a. At present, it is difficult to correct for the influence of obstruction and tortuosity factors on Ds. A fundamental study on the mass transfer mechanism of surface diffusion should be made by analyzing the characteristics of Ds, including the influence of restriction factors. Frequency Factor of Surface Diffusion. As illustrated in Figure 3, it is likely that Ds tends toward Dm with decreasing retention. Although it is unclear whether Ds coincides with Dm because of the uncertainty of the extrapolation, both values are close, suggesting a correlation between surface and molecular diffusion.8,21,22,24 Similar to surface diffusion, molecular diffusion is also assumed to be an activated process. The temperature dependence of Dm is usually analyzed by the Arrhenius equation.

( )

-Em Dm ) Dm0 exp RT

(10)

where Dm0 and Em are the frequency factor and the activation energy of molecular diffusion, respectively. Combining eqs 1 and 10 gives the following equation: 1486


[

]

Ds0 -(Es - Em) Ds ) exp Dm Dm0 RT

(11)

If Es is expressed as eq 8b, eq 11 is rearranged as follows:

[( )]

Ds Ds0 Qst ) exp Dm Dm0 RT

β

(12)

Figure 7 shows linear correlations between ln(Ds/Dm) and Qst for various stationary phases. Similar linear correlations between ln(Ds/Dm) and Qst were reported with different compounds and methanol/water solutions of different compositions (not shown). For example, the linear correlation between ln(Ds/Dm) and -Qst for the C18-silica gel (carbon content, 17.1%) is represented as follows:

( )

ln

Ds ) -0.12(-Qst) - 0.67 Dm

(13)

Equation 12 suggests that the intercept of the linear correlation between ln(Ds/Dm) and -Qst corresponds to the ratio Ds0/Dm0 if this ratio is constant, irrespective of Qst. From the results in Figure 7, the ratio Ds0/Dm0 appears to be about 0.5-1.5. However, β is 0.31 (0.12 × R × 298) from the slope of the linear correlation for the C18-silica gel (carbon content, 17.1%). This β value is smaller than that previously estimated from the slope of the linear correlation between Es and -Qst in Figure 5a, i.e., 0.59. The apparent linear correlations in Figure 7 are drawn by assuming that the ratio Ds0/Dm0 in eq 12 is constant for all the related plots. The difference between the two β values, 0.31 and 0.59, probably suggests that the ratio Ds0/Dm0 depends on the experimental conditions. Figure 8 illustrates the correlation between Ds0/Dm0 and K. A definitive trend is not observed, suggesting that it is difficult to perform a quantitative analysis of the entropy term. Whereas the apparent ratios Ds0/Dm0 are about 0.5-1.5 from the intercept of the straight lines in Figure 7, Figure 8 shows plots of Ds0/Dm0 scattered in a wider range, from ca. 0.01 to 1.0. These data were determined by analyzing the temperature dependence of Ds and Dm by the Arrhenius equation. The difference between these Ds0/Dm0 values is probably explained as follows. In Figure 5a, the Ehs value for the alkylbenzenes is about 17 kJ mol-1 from the intercept of the straight line for the C18-silica gel (carbon content, 17.1%). This is smaller than Em (21 kJ mol-1) estimated from the temperature dependence of Dm. Equation 12 is derived without taking into account the difference between Ehs and Em. The difference between the Ds0/Dm0 values in Figures 7 and 8 probably results from the difference between Ehs and Em. For the alkylbenzene homologues, eq 12 should be written as follows:

( )[ ( )]

Ds Ds0 Qst 4 ) exp exp Dm Dm0 RT RT

β

(14)

The additional parameter in the rhs of eq 14, i.e., exp(4/RT), is about 5 at T ) 298 K. The Ds/Dm values at 298 K and Qst of ethylbenzene are 0.16 from Figure 7 (open square) and -9.7 kJ mol-1 from Figure 4a (open square), respectively. When β is

Figure 7. Correlation of the ratio of the surface diffusion coefficient to the molecular diffusivity with the isosteric heat of adsorption.

assumed to be 0.59 as shown in Figure 5a, the ratio Ds0/Dm0 is 0.32, derived from eq 14. These values agree with those plotted in Figure 8 (open square). Similarly, the ratio Ds0/Dm0 of ethylbenzene on the C1-silica gel is estimated from Ds/Dm (at 298 K) ) 0.39, Qst ) -8.4 kJ mol-1, Em - Ehs ) 12 kJ mol-1, and β ) 0.62. Equation 14 gives the ratio Ds0/Dm0 as 0.025, which also agrees with that in Figure 8 (square with center +). These calculations are merely made by eq 14 based on the Arrhenius equation. However, this shows that the ratio Ds0/Dm0 can be estimated from Ds/Dm by using appropriate values of the related parameters. The scatter of the plots in Figure 8 may be due to errors in the determination of Ds0 and Dm0. For instance, when Ds at 298 K is 2 × 10-5 cm2 s-1 and Es is 10 kJ mol-1, Ds0 is 1.1 × 10-3 cm2 s-1. A similar calculation assuming Es ) 10.5 kJ mol-1 gives Ds0 ) 1.4 × 10-3 cm2 s-1. In this case, an error of 5% in the determination of Es causes an error on Ds0 of about 30%. Although these calculations are hypothetical, they suggest that accurate determinations of Ds0 (and Dm0) are more difficult than that of Es (and Em). The representation of Ds by the restricted diffusion model may be useful for the estimation of Ds0 from Ds or for the confirmation of the validity of values of Ds0 determined by analyzing the temperature dependence of Ds. The relationships between Ehs and Ebs on one hand and Es on the other were derived from the analysis of the correlation between Es and Qst under various RPLC conditions. Although the energy term in eq 1 could be interpreted with a reasonable accuracy, we showed here that, by contrast, a similar quantitative analysis of the entropy term was quite difficult.

