Correlation between Surface Diffusion and Molecular Diffusion in

Aug 24, 2001 - diffusion coefficient (Ds) and the corresponding molecular diffusivity ... that surface diffusion is fundamentally related to molecular...
0 downloads 0 Views 125KB Size
Download PDF
9202

J. Phys. Chem. B 2001, 105, 9202-9209

Correlation between Surface Diffusion and Molecular Diffusion in Reversed-Phase Liquid Chromatography Kanji Miyabe† and Georges Guiochon*,‡ Faculty of Education, Toyama UniVersity, 3190, Gofuku, Toyama 930-8555, Japan and 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 ReceiVed: February 14, 2001; In Final Form: July 17, 2001

Surface diffusion data previously measured for different compounds, mobile phase solvents, stationary phases, and temperatures in reversed-phase liquid chromatography (RPLC) were reevaluated to derive information related to the mechanism and characteristic features of surface diffusion. First, a comparison of the surface diffusion coefficient (Ds) and the corresponding molecular diffusivity (Dm) suggests that the dependence of Ds on the RPLC conditions is a consequence of that of Dm. There is a correlation between surface diffusion coefficients and molecular diffusivity. Second, we showed that Ds correlates with the adsorption equilibrium constant (K) and the isosteric heat of adsorption (Qst). These two parameters represent the intensity of the adsorptive interactions between the sample molecules and the surface of the stationary phase. Finally, the ratio Ds/Dm increases with decreasing K and -Qst and Ds becomes of the same order of magnitude as Dm when the adsorptive interactions become small. These results imply that surface diffusion should be regarded as molecular diffusion restricted by the influence of adsorptive interactions. Other information on surface diffusion was also obtained. The value of Ds extrapolated at Qst ) 0 is not equal to Dm. The difference between the two coefficients depends on the experimental conditions, e.g., the nature of the organic modifier in the mobile phase or the surface density of the C18 alkyl ligand on the stationary phase. Our results demonstrate that surface diffusion is fundamentally related to molecular diffusion.

Introduction Because adsorption takes place on a solid surface, porous materials such as silica gels are preferred to solid silica microbeads as adsorbents because of their larger specific surface area. However, porous adsorbents have a major drawback. The adsorbate molecules must migrate through the pores of the adsorbent particles, from the bulk fluid phase to the actual adsorption sites.1,2 Intraparticle mass transfer in porous materials has a significant influence on the performance of adsorbents. In many cases, surface diffusion plays an important role in intraparticle diffusion, particularly with the rather weak adsorbents used in liquid chromatography. Numerous studies have tried to elucidate the mechanism and the characteristics of surface diffusion.3-5 The fundamentals of surface diffusion are mostly studied from the viewpoints of the surface diffusion coefficient (Ds) dependence on the temperature and the amount adsorbed (q).3-5 The temperature dependence of Ds is usually discussed on the basis of the 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 universal gas constant, and T is the absolute temperature. An empirical parameter (R′) is often introduced to correlate Es with the † ‡

Toyama University. University of Tennessee and Oak Ridge National Laboratory.

isosteric heat of adsorption (Qst). For surface diffusion, R′ is usually smaller than unity.

Es ) R′(-Qst)

(2)

Combination of eqs 1 and 2 gives the following relationship

[

Ds ) Ds0 exp

]

-R′(-Qst) RT

(3)

Several models were derived on the basis of eq 3 to explain the dependence of Ds on q. Equation 3 is frequently used as the most basic equation for studying the temperature and the concentration dependence of Ds in gas-solid and liquid-solid adsorption systems.4,5 Obviously, eq 3 indicates that Ds should tend toward Ds0 when the interactions between adsorbate molecules and surface decrease, i.e., when Qst tends toward zero. Values of Ds0 between ca. 10-4 and 10-1 cm2 s-1 were previously reported in different liquid-solid systems.5-12 These data suggest that Ds could be several orders of magnitude larger than the molecular diffusivity (Dm) when -Qst is small, because Dm is usually of the order of ca. 1 × 10-6 to 2 × 10-5 cm2 s-1 in solutions.13 This conclusion is unreasonable because molecular diffusivity is not affected by adsorption at all while surface diffusion obviously is. This observation suggests that eq 3 should not be used to study surface diffusion when -Qst is small. Finally, there is no information regarding the conditions of validity of eq 3, not even what is the acceptable range of Qst. Equation 3 has another major drawback. It leads to a contradiction when trying to explain some thermodynamic

