Mass transfer kinetics on the heterogeneous binding sites of molecularly imprinted polymers

Chemical Engineering Science 60 (2005) 5425–5444

www.elsevier.com/locate/ces

Mass transfer kinetics on the heterogeneous binding sites ofmolecularlyimprinted polymers

Hyunjung Kima,b, Krzysztof Kaczmarskia,b,c, Georges Guiochona,b,∗aDepartment of Chemistry, University of Tennessee, Knoxville, TN, 37996-1600, USA

bDivision of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, 37831-6120, USAcFaculty of Chemistry, Rzeszów University of Technology, 35-959 Rzeszów, Poland

Received 26 January 2005; received in revised form 13 April 2005; accepted 14 April 2005Available online 29 June 2005

Abstract

The mass transfer kinetics of theL- andD-Fmoc-Tryptophan (Fmoc-Trp) enantiomers on Fmoc-L-Trp imprinted polymer (MIP) and onits reference polymer (NIP), were measured using their elution peak profiles and the breakthrough curves recorded in frontal analysis forthe determination of their equilibrium isotherms, at temperatures of 40, 50, 60, and 70◦C. At all temperatures, the isotherm data of theFmoc-Trp enantiomers on the MIP were best accounted for by the Tri-Langmuir isotherm model, while the isotherm data of Fmoc-Trpon the NIP were best accounted for by the Bi-Langmuir isotherm model. The profiles of the elution bands of various amounts of eachenantiomer were acquired in the concentration range from 0.1 to 40mM. These experimental profiles were compared with those calculatedusing the best values estimated for the isotherm parameters and the lumped pore diffusion model (POR), which made possible to calculatethe intraparticle diffusion coefficients for each system. The results show that surface diffusion contributes predominantly to the overall masstransfer kinetics on both the MIP and the NIP, compared to external mass transfer and pore diffusion. The surface diffusion coefficients(Ds ) of Fmoc-L-Trp on the NIP does not depend on the amount bound (q) while the values ofDs for the two Fmoc-Trp enantiomers onthe MIP increase with increasingq at all temperatures. These positive dependencies ofDs on q for Fmoc-Trp on the MIP were fairlywell modeled by indirectly incorporating surface heterogeneity into the surface diffusion coefficient. The results obtained show that themass transfer kinetics of the enantiomers on the imprinted polymers depend strongly on the surface heterogeneity.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Fmoc-L-tryptophan imprinted polymers; Frontal analysis; Mass transfer kinetics; Isotherm parameters; Tri-Langmuir isotherm model;Bi-Langmuir isotherm model; Lumped pore diffusion model; Intraparticle diffusion; Peak profiles; Isosteric heat of adsorption; Surface heterogeneity

1. Introduction

Molecularly imprinted polymers (MIPs) have becomepopular materials for the separation of enantiomers, due totheir high selectivity and to their chemical and physical sta-bilities. MIPs provide a high selectivity toward the moleculepresent in solution during their polymerization (template).

∗ Corresponding author. Department of Chemistry, University of Ten-nessee, 1420 Circle Drive, Knoxville, TN 37996-1600, USA. Tel.:+1 8659740733; fax: +18659742667.

E-mail address:[email protected](G. Guiochon).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.04.057

The most commonly used strategy to prepare MIPs is the useof non-covalent interactions between a target molecule (thetemplate) and some suitable functional groups. These inter-actions allow the formation of template-functional monomercomplexes in solution. These complexes are then immobi-lized into a polymer matrix by copolymerization with a highconcentration of cross-linking monomers. Complementarysize, shape, and functionalities toward the template in theMIPs can be obtained by extracting the template from thepolymer matrix after the end of the polymerization process.One of the major problems encountered when using

MIPs as chiral stationary phases is the serious tailing often

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5426 H. Kim et al. / Chemical Engineering Science 60 (2005) 5425–5444

exhibited by the profiles of the peaks, particular those of theimprinted molecule or template. The tailing of the peak ofthe template on the corresponding MIP have been attributedto the wide distributions of energies of the adsorption sitesand to slow intraparticle mass transfer kinetics on the MIP(Sellergren, 2001). If this is so, the chromatographic per-formance of MIPs could be improved by either preparinga more homogeneous surface or improving the accessibilityof the binding sites. Attempts to prepare a more homoge-neous surface have included chemical modifications of thebinding sites (Umpleby et al., 2001) and their imprintinginside dendrimers (Zimmerman and Lemcoff, 2004) whileattempts to achieve a higher accessibility of these sites weredirected toward developing different formats of imprintedpolymers to locate the binding sites only on the polymersurface (Sulitzky et al., 2002; Biffis et al., 2001; Perez et al.,2001; Ye et al., 2000; Yilmaz et al., 2000).Before defining a clear strategy for the improvement of

the performance of MIPs, it is necessary to acquire a bet-ter understanding of the influence of the chemical modi-fications made in their synthesis on the kinetic and ther-modynamic parameters of their surface. The fact that tworeports (Sajonz et al., 1998; Miyabe and Guiochon, 2000)only, to our knowledge, give proper kinetic information onMIPs speaks for the large knowledge gap in this area. Reli-able kinetic data are needed to design better MIPs and makebetter chiral stationary phases. Acquiring peak profiles ina broad concentration range and comparing these profileswith those calculated using appropriate kinetic models ofchromatography allows the determination of the parametersof the mass transfer kinetics on the stationary phase. Thecalculation of the peak profiles it is necessary to know theisotherm parameters, which can be measured accurately andreliably obtained by frontal analysis. Numerous mathemati-cal models are available to relate the mass transfer kineticsand the peak profiles in chromatography (Guiochon et al.,1994). When the mass transfer resistances are small, theyhave little influence on peak profiles and the equilibrium-dispersive model can be used. Otherwise, the general rate(GR) model, the lumped pore diffusion (POR) model, or thetransport-dispersive (TD) model can be used, depending onthe nature of the problem. The GR model is the most com-plex and general model of chromatography. It takes into ac-count all possible mass transfer resistances. However, thismodel requires the determination of a relatively large num-ber of parameters, such as the axial dispersion coefficient,the external mass transfer rate coefficient, and the effec-tive diffusion through the pores, including the external bedporosity and the adsorbent particle porosity. These param-eters are difficult to measure accurately. The simpler PORmodel requires the same number of parameters but its nu-merical solutions can be calculated much faster. The simpleTD model requires only the value of the axial dispersioncoefficient, the overall mass transfer coefficient, and the to-tal bed porosity, parameters which can be derived from afew simple measurements. However, this simple model may

lead to erroneous interpretations of the kinetic parametersbecause it is often too simple to account for all the complexphenomena involved in the particles of the stationary phasewhen the mass transfer kinetics is slow compared to the rateof axial dispersion (Kaczmarski et al., 2001).This work provides a quantitative analysis of the mass

transfer kinetics of the Fmoc-Tryptophan (Trp) enantiomerson a Fmoc-L-Trp imprinted polymer (MIP) and on its ref-erence non-imprinted polymer (NIP) in organic based mo-bile phases, at four different temperatures. It reports on theacquisition and the modeling of the isotherm data, on thederivation of estimates of the parameters of the mass trans-fer kinetics, and on the calculation of band profiles usingthe POR model. Finally, these results allow a better under-standing of the mass transfer mechanisms on MIP stationaryphases.