Some of the factors discussed earlier do influence the thermodynamic properties of surface diffusion. Accordingly, the correlations between Es and Qst shown in Figures 5a-c are observed. At present, the restricted diffusion model simply represents a fundamental framework for the theoretical analysis of the mechanism of surface diffusion. This model can provide an appropriate interpretation of the intrinsic characteristics of surface diffusion and of the dependence of Ds on the temperature, concentration, and retention.8,21,22,24 However, some parameters of the model cannot be strictly calculated. It is essential to advance fundamental studies regarding the structure and composition of the solutions in the vicinity of a hydrophobic surface, the retention interactions in RPLC, and the mass transfer mechanisms in porous materials. The accumulation of accurate experimental data on surface diffusion measured under various experimental conditions is also necessary. Systematic analyses based on the results of these fundamental studies will probably lead to a comprehensive interpretation of surface diffusion. CONCLUSION This study demonstrated an enthalpy-entropy compensation between Ds0 and Es and a linear free energy relationship between Ds and K for surface diffusion in a variety of RPLC systems. The mechanism of surface diffusion seems to remain the same under all the RPLC conditions used in this study, suggesting the possibility of a consistent interpretation of the surface diffusion mechanism. From the dependence of Ds on the retention, it is likely that Ds/Dm tends toward a limit close to unity when K tends Analytical Chemistry, Vol. 72, No. 7, April 1, 2000

1487

Figure 8. Ratio of the frequency factors of surface and molecular diffusion against the adsorption equilibrium constant.

toward zero, whereas Qst seems to tend toward zero. A similar trend was not observed for Es. The experimental results demonstrated the validity of the representation of Ds with the restricted diffusion model, confirming a correlation between Ds and Dm and the different trends predicted by the model in the correlations between Qst and K and between Es and K. Although linear relationships are observed between Es and Qst, as predicted by the restricted diffusion model, they depend on the experimental conditions, i.e., on the type of compounds studied and on the chemistry of the stationary phase surfaces. Attempts have been made to discuss the significance of some factors influencing the linear correlation between Es and Qst from the viewpoints of the hole-making and jumping activation energies, the obstruction and tortuosity parameters, and the frequency factor of surface diffusion. In conclusion, we identified numerous areas where improvements to the accuracy of the restricted diffusion model would be useful. Nevertheless, the usefulness of this model which provides for consistent interpretations of the mechanism and the characteristics of surface diffusion in RPLC was demonstrated. GLOSSARY ∆A

reduction of total hydrophobic surface area due to adsorption, cm2

As

surface area of a solute, cm2

C

solute concentration, g cm-3

Cs 1488

saturation concentration, g

cm-3


De

intraparticle diffusion coefficient, cm2 s-1

Dm

molecular diffusivity, cm2 s-1

Dm0

frequency factor of molecular diffusion, cm2 s-1

Dp

pore diffusivity, cm2 s-1

Ds

surface diffusion coefficient, cm2 s-1

Ds0

frequency factor of surface diffusion, cm2 s-1

Eap

adsorption potential, kJ mol-1

Eb

activation energy of interaction-breaking (jumping) process, kJ mol-1

Eh

activation energy of hole-making process, kJ mol-1

Em

activation energy of molecular diffusion, kJ mol-1

Es

activation energy of surface diffusion, kJ mol-1

∆Ev

evaporation energy, kJ mol-1

Evis

activation energy of viscosity, kJ mol-1

H

Henry constant, cm3 g-1

K

adsorption equilibrium constant, cm3 g-1

K0

adsorption equilibrium constant at 1/T ) 0 or Qst ) 0, cm3 g-1

KJ

numerical parameter of the Jossens isotherm

p

numerical parameter of the Jossens isotherm

q

amount adsorbed per unit weight of packing material, g g-1

Qst

isosteric heat of adsorption, kJ mol-1

R

gas constant, J mol-1 K-1

Greek Letters

Superscripts

R

ratio of Es to -Qst, -

m

molecular diffusion

β

ratio of Eb to -Qst, -

s

surface diffusion

φ

volumetric fraction of the organic modifier, %

s

µ1

first absolute moment, s

µ2

second central moment, s2

Received for review August 27, 1999. Accepted January 5, 2000.

Fp

particle density, g cm-3

AC9909913


1489

Restricted Diffusion Model for Surface Diffusion in Reversed-Phase

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