10.1021/jp010563c CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001

Surface Diffusion and Molecular Diffusion in RPLC

J. Phys. Chem. B, Vol. 105, No. 38, 2001 9203

properties of liquid-solid adsorption systems. Surface diffusion is considered as a mass transfer mechanism of molecules in the adsorbed state and is an activated process. The energy gain Es is required for the adsorbate molecules to migrate by overcoming the energy barrier between two close adsorption sites. However, Es should not be larger than -Qst because adsorbate molecules do not need to be completely desorbed to move between two close sites. The ratio Es/(-Qst) should be smaller than unity. In many cases of surface diffusion in gas-solid adsorption, values of Es/(-Qst) smaller than unity were reasonably measured.5,6 By contrast, values of Es larger than -Qst have been frequently reported for liquid-solid adsorption systems.6-8,11,12,14 Surface diffusion is not expected under such conditions because it would be energetically more advantageous for the adsorbate molecules to desorb first from the surface rather than to migrate along it. On the other hand, there are a few reports of values of Es smaller than -Qst in liquid-solid adsorption systems.9,10,15,16 Equation 3 provides no interpretation for these contradictory correlations between Es and Qst in liquid-solid adsorption. The mass transfer inside porous adsorbents is frequently accounted for by assuming parallel contributions of pore and surface diffusion to intraparticle diffusion.3,5

De ) Dp + FpKDs

(4)

where De is the intraparticle diffusivity, Dp is the pore diffusivity (lower than the molecular diffusivity because of the steric hindrance of tortuous pores), Fp is the density of the porous adsorbent, and K is the adsorption equilibrium constant. When K tends toward zero, the contribution of the second term in the right-hand side of eq 4 becomes negligibly small, irrespective of the value of Ds. Because the contribution of surface diffusion is small at low values of K, few detailed studies have so far been made on the dependence of Ds on the intensity of the interactions between adsorbate molecules and the surface of adsorbents, although the temperature and concentration dependence of Ds were abundantly studied, as described earlier.3-5 The main disadvantage of eq 3 is probably the insufficient number of studies made on the correlation between surface diffusion and the intensity of the interactions between adsorbate molecules and adsorbent surface. The derivation of another equation for Ds is necessary to understand better the mechanism and the characteristic features of surface diffusion. Previously, we analyzed surface diffusion data measured in RPLC.6-8,17,18 This chromatographic method is most suitable for fundamental studies of liquid-solid adsorption and of the related mass transfer phenomena because the experimental conditions, i.e., the properties of adsorbates, adsorbents, and solvents, can conveniently be chosen in a wide range.19 Our initial results suggested that Ds depends primarily on Dm and that Ds tends toward the corresponding value of Dm with decreasing adsorption energy.6,17,18 They also suggested a correlation between surface and molecular diffusion. This conclusion is different from the one arising from the conventional concepts on surface and molecular diffusion. So far, it was generally considered that surface diffusion and molecular diffusion are two different modes of molecular migration and that their mass transfer mechanisms differ profoundly. For instance, surface diffusion was frequently studied on the basis of the hopping model, the heterogeneous surface model, and the chemical potential driving force model.3-5 On the other hand, molecular diffusion has been discussed from different theoretical viewpoints of the hydrodynamics or the transition state theories.13

On the basis of the results of our previous studies, we considered surface diffusion as a molecular diffusion process, restricted by the potential field of adsorption within which it takes place. On this basis, we derived a surface-restricted molecular diffusion model as a first approximation of the mechanism of surface diffusion.6-8,17,18 This model allowed a comprehensive interpretation of the mass transfer mechanism of surface diffusion in liquid-solid adsorption systems and of its intrinsic characteristics. It provided an explanation of the contradictory correlations between Es and Qst in liquid-solid systems.6,17 It provided a consistent explanation of the dependence of Ds on the temperature and the amount adsorbed in liquid-solid systems, when the adsorption isotherms were accounted for by the Langmuir, Freundlich, and Jossens-type equations.6,20 However, further investigations of surface diffusion are needed in order to validate the concept and model just discussed because the available experimental data reported in previous papers6-8,17,18 are still too limited. The goal of this study is to present further proofs of the validity of the correlation between surface and molecular diffusion, using surface diffusion data previously measured under different experimental conditions (temperature, sample compounds, mobile and stationary phases) in RPLC. Experimental Section In this study, we reinterpret experimental data previously measured under different experimental conditions of RPLC. Only the information needed to understand the scope and range of validity of the data analyzed here is reported. Other details can be found in the original papers and related review.6-8,21-25 Apparatus. Pulse response experiments (i.e., elution chromatography) were carried out, using a high performance liquid chromatograph system. A valve injector was used for injecting small amounts of a sample solution into a column. The column temperature was kept constant by circulating temperaturecontrolled water around the column. The concentration of the sample compounds at the column exit was monitored with the ultraviolet detector of the HPLC system. Columns and Reagents. Six RPLC columns were filled with different packing materials derived from the same base silica gel. Four materials were chemically bonded to C18 ligands with different bonding densities and the other two were bonded to other alkyl ligands (C1 and C4). The carbon contents of the four C18-bonded silica gels were 6.6, 8.6, 13.7, and 17.1 wt %. The surface density of the C18 ligands was estimated between 0.59 and 3.2 µmol m-2, on the assumption that the typical density of silanol groups is about 8 µmol m-2 on the surface of silica gels.26 The average particle diameter of the base silica gel was 45 µm. This coarse-particle material was chosen in order to facilitate the derivation of accurate values of Ds. As described later, the contributions of the other mass transfer processes in the column and that of the extra-column volumes to the second central moment (µ2′) of the elution peaks must be subtracted to allow the derivation of Ds from µ2′. In this study, the influence of the other sources on peak broadening was minimized because µ2′ was relatively large. Methanol/water and acetonitrile/water mixtures were used as the mobile phase because these two organic solvents are the most popular organic modifiers in RPLC.19,27 The volumetric fraction (φ) of organic modifier was adjusted between 60 and 80%. This range of φ is typical for the concentration of an organic modifier in RPLC.19,27 For instance, it is the one usually selected for the initial evaluation of the efficiency of packed columns by most manufacturers. Alkylbenzene and p-alkyl-