2. Theory

2.1. Modeling method

The kinetics of mass transfer in the cross-linked poly-mer is relatively slow. Accordingly, a simple chromatog-raphy model like the equilibrium dispersive model (ED)(Guiochon et al., 1994) cannot be used. The best chromatog-raphy model is the GR model which takes into account allpossible sources of band spreading (axial dispersion, exter-nal and internal mass transfer resistances). The main dis-advantage of this model is the long CPU time needed forband profile calculations. In most cases, i.e., when the masstransfer resistances are not very low, the simpler PORmodelcan be used (Morbidelli et al., 1982, 1984). If Pe>100,St/Bi >5, replacing the GR with the POR model is legit-imate (Kaczmarski and Antos, 1996). Later, it was shownthat this option is also valid when the effective diffusivity isthe sum of the pore and surface diffusivity (Gubernak et al.,2004a). The same version of the POR model as the one usedin Gubernak et al. (2004a)allows a detailed investigation ofthe mass transfer resistances in a non-imprinted (NIP) and inan imprinted polymer (MIP). The condition of applicabilityof POR model is fulfilled since the Peclet number was al-ways greater than 8000 and the quotientSt/Bi greater than60 in our experiments.The modeling method consists of simulating the dynam-

ics of band migration in the column by calculating the bandprofiles under various sets of experimental conditions us-ing the POR model and determining the value of the intra-particle diffusion coefficient that minimizes the differencebetween the calculated and the experimental band profiles.This method of parameter identification gives the overallmass transfer coefficient(ki) from which the intraparticlediffusion coefficient(Deff) can be derived.The POR model consists of two mass balance equations,

one in the fluid phase percolating through the stationaryphase (Eq. (1)); the other in the stagnant fluid impregnating

H. Kim et al. / Chemical Engineering Science 60 (2005) 5425–5444 5427

the pores of the stationary phase (Eq. (2)).

�e�C�t

+ u �C�z

= �eDL�2C

�2z− (1− �e)kiap(C − Cp), (1)

�p�Cp�t

+ (1− �p)�q�t

= kiap(C − Cp), (2)

where�e is the external porosity,�p is the internal (or parti-cle) porosity (with�p = �t − �e/(1− �e), �t being the totalporosity),C andCp are the concentration of solutes in themobile phase and inside the particle pores, respectively,qis the concentration of the solute in the adsorbed phase (Cpandq are average concentrations in the pores),t is the time,z is the distance along the column,u is the superficial ve-locity of the mobile phase,DL is the axial dispersion coeffi-cient,ap=3/Rp is the external surface area of the particlesper unit volume of the particles,Rp is the equivalent parti-cle radius (it is assumed that all the particles are sphericaland take for value of their diameter the average of their sizewhich ranges between 25 and 38�m), andki is the overallmass transfer coefficient for the solute, which is obtainedfrom the modeling method and is given by

ki =[

1

kext+ 1

kint

]−1

(3)

with

kint = 10Deff

dp, (4)

wherekext is the external mass transfer coefficient, andkintis the internal mass transfer coefficient, derived from theestimatedki given by the modeling method.The effective diffusivityDeff is calculated from one of

the following two equations (Gubernak et al., 2004a):

Deff = �pDm�

+ (1− �p)DSq

Cp, (5)

Deff = �pDm�

+ (1− �p)DSdq

dCp, (6)

where� is the tortuosity factor, given by

� = (2− �p)2

�p.

In Eq. (5),q is the average surface concentration calculatedfrom the isotherm model using the average pore concentra-tion Cp calculated from Eq. (2). The derivative dq/dCp inEq. (6) was calculated directly from the isotherm model us-ing the average pore concentration of a substrate (Cp).Eqs. (5) and (6) express the assumption that the diffu-

sional mass flux inside the particles is accounted for by thesum of the molecular flux inside the fluid phase containedin the pores of the particles of the packing material and ofthe surface diffusion flux. Eq. (5) assumes that the drivingforce of surface diffusion is the gradient of chemical po-tential (Krishna, 1990; Kaczmarski et al., 2002), whereas

Eq. (6) assumes that it is the gradient of surface concentra-tion (Ruthven, 1984).The initial conditions for Eqs. (1) and (2) are

Ci(t = 0, z)= C0i (z),Cp, i(t = 0, z)= C0p, i(z),qi(t = 0, z)= q0i (z). (7)

The Danckwerts condition (Danckwerts, 1953) was cho-sen as the boundary conditions for Eq. (1)for t >0 andz= 0:

uf Cf,i − u(0)C(0)= −�eDL�Ci�z

(8)

with

Cf,i = C0f,i for 0< t < tp,Cf,i = 0 for tp < t

for t >0 andz= L:�Ci�z

= 0.

2.2. Calculation of the necessary parameters

The external mass transfer coefficient,kext, in Eq. (3)was determined, using the empirical correlation ofWilsonand Geankoplis (1996)which is valid in the range of Pecletnumbers considered in our measurements:

Sh= 1.09

�eSc1/3Re1/3. (9)

This equation relates the Sherwood number,Sh, a functionof the external mass transfer coefficient, to the Schmidt,Sc,and the Reynolds,Re, numbers. These numbers are definedby Reid et al. (1997)

Re = �sudp/�,Sc = �/�sDm,Sh= kextdp/Dm, (10)

where� and� are the viscosity and the density of the mobilephase in the pores, respectively.The molecular diffusion coefficient,Dm, was calculated

according to theWilke and Chang equation (Gidding, 1965):

Dm = 7.4× 10−8T(�AMS)0.5

�́V 0.6A

, (11)

whereT is the temperature of the column (K),VA is the mo-lar volume of the solute at its normal boiling point (calcu-lated from Le Bas correlation) (Reid et al., 1997),MS is themolecular weight of the fluid,́� is the fluid viscosity (cP),and�A is the association factor for the fluid which accountsfor the effects of the solute–solvent interactions on the dif-fusional flux and which is assumed to be 1 in this study.


The axial dispersion coefficient,DL (Eq. (1)) can be de-rived from Gunn’s equation (Gunn, 1987):

�eDLdpu

=[Re Sc(1− p)24�21(1− �e)

+ (Re Sc)2p(1− p)2

16�41(1− �e)2

]

×[exp

[−4�21(1− �e)

p(1− p)Re Sc

]− 1

]

× (1+ �2v)2 + �2v

2+ �e

�Re Sc, (12)

where�1 is the first root of the zero-order Bessel function,J0(x) (�1 = 2.4048), �v is the dimensionless variance ofthe distribution of the ratio of the local fluid linear velocityand the cross-section average velocity, a variance which, inthis work, is assumed to be equal to zero,� is the tortuosityfactor (for dispersion in a bed of spheres,� = 1.4), andp isa parameter defined inGunn (1987):

p = 0.17+ 0.33exp(−24/Re). (12a)

The POR model was solved using computer programsbased on implementations of the method of orthogonalcollocation on finite elements (Guiochon et al., 1994;Berninger et al., 1991; Kaczmarski et al., 1997). The set ofdiscretized ordinary differential equations was solved withthe Adams–Moulton method, implemented in the VODEprocedure (Brown et al.,). The relative and the absoluteerrors of the numerical calculations were equal to 1× 10−6

and 1× 10−8, respectively.

2.3. Isotherm models

The solutions of the POR model require an appropriateisotherm model accounting for equilibrium behavior of thesolute between the two phases of the system. The experimen-tal isotherm data of the Fmoc-Trp enantiomers on the MIPand those of Fmoc-L-Trp on the NIP were fitted to severaldifferent isotherm models at each temperature considered.The best isotherm models selected were the Bi-Langmuirmodel (Eq. (13)) for Fmoc-L-Trp on the NIP and the Tri-Langmuir model (Eq. (14)) for the Fmoc-Trp enantiomerson the MIP (Kim et al., in preparation):

q = qs1b1C

1+ b1C + qs2b2C

1+ b2C , (13)

q = qs1b1C

1+ b1C + qs2b2C

1+ b2C + qs3b3C

1+ b3C , (14)

whereqs1, qs2 andqs3 are the saturation capacities for thefirst, the second, and the third types of sites, respectively;andb1, b2 andb3 are the corresponding adsorption constants.The concentration in Eqs. (13) and (14) refers to the averageconcentration of a substrate in the stagnant fluid in pores.

3. Experimental

3.1. Chemicals

Fmoc-L-tryptophan (Fmoc-L-Trp) and Fmoc-D-tryptophan(Fmoc-D-Trp) were purchased from Novabiochem (SanDiego, CA, USA). 4-Vinylpyridine (4-VPY), 2,2-azo-bis(isobutyronitrile) (AIBN), and Ethylene glycol dimetha-crylate (EGDMA) were obtained from Aldrich (Mil-waukee, WI, USA). 4-VPY (60mmHg at 75◦C) andEGDMA (60mmHg at 120◦C) were distilled under vac-uum. Polystyrene standards with molecular masses rangingfrom 1780 to 1,877,000 were purchased from ScientificPolymers Products (Ontario, NY, USA). All other chemicalsand solvents were commercially available, of analytical orHPLC grade, and were used as is.

3.2. Preparation of the stationary phase and packing ofthe column

The composition of the polymerization mixture was:1.58mmol Fmoc-L-Trp, 4.74mmol 4-VPY, 18.96mmolEGDMA, 0.474mmol AIBN, and 5.4ml acetonitrile(MeCN). The amount of the solvent was 4/3 of the volumeof the monomers and the crosslinking monomers. The solu-tion was purged with N2 for 5min in a scintillation vial andpolymerized at 45◦C for 12h. After the polymerization, thebulk polymers were crushed, ground, and sieved to obtainparticles within the size range of 25–38mm. These poly-mer particles were slurry packed into a stainless column(100× 4.6mm). To remove any residues from the polymer-ization mixtures and the template from the polymers, thecolumns packed with the two polymers, imprinted (MIP)and not imprinted (NIP) were exhaustively washed withmethanol/acetic acid (4/1 v/v).

3.3. Apparatus

The isotherm data were obtained by frontal analysis, usinga Hewlett-Packard (Palo Alto, CA, USA) HP 1090 liquidchromatograph. This instrument is equipped with a multi-solvent system (tank volumes, 1 dm3 each), an auto-samplerwith a 250�L sample loop, a diode-array UV detector, acolumn thermostat and a computer data acquisition station.The microcomputer of this system was used to programa series of breakthrough curves. Large samples were alsoinjected in this instrument and the elution profiles recorded.

3.4. Experimental measurement of the breakthrough curvesand the porosities of the columns

3.4.1. Breakthrough curvesThe two pumps of the solvent delivery system were used

to obtain the breakthrough curves of the solutes. One ofthe pump delivered the pure mobile phase, the other pump


delivered solutions of the compound studied. The concentra-tion of the studied compound in the steady stream was de-termined by the concentration of the sample solution and theflow-rate fractions delivered by the two pumps at constantflow rate (1mL/min) for a set time interval. The concentra-tion range of the compound studied was between 0.005 and40mM. The frontal analysis (FA) experiments were carriedout at four different temperatures (40, 50, 60, and 70◦C)with acetonitrile as a mobile phase in the presence of onepercent of acetic acid, as an organic modifier. At each con-centration of substrate, breakthrough curve was recorded byallowing a sufficiently long delay time (60–120min) be-tween each successive curve to allow the re-equilibrium ofthe column with the pure mobile phase. The injection timeof the sample was between 10 and 40min to ensure that thecomposition of the eluate was the same as that of the plateauinjected at the column inlet. This was checked by observ-ing the plateau of the breakthrough for more than 5min.The signal was detected, depending on the concentrationrange, between 260 and 310nm to avoid recording any sig-nals above 1500 mAU. After a breakthrough curve is com-pleted, the column is regenerated by percolating the puremobile phase until the concentration of the Fmoc-Trp solutegets below a threshold. Then, another breakthrough curve isgenerated. Each such curve was repeated three times. Suc-cessive breakthrough curves at one temperature were madeby raising the concentration of the solute, so as to minimizethe consequences on the new data point of a small residualconcentration in the column.

3.4.2. Porosities of the columnsThe hold-up time for the MIP columns (t0) was mea-

sured by injecting a small amount of acetone from the auto-sampler into the column. The extra-column volume fromthe pump to the detector was measured by injecting a smallamount of the substrate from the pump into a zero dead-volume connector replacing the column. A value oftx =0.07 min was obtained. All the experimental data have beencorrected by subtractingtx . The total porosities (�t ) of theMIP and the NIP columns were 0.73 and 0.75, respectively.The external porosities (�e) of the MIP and the NIP

columns at 23 and 40◦C were measured from the retentiondata of polystyrene samples of narrow molecular size andknown average molecular mass ranging between 1780 and1,877,000 (Al-Bokari et al., 2002). Triplicate samples of10�L of each individual polystyrene standard dissolved inmethylene chloride (10mg/L) were injected into each col-umn at a flow rate of 1.0mL/min, using methylene chlorideas the mobile phase. Each data point reported in this work isthe average value of the three retention times measured. Theretention volumes of all the polystyrene standard sampleswere corrected for the extra-column volume from the auto-sampler. The logarithm of the molecular weight of eachpolystyrene sample was plotted versus the retention timeof the peak for the MIP and the NIP columns, as shown in

Retention Volume (mL)

0.5 0.6 0.7 0.8 0.9 1.0

Lo

g10

(MW

of

Po

lyst

yren

e)

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

23 0C

40 0C

(a)

External Pore Zone

Internal Pore Zone

Excluded Pore

Retention Volume (mL)

0.5 0.6 0.7 0.8 0.9 1.0

Lo

g10

(MW

of

Po

lyst

yren

e)

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

23 0C

40 0C

(b)

External Pore Zone

Internal Pore Zone

Excluded Pore

Fig. 1. Plot of the logarithm of the molecular masses of the polystyrenestandards versus their retention time at temperature 23◦C (©) and 40◦C(�) on (a) MIP; (b) NIP.

Fig. 1. The graph shows two straight lines, the steeper linecorresponding to the external pore zone (or macropores),the less steep line to the internal pore zone (or mesopores).The intersection point of these two straight lines gives thelargest pore diameter of the mesopore network. The exter-nal porosity of the column is derived from this intersectionpoint, using the following equation:

�e = VeVg

, (15)


whereVe is the retention volume of the excluded molecularmass (i.e., the retention volume of the polystyrene sampleat the intersection point) andVg is the geometrical volumeof the column, given byVg = �r2L wherer andL are thecolumn radius and length.Measuring the external porosities of the columns at tem-

peratures higher than 40◦C was impractical due to the lowboiling point of the mobile phase (i.e., methylene chloride).The values of the external porosities at temperatures higherthan 40◦C were extrapolated. The external porosities usedin this study are 0.368 and 0.418 for the MIP and the NIPcolumns, respectively.

3.5. Calculation of the isotherm data and isotherm modelselection

Adsorption isotherm data represent the amount of sub-strate (q, mmol/L) bound to the column in equilibrium witha given mobile phase concentration of this substrate (C,mmol/L). To calculate this amount from the FA data, thefollowing equation was used:

q = CVequ− V0Va

, (16)

whereVequ is the elution volume of the equivalent area ofthe solute,V0 is the column hold-up volume andVa is thevolume of stationary phase in the column. The value ofVequof each breakthrough curve was calculated using the equalarea method (Sajonz et al., 1996a,b). The value ofVa wascalculated by subtractingV0 from the geometrical volumeof the column.The experimental isotherm data were fitted to different

isotherm models and the Bi-Langmuir and the Tri-Langmuirmodels selected for the NIP and the MIP columns, respec-tively because they gave the best statistical figures. The bestnumerical values of the parameters of these models wereestimated from these best fits. To give the same weight tothe low and high concentration data, the data were fittedwith weights 1/q2, where q is the experimentally mea-sured amount adsorbed. The Marquardt–Levenberg method(Fletcher,) was used and the goodness of the fittings wereestimated by calculating the standard deviation of eachisotherm parameter, the Fisher parameters, and the residualsum of squares and the confidence level (Kim et al., inpreparation; Seber and Wild, 1989).