9204 J. Phys. Chem. B, Vol. 105, No. 38, 2001

Miyabe and Guiochon

phenol homologues were used as sample compounds. Uracil and sodium nitrate were used as inert tracers. Procedure. Information on the phase equilibrium and the mass transfer kinetics in the column was derived from the first (µ1) and second moment (µ2′) of the elution peaks, at different column temperatures and mobile phase flow rates, in the ranges 288-308 K and 1.0-2.0 mL min-1, respectively. These measurements were made either under linear isotherm conditions or under local linear conditions (elution of pulses on a concentration plateau). The calculation procedures are briefly described below. More detailed information on moment analysis is available in the literature.1,3,5-7,18,28-31 First, K was derived from µ1. Then, information on the mass transfer kinetics was derived from µ2′. Ds was estimated by subtracting from µ2′ the contributions to band broadening of the other mass transfer processes, axial dispersion, fluid-toparticle mass transfer, and pore diffusion. The band broadening contribution of the adsorption/desorption kinetics at the actual adsorption sites was assumed to be negligibly small. This assumption was experimentally validated for RPLC24 and is widely accepted in studies of mass transfer kinetics in similar adsorption systems used in RPLC.1,5 The contributions of axial dispersion and intraparticle diffusion were separated by taking advantage of the difference in the mobile phase flow rate dependence of the parameters of these two kinetic processes. The fluid-to-particle mass transfer coefficient was derived from the Wilson-Geankoplis equation,3,5,32 and the corresponding contribution to µ2′ was subtracted. The value of Ds was derived from De by subtracting the contribution of pore diffusion (Dp) to intraparticle diffusion on the basis of eq 4. According to the parallel pore model,5 Dp was calculated from Dm, the intraparticle porosity of the stationary phase particle, and the tortuosity factor of the pores. The Wilke-Chang equation was used for estimating Dm in methanol/water mixtures because of the popularity of the equation.13

Dm ) 7.4 × 10-8

(RsvMsv)1/2T ηsvVb,a0.6

(5)

where R is the association coefficient, M is the molecular weight, η is the viscosity, and Vb is the molar volume at the normal boiling point. The subscripts a and sv denote the solute and solvent, respectively. The Perkins-Geankoplis equation was used to calculate Dm in acetonitrile/water mixtures because no value of R is available for acetonitrile.13 Dm in neat acetonitrile and in water was calculated using the Scheibel and HaydukLaudie equations, respectively.13 Information on the intraparticle porosity and tortuosity factors was derived from the pulse response experiments made with the inert tracers. The influence of the extracolumn volumes on µ1 and µ2′ was measured by performing the same measurements without a column. This information was used to correct the experimental data. The influence of the peak asymmetry (tailing or fronting) on the estimation of the moments µ1 and µ2′ of the elution peaks was considered. There are several possible causes of peak distortion, e.g., heterogeneous mass transfer kinetics1 and column radial heterogeneity.33-35 In this study, only the radial heterogeneity of the column was regarded as a possible origin of peak skew. The possibility of heterogeneous mass transfer kinetics was ignored because the surface of C18-silica gels behaves as energetically homogeneous toward the compounds studied here.1,6 For these compounds, (1) the phase equilibrium on C18-silica gels is usually accounted for by the simple Langmuir isotherm, (2) Qst is nearly constant, irrespective of