3.6. Measurement of the band profiles

Experimental band profiles of Fmoc-L-Trp and Fmoc-D-Trp on the MIP and the NIP were measured at each tempera-ture at the time when the frontal analysis data were acquired.The injection of the substrates were done from the pump, atsix different concentrations ranging between 0.1 and 40mM.The band profiles were recorded at wavelengths of 260

and 310nm, depending on the concentration, and the ab-sorbance data (mAU) were converted into concentrations

-5 0 5 10 15 20 25 30 35 40 45

0

50

100

150

200

250

q (

mM

)

C (mM)

0 5 10 15 20 25 30 35

0

50

100

150

200

250

q (

mM

)

C (mM)

(a)

(b)

Fig. 2. Adsorption equilibrium isotherms at each different temperature:(a) Fmoc-L-Trp (solid symbols) and Fmoc-D-Trp (open symbols) on theFmoc-L-Trp imprinted polymer. The lines represent the best calculatedTri-Langmuir isotherm for Fmoc-L-Trp (solid lines) Fmoc-D-Trp (dottedlines). (b) Fmoc-L-Trp on the non-imprinted polymer. The lines representthe best calculated Bi-Langmuir isotherm.

(mM) by estimating the numerical coefficients of a secondorder polynomial, by minimizing the square root sum be-tween the left- and the right-hand side of the following equa-tion, usingnmeasured band profiles, wheren is greater thanthe polynomial degree (Gubernak et al., 2004b):

Cf tp =∫ +∞

0p1 × S(t)+ p2 × S(t)2 dt , (17)

whereCf is the injected concentration of the substrates,tpis the duration of the injection,S(t) is the detector response,a function of time,pi are the coefficients of the polyno-mial function. The detailed derivation of the equation usedto convert the detector signal to concentrations is availablein the supporting information. Although the short-term re-peatability of the band profile was satisfactory, the long-term


Table 1Tri-Langmuir model fittings for Fmoc-Trp enantiomers on the MIP with assumption thatqs is constant with temperature

Temp Fmoc-L-Trp Fmoc-D-Trp

(◦C) qs1 b1 qs2 b2 qs3 b3 qs1 b1 qs2 b2 qs3 b3(mM) (mM−1) (mM) (mM−1) (mM) (mM−1) (mM) (mM−1) (mM) (mM−1) (mM) (mM−1)

40 389 0.0322 7.75 1.52 0.342 110 422 0.0268 12.2 0.651 0.0636 97.1±20 ±0.0028 ±1.4 ±0.29 ±0.040 ±20 ±18 ±0.0018 ±2.4 ±0.11 ±0.022 ±49

50 389 0.0270 7.75 1.15 0.342 93.9 422 0.0222 12.2 0.542 0.0636 66.8±20 ±0.0023 ±1.4 ±0.22 ±0.040 ±16 ±18 ±0.0022 ±2.4 ±0.091 ±0.022 ±30

60 389 0.0223 7.75 0.992 0.342 66.9 422 0.0186 12.2 0.398 0.0636 22.0±20 ±0.0019 ±1.4 ±0.19 ±0.040 ±10 ±18 ±0.0015 ±2.4 ±0.064 ±0.022 ±13

70 389 0.0195 7.75 0.812 0.342 45.6 422 0.0158 12.2 0.368 0.0636 40.8±20 ±0.0017 ±1.4 ±0.15 ±0.040 ±6.2 ±18 ±0.0013 ±2.4 ±0.062 ±0.022 ±16

Table 2Bi-Langmuir model fitting for Fmoc-L-Trp on the NIP with the assumptionthat qs is constant with temperature

Temp Fmoc-L-Trp

(◦C) qs1 b1 qs2 b2(mM) (mM−1) (mM) (mM−1)

40 309± 25 0.0272± 0.0034 3.47± 1.3 0.931± 0.2650 309± 25 0.0230± 0.0029 3.47± 1.3 0.953± 0.2860 309± 25 0.0190± 0.0024 3.47± 1.3 0.740± 0.2270 309± 25 0.0155± 0.0019 3.47± 1.3 0.571± 0.16

one was poor. The column performance degraded markedlyafter a few months.

4. Results and discussion

4.1. Adsorption equilibrium

The adsorption isotherm data of the two Fmoc-Trp enan-tiomers on the MIP and of Fmoc-L-Trp on the NIP weremeasured by frontal analysis, using acetonitrile (1% aceticacid) as the mobile phase at different temperatures.Fig. 2shows these data for the Fmoc-Trp enantiomers on the MIP(Fig. 2(a)) and for Fmoc-L-Trp on the NIP (Fig. 2(b)), at dif-ferent temperatures. The symbols represent the experimentaldata. At all temperatures, the amount of solutes adsorbed islarger on the MIP than on the NIP and the amount of Fmoc-L-Trp adsorbed on the MIP is larger than that of Fmoc-D-Trp, also at all temperatures. The amount of solutes adsorbedat a given solution concentration decreases with increasingtemperature. It is noteworthy that the difference between theisotherms of the two enantiomers increases with increasingtemperature. This suggests that the resolution of the racemicmixture should improve with increasing the column temper-ature. This hypothesis will be addressed in future work.The solid lines inFig. 2represent the best isotherm model

for each system. These isotherm models were selected onthe basis of the results of the statistical tests (Fisher pa-

rameters and residues), of the calculation of the affinity en-ergy distribution, and of the comparisons of experimentalband profiles with band profiles calculated assuming thedifferent isotherm models considered (i.e., the Langmuir,the Bi-Langmuir, the Tri-Langmuir, and the Tetra-Langmuirisotherm models) (Kim et al., in preparation). After com-pletion of the selection process, the best isotherm model forFmoc-Trp enantiomers on the MIP was the Tri-Langmuirisotherm model, and the best isotherm model for Fmoc-L-Trp on the NIP was the Bi-Langmuir isotherm model. Thebest isotherm parameters for these systems are reported inTables 1and2. In each case, it was assumed that the numberof types of adsorption sites does not change with the tem-perature (Kim et al., in preparation; Svenson and Nicholls,2001).

4.2. Study of the mass transfer kinetics

Experimental overloaded elution band profiles were ac-quired for the Fmoc-Trp enantiomers on the MIP and forFmoc-L-Trp on the NIP at each different temperature, theconcentration of the injected samples being between 0.1 and40mM. These band profiles were compared to those calcu-lated using the model POR.Tables 3and4 report the valuesat each temperature of the different parameters necessaryto perform these calculations for Fmoc-Trp enantiomers onthe MIP (Table 3) and for Fmoc-L-Trp on the NIP (Table4). These parameters are the total porosity (�t ), the externalporosity (�e), and the internal porosity (�p) of the columns,their tortuosity (�), the molecular diffusion coefficient in thebulk phase (Dm), and the axial dispersion coefficient (DL).

4.2.1. Pore and surface diffusivitiesIn the calculations of band profiles using the POR model,

there are three different possibilities to interpret the overallmass transfer coefficient (ki).(1) First, the external mass transfer (kext) can be assumed

to be negligible, making pore diffusion (Dm in kint) essen-tially responsible for the mass transfer resistance. For theimprinted polymers, recent publications show that the con-


Table 3Values of parameters for Fmoc-Trp enantiomers on the MIP used in the POR model

Temp �ta �eb �pc Tortuosityd Dme×10−3 kext

f DLg

(◦C) (cm2/min) (cm/min) (cm2/min)

40 0.728 0.368 0.571 3.56 1.40 3.14 0.029650 0.728 0.368 0.571 3.56 1.84 3.77 0.023860 0.728 0.368 0.571 3.56 2.05 4.05 0.021970 0.728 0.368 0.571 3.56 2.29 4.35 0.0201

a�t total porosity.b�e external porosity.c�p internal porosity= (�t − �e)/(1− �e).dTortuosity= (2− �p)2/�p .eMolecular diffusion coefficient of the solute in the mobile phase (see Eq. (10) in Section 2.2).f External mass transfer coefficient.gDispersion coefficient.

Table 4Values of parameters for Fmoc-L-Trp on the NIP used in the POR model

Temp �ta �eb �pc Tortuosityd Dme×10−3 kext

f DLg

(◦C) (cm2/min) (cm/min) (cm2/min)

40 0.746 0.418 0.570 3.59 1.40 2.76 0.028050 0.746 0.418 0.570 3.59 1.84 3.32 0.022660 0.746 0.418 0.570 3.59 2.05 3.57 0.020870 0.746 0.418 0.570 3.59 2.29 3.83 0.0192

a�t total porosity.b�e external porosity.c�p internal porosity= (�t − �e)/(1− �e).dTortuosity= (2− �p)2/�p .eMolecular diffusion coefficient of the solute in the mobile phase (see Eq. (10) in Section 2.2).f External mass transfer coefficient.gDispersion coefficient.

tribution of the intraparticle mass transfer to the overall masstransfer is much larger than that of the external mass trans-fer (Sajonz et al., 1998; Miyabe and Guiochon, 2000). Inthis case, the effective diffusivity in Eqs. (5) and (6) (inSection 2.2) simplifies to the following equation with theoverall mass transfer coefficient expressed in the followingequation:

Deff = �pDm�

, (18)

ki = kint = 10Deff

dp= 10(�pDm)

dp�. (19)

(2) Second, both the external mass resistance and the porediffusion can be considered as negligibly small, so surfacediffusion dominates the other sources of mass transfer resis-tances. In this case, the effective diffusivity and the overallmass transfer coefficient can be expressed as follows:

Deff = (1− �p)DSq

Cp, (20)

Deff = (1− �p)DSdq

Cp, (21)

ki = kint =10(1− �p)DS

q

Cp

dp, (22)

ki = kint =10(1− �p)DS

dq

dCpdp

. (23)

Eqs. (20) and (22) give the effective diffusivity and the over-all mass transfer coefficient in the case in which the drivingforce for surface diffusion is assumed to be the gradient ofchemical potential. Eqs. (21) and (23) give the effective dif-fusivity and the overall mass transfer coefficient in the casein which the driving force for surface diffusion is assumedto be the gradient of surface concentration.(3) Third, it is possible to assume that the external mass

transfer is negligible but that both pore and surface diffusionare the dominant contributions to the mass transfer resis-tances in the system studied. This gives the following equa-tions:

Deff = �pDm�

+ (1− �p)DSq

Cp, (24)


C (mM)

0 5 10 15 20 25 30 350.6

0.9

1.2

1.5

1.8

DS *

10-6

(cm

2 /min

)

Dp *

10-4

(cm

2 /min

)

Dp *

10-4

(cm

2 /min

)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 5 10 15 20 25 30 350.6

0.9

1.2

1.5

1.8

DS *

10-6

(cm

2 /min

)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 5 10 15 20 25 30 350.6

0.9

1.2

1.5

1.8

DS *

10-6

(cm

2 /min

)

Dp *

10-4

(cm

2 /min

)

Dp *

10-4

(cm

2 /min

)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 5 10 15 20 25 30 350.6

0.9

1.2

1.5

1.8

DS

*10

-6(c

m2 /m

in)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

(a)

(c) (d)

(b)

Fig. 3. Plots of pore diffusion coefficient (Dp , ©), surface diffusion coefficient (DS(q/c), �), and surface diffusion coefficient (DS(dq/dc),$) versus feedconcentrations,Cinj , of Fmoc-L-Trp on the NIP at a temperature of: (a) 40◦C; (b) 50◦C; (c) 60◦C; (d) 70◦C. DS(q/c) represents the surface diffusiondriven by the gradient of chemical potential, andDS(dq/dc) represents the surface diffusion driven by the gradient of surface concentration of the substrates.

Deff = �pDm�

+ (1− �p)DSdq

dCp, (25)

ki = kint =10

(Dm�p

�+ (1− �p)DS

q

Cp

)

dp, (26)

ki = kint =10

(Dm�p

�+ (1− �p)DS

dq

dCp

)

dp. (27)

Eqs. (24) and (26) give the effective diffusivity and the over-all mass transfer coefficient in the case in which the driving

force for surface diffusion is assumed to be the gradient ofchemical potential. Eqs. (25) and (27) give the effective dif-fusivity and the overall mass transfer coefficient in the casein which the driving force for surface diffusion is assumedto be the gradient of surface concentration.In the following sections, the mass transfer coefficients

are denoted as follows, clearly to present our data:

1. The pore diffusion coefficient,Dm, derived from Eqs.(18) and (19) is notedDp.

2. The surface diffusion coefficient,Ds , derived from Eqs.(20) and (22) is notedDs(q/c).

3. The surface diffusion coefficient,Ds , derived from Eqs.(21) and (23) is notedDs(dq/dc).


2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C/C

0

time (min)

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

C/C

0

time (min)

2 4 6 8 100.0

0.2

0.4

0.6

0.8

C/C

0

time (min)

2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

C/C

0

time (min)

(a) (b)

(c) (d)

Fig. 4. Comparison of experimental and simulated peak profiles for Fmoc-L-Trp on the NIP for a temperature of: (a) 40◦C withDS(dq/dc) = 5.79× 10−6 (cm2/min); (b) 50◦C with DS(dq/dc) = 6.43× 10−6 (cm2/min); (c) 60◦C with DS(dq/dc) = 7.39× 10−6 (cm2/min); (d)

70◦C with DS(dq/dc) = 8.94× 10−6 (cm2/min). The y-axis is normalized by dividing the elute concentration with the injected concentration of thesamples. The injection time was equal to 1min. The following inlet concentrations were used: at 40◦C 0.116, 0.236, 2.35, 4.71, 15.2, and 30.4mM;at 50◦C 0.119, 0.238, 2.35, 4.71, 15.1, 30.1mM; at 60◦C 0.119, 0.239, 2.36, 4.71, 15.0, 30.0mM; at 70◦C 0.128, 0.255, 2.37, 4.71, 15.1, 30.4mM(peaks from right to left).

Note that, in these definitions,q andCp are replaced withq andC, respectively, for the sake of simplicity. However,whenq andC are used in the rest of this paper, they standfor the average surface concentration and the average poreconcentration of the compound, respectively (see details inSection 2.2).Calculations of the band profiles were made using the

POR model and the values of each of the diffusion and masstransfer coefficients in the mass balance equations were op-timized to minimize the differences between the calculatedand the experimental band profiles.Fig. 3 shows plots ofthese diffusion coefficients versus the concentration of thesample of Fmoc-L-Trp injected on the NIP at each tempera-

ture. In all cases, it is the pore diffusion coefficient (Dp) thatdepends most on the concentration, with, e.g., a decrease of56% when the concentration increases from 0.1 to 30mM

at 50◦C. On the other hand, the two surface diffusion coef-ficients exhibit a relatively low dependency on the sampleconcentration. The coefficientDS(dq/dc) is almost constantin the whole concentrations range whileDS(q/c) decreasesby ca 20% from 0.1mM to 30mM 50◦C.There are no fundamental relations betweenDp and the

concentration of the injected sample, especially at the lowconcentrations used in our experiments (the largest one was1% w/w). Similar empirical relationships between the masstransfer coefficients and the sample concentration have been


C (mM)

0 10 20 30 400.6

0.9

1.2

1.5

1.8

2.1

2.4

DS

*10

-6(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

2

3

4

5

6

7

8

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 10 20 30 40 500.6

0.9

1.2

1.5

1.8

DS

*10

-6(c

m2 /m

in)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 10 20 30 40 500.6

0.9

1.2

1.5

1.8

DS

*10

-6(c

m2 /m

in)

3

4

5

6

7

8

9

DP

DS (q/c)

DS (dq/dc)