q, and (3) Es is independent of q. The assumption of an apparent uniformity of the C18-silica gel surface was also supported by a theoretical analysis of the adsorption behavior of 2-phenylethanol and 3-phenylpropanol on a C18 phase, from a methanol/ water mixture.36 Finally, the influence of the width of the sample pulses on the moments µ1 and µ2′ of the elution peaks was corrected by assuming that the injection pulses had a rectangular profile. This effect was found negligible because of the extremely small size of the sample pulses injected. These corrections are responsible for the error made in the estimation of Ds. It seems that this error is of the order of several percent.6 Results and Discussion The behavior of RPLC systems depends on the composition of the sample, the packing material, and the mobile phase. The characteristics of the adsorption equilibrium and the mass transfer kinetics are affected by the properties of these three components.1,19,27 In this study, we reevaluated surface diffusion data previously measured under different experimental conditions characterized by the temperature, the nature and concentration of the sample compounds, the nature and concentration of the organic modifier in the mobile phase, and the length and density of the alkyl ligands bonded to the surface of the packing material. The surface diffusion data were analyzed by taking the correlation between Ds and Dm into account. The value of Ds was also correlated with K and Qst, parameters that depend on the adsorption energy of the sample components. Correlation of Surface Diffusion with Molecular Diffusion. Dm depends on the physicochemical properties of the components of the sample and the mobile phase and on the temperature (eq 5). In this section, the dependence of Ds on the properties of the compounds involved is analyzed on the basis of that of Dm. Influence of the Temperature on Ds. Parts a-c of Figure 1 illustrate the temperature dependence of Ds and Dm. Figure 1a compares the values of Ds for alkylbenzenes at 288 and 308 K with those at 298 K. As expected, Ds increases with increasing temperature. The values at 288 and 308 K are respectively smaller and larger than those at 298 K by a factor of about 1.2-1.4. Figure 1a also shows the correlations between the values of Dm for the same compounds estimated by eq 5 at the same temperatures. The solid lines represent the correlation between values of Dm at these temperatures. The dotted lines are the extrapolations of the solid lines. Although Figure 1a shows a slight scatter, most values of Ds are located on the dotted lines, suggesting that the temperature dependence of Ds is very close to that of Dm. The same conclusion was obtained for the p-alkylphenol homologues (not shown). Parts b and c of Figure 1 show similar results obtained under different conditions. In Figure 1b, the values of Ds at the same three temperatures, for three different values of the methanol concentration of methanol/water mixtures, are correlated with those of Dm. In Figure 1c, the values of Ds at the same three temperatures, on the surface of C18-silica gels having different alkyl ligand densities, are similarly correlated with Dm. Although there are some small shifts from the dotted lines in Figure 1ac, the temperature dependence of Ds is very close to that of Dm in all cases, irrespective of the type of compound, methanol concentration in the mobile phase, and alkyl ligand density on the adsorbent surface. Similar results concerning the relative temperature dependence of Ds and Dm were observed for

Surface Diffusion and Molecular Diffusion in RPLC

J. Phys. Chem. B, Vol. 105, No. 38, 2001 9205

Figure 2. Comparison of the surface diffusion coefficient with the molecular diffusivity of alkylbenzene and p-alkylphenol derivatives at three different temperatures.

Figure 1. Comparison of the temperature dependence of the surface diffusion coefficient with that of the molecular diffusivity in RPLC systems using (a) alkylbenzene derivatives, (b) methanol/water mobile phases of three different compositions, and (c) four C18 stationary phases of different ligand densities.

acetonitrile/water mixtures and for silica gels bonded to alkyl ligands of different length (not shown). The results in Figure 1a-c imply a correlation between surface and molecular diffusion. Influence of the Nature of the Sample Compounds on Ds. Figure 2 shows the correlation between values of Ds and Dm for alkylbenzene and p-alkylphenol homologues . Almost the same values of Ds and Dm were observed for compounds having the same alkyl ligand (R-) in the two homologous series, at each temperature. The data for Ds are very close to the dotted line, which is the extrapolation of the solid straight line. As in Figure 1a-c, this solid line is the extrapolation of the correlation between the Dm values for alkylbenzene and p-alkylphenol homologues , derived from eq 5. Thus, it is likely that a difference between the values of Ds for compounds having the same R- is due to a difference in their Dm. Finally, it is known that Ds varies with the compound concentration. The correlation between the concentration dependence of Ds and Dm is discussed later. Influence of the Composition of the Organic Modifier on Ds. Equation 5 indicates that Dm depends also on the mobile phase composition. Previously, we reported on the influence of the nature of the organic modifier on Ds in RPLC at different temperatures and φ’s6,17,18 and suggested the existence of a correlation between Ds and Dm. In this work, we studied the influence of the methanol concentration on Ds, taking into account the corresponding changes of Dm. Figure 3 compares the values of Ds at φ ) 60 and 80 vol % with those at φ ) 70 vol %. Most data points are located on or close to the dotted straight lines. The results in Figure 3 suggest that the variation of Ds arising from a change in φ result primarily from the corresponding variation of Dm. A similar result was obtained with acetonitrile/water solutions (not shown). It is obvious that Ds increases with increasing φ, regardless of the nature of the organic modifier (Figure 3). This is explained by the decrease in the mobile phase viscosity η. It is known that this viscosity decreases almost linearly with increasing organic modifier concentration for both methanol/water and acetonitrile/water solutions.1,37 The results in Figures 1b and 3 demonstrate that the influence of the mobile phase composition on surface diffusion and molecular diffusion are similar.

9206 J. Phys. Chem. B, Vol. 105, No. 38, 2001

Figure 3. Comparison of the dependence of the surface diffusion coefficient on the mobile phase composition with that of the molecular diffusivity in RPLC using methanol/water mixtures of three different compositions.