(a) (b)

(c)

Fig. 5. Plots of pore diffusion coefficient (Dp , ©), surface diffusion coefficient (DS(q/c), �), and surface diffusion coefficient (DS(dq/dc),$) versusinjected concentrations of Fmoc-D-Trp on the MIP at a temperature of: (a) 40◦C; (b) 50◦C; (c) 60◦C; (d) 70◦C.DS(q/c) represents the surface diffusiondriven by the gradient of chemical potential, andDS(dq/dc) represents the surface diffusion driven by the gradient of surface concentration of the substrates.

reported in other systems. A strong dependence of thesecoefficients on the concentration have been shown to arisefrom the selection of an inappropriate model of mass trans-fer (Kaczmarski et al., 2001; Gubernak et al., 2004a). Incontrast, the nearly constant value ofDS(dq/ds) in a widerange of concentration of Fmoc-L-Trp suggests that surfacediffusion plays a dominant role in the mass transfer resis-tances of the substrate on the NIP, and that the driving forcefor surface diffusion is the gradient of surface concentration.Fig. 4 compares the experimental and calculated band pro-files of Fmoc-L-Trp on the NIP at the four different temper-atures. For these calculations, Eqs. (21) and (23) were used,assuming that the external and the pore diffusion are neg-ligible, that surface diffusion is the dominant mass transferprocess, and that the driving force for surface diffusion is thegradient of surface concentration of the substrate. Finally, a

constant value ofDS(dq/dc) was used. At all temperatures, avery good agreement is observed between the experimentaland the calculated band profiles.Fig. 5 shows plots of each mass transfer coefficient ver-

sus the concentrations of Fmoc-D-Trp in the sample injectedonto the MIP column, at the four different temperatures. Inall cases, we again observe that the pore diffusivity (Dp)depends more strongly than the surface diffusivity on thesample concentration. This suggests that surface diffusion isalso the dominant contribution to mass transfers on the MIP.Except at 50◦C, the surface diffusion coefficient seems toincrease with increasing sample concentration and the valueof DS(q/c) changes slightly less than that ofDS(dq/dc) withincreasing sample concentration. For Fmoc-L-Trp the de-pendency of the surface diffusion coefficient on the sampleconcentration is much stronger than it is for its enantiomer


C (mM)

0 5 10 15 20 25 30 350.4

0.8

1.2

1.6

2.0

DS

*10

-6(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

1

2

3

4

5

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 10 20 30 400.4

0.8

1.2

1.6

2.0

DS

*10

-6(c

m2 /m

in)

1

2

3

4

5

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 5 10 15 20 25 30 350.4

0.8

1.2

1.6

2.0

DS

*10

-6(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

Dp

*10

-4(c

m2 /m

in)

1

2

3

4

5

DP

DS (q/c)

DS (dq/dc)

C (mM)

0 5 10 15 20 25 30 350.4

0.8

1.2

1.6

2.0

DS

*10

-6(c

m2 /m

in)

2

4

6

8

10

DP

DS (q/c)

DS (dq/dc)

(a) (b)

(c) (d)

Fig. 6. Plots of pore diffusion coefficient (Dp , ©), surface diffusion coefficient (DS(q/c), �), and surface diffusion coefficient (DS(dq/dc), $) versusinjected concentrations of Fmoc-L-Trp on the MIP at a temperature of: (a) 40◦C; (b) 50◦C; (c) 60◦C; (d) 70◦C.DS(q/c) represents the surface diffusiondriven by the gradient of chemical potential, andDS(dq/dc) represents the surface diffusion driven by the gradient of surface concentration of the substrates.

on the MIP, at all temperatures (Fig. 6). For example, in-creasing the concentrations from 0.1 to 30mM causes a 30%decrease inDp, and 180% and 160% increases inDS(dq/dc)andDS(q/c), respectively. These results show that, although amass transfer kinetic model based on Fick’s law can accountfor the behavior of Fmoc-L-Trp on the NIP, it is not appro-priate for the mass transfer kinetics of the two enantiomerson the MIP. Band profiles calculated using Eqs. (24)–(27),which assume that both pore and surface diffusion are thedominant contribution to the mass transfer resistances of theFmoc-Trp enantiomers on the MIP, are compared with ex-perimental profiles inFig. 7 for Fmoc-L-Trp on the MIP at50◦C. The agreements between the experimental and calcu-lated band profiles obtained in either case was not satisfac-tory, especially for the profiles in the middle concentration

range. Note that the first moments of the calculated and ex-perimental peaks (not shown) are nearly identical, suggest-ing that the reason for the discrepancies if of kinetic, notthermodynamic origin.

4.2.2. Isosteric heat of adsorption and activation energyfor surface diffusion on the MIPIn the previous section, surface diffusion was shown to

be the dominant contribution to the mass transfer kineticsof the enantiomers on the NIP and it was suggested that thesame was also true on the MIP while the contributions ofthe external mass transfer and the pore diffusion were bothnegligible. The mass transfer kinetics of the two compoundson the NIP was shown to be dominated by a surface diffu-sion process driven by the gradient of the surface concentra-


5 10 15 20 25 30

30

0.0

0.1

0.2

0.3

0.4

0.5

C/C

0

time (min)

5 10 15 20 25

0.0

0.1

0.2

0.3

0.4

0.5

C/C

0

time (min)

(a)

(b)

Fig. 7. Comparison of experimental and simulated peak profiles (as-suming that both pore diffusion and surface diffusion are dominantmass transfer resistances (Eqs. (21)–(24))) for Fmoc-L-Trp on the MIPfor a temperature of 50◦C: (a) Dp = 6.5 × 10−5 (cm2/min) andDS(q/c)= 5.5× 10−7 (cm2/min); (b) Dp = 6.5× 10−5 (cm2/min) and

DS(dq/dc) = 5.5× 10−7 (cm2/min). The y-axis is normalized by divid-ing the elute concentration with the injected concentration of the samples.The injection time was equal to 1min and the inlet concentrations were0.108, 0.233, 2.34, 4.69, 15.2, and 30.5mM (peaks from right to left).DS(q/c) represents the surface diffusion driven by the gradient of chem-ical potential, andDS(dq/dc) represents the surface diffusion driven bythe gradient of surface concentration of the substrates.

tion of the substrates. On the other hand, although the masstransfer kinetics of the two enantiomers on the MIP is dom-inated by a surface diffusion mechanism, this process (andespecially that of Fmoc-L-Trp) cannot be fully explained byeither the gradient of surface concentration or the gradient ofthe chemical potential of the enantiomers. The surface diffu-sion coefficients increase with increasing sample concentra-tion. Similar results were also observed for the phenylalanineanilide (PA) enantiomers on anL-Phenylalanine Anilide im-printed polymer in an aqueous mobile phase (Sajonz et al.,

1998; Miyabe and Guiochon, 2000). In this early work, theapparent increase in the surface diffusion coefficient with in-creasing sample concentration was explained using the het-erogeneous surface diffusion model.In this model, surface diffusion is an activated process in

which the temperature dependence ofDS is analyzed usingthe following Arrhenius equation:

DS =DS0 exp(−ESRT

), (28)

whereDS0 is the diffusivity at zero energy level,ES is theactivation energy of the process,T is the temperature in K,andR is the universal gas constant.The activation energy (ES) is assumed to be linearly cor-

related with the isosteric heat of adsorption (Miyabe andGuiochon, 2000), (Qst ), giving the following equation:

ES = �(−Qst ). (29)

Combination of Eqs. (28) and (29) gives the surface dif-fusion coefficient (Eq. (30)), which is a function of the isos-teric heat of adsorption:

DS =DS0 exp(−�(−Qst )

RT

)

=DS0 exp(−�T

(−QstR

)). (30)

The isosteric heat of adsorption is determined at a constantamount adsorbed (q) by the following equation:

−QstR

=[d(lnC)

d(1/T )