To date, surface diffusion is still considered as the migration of adsorbate molecules in the adsorbed phase on the surface of adsorbents and it is regarded as a phenomenon that is independent of the properties of the bulk solvent phase.4,5 Few papers ever discussed the influence of the solvent composition on Ds in liquid-solid adsorption. Most conventional adsorbents, e.g., activated carbons and polymeric resins, are used to extract organic compounds from aqueous solutions. Solvents other than water are rarely used. Few studies besides our own have been pursued on surface or lateral diffusion in RPLC.38-42 Ds of rubrene on a C18 phase was reported to increase with increasing concentration of methanol in methanol/water mobile phases, in the range 0 < φ < 20 vol %. Ds of iodine on a C1 phase increases with increasing φ of methanol between 50 and 75 vol %.40,42 However, no comprehensive interpretation of the characteristics and mechanisms of surface diffusion has yet been suggested. Influence of the Surface Density of the C18 Ligand on Ds. In Figure 4, the values of Ds on the first three C18 stationary phases (carbon contents, 6.6, 8.6, and 13.7 wt %) are compared with those on the fourth phase (17.1 wt %). Almost parallel plots of Ds are observed. The lower the carbon content, the larger the value of Ds, irrespective of the temperature (The same result, not shown, was obtained with acetonitrile.) Similar results were observed with the C1 and C4 bonded silica gel (not shown).25 Figure 4 shows also the values of Dm, which are on the straight line having a slope unity because Dm is independent of the nature of the stationary phase. By contrast with the results in Figures 1-3, the differences between the values of Ds cannot be accounted for only on the basis of Dm. The results in Figure 4 suggest that some factors other than Dm should be taken into account to explain the influence of chemical modifications of the stationary phase surface on Ds. Previously, we studied the influence on Ds of modifications of the surface chemistry of the silica gel.6,8 The hydrophobicity of the packing material depends on the length and density of the bonded alkyl ligands. The surface diffusion data were analyzed by assuming the same mechanism of surface diffusion, irrespective of the surface chemistry of the packing materials.

Miyabe and Guiochon

Figure 4. Comparison of the surface diffusion coefficient on four different C18 stationary phases with the molecular diffusivity.

However, the adsorption energy increases with increasing length or density of the alkyl ligands. It seems that surface diffusion becomes more restricted by an increase of the adsorption energy. This could be one of the factors alluded to above. This subject is discussed in the following section. Dependence of Ds on K and Qst. As shown in Figures 1-4, Ds depends on the nature of the sample compounds, the mobile phase solvents, the stationary phase surface, and the temperature. The previous discussion showed that the Ds depends primarily on the value of Dm and to a lesser extent on the surface chemistry. In the following subsections, the ratio Ds/Dm is correlated with the adsorption energy in order better to clarify the correlation between Ds and Dm. While measurements were carried out between 288 and 308 K, the ratio Ds/Dm could not be usefully plotted against K or Qst because K does not vary sufficiently within this narrow temperature range. However, Qst could be calculated from the temperature dependence of K, according to the van’t Hoff equation. Influence of the Nature and Concentration of the Sample Compounds on Ds. Figure 5a shows the dependence of Ds/Dm on K for some alkylbenzenes. Ds/Dm increases with decreasing K and the values of Ds/Dm scatter around a curved solid line (inset). This line was obtained as a nonlinear regression of the data points. The value toward which this curve tends when K tends toward 0 cannot be precisely known because the extrapolation includes a certain error. It is probably lower than 1 but Ds remains of the same order of magnitude as Dm when the adsorption constant becomes very small. The adsorption energy is related to Qst. Figure 5b shows a plot of Ds/Dm vs Qst/RT. Although a similar relationship between Ds and Qst was previously reported, the authors plotted Ds in various gas-solid systems vs -Qst/RT without taking into account the correlation between Ds and Dm.43 Figure 5b shows that ln(Ds/Dm) increases linearly with decreasing -Qst/RT. The results shown in Figure 5a,b indicate that Ds/Dm increases with decreasing adsorption constant and that the value measured for Ds is 1 to 2 orders of magnitude smaller than that of Dm. The analysis of the surface diffusion data of p-alkylphenols provides a similar conclusion (not shown). These observations imply that surface diffusion should be regarded as molecular diffusion restricted by the adsorption energy of the sample

Surface Diffusion and Molecular Diffusion in RPLC

J. Phys. Chem. B, Vol. 105, No. 38, 2001 9207

Figure 6. Dependence of the ratio of the surface diffusion coefficient to the molecular diffusivity of p-alkylphenols at different amounts adsorbed on adsorption equilibrium constant.

Figure 5. Dependence of the ratio of the surface diffusion coefficient to the molecular diffusivity of alkylbenzene derivatives on (a) the adsorption equilibrium constant and (b) the isosteric heat of adsorption.

molecules. It was suggested that the migration of adsorbate molecules in the small pores of an adsorbent is restricted by both a hydrodynamic effect and the adsorption of the molecules on the surface of the adsorbent.44 The variation of the effective diffusivity of the adsorbate in the pores was discussed in connection with a correlation between the diameter of the adsorbate molecules and that of the pores. It was also shown that the adsorption of the adsorbate further reduced its effective diffusivity. However, there was no detailed analysis of the restriction of surface diffusion due to adsorption of the molecules on the adsorbent surface. There is a linear correlation between the logarithm of Ds/Dm and Qst/RT (Figure 5b). The extrapolated ratio at Qst ) 0 is Ds/Dm ) 0.54 (in agreement with Figure 5a), suggesting that Ds is not exactly equal to Dm at Qst ) 0, even though the extrapolation is somewhat imprecise. Surface diffusion probably takes place in the vicinity of the adsorbent surface. In RPLC, the organic modifier is preferentially adsorbed onto the hydrophobic surface of the stationary phase.45,46 The concentration

of organic modifier is slightly higher close to the surface than in the bulk mobile phase. The structural and chemical environment in the monolayer region is different from that in the bulk mobile phase. This monolayer, consisting of alkyl ligands and mobile phase solvents, plays an important role as an actual stationary phase.47 The retention of the sample molecules takes place in this stationary phase. Obstruction of the lateral migration of the sample molecules increases with increasing density and/ or length of the alkyl ligands and interfers with fast mass transfer. The tortuosity of the surface may also influence Ds. These are probably the reasons why Ds at Qst ) 0 differs from Dm. The difference between the extrapolated value of Ds at Qst ) 0 and Dm could depend on some or all these factors. When a sample pulse is injected on a concentration plateau, the retention of the peak decreases usually, since most isotherms are convex upward.1 Assuming a Langmuir isotherm (q ) aC/(1 + bC)), which is always legitimate at moderate concentrations, the retention time on a plateau of concentration Cp is given by

tR ) t0 + tp +

k0′t0

(6)

(1 + bCp)2

where k0′ ) aF is the retention factor, F being the phase ratio. The apparent retention factor of the peak and the apparent equilibrium constant become

k′ ≈

k0′ (1 + bCp)

2

hence K ≈

K0 (1 + bCp)2

(7)

Figure 6 shows a plot of Ds vs K for peaks acquired with samples of p-tert-butylphenol and p-tert-octylphenol at increasing concentration from near-zero to 1 wt %. It was already known that Ds shows a positive concentration dependence.3-6,10,23,24 As in Figure 5a, Ds/Dm increases with decreasing K. The solid line is the best polynomial function, i.e., Ds/Dm ) c1/K + c2 + c3K + c4K2 (with c1 - c4, numerical coefficients) fitted to the experimental data of p-tert-octylphenol at the three different temperatures. A similar profile is observed for the data of p-tert-butylphenol. In this study, it was assumed

9208 J. Phys. Chem. B, Vol. 105, No. 38, 2001

Miyabe and Guiochon 7b) for data acquired with methanol/water mixtures at several methanol concentrations. As in Figures 5a,b, and 6, Ds/Dm increases with decreasing K and -Qst/RT. Similar results were observed for acetonitrile/water mixtures (not shown). Parts a and b of Figure 7 also confirm that the restriction of molecular diffusion in the potential field of adsorption, in the vicinity of the adsorbent surface, increases with increasing adsorption energy. These results are consistent with those in Figures 5a,b and 6. Although linear correlations are observed between ln(Ds/Dm) and Qst/RT, in Figure 7b, the slope and intercept of the straight lines seem to depend on φ. We can write

ln

Figure 7. Dependence of the ratio of the surface diffusion coefficient to the molecular diffusivity at three different methanol compositions on (a) the adsorption equilibrium constant and (b) the isosteric heat of adsorption.

that Dm values of the sample compounds were constant irrespective of their concentration because other experimental data suggested that the concentration dependence of Dm was small in the concentration range up to about 5%, despite the significant increase in the viscosity of the sample solutions.1 The results in Figure 6 are similar to those in Figure 5a. The concentration dependence of Qst is known to be small in RPLC.6,23,48 That of Dm is minimal. An increase in q being accompanied by an increase in Ds even though Dm and Qst are almost constant, irrespective of the sample concentration, we must conclude that other parameters exhibiting a concentration dependence must be taken into account. Previously, we discussed the influence on Ds of two other parameters, the slope of the adsorption isotherm and the adsorption potential, and proposed a new equation to analyze the temperature and concentration dependence of Ds.6,17,20 Influence of the Mobile Phase Composition on Ds. Figure 7 shows plots of Ds/Dm against K (Figure 7a) and Qst/RT (Figure

( )