]q=constant

. (31)

Fig. 8 shows the relationship between ln(C) and 1/T atconstant amount adsorbed,q, from which is determined theisosteric heat of adsorption,Qst , for Fmoc-L-Trp (Fig. 8(a))and for Fmoc-D-Trp (Fig. 8(b)) on the MIP, and for Fmoc-L-Trp (Fig. 8(c)) on the NIP. The isotherm parameters ob-tained for these compounds are used to calculate ln(C) atconstantq for each different temperature, as explained ear-lier (Tables 1and 2). An almost linear relationship betweenln(C) and 1/T is observed in all cases. The value ofQstwas derived from the average slope of these lines.Fig. 9shows the exponential relationship between the isosteric heatof adsorption and the coverage density for the Fmoc-Trpenantiomers adsorbed on the MIP, in agreement with previ-ous results (Miyabe and Guiochon, 2000; Szabelski et al.,2002). The sharp decrease ofQst (i.e., the amount of en-ergy released during adsorption) with increasingq whichtakes place with either enantiomer illustrates the energeticheterogeneity of the surface of the MIP. For Fmoc-Trp onthe NIP, there is almost no dependency ofQst on q (Fig.9, $ symbols), indicating a nearly homogeneous behaviorof the surface of the NIP for Fmoc-Trp. The homogeneousbehavior of the NIP surface was also demonstrated by the


0.0029 0.0030 0.0031 0.0032

-3

-2

-1

0

1

2

3

4

5

(a)

q =1 [mM]

q =5 [mM]

ln (

C)

1/T (1/K)

q =190 [mM]

0.0029 0.0030 0.0031 0.0032

-3

-2

-1

0

1

2

3

4

5

(b)

q =1 [mM]

q =5 [mM]

ln C

1/T (1/K)

q =190 [mM]

0.0029 0.0030 0.0031 0.0032

-3

-2

-1

0

1

2

3

4

5

(c)

q =1 [mM]

q =5 [mM]

ln C

1/T (1/K)

q =130 [mM]

Fig. 8. Plot between ln(C) and 1/T for determiningQst for (a) Fmoc-L-Trp on the MIP; (b) Fmoc-D-Trp on the MIP; (c) Fmoc-L-Trp on the NIP.

excellent agreement between the experimental and the cal-culated band profiles (Fig. 4), the calculations being madewith the assumption that mass transfer is controlled by sur-face diffusion driven by the gradient of surface concentra-tion. In contrast, the MIP surface is heterogeneous, as indi-cated by the strong dependency ofQst on q. For Fmoc-D-Trp, the initial decrease inQst is less strong than that forFmoc-L-Trp, indicating that Fmoc-D-Trp experiences a lessenergetically heterogeneous surface than Fmoc-L-Trp on theimprinted polymer.Contrary to previous reports (Miyabe and Guiochon,

2000; Szabelski et al., 2002), a single-term-exponential de-cay function cannot account for our data inFig. 9. This isbecause the maximum mobile phase concentration (Cmax)

used to calculateQst was 40mM, at which adsorption ishighly non-linear; while the previous studies used a lowerCmax of 3mM, at which adsorption is still almost linear.Thus, extensions of the single term exponential decay func-tion were used to correlateQst andq. Using a three-termexponential decay to fit the isosteric heat adsorption datadoes not significantly improve the goodness of fit comparedto that obtained with a two-term exponential decay. Theratio of the Fisher parameters for the three-term and thetwo-term exponential decay was only 1.50, lower than thecritical F value of 4.00. However, the ratio of the Fisherparameters for the two-term and the one-term exponen-tial decay was 100, far above the criticalF value of 9.24.Thus, the following equation was used to correlate the


0 50 100 150 200

1800

2000

2200

2400

-Qst

/R

q (mM)

Fig. 9. The dependency of isosteric heat of adsorption (−Qst /R), on sur-face concentration (q) for Fmoc-L-Trp on the MIP (©), for Fmoc-D-Trpon the MIP (�) and for Fmoc-L-Trp on the NIP ($). The solid linesrepresent best fit parameters calculated using Eq. (29).

isosteric heat of adsorption and the surface concentration ofsubstrate.

(−Qst )R

= p1 + p2 × exp(−p3 × q)+ p4 × exp(−p5 × q). (32)

The solid lines inFig. 9 correspond to this equation withthe best values of the parameters (i.e.,p1, p2, p3, andp4 inEq. (29)) inTable 5. For Fmoc-Trp on the NIP, the surfacebeing almost homogeneous, there is no need to include acontribution of the surface heterogeneity to the surface dif-fusivity to describe the band profiles.Eq. (32) was substituted into Eq. (30) to calculate the

value of the surface diffusion coefficient that incorporatesthe contribution of surface heterogeneity. This gave the fol-lowing equation:

DS(Qst )=DS0 exp(−�T(p1 + p2 × exp(−p3 × q)

+ p4 × exp(−p5 × q))). (33)

This surface diffusion coefficient was used in Eq. (21) (whichgives the effective diffusivity) and in Eq. (23) (which givesthe overall mass transfer coefficient) to calculate band pro-files for the Fmoc-Trp enantiomers on the MIP at each

Table 5Values of parametersp1, p2, p3, p4 andp5 for Eq. (32)

Isomer p1 p2 p3 p4 p5

Fmoc-L-Trp 1784± 1.0 724± 9.9 2.17± 0.058 232± 3.1 0.081± 0.0024Fmoc-D-Trp 1891.9± 0.46 156± 4.9 3.5± 0.14 55.2± 0.75 0.046± 0.0018

Table 6Values of parameters for Eq. (33)

T (◦C) Ds0 (cm2/min) −�

Imprinted polymer, isomer Fmoc-L-Trp40 — —50 1.57× 10−4 ± 7.7× 10−6 0.664± 0.008160 2.5× 10−4 ± 1.2× 10−5 0.731± 0.008970 1.21× 10−4 ± 8.8× 10−6 0.59± 0.013

Imprinted polymer, isomer Fmoc-D-Trp40 0.50± 0.010 1.960± 0.003550 2.12× 10−5 ± 2.3× 10−7 0.284± 0.001960 5.15× 10−5 ± 3.7× 10−7 0.42± 0.01270 0.500± 0.0071 2.017± 0.0024

Non-imprinted polymer, isomer Fmoc-L-TrpDs

a (cm2/min)40 5.79× 10−6 ± 3.2× 10−8

50 6.43× 10−6 ± 4.1× 10−8

60 7.39× 10−6 ± 5.7× 10−8

70 8.94× 10−6 ± 6.3× 10−8

aThe values ofDS for Fmoc-L-Trp on the NIP were estimated withoutexponential terms in Eq. (33), which account for surface heterogeneity.

temperature. The parametersDS0 and� were optimized forbest agreement between the experimental and the calcu-lated band profiles. The optimized values ofDS0 and� forthe Fmoc-Trp enantiomers on the MIP and ofDS(dq/dc) forFmoc-Trp on the NIP are summarized inTable 6. The valuesof DS(dq/dc) on the NIP was derived assuming a homoge-neous surface (no exponential terms in Eq. (32), see earlier).Figs. 10and11 compare experimental and calculated bandprofiles for Fmoc-L-Trp and for Fmoc-D-Trp, respectively,on the MIP at each temperature. There is a good agreementbetween the two sets of band profiles.The values ofDs0 and � estimated from Eq. (30) de-

pend on the temperature, as shown inTable 6. Accordingto the Arrhenius equation (Eq. (28)), however, these twoparameters should be independent of the temperature. Thedependency ofDs0 and� on the temperature observed canbe explained by two reasons. First, Arrhenius law may betoo simple. Second, the relationship betweenE andQst(Eq. (29)) is probably not linear. In this work, the relation-ship between the isosteric heat of adsorption and the amountadsorbed (q) was obtained by fitting the data inFig. 8 to astraight line. However, the plots are slightly curved. A moreaccurate (albeit less precise) description of the isosteric heatadsorption on the surface can be obtained if using narrower


0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

C/C

0

C/C

0

C/C

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (min) time (min)

time (min)

0 5 10 15 20 25 30

0 4 8 12 16 20

(a)

(c)

(b)

Fig. 10. Comparison of experimental and simulated peak profiles, calculated using Eq. (30), for Fmoc-L-Trp on the MIP for a temperature of: (a) 50◦C;(b) 60◦C; (c) 70◦C. They-axis is normalized by dividing the elute concentration with the injected concentration of the samples. The injection time wasequal to 1min. The following inlet concentrations were used: at 50◦C 0.108, 0.233, 2.34, 4.69, 15.24, and 30.5mM; at 60◦C 0.124, 0.277, 2.35, 4.70,15.1, 30.1mM; at 70◦C, 0.095, 0.207, 4.70, 15.1, and 30.1mM (peaks from right to left). The best values of parameters used to simulate these peakprofiles are summarized in Table 6.


0.0

0.1

0.2

0.3

0.4

0.5

C/C

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

C/C

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

C/C

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

C/C

0

0 5 10 15 20

time (min)

0 3 6 9 12 15 0 3 6 9 12 15

time (min)

0 5 10 15 20

time (min)

time (min)

(a)

(c) (d)

(b)

Fig. 11. Comparison of experimental and simulated peak profiles, calculated using Eq. (30), for Fmoc-D-Trp on the MIP for a temperature of: (a) 40◦C;(b) 50◦C; (c) 60◦C; (d) 70◦C. They-axis is normalized by dividing the elute concentration with the injected concentration of the samples. The injectiontime was equal to 1min. The following inlet concentrations were used: at 40◦C 0.096, 0.251, 2.83, 5.50, 17.6, and 35.0mM; at 50◦C 0.122, 0.249,2.35, 4.71, 21.0, and 42.0mM; at 60◦C 0.130, 0.260, 2.41, 4.82, 21.0, and 42.0mM; at 70◦C 0.122, 0.245, 2.36, 4.74, 23.6, 49.1mM (peaks from rightto left). The best values of parameters used to simulate these peak profiles are summarized in Table 6.


temperature intervals, using the following equation:

−QstR

=

ln(C1(T1))− ln(C2(T2))

1

T1− 1

T2

q=const

. (34)

Fig. 12 shows the results obtained when calculating withthis equation the isosteric heat of adsorption for the differenttemperature intervals considered (i.e., 40–50◦C, 50–60◦C,and 60–70◦C), for Fmoc-D-Trp on the MIP, as an example.As shown in this figure, a rather different dependency ofthe isosteric heat of adsorption onq can be observed, de-pending on the temperature interval. This suggests that theMIP surface is more complicated and that its behavior can-not be fully described by the empirical relationship used todescribe the dependency of the isosteric heat of adsorptionon q (Eq. (32)).Fig. 13 shows plots of the surface diffusivities obtained

for Fmoc-L-Trp (Fig. 13(a)) and of Fmoc-D-Trp (Fig. 13(b))

0 25 50 75 100 125 150 175 2000.000002

0.000004

0.000006

0.000008

0.000010

(a)

700C

600C

500CDS (

cm2 /m

in)

q (mM)0 25 50 75 100 125 150 175 200

0.000002

0.000004

0.000006

0.000008

0.000010

700C

600C

500C

400C

(b)

Ds (

cm2 /m

in)

q (mM)

0 25 50 75 100 125 150 175 2000.000002

0.000004

0.000006

0.000008

0.000010

700C

600C

500C

400C

(c)

DS (

cm2 /m

in)

q (mM)

Fig. 13. The dependency of surface diffusion coefficients (Ds ) on the amount adsorbed (q) for: (a) Fmoc-L-Trp on the MIP; (b) Fmoc-D-Trp on theMIP; (c) Fmoc-L-Trp on the NIP.

0 20 40 60 80 100 120 140 160 180

1200

1400

1600

1800

2000

2200

2400

2600

650C

550C

450C

-Qst

R

q [mM]

Fig. 12. Plot of the isosteric heat of adsorption (−Qst /R) versus theamount adsorbed (q) for Fmoc-D-Trp on the MIP. Each different symbolbelongs to different temperature intervals: 40–50◦C (©); 50–60◦C (�);60–70◦C ($).


on the MIP, and for Fmoc-Trp on the NIP (Fig. 13(c)) ver-susq. The surface diffusion coefficients (DS) of the enan-tiomers on the MIP increase with increasingq due to thesurface heterogeneity. The surface diffusion coefficient (DS)of Fmoc-Trp on the NIP is constant due to the relatively ho-mogeneous surface on the NIP. Comparing the values ofDSon the MIP and on the NIP shows that the surface diffusivi-ties measured for the NIP are larger than those found on theMIP at all temperatures. In all cases, the surface diffusivi-ties increase with increasing temperature. However, a lesserinfluence of the temperature on the surface diffusivities forthe template, Fmoc-L-Trp, on the MIP than for either Fmoc-D-Trp on the MIP or Fmoc-Trp on the NIP is observed.

5. Conclusion

The quantitative comparison between experimental andcalculated peak profiles permits a detailed analysis of themass transfer kinetics of enantiomers on imprinted poly-mers. The comparison of kinetic data derived for a MIP andthe corresponding NIP informs on the mechanism of theenantioseparation. The excellent agreement between exper-imental and calculated peak profiles validates the conclu-sions of the study. It shows that the rate controlling processof mass transfer for the enantiomers, whether on the MIP orthe NIP, is surface diffusion while the contributions of ex-ternal mass transfer and pore diffusion are negligible. Thesurface diffusion coefficients of the enantiomers on the NIPare independent of the amount adsorbed. In contrast, thesurface diffusion coefficients of the two enantiomers on theMIP increase with increasing amount adsorbed. The corre-lation betweenDs andq can be modeled correctly by incor-porating the surface heterogeneity into the surface diffusioncoefficients. Thus, our results demonstrate that the surfaceheterogeneity plays an important role in the mass transferkinetics on imprinted polymers. To improve the chromato-graphic performance of imprinted polymers (e.g. to reducepeak tailing), efforts should be directed at decreasing thedegree of surface heterogeneity rather than at improving thepore structures of MIPs.

Notation

ap ratio of absorbant particle external surface areato volume

b association constantBi Biot number (kextdp/2Deff )C concentration in the mobile phaseCp concentration in the stagnant fluid phase con-

tained inside poresdp equivalent particle diameterDm diffusion coefficient

Dp pore diffusivityDeff effective (or inside-pore) diffusion coefficientDL dispersion coefficientki overall mass transfer coefficientkext external mass transfer coefficientkint internal mass transfer coefficientL column lengthM molecular massPe Peclet number(uL/DL�e)q concentration of the solute in the stationary

phaseqs saturated amount absorbedRe Reynolds number(�udp/�)Rp particle radiusSc Schmidt number (�/�Dm)Sh Sherwood number (kextapL�e/u)t timetp injection timeu average superficial velocity of the mobile phasez longitudinal distance along the column

Greek letters

� tortuosity parameter�e external porosity�p internal porosity� viscosity� fluid density

Subscripts

i component indexinj injection− average values solidphasef inlet value

Superscript

0 initial value

Acknowledgements

This work was supported in part by Grant CHE-02-44693of the National Science Foundation, by Grant DE-FG05-88-ER-13869 of the US Department of Energy, and by thecooperative agreement between the University of Tennesseeand the Oak Ridge National Laboratory.

Appendix A. Supplementary Materials

The on-line version of this article contain additional sup-plementary data. Please visit doi:10.1016/j.ces.2005.04.057.

http://dx.doi.org/10.1016/j.ces.2005.04.057


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Mass transfer kinetics on the heterogeneous binding sites of molecularly imprinted polymers

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