(-Qst) Ds +δ )λ Dm RT

(8)

where λ and δ are the slope and intercept of the straight lines in Figure 7b, respectively. λ is related to the restriction to molecular diffusion due to the adsorption of the molecules on the surface of the adsorbent. The absolute value of λ must be smaller than unity because it is not required for the adsorbed molecule to desorb completely from the surface into the bulk mobile phase when the molecules diffuse on the surface. As shown in Figure 7b, the value of λ is about -0.3. δ accounts for the contributions to Ds due to the difference in structural and chemical environment in the region near the surface and in the bulk mobile phase, as described earlier. The difference in the values of δ suggests that this environment in a region close to the adsorbent surface depends on φ. A study concerning the physical meanings of λ and δ is in progress. Influence of the Stationary Phase Composition on Ds. Ds also depends on the chemistry of the surface, for instance, on the length and density of the alkyl ligands on the surface of the adsorbent. In a previous paper,8 we reported experimental data on the influence of K or Qst on Ds obtained with several silica gels chemically modified with alkyl ligands of different lengths and densities. This suggested that a decrease in K or Qst causes an increase in Ds/Dm and that the ratio Ds/Dm tends toward a value close to unity at K ) 0 or Qst ) 0, indicating again that Ds and Dm have the same order of magnitude when the adsorption energy is negligibly small. The correlations observed between ln(Ds/Dm) and -Qst/RT for the six RPLC packing materials studied (four C18-, a C1-, and a C4-silica gel) are nearly linear. Their slope and intercept depend on the density of the C18 ligand and on the length of the alkyl ligands. To elucidate in more detail the intrinsic characteristics and the mechanism of molecular migration in surface diffusion, the physical and chemical properties of the monolayer in which surface diffusion takes place should now be clarified. Conclusion The reevaluation of surface diffusion data previously measured showed a correlation between surface diffusion and molecular diffusion and that surface diffusion should be regarded as molecular diffusion restricted by the potential field of adsorption. These conclusions differ from the conventional concept that surface and molecular diffusions are two completely different modes of mass transfer and that there is no correlation between their mechanisms. By contrast with the diffusive migration of the sample molecules in the bulk mobile phase, the mass transfer of molecules diffusing along the surface of an adsorbent is more restricted because of the localization of the molecules in the monolayer.

Surface Diffusion and Molecular Diffusion in RPLC The new surface-restricted molecular diffusion model that we suggested to interpret the characteristics and mechanisms of surface diffusion consider that the parameter R′ in eq 2 is not constant, by contrast with the conventional assumption made so far, that it is constant and smaller than unity. The value of Es seems to be the sum of two contributions. One, needed to make a hole in the mobile phase, is independent of Qst. The other, needed for the adsorbate molecule to jump into this hole, has an activation energy proportional to Qst. This model and the new equation for Ds overcome several serious drawbacks of the conventional basic equation (eq 3) for surface diffusion. A comprehensive interpretation of the characteristic features and the mechanism of mass transfer by surface diffusion has not yet been completely accomplished. For instance, surface diffusion of an adsorbate takes place even if its retention is weak but there cannot be surface diffusion for unretained compounds. Conversely, when a compound is very strongly retained, we expect that its migration by surface diffusion be highly restricted by the adsorptive interactions. If K is infinitely large, molecular migration by surface diffusion should be extremely slow. The behavior in the regions where surface diffusion appears (at very low values of K) and disappears (at very large values of K) are unclear. It is likely that the variations of Ds are continuous because K and Qst, which represent the intensity of the adsorptive interactions vary also continuously. The results of this study consolidate our model of surface-restricted molecular diffusion6-8,17,18,20 and will permit its further development, leading to a new correlation between Ds and Dm. This topic is currently under investigation. Glossary De Dm Ds Ds0 Es F K k′ Lf M q Qst R T Vb

intraparticle diffusivity, cm2 s-1 molecular diffusivity, cm2 s-1 surface diffusion coefficient, cm2 s-1 frequency factor of surface diffusion, cm2 s-1 activation energy of surface diffusion, kJ mol-1 phase ratio ((1 - )/, with , total column porosity) adsorption equilibrium constant, cm3 g-1 retention factor loading factor molecular weight amount adsorbed, g g-1 isosteric heat of adsorption, kJ mol-1 gas constant, J mol-1 K-1 absolute temperature, K molar volume at normal boiling point, cm3 mol-1

Greek Letters R association coefficient R′ ratio of Es to - Qst δ intercept of the correlation between ln(Ds/Dm) and - Qst/ RT η viscosity, Pa s λ slope of the correlation between ln(Ds/Dm) and - Qst/RT µ1 first moment, s second central moment, s2 µ2′ Fp particle density, g cm-3 φ composition of the organic modifier in the mobile phase Subscripts a adsorbate sv solvent

J. Phys. Chem. B, Vol. 105, No. 38, 2001 9209 Acknowledgment. This work was supported in part by a Grant-in-Aids for Scientific Research (No. 12640581) from the Ministry of Education, Science, and Culture of Japan, by Grant CHE-00-70548 of the National Science Foundation of the U.S.A., and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. References and Notes (1) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of PreparatiVe and Nonlinear Chromatography; Academic Press: Boston, 1994. (2) Giddings, J. C. Dynamics of Chromatography, Part I, Principles and Theory; Marcel Dekker: New York, 1965. (3) Ruthven, D. M. Principles of Adsorption & Adsorption Processes; John Wiley and Sons: New York, 1984. (4) Kapoor, A.; Yang, R. T.; Wong, C. Catal. ReV.-Sci. Eng. 1989, 31, 129. (5) Suzuki, M. Adsorption Engineering; Kodansha/Elsevier: Tokyo/ Amsterdam, 1990. (6) Miyabe, K.; Guiochon, G. AdV. Chromatogr. 2000, 40, 1. (7) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 1999, 103, 11086. (8) Miyabe, K.; Guiochon, G. Anal. Chem. 2000, 72, 1475. (9) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 728. (10) Suzuki, M.; Fujii, T. AIChE J. 1982, 28, 380. (11) Awum, F.; Narayan, S.; Ruthven, D. Ind. Eng. Chem. Res. 1988, 27, 1510. (12) Ma, Y. H.; Lin, Y. S.; Fleming, H. L. AIChE Symp. Ser. 1988, 84, 1. (13) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. (14) Chiang, C. B.; Hidajat, K.; Uddin, M. S. Sep. Sci. Technol. 1989, 24, 581. (15) Muraki, M.; Iwashita, Y.; Hayakawa, T. J. Chem. Eng. Jpn. 1982, 15, 34. (16) Itaya, A.; Fujita, Y.; Kato, N.; Okamoto, K. J. Chem. Eng. Jpn. 1987, 20, 638. (17) Miyabe, K.; Takeuchi, S. J. Phys. Chem. B 1997, 101, 7773. (18) Miyabe, K.; Guiochon, G. Anal. Chem. 1999, 71, 889. (19) Krstulovic´, A. M.; Brown, P. R. ReVersed-Phase High-Performance Liquid Chromatography; John Wiley & Sons: New York, 1982. (20) Miyabe, K.; Takeuchi, S. AIChE J. 1997, 43, 2997. (21) Miyabe, K.; Takeuchi, S. Anal. Chem. 1997, 69, 2567. (22) Miyabe, K.; Takeuchi, S. Ind. Eng. Chem. Res. 1997, 36, 4335. (23) Miyabe, K.; Suzuki, M. AIChE J. 1992, 38, 901. (24) Miyabe, K.; Guiochon, G. Anal. Chem., in press. (25) Miyabe, K.; Guiochon, G. J. Chromatogr. A 2001, 919, 231. (26) Unger, K. K. Porous Silica; Elsevier: Amsterdam, 1979. (27) Snyder, L. R.; Glajch, J. L.; Kirkland, J. J. Practical HPLC Method DeVelopment; John Wiley & Sons: New York, 1988. (28) Kucera, E. J. Chromatogr. 1965, 19, 237. (29) Kubin, M. Collect. Czech Chem. Commun. 1965, 30, 2900. (30) Grushka, E.; Myers, M. N.; Schettler, P. D.; Giddings, J. C. Anal. Chem. 1969, 41, 889. (31) Grushka, E. J. Phys. Chem. 1972, 76, 2586. (32) Wilson, E. J.; Geankoplis, 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. (36) Quin˜ones, I.; Guiochon, G. J. Chromatogr. A 1998, 796, 15. (37) Colin, H.; Diez-Masa, J. C.; Guiochon, G.; Czajkowska, T.; Miedziak, I. J. Chromatogr. 1978, 167, 41. (38) Bogar, R. G.; Thomas, J. C.; Callis, J. B. Anal. Chem. 1984, 56, 1080. (39) Stahlberg, J.; Almgren, M.; Alsins, J. Anal. Chem. 1988, 60, 2487. (40) Wong, A. L.; Harris, J. M. J. Phys. Chem. 1991, 95, 5895. (41) Zulli, S. L.; Kovaleski, J. M.; Zhu, X. R.; Harris, J. M.; Wirth, M. J. Anal. Chem. 1994, 66, 1708. (42) Hansen, R. L.; Harris, J. M. Anal. Chem. 1995, 67, 492. (43) Hansen, R. L.; Harris, J. M. Anal. Sladek, K. J.; Gilliland, E. R.; Baddour, R. F. Ind. Eng. Chem. Fundam. 1974, 13, 100. (44) Satterfield, C. N.; Colton, C. K.; Wayne, H. P., Jr. AIChE J. 1973, 19, 628. (45) Jaroniec, M. J. Chromatogr. A 1993, 656, 37. (46) Jaroniec, M. J. Chromatogr. A 1996, 722, 19.37. (47) Tanaka, N.; Kimata, K.; Hosoya, K.; Miyanishi, H.; Araki, T. J. Chromatogr. A 1993, 656, 265. (48) Quin˜ones, I.; Ford, J. C.; Guiochon, G. Chem. Eng. Sci. 2000, 55, 909.